Buried Gas Transmission Pipelines: Temperature Profile Prediction through the Corresponding States Principle

MOHSEN EDALAT and G.ALI MANSOORIUniversity of Illinois at Chicago, (M/C 063) Chicago, IL 60607-7052, USA

Abik,lle;:t .,4 new analytic technique for the prediction of temperature profile of buried gas pipelines is developed based on the corresponding states principle. The resulting equations a.re tesud through prediction of the actual experimental data. It is demonstrated that the new technique can predict temperature profile quite accurately without usinl? any additional chart or table. This technique can also be used for prediction oi gas mixture temperature profile flowing in, a buried pipeline.

Knowledge about the variation of temperature along a gas transmission pipeline and the capability of its prediction is of major importance in the design and operation of such pipelines. There exist a number of analytic equations that are used in gas industry to predict temperature profile in buried gas pipelines. The Schorre equation (Schorre 1954) is one that has been used widely. Fonest (1978) int�rprered an,d :w.odi6ed the Schorre

equation to show that the Joule-Thomson coefficient µ varies with respect to distance. Coulter (1979) solved the energy equat�on by using the compressibility factor z and as­suming that heat capacity at constant pressure C

P remains constant along the pipeline, in

order to derive the temperature profile along gas pipelines. While t..11ere has been extensive activity in the development of such equations, !ittJe­

attention has been paid to the fact that the Joule-Thomson coefficient and the heat ca­pacity at constant pressure of a gas are not constants: They are functions of temperature as well as pressure. In the present report we introduce a new concept for the development of an analytical technique through the corresponding states principle

1 while considering

the fact thatµ and c,.

are functions of both temperature and pressure.

Derivation of the New Analytic Technique

Differential fonnultion of the first law of thermodynamics has been considered for a control volume segment dx of a pipeline which is

A(dq,_ldx) + pvA(dhldx) + q/11 = 0 (1)

where q._ is the axial heat conduction flux, his the axial enthalpy flow ra.te, and qr is the radial heat conduction flux. In derivation of Eq. 1, changes in kinetic and potential en­ergies across the control volume are neglected. The fi�t term in the above equation

Energy Sources Journal Volume 10, Issue 4, Pages 247-252, 1988

DOI: 10.1080/00908318808908933Print ISSN: 0090-8312; Electronic ISSN: 1521-0510

represents the axial heat conduction, the second term represents the axial enthalpy flow, and the third tenn represents the radial heat conduction across the control volume.

The Pecl�t number is defined as the ratio of the axial enthalpy flow and the axial heat conduction. One can neglect the axial heat conduction when the Peclet number is very large, greater than 100 (Arpaci 1966). Generally, in natural gas transmission pipelines ) as can be shown by any case study, the Peclet number is larger than 100. As a result, the axial conduction term may be neglected in su<::h cal.culations.

The Biot number Bi is defined as the ra6o of the internal and external resistance to heat transfer in a system. The Bi9t number for a gas transmission pipeline can be de­fined as

where hga,s is the heat transfer coefficient of the gas inside the pipeline, Rpipdine is the

radius of the gas pipeline, and K�otjd is the term.al conductivity of the solid surrounding the gas. Heat conduction of the natural gas can be conside-red radially lumped (Arpaci 1966) when the Biot number is less than 0.1. Study of heat transfer in buried natural gas transmission pipelines shows that the Biot number is always less than 0. 1 and therefore the following relation is used for qr:

(2)

where the overall heat transfer coefficient U i$ chosen to be (Eckert and Drake 1972)

U = 2k/D[l/ln(4z/D)] (3)

The enthalpy of gas is generally a function of temperature and pressure and it is de. fined as

dh = C p dT - µC p dP (4)

By substituting Eqs. 3 and 4 into Eq. i we can obtain

dT/dx + �(T - T8) = µ dpldx (5)

where� = TIDUl(pvACp). Furthennore, Eq. 5 can be reduced to a dimensionless form:

dq./dx,i,, +- {*qr == µ(P JT,::)(dP ,jdx*) (6)

where f'' = if>, qr -- (T - Tg)ITc , and x* = x/D. The Joule-Thomson coefficient µ .is

obtained from the following relation (Reid et a.I. 1977):

(7)

where

BJ jRTc = (0.1445 + 0.073wm) - (0.33 - 0.46w,n,)Tr-l - (0.1385 + 0.5wm)Tr -Z

- (0.012 + 0.0971-1-·'m)Tr - 3 � 0.0073wrnT, -s (8)

M Edalat and GA MansooriBuried Gas Transmission Pipelines: Temperature Profile Prediction through the Corresponding States Principle

Energy Sources J.10(4): 247-252, 1988

248

249

In the above expressions Brn is the gas mixture second virial coefficient, Pc and Tc are critical pressure and temperature, re:spcctively, and wm is the gas mixture acentric factor.By substituting Eq. 8 into Eq. 7 and upon lincarization of the resulting expression one obtains

Since the second- and higher-order tenns of the above expression have little or no effect on the calculation of the temperature profile they can be neglected for all practical pur­poses.

