Embed Size (px)
Citation preview
MASTER
DISCLAIMER
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.
BNL 51061 UC-59c
(Heating and Cooling-Research and Development - TID-4500)
A TRNSYS-COMPATIBLE MODEL
OF GROUND-COUPLED STORAGE
September 1979
W o r k Sponsored by t h e SYSTEMS DEVELOPMENT DIVISION
SOLAR APPLICATIONS OFFICE OF ASSISTANT SECRETARY
CONSERVATION AND SOLAR APPLICATIONS U.S. DEPARTMENT OF ENERGY I-
- -- ----__ DISCLAIMER I
This Wok war Dremrm as an acmunt of u a k wosred by an me%-+ of the United Stale Gowrnment. Neilher the United Staler Governmenr mr anv asency thereof. nor any of their employeer. mkerany warranty. exprerr or implied. ar swmer any lcgal liability or rerponlibilily for the ascuracy. mmeletenerr. or usefulnerr o f any informtion. a~wratus. product. or proceu dialorod. or reerwntr thal i ir use vauld mt infringe privately owned rights. Referem herein ro any w ~ i f i c mmmercial Produ". procerr. or service by trade -me. lrademrk. mnufaclurer. or otherwise. do- nal ne-rib mnfiitute or imply its endorsement. remmmendation. or fawrirq by the United Slam Governm'en! or any agency (hereof. The "ism and opinion9 of authors cxpreYed herein do not nKeuorilv nate or reflect $hose of !he United Slsles Governmnc or any agency thereof.
, - - - -- - - - - -
SOLAR TECHNOLOGY GROUP DEPARTMENT OF ENERGY AND ENVIRONMENT
B R O O K H A V E N N A T I O N A L L A B O R A T O R Y U P T O N , N E W Y O R K 1 1 9 7 3
DISCLAIMER
Thk book w a ~ prepared R F an arrnlint nf wnrk spnnsnrcrl by an agency of the United States Government. Neither the United Starm Government nor any agency therkot; nor any nf their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial prod- uct, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily consti~ute or il~lply its ~ C I I ~ U I S C I I I C I ~ L , I .CCOIII I I ICII~~~IO~, or tnvorlng by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States
. Government or any agency thereof.
Printed in the United States of America Available from
National Technical Information Service U.S. Department of Commerce
5285 Port Royal Road Springfield, VA 22161
Price: Printed Copy $4.50; Microfiche $3.00
ABSTRACT.
The u s e of t h e ground a s a source of heat' and a s a s t o r a g e element f o r low-grade h e a t i n s o l a r a s s i s t e d hea t pump systems is t h e s u b j e c t of an ongoing program of resea rch a t Brookhaven National Laboratory. A s p a r t of t h i s program, a computer model of ground-coupled s t o r - age was developed. Th is r e p o r t d e s c r i b e s t h e i n t e g r a t i o n of t h i s program i n t o t h e t r a n s i e n t s imulat ion computer program TRNSYS t o a l low t h e hour-by-hour s imulat ion of s o l a r hea t ing and cool ing systems which use ground cou- p l ing . The r e p o r t i s intended t o se rve a s a u s e r ' s manual f o r t h e TRNSYS-compatible ground-coupling sub- r o u t i n e s .
ACKNOWLEDGMENTS
The computer program GROCS was designed and w r i t - t e n by D r . P h i l i p Metz, and t h e s e c t i o n of t h i s r e p o r t t i t l e d Design of GROCS, taken from Ref. 8, was w r i t t e n by him.
- iii -
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In t roduc t ion 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design of GROCS 1 . . . . . . . . . . . . . . . . . . . . I n t e g r a t i o n of GROCS w i t h TRNSYS 3 . . . . . . . . . . . . . . . . . . . . . . . . A . Buried Tank Model i 3 . . . . . . . . . . . . . . . . . . . . . . . B . B u r i e d P i p e M o d e l ... 4 . . . . . . . . . . . . . . . ~ e t a i l e d Descr ip t ion of Buried Tank Model 5 A . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Approach 6 . . . . . . . . . . . . . . . . . C TRNSYS Component Configurat ion 6 . . . . . . . . . . . . . . . . . . . . . . . . . . D . Program Flow 7 . . . . . . . . . . . . . . . . Detai led Descr ip t ion of Buried P ipe Mode1 9 . . . . . . . . . . . . . . . . . . . . . . A . Nomenclature . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Approaeh ' 10 . . . . . . . . . . . . . . . . . . . C Flowing Sheet Appraximation 1 U
........ D . Correct ion f o r t h e Di f fe rence Between A . .. ' . . I2 . . . . . . . . . . . . . . . Blowing Sheet and ail Array of P i p e s . . . . . . . . .
..... i 3 . . . . . . . . . . . . . . . . . E . Two-Vay Flow i n t h e P ipe F i e l d . . . . . . . . . . . . . . . . F . TRNSYS Component Con£ i g u r a t i o n 13 . . . . . . . . . . . . . . . . . . . . . . . . . . G ProgramFlow 14 . . . . . . . . . . . ~ o d i f i c c l t i o n of GROCS f o r I n t e g r a t i o n w i t h TRNSYS 16 . . . . . . . . . . . . . . . . Data S e t f o r a Ground-Coupled Simulation 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . reference^ ; 20
. . . . . . . . . . . . . . . . . . Ground-coupling model schematic 2 Flow c h a r t f o r Lu i l ed tank subrout ine 8 . . . . . . . . . . . . . . . Flowing-sheet approximation t o pipe f i e l d . . . . . . . . . . . . . lo . . . . . . . . . . . . . . Geometry of f lowing s h e e t approximation 10 . . . . . . . . . . . . . Pipe f i e l d embedded i n semi-inf i n i t e mass 12 . . . . . . . . . . . . . Pipe f i e l d embedded between two boundaries 12 . . . . . . . . . . . . . . . PIOW c h a r t t o r bur ied pipe subrout ine 15 Flow c h a r t showing modi f i ca t ions to . . . . . . . . . . . . . . . . . . ground-coupling subrout ine GROCS 17
INTRODUCTION
The use of t h e ground as a h e a t source t o a h e a t pump and as a s t o r a g e
element i n s o l a r energy systems has received inc reas ing a t t e n t i o n i n r e c e n t
years. The evident d e s i r a b i l i t y of modeling t h e thermal i n t e r a c t i o n of
t h e ground with h e a t pumps and s o l a r systems has l ed t o t h e development of
a FORTRAN computer model c a l l e d GROCS (GROund Coupled Systems). 7 y 8 This r e -
p o r t d e s c r i b e s t h e i n t e g r a t i o n of t h i s program i n t o t h e t r a n s i e n t s imula t ion
computer program T R N S Y S , ~ developed a t t h e Univers i ty of Wisconsin, so t h a t
hour-by-hour s imulat ions of s o l a r systems us ing ground coupling can be per-
formed. The mot ivat ion, theory, and c u r r e n t s t a t e of t h e a r t of ground cou-
p l ing have been A summary d e s c r i p t i o n of t h e computer model
G R O C S ~ ' ~ is given below. F a m i l i a r i t y wi th TRNSYS is assumed.
