CORPS OF ENGINEERS, U.S. ARMY

FROST INVESTIGATIONS FISCAL YEAR 1953

FIRST INTERIM REPORT

by

Harl P. Aldrich, Jr.

Contract No. DA-19·016-eng-2314

TECHNICAL REPORT NO. 42

UNDER CONTRACT WITH

ARCTIC CONSTRUCTION AND FROST EFFECTS lABORATORY NEW ENGLAND DIVISION

BOSTON, MASSACHUSE'ITS

FOR

OFFICE OF THE CHIEF OF ENGINEERS AIRFIELDS BRANCH

ENGINEERING DIVISION MIUTARY CONSTRUCTION

. ~ .......

CORPS OF ENGINEERS, U. S. ARMY

FROST INVESTIGATIONS FISCAL YEAR 1953

FIRST INTERIM REPORT

.ANALYTICAL STUDIES OF

FREEZING AND THAWING OF SOILS by

Harl P. Aldrich, Jr.

Henry M. Paynter

Contract No. DA-19-016-eng-2314

TECHNICAL REPORT. NO. 42

UNDER CONTRACT WITH

ARCTIC CONSTRUCTION AND FROST EFFECTS LABORATORY

NEW ENGLAND DIVISION

ARMY- NED BOSTON, MASS

BOSTON, ~SSACHUSETTS

FOR

OFFICE OF THE CHIEF OF ENGINEERS

AIRFIELDS BRANCH

ENGINEERING DIVISION

MILITARY CONSTRUCTION

JUNE 1~53

~I

-. ~

--~-

.· ~ ....

\

SECTION

1-01

1-02

1-03

2-,01

·r·

.· . ' ~. . . •' . ~. ~

FIRST INTERIM REPORT

AJIALYTICAL STUDIES :, :")>·?'

OF

FREEZING AND THAWING OF SOILS

TABLE OF. CONTENTS

TITLE

LIST OF FIGURES

LIST OF TABLES : ' ....

NOTATION

SYNOPSIS

PART I .INTRODUCTION

CONTRACT

SCOPE

PERSONNEL

PART II ANALYTICAL STUDIES

MA THEMA TICAL. FORMULATION

a. S:y"'lopsis

b. Fund~~ental Principles

c •. Differential Equati.ons

d. Approximate Equations

. ~ . . . ·-

PAGE

i

ii

iii

v

1

2

2

3

3

3

6

:. / . /

./

2-Q2

3-Gl

3-02

e. Fluid and Electric Analogies

£. Practical Methoqs of Computation

(1) Hydraulic Model

(2) Electronic Analog Computer

(3) IoBoMo Solutions

(4) Hand Computations

DEPTH OF FROST ~RATION

a. Synopsis

b o Review of Literature

10

12

12

13

18

20

22

22

22

c. Derivation and Interpretation of Rational Formul~ 24

do Comparison of Rational Formula with Other Formulas 28

( 1) General 28

{2) Statistical Studies 31

e. Li.mi tations and Applioabili ty of· Formulas 35

f o Depth-Time Curves 37

go ldultilayered Systems 43

PART III DEPTH OF FROST PENETRATION NOMOGRAPH

DESCRIPTION OF .NOMOGRAPH

DIRECTIONS FOR THE USE OF 'mE NOMOGRAPH

a. General

b. Homogeneous Soil

Co Stratified Soil

do Negative v0

45

46

47

47

47

50

/

"· ........

~ ... -_,.

-. .-

;-: ,

3-o4 '

4-0l

4-o2

4-03

4-o4

UMITATIONSg ASSUMPriONS~ AND OTHER SPECIAL CASES

a. Limitation on k

b. Allowance for Frost Heave

c. Thaw

d. Surface Transfer Coe-f'f~ient, n

e. Limitations Due to Stratification

CONCLUSIONS AND RECOMMENDATIONS

PART IV THAW-CONSOLIDATION PROBLEM

SYNOPSIS

PHYSICAL NATURE OF THE PIDBLEM

MATHEMATICAL FORMUlATION

SOLUTION TECHNIQUES

a. HYdraulic Model

bo Numerical Methods

(1) Hand Computations

{2) Machi~ Computations

BIBU OGRA.PHY

APPENDIX A

APPENDIX B

APPENDIX C

DERIVATION OF A RATIONAL FORMUlA FOR THE DEPI'H OF FROST PENETRATION

TABLES OF I o~ B o Mo SOLUTIOBS

NUMERICAL SOLUTION OF A THAW-CONSOLIDATION PROBLEM }

50

51

51

52

52

54

54

56

56

57

6o

60

64

.:· .... ,..·

Figure No.

1.

2.

4.

6.

8.

10.

11.

A. B.

A. B.

A.

B.

LIST OF FIGURES

Title

Internal Ener~y Diagram for Water

Schematic Block Diagram, Electronic Analog Frost Computer

Assembly "F" for One Layer Interconnection of "F'' Components

Circuit Diagram~ Single "F" Component, Electronic Analog Frost Computer

Thermal Conditions During Frost Penetration

Correction Coefficient in the Modified Berggren Formula

Comparison of),..d ,~Curves

):.,t::(,~ Curves for Equation (2-29) )., ,of,~ Curves for Other Equations

Dispersion of Computed Frost Depths Using Modified Berggren Formula

Shape of Depth-Time Curves

Curves Used For Determinations of the Depth of Frost PeneL~ation

Semi-Empirica~ Gurves for Correction Factor A

Approximate Depth of Frost Penetration Below Pavements with Granular Base Courses

Pressure-Void Ratio Curve, Thaw-Consolidation

S:i.mp lified Hydraulic Model for Thaw-Consolida­tion

i

Page No.

5

15

17

26

30

33

42

48

58

62

.,

LIST OF TABLES

Table No • Ti. tle Page No. ..... , · ..... ~ . .

I Thermal-Fluid-Electric Analogies 11 ,.~

II Sample Hand Computation 21

III Shape of Depth-Time Curves 41

ii

_.:

-. -.r

NOTATION

'!he nomenclature used in this report agrees generally with that reconunended

iri·'::n'sch·l-,Lt~oh~nics Nomenclature", ASCE M~~al bf Engineering- Practice No. 22,

1941. - ~- '

A partial list of recommended s·ymbols- an{\ .. new symbols is given· below: -

S:Ymbol __ , •''

. a \ ·•.:.

Cs

rei'';; •. j,,

• •. r :~.. ·, •· • • •

.F

L

n

A-S

t

u

v

v 0

w

Descrintion ... -

)hermal Diffusi vi ty

Specific Heat of Soil Solids

·volumetric Heat Capacit~r of Frozen Soil.

· Volumetric Heat Capacity of Unfrozen Soil

Freezing Index

Thermal Conductivity of Frozen Soil

Thermal Conductivity of Unfrozen Soil

Latent Heat of Fusion of Soil 1'~oisture

Surface Transfer Coefficient

Rate of Heat Flow

Finite Distance

Tins (For frost penetration formulas and nornot;raph: Duration of .ltreezing Index)

Internal Energy, total heat content

Temper a tur e

Degrees F by which mean annual temper~ ature exceeds freezine point of soil moisture

Water Content of Soil

iii

Units · .

sq.ft. per hr •

BTU per lb.

BTU per (cu. rt.X °F)

BTU per {cu.ft.)( °F)

Degree Days

BTU per (ft. )( °F) {hr.)

BTU per (ft.) ( °F) (hr.)'

BTU per cu. ft.

(Dimensionless)

BTU per (sq.ft.)( hr.)

ft.

hr.

(days)

3TTJ per cu. ft.

Symbol

0(

Description

Depth of Frost Penetration

Finite Dista~ces in the x, y, and z· directions·

Thermal Ratio

a 6. t £:::. s 2

Correction Coeffici~nt

Fusion Parameter

iv

Units

feet (or inches)

ft •.

(Dimensionless)

(Dimensionless)

(Dimensionless)

(Dimensionless)

r----

SYNOPSIS '- ... ·. ! .. . ~ ~ '. '

.. This·- report': contains; the results' of studl.es<relate·d 'tO ·~the·::.predicti:on::

, called thaw-oon.solidation· problem has been stUdied ana the ·results: are ':re·por•

··-ted by: the writers.· ,A••,

Part II is devoted to the presentation of a mathematical formulatlon 1 of

the problem derived from first principles of heat transfer. The fluid and

electric analogies to ther.mal problems are treated in some detail since they

-are of especial value not only for visualizing the thermal problem but as

important aids in solving complex situations. Furthermore~ because a closed-

form analytical solution is impossible except for the simplest cases 8 conai-

derable effort has. been devoted to a discussion of practi~al methods of com-~

putations such as numerical and machine solutionso Descriptions of an hy-

draulic model and an electronic analog computer applicable to freezing and

thawing problems are presentedo

The second section of Part II deals with the writers' rational formula

for the depth of frost penetrationo Results of statistical studies involving

actual frost penetration data are describedu The effects of cyclic variations

in surface temperature and of multilayered systems on the depth of freezing

are presentedo

Part III deals with all phases of the writers' depth of frost penetration

nomograph including instructions for its use and its limitations and assum.p-

tionso A copy of the nomograph is included on the inside of the back cover of

this report.

v

The final part of this document -treats specifically the thaw-consolidation

pro:Olem which ar.ises during the spring me,lting period and with which pavement

weakening and .possible. failure are. associated. The problem :is ·still in t!le

fonnula tion stage·. Field. investigat-ions. of the thaw-consolidation process and

a.n hydraulic analog apparatus are needed at this time to advance further~ our

knowledge of· the problem.,

-. ·,.r

PREFACE

g.~;tt()P .. w!l.~,~hJf~s .the . .follo~l?-K-111 timate a:ims :: .. '•. . ' .. ' . '~ . .• . ' .. · .. .. :. ' ' .· •... ·' ' .· ,. '"' . ... . . ""' . . . .

. .. ;, ~ao .. ,, J?evel~pment. of.~ improved. methods,., 9f:· c,.omputing d.epth .an~· rate

?f..~r~ez~n-~ and thawing . .in soils.,.

' . ,. _bo Dete:nnination of the relationship. beb,reen_ the 1oad~_support-

it:l.g ,ca_pacity_ of soils. du~_l.ng thawing and the pertinent affecting· variables

:·(herein c~l~ed the "thaw=consolidat~on" problem}, and the developm~nt

()r ,improv~d enginee~in g design criteria therefromo

~ct,ive._work on the program was initiated by the Arctic Construction

rm.~. Frost Effects Laboratory in. May 1952o At that time. getiera.l outl.in~fJ.

of the needed studies _were prepared~ These outli.nes anticipated that

t~~. studies would be made in two distinct· phases: (a) .Theoretical and

ana}yti.cal, inves~igationss ·.to -be carried out by contract9 (b) Pra~tical

app1i:eat_io~. o:.f, the resul ts9 th~<?ugh formula~i on of ~gineering design

criteria,~~ .. to b_e carried, out by Arctic Construction and Frost Effects

.Labor,~tory personnel~ The outline of the basic. physical concept- of the

.: .. thaw~~Oil;$Olidation .phenomenons. as given in the outline of .desired· works was

prepared by Mro Ko A~ Linell, Chief9 Arctic Construction and Frost ~facts

Laboratory, on basis of previous personal observation in the field~

After review of the initial Arctic Construction and Frost Effects

Laboratory outlines$ Drs" He.rl Pt' Aldrichs Jrc and Henry M;: .Paynter of the

Massachusetts Institute of Technology submitted a definite proposal

presenting their proposed methods for making the desired theoretical and

analytical studies, A contract. was prepared9 and the studies reported

herein were carried out during Fiscal Year 1953 e

· The report its:elf', which was prepared by Drs. ·Aldrich and Paynter,

is self-explanatory of their work under the contract. During the course

of' the studies, the contractors held frequent conferences with Mr. James F~

Haley, Assistant Chief', Arctic Construction and Frost Effects Laboratory,

at which agreements.were reached from time to t~e concerning the direction

of' the continuing studies. Under direction of Mr.· Haley, the Arctic Con­

struction and Frost Effects Laboratory also conducted laboratory tests under

controlled conditions to check the validity of' the frost penetration formulae

developed by Drso Aldrich and Paynter and made· stati·stical studies compar-

ing the theoretical frost depths with depths determined from field measure-

ments during·previous frost investigational programs.

In the present report Drs. Aldrich and· Paynter analyze in detail the

penetration of freezing below the ground surface and present results of

their initial studies of the thaw-consolida.tion problem. The investiga..;,

tions ltill be continued by Drs~ Aldrich end Paynter in Fiscal Year 1954;

it is planned to make a more comprehensive attack upon the thaw ... consolida•

tion problem and to extend the present study of freez·ing to a ,consideration

of the methods of computing depth and rate of thaWing.

-­..

·.· ,:_. .. ,. ~

1-01. CONTRACT. -

On 23 June 1953, the writers entered into a one year contract with -~·~ ,· .

the Corps of Engineers, New England Division for the frost studies presented

in the following pages. This is the final report covering all phases of' that

contract, No. DA-19-0l6-.ENG-2314o

The fo"llowing paragraphs, which are quoted from Appendix "A11 of the

contract, describe the investigations which the writers agreed to conduct: 'i.' .•

1. "The Contractor shall- prepare nomor,raphic charts for use

in estimating the depth of frost penetration beneath paved and unpaved surfaces,

and'shall make a comprehensive theoretical analysis of the supporting capacity

of thawing soils to determine the relationship between rate of thaws consolida-

tion and strength.

2. During the first phase of the investigation, the Contractor

shall review existing literature ·on mathematical methods of computing frost

penetration with the objective of determining which of the various formulae is

most suitable. The Contractor shall then prepare charts to permit the solution

of this formula wit~out the need of ~aborious computations and still include the

following variables affecting the depth of freezing, air freezing' index, mois-

ture content of soil, thermal properties of soil (i.e,., thermal conductivity,

~~ specific heat, and latent heat), f'reezine; point of soil moisture and surface tem­

perature transfer coefficient.

The Contractor shall investigate more accurate methods of

analyses of frost penetration problem with a view to numerical and machine solu-

tions of the resulting equations, giving due considerations to the effect of such

factors as surface insulation and soil stratification. During the second phase

of the investigation the contractor shall, by means of theoretical mathematical

analyses, attempt to develop practical equations and analyses to express the

void ratio or saturated water content of any type of soil as a function of time . ...

du~ing the frost melting period."

1-02. SCOPE. -

While the proposed scope of this investigation is outlined under Sao-

tion 1-01, in actuality the scope has been considerably broader. ·Principal

additions covered in this report are (1) the presentation of a new formula for

predicting the depth of frost penetration and (2) results of statistical stud-

ies conducted to verify the formulae The SYNOPSIS serves to pcint out other

factors considered within the scope of this investif:ati on.

1-03 • PERSONNEL. !,':,..

The contract studies and the preparation of this report were carried

out primarily by the writers referred.to as the Contractor in the contract.

Several students at the Massachusetts Institute of Technology shared in con-

ducting statistical analyses, performing computations, and preparing drav;ings

for monthly progress reports and for the nomograph.

-2-.

. .

PART II. ANALYTICAL STUDIES:

'.L

2-01. MA THEMA TICAL FORMULATION o -.... :'"

a. Synopsis_. ""'.This art.i_cle treats the mathematical ba3is for

studies of thermal diffusi?n~ depth o~ frost and thaw penetration, a~ well

_as the seepage diffusion concepts required for analytical studies ?f_the

thaw-consolidation problem~

After stating the first principles from 't"'hich all . developmen·ts ,. _:,

begin, both exact differential equa·tions and approxima·ta difference eq~e,.- .

tions are given which embody these principles. Moreover9 practical and

fruitful fluid. .and electri•.} analogies to the thermal problem_ a:-e outli~ed. , .. ·.

In. order to over;~ome the handicap to analys~s impcsed })y the C_()m-

plex latent. heat condition;::;, several pr.acti•::al machine and hand m~thods o~.

computation are presented.,.

bo Fundamental Principl.;;:so .... At SI,lY p(_;in:t and instgnt :in a.

thermal medium, such as soil: tha local rate of change of total heat ''on-

This amounts to a statement that: heat is conser.veq and in the absence of

heat sources, its flow must satisfy the continuity equation~

du at + .divq e . 0 o o o ~· ( 2.' -~-- I ) ·

·Moreover!!. this directed heat flow """t· is exper:Lrrientaiiy ·fol.md to·:

be proportional to the negative temperature gradient,· vrl. th the· fhct8r'· dr···

proporti~nali,ty defined as the thermal conductivit-,{ .·k of 'the substance.

This heat flow law thus yields the seovnd fundammltal equation~

__.,.. ~

q = -k grad v "000 (2- 2)

where v is the instantaneous temperature field in the materialo

These two principles were first set forth by J. B. J~ Fourier in

his classic treatise on the "Theorie Analytique de Chaleur" in 1822J and

they form the basis of all rational investigations of heat conduction

phenOJnenao

The heat ~~ontent u of a substance is found experimentally to

depend primarily upon its temperatUre v, and one finds under constant

volume condi tiona, at least9 tha·t from a consideration of energy conserva-

tion~ it must vary only with the temperatureo For example, the experimental

curve of the internal energy u of water as a function of temperature v is

shovvn as F:i. ~;ure lo Three distinct.') effectively line~r :3egmen ts are dis­

tin~Jished, ;s marked by the numbers I, II and IIIo

The region I denotes the fluid state of' water in which the tem-

perature v increases approximately in propor-tion to the increase in stored

heat Uo This proportionality or slope we may define as the volumetric

specific hea·t; (under constant volume) C in the unf:("ozen or liquid phase, or: u

c r::: au J u ov

1m.f'roze.a

In the_ region marked !IID the substance (water) is in the frozen or solid

phase and the slope of the curve may be measured by the frozen volumetric

-~ ... -

-,

. . . It ..

. ....

TEMPERATURE

FREEZING.

TEMPERATURE

:nr FROZEN

I. UNFROZEN

INTERNAL ENERGY

u

-~:- INTERNAL ENERGY DIAGRAM FOR WATER

FIGURE

. .

.....

·· ~' ; ... The .. flat· or horizontal portion II of the internal energy curve dis-

plays the heat content which must be released or absorbed to prot:luce the··

phase changeJJ and which is defined as the latent heat of fusion L of the ma-

terial.

As stated, the curves I and III are not exactly straight, but the

deviations are usually negligible such that the mmall errors introduced by,

assuming C and C both constant are consistent with the assumption of ne~­u f

ligible unit volume change, upon change of phase.

Under these conditions, except in the region of the fusion (or

freezing) temperature., the variation in u with ~illle may be· directly related

to the variation in v, as the following~

Ou 0 t = c

ov dt

ooo~(2 ~ 5)

where C becomes Cu or Cf depending on whether v is above or below freezing,

respectively•

· ·Substituting·this last expression, toge.ther with Equation (2- 2),

into· Equation (2 - 1) yields::

C ~: + div (-k ~ v) = 0

c. Differential Equationso - If the substance is homogeneous

such that C and k are constant throughout a region, and is entirely either

above or below the fusion temperature~ then the following differential equa-

tion holds for the temperature field v: · .. dv·· ,

o--a-t = k vr 2 v

This las+ expr·ession may be re=arra"lged. to give the thermal diffusion equa-

tion~

0 v - 2 - -a\J v at oooo(2 cs 8)

in 'Whioh a = k/C is called the ·thermal dii'fusi vi ty. For three-di."'lleneional

(~, y, z) problems, this equation would be writteng

av 0 2v

2 2 ( d v a at = a 2 + + v

Ox a·2 a 2 y z

For ~~a-dimensional (x9 y) situations it becomes~ 2 d 'V\

ol) while for one=dimensional (x) problems there remains:

0 v d'2v ~ = ta~

c)x_

) o ••• (2 - 8a)

oooo(2- 8b)

do Approximate Equationso - While the expressions in the pre-

vious sections describe_the frost penetration problem in a mathematically

correct fashion., exact solutions can be fou.'ld only for a small number of

idealized cases, due to the complex conditiong of the latent heat transfer

and other ef•.t•ects o

When such conditions arise in engineering problems, it is the . . :

customary practice to resort to largely empirical techniques to obtain

reliable prediction. fonnulasu However, a useful alternative follows from

the substitution of an approximate system of simpler· mathematical equationss

which may either be solved directly in closed form.? or which may be solved

by various forms of computation or analcgieso

....

.. _, . ._

.. .

-"'- :_.-.

·.' In particular;;; it·. is. often. found effective to· replace the con­

tinuous :·conduction· equation (written in one.;;;.dimensional; f'orm' for simplicity):

q = -k : ; •••• ( 2 - 9 )'

by a "~umped" approximation for a finite layer n of thic~ess 6 x = dn, •

in the form: ·

~ = ... k n

'Y v ( n+l - n)

du or, by defining a thermal resistance ~ = dz/kn 1 as:

q = n +

•••• (2- 10)

The thermal continuity equation is here·considered to hold for ltnnps of

material concentrated at the terminals or end points of a series of' such

resistors; the relation between the net heat input and·the stored heat for

the n-th such lump would then be given by the equation~

du n

dt ·~o o o(2 ~ '12)

and the relation between the temperature vn and hea·t un' is indicated by the

form:

: : < ' I ~ .. !

where the f'lm.ction fn resembles the curve of Figure 1 •.

For hand computations and digital computing machines the oontinu-

ous thennal continuity Equation (2 ""' 12) may be expressed discretely in

ter.ms of the finite difference approximation:

dti u n ""'" n ( t + ~ t) ,~ · ~(t)

~d-t--~--~-~--t--~----~~-· .. o io o (2 ; ;.. 14)

to give:

In terms. of a three ... dimensional spaoe lati;ioe, the differential

·Equation (2c:o8a)· may be approximately represented .by the fini.te difference

equation:

v - v At o At

wheres vAt = v(X 9 y,i! 1 t + 6.. t)

v0 = v(x, y,j!, t )

vl = v(x +Ax, YsZ.$ t)

. v2 = v(x-Ax, y •. z!) t)

·V .· = v(x,. Y +Ay, c' t) 3.

v4 = v(x, y-Ay, i! ·t) "

v5 = v(.x, y,z+t:::..r,, t)

V· = v(x, y, ~-Ar.., t) 6

+ ._v 5_-_+_(:_6_r:_~_2_~_o_J 0 0 • 0 ( 2 co 16)

· If 6.x = a y =a~= !!:. s, this may be abbreviated to the form::

vAt- vo = (at.:)- (~1 + v2 + v3+ v4 + v5 + v6- 6vo)

A. S oooo(2"" 17)

2 or, with ~ = aA. t/~ s , to the expression~

6 . . v = Q ~ v + (1=6 ~ hr ~t Y L k . . o

oo "0 (2 .... 17a)

k=l \

In the same way, Equation ( 2 .... 8b) beoomes ~

0 0 " 0 ( 2 '"" 17b) v + (1.-...J,A)v k "+t' 0

0

-·-. -

~ .

•.

-....

