1249IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 5, May 1983
ON HEAT TRANSFER BETWEEN CABLE AND ITS SURROUNDING PIPE OR DUCT WALL
Y.T. TsuiInstitut de recherche d'Hydro-Quebec
Varennes, Quebec, Canada
c CableAbstract
There are three modes of heat transfer, namely,conduction, convection and radiation, between cable andits surrounding wall. An up-to-date but simple heattransfer calculation for each mode is presented in thispaper. The exact knowledge about three modes of heattransfer is important not only for transient load and/or cyclic load analysis but also for calculating theload capability of cables.
NOMENCLATURE
A Area, cm
c Specific heat at constant pressure, Joule/gm Cp
D Diameter, cm
g Gravity constant = 981.7 cm/sec2
Gr Grashof number = v (L AT), dimensionless1- -r
h Heat transfer coefficient, Watt/ C
k Thermal conductivity, Watt/cm C
L Characteristic length, cmk q
Nu Mean Nusselt number = -q - , dimensionlessc 1
k qcnCII condPr Prandtl number = ks dimensionlessk
q Heat flux per unit length of cable, Watt/cm
R Radius, cm 2p agcp 3
Ra Rayleigh number = Gr.Pr = p k (L AT), dimension-less
T Temperature in C or K
(0-6 Coefficient of volume expansion (°C)
p Viscosity, gm/(sec) (cm)
£ Emissivity, dimensionless
p Density, gm/cm
a Stefan-Boltzmann constant = 0.00567 x 10 , Watt/(cm2) (K)4
Subscript
1 Inner cylinder or the shield (concentric neutral)
2 Outer cylinder or the pipe or duct
82 SM 374-7 A paper recommended and approved by theIEEE Transmission and Distribution Committee of theIEEE Power Engineering Society for presentation at theIEEE PES 1982 Summer Meeting, San Francisco, California,July 18-23, 1982. Manuscript submitted January 11, 1982;made available for printing April 19, 1982.
cond Conduction
conv Convection
eq Equivalent
r Radiation
INTRODUCTION
In a casual review of my colleague's report [1],it came to my attention that Neher and McGrath [21published an article entitled "The Calculation of Tem-perature Rise and Load Capacity of Cable Systems," in1957. This article has served and continues to serveas a basis for the calculation of the load capabilityand for the preparation of the IPCEA-NEMA ampacitytables. For heat transfer calculations between cableand its surrounding pipe, this article used the pro-cedure developed by Neher in his early publications,notably [3,4,51.
At the time of publication in late Forties andearly Fifties, Neher's thermal analysis was excellent.During the intervening three decades, much progresshas been made in heat transfer, particularly in con-vection. Hence there appear to be room for improve-ment for his radiation and convection analysis. Nev-ertheless, his conduction analysis remains to be per-fect. The purpose of this paper is to present a simple,practical but up-to-date heat transfer calculation be-tween cable and its surrounding pipe.
Following Buller and Neher [3], we idealize thecable and its surrounding pipe as two horizontal con-centric cylinders with isothermal surface temperaturesT and T2 respectively, as shown in Fig. 1. We fur-ther assume infinitely long cylinders with radial sym-metry. Thus the three-dimensional problem boils downto one-dimensional problem with the radius as the solespatial variable. For a single cable, the diameter ofthe inner cylinder is taken to be the diameter of thecable, i.e. D1 = Dc. For three cables in the pipe, it
Fig. 1 Heat Transfer Between Two Concentric HRoriz-:ntal Cylinders of Isothermal Surfaces With Airor Oil as Intervening Medium.
is customary to base D1 on the circumscribing circleof the cables in triangular configuration, D1i 2.15Dc. For two cables, the relationship D1 - 1.65 Dc issatisfactory.