DESIGN OF GROCS
The realm of i n t e r e s t i n g underground h e a t f low problems t h a t can be solved
without a l a r g e d i g i t a l computer i s l i m i t e d . Thus, t h e need was seen t o do un-
derground h e a t flow modeling on such a computer. The u s u a l method of numeri-
c a l l y solving d i f f e r e n t i a l equat ions such a s t h e h e a t f low equat ions on a com-
puter i s t o conver t t h e d i f f e r e n t i a l equat ions t o f i n i t e d i f f e r e n c e equat ions
which a r e solved on a "mesh" of equa l ly spaced po in t s . The mesh spacing sets
a s c a l e which determines t h e accuracy of t h e s o l u t i o n . It i s n o t p o s s i b l e f o r
t h e s o l u t i o n t o have meaning over d i s t a n c e s smal ler than t h e mesh spacing. I n
3-dimensional systems, t h e number of mesh p o i n t s grows a s t h e cube of a system
dimension so t h a t mesh-based 3-dimensional programs u s u a l l y r e q u i r e l a r g e
amounts of computer time, and t h i s i s undes i rab le , e s p e c i a l l y because of our
i n t e n t i o n t o i n t e g r a t e GROCS with TRNSYS, t h e ' u n i v e r s i t y of Wisconsin s o l a r sys-
tem s imulat ion program, which is a l ready q u i t e l a r g e .
Therefore, ins tead of mesh po in t s , GROCS s o l v e s t h e h e a t flow f i n i t e d i f -
ference equat ions over a system of "blocks" of e a r t h . Each block i s a volume
of e a r t h whose s i z e and shape a r e determined by a hand-drawn model. One such
model i s i l l u s t r a t e d i n Figure 1.
A block-type model ha3 t h r e e advantages:
1. Useful problems can be solved wi th a s h o r t , s imple, and economical
computer program.
2 . Adequate accuracy can be obta ined. Na tura l ly occurr ing ground in-
homogeneities l i m i t t h e accuracy of any model t h a t r e l i e s on bulk thermal
. . r
Figure 1. Ground-coupling model schematic
p r o p e r t i e s t o about 10%. Therefore, i t i s a waste of time and money' t o use
f i n e mesh models more a c c u r a t e than t h i s l i m i t .
3. New ground-coupling conf igura t ions can be s tudied by c r e a t i n g new
hand-drawn models, a process which t a k e s a few hours.
GROCS uses two d i f f e r e n t k inds of b locks , c a l l e d "rigged blocks" and
" f r e e blocks." The rigged blocks surround t h e f r e e blocks ahd provide the
necessa ry s p a t i a l boundary cond i t ions . The temperatures of t h e rigged h1.nr.k~
are dktermined a t each t imestep by a f u n c t i o n subprogram c a l l e d TINTERP which
r e q u i r e s a t a b l e of exper imental ly measured average ground temperatures a t a
llumbcr of d i f f e n e u ~ dep ths sparixiing t he por t ion of t h e ground t o be s imulated, .
f o r each month of t h e year. At every t imestep in.GROCS, t h e subprogram i s
t o l d t h e time of year and t h e dep th of t h e c e n t e r of t h e block whose tempera- . .
t u r e i t is t o compute. TINTERP then d e t e r m i n e s ' t h e temperature of t h e block
by l i n e a r l y i n t e r p o l a t i n g wi th r e s p e c t t o time and depth between the r e l e v a n t
t a b l e e n t r i e s .
The f r e e block temperatures a r e i n i t i a l l y determined by spec i fy ing them
as d a t a input o r , i f a d e f a u l t va lue i s s p e c i f i e d (which i s a timesaver) , by
TINTERP a s descr ibed above. A t a l l f u t u r e t imesteps , however, t h e f r e e blocks
have t h e i r temperatures determined by t h e i r thermal i n t e r a c t i o n wi th each o t h e r
and wi th t h e rigged blocks, and by h e a t i n p u t s placed i n them t o s imulate t h e
e f f e c t of a s o l a r heat ing system and load.
A number of phys ica l parameters of t h e model must be s p e c i f i e d f o r inpu t
t o GROCS. These inc lude t h e numbers of f r e e and r igged blocks,. t h e i n i t i a l
temperature o r t h e d e f a u l t temperature, t h e volume and volume h e a t capac i ty
(Cp) f o r each f r e e block, t h e depth of each block a l l nonzero h e a t t r a n s f e r
a r e a s , and center-to-center d i s t a n c e s of ad jo in ing blocks .
INTEGRATION OF GROCS WITH TRNSYS
I n o rder t o merge GROCS s u c c e s s f u l l y i n t o TRNSYS i t was necessary t o make
provis ion f o r t h e s imulat ion processes going on i n each program t o proceed
without i n t e r f e r i n g wi th t h e o t h e r , and y e t t o a l low appropr ia te thermal i n t e r -
a c t i o n s t o occur between them. To make t h i s poss ib le , subrou t ines were wr i t -
t e n i n t h e same format a s o t h e r component TYPE subrou t ines i n TRNSYS, which
model s o l a r components and bu i ld ing l o a d s and i n some c a s e s perform v a r i o u s
a l g e b r a i c manipulations necessary t o a s imulat ion. These subrou t ines , TYPE33
f o r t h e bur ied tank and TYPE32 f o r t h e bur ied p ipe f i e l d , i n t u r n communicate
wi th GROCS. During each s imulat ion t imestep, the TYPE subrou t ines a r e c a l l e d
i n t u r ~ i by a c e n t r a l "command" s u h r n l l t i n e i n TRNSYS c a l l e d EXEC. A s each TYPE
subrout ine i s c a l l e d , i t s i n p u t s a r e ad jus ted t o r e f l e c t t h e ou tpu t s of pre-
v ious ly c a l l e d subrout ines , and a f t e r i t f i n i s h e s process ing, i t s ou tpu t s i n
t u r n a f f e c t t h e inpu t s of subrou t ines t o be c a l l e d l a t e r . After (usua l ly )
s e v e r a l passes through t h e s e t of subrout ines which c o n s t i t u t e t h e system t o
be simulated, t h e i n p u t s and ou tpu t s of a l l t h e subrou t ines converge t o a s e t
nf mutually c o n s i s t e n t va lues .