::L.l.5. the ope dimensional Equation (2-8c) becomes:

2 =(3 .L •••• (2- 17o)

k=l :- ·- .. ,_

= fJ (v1 + v2~ + (I~2~)v0 ' ' : ~ -

It: :has been demonstrated that certain stability limit~' impose conditions on ··.(

the ma.ximuin values of ~. whi.ch can_ b~ used for. computation (1) (2).•

e. Fluid and .Electric. Analogies. - It ~s of. signific~t .. W.ter~st

tc·: :.::;I!lpare the thermal principles and equations to the analogous set of equa-

ti:)~J.S g~ver?l~g the diffusion of water pres:3ures in a ~onfil:lf:d fluid medilml. . . •, . . . . \ : .. -~·

This ana'fogy giv:es rise to a sim.ply realizeable physical model" a:·s·.~_e-~~~~~?~

i:n ;,$epti.ons 2-01-f-(1) and 4-04-a belovv as well as in References 3 and 4o • ' ~ r ' ' ' ' -·,

Mo:r-eove_r-_, the same. equations hold for the diffusion of <?u.~rEf}nts" an~ ··.) , ..

vci tages in an electrical conducting medium which has a fixed cap~ci tanpe ~ . . . . : . '\· : :) :- i :. . . . .l~:~ ~ '

This, too~ may be rea.li zed in simple one and two-dimensional . ,, - '-"' ~- . .. ' . . . ' . ~ •' . , .. , ...

passive ele·~tr-ice.l circ-u.its, as described in References 5, 6 ar.~.d 10 in ad-

di tion to the lumped equivalent as represented by the electronic analog

ccr:putcr outlined in Section 2-Ql,·f-(2).

Tr.o.ese analogies may best be S';l.IDI1larized :in tabular form as shm.m. in

Table I. ThE- ~1ractioal value of such analogies becomes obvious when the

m';;.~.h greater manipulative flexibility of the fluid and electrical systems

ar6 considered. . Mere over!' instrumentation is significantly si1npler and

~vre precise in the analog media than in the prototype thermal system.

However, one feature is distinctively absent, under normal conditions, in

the fluid end electrical systems that indeed characterizes the present

t!:erntal problem and makes studies difficult~ the presence of state changes

*N~"T1bers in parenthesGs refer to Bibliography

-10-

TABLE I

THERMAL - FLUrD - ELECTRIC ANALOGIES .. ~

. '

MEDIUM :

ITEM THERMAL. FLUID ELECTRIC

A-Variables (1) Heat u Volume s Charge Q

(2) Heat Flow q Flow Q Current i

(3) Temperature v Hea~rj_, H Voltage e

B-Principles:

· Continuity (1) ~ + yr.q = 0 ~s +v·.Q = o d_g + \j. i = 0 -Qt ot ot ConductiVity (2) q = -kV'v Q = -kV'H i = -(f "'Ve

f.

Capacitance (3) Aeu \

= Cdv dS = A dB dQ = Cde

:-~

... 11-

,', ' . . '

and the concurrent latent heat effectso The analogy to this situation 't , . ,·,

would reqtl.ire the equivalent capacitances to remain temporari·ly infinite

.. . ::. ' ' -~ .: j· . ' . : . ·. ' . . ' . ·. ' .

until a corresponding ~ount of fluid volume or electrical charge had been ..

-.. transferred to or from the elemento How this feature is secured in the

hydraulic analog and in the electronic analog is described briefly in

Sections 2-01-f-(1) and 2-01-f-(2), respectivelyo

. f. Practical Methods of Computationo - Due to the ccmplexity of ' :: --. ~ .. ~~. . , ..

the latent heat conditions in addition to the normally heterogeneous nature

of the soil, and the randomness in the surface temperature distributions,

it is not possible to obtain very many practically significant solutions

to these problems in closed, mathematical for.mo

In order to over~ome this handicap to rational analysis~ several

useful machine and hand computation methods are presented in t(he following

section which are based on the principles outlined in the previous sections.

{1) Hydraulic Modelo = By applying the analogy relationships

of Section 2-0l .... e and the "lumping" approximations of Section 2 ... 01-d, one

finds that it is possible to solve a wide variety of thennal problems to an \:. ;' -~ .

excellent approximation by means of a simple hydraulic model~

Basically, the hydraulic analog would consist of a series of small

vertical chmnbers or wells connected to each other at the bottom through

laminar tubes or orificeso Each chamber represents the heat content, both

volumetric and latent, of a lump or layer of soil at a particular deptho : .~ · .. -; ' ': . . ' ,. :

Distance along the model corresponds to depth in the soilo The surface .. ... -... , ..... ' .-~ · .

level of the liquid in the well is analogous to the te.mperatureo .To. simu-•1:'

~· i. ::.

late the latent heat content of the layer an expansion chamber of suitable ,·,·.

-~ ;

area would be provided at a level representing the freezing temperature of

the soil moisture. The size of this expansion, as well as the size of the . ' ~

vertical Chambers. would be adjustable to correspond to the variations in ... latent heat and volumetric heat, respectively. More specificallys the

sectional areas of the chambers are directly analogous to the C and L

values, since storage volume S represents heat u and level H represents

temperature Vo

The interconnecting capillaries or laminar orifices are

directly analogous to the conductive paths with the fluid conductance rep~

resenting the ther.mal conductanceo These conductances can be made adjust-

a.ble in a simple fashion so as to readily parmi t wide variations in the

values of k between layerso

The surface temperature variations are directly represented

by variations in the level of an overflow-source tank at the end of the

· model which corre.sponds to the ground surface?

The flexibility of .such a scheme is directly evident~ more-

over, the same model can be used to represent consolidation~ seepage now and other diffusion problemso Such a model had been proposed several years

ago by. Barron (4) and has been successfully used for latent heat problems

by Backstrom (7). For diffusion problems associated with consolidation

phenomena such a model has been effectively employed by one of the authors

of this report (3). ....,.-

(2) Electronic Analog Computoro =The hydraulic model just· ._, ~· ...

described becomes complex for 2~dimensional problems and 3-dime.nsional -· representations seem entirely impractical, due principally to the difficulties

of interconnecting the chambers in a manner which will still per.mit direct

observation.

·· .. __,_

- ,._.

These disadvantages are not present in an electronic com-

putor operating on essentially analogous principles o A 2-dimensional

array of elements can be mounted on relay racks covering a relatively small

amount o£ wall space.

The basic computing elements can be designed to solve the

"lumped" approximate equations - the same as those which govern the be-

havior of the hydraulic modelo For the sake of simplioi ty, only the

circuits for a one-dimensional homogeneous oamputor will be outlined

here, although, as stated, this method readily generalizes to the 2 and

3-dimensional oases.

In tenns of operational blocks 9 the basi.c structure of

the computer components representing one lump or layer of soil is shown

in Figure 2A where .the letters stand for the operations indicated beside

the drawing. In particular~ the "Z" .... component repre_sents the latent

heat effect and has an input ~ output characteristic similar to Figure lo

Thus each layer of soil is represented by an assemblage

of computing components Fwhich are interconnected as shown in Figure 2B

to represent the entire soil masso

This particular type of electronic computer is referred

to as an active representation in which all variables are represented by

analogous voltages in order to expedite the instrumentation and inter~

connection. Thus ground surface temperature fluctuations are represented

by corresponding input voltage variations applied to the elements repre-

senting the top layer of soilo

·-.

.......

NEIGHBORING ( TEMPERATURES ~

~- - - - - - - - I -.- -,

SYMBOL NAME OPERATION

A Adding Output

I : I v­t z I

Component =r(lnputs)

c Coefficient Output Component = Cl({lnput)

,

J Integrating Output Component =/(Input) dt

z Inert Zone (Similar to Component Figure I )

A. ASSEMBLY 11

F11

FOR ONE LAYER

v: n

v;.+2 .,._.....,. ..._ __ ........ B. INTERCONNECTION OF

11F

11 COMPONENTS

SCHEMATIC BLOCK DIAGRAM

ELECTRONIC ANALOG FROST COMPUTER

FIGURE 2

-15-

.....

This particular representation is ~herefore a so-called

"active" analog, in which all variables are voltages; in the absence of

latent heat components, such computors are discussed in References 8 and

9. This type of analog computor is distinct from the more conunon "passive"

analogs (5) (6), in which temperature (v) is represented by a voltage (e),·

but heat flow (q), by a current (i)o.

Such a computor can be made to operate on so-called "fast-

time" to permit display on a cathode ray oscilloscope, or it may be.run

at "slow-time" (over·a period of seconds or.minutes) to permit recording

on a single or multi-channel chart oscillographo . Actual surface tempera-

ture observations may be traced by a stylus input device which drives the

computer, permitting direct check against field temperature and frost -

thaw depth measurementso

The internal circuitry of a single "F" component is indi-

cated in Figure 3o The input resistors (ROY and the feedback resistor l.

(Rr) correspond to the thermal resistanceo. The feedback capacitor (C)

corresponds to the volumetric ·heat of the soil layero The dropping re=

sistor (Rd) determines the ratio of latent heat to volumetric heato

Since the computor may be arranged so that these circuit elements can be

varied, it is possible to represent variations in soil thermal properties

in a routine fashiono

The only changes in the ~bove circuit required for 2 and

3-dimensional representation would be to provide four and six input resistors

- . . . . : ,.,_..._

R. I

R. I

R. \

:!E s

CIRCUIT DIAGRAM

SINGLE .. F" COM.PONENT

ELECTRONIC ANALOG FROST COMPUTER

FIGURE 3

-1.7-

. .

.·.·:,:

(3) I .. B .. M.. Solutions o .,, As a means of obtaining reliable .. i.i~ ... . . ·.· ''•-. .:: ."~. ~--~! ;-· ·. ~ ' . . . ·-·

rational data on the freezing and thawing process as par:t; of the second " ... :~ -~ ' ~ .... ' -~ ,' ~ ... .. :· .. : ... ' t': ~ . ~ ~ . . "• : .;::

phase of the contract~ and due to the non-availability of the: ~~~raul~~ ..

and electronic analog equipment just referred to~ a numbe~ .. of computed

s·tudies were undertaken using the VIoBoMa Card Pro~ra.mmed Calculator of

the M .. I.T. Statistical Services Division. . .....

These studies had three main purposes 9 as follows.~

(1) Verification of.the difference equations as outlined

in Section 2-01-d, including the effects of latent heato ·

(2) Determination of the effect of the sh~pe of the sur-

face temperature-time curves on the total depth of freezeo

(3) Investigation ofthe shapes of the depth~time curves

in all essentials the-procedures outlined in Section 2-0l=do ~owever~

to the best of the authorsY knowledge no numerical solutions have, been

published involving the diffusion process when latent heat effet:~s are ' .....

significant.

In order_to ascertain the precision ?f the ~~erical ap-

proximation procedure it seemed desirable to run check solutions to com-

·.

·j,., pare with the rational analysis of Section 2 .... Q2 .... co Accordingly, twelve

test solutions With. various . values of o( and fo'. wer~ .. carri~d ou% on' the' .c.<~~,.

.;:x; ... B~M;.· ~qu:i:pmerito · These-· ·are- presented as Table's · B-I throu·g:n··;B~XII~"' ::·in.l

c:t'tul'ive, :fu Appendix· ,B, : and: iasSuiile a< step~:·charige in surfaC'e:._ t'emp\e'ra:tU.re~ ,,,;. .. :'

.. ~ ' :

Two I.B.~ solutions given as Tables B-XIII and B ... xrv in Appendix B.9 dis-

play the efrects of a sinusoidally va~Jing surfac6 tamperatureo They

refer to purposes listed as items (2) and (3) above and are considered in

Section 2-02-fo

These fourteen solutions, as well as se"V"eral othe:!' check

and trial solutions not presented9 were obtained by automatic computation

of the following set of equations and eonditionsg

s ::: s + ~ I + e 2e k) \e 1

n~ ,~+1 n, k n- , k n+l, k n, 0 0 u tJ ( 2 .... 18)

where e = s ... 1 for s k> 1 n, k n· k n,

~

e~ = 0 for 0 ~ s ~ .,,

k k J.

. .u. .9 na

e = s f"o:r· s < 0 n.!' k n, k .,., k ··:;

The set of condi t.ions ( 21jl.9) defines a normalized. ~re~zi:m.

of the internal energy - temperatttre relation simila!' to Fi~J.re L .. wi t.h e

being a meas~.lre of the temperature ·v and S a measure of the hea.+ con tent u.,

Rendering the relation in this simple f.:;nn permitted a sai:;.i;::;f·9.~t·:~:r-y pr~=

gramming of the calculation~ However$ it makes ne~essary th6 propex· scalL ...

ing of variables and interpretation of results, as indicated below and in

Appendix Bo

as

The dimensionless parameter ~ is defined in Section 2~0l~d

? = a ~t ~02

where a is the (thermal)dif.fusivity, ~.t, the time increment, and ~ s ,

the depth incremento For stable calculations. in the ona=dimensi.onal oasa. ~

must be taken equal to or less than Oo50i Tl1e values ch~:sen for the. various

I.B.M~ sclutions are indicated in Appendix Bo -19-

• I

. .

~ .

' :.·

Th~'- 'i-~f~f:{~h;' ~~tween the' ~ciip\it~d e - 'value~ 'and the actual •"' • ' -. •••. :t l

, •• "' • I .I ';_ -:. ~ ..... }'

' •• ~-· ( 2 :: .. ~, .2<>} ... , .

~ • '·: ' L.

whe~-~-;~ .. =. F/T. is the a~i?U~~- or equivalent. step change in surface tempera-·.· ............ · ... ~s· -·

ture' 'below freezing and ' In other words~ the e - values = Cv /Lo ' s .

bear the same relation to the parameter ~ that . the v - values bear to

. V:s• _ The time .. t oorrespondin·g to any k - value is given by the defining

· relation: ·

t = k~t ••.• 0 (2 ... 21)

· ·. ·.while· the depth x corresponding to any n - value is gi veri by the definition:

· x = nAs ·~41011(2- 22)

Thus instantaneous temperature _gradients as well as tempera­

.. · _·,fure-t'ime curves 0~ be plotted from the values gi van 'in Tables B-I to

' v.:.xrv 'just as' for actUal' thermocouple datao Such . curves were dravm by' the

authors and the oonolusions are given in Section 2=02~fo I

{4) Hand Computationso ... Using procedures very similar to

those outlined in the previous section, it is possible to carry out numeri-

cal solutions using a slide rule or. desk calculator. In fact, ~or certain

oases completely graphical procedures are p~aotioalo

Efficient tabular forms for such numerical solutions are

detailed in the available li-terature (~1) (12) (13). (14) while the Binder

graphical procedure for heat flow problems is outlin_ed by Ja1cob ( 15) o

TABLE II SAMPLE HAND COMPUTATION OF FROST PENETRATION

(A) NUMERICAL CALCULATIONS:

~· 0 1 2 -- . ... -

0 -1.40 C>< . -1.48 C>< -1.50 [X -1.o6 - o42 ... o90 ..... 36 - ·19. ~ .32

- .34 - .34 - .58 - o58 .. o71· - o71· 1

- .46 - ol8 - .58 ... ct23 ~ o71 ... o28

+ .,12 +1.12 0 + o97 0 + o81 2

- .08 - .03 ~ .18 - o07 - .12 - o05

+ o20 +1.20 + .18 +lol8 + .12 +lol2

3 - .02 - oOl - o02 - oOl - o06 - o02

4 + .22 X + .20 ·x + .18 IX

(B) KEY TO CALCULATIONS:

k k+l

n

n+1

3

-lo4B >< - o73 - ')~

..... 75 - o75

~ o75 .... ~30

0 + o58

.,. o09 - .o4

+ .~ +1~:~09

.,. o05 .,. o02

+ o141><

.. .

. ..

~- ... "'

. a. :· ~{_8mc:)PS~s·~s~/0.~,:.Th~1S;z:,a~1?t,~11~; ~.R.~~(lJA.e&,;f,:~~e }'!~:.~:~~~;~·J::..c~~Y~.lopmant of . .

a. rational formula to predict the depth of frost penetration; which takes into

cJJ:~id·e~~tion nearly all of the factors of primary signif_~ca.ri·~~e to· the prob-

lem. The nomograph included in this~.inv:estigation .. employs- this formula as

. This formula is premised on ootalnirfg···-ehe~-Jixac-t· ·s-olution to an r::: ·-.:· :~. ~: ~

! "-·(~·- ... ·•;>:.,.._, -~-...--~-~.:,. .. ·.:..- ......... _,..,; --· • ., .... :.·..;~.~· ....... i" ...

id~iii"zed .-problem which is sufficiently r~pr~,sentati ve. of actual ·physical 1... • ...

' '

present the results of their statistical studies of a~tual frost penetration

da"t1~.·-: :It . .5~-s .COll.Ql~de,d;,.:tQ.at, .. ,~R-!3 ,.r~ti.onal ~.formu,la ,; gen~ral];y: __ yields better re-·:o.i,..,,..,_.~·····-- .~ .·I.·~: .• · .. ~)\~'- , .• ·..._ ,_,..t'.-. l ,:·r! ..... • •-r..,.".~···· ~·,_...·~ -.•. •'" ... ->,.,1 .· ••• ! .. ~· .. ..- .. •. ,.,. '-"1'' .. "'• ·- '-·

P~E!~~·~r;.~.~-~9~'~\~~i ~h~·.:-.}~~~,~~;.f't. f)~~~z}!lg ... L:~[l_,:,~P.a~!f·~:~~-:tr.~~~-a~}Y,~,;;.~~!3 ~v}:t;ila~~!.r~

system is treatedo ', '-· ~ : ... . ..

·b .•. , .. R:~:vJ~- of~. ~itera:ture.. ,-~ :As; .. ,a .. .fi .. rs .. t, ~~,t~p.: ~;n, tl1~ s~.·~~tudies, · .. :. ~--· . . ' ·--~-·" --·. ~. '"·-~ ·-- . ,.: .... c.~-- ... · ....... ·.·. .... .. ·- .. "": .

exis~~t:l;g J.;i..:b~r._a.~r;~. p~rj;a~~ing, .t9., p;r::~dic:i{;iQJ?.· .o,f()·~he. 9~pt1;1 "qf f~ost. ,.pen.etr~-~ .,.,.....,._,; ... r.;._.,:~ ., ..... ~ •. ..._.-- ·"'· , ........... ~.-....... ::t .. - r·,._\.. .•. l .... '/-·-~.,,... ,.1'-...1• .... .-.......... _ "':'...,.1 ......... ~ •. \ •• :-~ -.• -... _ ... .

~~PP, ~~t "'rev~~~.P.·- .. :-FTh_e_i :.Prino~pl~:-. -~J?j~c:t±;,k..:.wa~, ... ;G9 .Q.e.t~.~n,~ _the. r;t9st su~,t~ ~ ... .. !.';._··./ •: .. ) . .! ••••••. ·-·.· ·-· ....... -~ .. '.·~:. -···-···- . '.··· •• : .......... - • .... :!.""' ·- ... .-~.·.·-~-i .. • ..... · . .;., ........... · .. 1.•· .••. - (, . ·- _ .... -;' ;:; 1,< ._._.,1 \

ab~~ .. fq~}._a: :9_~.· W~_~ph_. ;tp,_:bB:~~c· ~1,1~ 1 ,.:t;l(?m,~graP,J:l~·- :S9.~~i,4t31:"8;~,~~n.; 1V.8;~ .. , g~ v;~n,,;f;? .. t~ee .... ~ .•. (: •,.,.; ,•.,, .. ··.·,.: -.. ~·-·H '"' \,._•'t, ··~··' .,,,'-•.••: ',,' ,, \f ,..,..,,., -~-. ••••-,,J ~ ). ... ~..., ........ ,_ .... _.,? .... , .. , ... '\,,,,\-; •'oo:o ...... \~ ....... ,.:.~~ ,,',<•~ •,

criteria: . 1'~ ...

ca2.!2~ rr..rs.r-:~

(3) T.he statistical reliability in predicting depths ·of

frost penetration as compared to field observations.

In particular, four -formulas were considered, namely~

X =V~F. eo a e (2 - 23)

X =VL + l:J:BkF ~'{,(C8 + OoOlw)

=~ 48kF X .F

L + c {v +-) 0 2t

octao{2- 25)

X =J--24k_F __ _ L + C (v

0 + ~)

oooo(2- 26)

The first equation is the Stefan equation which considers latent

heat only. The second equation is the so ... called "Minneapolis" fornrulao The

third and fourth express"ions are formulas which have been studied by the

Frost Effects Laboratory of the New England Division of the Corps of Engin-

eers. These last three consider certain of the significant volumetric heat

effects by making various rough approxLmationso

· After a prelimina~ analytical and statistical review of the above

four formulas, the writers concluded that Equation (2 ... 25) was the most

sound. other statistical studies(l6) comparing actual depths of frost pene-

tration against those computed from the above equations had indicated that

Equation (2 - 25) gave reasonably reliable resultso

However, the writers were not entirely satisfied vnth this equa-

tion, since it did not account for the volumetric heat effect i.."l. the unfrozen

ground, which causes a substantial reduction in frost depths in the more

-23-

. .

- ..

"', .l I .~,

r ~~f~ri~~h~~;:~ ffl1W?;1,~~ ~~j~~ ~e;v~-~-,~P~?-_1J -~~: ~~~~~ 4i:.:~P9:~~~·~::Btp~ .}?;~·: ~id.:ffi1:ti_~~:

cal in nature to a formula developed by w. Po Berggren (lB) ,_ ,wh~"Q.h. h~Jt ,.b~,e:tl-,··~·· ·; w'-•...,.1 l'J ... _.'I..* !, ,\ ~.' ·~~"'-J',, ,,! .1•~'-:.."'~··f?

expres s~~~.r .. :?f .~~~? :f;?~h;) x~.;:;:; :;);_'f

X=2Bv;:t i '!. ,., ·. (' 'I ( 19 ) by- ~Shehn:em'· ·.· '· ·· •

27)

\f.;~~~:~~:.:c:~-~ ';~~~~~t~?~~~;;~( ·~~~{:{~f~·: o~!{~~~P:.~Pd<:f~:~::.!c-~::,:::·?J;~_):I~E~??J!:Bt( =f~port by

Shannon in the reference cited had caused the writers to be initially.mis-_·l ·:·:-·.:-.'~:..·•:'\' ~'ii<···;.',·.;~ ~:-> ··?- ~-;::,::: ~--~'!.:.k ';,7:/'~'J::~:\::'· '}.:~:.~· (~:.Y:.~·:;~l:'.f';"J~_:~f;· ~- c'. :;·;:::·:~:t D'~-,~~~~.r.:· -~j

lead;, the 5o% d~scr~panqies mentioned .. ~P. .. connectio,l). nt~ .?b~o:p. ~ s~ .Y.~19ul..~--. !tX>~~t·.~.t ~'}(J .. !. ~ :: \"!;! .. ·~-~· . .. :~ .. :..r: f):·.~t t< .. ; -~--~:~~~.:.~~~:·-:'~(1~3' .. :- '.t_ :k.!. ;: .. ;.~·:Jl~;_~, o{.J~:lr'-'.t.'3t} 2;-ti~ ...... ~!~.: ~.-~-~ ·> ~r·:··f-..-.· ~:; ··1J . . "5'.

tions must be attributed largely to -~~-~-~~q~!l~:e,, ~a.t,!l,. ~,_,,~lle.:"~p.~~p-1 .)>r.op~rt~,~s h :...{;.,..,;_,#~-E........ -~·~--· ,.;!~J : j-,,.', ~'3-i- . ._ .... 4 >" J ~- '.. p-~ .. __ ·:. .. ~ <" ·'" ... .:.. .I., . . ~·· ,:..;... .. -.

of the soils consideredo

The statistical studies 9 undertaken by the writers ~~i· . .-f!>~lN~.; .-::.:·· --- ,.(~~~ =--~ ;_--::;_\_~,.. j

Frp~~ Et:.fE(cts Laboratory as reported in Section 2-02...:d.!"{i2)j .. t.estify strongly ~'- ·::·~ ~-.: -.~ ~ ·' -~ --~' . . '~~ ... ··c ..... ,;.;;--·~~ ·~i :/ \.,J .-. _.._.._

to the general reliability of this modifi.ed~ B~_rggren formulao .. . .. , . · (

1i¢· .. ~i .. n·~-----~~>~.' . .r;-~:~~:~.~:~- .. ~: .. ·\~ i:.~-,. · __ J._=\ .• ~.-.-.z;;.·:-~ (_;:-=i."!..~~~ fJ(.).i-.~-~~-.) -/-;·:_ .. : ... - .. ~->~~- .:·r.{} r:}·l_:; .. ~-

. c •.. _,Deriv~tJ,on and. In);erpreta~ion of Rationa_l: Formula.~ .. ~ A: de,. .. ~;;;./',_\.' ~ .. !.:.: .. ~: ;.~ .... '.....:·-~-~- ; )...~. ·.~· -~~·...-•,.,. ~ _:_.:; .. i,_ ,j_:_-:_J _.,..· ~·~·-:· ' •• ·;._ • .._';,_' ,.· < -;)' ·-. .·_: __ J..,·~~;. ... _ ,_ .... , .•. '.

penetra~i<?~ is gi·ven .in App~ndi~ .. Ao... ~~ ~ost u~~ful_ £om fo~ .usi)l.g t_h~ ....... , :·,::'~~~(,\) ~~~i~5-~ .... :~-;~ .. ~--; :;).'.I:-~: ... ,.,~\- .;:: .. >:~--.~: ·-·~-·,3 >~:":·..t:-~~:)· ... ~~3't.:J. ~:;~..:_ ... _ ~ .. rJ.<).f~~.~~..~:.'"£ ... .t.~ ;-~ .. \,._,··;;.~~-..~..;· ~~~~; ~"'.fC;._.,_-.J~-~r.·J1 ·:

foriin:tla is given pelow as ~quation~-·-{2- ~)o_ .. , . , . .- . . r!:;:: 4) ·-:~~~ .. :~:-- .-.~~~:.·t~~.~:y i~::~:~~:':::~·.:.-~:.~ '):~ ~-,~1<--~~j:·~ ·.:. ~~~-t~~:~·:-·)·-<.: .. '·.:;· .·.~-B·!}.'~::.:.:l::.~~~--.~ 4: .{~-: .. ·i .... ·~-< .~--~~~.:··j~- ~-~I~.~··:··i·.\··~:

BA;~.--·~d~~-~~r.?e t~ ~g~~~~i,~.;t'<:_:~t·,;:~~;-·~s·~~-~g fll:~t.~te~ s~pf~ !-~:-.·~ .~~~·7-

infini te mass of uniform properties and having ~~~~~-~_ly $.<}mifS?,;:m tell!J?~ra~e • • ·- .· ""'" ... ~.~ ~:..J~.-~:qt)_,~ .l"·~, .. -:l" lt ··~ ·-·.}

Iti. ~~ :U§~~~r, assumed that the surfac~ t}~~~er~tur~~.--~ i~:···~~~~.el£~~"1gh~ged from

its initial value v0

above freeziri-g to a tifutperature v8

below freezing.

