Modified Values for Geometric Factor of 00193924

7/28/2019 Modified Values for Geometric Factor of 00193924

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IEEE Transactionson Power Delivery, Vol. 3, No . 4, October 1988 1303MODIFIED VALUES FO R GEOMETRIC FACTOR OFEXTERNAL THERMAL RESISTANCE OF CABLES IN DUCT BANKS

M.A. El-Kady , enior Member, IE EEG.A. Anders, Senior Member, IE EE J. Motlis, Member, IEE ED.J. HorrOcks, Member, IEEEOntario Hydro, Toronto, Canada

Abst ract - In a previous paper [I], the authors described a tech-nique fo r calculating the geometric factor Gb or extended rangesof the heightlwidth ratio of the duct b ank or backjill, which enabledthe application of the very popular Neher-McGrath method to awider range of cable configurations. However, the technique of[ I ] was, like other existing techniques, based on the assumptionthat the duct bank surface represents an isothemal boundarywhich may result in some errors in the derived Gb-values. In thispaper, a novel finite-element based technique is described for cal-culating modified values of the G b-factor based on the actual heattransfer mechanism (c onstan t total heat flux) around the duct banksurface. The technique was applied to several cable configura-tions yielding more accurate Gb-values.

1. INTRODUCTIONIn the most widely used Neher-McGrath method [2] of cal-culating the extemal thermal resistance between cables in duct

banks (or backfills) and ambient earth, approximate formulas forthe geometric factor Gb of the duct bank are used. Two limitationsexisted in the applications of these formulas. First, the formulasare valid only for a limited range (1/3 to 3) of the heighdwidthratio of the duct bank, a limitation which causes difficulties whencalculating extemal thermal resistances of some cable systems.Second, these formulas are based on the assumption that the ductbank surface represents an isothermal boundary [3]. The tempera-ture distribution around the perimeter of the duct bank is notuniform; the bottom portion of the surface is always hotter than thetop portion. The differences in temperatures can, in some cases, belarge and the isothermal assumption can be very restrictive leadingto some errors in computations.In Reference [l], a technique was presented to overcomethe fi s t limitation. Extended values of the Gb-factor were derivedwhich are applicable to wide ranges of the heighdwidth ratio of theduct bank. These extended values were displayed in a tabular form

for direct use in the Neher-McGrath analysis.In this paper, the authors investigate the second limitation,namely the isothermal assumption for the duct bank surface. Afinite-element based technique is described which simulates theheat dissipation pattern of the duct bank surface as it actually oc-curs in practice; that is, a constant total heat flux surface (the totalheat produced by the cables and flowing through the surface perunit time in the steady state is constant). Via extensive finite-element program runs and subsequent analyses, modified values ofthe Gb-factor are derived and tabulated for some practical cableconfigurations as will be described in the subsequent sections.

57 SX 591-1 A paper recommended and approvedby t h e I E F F I n s u l a t e d C o n d u c t o rs C o m mi t te e of t h eIEEE P ow er E ng ine er i ng S oc i e t y f o r p r e s e n t a t i o n a tt he IEEE/PES 1937 Summer f le et ing , San Fr an ci sc o,C a l i f o r n i a , .J u lv 12 - 1 7 , 1 9 87 . Y a n u s c r i p t s u b m i t t e d.January 27, 1987; made a v a i l a b l e f o r p r i n t i n gA p r i l 2 1 , 1987.

The purpose of this paper is mainly to investigate thevalidity of the isothermal duct bank surface assumption and topresent the computational technique which enables a more accuratecalculation of the Gb-factor. The paper shows only some resultsfor selected cable configurations which serve the purpose of illus-tration and should not be generalized for other cable configura-tions.