A. Buried Tank Model
A s t h e s imulat ion of t h e ground-coupled tank i s conceptual ly simpler than
t h a t of the buried c o i l , i t w i l l be considered f i r s t . During t h e course of
simi.lla t.ing a given t imestep, TRNSYS "sees" a mixed tank of a s p e c i f i e d volume
w i t h t h e same i n p u t s and o u t p u t s as a convent ional tank. After convergence
. has been obtained f o r t h a t t imestep, however, t h e f lows of h e a t between t h e
t ank and i ts surroundings are c a l c u l a t e d d i f f e r e n t l y . During t h e i t e r a t i v e
process w i t h i n a. t imestep, thermal i n t e r a c t i o n occurs between t h e tank ( t i n
F igure 1 ) and t h e rest of t h e system, he re diagrammed 'schematically a s a co l -
l e c t o r , a n energy t r a n s p o r t module, and a load. At t h e end of t h e t imestep,
t h e h e a t t r a n s f e r r e d between t h e tank a n d ' t h e conceptual blocks of e a r t h (a)
a d j a c e n t t o t h e tank a r e c a l c u l a t e d by TYPE33 and made a v a i l a b l e t o GROCS.
Then GROCS is c a l l e d by TYPE33; i t computes t h e h e a t f lows between a l l p a i r s
of b locks , including both t h e ad jacen t b locks (a ) , t h e surrounding nonadjacent
f r e e blocks ( s ) , and t h e rigged blocks ( r ) . The use fu lness and accuracy nf
t h i s model depend on t h e use of a t imestep s h o r t enough so that t h e change i n
t ank temperature dur ing any t imestep i s much less than t h e mean ternperat~~re.
d i f f e r e n c e between. the t ank and ad jacen t blocks. For s t o r a g e t o c o l l e c t o r 3 2 2 .
a r e a r a t i o s around 0.3 m /m c o l l e c t o r (8 g a l l f t c o l l e c t o r ) , t h e maximum tank
temperature change dur ing a 15-min t imestep wi1.l be around 0 . 3 O ~ ( 0 . 6 ' ~ ) .
S to rage volumes of t h i s s i z e o r l a r g e r a r e normally considered f o r t h i s a p p l i -
c a t i o n .
B. Buried P ipe Model
The s imulat ion of a s e r p e n t i n e a r r a y 06 buried pipe prPqcnts problms
n o t encountered wi th t h e bur ied tank. Most of t h e s e problems a r e r e l a t e d t o
t h e f a c t t h a t t h e t y p i c a l p ipe diameter i s small enough t o r e q u i r e ve ry small
block s i z e s , and t h e r e f o r e t imesteps much s h o r t e r than t h e 15-min t y p i c a l f o r
TRNSYS runs . St ra ightforward modeling of t h e pipe along t h e same l i n e s a s t h e
tank would r e q u i r e i n o r d i n a t e amounts of computer time. For t h i s r eason it
was decided t o model t h e p i p e f i e l d a s a t h i n shee t of water flowing i n t h e
plane of t h e p ipe f i e l d .
The h e a t f low from a s h e e t w i l l no t i n genera l be t h e same a s t h a t from \
a s e t of p ipes . The shee t assumption, a s descr ibed so f a r , does no t t a k e i n t o
account t h e r a d i u s of t h e p ipes o r t h e d i s t a n c e between the^. However, i t i s
p o s s i b l e t o t ake these f a c t o r s approximately i n t o account, a t l e a s t f o r quasi-
s t eady-s ta te h e a t flow, by reducing t h e e f f e c t i v e h e a t t r a n s f e r a r e a between
t h e plane shee t of water and t h e ad jacen t blocks of e a r t h , making it l e s s than
t h e g e o u e t r i c a l s u r f a c e a rea . The determinat ion of t h e c o r r e c t f a c t o r by which
t h e h e a t t r a n s f e r a r e a must be reduced i s discussed below.
Because t h e amount of water r e s i d e n t i n a p ipe f i e l d i s g e n e r a l l y one t o
two o r d e r s of magnitude l e s s than t h a t i n a bur ied tank, it i s n o t f e a s i b l e t o . .
d e f e r t h e accounting of temperature changes wi th in t h e p ipes t o t h e end of t h e
timestep: Ins tead , thd '.heat flow from t h e water t o t h e .ground i s c a l c u l a t e d
on she b a s i s of t h e i n l e t water temperature and t h e temperatures of t h e e a r t h
blocks ad jacen t t o t h e p ipe f i e l d . The change i n water temperature along t h e
flow path, as it t r a n s f e r s h e a t t o t h e s o i l , i s accounted f o r t o f i r s t order . '
A s wi th t h e b u r i e d tank model, c a l c u l a t i o n of h e a t t r a n s f e r between b'locks of
e a r t h by GROC'S .is defe r red t o ' t h e end of t h e t imestep. '
. . . .
DETAILED DESCRIPTION OF BURIED TANK MODEL .
A. Nomenclature ,
Symbol D e f i n i t i o n Uni ts S I - English
Area of i t h ad jacen t block of e a r t h m 2
f t 2
Ai
C S p e c i f i c hea t of tank f l u i d kJ/kg-OC Btullb-OF P f
hi Distance from tank edge t o c e n t e r of i t h
ad jacen t block m
k Thermal conduc t iv i ty of t h e ground W/hr-m-°C Btulhr-f t-OF . . . , ,
fib Mass flow r a t e t o o r from h e i t source kg I h r l b / h r
$ Mass f low r a t e t o o r frpm load
Qenv Net r a t e of h e a t l o s s from tank t o e a r t h G / h r Btu/hr
Q i Rate of hea t l o s s from tank t o i t h
ad jacen t block W l h r Btulhr
Qtank Rate a t which energy i s removed t o supply
t h e Inad W l h r Btu/hr
Th ~ A ~ e r a t u r e of f l u i d from 'heat source ' O C OF
Ti Temperature of i t h ad jacen t block of e a r t h "C OF
T~ Temperature of f l u i d re tu rn ing from load O C OF
T t k Tank temperature C OF
V Tank Volume . . m 3 3
f t .