The equations which :nust hold for this problem are the diffusion

equations (2 - 8·:;~) in both the frozen and unfrozen soil as indicated ili

Figure 4. The latent heat property becomes a continuity oondi tion which

states that the rate of flow of heat in the frozen soil at the frost inter·-

faoe must be equal to the sum of the rate of heat flaw in the unfrozen soil

at the interface and the heat given off at the interface when the soil

moisture freezes.

The solution of these equations may be written in the form:

I)O eo ( 2 - 28)

For practical computatior1s for the depth of frost it is generally

as~ed that v represents the avarage surface temperature below freezing s

during the free~ing periodo '!".nus~ if F represents the air freezing i!ldex ·

and nF the surface freezing index~ then~

nF = v8t

an~ therefore:

x=AV~F 0 0 0 0 ( 2 = .29)

wbich ~dll be called the modified Berggren formulae

The ter.m )\ , a correction coefficient modifying the square-

root ter.m, which is merely the familiar Stefan equation~ is a complicated

£'\motion of three dimensionless parameters 0( ~ ~ and 0 It has been

shown that for all practical purposes may be assumed equal to loO in

which case )\. is given graphically by the curves in ~gure 5 as a function

or 0( and ~ where~ ·

thermal ·ratio, C( = v0 t = nF

v 0

nvs

CnF and fusion parameter, ~ = -Lt

0 \) 0 0 (2 - 30)

oooo(2- 31)

,-- .;.,

r./" -_

• 4

.. \ .

- ..

•.. .~- ...

THERMAL PROPERTIES TEMPERATURE

.,

THERMAL CONDUCTIVITY: kf VOLUMETRIC HEAT: Cf DIFFUSIVITY: Of = . kf /Cf

,BELOW FREEZING

FROZEN SOIL

UNFROZEN SOIL

LATENT HEAT CONDITION

k ~Vf = L dX + k ~vu f ~X d l u ~X

THERMAL PROPERTIES

THERMAL CONDUCTIVITY: ku VOLUMETRIC HEAT: Cu DIFFUSIVITY: au = ku /Cu

DEPTH X

ABOVE FREEZING

I

I DEPTH OF FROST PENETRATION:

X

INSTANTANEOUS TEMPERATURE IN

I UNFROZEN SOIL: Vu

I I I I I I I I

THERMAL CONDITIONS DURING FROST PENETRATION

FIGURE 4

-26-

. . .. ;.

w •

~ --

. l '\.

A: .. 1-z w 0 LL LL IJJ 0 0

0 1.0

0.9

0.8

0.7

~ 0.6

·1-0 IJJ a: 0:: 0 0.5 0

0.4

0.3

FUSION PARAMETER, 11-

0.1 0.2 0.3 0.4 0.5

-~ -~~ f"'

~ I I I

'~ ·"' ........ .......... ~THER '- ......... MAL RATIO _ -

'~ ~' ' ........ ~ ......... , a_ o -

"" ........ -\~ ~'

"'Il

' ' ...... ........

~ llir... ......... ......

~\ '"""" ~ ' "" ~ ""-.... -r--. ........ -.....

11\ ~ ' ,...

' ..... ,-..... """'iiiii

~ ~ ........ ~

..._,. 1-- 0. 2""' '\ ' '""

~, """'ii r--.... . ~ ....... ........ I I

\~ ~ ' "'-, ..._, ' ...........

.......... ........ ........ 1\ \ ~ ' "" "

-......_. ~

...... ........ I I ....... 0.4 ... ' \ \ '- "" '- ""~iii~~;

' .......... .......... I I

\' \ "'" ' ~

' -......... .......... -........

\ ~ \. "'-~ ' ..... ~ ..,. --., I

-....... ......... o.e .... \\ \ ~ ' 'lo.. .......... -......._,.. I I

\ ~ \ "'- ' ~~

........ .......... r--- I I

\ \ \ "'" """ "--......... ..........

o.s.:. I ' \ \ '"'' '

~

..... Q-::1, L \ \ '

""''iiii

' ~ ·0 ...

\ l. \ \ ~ ' "" ' ,, \ ,

\. // "' '-.. ""' ' - ~

\ \ \ ~ /

' ........

' \. ' ...... -...

'\ \ \ ~ .... ,

r--. I. S .;;

' ' ..,~

'~ ~ .........

~

\ ~ \ \ ~ ......

' ~ \ \ \ r'\ ' ~ f' ~2 ..

~ ~ ' ~ '-... ·0 ...

\ ' " ' "" ~ ~ \.. ~ "'---... """"""' ' '•,.

'\ \. r"-. ..._, ........ .......... -.....

' "'" ' "' "-.. ' ~" -\. " "" "" '-. ' ""' .• ·0 ....

"~ ~

'" ' .............

...........

'"''" ' "" ....... ~ "- -.........

~

"'- ' IQ--:: s 4.o ._ ~0-~e:o •' I .,, I ·JO .......

CORRECTION COEFFICIENT IN THE

MODIFIED BERGGREN FORMULA

FIGURE 5

-27-

- ·--~

. ., ~ :.

. .'"'J . ,;•

The thennal ratio o( is a measure of ~P.~,.:.--F.~~.i8~?9t\: t~t~·;:; inti?~at~h~~

gr?~d.,.t~I?~r~ture, or in pz:_actice., the mean annual tempe;~~'fur.~· a~~;Ye the .~=.., r;· ~

freezing. point, to the time-average surface tem~erature below freez~ng during j

the;.~'treez:i.ng,. period. It is apparent-~'-t:hen-,·~--that· .. :·~-·-0<~·~: ·d;~,;~as:s irf .. the colder

climates. In Parts of Alaska 0( ma~"b~ :e~~' :r\~~tu~il~ negative.

< : 'f'h~ :. fusion parameter fo· is.~-~-·~~~-~~.~~.~~§~~;_:~h~.--~_!1~-~t removed in the :· ~

frozen soil below the freezing point as::~c6mpa·~~d ~ t'o-:~'1-;he ia~~nt ~eat of the . . ~ ; ..

. ~ ./; f-.::,''::· ·• ~- ;.~ •.. ~-·~ ,~,-~~.:. -..J .. ;'~"' :::.; ,;.·· . .._ ; .. ·.·~:::. . ~~ . \' ...... ; ":-'-·.... ·-q• ...... '; ;.., ~~· . .t.-:.-:- ~-~·:··':

soil moisture. As the ·ra.t·ent. heat .. term becomes·· very large, '# ., approaohe·if'

zero·:·z·, ··rt .. ·; i ~ s-·~~~; ~-fttthcirifr6~~~- that·· ,A.;,-;~_··~~·fik~ littk:ih~~;: tr~~i~~g index

and iS. ther~fore' ·targe~t :iii the ar6t':{~-i: clim'~t~~~ 0 ·- . l' ;;}' ··L-i ~I·'Jc, ·'·-'··; ::

... ·- - } .. ~.. ., . !' "!-:: ..... ·- -.!o~ . ·! .,.,, ... :~ .. ~-~--~~-- ~\· ....... :~~ .. ;., ---. -{·, ~~ ~~--~· • . ~.··:,·,N.:. . . : := .·"': • .>:; >:.-·r·.(~ ... ; d·:· Comparison. of Rational·· Fom~la -m. tli' bther Formulaso -~ A com~.:.

~~rf~bn of'''~quation '···(2 L~ •:ijf'·lri_·th bth~f to~Ji~~i:·p~opcis,~d ~d:-~~i~tt~;;lg-~iJi~ti:·_';

... ;~~: ... ;'~l~~~u· )_./·:_~·;: ~1 -~r~·. ~-~.:~~---_;~--- ·:,,;:.·_.} · · ... -;<~ ·-;,_:-~~~~--;~~~ -.. r:.~ .~.s~ ;:> ·._;·-.~--~c~~l:* .... t~-J-~·~-~·.:~ ~ ;~ -~.'-.. :~_:· .. :1·;:;- .. :~.:·y'·_:.~ -· .. <~~i pute the depth of frost ·penetration will be made at ·tvro levelso First;,····the

•. ass~pt.iori~: riad~ i'Ji"··~adh derivatio~ ;-~r~'··6~tlingd.z. ~a ~:.: g~ii~r~l e~~luation ~r

th~·,··~rf~ct.;"l~-i- ~h~s~ :~,·~g~~ptiori~·':' 6ri ca~:Pilt~d: '·fa~riit~ is~:r pr~~cihtgd.o ·:.: .. "sec6Iia~·~·>

reslitt~ b£:. st~t'i~~ti~kl' -~~dies '\i~irig·;·~gtli~i-''fr6~t''':peri~t·r~ti6n d.l:{f~~ ·af·g~ ~;·;:;;~:·j·

( 1) GeneraL. - Eq~tit:f6ris :: (2~ ~''~§-)''~ ( 2~~0

--·,g);· :;·Jild';~'f2::; !:·126) ;:.~g;.:t

Section 2 .... o2-b· will be ·used _:r6f:·this-:6omp~fi~~A. ·., ·:-;¥t·~·c·~ b~;.)~h·ci~.:-1that a~l

:/j\';._'i~-- -:. '- -~·-·,'.· :~·.: _···.·,·,~:.~ .·t.·:.~';··· . -_~. ·H···--[~·:, ,·:;.:>_,_ ~-~-.~--:~_,-; -~_...~}l···.:".;'-;;_:; ;:.,::; .. ,""_~.:.~;·_~.r..·.-..- :-_-:~:-.. ':A:r:· ... ~-.._;·-~:_.i;.f'; three expressions can be rearranged and ·Wri tteri in terms . of a correction- co-

effib':{~~:{·~~i-~~ is' som~ f~ction 6r .·.· ·2:(·-' ... ;,~d::· ';;d] .,'fa~ ;:a~'firi~d:·y~ ''·:th~frd' . . .

, ... , r•··,;J- ,.l'" .-., ., '"\,

p~e~i;~~ s~otion: Thus~

.,:,::·'::::,.::i:··'~:·~~.~~: ·,·

~

·where A is given by the following expression for for·mula.s (2 = 23) 3 {2 - 25)

and (2 - 26) respectivelyg

Al = l.o oooo(2 ... 32)

A2 = 1

\)1 ooo&{2- 33) A.-{ a(+ Oo5) +

A3 = 0.707 0 0 t.& ( 2 ~ 34) \} 1 + ~ ( 0( + Oo5)

Curves for A.l• A2 and )\. 3

are sho'Wll in Figure 6Bo

Since the Stefan equation (2- 23) ignores the volumetric heat

given of:f as the soil temperature is lowered to and belcw the free:zin.g

point, ~ is equal to zero and therefore. )\.. = 1.0 for all o( o

Computed- depths of frost penetration based on this formula will generally be

too greato This is especially true in temperate regions where actual values

of A may be 0.6 or smallero From Figure 6A it is seen that in temperate

Kansas, for exampleD the correction coefficient 'i:;J perhaps Oo55 whic?- means

that the actual depth of frost penetration would be about 55% of the value

determined from the Stefan equationo O.a the other hand9 in Alaska. the equa~

tian would be expected to yield good resultso 0

Equations (2 - 25) and (2 - 26) consider the effects of volumetric

heat under various assumptions a They should be expe<?ted to yield somewhat

better pre.dictions than. the Stefan equation~ It can be seen by comparing

Figures 6A and 6B that for small o( equation (2 ... 25) gives good results

while equation (2 .... 26) with a much smaller A · will, predict too shallowo

An angineer in Kansas or even Nebraska where C>( is large would find that

equation (2 - as) was very good while equation (2 ..;. 25) predicted too deep.

-29-

.......... .;-

•. -

... ~.

. .. - ..... ...

. -··'

FUSION PARAMETER J.L

0 0.2 0.4 ·0.8' 0.8 1.0 ~~---+-___ _..., ______ .,.__ ___ -w

.,<, '

.....-: z· LI.J 0' 0.8 LL. LL. L&J 0 0'

z 0' 0.8

..... 0 IJJ a:: a:: 0 0

NORTH DAKOTA

a • o--ALASKA

SOUTH EQUATION ( 2- 29) DAKOTA

NEBRASKA~ a:::,

0.4~------------------------------~

A. X, a, JL CURVES FOR EQUATION ( 2- 29)

"' .~

J.L 0 0.2 0.4 0.6 0.8 1.0 ~---...,._ ___ _..,.,..... ___ +--_____ ..,.

0.8

, ...... --,, .......... ----' ', .......... ' ' ..... ' ' ..........

EQUATION (2-23) (STEFAN)

a.o

EQUATION ( 2- 25)

,-<- 0.6 ' ' ...... ' ' .... ' ' EQUATION (2-26)

-a .. 1

..... ...... ', .............. .....

0.4-----------------------------------

fL = CnF Lt a = v;,t

nF

B. X, «, p. CURVES FOR OTHER EQUATIONS

COMPARIS.ON OF A, cr, }l CURVES

FIGURE 6

/

The statistic~f,('?~"iflAt~~,-~ ~~~·?r~J?e~ ~~hi:t~e:~rf<?!l?,~f?n~~c~;ol?:~i1.@i.al!ftP!-11Y

confinn the. above statementsct - . . ,._ ··-M: .•. , ·-·, ··c.'"' ~ • ...,.-t. ... -~·- ~·-··· ····:"."l·-~~ >; • __ .,,, __ ~,- ..... -~ ·. ·- -.:_~;.. -~:::.~J~~·;:;.~r.-ff;~i~ ~J~!:.4j~l~~ --~-~~:-~~:~~1~-~~,-,_-~ xi::.~

.~ .. ~~.\-~.._:;.'J.(.<. -::.-./~ . 1.~<ttJ3.(.;,._·~ ~.:~·-·,.• .• ·-t~-!~} .•. _;'·:_.•,.~:t 3--ll ;._,).';.I.~:-·-:..; ,l,;- 1 :- ~ -- - 1_

(2)

a:t .. Q,O~P.~·i1lg. pr~dt9.ted .d~pth~ .-9f :fr~st .pen:et~ation ,-gi yen,._by:,.!~h~f-;modified t.'·-~ ... t . _.-;.. .... .; ~~ .. -· .(•l:' ~ J -_ ... •. - _..,. ~~. ~ •. , ' ·'.. ~ •. ·' ._ ,._, ..... ).· • - •.•.• ·~-.._...:-.. ............. ~.\.

Be:~,~_f?ren.:\to~~~a (?.~·.-7 ._,fQ.). :vn:~h .actuEil ~ge,p~}J,s.~d(e~~~ip,ed,.:by~-tl;e:r;mg~,~~pl,~ r~E;)~d­

ings_ .. and .;test pi.ts,~ .r· T.q.ese results .are compared ~~~h,_.predictEJ),d.· .r~.sult~,r'<us.~g :·_,,,.; ___ .-:: .. ···' \• .-· ·: ... · .• • . ' .. ·' • -: ··.·-,. ·- ,·. -_ ••• . )_ •·•··•··• --~.,.,,J .... ..... ~ .. (,~··-.··· ··-·--: ...

Equations,.( 2 -;-. :25) .. atfct..(? .... , 26)·~ the. ,l~_tter JnQl\ld~g the ,p~VEJpiez?.t ,-th.~;ck;qe.s;s. :,.-1;. ,~..; 1,..._,. _. -~ ,_' .. · •.. :- •• J--·4 ·~ •· ·- .~ • ·' _._, • • .. •• . , .! . ., • ..! -·) *-. . ......... -- 3r~""'!''' _..,,_,..""" ·'··•- ...., ........ """" •.•• \1- .. \. ••

,;.!\Y:P. _s.wdie~.- w~:re._po_nd~q~~-9: by the-,;writ~::r~· .A::v~:t:~ge v,~l'l:li~~.":9.t, O~r_~L : .. :,1r_· ._;_ .• :::J ~ ',) •. ..:. .. ~-· · .. · .:'•J _', ... : .· .~#:.. •

1 ll.' ·'1', ~ ... 4 .. •• .·-. ~ .,, ·:.""'"' ~ --~"'-' ~' --: __ , :. l -~· -1--.J ·"' ~ •· "' ..• ·...! ·.·.·~···

........ H .. -~4...lf?_~[._g~v.-~.- -~~ .!"~ . .t:~!ez:l~e ... (~_~) ... h.~_y_e.J?.~~- ~.e.<i~--~--~---~Q.4it.j,_QP..s~-:-th~--.air_ __ tr~~~z~ . l

~- ... ___ ing .in <lex :·.h~,~-·J;ien_ us.ed .. f.or, .. both_ ... s.:bldies •. -~· .. ~:_._~_:.- -~~-..... 4 •• _ .. -·-·

v .. ~~- _-#~: . ~-· ; ~ .·: ' /':,_ ~ . ·: c .: ·::~ J• • t )~, ~

· · ··· 'The first .. analysis,· involving 'the :riiodif'fed 'Berggren formula only, 1

. "''~i~' ~~~ji ~~-':ki~-~9-~~:\~~ii~ Q~. ~~~~~~ ·bis;_~;..~~-~~ :i·~~~£~i~~4ep ths • The

· ,, lc ~~~lc>i~;g! ~~s~l_;t~Y~~~~-,_._._·: -~a-.. ~.Y. :: .. :~~·~ ___ i;i~ __ ¥r£~~1 d:,·-:_. __ JJ~;~c_:.~J.'~~-~.:·"~l]_,f~~:·:5~u.~:_.;~-~ .:.~.~_ ... ,._,~~-~·-·.-.... ~,.,_,_,:". -~_,_,_':'_!_· :·~ ...... --;::,_... .. "''.l-'~--~,,.._,,.,"-'~-"'~•'"'-... .,.-..rr-.:,... • ..... ~ ~···-.....,~:-:-~ --.. ·. ~..,_-p ., ..... ~ .·4·· -_.~ '1.' .- • __ _ _ ...__ • _ •. ~

' s"bt.~~?:s c~nducte~~- -~~~" the Fr;ost E.ffe_cts La9oratory ~;~.<t·~reported :! q~::(:p·~g~:: )1?_ ~

'

o£:· reference (16) ~ ... j.>,' .. ' : ·'·:: i

Mean "~· . +"1'5 ~0%- --~,-~-~ ~-~"" -----

-:.~.,:'1.){1_~!: ···>' . .+.'·'7~~:~%

~~) (: ::1·1 ~~ .. ~- -~-.-: -~'-· .__ •• C\ .... -

Mode . (.'" ·i'. -~ ~ .. :t

-r. ··st8.nda~d n~~ati'on . _ .. ., .. _

.. +"" 6.45{ ~-"'< .~ .. ~~--~1:~:"~".-·.: .. x)-:~1·:·~-/ . ;:~::-- r.~ ... ; .

SUm of Squares

.... ~=· .. "·-- --~- ..... , •• , ........ ~-H.·~-,~---'··-~~·~··· "'·-c-.r-r.o •.. of. .... pre.dicted -~~it:.~~~:~dth;=_''26:S: ..... :~_'-~~· ... :· .. :·.-~=-"'···--~--~~···--·"·"~--.,.-~,.~,·,<·~ . in +6 in. <?J\.;o...:bs~~rv~:d:· <l.eit:th, ::~::· 7··::::; :~~~·

• .::., . ..:.. -~--·--·-'"'~.o ......... ~·.,.. .~;:..-· • .__,...-_·.--::c.¢;.'~-"'--.. ~·-=·······: .... ..)..:>.;:T: ... :,. ...... ';~;....·.·.·.-~-....... -.-,;.-;· .. ~-j> .... -...:

The graphical representation of the dispersion is shown in Figure 7·)

This may be compared with plate 5 of' reference (16) •.

In general these results do not shaw a prediction which is signifi-

cantly superior to the other equationso

The. second analysis is based on inches variation between observed

and predicted depths, ·comparing Equation (2 .. 25), Equation (2 - 26) (with

pavement thiclm.ess added) and Equation ( 2 "" 29) ~ For this study the dat9.

have bee:1 arr~'"lged in two groups, one with the 2l.1- highest values of' the

actual correction coefficient .A a* ( o915 to o 715) and the other with

the lowest ·24 (o714 to o2]5)a The following results we~·a obts.ined.g

Group I G!"oup II

,A a(o915 to .. 715) A ..... , ... 1f· t~ o~5) ' c i . .::.'"-j. ..,(.; a·· ..

A'Vgo Obso Depth:-:: 52o3 inched Avgo ObE"o Depth ~ 40o9 inches

A-vgo Deviation Standard D~lVo Aor;g, Deva ±":rom Stdo De~o from Observed Depth from the A~Go otar-Vd~ Depth frmo the Av_gQ

Eq:) (2 - 25) .+5o4" +41)3" +19o219 +9o819

Eq. (2;..., 26) -4o4" +5~9" + 7ol11 +Bc-4"

Eqo (2 ""' 29) -0~4" +1. 5" +12o2" +8::1 11 --L+o~ -

It is· apparent from the above table that in. ger.~.eral fc·r high A values Equation (2- 25) predicts frost.depths wl1ioh are too great while

Equation (2 ~ 26) predic~s too shallawo For low ~ D all equations pre­

dict too deep., Equation (2- 25) predicts about 5o% too deep whilfJ Equation·

(2- 26) is less than 20% in erro:r·o Thea~· re~~i.ltsD thenD _confi:r."'n'l ·t.~he general

comparisons made in the pre1dous se~tiono

cbserve~-~~gth o_!_!!~_e_~_·i_r.i._fb_. __ _ 48kF

- Ij

. .