2. METHOD DESCRIPTIONA s outlined in the Introduction, the purpose of this paper isto investigate the derivation of improved values of the geometricfactor Gb which takes into account (at least in an approximate way)the variations of the temperature along the duct bank perimeter sur-face. In Reference [11, extended values of the Gb-factor for wideranges of depthheight and heighdwidth ratios of the duct bank (orbackfill) were provided assuming that the duct bank surface is anisotherm. While a more complex finite-element based methodol-

ogy was used to derive those extended values of the G,-factor, thefinal results were displayed in a simple tabular format which canbe used directly in the context of the conventional Neher-McGrathcomputational procedure. A similar procedure is also followed inthis paper. A finite element based methodology will be used toderive "modified values" of the Gb-factor which takes into accountthe non-isothermal conditions of the duct bank surface. However,once these new, improved G,-values are derived, they can be dis-played in a simple tabular form for direct use in the Neher-McGrath procedure.The idea of the methodology used in this paper is based onthe fact that the sum of the heat flux through all segments of theduct bank surface is constant. That is, the closed rectangular ductbank surface (which contains the cables as heat sources) releases atotal constant heat flux rather than being an isothermal surface.This leads, as illustrated in Figure 1, to higher temperatures at the

0.5 1 . 0 1. 5 2.0 2.5 ( m )I I I

\

. . . . . . .

. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .1..:@:. . . . . .: : :: , , , ji. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . .UI S O T H E R M A LCONTOU RS

T = 3OoC= 45oc

1 m )Figure 1. Isothermal contours surrounding cables in a duct bank

088 5-8 977 /88 / 1OOO- 1303$ 01.OO0 988 IEEE


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1304bottom surface of the bank than at the top surface. Moreover, thetop surface approximates an isothermal condition better than thebottom surface. This is primarily because the top bank surface iscloser to the flat ground isothermal surface, and because the cablesoften span most of the horizontal dimension of the duct bank inmost practical configurations.

Consider the thermal circuit configuration of Figure 2where the cable bank is represented by a rectangular cross-sectional surface C of height h and width w. For this configura-tion, the total thermal resistance between the duct bank surface andthe ground ambient is given by Equation (1):R =-p (T, - T,) /I (aT/an)ds (1)C

I S O T HE R M A L S U R F A C E S,T, tr----"- LbC O N S T A N T T O T A LH E A T F L U X S U R F A C E C

Figure 2 . Thermal circuit configurationwhere p is the thermal resistivity of the medium, C represents theduct bank surface and a p n denotes differentiation along thenormal to C. Note that Equation (1)can be written approximatelyas

ATiR = - p / [ - Si/(T,i-T,)I (2)id, AI+where, as shown in Figure 3 , Ti is the temperature of segment ialong the first finite element grid layer surrounding the duct banksurface as was described in [l] , TCi is the temperature at the ductbank surface C of segment i, and I, is the index set of segmentsalong surface C. By choosing ASJAni =1for all i, Equation (2)reduces to P

2% icI,R = - Gb=-P/[C (Ti-Tci)/(Tci-Ts)I ( 3 )Hence,

(4)

Important Rema rks1.

2.

3.

4.

5.

Equation (4) provides the new value of the Gb-factor n termsof the temperature results from the finite element analysis.Note that if the duct bank surface is in fact an isotherm thenTci=T, for al i in Equation (4) leading toGb=2% r, - TJ /which is the same result obtained in [l],as is expected.Since the values of Ti and Tci depend on the specifictemperature distribution on the surface of the duct bank (orbackfill), the new Gb-values will also depend on thistemperature distribution. That is, they depend on the specificconfiguration and arrangement of the cables inside the ductbank.In the present work, new improved values of the Gb-factorare derived and tabulated for several practical cable con-figurations as will be presented in Section3.In Appendix I, an approximate technique is also presentedwhich may be used for other cable configurations. The ap-plication of this approximate technique does not require fullfinite element analysis. It does, however, require the displayof some isothermal contours around the cable bank.