~ n t e r n a l energy change of tank kJ Btu
A t TRNSYS s imulat ion t i m e s t e p . . , . h r h r
f ' Tank f l u i d d e n s i t y kg /m3 l b / f t 3
B. Approach
The bur ied t a n k model .is intended t o be i n t e g r a t e d . i n t o t h e TRNSYS program.
Subrout ine TYPE^^ ,". th.e, bur ied t ank subrout i n e , i s designed t o .appear t o TRNSYS
e x a c t l y l i k e a, fully-mixed tank, t h a t is , a TYPE4 tank wi th one DERIVATIVE.
The d i f f e r e n c e l i e s i n t h e f a c t that i n TYPE?? t h ~ heac loooca f r o ~ u Llle ~ ; i l ~ k
a r e c a l c u l a t e d on. t h e b a s i s of t h e h e a t t r a n s f e r t o t h e ad jacen t b locks of ..
e a r t h i n t h e GRDCS model. The r a t e of h e a t l a s s t o each ad jacen t b lock I s
given by
The n e t r a t e of heat l o s s t o a l l ad jacen t b locks is then
where n i s t h e number of b locks ad jacen t t o t h e tank.
C. TRNSYS Component Configuration
PARAMETER NO. DESCRIPTION 1
1 V - Tank volume
2 C - s p e c i f i c h e a t of tank f l u i d P f
3 f
- f l , u i d d e n s i t y
4 Block number of f i r s t ad jacen t block of e a r t h l a GROCS
5 A1 - h e a t t r a n s f e r a r e a t o f i r s t ad jacen t block
6 hl - d i s t a n c e from tank t o cen te r of fi.r.st a d j a ~ u n t bloclc
7, 8, 9 Same a s 4 , 5 , 6 f o r second ad jacen t block
Addi t ional s e t s of 3 parameters f o r each ad jacen t block, up t o 10 blocks.
INPUT No . UESCRIPTPON
1 Th - temperature of f l u i d from h e a t source
2 % - mass flow r a t e from heat source
OUTPUT NO.
1
TL - temperature of f l u i d r e t u r n i n g from load
5 - mass f low r a t e from t h e load
DESCRIPTION
Temperature of f l u i d t o hea t source (= Ttk f o r mixed t ank)
$ - mass flow r a t e t o h e a t .., source .. . . , . .
.Temperature of f l u i d t o b a d (= Ttk f o r mixed tank)
- mass f low r a t e t o load
Q m v - r a t e of? energy l o s s t o t h e ground
Qtank - r a t e a t which energy is removed t o supply t h e load
AE - i n t e r n a l energy change of t h e ' t a n k
Q I A t - energy t r a n s f e r r e d t o f i r s t ' adjacent block of e a r t h i n GROCS model
Q i A t - energy t r a n s f e r r e d t o i t h a d j a c e n t block
DERIVATIVE NO. . DESCRIPTION
1 Ttk - t ank temperature
D. Program Flow
A flow c h a r t f o r subrou t ine TYPE33 is given in F igure 2 . The subrou t ine
f i r s t checks f o r consis tency of t h e numbers of INPUTS, PARAMETERS, OUTPUTS,
and DERIVATIVES with those requ i red by t h e subrou t ine . It then asks whether
t h i s i s t h e f i r s t c a l l of t h e subrou t ine f o r t h e c u r r e n t TRNSYS s imulat ion run
(MODE = -1). I f it i s , GROCS is c a l l e d immediately t o read i n t h e d a t a needed
by GROCS t o desc r ibe t h e ground model, and then c o n t r o l i s re turned t o t h e
c a l l i n g subrout ine EXEC in TRNSYS. ' .
I f i t is a l a t e r c a l l than t h e f i r s t , t h e program t a k e s t h e va lues of t h e
INPUTS and PARAMETERS from t h e X I N and PAR a r r a y s and g ives them ind iv idua l
v a r i a b l e names. It then c a l c u l a t e s der ived q u a n t i t i e s : t h e t o t a l heat capac i ty
of t h e tank, t h e hea t c a p a c i t y f low r a t e t o t h e hea t source and t h e l o a d , and
t h e r a t e of heat withdrawal from t h e tank t o s e r v i c e t h e load .
Next t h e h e a t l u s s e s t o t h c ad jacan t b ln rks of e a r t h a r e c a l c u l a t e d , and
added t o ob ta in t h e n e t heat l o s s from t h e tank t o t h e ground. The rate of
change of .tank temperature is c a l c u l a t e d nex t . Th i s w i l l be used by TRNSYS
t o c a l c u l a t e t h e new tank temperature a t t h e end of t h e t imestep, when a l l i t e r -
a t i o n s have been completed. The change in i n t e r n a l energy of t h e t ank from
DATA CONSISTENCY
CHECKS
CALCULATE QUANTITIES DERIVCD FROM INPUTS
AND PARAMETERS FROM ARRAYS
CALCULATE HEAT LnSS RATFS t Q BLOCK6 ' ADJACENT TO TANK
CALCULATE RATE OF CHANGE OF TANK
TEMPERATURE
PLACE OUTPUTS IN
CALCULATE HEAT TRANSFERRED TO
ADJACENT BLOCKS AND PLACE IN OUTPUT ARRAY
t YES
IN -RROl.lNU H t A l FLOW
RETURN c3 Flgiire 2. Flow chart for buried tank subroutine '
t h e beginning of t h e s imulat ion is c a l c u l a t e d nex t . :.The output v a r i a b l e s a r e
now placed i n t h e output a r r a y .
I f t h e c u r r e n t c a l l t o TYPE33 i s n o t t h e i a i t c a l l of t h e c u r r e n t t imestep,
c o n t r o l i e re tu rned t o EXEC. I f it is (MODE = 2) , then t h e v a l u e s of h e a t
t r a n s f e r r e d t o t h e ad jacen t blocks of e a r t h a r e added, t o t h e a r r a y Q I N , which
is i n COMMON wi th GROCS ,, and then GROCS is c a l l e d t o c a l c u l a t e t h e hea t f lows
among a l l t h e blocks of e a r t h , given t h e thermal i n p u t s from t h e t ank a s c a l - I
cu la ted above. Control i s then re tu rned t o EXEC.