.........

BASED ON A SAMPLE OF 45 OBSERVATIONS

m15~----------~------~----~----~----~ z 0 ·-.... c > ~ 10 Cl)

CD 0

-40°/o -20% 0 +20°/o +40°k +60°/o · +80°/o

LESS THAnnREATER THAN

PERCENTAGE DEVIATION FROM OBSERVED VALUES

DISPERSION OF COMPUTED FROST DEPTHS

USING MODIFIED BERGGREN FORMULA

FIGURE 7

.1 • .., ••• • .. ,

To be more .specific,_ 9!J..e m~y. r;eadily see .. from::.the. above table .. :.:>"~:-

that :·for _:~he _first .24 points .. (Group: I) the modifi~d ~e:r:-ggren· _fo~~•A2·.,- ,28)

predicts the depth of penetration within an averag~. d~vi-at:iqn:: o:t\:-0.~4··i:Q.9h~s

-~ .. ~ .. o~t. of a: gt:"Oup av~rage observed· depth .of 52 mches, .wi~ _a .. :.Pro1;>~ble error

.~f +3 •. 0 .~ches (0.6745 x 4.5). _Thus the average_ deviation _fgr.-.th~ ·data ~of·

Group; I ._is less .than. 11o of the average depth a;nd is l;>etter . thflt1 _10 times ...

( small~r fo~ the new fonnula- than for Equations ( 2 - 25) <?.r .(2:,-:-.. 26) •.. ,.. ~~ .·

~eei.tl~ ,th~t, on the average, .for values _of A. between lo():.an~,:9~797 c(say):1 ,

()ne --~ay .e:xp~ct Equation (2 - 29) to predict about 82 per ·ce.ni?, ._ojJ~:;t}l~.~,·~'Qs~~:~

va-pions within 6 inches- of the actual values. For _example,.:.~ Group, J:· o.;f, , ..

the data studied, 21 out of 24 points or 87 per cent were. wi ~hin. (,,. in()h~s~, . .-

For the second 24 pqints (Group II), Equation. (2 ~ .·~), .. :predicts

1;};1~: depth of penetration within an average deviation of: +~.2.2. in:ch.~~ ifo:qt .. .9f

a group average observed depth of 41 inches ,.r.i th _a probable ... ~rror:-9~ :\+5.~ ~ ....

inches._,. Then, on the average, for values of ..,A less t_han Oo707 s one may

expect the new formula to predict only about 22 per cent of the .o:Qse;ryat;q~,s

w:ithit?- .6 .-inches of ~}le observed values. For example, . in Group II,. 5 out of

24 points or 21 per cent were within 6 inches.

Since, in Group II, for all the fo:nnulas, .both ~~ average devia-

. -tions ~.d the probable errors. are significantly large~ one is leQ. to ~u~~ .. ; .,_

pect that at least part of the fault lies with the asswned data. Fro~.-~ · ~·,1·

sup_e~f.~c_ial analysis of_ this group~ ~ t would appear that ~he: actUal values

of th~ the:nnal conductivity k are in some cases s~gnificantly ._di£:f.~;~E:):p.t

from the value asswned for this and the previous _studies repo.;t-ted, i~ -::·_·

Reference ( 16). Moreover, the assumption of ku = kr seems to be appreciably

-34-

in error .f'or certain of these points~ These conclusicns would suggest

that for those ... cases· in ·which· the approxi.Iiiate .values of . rx arid# point

·to· values of- -A . . .

less than b~ 101' a more refi:ued determinatio~/of reliable

·soil oo:nstants ·is in· orderc; · .·

·Following a ·suggestion by the viriters.9 ·the Arcti(~· Const::;·uction ·an.d

'j Ft.· ost Ei'f'ects Laboratory made new statistical studies· using for ea:ch case an

I average thermal coriduCti·vity k detennir.:e<i f'rom the water conten-t omd dry·

j density ·of ·the· soilo · Results of· these studies showed that EquatioD. (2 .:~ 29)

I I I f

I 1 i \ \ -. _ _.......

gave predicted·· results superior to other fonnulas o The· Arctic Construct'ion

and Frost· Effects Laboratory therefore approYed the '\"1l''itersl. r·ec.orr.IIl'le~J.dation

that the n::-;:mogi'aph ·called for· in the first; phase of this resea.r(Jh be based

on the ·:"ational formula (2 = 29) developed by the writers:.

eo .Limitations and Applica.b~li ty of Fonrrulaeso '""' B.sfo:.re discussing

the limi ta.tions of Equ·ation (2 -~ 2=)) it is of value tc li~~t agai.r1 the ba.si.c

assumptions made in thG derivatiol:i of the fonnulti.:;

(1) The soil mass was ·considered t•.) be ·homogenc:H.::'!·u~ and· one-

dimensional~

(2) The initial temper-ature L"l the soil mass wa.ci a.ssmned ·tt.:::-

be everywhere constant at the ·value v above·.freez.ingo

(:;) The surface teniperat;ure was asstmJ.ed to c~'lenge suddenly

from v 0

above· f:reezing to "tJ' s

valueo

below freezing.., and· remain s·t'eady ~t.~ thi~ lat..t-:er

(4) 'f.he effec:.t t'f lat;ent heat was 0onsidet·ed. tc~ in.t:todu~:e a

heat source (or sink) at th-s moving freezing point interface.:: with a heat

flow proportional to the. speed of motior=.o

"·' r

(5) Under the above conditions en exac.t ,solution .. was-ob-- · ; .... . . ,., ' ' . .. . . :·~ . . .. . . ' .

tain~_4, takin_g .i~~;() ac~ount .. bot~ .. voh~e~r,ic.-hea,.t_, an,d:-la,:~.t:m"t; :·l,le~t;s as well as

the ;v:ar~_ations ... in-, :t;he.!7Ilal. properties upon. freezing. of the ;s.oi:+ n1ots:tu]i.e •

. This exact formulation of the above ide.alized case was expressed. . . - _.·.. . . . ' - .·. . -· ' - . . .. . ··'•.

in tenns of. -three parameters 9 namely th_e thermal ratio,&(:: the fusi<?n .;

·.P~~eter,_~ ,._,and the root diffusivity ratio, • These sam~. con~ .

stant~ .wop.ld govern the behavior in. soil freezing probl~ under the: m-g_ch:

more. compl.:ex con~itiqns encountered. in ac;tual practice. Thus it may .:.be,; ..

eipected ~hat .·the depth . of frost penetration in these _actual cases. is ·af.'"'": ·· :.

fect~d . in :much the s~e way by. 0( , ~ , and

ca~e for which the fonnula was derived •

, as the ide ali z.e<i:

. However, the failure to realize these simplifications may ,pP():duc~,

si~ificant .d~vergences; therefore~ the most important of these deviations

are outline4 .in _the following paragraphs.

The actual s~rface temperature variations are.not uniform over the

surface area, but rather are appreciably affected by ground cover, p~:v.:~me~t ..

type~· :~~~, .• ,_ as w~ll as by local variations in .wind velocity, shielding, . )

snp,~.C0'1!,er, etc. For.any measure of success using O!le~dimensiona;t ~~:t,~o4~,·.·.

of prediction, considerable care must be tak~ to derive a set of. con.si~;;~ent

and t:ruly- r~presen tati ve conditions and equal c~re must be taken in the inter-

preta;t?ion.: of -;fqrm;ula. results for actual .. cases • t't, :,,

,_. .A1·1., :aotue,.l soils. are layered or: otherwi~e. v:ariegated to ~, ;~.~a~.~r:

or 1~~~.81\· :extent .. ~4 :the· as.s~ption of .homogen.eity must be. aco..ep_ted: ~~;.~~~\~";':·.

mere approximation. As the variations from layer to laye:r, .b .. ec.pJr1e: mot~:::·,·:/

pronounced the formula. predictions may become seriously in error lm.less

-36-

deliberate care is taken to compensate for the layering as outlined in Sec•

tions · 2-02-g and -3-02-c ,

!he initial tempel"•~re in the soil nia~s upon the start ot fr~e;.

ing is :n.-ot uniform at a value v above freezing, but rather varies in some 0

non-tmifonn way from the surface temperature down to a temperature comparable Q

to the mean annual temperature at a point sufficiently deep in the soil. Rep-

reseri.tative of such variations are the Va.lues at the start of f'reez'ing given

in Tables B-XIII and B-Xrv.· However, as testified by the I. B.M. solutions

themselves, as well as field data ·and the analyses. of Section 2-02-f', the

effects of such non-uniformity of tem_perature on the correction factor, A. 1

are not ·appreciable so long as v is taken . as the me8.l"'l S.."'Ulttal temperature. 0

Much of the same reasoning applies for the shape of ·the ·surface ·

temperature-tL~e curve and its effects on penetration depths, as discussed

in Section 2-02-f'. Again, as long as thA equivalent step change vs is taken

as the mean ·temperature below freezing during the freezing period, a good

agreement exists between actual seasonal depths and 'those predicted by the

rormula. Other limitations and modifications of the formula as applied to ·

practical problems or prediotien are discussed in connection with tke no~ov

graph in Sections 3-02 and 3-o3.

f. Depth -Time Curves~ - While Equation ( 2 - ~) gives consistent

pFe«S~ctions: of seasonal depths of frost penetration, attempts to calo~lo.te

the depth-time curves based on· partial freezing indices are likely to en•

co1m.ter substantial errors. This may be demonstrated roughly as in the

following paragraphs. -31;..

·If the soil is initially at a unito~_temperature ~0 and a sur­

race temp_~rature variation.v (t) below_treezing is applied, then to a orude . ··'·'·' ·.. . ... , .. ·s . . . , .. . . ... . .

fir_~.:'t ~p~ro~mation the . f()ll()wing equation ~ay be shown to hold:

;;··, '. ,.

• e • • ( 2 - 35)

This equation accounts for the latent heat eff~ot, and the most

si~ificant terms due to volumetric heat effects. ·., ~.~l~:. ,, ,<. ·~:-: . : I, ~ ' .~

It may be normalized to

the form:

._¢_2(t) ' = • 0 0;) (2 - 36)

Sl . E-

--.·- < '+Pt: >de 0

1

1+~ s ( ~) d~·· U lTpE_ ·

0 '

0 0 0 0 ( 2 - 38)

;here ··cr··= O(jA._ = CuvofL ·

·p -~ l ~ /1 + IK,M-. 2

G.(_?:')= v /(F/T) .·' . s ·,.

T ~ Duration of Freezing Index (in DAYS) .. T ..

F = J 0

v 8(t) dt = Freezing Index (in DEGRii:E-DA.YS)

~ t- • • _ •. : e _:: t/T ·= Relative Time during Freezing Period

(i.e•:~= 0 corresponds to start of' freeze. f • • ~--·~ '., • corresponds to end or freeze.)

Then?~ =X (~)/Xm measures the relative SHAPE· of' the· depth-time, curve •... ·

From th~. definitions above it is necessary that: < •• < r' € d~ ; o . . . . . .

)C> .

whence it becomes evident that the £'unction. E( c) , also, merely

measures the- SHAPE of the surface temper-ature curve during the freezing

period. . Thus we may investigate simply usirig Equations ( 2 = 36) and

( 2 ... 38) the . effects . oii tlJ.e depth~time curves . of various surface tempera-

ture time curves o.

The, simplest such case is that of the constant step. changes o

In this case E = 1.0 = constant throughout the period and the in­

tegral for A becomes

' [' cl~ l+tS () .l+p

00 0 .(2 - 39)

Upon substituting the defining constants tor the parameters 5

and p 3 this e~ression for A is seen to give the value corresponding

to Equation (2 - 25), ~amely~

~I+ «.(~+0.5)

Thus the problem of investigating the effects of the S1.:trf'ace

temperature curve on the frost penetration depth curva may be .factored into

two parts, namelyg

(a) The SHAPE effect (¢)

(b) . The MAGNITUDE effect (,\_)

Equation oooo(2- 36)

Equation oooo{2- 38)

It is easy to show that f'or e:ny shape E (~) symmetric about

the midwinter point (} ~ o·os;· wli:ich is very nearly true for all practical

instancesJ) then A may be very closely related to th~ -me~ value E = loO

by the follOwing expression~

' + cr- l+pE -39-

._· I -(t+tSXI+f'~

.,- .. "'

. . . .. - . ..

This fact would indio~.-~~- th:~t ... the, ... :spap_e_ of .. j:;:Q,e surface temperature ' • '. .. • < ... •• • • " ~. ~· : • • -~ :· • • • ·l .•. , .·

curve during the freezing season ha~, .. 9nly a:.:very- slight effect on A. , and

therefore on the total .~ep~~ .Pf ,freezeo ... This_ .. hy:p.othesis .has b~en substantiated

both from field measurements and by the L.BoM. studie·s referred to in Section

2-01-f-(3).

On·· the other h~d!l a slight effect on the shape of the depth-time

curve· does. exis't and may be explored by substi tu.ting the above expression

for )\. into the· equation for ¢ to give the form:

The integral· may b~ approximated by e. finite sum in the fo:nn~

.. ¢K. = J(l + f')h I . . K. ·~~ ~(2 - 43)

where .[' ed~ =-l.o 0 .

is· replaced by the and where

condition : h [ .E k. ,:- 1.0 . . I< .

magnitudes not much greater than Oo2o

are of o .Practical. values pf p Accordingly, the table below has been

calculated with h = 0.1 and values ol" € K and p as indicated, in

order to shaw the effect of p on ¢ o

. From Table IIIs it is clear that the parameter p has no sig­

nificant effect upon the shape of the depth-time curve over its physically

practical range. This permits one to draw valid conclusions as to the shape

o~ the depth-time curves from the simple case -where only latent heat is im­

po~ant ( {> •o). Since ¢ 2 is then merely proportional to the freezing

index, such oases may be explored very· quickly. Conclusions based upon this

approximation are shown in Figure Bo -40-

TABLE III

SHAPE OF DEPTH-TIME CURVES

VALU!S OF ¢K BASED UPON A PARABOLIC DISTRIBUTION' OF eK

k el<. VALUES OF p 0 0~1 0.2

1 Oo29 . 0.,17 0.,17 Oo18

2 0.76 0.32 0.32 0.33

3 1.12 0.47 Oo47 · Oo48

4 1.36 0.,59 o.6o ·o.6o

5 lo47 Oe71 Oo71 Oo71

6 1o47 .o.ao. 0.81. . 0.,80

7 1.36 Oo88 Oo89 Oo88

8 1.12 Oo95 0.94 Oo94

9 0.76. 0~99 Oo99 Oo98

10 Oo29 loOO loOO · loCO

.. ·;

...

Oo4

Ool9

. Oo35

0.48

o.6o

Oo70

Oo80

Oo88

0.94

0~98

loOO

.,r:· ....

. .. __

APPROXIMATELY PARABOLIC

---- APPROXIMATELY LINEAR

TIME

---THAWING CURVE:

{b) X

FREEZING CURVES:

(a) FOR PARABOLIC TEMPERATURE: APPROXIMATELY PARABOLIC WITH .. VERTICAL AXIS THROUGH 83

(b) FOR STEP TEMPERATURE:

APPROXIMATELY PARABOLIC WITH HORIZONTAL AXIS THROUGH A 2

APPROXIMATELY LINEAR

SHAPE OF DEPTH- TIME CURVES

FIGURE 8

-42-

. ...

. ; -

\ . ,_

t.,

:>'~·· .. :Mul·til~ye~ed _Systeni_s.: b I.n .. (3.rlY; g~~~ .le.y.ez:, ~~, _the._,?:nat..~r:~!e~

.Jus heat flow, assuming a linear· gradient, is given by Equation,_(_~ - 10) or:

, . ; q~·. ·: --km 1:::. Ynr ,,·:: .:·. • • ( 2--·~ ... 44) .. ·. , ~:

',",

which may' be rearranged to solve for the temperature difference L."'l _terms of __

flm'f and th~ thermal re~istance. of the layer, Rm . = . dn/km" to give:

Av = R q m m m

•••• ( 2 - 45)

For an entire series of layers, from the surface to layer n, the total tem-

perature differential may be given by:

=

If the n - th layer is at the freezing temperature, ~hen _vn = o.

Moreover, if the o;nly source of h~at .is considered to be the cooling of. the

layer from. v to the freezing point and subsequent revival of the latent 0

heat Ln, then the total heat removed to freeze the layer n is gi7an by:

C.t'

d n

•••• (2 - 47)

Thus the equa.ti on for the freezing of this layer can be wri tt~:

dU a = --lL. o ••• (2 ... 48) '"Il dt

v = _s_ • dt

R . t

-43:.

•••• (2 ... 49)

=fttn In terns of the partial freezing index F v dt and realizing .n s

n-1 · n-1 that Rt = ( L R) + l R , the expression may finally be written as:

2 n . .

F = U n n

.R •••••••••• + -: J •••• ( 2 ... 50)

For a double-layered system, then, the total freezing index will be:

u u2 F = F + F ~ R (_! + U2) + R •••• (2 - 51) 1 2 1 2 -2 2

R since Fl = ul ..l.

2

2 For a triple-layered system this gives:

1 1' 1 . F = Rl (2 Ul + U2 + U3) + R2 (2 U2 + U3) + 1) (2 U3)

0. 0. ( 2 - 52) For the general case this becomes:

[ !u 2 k

•••• (2 - 53)

This may also be Written:

[ !R + 2 k

k-1

L m=l

•••• (2 - 54)

These last equations are employed in Section 3-02-o in connection

with the applicability of the nomograph to multilayered systems. In this

instance % = ~ dn accounting for latent heat effects only • since the

volumetric heat e ffeots are contained in the effective A value.

-44-

.•

. ~._

... ~.·

. •,

PART IIIo CONSTRUCTION AND USE OF NOMOGRAPH

3-oi. :·DESCRIPTION OF THE l{OMOGRAPHo ~

:A copy· of the nOJ:i:J.ograph as prepared by the authors as part of

this researdh: is included in an envelope on the inside of ·the back cover •

This section of the report will serve ·to explain its physical makeup~ to

cite ditections for its use, and to" discuss its limitations and ass~p_;

tions.

:; The nomograph ·consists principally of three figures •.. ·In addi-

tion, ··a ·table of notation, a list of'· equations, a.."1.d :an example of its ·

use ·are included on the printed sheet.

Figure 1:' Figure 1 is a composite of noinogre.phs already pub-·

lished (2o, plate 16) ·for the determination of the volumetric heat of

the :'unfrozen arid frozen soil Cu and· Cr~ and for the evaluation of the

latent heat Of ftiSiOn l,e One modifiCS:ti:On haS been made_, h0W9Ver I that

of assuming a ·value of 0.17 BTU/lb for the specific heat of dry solids

(instead'.of 0.20) in the equation for c. The value of 0.17 appears to

represent a more reaiistic figure for soil temperatures near the freez-

ing point (211 page 71).

Figure 2: Figure· 2 presents curves for· determining the thermal

conductivity of the frozen soil, kf' _and unfrozen soil, ku. These curves

are the results of in-vestigations made at the University of Minnesota

under the ·immediate supervision of Professor Miles So Kersten (21, pages

86, 87, · 88, 90). The ·writers have chariged· the form of presentation from

that or:iginatly given but the o'oriductivity vaiues, which are 'dependent

priDiar.i1y:·on: :water content arid dey density:, have not ·been.: altered. ·~451D

Figure 3: The Depth cf Penetration part oi" the nomograph may

be considered to oonsist·of three oomponents: (l).the portion above the

0(,~, )\curves solves the equations for the fusion parameter~ and

the thermal ratio t(; ( 2)· the nest of curves represents the relation­

ship between o( ~fo and the correction coef.ficiant · ; (3) the portion

·to the ~ight of the curves solves the equation for the depth of penetra­

tion X. Origin of the equations endo( t>ft D A curves is presented in

Appendix A.

Table of Notation 8 The symbols used on the nomograph are

generally consistent with prevailing nota·tior.~., There are two impo:t>tant

exceptions9 howev~r: {1) "n" haa been used for the su~face tra:r.Lsfe:ro co-

efficient a...Yl.d (2) "v ".., which commonly refers to the mean an.'lual tempera-. . 0

ture9 is the number of degrees F by whi·~h the mean. ar.a1ua:i temperature

exceeds the freezing point of soil mois~ureo For exampleD if the freez­

ing point of soil moisture is taken ·equal to 31 °F~ then!' ·11·0 = (mean

annual temperature) ~ 31°o

List of Eguationsg All equations given in.this list are either

well known or are derived in Appendix Ao The nomograph is p!"imar·ily a .

graphical solution of these seven equationso

Example of the Use o.f the Frost Penetration Nomogra.phg This

example is discussed in some detail ur1der the following sectiono

3·02. DIRECTIONS FUR THE USE OF THE N'OMOGRAPIL. .,.

The following paragraphs present oonoise directi~~s for the

use of the frost penetration nomographo These directions are based o~

the writers' present state of knowledge regarding the effect of important

variables. Limitations and assumptions are given in the following seotiono

-·' .

. .. '

. ,, ....

".,.\ *'-

.; ,J ..

\ 'l ....

a. General. - Before the nomograph can be used the following

terms must first be calculated or estimated.

(1) Climatic conditions of the locality; v , F, end t. 0

These values may be determined directly from weather_ bureau records or

estimated fram contour maps. (For example, Reference 22, page 149).

( 2) Type of Surface; Until further data are available,

use n = 0.9 for all pavement surfaces. The basis for this assumption

is discussed in Section 3-03-d.

(3) Soil Properties For each soil stratum within the

depth subject to freezing; w,G(0 , and the soil type (whether predomi­

nantly sandy or clayey). A soil which is not distinctly either type may

be treated as explained in Section 3~03-a.

b. Homogeneous Soil .... This case is represented by the "Example

of the Use of the Frost Penetration Nomograph". One can enter, With a

soft penci~ data for the case at hand in the blanks provided in the table.·

On the nomograph, an arrow pointing into a scale indicates a value which

is entered. ~ arrow pointing away from the scale represents a number

which is read out for use elsewhere. Each line connecting nomographic

scales, has a sequence number and two arraws. The latter indicate the

direction of travel required to obtain the anmwer for that step.

c. Stratified Soilo - The following steps for this solution

are suggested:

Step 1 Determine the approximate depth of frost penetration

X from. Figure 9.,.B of ·this report.

Step 2 Determine the approximate correction factor)\ from

Figure 9-A of ~his· report. Use a weighted average

value of water content within the estimated depth

of freezing x. =47CJ

.; "~

. . ....

... . ·- ....

A

1.0

0.9

0.8

0.7

0.6

0.5

0.4 0

.... I,--~

., ~ ~

"' , "'

---""""" --0/o ...

_..,. """"" ~

"'"""""" ~ -l...,.,oo ..... ~ -..... __.

'fl ~ z61 I ~ _..,. ,.... ,. ~ "'""""'" ,.

~ ,. I I _, ,.,

"""""' ~ ,. ~

~ ~ I I 'o O#o I ~ --- ~

,. -..,r$\ I """

, """'

~ , 1 lf)oto """' ~ .... ..r"' t "" ~

, ,. ttl" .,

~ ~ I 'oto -

.1-~

. ~-

-~ ~

~ ~

500 1000 1500 2000 2500 3000

SURFACE FREEZING INDEX , nF

A. SEMI:..EMPIRICAL CURVES FOR CORRECTION· FACTOR ~

70 / ~ ~~

.