C (T, - Ti)]id,

The new values of the Gb-factorcan be higher or lower thanthose obtained in [I] for an isothermal duct bank surface. Inother words, the new G,factor may be more conservative orless conservative than the old (isothermal-based) Gb-factor,depending on the specific configuration of the cable system.This is clearly shown from the derivationsof Appendix I1 fora two-portion duct bank surface model (top and bottomhalves).The Gb-factor is assumed, in principle, to be dependent onlyon the cable system configuration and independent of the to-tal heat flux from the duct bank. This was co nf me d by thesensitivity analysis results carried out by the authors, inwhich different levels of cable loadings were used. Thevalidity of this assumption is also related to the observationthat, for most cable configurations, the shapes (not thevalues) of isothermal contours around the duct bank remainessentially the same for different levels of heat flux.The accuracy of the Gb-values calculated using Equation (4)would naturally depend on the level of detail of the finite-element analysis employed. Because a very extensive set offull finite-element analyses is required for each cable systemconfiguration, the authors did not use fully detailed finite-element grid modelling in the results shown in the next Sec-tion. The authors, however, believe that these results are ofsufficient accuracy for comparison and investigation pur-poses.

3. RESULTSThe finite-element based technique described in Section 2was implemented to evaluate the geometric factor Gb for severalcable systems. The calculations were performed using a finite-elemendsensitivity analysis program at Ontario Hydro. Figures4(a) to 4(f) show the results obtained. It is important to reiteratethat these results are applicable only to the particular cable arrange-ments considered.

Figure3. Finite-element grid structure


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b'

6

5

4

3

2

1

0

Gt:

2 . (

2 .

2 .

1.

1.

0.

h LbI I

J W L

D U C T # 1

Lb w IqJl Nw Gb Gob1.2 0.6 0.6 2 1.0 1.48 2.002.4 0.6 0.6 4 1.0 2.40 2.933.6 0.6 0.6 6 1.0 2.88 3.671. 2 0.6 1.2 2 0.5 1.26 1.512.4 0.6 1.2 4 0.5 2.22 2.213.6 0.6 1.2 6 0.5 2.56 2.67

Notes: 1. Values of Lb, h, w are in metres2. GObrepresents values of Gb from [11 based on theisothermal duct bank surface assumption

I I I I I I I I ,0 1 2 3 4 5 6 7 8 Lb/h

F I G U R E 4 ( a )MODIFIED V A L U E S O F G b - F A C T O F ? FO R C O N F I G U R A T I O N #1

-L Lb W L& h/W Gb Gob

1.2 0.6 1.8 2.0 0.33 0.86 1.322.4 0.6 1.8 4.0 0.33 1.84 2.083.6 0.6 1.8 6.0 0.33 2.56 2.541.2 1.0 1.8 1.2 1.34 0.99 1.482.4 1.0 1.8 2.4 1.34 1.86 2.353.6 1.0 1.8 3.6 1.34 2.56 2.82

h E me k !DiJ CT # 2

= O. 3 3 Notes: 1. Values of Lb,h, w are in metres2. GObrepresents values of Gb from [11 based on theisothermal duct bank surface assumption

I I I I I I I I *1 2 3 4 5 6 7 L b / h

F I G U R E 4 ( b )M O D I F I E D V A LUE S O F G b -FA CTO R FO R CO N FI G UR A TI O N #2


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GbA

5 -

4 -

3 -

2 -

1 -

1.2 0.6 1.8 2.0 0.33 1.14 1.322.4 0.6 1.8 4.0 0.33 1.92 2.083.6 0.6 1.8 6.0 0.33 2.72 2.541.2 1.0 1.8 1.2 0.56 0.95 0.992.4 1.0 1.8 2.4 0.56 2.00 1.735.4 1.0 1.8 5.4 0.56 3.80 2.62

Notes: 1. Values of Lb,h, w are in metres2. Gob represents values of Gb from [11 based on theisothermal duct bank surface assumption

0 1 I I t I I I I I )5 6 7 %/h1 2 3 4

F I G U R E 4(c)M O D I F I E D VA L U E S O F G b -F A C T O R F O R C O N F I G U R A T I O N # 3

3

L W - 4D U C T #4

0

L b h w Lbhl ww Gb Gob

1.81.82.41.82.43.6

Notes:

1.8 1.5 1.0 1.2 1.12 1.241.2 1.0 1.5 1.2 1.83 1.631.2 1.0 2.0 1.2 2.06 2.061.43.8 1.0 1.0 1.8 1.51.8 1.0 1.33 1.8 1.81 1.701.8 1.0 2.0 1.8 2.26 2.24