DETAILED DESCRIPTION OF BURIED PIPE MODEL
A. Nomenclature
Symbol D e f i n i t i o n Uni t s S1 - Engl ish
*i Area of i t h ad jacen t block of e a r t h in
t h e GROCS model m 2 f t 2
B1, B Constants used in 'determination of p ipe f l u i d temperature change - -
C Spec i f i c h e a t of p ipe f l u i d k~/kg-OC Btullb-OF P f
hi . Distance from p ipe f i e l d plane t o cen te r
of i t h ad jacen t block m f t
k Thermal conduc t iv i ty of t h e ground kJ/hr-m-OC Btu/hr-ft-OF
L Length of each p ipe i n a p lace p a r a l l e l a r r a y m
. . . . i. Mass ' f low ' r a t e through p ipes ' 'kglhr l b / h r , .. . - . , . . 4 , ' Rate of heat t r a n s f e r from pipe f i e l d t o
: a p a i r of ad jacen t blocks k.J/hr Btu/hr
6, '
' Rate of 'heat t r a n s f e r from 'p ipe f i e l d t o i t h adjace~it block , k J / h r Btu/hr
R , Radius . of p ipe m f t
R* Thermal r e s i s t a n c e of a half-block "C-hr/kJ OF-hr/Btu
s Separat ipn between ad jacen t p ipes m f t
TE Pipe f l u i d temperature af t e r passing between a p a i r of ad jacen t blocks O C OF
Ti Temperature of i t h ad jacen t block O C OF
Tin I n l e t temperature t o p ipe f i e l d O C OF
To Pipe f l u i d temperature be fore passing between a p a i r of ad jacen t blocks "C OF
Tout Out le t temperature from ' 'pipe f i e l d O C F
- . Yi . Length,of a . p a i r of ad jacen t blocks i n '
' the d i r e c t i o n of f low m f t
A t TRNSYS s imula t ion t imestep h r h r
r\ Correct ion f a c t o r t o h e a t t r a n s f e r area from p ipe f i e l d t o ad jacen t b locks -
B. Approach
The major approximation made i n t h e TYPE32 bur ied p ipe model i s the ' sub'- '
s t i t u t i o n of a f lowing plane s h e e t of f l u i d f o r t h e bur ied p ipe f i e l d (Figure
3 ) . This approximation was introduced above. The method f o r c o r r e c t i n g t h e
model to t a k e i n t o account t h e p ipe r a d i u s and spacing i s considered i n more
d e t a i l below. The incorpora t ion of t h e £.lowing s h e e t approximation i n t o t h e
model' i s now discussed .. C. Flowing Sheet Approximation
A c r o s s s e c t i o n of the geometry o f t h i s approximation i s shown -irk Figure 4 .
For s i m p l i c i t y , on ly two ad jacen t blocks of e a r t h are shown, al though t h c model
w i l l a l s o accommodate m u l t i p l e p a i r s of ad jacen t blocks. The ad jacen t blocks
. . m u s t be set up i n p a i r s , one on each s i d e of t h e flowing s h e e t , and both mem-
b e r s of a pair must have t h e same leng th YE i n t h e d i r e c t i o n of flow. Di f fe r -
e n t p a i r s may have d i f f e r e n t l eng ths , a s long a s both members of each p a i r are
d i r e c t l y oppos i t e one another and have t h e same length .
ADJACENT BLOCKS
J' 0 F
EARTH
1 FLOWING SHEET
OF FLUID
Figure 4. , Geometry of flowing sheet approximation
. . . . .
Knowing t h e temperature T6 a t t h e beginning of t h e f low and . the mass f low
r a t e (which a r e INPUTS t o TpE32)., the.. d i s t a n c e s hl and h and t h e h e a t t r a n s - 2
f e r a r e a A (which a r e PARAMETERS), and t h e block temperatures T and T2 (which 1
a r e obtained from GROCS), one can c a l c u l a t e t h e p ipe f l u i d temperature T E a f t e r t h e f l u i d h a s passed by t h e p a i r of e a r t h blocks , and t h e r a t e s of h e a t
f low i n t o (or from) t h e ad jacen t blocks. A f i r s t - o r d e r assumption is made, . . .
. . . .. . . IT^ - TEl << max(lTo - T ~ I , I T ~ - ~ ~ 1 ) . (3
t h a t is, t h e temperature change of t h e f l u i d is small compared wi th t h e temper-
a t u r e d i f f e r e n c e between t h e f l u i d and t h e ad jacen t blocks. I f t h i s i s so ,
then t h e r a t e of h e a t l o s s from t h e f l u i d is given by
I L
This rate of h e a t l o s s must be accounted f o r by t h e temperature drop i n t h e
f l u i d i t s e l f :
6 = c (T - TE) . pf 0
Equating t h e t w o express ion f o r 4, and def in ing t h e dimensionl&ss c o n s t a n t s B1
and B2:
kA,
one o b t a i n s
T E = To + B1(T1 - To) + R2(T2 - T o ) . (8)
From t h i s express ion t h e h e a t f low r a t e s t o . t h e ad jacen t blocks can now be
ca lcu la ted from Eq. (4) .
I f a second p a i r of adjacent blocks is used, then t h e v a l u e of TE obtained
from t h e f i r s t c a l c u l a t i o n is used a s t h e i n i t i a l temperature T f o r t h e sec- 0
ond. The f i n a l temperature T from t h e l a s t p a i r of blocks i s t h e o u t l e t tem- E
perature from t h e p ipe f i e l d as a whole.
D. Correct ion f o r t h e Difference Between a Flowing Sheet and an Array of P ipes
The approach t o making t h i s c o r r e c t i o n is t o consider t h e thermal resis-
tance of a half-block of e a r t h wi th a cons tan t temperature boundary on one
s i d e and a planar a r r a y of p ipes on t h e o t h e r . This r e s i s t a n c e i s compared
wi th t h a t f o r t h e same haif-block of e a r t h wi th t h e same constant-temperature
boundary, b u t now with t h e s h e e t of f l u i d on t h e o ther s i d e i n p l a c e of t h e
p i p e a r r a y . I n genera l , t h e r e s i s t a n c e i n t h e c a s e of t h e plane shee t is t h e
lesser of t h e two. To c o r r e c t f o r t h e d i f f e r e n c e , t h e h e a t t r a n s f e r a r e a from
t h e shee t t o t h e ad jacen t block is reduced by a f a c t o r equal t o t h e r a t i o of
t h e s e r e s i s t a n c e s , so t h a t t h e r e s i s t a n c e f o r t h e flowing shee t case , wi th t h e
reduced h e a t t r a n s f e r area, is t h e same a s f o r t h e a c t u a l p ipe a r r a y .