~ """""" v ~ ./

/~

60

50

X 4o /

~v ~

I THIS CURVE FROM_

~/ _REFERENCE 24

I -

[/

30

20

. 10

0 0 500 1000 1500 2000 2500 3000 .

SURFACE FREEZING INDEX, nF

.8. APPROXIMATE DEPTH OF FROST PENETRATION BELOW PAV~MENTS WITH. GRANULAR BASE COURSES

CURVES USED FOR DETERMINATIONS OF

THE DEPTH . OF FROST. PENETRATION FIGURE .9

-48-

·-

. '-

. ... -

. ,.·

'':. .t~··. : .

riomo grapho

' notnog·rapho ·

Step 5 Compute the effective K from· the formula~. L

L d ~+

2 (L) k effo

L d + T. ) 1 1 • o o o o' • •-ndn . . '; : • ..

+ •••••••• Lnd~)

+ dn (~~)] k -n 2

step 6 Use any effective value for k and determine an ef-

fective L. Suggestion: use k = loOa then

Step 7 Enter the portion of Figure 3 to ·the ri.ght of ·the

· 0( ~ ~ $ )\.. curves on the fros··t penetration ··noma- ..

graph with values determined.· in Steps ·2 and·· 6· to:·.:

obtain a corrected depth of frost penetration Xo·'.

If X differs appreciably from that· determined in Step I, then:

step 8 Adjust the value of A ~-rom Step 2o · ·

Step 9 Correct the value of· (1 ) . · from Step ·5~· ·:·~ -·' ... <·. · . . . . k err.

St~p JD .{edeterm:in e X .. as ·in Steps 6 and 7.

Repeat the above steps, if necessary.

-49=

d. Negative v0 (Case where mean annual temperature is belmv

the freezing poini; of soil moisture) .... For the homogeneous case.Y pro­

cede with steps 11 S 1 6, 11 and 8 of the "Example of the Use of' the Frost

Penetration Nomograph" 1 then use A= 0.9 to complete the computation

with steps 13, 14 and 15. For the stratified case~ use A= 0~9 in

Step 2 of' Section 3-03-c.

3-03. LIMITATIONS, ASSUMPTIONS3 AND OTHER SPECIAL CASESo ... It is

obvious that the aocuraoy of' the nomograph for predicting the depth o£

frost penetration depends to a large extent on the depandabili~· of the

data used in the computation. This situation is not an un.fa.mi.liax· one

in soil engineering problem~. If' the soil properties of water content

and dry density are at best only crude approximations or if the climatic

conditions are not too well known then the nomograph 8hould probably be

disregarded. In this case, simplified procedures uti:izing curves such

as those in Figu.res 9-A and 9-B are indicated.

The simple curve of' Figure 9 ... B which represents a· first approxima.- -

tion, may be expected to yield predicted depths of frost penetration

below pavements having granular base courses, diff·?ring not. mere than

50 per cent from. the actual if' the freezing index is greater than 500

degree days. A second stage approximation may be made by utilizing the

semi-emperical curves of Figure 9 ... A to determine a value of the correc"·

tion coefficient )\ for use in. the equation

X = 12Aif48 la!.F . L

-50-

-·

· .....

.. '

....... ~ .

.. -

.. 0

which can 'be solved· by the ·portion of the nomograph' to the right of ·the···

0( • ~- 1 . )\ curves •. ·The contractors feel· that Figures 9-A::and ;9-B:·: .. :··. ·

might very well be added to the nomograph at some future· date. '

The· frost penetration nomograph is, of course, ·subject ·to all or·

the limitations and·assmnptions used to derive the rational-formulao

These are stated and discussed in Section 2-02-e. The importance· of ·

these ·assumptions· as a whole can best' be shown by statisticalc;studies

comparing predicted and actual depths of frost ·penetration ... The ·reader·

is -referred to Section 2-02-d..;.( 2).

a. Limftation on ko ... According to Professor Miles Kersten.(2l)l

the thermal conductivity values from Figure 2 of the. nomo~raph are good

to ~ 25 per cent. ·,.

--"' If the soil under consideration· is a well graded· one· -which is

neither truly sandy rior a silt or clay then the conductiv'i ty may be·· es··

timated by interpolation be~veen th~ dashed and solid curves for the

given water content and dry density. No specific rules can be offered

at this time for the interpolation. To quote Professor Kersten_, "I··my-·".

self,sometimes study the- test results on the individual soi~s ·in Ap­. (21)

pendix -1 : · to find a soil similar to the one for which k i·s desired."-· .,

b. Allo,,rrnnce for Frost Heave. ·- If. freezing temperature·s· pane~:

trate irito a frost susceptible soil which is part of an open system; ::ice-"·

lenses may form and cause considerable heave of the ground surface~ The· .

maximum· depth of frost penetration in this instance will be smaller· than··

for a corresponding case with no ice segregation. In effect, the-water·

content of the soil is increased as 1rnter flows upward to the zone of

-5lc:a

freezing. This, in turn, increases the latent heat of fusion which is

the most important soil characteristic affecting the depth of frost pene­

tration. Since·x~~ it follows that the maximum depth of frost pene­

tration will be smaller.

If the amount or frost heave due to lanse growth can be reason-

ably esttmated, an equivalent average water content, greater than the

normal, and a corresponding dry density, smaller than the normal, may

be determined tor the soil in the troz~. state. T.hese values may then

be us_ed in the nomograph to predict the depth of frost: penetration.

c. ~· - The nomograph a.t the present· time is not directly

applicable for use in predicting the rate at which thawing ccours during

the spring months. The writers will study this case under a. Corps of

Engli1eers contract during the Fiscal Year 1953-1954.

Carlson and Kersten (23) have shown that the depth of thar....ng

below pavements in Alaska may be accurately predicted assumiugA = 1.0

in the formula

where n· = 1.4 and I represents the air thawing index.

d. Surface -Transfer Coefficient~ n. - ~e writers have selected

for use at this time a val9-e of n equal to q.9 for all pavement surfaces.

There can be little question, however, that n for bituminous concrete

pavements is smaller than for Portland cement pavements ·because of the

increased absorptivity for solar radiation- in the former.· The Frost Ef­

fects Laboratory (l6) has, in facts used values of n equal to Oo75 and.

0.90_, respectively, for the tvro pavement types. These values wer,e

-52-

-" .

- ..

.. ...

<;!,, • .. --

arr$v_eq;: a:t-,'through an analys_is e>f' thermo9.9:t.tpl.e·_;m~-~su.X:emeilt.f().n -the two

t-y:pe:s of~ ·pa:trements at three' site Tocatioris/ :Neve·rthe.les.s,' . re·sults3 6t~ _; -­

recent': ·sta5ti·stical ·studies' by ·the>wri ter·s 'ahd:'tha·· :Frost Effe:cts···Labor:a;.,;·i':·c::'.

tory:usi:trg·.'the~ modifi-ed_ Berggre~ formula, se·ctio:rf -2~02-:a-.('2}~ ifidfcfitEf--' ' ....

that COmputed' depthS Of freezing ara generally tOO small when .:p'avenfantL~

freezing indices equal to o. 75· ·and 0.9 are used. furthennore, 'there is·

less ;·scattering of statistical results ·when the air freezing' index 'is . :·.!

analys:·a~s ·are: available, a value of n equal to· 0.9 be· used."·

·:Th~' surfac'e freezfug index for other surface typos'· ~inbluding: ..•.... : ',-:

trees, moss, grass., loe.t-rn, etc., are undoubtedly considerably smaller

than 0~9. ·· The wri tars will not venture a recommendation' at· thi:s: time

for t:hes'e surface types. Some· data are ·available in Reference::(16)'~ :-,_

:;;The' following factors are lmown to affect' the-- surface-: t;emp'era;..' -,.··

ture transfer-phenomenon:

(1) Ambient air temperature; . ~. . :

(2) Surfac·e connective heat- transfer coeffi¢t<3nt; ·

. - {3) Solar radiation:; direct and diffuse; ..

(4)' Absorptivity for Solar radiation;

• : ~ ; • .... • j

': ,:.: (5) Long·~wave radiation .(a function o.f · c-iou.dili:ess··: andi

· · ·hu1·nidi ty) ; __ ·

-' (7) · ·Preoipi tationo

and expect to .report on their importance a~ a part of researches for the

Corps of Eng:ineers during the period 1953-1954.

e. Limitations due to Stratification. - A procedure for apply- ·

ing the nomograph to layered soils has been given in Section 3-02-c. This

techniq'Ue should be restricted to cases which lie between the e;xtremes

of very small and very large differences in thermal properties from layer

to layer. In other words, i.f only very slight variations occur in soil

type, dry weight and water content, simple average soil constants may

be determined and the nomograph used as detailed in Section 3-02-b. On

the other hand, if the stratification occurs with extreme changes in

properties, the procedure of Section 3-02-c will lead, to only very ap-

proxL~ate results, which, however~ should be as reliable as those com-

puted by any other technique now available.

l~ore accurate determinations of this last case would require

use of. the more elaborate methods outli11ed under Sectj_on 2-0l-f-(4), in

which event, the nomograph would be of little use. If such procedures

e ( ' are employed, the number of depth nodes the range of n) should be at

least tvdce the number of distinct layers.

3-04. CONCLUSIONS .AND. RECOMMENDA TIONSo - The nomograph as presented

in the previous sections is felt to be a useful a.."''J.d rapid computational

aid.

It is reconunended for users of the nomograph that a record be kept

on the A - o<-~ curves of all .val:1es. of A obtained from actual

depth measurements, wheneyer sufficient data are available for such

checks. This permits the nomograph to be used in.a semi-empirical

fashion for more reliable predictions in a given locality.

.. ... ..

... • II -

- . ..,.

"' ........ ,.~

..... .,.

It is further recommended that a table of.n values for different

surface type~ be included on the nomograph when more reliable infor.ma-

tion becomes available. In addition, it is s~ggested that curves similar

to thos~ shoWn. in Figure 9 be· added to the nomograph.. These additions

along with thaw considerations will further the objective of making the

nomograph a complete entity for determining the depth o£ freezing and

thawing.

\

> '·

-55-

..........

r PARr ~IV o THAW-CON SOLIDAT!Ql PROBIBM

4-ol. SYNOPSIS o -

In the following section, the physical nature of the so-called

thaw-consolidation problem is discussedo This is followed bv a mathema-. -;-=.:_ ::·· ;_ ·:

tical formulation of the problem and a discussion of solution techniques.

It :is apparent that a :rigorous analytical solution of this

problem is i~possible without questionable simplifying assumptio.nso "I .:~· • • i. ~} ~·:~ "\~~·•:.• '";~ :_. q : •• : ,;_ .~ • ,·' .:"

Numerical methods by hanq or I.B.M. a~d hydraulic models offer_ the beat

possibilities for practiqal solutions o Important considerations bear~ng

upon the solution of this proble~ are discussed in earlier sec~ions~ . ' : ' .: ..... '!·:· ..

principally under Practioal Methoos of Computation (Section 2-01-f) and

Depth-Time Curves ( Secti em 2-Q2-f). '· i. :- ~ '·: :

It is recommended that this problem continue to be explored in

a limited way with addi tiona! I oBeMo and band numerical solutions but

that the bulk of the work be postponed until an hydraulic analog is

constructed along the principles outlined in Section 2-Ql-f-(1).

4-o2. PHYSICAL NATURE OF THE PROBLEM. -

If ~elow-freezing temperatures penetrate into a frost-susceptible

soil, significant ioe segregation is likely to occur o ~ thi~ state~ the_

soil stra.tum contains an excessive amount of water. More specifically,

the soil exists at a higher average void ratio an~ water content than it •'

norma_lly w9uld under the weight of the overlying material. · .·:. :. ' .,_ ... ~ . . . . . ·, . . . . . . . :

tn .the spring, as warm weather_ a~p.z:oaches, thawmg progresses

fro~ .g~~und. s~faoe -~o~w~rd into the .s~pe.,rp~_~rated soil. _ _In eff',e~t, -.':"_:,·: ~~.::-~· ~- .. · .. • ·-'···· . .:. : .. ~~~: : ··. ' .. ·- .. -' :·.-.-1\ .. ·· .

tor each unit o£ depth the thawing re .. leases a ph~rge of water which was ·. ':::·; .. ; '' ' .:" ·,;,<_,· ·.~:-·:·. . ·.6· ~~·.· ..

-5 -

!/''

.t.

stored in the soil in th~ form of ice lenseso This excess water must

initially flow vertically upwards if' a one-dimensional case is visualized,

through the thawed soil to a tirainage surface. The rate at which the

dissipation occurs depends principally on the permeability of the over-

lying soil and the distance the water must travel to a drainage surface~

After the soil stratum has completely thawed~ the excess water is eventu-

ally entirely dissipated through a seepage diffusion process. The soil

has adjusted~ then to a stable void ratio under the weight of the overlying soil.

4-o3 o MATHEMATICAL FOIWIATION. -

Before formulating the equations and boundary conditions applying

to this problem it is important to iL:vestigate the pressure-void ratio ·

relationship of an element of soil during the thawing process.

In the classical one-dimensional consolidation theory a straight

line relationship between void ratio and pressure is assumed. It is

apparent that for frost-heaves of appreciable magnitude the void ratio of

the frozen soil will increase to a value far greater than that given by

the usual straight line relationship extrapolated to zero pressure. The

straight line relationship thus nee~s clarificationo

let e2

and P-2' Figure 10.~~ represent the void ratio and inter-

granular pressure, respectively, in an element of soil before ice-segregation

occurs. Point A thus represents an equilibrium state under the weight of

the overlying soil. Line CD through point A represents a portion of the

· de · pressure-void ratio relationship of the element o Its local slope.11ap ~ J.S

thus the coefficient Ot compressibility ~·

When significant ice segregation occurss the void ratio" in effect

incN&ses to some value e1

o Most of the load is actua11y· carried by the ice.

""57c

. ;.

. -.... ...

I:~.· q

VOID RATIO

e

0 p2 INTERGRANULAR PRESSURE p

PRESSURE ... VOID ·RATIO CURVE .. ·THAW- CONSOLIDATION

FIGURE I 0

-58-

.... -.

..... -

I o

T:"" suddenly thavved, the intergranular pressure would thus be very nearly ; .;.

zerj as the element begins to consolidate. Assume, then, that point B.

rep;~~~~t~ '\h~ s~il b~t~ .in the frozen stat~ -~d innnediat~t;'·'~;~~; thaw. : !• '',I .•..• ~ .. '~

.• .· Afte.r thawing, the element follows some path from B 'to A as it

.:· I ' ,·.._ , ~-.' : , !- , •. ; • •' ~ ' ,

cons_olidates under the weight of the overlying soil. The path is probably - ... _.

,. : ·,· -:

given by a curve similar to that indicated in Figure 10 by the heavy dashed

line.

As a· first approximation, it is reasonable to a:ssume that the ·' . ..' \ . . • , '~;·· . ·. ci· · -~,; ·;

element follcn~s the path BCvA. Under this assumption the physical seepage.

and consolidation process may be described as follows. On first thawing

· .. _.., ·•; . ,. ,. ' ',.' . .

Flow occurs throughout this period under a constant excess water :\ -·· .'

· .. :-pressure p

2 since the intergranular pressure is zeroo In consolidating

' ' from e3 to e 2 the soil follows c~A (with av = constant, determined~t A)

as the excess pressure is dissipated according to the unsteady flow ,.·

diffusion process, governed by the ' ,"' . ~ - . ":· ~- . _:.. ;. .- ..

~-~ •• .i ~· • ~- .• " •

Q = . ~k d H a.x ~ s dQ ·qt + e;x. AS =

= 0

. l+e

fundamental equations!

oooo(4~2)

• ~~i ~t :" .. J·~..... ·)<"~ .)' .. - •' t -·.:·.- ·,' . .i. '" ~ ' ' ~: =:~' .~, ·:.~.~ :(: :~~ ~·~; ,: ··:. which may be combined into the classical diffusion equation for the

hydrost~tic ex~ess pressure u, as' ov ~;~2,·:=··~- ~ .... k (l+e) where c = ·

v ~ Yl' av ,..,9oa·

··'-_';:··

4-04. SOLUTION TECHNIQUES. -

Even with the simplifying assumptions as stated above, the thaw-

consolidation p~oblem as for.mulated in the previous section is not amenable

to closed for.m ~alytical solution.

This difficulty suggests the application of methods similar to

those outlined in Section 2-01-f; these are.detailed in the sections to follow.

a. Hydraulic MOdel. - The use of an hydraulic model represents one

of the most efficient methods of solving the thaw-consolidation problem. In

addition, the model presents a graphic picture of the state of consolidation

within the soil stratum at any time. The relationship of an hydraulic model

'to the thermal diffusion problem with latent heat is described in Section

2-01-f-(1). Its relationship to the consolidation process is presented in

References 3 and 4. Only the direct application of the model to the thaw-

consolidation process wi 11 be discussed here.

A simplified hydraulic model for the one dimensional thaw-

consolidation process is shown in Figure 11. This particular model is

based on the assumption that the soil is homogeneous and that each soil

element follows a path similar to line BC'A, Figure .10, as it. consolidates

following·thaw. It is further assumed for this discussion that the co-

efficient of penneabili ty k remains constant during the consolidation process.

As a result of ice-segregation in the frozen portion of

the soil en excess amount of water is stored in each layer. This excess

water is represented by the volume of water. in the rese.rvoir and o orre-

spending standpipe. The water volume in the reservoir· represents the void

ratio change e1 - e3, Figure._lO, while that in the standpipe corresponds to

-6o-

.· _, ·-..., ~.-

- •• '4o.

.v.'

....

• J

......

. . ..

e3

e2 o 'After thaw, the initial excess pressure u1 in each soil element

is equal to the effective weight of the overlying soil. This assumption

fixes the :ini;tial water level in the standpipes.

When thawing penetrates a distance~ x, the excess water

~tored in the first layer of soil is released. In essence, valve A is

opened and steady-flow commences toward the adjacent tank r under a head

differential equal to ui2

• As soon as the reservoir is empty unsteady

flow from the standpipe occurs under oontinously decreasing head.

Successive valves are opened as each layer thawso When the

depth of thaw equals X , the soil within the unfrozen lower portion of the . f

stratum takes part in the diffusion process and swells as water flows ver-

tically downward toward the bottom drainage· surfacee On the model this

cdndition is initiated by opening the last valve E.

Any specific thaw-con so lid ati on problem may be treated

using two separate hydraulic models.- The first· model is set up for the

thermal problem of thaw while the second, similar to that in Figure 11~

represents the accompanying consolidation processo The first model in-

dicates the time when each valve in the second should b.e opened.

: It is theoretically possible to constrt.1.ct the model such

that it can represent complex pressure-void ratio functions during thaw~

variations in permeability v.ri th void ratio, and other features of stratifi-

cation and non-homogeneityo These ref~nements may be incorporated when

knowledge regarding the actual thaw-consolidation characteristics of soil

. ~-\ has been advanced. Laboratory and full '3cale field tests are needed for

this purpose •

' -.

X

\..,..·: -- I·NITI'ALLY __ ........,...,___ UNFROZEN_.....,. FROZEN

x, ~X r-

: ..--RESERVOIR

:----

A

. . .. . . . . .

2

----­U.--·

8 c

3

UNFROZEN

-:· :·d;ti·. :-:.

4

J

D E

5 6 ·1

-+-STANDPIPE

~CAPILLARY I TUBE

i + r- GROUND SURFACE

• . . . • . . • • . . . I

. ·. . cOARSE . sAN 0 ..... ·. ·. . : ~ . : ··.' · · · · · · · · · inWi · ·,,iiiJStl · · · ·

SIMPLI~FIED HYDRAULIC MODEL

FOR THAW• CONSOLIDATION

FIGURE II

-&2-

ANALOG

DATUM

b. Numerical Methods. -Numerical methods of solution

applicable to the thaw-consolidation problem parallel the methods dis-

cussed in Sections 2-ol-d~ 2-01-f-(3) and 2-01-f-(4). They can be

divided into two types, hand computations and machine computations.

(1) Hand Computations. - The technique of solving a

problem of this type is best illustrated by an example. Such an example

is presented in Appendix c.

(2) Machine Computations. - The use of automatic

digital computing equipment such as the I.B.M. Card Programmed Calculator

represents a rapid but relatively costly method of obtaining solutions

to the thaw-consolidation problem. Technique~ very similar to those

described in Section 2-01-f-(3) would be used.

. '

· ·#

-~ " ....

BIBLIOGRAPHY

1 • . Leutertl Wo I non the Convergence of ·unstable Approximate Solutions of the Heat Equation to the Exaot Solution", Journal of Mathematics and Physics, Volume 30s Number 4, January 1952.

2. O'Brien, Go Go, Hyman~ Mo Ao, and Kaplan, S., "A Study of the Numerical Solution of Partial Differential Equations", Journal of Mathematics and Physics, Volume 29, Number 4, January 195lo

3. Aldrich, H. P. 1 Jr., "Analysis of Foundation Stresses and Settlements at the Hayden Library", Mass. Institute of Technology, Sc.D. thesis, 1951 9 unpublished.

4. Barron, Ro A., "Viscous Flow Tube Model", Proceedings Seoond International Conf. Soil Mech. and Foundo Eng., Vol. III 1

1948, Paper IIgl3o

5o McCann, Go Do and Wilts, Co Ho~ "Application of Electric Analog Computors to Heat-Transfer and Fluid-Flow Problems", Journal Applied P~ysics .9 Vc;lo 1.6_, No. 3, Sept. 1949o

6. Avrarni, M. and Paschkis, V., Bulletin American Phyo Soc., Volume 16, 1941.

7. Backstrom, Matts, "Avkylningsforlopp och a.ndra temperaturvan­dringar yid komplicerade kylproblem, belysta med hydrauliak raknemaskin" 3 Tidskrift vvs No. 8, Aug. I 1948.

a. Haneman, v 0 So I Jr 0 a and Howe I Ro Mo .9 "Solution of Partial Differential Equations qy Different Methods Using the Electronic Differential Analyzer'', Engineering Research Institute, AIR-1 9 Depto of Aeronautical Eng. 1 University of Michigan, Ann Arbor, Oct. 1951.

9. University of California Dept. of Engineering, Los Angeles~ "Proposed Solution of Partial Differential Equations with Electronic Analog Equipment", Computation Laboratory Notes.

10. Jakob, Max, "Heat Transfer", John Wiley and Sons, Inc., New York, -' 1949, Chapter 20.

11. Scott, R. FoJ} nNumericalAnalysis of Consolidation Problems", Mass. Institute of Technology, s. M. Thesis, 1953, unpublishedo

12. Jakob. Max. "Heat Transfer"» John Wiley and Sons., Inc.~ New York, 1949, Chapter 18.

13. Milne~ w. Eo~ ~umerical Solution of Differential Equations", John Wiley and Sqns, Inc., New York. 1953. Chapter 8.

14. Salvadori, M. G. and Baron. M. L., "Numerical Methods in Engineering" • Prentice-Hall, Inc. 8 New York, 1952, Chapter 5o

15. Jakob, llax 8 "Heat Transfer" f) John Wiley and Sons., Inc., New York, 1949, Chapter 19o

16. Corps of Engineers. ~eport of Pavement Surface Temperature Transfer study" 8 Frost Effects Laboratory, New England Division, Boston, Mass • ., June 8 1950.

