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    12 IEEETransactions on Power Delivery, Vol. 11,No. 1 January 1996Refinements to the Neher-McGrath Modelfor Calculating the Ampacity of Underground Cables

    Sally M. Sellers W. Z. BlackThe G eorge W. W MSchool of Mech anical EngineeringGeorgia Institute of TechnologyAtlanta, Georgia

    Abstracf - The Neher-McGrath method has been widelyaccepted as an accurate and relatively simple way to calculatethe ampacity of underground cables. It is based on a numberof assumptions that simplify the mathematics, but at the sametime limit the accuracy and scope of the model. Furthermore,the model relies upon correlations that are now dated, becausemore up-to-date and accurate heat t ransfe r correlations arenow available.

    This paper examines improvements to the Neher-McGrathmodel in three different areas: a more accurate expression forthe mutual heating parameter that accounts for unequalheating among the cables; improved correlations for thethermal resistance of a fluid layer that exists in pipe-typecables and cables installed in ducts; and a more accuratemodel for the thermal resistance of concrete duct banks withnon-square cross-sections.

    Example installations are then considered to illustrate howthe incorporation of these changes will affect the ampacity ofthe installation. The refinements suggested resul t in a morecomplex mathematical algorithm and require morecomputational time, but the changes can result in significantlydifferent ampacity values than the ones provided by the Neher-McGrat h model.

    95 WM 015-8 PWRDby the IEEE Insulated Conductors Committee of t heIEEE Power Engineering Society f o r prese ntat ion a tthe 1995 IEEE/PES Winter Meeting, January 29, t oFebruary 2, 1995, New York, Np. Manuscript submitte dJul y 28, 1994; made avai lab le for pri nt in gNovember 23, 1994.

    A paper recommended and approved

    I. INTRODUCTIONThe first models proposed for calculating the currentcarrying capacity of undergroun d cables date back to the late

    1800's and early 1900's [l, 21. The early ampacity models,as they are now known, remained rather crude and highlysimplified until Neher and McGrath [3] published acomprehensive paper that, for the first time, considered allrelevant factors that influence the temperature of moderncable installations. Since its publication in 1957, the Neher-McG rath technique has been widely accepted as the s tandardfor determining the current-carrying capacity ofundergrou nd cable installations.The mathematical model outlined in the Neher-McGrathpaper is based on numerous assumptions, some of whichwere necessitated by the computing capabilities of the1950's. The heat transfer theory and correlations that areincluded in the model were the best available at that time.However, more ac curate results are available today and thesemore recent developments should be incorporated into anupgraded and imp roved version of an undergrou nd ampacitymodel. The goals of this paper are to objectively examineseveral improvements to the Neher-McGrath model, toincorporate recent heat transfer correlations, and to removeseveral assumptions considered in the N eher-McGrath paperthat limit th e accuracy of the calculations.While the suggested changes result in a more complexampacity model, they can be easily incorporated into themathematics without a significant increase in effort,particularly considering the computational capabilities oftoday's personal computers. The factors considered in thispaper will not only improve the accuracy of the existingampacity model, but will also extend the application of themodel so that it will apply to a larger foundation ofgeom etries and installation practices. In some cases thechanges to the model suggest an increase in the existingampacity values and in others, the changes will reduce theampacity.

    0885-8977/96/$05.CN 0 1995 IEEE

    http://0885-8977/96/$05.CNhttp://0885-8977/96/$05.CN
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    13The refinements suggested in this paper have beenincluded in an ampacity program called TOAD (TrenchOptimization and Design) that was written as part of anEPRI (Electric Power Research Institute) contract. TheTOAD program has a wide range of capabilities, includingthe ability to calculate the ampacity of cab les surrounde d bya backfill material. By varying the backfill resistivity andthe amount of the backfill used around the cables, anengineer can quickly optimize the installation so that itachieves a maximum am ount of power delivered for a fixedinstallation cost. The TOAD program is part of EPRI'sUnderground Tran smission W orkstation software package.