So lu t ions f o r t h e r e s i s t a n c e of e a r t h around a row of p ipes have been
foundlo f o r two cases of inrerLst. I n ;he f i r s t case , t h e p ipes a r e embedded
in a semi- in f in i t e mass a f i n i t e d i s t a n c e from a cons tan t temperature plane
tiru~lclary (Figure 5) . The thermal r e s i s t a n c e 'in t h i s c a s e , f o r a s i n g l e p ipe ,
i s given by
R* = 1 2rLk ,,I$ .in. (2")] hwere k i s t h e therma1,conduct ivi ty of. t h e s o i l , L is t h e l eng th of t h e p ipes ,
R i s t h e pipe' rad ius , s i s t h e pipe separa t ion , and h is t h e d i s t a n c e from t h e
p ipe f i e l d plane t o t h e constant-temperature boundary. . .
Figure 5. Pipe field embedded in semi- Figure 6. Pipe field embedded infinite mass between two boundaries
I n t h e second case , t h e pipes a r e embedded i n a s l a b of e a r t h , e q u i d i s t a n t
from t h e two plane boundaries both at t h e samc temperature (Figure 6 ) . I n t h i s
' c a s e , t h e thermal r e s i s t a n c e f o r a s i n g l e p ipe i s given by
For our a p p l i c a t i o n , t h e f i r s t s o l u t i o n would be appropr ia te i f t h e temperature
of one ad jacen t block of e a r t h i n theGROCS model were equal t o t h a t of t h e p ipe
f i e l d and t h a t of t h e o ther were d i f f e r e n t . The second s o l u t i o n would b e ap-
p r o p r i a t e i f t h e temperatures of both ad jacen t blocks were equal t o each o ther
but d i f f e r e n t from t h a t of t h e pipe f i e l d . A s o l u t i o n in which t h e 'tempera-
t u r e s of t h e ad jacen t blocks may vary independently would be d e s i r a b l e , bu t i n
- 12 -
most s o l u t i o n s t h e average temperatures of t h e two ad jacen t blocks a r e n o t
very d i f f e r e n t , and t h e r e f o r e use of t h e second s o l u t i o n i s recommended.
Since t h e s imulat ion model treats t h e two ad jacen t b locks independently,
w e . t r e a t t h e problem a s one of two r e s i s t a n c e s i n p a r a l l e l , each twice a s g r e a t
a s t h a t given i n Eq. (10):
R* = - En [$ s i n h (%)I . aLk
This r e s i s t a n c e is t o be compared with t h a t of a rec tangu la r s l a b of th ick- . , .
ness h,.. a r e a sL, and conduc t iv i ty k:
The f a c t o r 0 by which t h e h e a t t r a n s f e r a r e a between t h e plane shee t of
f l u i d and t h e ad jacen t blocks must be reduced is then t h e r a t i o of t h e r i g h t -
hand s i d e s of Eqs. (12) and (11) :
Note once again t h a t h is t h e half-width, n o t t h e f u l l width, of an adjacent I
block. E. Two-Way Flow in t h e Pipe F ie ld
Cer ta in a p p l i c a t i o n s , such a s a ground-coupled h e a t pump without s o l a r in-
put , may involve flow of t h e f l u i d wi th in t h e p ipe f i e l d i n one d i r e c t i o n only.
For o ther a p p l i c a t i o n s , such a s t h e depos i t ing of s o l a r h e a t o r a i r -cond i t ion ing
r e j e c t h e a t i n t h e ground, it may be d e s i r a b l e f o r t h e f l u i d t o f low i n one d i -
r e c t i o n a t some times and i n t h e o ther d i r e c t i o n a t o ther times. I n t h i s c a s e
i t is necessray t o s e t up two UNITS, each of which i s TYPE32. The second UNIT
would have t h e same s e t of ad jacen t blocks a s t h e f i r s t but i n r e v e r s e o rder .
The o ther components of t h e s imulat ion a r e then set: up t u c a l l whiehcvcr of
t h e TYPE32 components r e s u l t s in t h e appropr ia te d i r e c t i o n of flow.
F. TRNSYS Component Configuration
PARAMETER NO. DESCRIPTION
1 C - s p e c i f i c heat of f l u i d i n pipes P f
2 Rlnck number of f i r s t adjacent block of e a r t h i n GROCS
3 Y1 - Length of f i r s t adjacent block
4 A1 - Heat t r ans f e r a r e a from p ipe f i e l d t o f i r s t ad jacen t block
5 hl - Distance from p i p e f i e l d p l a n e t o cen te r of f i r s t ad jacen t block
6-9 Same as 2-5 f o r second ad jacen t block
Addi t iona l sets of 4 parameters f o r each ad jacen t block, up t o 1 0 blocks.
INPUT NO. DESCRIPTION
1 T i n
- i n l e t temperature t o p ipe f i e l d
2 $ - mass flow r a t e of f l u i d through pipes . . OUTPUT NO. . . DESCRIPTION ' .
- o u t l e t temperature from p ipe f i e l d . Tout
2 $ - mass f low r a t e
3 4 1 ~ t - energy t r a n s f e r r e d t o f i r s t ad jacen t block of e a r t h i n GROCS model .
2 + i Q i A t - energy t r a n s f e r r e d t o i t h ad jacen t block
DERIVATIVE NO. DESCRIPTION
1 A dummy d e r i v a t i v e which is not used in t h e program and which c a ~ have any i n i t l a 1 value. ~ t s presence is required tu cause 'EXEC t o c a l l TYPE32 i n MODE2 a t t h e end of t h e t imestep. Only subrou t ines having DERIVATIVES a r e c a l l e d i n MODEZ.
G. Program Flow
A' f low c h a r t f o r subrou t ine TYPE32 is given in Figure 7. The subrout. ine-.
f i r s t checks f o r consis tency of t h e numbers of INPUTS, PARAMETERS, OUTPUTS, .. r.,
and DERIVATIVES wi th those requ i red by t h e subrout ine . It then asks whether ':
t h i s i s t h e f i r s t c a l l of t h e subrout ine f o r t h e c u r r e n t TRNSYS s imulat ion run
(MODE = -1). I f i t is, GROCS i s c a l l e d immediately t o read i n t h e d a t a needed
by GROCS t o d e s c r i b e t h e ground model, and then c o n t r o l is re tu rned t o EXEC.