Heat capacity is generally a function of temperature and pressure (Reid et al. 1977)

(10)

where

and

4Cp(P, T) = P�[(0.66 - 0.92wm)rr- 2 + (0.83 + 3wm)Tr - 3

+ (0.145 + 1. lwm)Tr �4 + 0.526wmTr �9

where C,.O(n is the ideal-gas heat capacity and dCp(P, T) is not ideality deviation from ideal gas heat capacity. Equations 9 and 10 have been substituted into Eq. 6 and upon integration of the resulting equation, the following expression has been derived for the temperature profile:

T2l(T2 - T1)t'n[(T - T-i)l(T0 - T2)] - T1l(T2 - T1)t'n[(T - T1)/(To - f1)]

where = "ltDal(m0b)in[al(a + bx)] (11)

T1 � - c l/2 + T /2 - [(Tg

- c1)2/4 + c2]0

·j

T2 = -c112 + T J2 + [(Tg

- c1)2/4 + c2]0·5

c1 = (0.34 + 1.30wm)RT /P0(dPldx)m0l(TrDU) c2 = (1.28 + 2.0lw.m.)RT/·IP0(dPldx)m0t('nDU)a == a1 + aiP b = az(dPtdx)

a 1 = Ci(T)£12 = DC ,,(P, T)/p

Equation 11 can be used to obtain the temperature in a pipeline at a specific distance. In case the temperature is known, Eq. 11 can be written explicitly with respect to distance in the following fonn:

x = a/b{[exp(m0b/rrDa)]T2/(T2 - T1).f n[(T - T2)/(T0 - Ti)] - Tif(T2 - T1) X fn[(T - Tl )/(T0 - T1 )]-1 - l} (12)

M Edalat and GA MansooriBuried Gas Transmission Pipelines: Temperature Profile Prediction through the Corresponding States Principle

Energy Sources J.10(4): 247-252, 1988

250

This equation can then be used to locate a point in the pipeline that would posses a specified temperature_

Calculations and Results

Equation 11 is nonlinear with respect to temperature. To obtain temperature at a specific distance ,,fa gas pipeline the r,.�ewton-Raph.son (or any ot1:ler) regressi(lU method can be used to solve this equation numerically. Equation 12 is used to calculate the distance for a specific temperature.

The proposed analytical technique can predict temperature profiles more precisely than the other existing techniques. Figures 1 and 2 show that for the same environmental conditions, the analytical technique presented here gives more accurate results than other existing techniques. lo. addjtj_oQ., since the present technique is based on the theorem of corresponding states, it is readily extended to the mixtures. Since the temperature profile prediction technjquc developed in this work is quite acl:urate it can be expected that one would be also able to predict the pressure drop in a gas pipeline accurately. For thi�; purpose the present technique must be joined with an accurate gas-flow-rate equation.

Nomenciature

A

Ai, A2, A3i A4

Bm cl!D

h

k mO

Pipe cross section (m2)

Ideal gas heat capacity coefficient The gas mixture second virial coefficient Heat capacity (cal/mol .K) Pipe diameter (m) Enthalpy of fluid Soil thermal conductivity (calls m2 K) Molar flow rate (molls)

130

Winter

100

OF

0 Schorre

Coulter

Corr. State 80

70 79

sor--=--

----=--=��� Q

10 30 50 80 Miles Figure 1. Temperature profile in a buried gas pipeline as a function of distance considering a winter ground temperature T

g of 60 "F.

M Edalat and GA MansooriBuried Gas Transmission Pipelines: Temperature Profile Prediction through the Corresponding States Principle

Energy Sources J.10(4): 247-252, 1988

OF

25 l

150

120

1nn

·��[ T:10 30

Summer

0 Schorre

- Coulter

50 80 Miles Figure 2. Temperature profile in a burled gas pipeline as a function of distance considering a suro.mcr gi:o11.11d tc�pcm1ture 78 of 80 "F.

'!I>

f'c qx qr R r

To r.

T�

u

V X dp!dx.

p µ.

Wm

Pipe prhneter Reduced pressure Axial heat conduction flux Radial heat conduction flux Universai gas constant (cal/mol K) Gas temperatu.re (K) lnjtfal gas temperature (K) CriticaJ temperature (K) Reduced temperature (K) Ground temperature (K) A vi::rase temperature (K) Overall heat transfer coefficient (cal/i; m2 K) Velocity of fluid (mis) Pipe length (rn) Avei:age pl'Cssure drop (atm/m) Density of fluid (moVm3) Joule-Thomson coefficient (K/atro) Acentric factor

Acimowiedgments

The authol'S thank K. Adham of Sharif University of Technology foi: performing some of the computations reported in this paper. This research is supported in part by the Gas Research lnstitute, Contract No. 5086-260-1244.

References

Arpaci, V.S. 1966. Conducriqn_ heat transfer. New York; Addison-Wesley. Coulter, D.M. 1979. New equation acc�•ratcly pri;:dic:ts flowing ga� temperature. Pipeline Industry

(May), 71-73,

M Edalat and GA MansooriBuried Gas Transmission Pipelines: Temperature Profile Prediction through the Corresponding States Principle

Energy Sources J.10(4): 247-252, 1988

25.2

Ei;:kert, E.R., and R.M. Drake, Jr. 1972. Arw.lysis of heat a11d maH tran.sfe.r. New York: l\k:Graw­Hill.

Forrest, J.A. 1978. lnteqJI:cting the S,:,borre gas temperature equation. Pipeline Industry (Feb.), 58-60.

Reid, R.C., T.K. Sher.vood, and J.M. Pn1.usnitz. 1977. The properties of gases and liquids. Nc;:w York; lvlcGraw-Hill.

Schnrre, C.E. 1954. F1ow tornperature in a gas pipeli.11c. Oil and Gas Journal (Sep.), 64-68.

M Edalat and GA MansooriBuried Gas Transmission Pipelines: Temperature Profile Prediction through the Corresponding States Principle

Energy Sources J.10(4): 247-252, 1988