Radiative Heat Transfer
Reference is made to Fig. 1. Following Holman[6], the radiative heat flux qr between two concentriccylinders is as follows:
CaA (T 4-1 1 T2)1 1(1A - 1)
(1)
where a, s, A and T stand for Stefan Boltzmann con-stant, emissivity, area and absolute temperature andsubscripts 1 and 2 denote inner and outer cylinder re-spectively. By assuming £1 = £2 = 1, equation (1) sim-plifies to:
"4 4qr = Cr A1 (T1 T2 (2)
Introducing radiative heat transfer coefficient
4_ 4aT(T1 - 2
hr T , then equation (2) can be linearized1 2
as follows:
qr =r T1 T2 (3)
Neher et al employed a linearized equation of dif-ferent version, the origin which dates back to thefirst edition of the famous McAdam's book published inthe Thirties.
Now one can see that the ratio between the exactexpression given by equation (1) and the approximateexpression given by equation (2) is as follows:
gE1 A2 (E2 l)} (4)
Conductive & Convective Heat Transfer
Reference is made to Fig. 1 again. FollowingHolman [6], the conductive heat flux qcond between twoconcentric cylinders is as follows:
27r k (T - T20212 (5)qcond D2(5
Q,n-
where D, k and T denote diameter, thermal conductivityand temperature respectively. Equation (5) is exactlythe same equation employed by Neher.
Experimental and theoretical research on naturalconvective heat transfer in horizontal annuli have beenvery active in recent years due to its wide applica-tions ranging from nuclear reactors, solar collectors,thermal storage systems to underground transmissioncables. A complete review of this subject is certainlybeyond the scope of this paper. Instead we shall pre-sent a pragmatic but modern method of computing naturalconvective heat transfer in horizontal annuli precededby a few words of explanations.
For natural convection in horizontal annuli, ithas been shown [9,10,11], via dimensional analysis ofthe governing differential equations, that the meanNusselt number or heat transfer can be correlated as a
function of the Rayleigh number, Prandtl number and ageometry or aspect ratio. The writer is fully cogni-zant of the fact that many different techniques of cor-relating the mean natural convective heat transfer inhorizontal annuli have been proposed [9,101. In thispaper, we follow current practice of expressing themean Nusselt number in terms of an equivalent conduc-tivity. In fact, the use of equivalent conductivityis not new, it was first introduced by Beckmann [12]about half a century ago. The equivalent conductivityis defined as follows:
k qeq conv -k q- Nu
cond(6)
For simplicity, we use Kuehn-Goldstein's [11,13]correlations:
For air
keq = 0.159 Ra 0.272 4 4kRa for 2.1xlO < RaL4< 9.6 x 10 (7a)
0.200 Ra 0.25 10--21=0.200 Ra ~~~~~~~~~~~~~~~~~~~~~forRa< 2.1 x1k 14 L (7b)
For water
keq 0.238 4 5eq-0.23 Ra for 2.3 x10 <Ra <9.8x 10 (8)k L4 L
It is to be noted that, in equations (7a), (7b)and (8), the characteristic length is taken to be thegap of the annulus, i.e., L = i (D2 -Dl).
There are plenty of reasons for choosing Kuehn-Goldstein's correlation over many others. Chief amongthem are: (1) They have performed both experimentaland theoretical studies and they have checked theirexperimental findings with their theoretical ones. (2)They have made a literature survey on this subject mat-ter prior to 1976 and they have compared their resultswith others meticulously. (3) In contrast with otherexperimenters who measure the overall heat transfercoefficient, their measured heat transfer coefficientsare local values because they employ the optical meth-od via Mach-Zehnder interferometer. It is well knownthat optical methods do not disturb the flow field,thus yielding better results. (4) Due to the factthat they obtain the overall (mean) heat transfer byaveraging the experimentally determined local values,they are able to set up experiments to study the ef-fect of eccentricity on the local heat transfer. Theconclusion is that, although the local heat transfer isgreatly influenced by eccentricity, yet the mean heattransfer does not change more than ten percent wheneccentricity to gap ratio varies from zero to two-thirdin both vertical and horizontal directions. (5) Last,certainly not the least, their experimental range ofinterest nearly coincides with ours, i.e. air, Pr = 0.7,RaL = 2.2 x 102 to 7.7 x 107 and L/D1 = i (D2 - D1)/'D1= 0.8. This is of paramount importance because themean heat transfer or Nusselt number is a function ofthe Grashof number, Prandtl and its aspect ratio, i.e.