I f i t is a l a t e r c a l l than t h e f i r s t , t h e program takes t h e va lues of t h e
INPUTS and PARAMETERS from t h e X I N and PAR a r r a y s and g ives them ind iv idua l
v a r i a b l e names. Up t o t h i s po in t t h e flow c h a r t f o r TYPE32 is t h e same a s t h a t
f o r TYPE33.
The program now checks that both adjarent hlocks of each p a i r f a c i n g each
o t h e r have t h e same l e n g t h i n t h e d i rec t ion a€ fl.nw. Tf nnt, an e r r o r condi-
t i o n i s s igna led and execution terminates . I f t h i s test is passed, t h e program
a s k s whether t h e inpu t mass flow r a t e is zero . I f i t is, t h e h e a t f low r a t e s
t o a l l a d j a c e n t blocks a r e s e t equa l t o zero. The o u t l e t temperature i s set
equa l t o t h e average temperature of t h e ad jacen t blocks. This is important.
I f t h e o u t l e t temperature i s set equal t o t h e i n l e t temperature under these
cond i t ions , i n c o r r e c t c o n t r o l of t h e a c t i v e components of t h e system can re-
s u l t i n t h e s e s imulat ions .
DATA YES CALL GROCS L r CONSISTENCY TO READ IN
CHECKS DATA
I
GET INPUTS, PARAMETERS
FROM ARRAYS
FLOW RATE CALCULATE HEAT FLOW RATES TO
I HEAT FLOW RATES I CALCULATE OUTLET
OUTLET TEMPERATURE EQUALS AVERAGE
GROUND TEMPERATURE
CALCULATE HEAT TRANSFERRED TO
ADJACENT BLOCKS AND PLACE IN
OUTPUT ARRAY
TIMESTEP IN-GROUND HEAT FLOW
Figure 7. Flow chart for buried pipe subroutine
I f t he i n l e t mass flow r a t e is nonzero, then the o u t l e t temperature and
t h e heat flows t o t h e adjacent blocks a r e calculated i n t h e manner discussed
above.
I f the cur ren t c a l l t o TYPE32 is not t he l a s t c a l l of the cur ren t timestep,
con t ro l i s returned t o EXEC. I f i t is (MODE = 2 ) , then the values of heat
t ransfer red t o the adjacent blocks of e a r t h a r e added t o the a r ray QIN, which
is i n COMMON 'kith GROCS, and then GROCS i s , . c a l l e d t o ca l cu la t e thh heat flows
among' a l l the blocks of ear th , given the thermal inputs from the tank a s ca l -
culated above. cont ro l is then returned t o EXEC. .
A dera i led descr ipt ion of tRe theory and implementation of t he GROCS pro- 7
gram has been given and w i l l no t be repeated here. Certain modif i c a t i s n s t o
the program were necessary i n order t o i n t eg ra t e it i n t o TRNSYS, and these a r e
now described. A flow cha r t showing t h e modifications i s given i n Figure 8.
It is des i rab le t o s t r u c t u r e the GROCS program so that more than one TYPE32
or TYPE33 subroutine could be included i n t he simulation. This may be needed
t o accommodate two-way flow i n the buried pipe f i e l d , a s discussed above.
Tank-coil combinations may a l s o be of i n t e r e s t . On t h e f i r s t c a l l t o each
TYPE32 o r TYPE33 u n i t , GROCS is ca l led t o read in t h e da t a def ining the blocks
of e a r t h i n t h e model. Even though there may be more than one ground-coupled
component, these da t a should be read i n only once. Similar ly, a t t he end of
each timestep, each TYPE32 o r TYPE33 u n i t ca l cu la t e s t he heat t ransfer red t o
t h e adjacent blocks and then c a l l s GROCS t o perform t h e heat flow computations
between blocks. Again, t h i s heat flow computation should be done only once
per timestep.
Accordingly, a va r i ab l e JFLAG i s used t o count t he number of TYPE32 o r
TYPE33 components i n t h e simulation, and a sernnd variabla LCALL counts t he
number of times GROCS has been ca l l ed a t t h e end of each .timestep. These two
varfables are i n i t i a l l y s e t . equa1 t o zero. Each time GROCS is ca l led on the
infrial pass through the component subroutines (MODE = - I ) , JFLAG is increased
by one. Data a r e read i n only on the f i r s t c a l l , where JFLAG = 1. On fu r the r
c a l l s t o GROCS i n MODE = -1, JFLAG becomes grea te r than one, and data a r e no t .
read in again. The value of JFLAG a t t h e end of t h i s f i r s t pass is equal t o
t he number of TYPE32 o r TYPE33 components. MODE is now increased t o 0 (and
subsequently can take on values of 1 and 2 ) , so t h a t no fu r the r incrementing
of JFLAG takes place.
ENTER a JFLAG, LCALL SET TO ZERO INITIALLY
RETURN
RETURN
RETURN
LCAL L r; HEAT FLOWS
RETURN
F i g u r e 8. Flow c h a r t showing m o d i f i c a t i o n s t o ground;coupling s u b r o u t i n e GROCS
The on ly f u r t h e r c a l l s t o GROCS t a k e p l a c e i n MODE = 2, a t t h e end of a
t imestep. Here it ' i s d e s i r e d t o le t a l l of t h e TYPE32 and TYPE33 components
add t h e i r heat i n p u t s t o t h e i r r e s p e c t i v e ad jacen t b locks , and then le t GROCS
c a l c u l a t e t h e mutual h e a t f lows in t h e ground only once f o r t h a t t imestep. On
each cal l t o GROCS, LCALL is inc reased by one. As long' a s LCALL remains l e s s
than JFLAG, t h e r e a r e s t i l l a d d i t i o n a l TYPE32 o r TYPE33 subrou t ines which have
y e t t o add t h e i r h e a t i n p u t s t o t h e ground. When LCALL equals. JFLAG, a l l t h e s e
subrou t ines have been accounted f o r , and GROCS may now do t h e hea t f low c a l c u l a -
t i o n s . F i r s t , however, LCALL i s r e i n i t i a l i z e d t o ze ro t o start t h e count f o r
t h c nex t t imestep. . .. : . , . , ' . . . .