1. Values ofb, , w are in metres2. Gobrepresents values of Gb from [l] based on theisothermal duct bank surface assumption

1 2 3 4 L b l h

F I G U R E 4(d)M O D I F I E D V A L U E S O F G b - F A C T O R F O R C O N F I G U R A T I O N # 4


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~

1307

27

0 . 5I I l l

DUCT # 5

_ _ _ ~1.2 1.2 1.8 1.0 0.67 0.82 0.952.4 1.2 1.8 2.0 0.67 1.62 1.681.8 0.6 0.9 3.0 0.67 1.81 2.111.35 1.35 1.8 1.0 0.75 1.00 1.011.8 0.9 1.2 2.0 0.75 1.89 1.762.7 0.9 1.2 3.0 0.75 2.43 2.20

Notes: 1. Values of Lb, h, w are in metres2. Go b epresents values of G b from [13based on the= 0 . 6 7 isothermal duct bank surface assumption

0 1 2 3 Lb/h

FIGURE 4 ( e )MODIFIED V A L U E S OF Gb-FACTOR FOR CONFIG URATION # 5

DUCT # 6

Lb w L& NW Gb Go b1.5 0.75 3.0 2.0 0.25 1.1 1.121.5 0.6 2.4 2.5 0.25 1.41 1.313.6 0.6 2.4 6.0 0.25 2.56 2.292.4 1.26 3.0 2.0 0.42 1.79 1.445.0 1.26 3.0 4.0 0.42 2.49 2.196.0 1.00 2.4 6.0 0.42 3.17 2.66

Notes: 1. Values of L,, h, w are in metres2. GO represents values of Gb from [11 based on theisothermal duct bank surface assumption

I I I I I I w1 2 3 4 5 6 L b / h

FIGURE 4 ( f )MODIFIED VALUES OF Gb-FACTOR FOR CONFIGURATION A6


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13084. DISCUSSION AND CONCLUSIONS

1. The results of Figures 4(a)-(f), when compared with thosederived in [l], show that in most cases the assumption ofisothermal duct bank surface yields an error in the calcula-tion of the Gb-factor. For example, for configuration #1, theerror is approximately -35% for L,,/h=2 and h/w=l, andreduces (in magnitude) to -27% for L&=6 and the same h/wvalue. For configuration #3, however, the error decreasesfrom 15% to 7% as L&I increases from 2 to 6 and WwG.33.It should be noted that a certain percentage error in the Gb-value normally translates to a smaller percentage error in theultimate cable ampacity.The results obtained also show that the new Gb-valuescan beless or greater than those based on the isothermal assump-tion, as was discussed in Remark 3 of Section 2, dependingon the specific cable configuration and the heat transfer pat-tern around the cable bank. In approximately 60% of thecases analyzed, the values of the Gb-factor were higher thanthose derived in [l] for isothermal duct bank surface. Fromthe results of Figures 4(a)-(f), t is generally observed that theisothermal assumption yields less conservative values of Gb-factor for configurations #1 and #2 and more conservativevalues for configuration#6.The results of Figures 4(a)-(f) are applicable to other cablesystems having the same configurations. They are indepen-dent of cable loading. They are, however, dependent on therelative locations of the cables inside the duct bank or back-fill.

2.

3.

APPENDIX I

4. While the values of the Gb-factor displayed in Figures 4(a)-(0 are applicable only to the associated configurations, thetechnique presented is general and Equation (4) can be ap-plied to any other cable configuration.As was mentioned earlier, the purpose of this paper was toinvestigate the effects of the isothermal surface assumptionon the derived values of the Gb-factor, and to present asuitable computational technique for more accurate calcula-tions of the Gb-factor. It is hoped that some coordinated ointstudies will be conducted in the future by various researchersand institutes to derive and display full tables of modifiedGb-values applicable to most cable configurations and ar-

' rangements. While the efforts contained in such studieswould be extensive, as they would involve massive fiiite-element runs, the final result tables and diagrams would be ofinvaluable assistance to cable engineers.

5.