    11. MUTUAL HEATING FACTORThe first improvement to the existing ampacity modelinvolves the calculation of the mutual heating factor. Sincethe ampacity model assumes a one-dimensional heat transferproblem in which the temperature in the cable andsurrounding environment varies only with radial distancefrom the cable, two-dimensional effects that exist must behandled in an approximate fashion. Therefore when theinstallation consists of more than one cable and two-dimensional heat transfer occurs in the earth portion of thecircuit, the resistance of the earth must be modified toaccount for the temperature rise at one cable resulting fromheat generated in all other cables in the installation. Thiseffect, which is usually referred to as mutual heating, isincorporated into in the factor F, which occurs in theexpression for the earth resistanc e term ((44) in [3]).

    By utilizing the principle of superposition, and byrestricting short-circuited shield operation to three-conductor cables and three, single-conductor cables in aduct, therefore equal heat generation in each cable, Neher-McGrath show that the mutual heating factor is only afunction of the installation geometry. This result greatlysimplifies the model, because the mutual heating factor isknown on ce the cable geometry is specified. However, whenthe heat generated in all cables is not equal, the mutualheating parameter is also a function of the heat generationrates as well as the geometry. Therefore, when the heatgeneration in the cables vary, as it does when the circulatingcurrents in the metallic shield are unequal, the value for Fshould be mod ified appropriately .When the effect of unequal heat generation rates areincluded in the mutual heating parameter, the value, asderived in [4], ecomes

    where F, represents the mutual heating value for cable i .The symbols used in this expression are the same ones usedin [3] and they are defined in the nomenclature section.Equation (2) is identical to (46) in [3] when the heat transferrates in all cables are equal; that is when qi=qj. However,when the h eat transfer rates differ, the value for F calculatedfrom (2) will differ from the values proposed by Neher-McGrath.implementation of this new expression for the mutualheating factor obviously adds to the complexity of thethermal model. Since the value for F is not only a functionof geometry, but also a fu nction of the heat g enerated in eachcable, its value cannot be determined until the current ineach conductor and shield is known. Therefore a value forthe mutual heating factor can only be determined by usingand iterative scheme which assumes an approximate valuefor F and calculates an approximate value for the ampacityof each cable. Then the app roximate ampacities are used todetermine a new value for F and a corresponding new valuefor the cable ampacities. This iterative process continuesuntil the ampacity values converge.The change in the calculated ampacity value that resultsfrom the improved model for the mutual heating parameteris obviously only significant if the cables experience areasonable difference in heat generation rates. To assesshow the newly proposed mutual heatin g value influences theampacity of buried cables, seven different common cableinstallations are considered and are summarized in Table I.These arrangem ents were considered because expressions orthe circulating currents are known for these geometries [5].In each of the seven cases, the metallic shield is short-circuited so that currents induced in the cables are unequal.The example selected was a 1500 kcmil, aluminum, single-conductor, 138 kV cable, with a 1/12 neutral copper shield,and ope rating at a loss factor of 1.0. The cables were directburied with the shallowest one at a depth of 91.44 cm (36inches) in a soil with a resistivity of 90 "C cm/W. Themaximum operating temperature of the cables was 90 "C tan ambient temperature of 25 "C.Figure 1 plots the percent difference in the earth thermalresistance calculated with the mutua l heating factor, F, whenis does not account for unequal heating and when it isevaluated by (2) as a function of the ratio of cable spacing toouter diameter of the cable, dd/. Negative values in Fig. 1mean that the thermal resistance calculated by assumingequal heat generation in all of the cables is greater than thevalue assuming unequal generation. The differences inearth thermal resistance values are greater for small cablespacings and double, three - phase, opposed installations

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    14TABLE I.