DATA. SET FOR A CROUNDJ~,COUPLED SIMULATION
The d a t a required bj Lllt: TYPE52 "and T ~ ~ l i . 3 3 s ~ i hrrzi.lt ines in TRNSYS have been
descr ibed above. The d a t a requ i red by GROCS a r e descr ibed here . The GROCS d a t a
c a r d s fo l low t h e TRNSYS s imula t ion c a r d s , i n t h e following order :
TRNSYS c u n t r o l ca rds
SIMULATION card
Opt ional c o n t r o l c a r d s
UNIT-TYPE cards
and assoc ia ted
PARAMETERS,
INPUTS, and
DERIVATIVE3 cdrds
END card
GROCS c o n t r o l c a r d s
GROCS c a r d , .
NDEPS c a r d
DEPTHS card
TABLE c a r d s
START 1ICkI.E card
CONDUCTlVlTY card
NUMBER OF BLOCKS card
FREE BLOCKS c a r d s
RIGGED BLOCKS card^
NUMBER OF INTERFACES card
HEAT TRANSFER c a r d s
PRINT INTERVAL card
.CARD ( S)
GROCS
NDEPS
DEPTHS
TABLE
START TIME
CONDUCTIVITY
NUMBER OF BLOCKS
RIGGED FLOCKS.
NUMBER OF INTERFACES
HEAT TRANSFER
DESCRIPTION :. FORMAT
The £ i v e l e t t e r s GROCS i n columns 1-5. A5
The number of depths i n t h e t a b l e s used t o repre- I10 s e n t f ar-f i e l d (undisturbed ground) temperatures . Up t o 8 depths may be used, i n order beginning wi th t h e shal lowest .
The depths f o r which f a r -f i e l d temperatures a r e 8F10.2 given.
Twelve ca rds , one f o r each month. The monthly - - " 8 ~ 1 0 ; 2 average temperature of undisturbed ground is required f o r each of t h e depths given on t h e DEPTHS card, i n o rder of depth beginning wi th t h e shal lowest . These temperatures a r e used t o c a l c u l a t e t h e . r igged block temperatures a t a l l t imes and t h e f 'ree block temperatures a t t h e i n i t i a l time.
The i n i t i a l hour and month of t h e s imulat ion. F10.2, The hour i s measured from t h e beginning of t h e I10 month.
The thermal conduc t iv i ty of t h e ground. F10.2,
The number of f r e e blocks , and t h e number of 2110 rigged blocks.
One ca rd f o r each f r e e block, conta ining: 110, (1) The' block number ' 4F10.2 (2) The i n i t i a l block temperature (3) The depth of t h e c e n t e r of t h e block (4) The volume of t h e block (5) The volume h e a t capac i ty pCp of t h e block
Note: I f t h e i n i t i a l block temperature is s e t equal t o zero , t h e TABLE of undisturbed ground temperatures w i l l be used t o f i n d an i n i t i a l value .
One card for each r igged block conta ining: (1) The block number (2) The depth of t h e cen te r of t h e block
Note: The sequence of block numbers begins wi th 1 f o r both f r e e and r igged blocks.
The number' of hea t t r a n s f e r su r faces between p a i r s of blocks. This includes free-block- to-free-block and f ree-block-to-r igged-block i n t e r f a c e s .
One card f o r each h e a t t r a n s f e r i n t e r f a c e be- tween blocks conta ining:
(1) The block number of t h e f i r s t block (2) The block number of t h e second block
Note: The block number f o r a r igged block must have added t o i t t h e number of f r e e blocks . Thus, i f t h e r e a r e 18 f r e e blocks , r igged block number 1 becomes block 19 on t h i s card.
(3) H e a t t r a n s f e r a r e a between t h e i n t e r - f a c i n g blocks
(4) Center-to-center d i s t a n c e between t h e i n t e r f a c i n g blocks
PRINT INTERVAL The t ime i n t e r v a l i n hours between success ive F10.2 p r i n t o u t s from GROCS. A t each such i n t e r v a l , GROCS p r i n t s t h e heading, "THIS I S GROCS SPI3AKINGy1' t o d i f f e r e n t i a t e t h i s output from f hat generated hy TRNSYS , f crllowed by :
(1) The hour, month, and year of t h e simu- l a t ion
(2) The temperature o f each of t h e blocks (3) The n e t heat inpu t t o each f r e e block
from ground c o i l s and t anks (4) The Lutal n e t h e a t input t o a l l f r e e
hlocks
'REFERENCES
1 1. Metz, P. D., Experimental r e s u l t s from t h e s o l a r ground-coupling research f a c i l i t y a t Brookhaven Nat ional Laboratory, i n Proc. 1979 I n t . So la r Energy Soc. Congr:, At lan ta , May 1979, Pergamon Press , t o be published.
2. Bose, J. E. e t a l . , F i e l d t e s t r e s u l t s of s o l a r a s s i s t e d h e a t pumps us ing e a r t h c o i l coupling and s t o r a g e , I b i d .
3 . , Esbeaseu, T. V. and Mj-hhonen, M., A low energy house in Sweden heated by a s o l a r energy system with h e a t pump, I b i d .
4. Metz, P. D . , Design, cons t ruc t ion , and operat ion of t h e s o l a r a s s i s t e d h e a t pump ground-coupled s t o r a g e experiments a t Brookhaven Nati.nna1. Toah- o r a t o r y , presented a t 4 t h Annu. Heat Pump Technol. Conf., Oklahoma S t a t e U., S t i l l w a t e r , A p r i l 1979.
5. Ja rd ine , D. M., Phase I1 Phoenix/City of Colorado Spr ings s o l a r a s s i s t e d h e a t pump p r o j e c t , i n Proc. 3rd Annu. Solar Heating and-cool ing R&D Branch Cont rac to rs ' Meet., Sept; 1978, U.S. DOE CONF-780983, p. 241, 1979.
6. Nicho l l s , R. L. Comparisons of deep wel l and i n s u l a t e d shallow e a r t h s t o r - a g e of s o l a r h e a t , S o l a r Energy - 20, 127 (1978).
7. Merz, P. D . , A s imple computer program t o model three-dimensional under- ground h e a t f low with r e a l i s t i c boundary r .nnd i t ions , Brookhaven Nat ional Laboratory r e p o r t in prepara t ion .
8 . Andrews, J. W. and Metz, P. D . , Computer s imulat ion of ground-coupled s t o r a g e i n a s e r i e s s o l a r a s s i s t e d hea t pump system, i n Proc. 1979 I n t . So la r Enerpy Soc. Congr., At lan ta , May 1979.
9. Klein, S. A. et a l . , TRNSYS, a Transient Simulation Program, U . of Wisconsin- Madison, 1976.
10. Kutateladze, S. S . , Fundamentals of Heat Transfer, p . 93; Academic Press, New York, 1963.