ACKNOWLEDGEMENTSSome of the finite-element programs used in this work weredeveloped jointly with the Canadian Electiical Association (CEA)under Contract No. 138D-375. Also, some of the theoretical devel-opments have been sponsored by the Natural Sciences andEngineering Research Council of Canada (NSERC) under ContractNo. A1708.

REFERENCES1. M.A. El-Kady and D.J. Horrccks, "Extended values for ge-ometric factor of external thermal resistance of cables in ductbanks," IEEE Trans.on Power Apparatus and Systems, Vol.PAS-104, 1985, pp 1958-1962.2 . J.H. Neher and M.H. McGrath, "The calculation of th etemperature rise and load capability of cable systems," N E ETrans (Power Apparatus and Systems), Vol. 76, 1957, pp752-772.3. Discussion on [11by N.R. Spencer and G.A. MacPhail.

Approximate Calculation of Gb-Factorfor General Cable ConfigurationsThe approximate method proposed here is based on analyz-ing the isothermal contours, around the cable to determine thequantities AT @+ and T~ of Equation (4). These isothermal con-tours are normally plotted for complex cable configurations usingfinite-element programs. The temperature gradient ATi/Ani can be

evaluated approximately as (Tc Tci)/Ani as shown in Figure Al,where T, is the temperature of an isothermal contour close to theduct bank surface. The values of Tci at various segments of theduct bank surface can be obtained approximately by using twoisothermal contours and interpolating (or extrapolating) ogarithmi-cally as depicted in Figure Al. The values of Tci are evaluatedfrom the relationship4 %Tci=T,+(T, T,) In - In -ai a,

where T, and T are the temperatures of the two isothermal con-tours, and a, an8 are approximate radii as shown in Figure A l .Note that the fom$a (Al) is based on the assumption that the twoisothermal contours, r and q, represent two coaxial semi cylinders.

r------------

Figure Al . Approximate calculation of G,-factor

APPENDIX IIComparison of Old and New Gb-Factors

Consider an approximate thermal model of the duct bank inwhich the duct bank surface is assumed to be formed of twoisothermal portions of different temperatures, namely, the top por-tion of temperature T,, and the bottom portion of temperature Tc2The thermal resistances R, and R2 between the top and bottom por-tions respectively, and the ambient are given byTcl - Ts Tc2 -TsRI =- ndRz=-

i Q2where Q1 =XQ and Q2= (1-X)Q represent the heat flux throughtop and bottom portions, respectively. Assuming that Tc2 =Tcl+ATc, then,

Tcl - Ts (Tcl - Ts)+AT,RI = and R, =XQ (1-X)Q


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1309and the to t a l t hermal res i s t ance R i s t he para l l el comb ina t ion o f R1and Rz, t ha t i s

w h e r eAT,

Tcl - Tsc p = __

Now , the "o ld" t hermal res i s t ance RO based on an i so thermal duc tb a n k s u r f a c e is g i v e n b y

therefore,R T,,-T, 1 + ~ p- = - . ~Ro T,- T, 1+hcp

N o t e t h a t T, i s t h e a s s u m e d i s o t h e r m a l t e m p e r a tu r e i n t h e cal-cu la t ion o f Ro and h is t he f rac t ion o f t he t o t a l hea t f l ux escap ingth rough the top po r t ion . S ince in p rac t ica l s i t ua t ions , T,, >T,, (orcp> 0.0) a n d 1.0>M.5, then, i f we a s s u m e t h a t T, corresponds t oT,, ( a s m o s t of t he hea t escapes from t he t op po r t ion of t h e d u c tbank surface c loses t t o the i so thermal ground surface), then

R l+cpRO I + h ~- >1.0

Therefo re , one would expec t t ha t fo r such cab le sys t ems , t henew to t a l t hermal res i s t ance is h i g h e r t h a n t h e o n e b a s e d on th ei so thermal as sumpt ion . T h a t i s , t h e n e w v a l u e s o f G b - f a c t o r aremore conserva t ive than the old (isothermal-based) values. On th eo t h e r h a n d , i f we assume t ha t T, cor responds to (T, +TCz)/2or th ea v e r a g e o f t o p and bo t tom t empera tu res , t henR l+cp-

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Modified Values for Geometric Factor of 00193924