    INSTALLATIONS CONSIDERED FOR MUTUALHEATING COMF'ARISON

    which are conditions that magnifv the difference incirculating currents.Figure 2 shows the percent difference in calculatedampacity for the same conditions used to plot the curves inFig. 1. These curves show that the percent differences inampacity are all positive, or in other words, the ampacityvalues predicted by the computer program using the Neher-McGrath model are less than the values that result when (2)is used to calculate the mutual heating parmeter in th eTOAD program . Therefore, for each case considered, theNeher-McGrath model provides a conservative value for th ecable ampacity and the new method for calculating theampacity suggests that th e current in all of the cables cari beincreased beyond the Neher-McGrath value withoutthermally overloading the cables.

    111. RESISTANCE OF FLUID LAYER FOR CABLESINDUCTS AND PIPESWhen cables are placed in d ucts or conduits, the thermalresistance of the fluid layer that surrounds he cables must bedetermined in order to calculate the ampacity. Neher-McGrath presents two expressions for the resistance of thislayer which are simplified forms of an earlier expressiondeveloped by Buller and Neher 161. These expressionsconsider all three modes of heat transfer; conduction,convection, and radiation, behveen the surface of the cableand the inside surface of the duct. The total resistance of th e

    medium can then be determined by add ing the resistances inparallel.

    Spacing/Outer Diameter of Ceble (s/d,)2.5 3 3.5 4 4.5

    0 : I

    2 15'i __c_ 3cables,righttriangle__E__t_ 3 hori zontal cables- vertical cablesFig. 1. Percent difference inearth thermal resistancedue to unequalheat generation.- cables, right triangle

    _c)_. horizontalcables--+- 2dgt3x2,opp.phasel2 T

    10

    8Ena 6Gi -4+- 2x3 &3x2, rev.phase- verticalcables

    2 48E

    A11

    2.5 3 3.5 4 4.5Spacing/Outer Diameter of Cable (ddj )

    Fig 2. Percent difference in ampacity due tounequal heat generation.

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    15-1 The constants A , B , and C are presented in Table VI1 in

    (3) [3] for a variety of installation conditions. Neher andMcGrath further assume a mean temperature, T,, of theintervening medium of 60 "C which reduces (6) to thefollowingform.

    ++I 1 1Reon, &d il

    The effective resistance of the air layer can be written interms of the heat transfer, q , and the temperature differencebetween the outside surface of the cable and the insidesurface of the duct, AT . (7)3 0 . 4 8 nn ' A'( 0 ; / 2 . 5 4 ) +B'R, =

    To simplify this expression and provide an expressionthat can be used to calculate the ampacity, severalassumptions were applied by Neher-McGrath. Whendeterminin g the rad iative heat transfer, the emissivity of theouter surface of the jacket and the inner surface of the ductare assumed to be equal. Also, the surface area of the insid eof the duct is considered to be much larger than the surfacearea of the cable. Buller and Neher [ 6 ] present the followingexpression (modified for metric units) for the thermalresistance of a gas or air layer between the cable surface andthe inner wall of the duct.

    0.0457Di3/4 AT'J4PIJ21.39+D;/d,, =30.48nn'

    1-1

    A similar expression was also presented in [ 6 ] for oil-filledpipes.Equation (5 ) is the basis of (41) in the Neher-McGrathpaper. However several additional assum ptions are used toexpress the equation in a more general form and therebysimplify the task of determining th e thermal resistan ce of thefluid layer. First, a value for AT is assumed so that iteratingfor a correct value for the temperature is unnecessary. Avalue of 20 "C is chosen for all installations except gas-filledpipe-type cables at 20 0 psi, where 10 "C is assumed [3].Second, the range of 0 : is restricted to cable diametersbetween 1 and 4 nches when the cables are installed inducts and between 3 and 5 inches for pipe-type cables. Thecomplexity of the equation is further reduced by usingseveral empirical relationships resulting in (4 1) in Neher-McGrath, which is (modified for metric units),