qcn kLv= Nu = eq = f (Pr, Gr1,5D)qcond k
1(9)
where Gr = RaL/Pr.
SAMPLE CALCULATION
After setting up the simple but up-to-date for-mulas to computer each mode, i.e. radiation, conduc-tion and convection, of heat transfer between cable andits surrounding wall, we shall perform a sample calcu-lation by taking typical physical data from Ref. 1.
T, = 800C = 353 K1
T2 = 20 C = 293 K
D = 3.58 cmc
D = 2.15 D = 7.7 cm1 c
D = 11.43 cm2
T = 1 (T + T2)= (80 + 20) =500C
AT = T - T =80 - 20 = 60 C1 2.
El = 0.60 from Table 2
£2 = 0.94 from Table 2
a = 0.00567 x 10 Watt/(cm ) (K)Ra -3 o -l 03 = 7.183 x 10 cm (C) ,from Table 1 at T 50 C
L AT
k = 2.770 x 10 at T = 50 C from Table 1cm0C
L = CD - ) =- (11.43 -7.7) = 1.865 cm2 2 1 2
Table 1- SOME PHYSICAL PROPERTIES OF AIR [7J
Table 2 - TOTAL EMISSIVITY DATA [8]
Material Temperature C Emissivity
Oxidized copper 20 to 200 0.60
Concrete 0 to 93 0.94
To compute the radiative heat flux per unit lengthof the cylinder, we employ equation (1) and appropri-ate data listed above as follows:
Cr7D(T4 4
a 1 1 - T2 )
'r =1 D1 1 _1
1251
(.00567 x 10 9) (tr x 7.7) C3534 - 2934)7.7 /1+ _
Q0.60 11.43 0.94
=11118= 6.53 x 101 Watt1.713 cm
To compute the conductive heat flux per unitlength of the cylinder, we use equation (5) in con-junction with appropriate data listed above:
2 ir k (T1 - T2)1 2qcond D2
Qn -D1
-42 rr (2.77 x 10 4) (80 - 20)qcond 1:1.432n
= 2.644 x 10 1 Wattcm
To compute the convective heat flux per unitlength of the cylinder, it is advisable to obtain theRayleigh number first in order to determine whetherequation (7a) or (7b) should be used. In this case,RaL = 71.83 (1.865)3 (60) = 2.796 x 104. So we employequation (7a):
qonv = k = 0.159 RaL
qcond kL
= 2.58
qconv = .8qcond= 2.58 (2.644 x 10 1
-6.82 x 10- Wattcm
To sum up:
DISCUSSION & CONCLUSION
In the preceding section, we have presented a
simple and clear-cut method of calculating each ofthree modes of heat transfer. The exact knowledge ofheat transfer will not only play an important role intransient load and cyclic load analysis but also im-prove the calculation of the load capability of cables.This improvement surely carries economical overtonesto all utility industries.
The convective heat transfer calculation is basedon atmospheric air. Should pressurized air be employ-ed, one needs to multiply the Rayleigh number 2by thesquare of pressure ratio, i.e. (ppress'd/Patm) . For
the Rayleigh number contains p2 which varies proportion-ally to p2 (in other words, ideal gas law applies here),while the other physical quantities in Ra nearly do notchange with pressure. This statement can be applied toother gases such as SF6.