17. Carslaw, H. S. and Jaeger 9 Jo C. 8 "Conduction of Heat in Solids~, Oxford at the Clarendon Press, London, 1947.

18. Berggren, W. Poa "Prediction of Temperature-Distribution in Frozen Soils", Transactions, American Geophysical Union, Part III, 1943o

19. Shar.u."lon.~~ W. L., "Prediction of Frost Penetration" 8 Journal of the New England Water Works Association 8 Dec. 1945o

20. Corps of Engineers, "Addendum No. 1 1945=1947 to Report on Frost Investigation, 1944=1945" 8 Frost Effects Laboratory.~~ New England Division., October 1949.

21. Corps of Engineers, "Final Report Laboratory Research for the Determination of the Thermal Properties of Soil" 8 conducted by the Engineering Experiment Station 9 University of Minnesota, June 1949.

22. Highway Research Board, Special Report No. 1, '~rost Action in Roads and Airfields"• Washington.~~ Do C. 9 1952o

23o Carlson 8 H. and Kersten.~~ M. SoD ''Calculation of Depths of Freezing and Thawing Under Pavements"• Paper presented at the High~~y Research Board Meeting~ Washington~ Do C.~

January 1953o

24o Linell a Ko Ao ~ "Frost Design C:ri teria for Pavements", Paper presented at the Highway Research Board Meeting, ~Yashington, Do Co 8 January 1953o

- . ..... ·-

...... -

,. .. --.. ·•

- .....

I

r

I

. ,.. .. - ...

. ·-..

25. Carlson, H.~ 11Calculation of Depth of Thaw in Frozen Ground", Highway Research Board Special Report No. 2, Washington, D. C., 1952.

26. Jumikis, A. R., "Theoretical Treatment of the Frost Penetration Problem in Highway Engineering"~ University of Delaware~ Newark, Del., May 1952.

27. Ruckli, R., "Der Frost im Baugrwtd", Springer-Verlag, Vienna, Austria, 1950.

28. Aldrich, H. P. Jr.~ and Paynter, H. M., Discussion of •calculation of Depth~ of Freezing and Thawing Under Pavements'', by Carlson and Kersten. Discussion presented at Highway Research Board Meeting, Washington, D. c., January 1953 o

--

... -

-; . .. _, . ....

- .. r

Appendix A

DERIVATION OF A RATIONAL roRMULA roR · THE PREDICTION OF FROST PENETRATION

A-01. BASIC ASSUMPTIONS. -

We seek a solution to the thermal problem or a semi-infinite soil

~ass of uniform properties and at a uniform initial temperature subjected to

changes of temperature at the surface, which are assumed uniform over the

surface extent to yield a one-dimensional problem • .

~gure 4 illustrates the nomenelature a..'"'l.d significan-t variables

in this situation. FUrt her as sumptions are best listed in the .form of

specific conditions as fu llows ~

Condition I - AT THE GROUND SURFACE

It is asswned that the surf a(;e temp6rature is sudde!ll y eha:-1gcd.

from an ini tial t emperature v0

abo ve freezing to a temperature v8

below

freezing. This temperature value is then maintained constant and uniform

over the entire surface.

Condition II - IN THE FROZEN SOIL

It i.s as surned t hat t he d.::.f':f\ wi on eq·..lati c;n ~

with a.r=k/Cr measuring t he dif'fusivity of the frozen soil i s satis­

fied throughout the frozen S·)i l mass., subject to the surface temperature

condition (I) and the lat ent heat condition (III) at its boundaries.

Cond:i tion III - AT THE MOVING FROST INTERFACE

It is as surr:.ecl t hat at the i nterface betw-een the frozen soil

(above) and the u.'>lfrozen soil (bel.ow) t he temperature remains c .. cnste.nt

A-1

at the freezing point of the soil moist~reo It is further assumed that

the heat flow upward just above the in7.erface in the frozen soil is equal

to the sum of the heat flow just below the interface in the unfrozen soil

plus the heat flow due to the removal of the latent heat of fusion of the

soil moisture as it freezeso

Condition IV - IN THE UNFROZEN SOIL

It is assumed that the diffusion equation: 2 dvu

au'\[ vu = o t

with au = kufCu measuring the diffusivity of the unfrozen soil, is sat­

isfied throug~out the unfrozen regions s~bject to the latent heat condi-

tion (III) at the frost interface (the uppe~ boundar.r of the unfrozen

soil) and to the lower boundary condition that, a~ all times$ the tempera-

ture approaches the initial te:nperat~re .:f"or sufriciently great depthso

A-02. ~mRAL SOLUTIONo ~

Following the notation of Figure 4, t he 8o:!.ution for v f' wrd.ch

satisfies condi tiona I and II me.:v be 11vri tten:

= -vs

while the solution. for vu -satisfying condition IV may be written:

v - v + B G .... erf ( X )l . c:. o (A-2) u o L- 2 VSut J

At the interface (x = X), in order to satisfy condition III.

where the freezing point is _to.ken arbitrarily as zero degrees~> it is nee-

essary that

= =

. . - •-:'

.. ... --_,.

from which:

•••• (A-4)

... •••• (A-5)

. !#- But since these last conditions must be satisfied for all values of time,

then

X/ {t = (f = eoru;tant •••• (A-6)

or

X =(\[t •••• (A-7)

Thus the constant 't depends upon v , "f.i and the thermal coefficients. s 0

The manner of this dependence may be fo~~d by considering the latent heat

requirement of condition III. This thermal continuity condition relates

the temperar~re gradients each side of the interface to the rate of move-

ment of the interface in the form:

k d vf - 1r a vu = f~ -u ox

dX L-;,rr- •••• (A-8)

which becomes, upon substituting the expressions (A-4), (A-5) and (A-7),

performing appropriate differentiation~ and simplifying:

erf( '( /2 Va£) -~

e u = !:_~ • ~ - err ( b' /2.fa;""8 2 ·

•••• (A-9)

Furthennore~ by :making the substitutions:

o(= v cjvc - 0 sf •••• (A-10)

· • • •• (A-ll)

A-3

o•oo(A-12)

z :: 0/2 vat

it is possible to re1~ite Equation (A=9) in a simplified non~dimensional

fonn., namely~

0( - -~

_e_~_2_z -2 __ l "~Tf z (1 c. ez-fo z) ~,

If all the tenns in Z are then carried to the right hand side,

one has a direct relation betweetlft and eX'~ ~ D and Z

=iff z I ["'~ z'2 ~ o( erf Z 8

in the fonng

e

~2 2 -""'0 z

This last may then be inverted to obtain a tran:3 cender:.ta.l relai;ic~n for Z

in tenns of the parametersO(.; 8 s a.n<YLD which may be solved graphically

to give, in symbolic for.mg

oeoo(A-16)

Thus using Equation (A=l3), one may obta..i.n~

ll= 2{S;o Z oooo(A-17)

and, finally, for the f'rost penetrat ion depth~

ocoo(A-18)

It is found convenient to rearrange this last expression so that the

radical is expressed in ·terms of',A.L in the fonn~

X =~~~ J "2ft Jart

.. __ ,

.. .

' .

_, .... _

.. ..._ . ... r

By defining the · new dimensionless correction coefficient )\..as

A= J~2 = A<ol'. b ·J'--) •••• (A-20)

and by substituting under the radical in Equation (A-19) for the physical

variables, one finally obtains for X, the general expression~

•••• (A-21)

for time measured in days. The correction coefficient )\is, again, a

function of the three dimensionless parameters 0(, S , ~ • A-03. PHYSICAL SIGNIFICANCE OF TERMS AND PARAMETERS. -

a. Surface Temperature Change ( v } • - In c-rder- to apply E':Jua­s

tion (A-21) to act·:~l cases in which the surface tempereture does not

chan~e suddenly as assUffied in this derivation, the average temp~ra~Qre

below freezing during the freezing period is taken as v • Thus, in terms s

of the freezing index F (in degree-days) and the freezing period t (in

days}, the surface temperature v8

may be 1vritten ~

v s

= F/t

or, solving for F,

F = v t s ~- ... ;. (A-23)

The effects of the actual shape of the freezing index vs~ time

curve -- as compared with the assumed linear shape -- are not significant

so long as the freezing period is not too short nor the freezing index too

small (i.e., cases in which the depth of freeze would normally be fairly

small.) This problem is treated in Section 2-02-f •

b. Thermal Ratio (0(). - The thennal ratio{)(, defined as

0(= v c /v cf' 0 u s ... • • • • (A-24)

A-5

!!lea~ures the ratio of heat stored initially in the unfrc~ zen soil to the

b.1::1t loss i ,_., the frozen soil c If it may be assumed, as vri th the existing

formulas~ that the difference between the \"'Olumetrio heats CU and Cf iS

not us-:..1all;}r significant, the thermal ratiocXmay be written.

, ;, •• (A-25)

1vhich is the ratio of the initial ground temperature (or mean annual tem­

perature) above the freezing point to the average surface temperature be-

low f!"eezing during the freezing period.1P

In terms of the freezing index F ~d the freezing period t,

this may then be written

•• ,.(A..,26)

o o Diffusi vi ty Ratic ( b ) o .. The (r-oot) dif'fusi vi ty ratio 8,

measures the ral a·ci ve values of diffusi ·vi ty a ~lc in tha fr~zen and

unfr-tr~. ~x,. ;~ =')ilz c It is clear that for most soils of low moisture content

b is a:;:::proxlmate::~r unityo Tabulated herewith are typical values of 0 for

representative soil types and moisture contentso

A-6

•'

....... ...

.... ·. .. . .,

...

...

-..... •. "' r

.. - - ~ ·,- - ~-~ - .. - - - - ~ - - - ._

TABLE A - I

TYPICAL VALUES OF THE DIFFUSIVITY RATIO

8 = VkfC/kuCf

Assumed Dry Density = 110 lbs/rt3

Moisture Content Values of b %

0 5

10 15 2)

Si 1 t or Clay Sand

1.oo 1.07 1.15 1.22 1.;o

1.00 0.92 1.14 1.31 1.50

Source of Data for k and C: "Final Report, Laborat~ry Research for the

Determination of the Thermal Properties of Soils" c. of E., St. Paul

D[strict, June 19~.

The effect of variations in 8 may be fmmd directly f'rom the

)\._ 0(-~ graph, Figure 5, by making use of an equivalent 0( -value given

by the expression:

o<5 .. ';. [-l_-_e_rr-=~~fii2~r=2 =-l e - <S2

-l> A:- .... (A-2a> 1 - erfSAyA/2 J

which can be approximated by the empirical equation:

$ - 1 --1 2 "1 +t)( J • • • • (A-29)

A comparison of exact values of)\.. determined from Equations (A-15) and

(A-20) with estimated values using Figure 5 . and Equation (A-29) shows only

neg1igi b1e ~iff'.~rences for all prac_tical values of ~.

A-7

'!'he following table indicates the relationship between()( • S

and 0<'~ as given by Equation (A-~).

VALUES OF 0(

0 1 2 3 4

TABLE A - II

EQUIVALENT THERMAL RATIO ( «s ) Values of ~ • ~ r1 + 5- l l

. - '2> . ~ l 2 yl ?.D{ J

VALUES OF 8 0.9 loO 1.1 1.2

0 0 0 0 l.(J{ 1.00 Oo94 o.~

2.16 2.00 lo87 1.76 3o2'J 3.00 2.80 2.62 4.35 . 4.oo 3.72 3oLt7

lo3 1.4 1.5

0 0 0 o.as 0~82 0.79 lo67 1.59 1.53 2.48 2o36 2.25 3.28 3.11 2.97

Since it has been demonstrated that a 1% error in water content

produces only about a 1.5% change in the value of S , the above tabulation

in conjunction vri th Figure 5 would indicate less than a 1% change in the

oa.mputed depth of penetration due to thi6 variation. Moreover, typical

soils have (5 values of the order of 1.15, which would indicate an effect-

ive value of l)(s which is roughly 15% smaller than that computed as­

suming 8 = 1.0, yet this effect can produce a variation inA and there-

fore a change in predicted depth of less than 5% for typical thermal con-

ditionso

Since inclusion of a tenn involving b would increase the com-

plexi ty of represen ta·bion of the .formula in graphical form, and would

further increase the tenden.:ry t o predict depths of penetration whioh are

• f

..,. .. __

....

..... ~

.. -

too deep, it seems reasonable to base at least prelimina~ calculations

on the assumption that8 = 1.0~

Consistent, then, with the previous assumption that C • Cu • Ct

and ~ = (k£ ) ( en) 1 0 th ld f 11 t h di o ~ = • , .ere wou o ow hen t e necessary con -ku cf

tion that k = k = k • In practical applications, this would suggest f u

the use of average values of C and k.

d. FUsion Parameter (A_). - The fusion parameter~, defined

as •••• (A-30)

measures the heat removed in the frozen soil (below the freezing point)

compared to the latent heat of the soil moisture. This characteristic

may be written in terms of F, t, and C in the form

~ = FC/Lt •••• (A-31)

When~= 0, the only significant soil properties affecting the

depth of frost penetration are the latent heat L, and the thermal conduc-

tivity, kf; on the other hand~ as~becomes large, the stored heat in the

soil volume becomes proportionately more significant.

e. Correction Coefficient (Jl). - If Equation (A-22) is sub­

stituted into Equation (A-21) we obtain the more familiar expression

X =A J4~F .... (A-32)

and)\ , therefore, becomes a correction coefficient applied to the calcu-

lated depth of penetration due to latent heat only.

The value of A may be found from Figure 5 which shows the rela­

tionship between 0( ,_,JJ. and ~ for the case of 8 a 1. Data for this figure

were obtained by assuming values of z and solving Equation (A-15) for~ •

On.ce,A. was obtained A followed directly from Equation (A-20).

A-9

I

COMPARISON WITH BERGGREN FORMULA. -

As presented in Section 2-02-b. Shannon (l9)has written the

(18) Berggren formula in the form:

X = 2B J aft •• •• (A-33)

By comparison with Equation (A-18) of Section A-02, it is clear that the

term B in Equation (A-33) is identical with the ter.m Z in Equation (A-18).

Thus the only significant difference between the older Berggren formula

and the present rational formula lies in the inversion of the nuclear

radical term to permit expressing the correction factor~ to the depth

of frost penetration that wuuld occur if only latent heat were present,

rather than the correction term B _ Z applied to the depth occurring if

only volumetric heat is significant. Since for all practical problems

the effects of L are large compared to those of C (i.e., the~ values

are low), this seems to be the more useful for.m for the same fundwmental

solution.

For the above reasons, it would seem fitting to refer to the

rational formula proposed here as the "modified Berggren formula "·•

A-10

. ..

... -. . .

. . . ·· ~

~ -....

APPENDIX B

TABLES OF I ... Bo M.. SOLUT~ONS

B-01. GENERAL. -

As outlined in Secticn 2-01-f-(3)! part of this investiga-

tion wa~ concerned 1rnth obtaining precise numerical sol1·1tions to the

difference equations which represent the thermal transients in the

ground during the freezing and th~wing prc:eesses & 'Et1ese equations are

presented and explained in .the section cited.

Solutions to these equations were run on the IaB.1L Card Pro-

grarnmed Calculator of the M ... I.T ... Statistical Services Division, with

values and conditions indicated in the follo1vL."'1.g tabule.tio~.,

B~02. INDEX TO SOLUTIONS. -

SURFACE PARAMETERS fu\NGE COEFF. TABLE TE?vfi'ERA TI.:.:: ~

VARIATION _.,AA.., o( h K. (3 B-I step Oo5 0+* 8 24 Oo25 B ~~II Step 0.5 1.0 8 24 0.25 B-III Step 0.5 2.C 8 24 0.25 B-IV Step U:.5 3.0 8 24 0.25 Bn•V Step Oo5 4.0 Q 24 0.25 u

B-VI Step Oo5 5o0 8 ~L 0.23 '-"+

B-VII Step loO Q+Jt: G 24 0.25 B-VIII Step 1.0 1.0 8 24 Oo25 B-IX step 1.0 2.0 8 24 0.25 B-X Step 1.0 3.0 8 24 0.25 B-XI St~p 1.0 4.0 8 24 S.25 B-XII step 1.0 5o0 8 24 (:, ·:)!::

'<w~ O '-_..~

B-XIII Cyclic 1.0 0 7 108 o.4o B-XIV Cyclic loO loO 7 108 o.4o

•Temperature initially at the freezing point, soil ~~fro~e.no

B-03. TABULATED VALUES. -

The tabular entries in the tables give the value of e . cor­n,.K:

responding to each value of' n and k. These values for· n = o correspond

B-1

to the surface temperature variation, while the remaining values repre-

sent subsurface temperatures. The transformation of these dimension-

less values, together with the time measure (k) and depth measmre (n),

are explained in Section 2-01-f-(3). . . -· .

-. -

.• ._ .. "'

- - ~ ....

-- . -

_, ,_.,

. ..._ .

k

VALUES

0

1

2

3

4 5 6 ·

7

8

9

10

11

12

13

14 1.5

16

17

18

19

20

21

22

23

24

0 1

-0.50 0

-0.50 0

-o.so 0

-o.so 0

-0.50 0

-o.so 0

-0.50 0

-o.so 0

-0.50 0

-0.50 -0.12

-o.so -0.19

-0.50 -0.22

-0.50 -0.23

-0.50 -0.24

-0.50 -0.25

-0.50 -0.25

-o.so -0.25

-0.50 -0.25

-0.)0 -0.25

-o.so -0.25

-0.50 -0.25

-0.50 -0.25

-0.50 -0.25

-0.50 -0.25

-a.so -0.25

TABLE B-I .P- = o. 5 oc = 0+

n VALUES

2 3 4 5 6 7 8

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 '

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 · 0 0 n

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 ·o 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 o - 0 0 0 0 0

·-·"' ~

k

VALUES

0

1

2

3

4

s 6

7

8

9

10

11

12

13

14 15 16

17

18

19

20

21

22.

23

24

0 1

-0.50 0.50

-0.50 0.25

-0.50 0.12

-0.50 0.05

-0.50 0

-0.50 0

-0.50 0

-0.50 0

-0.50 0

-0.50 0

-0.50 0

-o.so 0

-0.50 0

-0.50 0

-o.so 0

-0.50 0

-0.50 0

-0.50 0

-0.50 0

-0.50 0

-0.50 -.06

-0.50 -.10

-0.50 -.12

-0.50 -.14

-o.so -.15

TABLE B-II ~ = o.5 oc = i.o

n VALUES

2 3 4

0.50 o.so 0.50

0.50 0.50 0.50

0.44 o.so 0.50

0.38 0.48 o.so

0.32 0.48 o.so

0.28 o.h6 0.49

0.26 0.46 0.49

0.24 0.46 0.48

0.24 0.45 0.48

0.23 0.45 0.48

0.23 0.45 0.47

0.23 0.45 0.47

0.22 0.45 0.47

0.22 o.lili 0.47

0.22 0.44 o.h6

0.22 O.h4 0.46

0.22 0.44 0.46

0.22 0.44 0.46

0.22 0.1.:1 0.46

0.22 o.!~. 0.46

0.22 0.44 0.46

0.20 0.!!4 0.46

0.19 0.43 0.46

0.17 0.43 0.45

0.16 0.42 0.45

5 6 7 8

0.50 o.so 0.50 0.50

o.so 0.50 0.50 0.50

0.50 0.50 0.50 0.50

0.50 0.50 o.so 0.50

0.50 0.50 0.50 o.so

0.50 0.50 0.50 0. )0

0.50 0.50 o.so 0.)0

0.40 0.50 0.50 0.50

0.49 0.50 o.so o.so

0.49 0.50 o.so 0.50

0.49 0.50 o.;;o o.so

o.L~9 o.so o.so o.so

0.48 0.49 0.50 o.so

0.48 o.L.9 0.50 o • .so

0.48 0.49 o.so 0.50

0.48 O.h9 o.so o.so

o.h8 0.49 0.50 0. )0

o.h8 o.L1.9 0.1.~.9 o.so

0.48 0.49 O.h9 0.50

0.47 0.48 0.49 0.50

0.47 0.48 o.h9 0.50

0.47 0.48 0.49 0.50

0.47 0.48 o.h9 o.so

o.L.7 0.48 o.L.9 0.50

0.47 0.48 0.49 0.50

TABLE B-II

TABLE B-III ~ = o • .5 oc = 2.0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0

1

2 Ill } .

3 SOLUT ~ON FOU rn TO BE DEEi'EC'I IVE

4 5

6

7

8

9

10

11

12

13

14 15

16 -

17

18

19 .. ..... _,

20

21

22

23

24

TABLE B-Ill

TABLE B-IV ~ = 0.5 ot: : 3.0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -0.50 1.50 1.$0 1.50 1.50 1.50 1.50 1 • .50 1.so

1 -o.so 1.00 1.50 1.)0 1.)0 1.)0 1.so 1.50 1.50 ' ' \

~: .... 2 -0 • .50 0.75 1.37 1.50 1.50 1.50 1.50 1.50 1.)0

3 -o.so 0.59 1.25 1.46 1.So 1.50 1.)0 1.50 1.)0

4 -0.)0 0.48 1.14 1.45 1.49 1.)0 1.50 1 • .50 1 • .50

5 -0.50 0.40 1.05 1.43 1.4.8 1.49 1.50 1.50 1.50

6 -o.so 0.34 0.98 1.41 1.47 1.49 1.49 1.50 1 • .50

7 -0.50 0.29 0.93 1.40 1.46 1.49 1.49 1.L!.9 1.50

8 -o.so 0.25 0.89 1.39 1.4S 1.L.8 1.49 1.49 1.50

9 -0.50 0.22 0.86 1.38 1.44 1.48 1.49 1.49 1 • .50

10 -0.50 0.20 0.83 1.37 1.44 1.47 1.L!.9 1.L9 1.so

11 -o.so 0.18 0.81 1.37 1.LJ 1.4.7 1.49 1.49 1.50

12 -0.50 0.17 0.79 1.36 1.42 1.46 1.48 1.49 1.50

13 -o.so 0.16 0.78 1.36 1.42 1.h6 1.L.8 1.49 1.50

14 -0.50 0.15 0.77 1.35 1.41 1.L6 1.48 1.49 1.50

15 -0.50 O.lh 0.76 1.35 1.L1 1.45 1. L~ 7 l.L.9 1.so

16 -0.50 0.14 0.75 1.34 1.40 1. L!.L~ 1.47 1.48 1.50

17 -o.5o 0.13 0.7.5 1.34 1.40 1.44 1.47 1.L.8 1 • .50

18 -0.50 0.13 0.74 1.33 1.39 1.44 1.4.6 . 1.48 1 • .50

-.-~~,. 19 -0.50 0.12 0.74 1.33 1.39 1.43 1.4.6 1.46 1. 50

20 -o.so 0.12 0.73 1.33 1.39 1.43 1.46 1.48 1.50

21 -o.so 0.12 o. 73 1.32 1.38 1.43 1.46 1.48 1.50

22. -0.50 0.12 0.73 1.32 1.38 1.42 1.45 1.47 1.50

23 -o.5o 0.12 0.72 1.32 1.37 1.L2 1.45 1.47 1.50

24 -o.so 0.11 0.72 1.32 1.37 ~.42 1.45 1.47 1.50

TABLE B-IV

TABLE B-V .~" = o. 5 oc: :: 4.0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -o.5o 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00