Watt/cm Percentage
Conductive heat flux, q 2.64 x 10 16.5%
Convective heat flux, qcon 6.82 x 10 42.7%
-1Radiative heat flux, q. 6.53 x 10 40.8%
1252
Very often, one concerns the effect of eccentric-ity on convective heat transfer. As pointed Qut pre-viously, this question is settled by Kuehn and Gold-stein's experiments [13] that tremendous changes of ec-centricity do not alter the mean convective heat trans-fer by more than ten percent.
Although equations (7a) and (7b) are correlatedfor air, they are also valid for all other gases in-cluding the pressurized ones because all gases havePrandtl number in the range of 0.7 to 1.0. For natu-ral convective heat transfer in oil, it is not advis-able to use equation (8) because water and oil havevery different Prandtl numbers. It is much preferableto employ Boyd's correlation [14], which, although com-plicated, does cover a wide range of Prandtl numbers.
REFERENCES
[1] M. Kellow, "The Forced Cooling of DistributionCables in a Duct Bank; Analysis of Cable Ampaci-ty and Thermal Performance," prepared by Hydro-Quebec Research Institute for Canadian ElectricAssociation, CEA #77-38, April 1980.
[2] J.H. Neher and M.H. McGrath, "The Calculation ofthe Temperature Rise and Load Capability of CableSystems," AIEE Trans., Vol. 76, pp. 752-772, Oct.1957.
[3] F.H. Buller and J.H. Neher, "The Thermal Resis-tance Between Cables and Surrounding Pipe or DuctWall," AIEE Trans., Vol. 69, pp. 342-349, 1950.
[4] J.H. Neher, "The Temperature Rise of Cables in aDuct Bank," AIEE Trans., Vol. 68, pp. 540-549,1949.
[5] J.H. Neher, "The Temperature Rise of Buried Cablesand Pipes," AIEE Trans., Vol. 68, pp. 9-21, 1949.
[6] J.P. Holman, Heat Transfer. McGraw Hill, 2nd ed.,p. 22 and pp. 238-9, 1968.
[7] W.H. McAdams, Heat Transmission. McGraw Hill, 3rded., p. 483, Table A-25, 1954.
[8] G.G. Gubareff, J.E. Janssen, and R.H. Torborg,Thermal Radiation Properties Survey. HoneywellResearch Center, Minneapolis, Minn., 2nd ed., p.74 and p. 197, 1960.
[9] M. Itoh, N. Nishiwaki and M. Hirata, "A New Meth-od for Correlating Heat Transfer Coefficients forNatural Convection in Horizontal Cylindrical An-nuli," International Journal of Heat and MassTransfer, Vol. 13, pp. 1364-1368, 1970.
[10] T.H. Kuehn and R.J. Goldstein, "Correlating Equa-tions for Natural Convection Heat Transfer BetweenHorizontal Circular Cylinders," International Jour-nal of Heat and Mass Transfer, Vol. 19, pp. 1127-1134, 1976.
[11] T.H. Kuehn and R.J. Goldstein, "An Experimentaland Theoretical Study of Natural Convection inthe Annulus Between Horizontal Concentric Cylin-ders," Journal of Fluid Mechanics, Vol. 74, pp.695-719, 1976.
[12] W. Beckmann, "Die Wdrmenbertragung in zylindri-schen Gasschichten bei nattrlicher Konvektion,"Forchung Geb. d. Ingenieurwesen 2, Band 5, pp.165-178, 1931.
[13] T.H. Kuehn and R.J. Goldstein, "An ExperimentalStudy of Natural Convection Heat Transfer in Con-centric and Eccentric Horizontal Cylindrical An-nuli," Journal of Heat Transfer, Vol. 100, pp.635-640, 1978.
[14] R.D. Boyd,"A Unified Theory for Correlating SteadyLaminar Natural Convective Heat Transfer Data forHorizontal Annuli," International Journal of Heatand Mass Transfer, Vol. 24, pp. 1545-1548, 1981.