1 -o.5o 1.37 2.0Ci 2.00 2.00 2.00 2.00 2.00 2.00

2 -0.50 1.06 1.8!~ 2.00 2.00 2.00 2.00 2.00 2.00

. '. 3 -0.50 0.87 1.68 1.96 2.00 2.00 2.00 2.00 2.00

4 -0.50 0.73 1.55 1.94 1.99 2.00 2.00 2.00 2.00

s -0.50 0.63 1. L~L: 1.91 1.98 1.99 2.00 2.00 2.00

6 -0.50 0.55 1.35 1.89 1.96 1.99 1.-99 2.00 2.00

7 -o.so 0.49 1. 29 . 1.88 1.95 1.98 1.99 1.99 2.00

8 -o.so 0.44 1.23 1.86 1.94 1.90 1.99 1._99 2.00

9 -o.so 0.40 1.19 1.85 1.93 1.97 1.99 1.99 2.00

10 -o.so 0.38 1.16 l.84 1.92 1.97 1.99 1.99 2.00

11 -o.5o 0.3.5 1.13 1.83 1.91 1.96 1.98 1.99 2.00

12 -0.50 0.34 1.11 1.83 1.91 1.95 1.98 1.99 2.00

13 -0.50 0.32 1.10 1.82 1.90 1.95 1.98 1.99 2.00

14 -0 • .50 0.31 1.08 1.81 1.89 1.94 1.97 1.99 2.00

1.5 -0.50 0.30 1.07 1.81 1.88 1.94 1.97 1.98 2.00

16 -o.so 0.30 1.06 1.80 1.88 1.93 1.96 1.98 2.00

17 -o.so 0.29 1.0.5 1.80 1.87 1.93 1.96 1.98 2.00

18 -o.so 0.28 1.05 1.79 1.87 1.92 1.96 1.98 2.00

19 -0.50 9-28 1.04 1.79 1.86 1.92 1.95 1.98 2.00

20 -0.50 0.28 1. 04 1.78 1.86 1.91 1.95 1.97 2.00

21 -o.so 0.27 1.03 1.78 1.85 1.91 1.95 1.97 2.00 .. I•

22 -a.so 0.27 .1.03 1.78 . 1.85 ·1.90 1.94 1.97 2.00

23 -0.5'0 0.27 1.03 1.77 1.84 1.90 1.94 1.97 2.00

24 -o.so 0.27 1.02 '1. 77 1.84 1.90 1.94 1.96 2.00

TABLE B-V

TABLE B-VI p = 0 • .5 oc = 5•0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -0.50 2.50 2 • .50 2.50 2 • .50 2.50 2.50 2 • .50 2.50

1 -0 • .50 1.75 2.50 2.50 2.50 2.50 2.50 2.50 2.50

.. ~ . 2 -0 • .50 1.37 2.31 2.50 2.50 2.)0 2.50 2.50 2.50

3 -o.5o 1.14 2.12 2.L5 2.50 2.50 2.)0 2.50 2.50 - r

4 -0.50 0.98 1.96 2.42 2.48 2.50 2.50 2.50 2.)0

s -0.50 0.85 1.83 2.39 2.47 2.49 2.50 2.50 2.50

6 -0.50 0.76 1.72 2.37 2.46 2.49 2.49 2.50 2.50

7 -o.so 0.69 1.64 2.35 2.44 2.48 2~49 2.49 2.50

8 -o.so 0.63 1.58 2.34 2.43 2.47 2.49 2.L.9 2.50

9 -0.50 0.59 1.53 2.32 2.42 2.47 2.L.9 2.49 2.50

10 -0.50 0.55 1.49 2.31 2.41 2.46 2.48 2.49 2.50

11 -o.so 0.22 1.46 2.30 2.40 2.45 2.48 2.L.9 2.50

12 -0.50 0.50 l.L4 2.29 2.39 2.45 2.48 2.49 2.50

13 -0.)0 0.49 1.42 2.29 2.38 2.LL 2.47 2.L~9 2.50

14 -o.so o.L.7 1.40 2.28 2.37 2.43 2.47 2.L8 2.50

15 -0.50 0.46 1.39 / 2.27 2.36 2.43 2.46 2.48 2.50

16 -0.50 0.4.5 1.38 2.26 2.36 2.h2 2.4.6 2.48 2.50

17 -0.50 0.45 1·.37 2.26 2.35 2.L1 2.45 2.48 2.50

18 -0.50 0.44 1.36 2.25 2.34 2.41 2.45 2.47 2.50

19 -0.50 0.44 1.35 2.25 2.34 2.40 2.44 2.L.7 2.50 .. • I"

20 -0.50 0.43 1.35 2.24 2.33 2.40 2.L4 2.47 2.50

21 -o.so 0.43 1.34 2.24 2.32 2.39 2.44 2.47 2.50

22. -0.50 0.42 1.34 2.23 2.32 2.39 2.43 2.46 2.50

23 -0.50 0.42 1.33 2.23 2.31 2.38 2.43 2.46 2.50

24 -0.50 0.42 1.33 2.23 2.31 2.38 2.42 2.46 2.50

TABLE B-VI

TABLE B-VII .P- = 1.0 oc: 0+

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -1.00 · o 0 0 0 0 0 0 0

1 -1.00 0 0 0 0 0 0 0 0 1 ..

2 -1.00 0 0 0 0 0 0 0 0

.... 3 -1.00 0 0 0 0 0 0 0 0 -

4 -1.00 0 0 0 0 0 0 0 0

s -1.00 -0.25 0 0 0 0 0 0 0

6 -1.00 -0.38 0 0 0 0 0 0 0

7 -1.00 -0.44 0 0 0 0 0 0 0

8 -1.00 -O.L~7 0 0 0 0 0 0 0

9 -1.00 -0.48 0 0 0 0 0 0 0

10 -1.00 -0.49 0 0 0 0 0 0 0

11 -1.00 -0.50 0 0 0 0 0 0 0

12 -1.00 -0 • .50 0 0 0 0 0 0 0

13 -1.00 -0.50 0 0 0 0 0 0 0

14 -1.00 -0.50 0 0 0 0 0 0 0

1.5 -1.00 -0 • .50 -0..-12 0 0 0 0 0 0

16 -1.00 -0.53 -0.19 0 0 0 0 0 0

17 -1.00 -0.56 -0.23 0 0 0 0 0 0

18 -1.00 -0.59 -0.2) 0 0 0 0 0 0

19 -1.00 -0. 61 -0.27 0 0 0 0 0 0

20 -1.00 -0.62 -0.29 0 0 0 0 0 0

21 -1.00 -0.63 -0.30 0 0 0 0 0 0

22 -1.00 -0.64 -0.31 0 0 0 0 0 0

23 -1.00 -o.65 -0.31 0 0 0 0 0 0

24 -1.00 -0.65 -0.32 0 0 0 0 0 0

TABLE B-VII

TABLE B-VIII

~ = 1.0 oc = 1.0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1 -1.00 o.so 1.00 1.00 1.00 1.00 1.00 1.00 1.00

2 -1.00 0.25 0.88 1.00 1.00 1.00 1.00 1.00 1.00

3 -1.00 0.09 0.76 1.97 1.00 1.00 1.00 1.00 1.00

4 -1.00 0.09 0.64 0.95 0.99 1.00 1.00 1.00 1.00

5 -1.00 0.09 0.56 Q.93 0.98 1.00 1.00 1.00 1.00

6 -1.00 0.09 0.51 0.92 0.97 1.00 1.00 1.00 1.00

7 -1.00 0.09 0.49 0.91 0.97 0.99 1.00 1.00 1.00

8 -1.00 0.09 0.47 0.90 0.96 0.99 1.00 1.00 1.00

9 -1.00 0.09 0.46 0.90 0.95 0.98 1.00 1.00 1.00

10 -1.00 0.09 0.46 0.90 0.94 0.98 0.99 1.00 1.00

11 -1.00 0.09 o.hS 0.39 0.94 0.97 0.99 1.00 1.00

12 -1.00 0.09 0.45 0.89 0.93 0.97 0.99 1.00 1.00

13 -1.00 -0.14 o.h5 0.89 0. 93 0.97 0.99 0.99 1.00

14 -1.00 -0.21 0.41 0. 89 0.93 0.96 0.98 0.99 1.00

15 -1.00 -0.25 0.37 0.88 0.92 0.96 0.98 0.99 1.00

16 -1.00 -0.28 0.34 0.87 0.92 0.96 0.98 0.99 1.00

17 -1.00 -0.30 0.32 0. 86 0.91 0.95 0.98 0.99 1.00

18 -1.00 -0.32 0.30 0.86 0.91 0.95 0.97 0.99 . 1.00

19 -1.00 -0.34 0.28 0.85 . 0.90 0.95 0.97 0.99 1.00

20 -1.00 -0.35 0.27 0.84 0.90 0.94 0.97 0.98 1.00

21 -1.00 -0.36 0.26 0.84 0.90 0.94 0.97 0.98 1.00

22. -1.00 -0.36 0.25 0.84 0. 39 0.94 0.96 0.98 1.00

23 -1.00 -0.37 0.24 0.83 0.89 0.93 0.96 0.98 1.00

24 -1.00 09.37 0.24 0.83 0.89 0.93 ' 0.96 0.98 1.00

TABLE B-VIII

TABLE B-IX ~ = 1.0 oc = 2.0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -1.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00

1 -1.00 1.25 2.00 2.00 2.00 2.00 2.00 2.00 2.00

2 -1.00 0.88 1.81 2.00 2.00 2.00 2.00 2.00 2.00

3 -1.00 0.64 1.62 1.95 2.00 2.00 2.00 2.00 2.00

4 -1.00 0.48 1.46 1.92 1.98 2.00 2.00 2.00 2.00

5 -1.00 0.35 1.3J 1.89 1.97 1.99 2.00 2.00 2.00

6 -1.00 0.26 1.22 1.87 1.96 1.99 1.99 2.00 2.00

7 -1.00 0.19 1.14 1.85 1.94 1.98 1.99 1.99 2.00

8 -1.00 0.13 1.08 1.84 1.93 1.97 1.99 1.99 2.00

9 -1.00 0.09 1.03 1.82 1.92 1.97 1.99 1.99 2.00

10 -1.00 o.os 1.00 1.81 1.91 1.96 1.98 1.99 2-.00

11 -1.00 0.03 0.97 1.80 1.90 1.9S 1.98 1.99 2.00

12 -1.00 0.03 0.94 1.79 1.E9 1.9S 1.98 1.99 2.00

13 -1.00 0.03 0.92 1.79 1.88 1.94 1.97 1.99 2.00

14 -1.00 0.03 0.91 1.78 1.87 1.93 1.97 1.98 2.00

1.5 -1.00 0.03 0.90 1.77 1·.86 1.93 1.96 1.98 2.00

16 -1.00 0.03 0.89 1.77 1.36 1.92 1.96 1.98 2.00

17 -1.00 0.03 0.89 1.76 1.85 1.91 1.95 1.98 2.00

18 -1.00 0.03 0.89 1.76 1.8h 1.91 1.95 1.97 2.00

19 -1.00 0.03 0.88 1.7S 1.84 1.90 1.94 1.97 2.00 • - l1 - ..;

20 -1.00 0.03 0.88 1.75 1.83 1.90 1.94 1.97 2.00

21 -1.00 0.03 0.88 1.74 1.83 1.89 1.94 1.97 2.00

22 -1.00 0.03 0.88 1.74 1.82 1.89 1.93 1.96 2.00

23 -1.00 0.03 0.87 1.74 1.82 1.88 1.93 1.96 2.00

24 -1.00 0.03 0.87 1.73 1.81 1.88 1.92 1.96 2.00

TABLE B-IX

TABLE B-X ./4- = 1.0 oc = .3.0

k n VALUES

VALUES 0 1 2 3 4 5 6 7 8

0 -1.00 3.00 3.00 ).00 3.00 3.00 3.00 3.00 3.00

. ' . 1 -1.00 2.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00

2 -1.00 1.50 2.75 3.00 3.00 3.00 3.00 3.00 3.00

.I , 3 -1.00 1.18 2.50 2.93 3.00 3.00 3.00 3.00 3.00

4 -1.00 0.97 2.28 2.90 2.98 3.00 3.00 3.00 3.00

s -1.00 o.eo 2.10 2.86 2.96 2.99 3.00 3.00 3.00

6 -1.00 0.68 1.97 2.83 2.94 2.99 2.99 3.00 3.00

7 -1.00 0.58 1.86 2. 81 2.93 2.98 2.99 2.99 3.00

8 -1.00 o.51 1.78 2.?9 2.91 2.97 2.99 2.99 3.00

9 -1.00 0.45 1.71 2.77 2.89 2.96 2.99 2.99 3.08

10 -1.00 0.40 1.66 2.75 2.88 2.95. 2.98 2.99 3.00

11 -1.00 0.37 1.62 2.74 2.86 2.94 2.98 2.99 3.00

12 -1.00 0.3h 1.58 2.73 2.85 2.93 2.97 2.99 3.00

13 -1.00 0.32 1. 56 2.72 2.84 2.92 2.96 2.98 3.00

14 -1.00 0.30 1.54 2.71 2.83 2.91 2.96 2.98 3.00

15 -1.00 0.28 1.52 2.70 2.82 2.90 2.95 2.98 3.00

16 -1.00 0.27 1.50 2.67 2.87 2.89 2.98 2.97 3.00

17 -1.00 0.26 1.49 2.68 2.80 2.89 2.94 2.97 3.00

18 -1.00 0.26 1.48 2.67 2.79 2.88 2.93 2.97 3.00

19 -1.00 0.25 1.47 2.67 2.78 2.87 2.93 2.96 3.00

20 -1.00 o. 2}~ 1.1~6 2.66 2.78 2.86 2.92 2.96 3.00

21 -1.00 0.2h 1.46 2.65 2.77 2.86 2.92 2.96 3.00

22. -1.00 0.23 1.45 2.65 2.76 2.85 2.91 2.95 3.00

23 -1.00 0.23 1.44 2.64 2.75 2.84 2.91 2.95 3.00

24 -1.00 0.23 1.4h 2.64 2.75 2.84 2.90 2.94 3.00

TABLE B-X

, . .

•• <¥ ,..

.. - , •

k

VALUES

0

1

2

3

4

5 6

7

8

9

10

11

12

13

14 15 16

17

18

19

20

21

22

23

24

0 1

-1.00 ·4.00

-1.00 2.75

-1.00 2.12

·-1.00 1.73

-1.00 1.46

-1.00 1.25

-1.00 1.09

-1.00 0.98

-1.00 0.88

-1.00 0.81

-1.00 0.75

-1.00 0.71

-1.00 0.67

-1.00 0.64

-1.00 0.62

-1.00 0.61

-1.00 0.59

-1.00 0.58

-1.00 0.57

-1.00 0.56

-1.00 ·o.55

-1.00 0.55

-1.00 0.54

-1.00 0.54

-1.00 0.53

TABLE B-XI

_p. = 1.0 oc; 4 0 •

n VALUES

2 3 4 5 6 7 8

4.00 4.00 4.00 4.00 4.00 4.00 4.00

4.00 4.00 4.00 4.00 4.00 4.00 4.00

3.68 4.00 4.00 4.00 4.00 4.00 4.00

3.31 3.92 4.00 4.00 4.00 4.00 . 4.00

3.10 3.88 3.98 4.00 4.00 4.00 4.00

2.88 3.82 3.96 3.99 4.00 4.00 4.00

2.71 3.79 3.93 3.98 3.99 4.00 4.00

2.58 3.76 3.91 3.97 3.99 3.99 4.00

2.47 3.73 3.89 3.96 3.99 3.99 4.00

2.39 3.71 3.87 3.95 3.98 3.99 4.00

2.32 3.69 3.88 3.94 3.88 2.99 4.00

2.27 3.67 3.83 3.93 3.91 3.99 4.00

2.23 3.66 ' 3.82 3.91 3.96 3.99 4.00

2.20 3.65 3. 80 3.90 3.96' 3.98 4.00

2.27 3.63 3.79 3.89 3.95 3.98 4.00

2.15 3.62 3.77 3.88 3.94 3.97 4.00

2.13 3.61 3.76 3.87 3.93 3.91 4.00

2.11 3.60 3.75 3.86 3.93 3.96 4.00

2.10 3.59 3.74 3.85 3.92 3.96 4.00

2.09 3.58 3.73 3.04 3.91 3.96 4.00

2.08 3.57 3·. 72 3.83 3.90 3.95 4.00

2.07 3.57 3.71 3.82 3.90 3.95 4.00 2.06 3.56 3.70 3.81 3. 89 ·3.94 4.00

2.06 3.55 3.69 3.80 3.88 3.94 4.00

2.05 3.55 3.69 3.80 3.88 3.93 4.00

TAlLE B-XI

TABLE B-XII ~ = 1.0 oc = 5.00

k n VALUES

• VALUES 0 1 2 3 - 4 5 6 7 8

0 -1.00 s.oo 5.00 5.00 5.00 5.00 5.00 5.00 .s.oo

. t .. 1 -1.00 3.50 5.00 5.00 5.00 5.00 s.oo 5.00 5.00

2 -1.00 2.75 4.62 5.00 .s.oo 5.00 5.00 5.00 5.00

. ., ,. 3 -1.00 2.28 4.25 4.90 5.00 5.00 5.00 5.00 5.00

4 -1.00 1.95 3.92 4.85 4.97 5.00 5.00 5.00 5.00

s -1.00 1.70 3.66 4_.79 4.95 4.99 5.00 5.00 5.00

6 -1.00 1.51 3.45 4.75 4.92 4.98 4.99 5.00 5.00

7 -1.00 1.37 3.29 4.71 4.89 4.97 4.99 '

4.99 5.00

8 -1.00 1.26 3.17 4.68 4.87 4.95 4.99 4.99 5.00

9 -1.00 1.17 3.07 4.65 4.84 4.94 4.98 4.99 5.00

10 -1.00 1.10 2.99 4.63 4.82 L.93 4.97 4.99 5.oo -

11 -1.00 1.05 2.93 4.61 4.80 4.91 4.97 4.99 s.oo

12 -1.00 1.00 2.88 4.59 4.78 4.90 4.96 4.98 5.00

13 -1.00 0.97 2.84 4.58 4.76 4.88 4.95 4.98 s.oo

14 -1.00 0.94 2.81 4.56 4.75 4.87 4.94 4.97 5.00

15 -1.00 0.93 2.78 4.55 4.73 4.86 4.93 4.97 s.oo

16 -1.00 0.91 2.76 4.53 4.72 4.84 4.92 4.96 5.00

17 -1.00 0.89 2.74 4.52 4.70 4.83 4.91 4.96 5.00

16 -1.00 0.88 2.72 4.51 . 4.69 4.82 4.80 4.95 5.00

19 -1.00 0.87 2.71 4.50 4.68 . !~ .• 81 4.89 h.95 5.00

20 -1.00 0.86 2.70 4.h9 4.67 4.80 4.89 4.94 5.00

21 -1.00 0.86 .2.69 4.48 4.65 4.79 4.88 4.94 5.00

22. -1.00 0.85 2.68 h.47 4.6!~ 4.78 4.87 h.93 5.00

23 -1.00 0.85 2.67 !~ .46 4.63 4.77 4.86 4.92 5.00

24 -1.00 0.84 2.66 h.46 h.62 4.76 · 4.85 4.92 5.00

TABLE B-XII

k

VALUES 0 1

0

1 0.26 0.10

2 0.75 0.23

3 I 0.96 . 0.36

4 1.15 0.49

s 1.30 0.63

6 1.40 0.75

7 1.48 0.84

8 1 • .50 0.93

9 1.48 0.98

10 1.hO 1.01

11 1.30 1.01

12 1.15 0.98

13 0.96 0.93

14 ' 0.75 0.85

15 0.51 0.75

16 0.26 0.62

17 0 o.h7

18 -0.26 0.32

19 -0.51 0.16

20 -o. 75 0 ... 21 -0.96 0

22 -1.1) 0

23 -1.30 0

24 -1.40 -0.34

TABLIS B~II ~ = 1.0 oc = 0

n VALUES

2 3 4

0 0 0

o.o4 0 0

0.10 0.02 0

0.17 0.04 0.01

0.25 0.08 0.02

0.33 0.12 0.04

o.L.1 0~17 0.06

o.h9 0.22 0.09

o.S6 0.28 0.12

0.61 0.33 0.15

0.66 0.37 . 0.19

0.68 0.41 0.22

0.69 0.44 0.25

0.69 0.47 0.28

0.66 0.48 0.30

0.62 0.48 0.32

0.56 0.47 0.33

0.49 0.45 0.33

0.41 0.42 0.33

0.31 0.38 0.32

0.21 0.33 0.30

0.17 0.27 0.28

o.ll~ 0.23 0.25,

0.12 0.20 0.22

5

0

0

0

0

0

o.o1

0.02

0.03

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.19

0.20

0.21

0.21

0.21

0.21

0.19

0.18

6 7 8

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0.01 . 0

0.02 0

0.03 0

o.o~. 0

0.05 0

0.06 0

0.07 0

0.08 0

0.09 0

0.10 0

0.10 0

0.10 0

0.11 0

0.11 0

0.10 0

0.10 0

TABLE B-XII I Sheet 1 of 5

. .. -

, · .

k

VALUES

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

0 1

-1.48 -0.58

-1.50 -0.70

-1.48 -0.74

-1.40 -0.74

-1.30 -0.71

-1.15 -0.66

-0.96 -0.68

-0.75 -0.63

-0.51 -0.56

-0 •. 26 -O.Lh

0 -0.31

0.26 -0.16

0.51 0

0.75 0

0.96 0

1.15 0

1.30 0.31

1.40 0.58

1.48 0.68

1.50 0.73

1.1!.8 0.74

1.40 0.74

1.30 0.79

1.15 0.81

0.96 0.78

TABLE B-XIII p. = 1.0 oc:; 0

n VALUES

2 3 4

0 0.18 0.20

0 0.11 0.18

0 0.09 0.14

0 0.07 0.12

0 o.o6 0.10

-0.21 0.05 0.08

-0.29 0 0.07

-0.33 0 0.04

-0.32 0 0.03

-0.29 0 0~02

-0.24 0 0.02

-0.17 0 0.01

-0.10 0 0.01

-0.02 0 0.01

0 0 0.01

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0.21 0 0

0.34 0 0

0.39 0 0

0.40- 0 0

5 6

0.16 0.09

0.15 0.08

0.13 0.08

0.11 0.97

0.10 o.o6

0.08 0.05

0.07 o.oh

0.06 0.04

0.04 0.03

0.03 0.02

0.03 0.02

0.02 0.01

0.02 0.01

0.01 0.01

0.01 0.01

0.01 0

0.01 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

7 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

TABLE B-XIII -iheet 2 of 5

------------------------------~~~~B-~XIn

~ :1: 1.0 oC = 0

k n VALUES

VALUES 0 1 2 3 4 ... so 0.75 0.70 0.39 0 0

51 0.57 0.60 0.36 o.os 0 ' . - ' " 52 i

I 0.26 0.47 0•33 0.15 0.02

I

53 i

I 0 0.33 0.31 0.17 0.07

54 -0.26 0.19 0.26 0.19 0.09

55 -a.S7 0.04 0.20 0.18 0.10

56 I

-0.75 0 0.13 0.16 0.10

57 -0.96 0 0.09 0.13 0.11

58 -1.15 0 0.07 0.10 0.10

59 -1.30 -0.15 0.06 0.09 0.09

60 -1.40 -0.53 0 0.07 0.08

61 -1.48 -0.67 0 0.05 0.07

62 -1.50 -0.72 0 0.04 0.05

63 -1.48 -0.74 0 0.03 0.05

64 -1.40 -0.74 0 0.02 0.04

65 -1.30 -0.71 -0.29 0.02 0.03

66 -1.15 -0.78 -0.33 0 0~03

67 -0.96 -0.75 -0.38 0 0.02

68 -0.75 -0.69 -0.38 0 0.01

69 -0.51 -0.59 -0.35 0 o.o1

70 -0.26 -0.46 -0.30 0 0.01

.. 71 0 -0.32 -0.25 0 0.01

72 0.26 -0.16 -0.18 0 0

73 0.51 0 -0.10 0 0

74 0.75 0 -0.02 0 0

5 6

0 0

0 0

0 0

0.01 0.

0.03 0

0.04 0.01

o.os 0.02

0.06 0.03

O.O'l 0.03

0.06 0.03

0.06 0.03

0.06 0.03

0.05 0.03

0.04 0.03

0.04 0.02

0.03 0.02

0.03 0.01

0.02 0.01

0.02 0.01

0.01 0.01

0.01 0.01

0.01 0.01

0.01 0

0 0

0 0

7 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

TABLE &-XIII

Sheet 3 of 5

k

. - VALUES 0 1 •

75 0.96 0

76 1.15 0

_I . ...

77 1.30 0.30

78 1.1.!0 o.L.8

79 1.h8 0. 68

80 1.50 0.73

81 1. 48 0 . 75

82 1.ho 0.7L

83 1.30 0.79

84 1.1) o. D1

85 0.96 0.78

86 0.75 0.70

87 o.s1 0. 6-.J

88 0.26 o. h7

89 0 0.32

90 -0.26 0. 16

91 -0. 51 0

92 -0.7.5 0

93 -0. 96 0

·· $1 94 -1.15 0

95 -1.30 -0.24 ... 96 -1. h0 - 0 • .56

97 -1.LL8 -0. 67

98 -1.50 - 0 . 73

99 -1.48 -0.75

TABLE B-XIII

,<~- = 1.0 oc: 0

n VALUES

2 3 4 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 .21 0 0

0 .34 0 0

0. 39 0 0

0.40 0 0

0.39 0 0

0.36 0 0

0.31 0 0

0.2.5 0 0

0.18 o . oc 0

- 0.13 0. 09 0 . 03

8. 06 0 . 08 0.04

o.os 0 . 06 0 . 05

0.03 o. os ' 0 . 04

0.03 0.04 0.04

0 0. 03 0 .03

0 0. 02 0 . 03

0 0 . 02 0 . 02

0 0. 01 0 .02

5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0. 01

0 . 02

0 . 03

0.02

0 ,02

0.02

0 . 02

0 .02

6

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.01

0.01

0. 01

0.01

0 . ()1

0. 01

7 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

TABLE B-XIII Sheet 1-t of 5

k

VALUES 0 1

100 -1.40 -0.74 . .

101 -1.30 -0.76 ,•' -.4 -.. 102 -1.1.5 -0.80

103 -0.96 -0.77

104 -0.7.5 -0.70

10.5 -0.51 -0.59

106 ; -0.26 -0.47

107 0 -0.32

108 0.26 -0.16

I

I

TABLE B-XIII

,# =1.0 oc: : 0

n VALUES

2 3 4 -0.12 0.01 0.02

-0.32 0 0.01

-0.37 0 0.01

-0.39 0 0.01

-0.38 0 0

-0.36 0 0

-0.31 0 0

-0.2.5 · 0 0

-o.l8 0 0

5 0.02

0.01

0.01

0.01

0.01

0

0

0

0 '-

6 7 8

0.01 0

0.01 0

0.01 0

0.01 . 0

0 0

0 0

0 0

0 0

0 0

TABIE B-XIII Sheet 5 of 5

k

VALUES 0 1

0

1 1.86 1.17

2 2.25 1.38

3 2.60 1.60

4 2.92 1.83

s 3.17 2.05

6 3.34 2.24

7 3.46 2.41

8 3.50 2.54

9 3.L6 2.64

10 3.34 2.62

11 3.17 2.68

12 2.92 2.64

13 2.61 2.55

14 2.25 2.42

15 1.86 2.24

16 1.43 2.03

17 1.00 1.80

18 0.57 1.54

19 0.14 1.26 ... ... ' 20 -0.25 0.98

. ., .- 21 -0.61 0.70

22. -0.92 0.44

23 -1.17 0.19

24 -1.34 0

~ = 1.0 oc: 1.0

n VALUES

2 3 4

1.00 1.00 1.00

1.07 1.00 1.00

1.16 1.03 1.00

1.29 1.07 1.01

1.h2 1.13 1.03

1.56 1.21 1.06

1.69 1.29 1.10

1.82 1.37 1.15

1.93 1.46 1.20

2.02 1.54 1.25

2.10 1.62 1.31

2.14 1.69 1.36

2.16 1.74 1.41

2.15 1.78 1.46

2.11 1.80 1.50

2.04 1.80 1.53

1.94 1.79 1.55

1.82 1.75 1.55

1.68 1.70 1.55

. 1 • .52 1.63 1.53

1.35 1.55 1. so 1.17 1.45 1.46

0.99 1.34 1.41

0.81 1.23 1.35

'

5 6

1.00 1.00

1.00 1.00

1.00 1.00

1.00 1.00

1.00 1.00

1.01 1.00

1.03 1.01

1.05 1.01

1.07 1.02

1.10 1.03

1,13 1.05

1.17 1.06

1.20 1.08

1.24 1.10

1.27 1.12

1.30 1.13

1.32 1.15

1.3h 1.16

1.35 1.17

1.36 1.17

1.35 1.18

1.34 1.18

1.32 1.17

1.30 1.16

7 8

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

TABLE B-XIV Sheet 1 of 5

k

VALUES 0 1

25 -1.46 0 .I I

26 -1.50 0

27 -1.46 0 • . . 28 -1.34 -0.32

29 -1.17 -0.41

30 -0.92 -b.43'

31 -0.61 -o.37

32 -0.25 -0.26

33 -0.14 -0.10

34 0.51 . 0

35 1.00 0

' 36 1.43 0

37 1.86 0.54

38 2.25 0.94

39 2.61 1.26

40 2.92 1.~~

41 3.17 1.81

42 . 3.34 2.04

43 3.46 2.23

••• 44 3.50 2.39 . 45 3.46 2.49

- 46 3.34 2.56

47 3.17 2.57

48 2.92 2.54

49 2.61 2.46 '

TABLBB-XIV

~. 1.0 oc:: 1.0

D VALUES

2 3 4

0.65 1.11 1.28

0.57 0.99 1.21

0.51 0.91 1~13

0.47 0.84 1.06

0.30 0.78 1.00

0.21 0.68 0.~5

0.14 0.60 0.88

0.12 0.53 0.82

0.13 0.48 0.76

0.18 0.45 0.71

0.22 0.45 0.68

0.22 0.45 ·o.65

0.22 0.44 0.64

0.43 0.43 0.62

0.64 0.51 0.61

o.84 0.60 0.63

1.03 0.71 0.67

1~21 0.82 0.72

1.39 0.94 0.78

1.54 1. cfs . 0.85

1.68 1.17 0.92

1.80 1.28 1.00

1.89 1.37 1.07

1.96 1.46 1.14

1.99 1.53 1.21

s 6

1.26 1.15

1.23 1.14

1.18 1.12

1.13 1.10

1.09 1.07

1.05 1.05

1.00 1.03

0.97 1.00

0.92 0.99

0.88 0.97

0.85 0.95

0.82 0.93

·o.8o 0.91

0.78 0.90

0.77 0.89

0.75 0.88

0~76 0.88

0.77 0.88

0.79 0.88

0.82 .0.89

0.86 0.91

0.90 0.93

0.95 0.95

1.00 0.97

1.04 0.99

7 8

1.00

1.·oo

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

. 1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

TABLE B-XIV Sheet 2 of 5

I·

k

VALUES 0 1 -

so 2.25 2.33

• • 51 1.86 2.16

I'- 52 1.43 1.96

53 1.00 1.73 .. ·" 54 0.56 1.47

55 0.15 1.21

56 -0.25 0.93

57 -0.61 0.66

58 -0.92 0.3~

59 -0.17 0.15

~ -1.34 0

61 -1.46 0

62 -1.50 0

63 -1.46 -0.94.'

64 -1.34 -0 •. 40

65 -1.17 -0.45

66 -0.92 -0.46

67 -0.61 -0.39

68 -0.25 -0.28

69 .' 0.14 -0.12 - .. ., . 70 0.57 0

71 1.00 0

72 1.43 0

73 1.86 0.45

74 2.25 0.9~

1AD1JI:I o•li~ v

~. 1.0

n VALUES

2 3 4

1.99 1.59 1.27

1.97 1.62 1.33

1.91 1.64 1.37

1.82 1.64 1.40

1.71 1.62 1.42

1.48 1.48 1.42

1.43 1.52 1.42

1.26 .1.44 1.40

1.09' 1.35 ' 1.36

0.92 1.25 1.32

0.74 1.15 1.27

0.61 1.03 1.20

0.53 0.93 1.14

o;L8 0.85 1.07

0.42 0.79 1.01

0.24 0.73 0.95

0.16 0.62 0.90

0.10 0.55 0.83

0.08 0.48 0.78

0.10 0.44 0.72

0.14 0.41 0.68

0.20 0.41 0.64

0.20 0.42 0~.62

0.21 0.21 0.61

0.40 0.41 . 0.60

s 6

1.09 1.02

1.13 1.04

1.17 1.06

1. 21 . 1.08

1.23 1.10

1.25 1.11

1.27 1.12

1.27 1.13

1.26 1.13

1.25 1.13

1.23 1.13

1~20 1.12

1.17 1.10

9.13 1.09

1.09 1.07

1.05 1.05

1.01 1.03

0.97 1.01

0.93 0.99

. 0.89 0.97

0.86 0.95

0.82 0.93

0.80 0.92

0.77 0.90

0.76 0.89

7 8

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

TABLE-B-XIV Sheet 3 of 5

k

VALUES 0 1

75 2.61 1.25

76 2.92 1.54

11 3.17 . 1.80

78 3.34 2.03

79 3.46 2.22

80 3.50 2.38

81 3.46 2.49

82 3.34 2.55

83 3.17 2.56

84 2.92 2.53

85 3.61 2.45

86 2.25 2.33

87 1.86 2.16 ;

88 1.43 1.96

89 1.00 1.73

90 0.57 1.47

91 0.15 1.20

92 -0.25 0.93

93 -0.61 0.65 .

94 -0.92 0.39 - .. ·• • 95 -1.17 0.15

96 -1~:34 0

97 -1.46 0

98 -1.50 0 :

99 -1.46 -0.04

•rABLI5 B-XIV

~ :. 1.0 oc = 1.0

D VALUES

2 3 4 s 0.61 0.48 0.59 0.75

0.81 0.58 0.·61 0.74

1.01 0.68 0.65 0.74

1.20 0.80 0.70- 0.76

1.37 0.92 0.76 0.78

1.53 1.04 0.83 0.81

1.67 1.15 0.91 . 0.85

1.79 1.26 0.98 0.89

1.88 1.36 1.06 0.94

1.95 1.45 1.13 0.99

1.98 1.52 1.20 1.04

1.98 1.48 1.26 1.08

1.96 1.61 1.32 1.13

1.90 1.63 1.36 1.17

1.82 1.63 1.39 1.20 ,_.

1.71 1.61 1.41 1.23

1.57 1.57 1.42 1.25

1.42 1.51 1.41· 1.26

1.26 1.L.4 1.39 1.26

1.09 1.35 1.36 ' 1. 26

0.91 1.25 1.31 1.25

0.74 1.14 1.26 1.23

0.60 1 •. 03 1.;20 1.20

0.53 0.93 1.13 1.17

0.48 0.85 1.06 1.13

6

0.88

0.88

0.87

0.87

0.88

0-89 - \

0.90

0.92

0.94

0.96

0.99

1.01

1.04

1.06

1.08

1.10

1.11

1.12

1.13

1.13

1.13

1.13

1.12

1.10

1.09

7 8

1.00 .

1 •. 00

1.00

1.00

1.00

1.00

1.00

1.00

, 1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

TABLE ~XIV Sheet 4 of 5

k

VALUES 0 1

100 -1.34 -4.02 . .

101 -1.17 -0.45

102 -0.92 -0.46

103 -0.61 -0.40 • - .

104 -0.25 -0.28

105 .Q.15 -0.12 ·-

106 0.57 0

107 1.00 0

108 1.4J 0

i

wr-ABLI5 B-XIV

~ •1.0 oc = 1.0

n VALUES

2 3 4 0.42 0.79 1.00

0.24 0.73 0.95

0.16 0.62 0.90

0.10 0.55 0.83

0.08 0.48 0.77

0.09 0.44 0.72

0.14 0.41 0 .. 68

0.19 0.41 0.64

0.20 0.42 0.62

5

1.09

1.05

1.01

0.97

0.93

0.89 I

0.85

0.82

0.79

-

6

1.07

1.05

1.03

1.01

0.99

0.97

0.95

0.93

0.92

7 8

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

TABLE B-XIV Sheet 5 of 5

. '-

f~

•

~-.,. "

""

APPENDIX C

NUl~RICAL SOLUTION OF A THAW- CONSOLIDATION PROBLEM

It is intended to show by the following example a general com-

putational procedure for solving a thaw-consolidation problem numerically.

The physical nature of the problem and its mathematical fonnulation are

discussed in Part IV. For the sake of clarity in solution technique, con-

ditions for this example have been simplified ~ They may not represent a

realistic case.

C-Ol. CONDITIONS AND ASSUMPTIONS._ ~

Assume for the purpose of this example that the following con-

ditions and assumptions hold :

(1.) The soil profile applying t o this problem is the sarn.e as

that shovm. in Figure 11. The sketch given below shows the profile at a

time when the clayey silt just begins to thaw. This situation is denoted

as zero time.

0

9" 14"

48"

78"

90"

' ·, . I I

I I ' 1 I

UNFROZEN

.,

' A

. , I J / • ·' , • • • I ' , , , , • • I ~ I t • • • • 1

. : ~I . I '~ ' \ • I ' , - · - GRAVEL ' , .. , ' .. ' 1 .

CLA.YEY SILT

8

' ' I

I COARSE .. \ ,. I , SAND .

C-1

(2.) The clayey silt has the following properties in t~ un-

frozen consolidated state:

¥t = 120 lb. per cu. ft.

el = 0.9 at x = 0

k a 3.5 X 10-S ft. per sec.

a = o.l sq. ft. per ton v

cv • 2.1 x 10-5 sq. ft. per sec.

It is assumed that these properties, with the exception of e 1, do not

change appreciably after ice segregation and during subsequent consoli-

dation.

(3.) Ground surface has heaved a total o£ 3.6", 1.6" of which

can be accounted for by volume expansion as the soil moisture freezes.

The remaining 2" results from lanse growth in the clayey silt which is

assumed to be distributed uniformily with depth in the frozen zone.

(4.) The average rate of thawing is linear at 1/16" per hour

and the soil follows a path similar to line BC'A of Figure 10 during

consolidation. A discussion of this simplified case is presented in

Part IV.

C-02. SOIL CONDITIONS PRIOR TO THAWING. -

The intergranular pressure p1 at the top of the clayey silt

is approximately 0.055 tons per sq. ft. if typical values of the effective

unit weight of gravel are used. The intergranular pressure at any depth x

is then given by the expression:

px = p1

+ ll p = 0.055 +

= 0.055 + o.029x

120 - 62.4 2000

(%)

•••• ( c-1)

'

. -·•

From condition (2,), it follows that the void ratio at any depth

x before freezing is:

+

=

a flp v

•••• ( C-2)

This void ratio at any depth is indicated by point A in Figure 10.

From condition (3.), it may be shown that the void ratio in-

crease ~e as a resalt of 2-inch, ice lense growth is:

Ae= ,2( l+e)

34 = 0.112 •••• ( C-3)

The void ratio wi thin t he fr ozen portion of the clayey silt i:rmnediately

after it t haws is theref ore :

=A e + e Ax

= • u. ( C-4)

vrhich is equivalent to point B i n F.i..gure 10. TI!e 7 c id !_4&.tio e or!'.espond-

i ng to point C' in the same f i 6Jre is equal to 0.9055 for all depths.

For the numerical s olu ticn_, the clayey silt will be consider·9d

to consist of 8 layer::; each 8 inches thick, a subdivision identical to

that made in Fiture 11 for t he hyd!"aulic model. According to the condi-

tion thg.t the soil is represented by point B in Figure 10 immediately

after tha·wing, the initial hydrostati c excess water pressure u is numeri­x

cally equal to p given by Equation (C~l);) The excess pressure remains X

equal to this value until the void ratio at any poi::1t decreases to Oe~9055

at which time the gradual transfer of overburden load from the pore vmter

to the grain str-..~.cture corr.a:nences ~

The follovrinG t aol.e appli e :.:.: th·~!'.i. to the conditions of this

problem:

C-3

,. • ,J

initial ~X 8x Point Depth X u before immediately

n inches ft. tons per freezing after thawing sq. ft.

1 14 0 0(0.055) 0.9000 1.0120

2 22 o.67 0.074 0.9019 1.0139

3 30 1.33 0.094 0.9039 1.0159

4 · 38 2.0 0.113 0.9058 1.0178

5 46 2.67 0.132 0.9077 1.0197

6 54 3.33 0 0.9C$1 0.9CY)7

7 62 4.0 0 0.9116 0.9116

8 70 4.67 0 0.9135 0.9135

9 78 5.33 0 0.9154 0.9154

C-03. 1nJMERICAL SOLUTION. -

It has been demonstrated in Section 2-01-d that the finite dif-

ference approximation of the diffusion equation for heat flow may be

written as Equation (2- 17c) for the one dimensional case. The oorree-

pending equation for consolidation becomftS:

+

If we select a value of ~ equnl to 1/3 for this solution then

Equation (C-5) becomes:

u ::: n,k+l

1

3 (u + ·U + ·1 )

n-l,k n,k n+l,k ••• " ( c-6)

, ,. Since: '

•••• ( C-7)

and since we have selected As = 8" then~

C-4

,

-.

..

. ~-

·~ • t

~ -

_!_ ·-8 2 = _L('I2)

2.1 X 10-5

= 7100 sec.

Say ~ t = 2 hrs. (time interval between steps)

Equation (C-6) applies while the soil is consolidating along a

path given by line C'A of Figure lOo Along line BC' the excess pressure

remains constant while the void ratio decreases.

The finite difference approximation for void ratio change may be

written:

+ ~ ( .,. '!..j.n+l k

: !J

+ u - 2u ) : n-l,k n,k • .... ( C-8)

which is the comparable to Equation (2-18) of Section 2-0l-f~· (3) .

Equation (C-8) may be rearranged~

Ke + ~ (ll:n-1 k n,k ,

and when (; = 1/3 it may be written ~

KAe = 1 (Au ... ~Au) 3 1 2

'Ltn k) .., ~ (ti ·~ u ) s n,k n+l,k

.... " ( c ... 9)

We must now find K before preceding with the numerical solutiono

In each t\vo-hour period the volume of water 6.. Q flowing from an

8,-..inch layer is given by Darcy -~ s Lawt

~ .H AHl

~Q = Q. .., Qout = -~k __ 2 7200 + k_ J.n ~s ~s

= 7200k <l1Hl ... 6_H2) As

7200

= 7200k 1 As '52:4 ( Aul - /),.u2)

= 6 X 10~6 (A u1 --~u2) cu.

since:

b_e ~v ~Q = __J_ = v V( 1 ... n)

s 6 = 6 x 10- (Aul Au2}

8 (1 c.~) I2

1.9

= - 17.3 X 10-6 (~ ul -~u2 )

then it follows from Equation (C-10) that~

1 K = ; = 9.6 tons per sqo fte

17;3 X 10~6 X 2000 The numerical solution will be carried out in a tabular for.m

simil~r to that shown in Table IIo Notation for this example is as follows:

k k+l

u n ... l,k Ken ... l,k

n-1 u n-l.,k -u

llsk ~ (u -u n-l,k n,k

)

u Ke k n,k n,

n

u -u n,k n+l,k ~ (u -u )

n,k n+l,k

u Ke

n+1 n+l,k n+l,k

- \

•""" ,

-..

Equations ( C-6) and ( C-9) govern the numerical process which is now presented. {''

lftJNilliCAL SOLUTICM OP A THAW - OOifSOLIDATIOI PROBLDI

N 0 1 2 63 6ij 6S ' 0 2 ..... , ..... 126 bn. 128 hn. 1)0 bn. 0 1/8 1ft. 1/4 ia. 7-7/8 iu. 8 iu. 8-1/8 1a..

'

a

1 ~ J 9.7lS2 0 I 8.6boo T

0 I 8.6boo 0 I a.~oo 0 I 8.6boo 0 I a.Q.oo -o.o?b -0.'»47 -o.074 -o.0247

o.~ I 9.7087 o.07b ,9.7334 0.074 J 9.7331& o...m.l&. I 9 • n34 o.,.,b I 9. n34 0~- J 9. 733k 2

0.094 19.7526 0~ J 9.7S26 o.l)94 I 9. 7526 PlOWCa.HDC .. 9. 73J4,t( -o.o247) 3 ti'OII Point 2 • ? • 7087 follow

toward aurtaee troa Equati~c-9

4 o.u3 , J 9. 7709 0_._11_.1 J 9.7709

J 9.7691 J 9.7691 o.n2 0.132

N s 127 128 254 hn. - 256 hn.

0 I _Q_ J 15·7/8 iu. 16 ina. 6

0 J 8.a.oo 0 J 8.6boo 1 I J 0 ·o 0 0 1 0

0 Je.6582 0 J 8.6582 I J t 0 0 ..0.094 8 -o.0313

0.094 J9. 7526 0.1:'94 j9. 7S26 I J J 0 0

9

flow eo-.nee1

REMARKS lo fiow OCCUI'8 unW the depth ot thawing It troa Point. 3 penetrate• 8 incbea toward eurt ace

/

66 lOS 132 ..... 210 bn.

8-1/4 1 ... 13•1/8 1•.

0 J 8.6boo 0 J e.~ ...0.0'71. -o_.l)2~ I -o.mla -o.0247

o .. ~J& 19 .66ho Q~74 I e. 7087

129 130 258 hn. 2&> hn.

16-1/8 ina. 16-1/4 1•.

0 J 8.6boo 0 J 1.61.00

-o.O)l3 -0.010b -o.Oh1&. -o.Ol)9

o.o:n3 1 8.6582 o.Qhl!Jj 8.6687

~.0627 -o.0209 -o.oS22 -o.o111a

0.094 j9. 7213 0.094 J9.'100L

' 1 I A J (~ $­, ,

lo6 212 hn.

13-1/4 iu.

0 I e.a.oo

-o.O'IIJ '-o.,'l2Ja

o.o7b J e.fR28

, ..-. I J ~.

10'7 21la ....

13-)/8 1•.

0 I ,.&oo ~.1)247 -o.oo82

o.0247 J 8.6681

Yoid ratio at Pbint 2 ba8 re*lhed 0.90SS and u.e••• l'I"M• 1un at Point 2 begine to died• pate accordi--'!~. to Equation ( c-6'

131 132

262 ""· 2a. hn. 16-3/8 ina. 16-1/2 iMo

0 J 8.64M 0 18.64oo

~.nbS) -o.OlSl -o.~64 -o.olSS

o.IJ4S3 j R.6722 o.Oit61& le.68)J ~.Oit87 -o.0162 i-O.Ob76 -o.Ol9J

0.094 j9.68)0 0.094 1?.6668

lote that • aoU at. Point. 2 . ... u. •• -veter ~roa Poiat. 3 becine to flow ~~DWal"d.

108 n, .... ll-l/11•.

0 I 8.64oo

k> ooe2 -o.oou o.0082J e.6!i99

a 1• ~--t. , tllat the uc••• ! pre .. ure at. Poiat 2 11111 be ••••nt.iall7 dh-lipated when thlo thawing pene-~·~-t,Q~

1)) 266 hn.

16-S/8 t..

() J e.a.oo

J 8.68)7

J

The eolutioa pro~ cede~ in lite junMI' u eaell ne:w la,.r th-aDd et.arte to jeouolldate.