    The constants A' and B' are also listed in Table VI1 [3].According to this expression , the thermal resistance betweenthe cable surface and the inner wall of the duct is only afunction of the outer diameter of the cable, the number ofconductors within the duct, and th e thermal prop erties of theduct which are incorporated into the values for the constantsA' and B'. The assumptions used in the derivation of (7)greatly simplify its form. However they also reduce itsaccuracy and even can produce erroneous results. Forexample, (7) suggests that the resistance of the liqu id layer isindepend ent of the thickness of the fluid gap.To improve the accuracy of the expression for thethermal resistance, the Buller and Neher expression isrevised by eliminating the restrictive assumption s and us ingmore up-to-date convective correlations. The three m odes ofheat transfer are considered individually and (4 ) is used todetermine an expression for the total thermal resistancebetween the cable surface and the inner wall of the duct.The cable installation is modeled as concentric cylinders.Also, the outer surface of the cable and the inner surface ofthe duct are assumed to be isotherma l. These assumptionsare not precisely met in practical installations , especially fornon-metallic ducts that have a very high thermal resistivityand can be hot where they touch the cable and cool in otherregions. Metallic pipes have a lower thermal resistivity and

    therefore the temperatu re of the pipe is more uniform. Theinstallations are also assumed to be infinitely long andtherefore the heat from the cable is transferred only in theradial direction.The first mode of heat transfer considered is conductionacross the layer. The exp ression for thermal resistance dueto conduction between concentric cylinders is used in anunmodified form but it is converted so that it is expressed interms of metric units.

    AT

    Next considering the radiant heat transfer that occursacross the gap, the Stefan-Boltzmann equation [7] is used todetermine the h eat transfer between concen tric cylinders,(6)3 0 . 4 8 nn'A

    1 +( B +CT,) (D5/2 .54)R, =where T, is the m ean air temperature in the duct.

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    16

    v - kinematic viscosity(an%)

    4o [ ( T J +273) - Tuqrad =AT 1A T [ - +-

    (9)j A j A j F j - a EaAa

    0.1880 0.01303 0.8278

    where p represents the reflectivity of the surfaces.Simplifling the expression by assuming gray, difusesurfaces and also considering the possibility of multipleconductors within th e enclosed duct results in the followingexpression.

    1. 7 8 1 ~ 1 0 - ~ ~;[(T .c 2 7 3 4 -(Tu +273)4qrud -This equation is applicable for installations which involveair or gas layers. For oil-filled pipes, the effect of radiationis negligible.The third mode of heat transfer is convection in the fluidlayer. The thermal resistance of the air layer can beaccurately calculated by using a more recent co rrelation forthe convective heat transfer coefficient between concentriccylinders. Raithby and Hollands [SI have proposed acorrelation that is valid for concentric, isothermal, long,horizontal cylinders in which the diameter of the innercylinder is large compared to the width of the air gap.Written in terms of dimensionless groups, this expression is

    4conv - 2nkeffAT l n ( d a / D ; )

    wherepr

    0 . 861 +Prkea = 0 . 3 8 6 k [ ] (Rai)114

    gp( T~ - r , ) a 3 PrV 2

    Rag =

    6 = * ( d a - 06).Substituting these expressions into (1 1) for the air layer andsimplifying results in the following expression.

    4conv =2.4253[ P r_ _ _AT nn'p, 0.861+Pr

    The thermal properties that occur in (12) are assumed to beconstant over the small range of expected operatingtemperatures. Appropriate properties for air, gas, and oil arelisted in Table 11.To illustrate the difference in thermal resistance of thelayer between the cable and duct that results from usingNeher-McGrath's equation (41A) and th e value predicted bythe new expression, a typical cable installation is consideredand the thermal resistance values are plotted in Fig. 3 . Thisfigure assumes a 250 kcmil, copper, single-conductor, 35 kVcable with a 12, #14 AW G copper wire shield buried in a0.635 cm (0.25 inch) thick fiber duct, at a depth of 91.44 cm(36 inches), a soil thermal resistivity of 60 "C c d W , andoperatingwth a loss factor of 1 .0. The maximu m operatingtemperature of the cables was 90 "C, while the ambienttemperature was 25 "C. For this installation, both the ductand the cable surface are assumed to have an emissivity of0.95. The thermal resistance values are plotted a s a functionof the ratio of outer diameter of the cable to inner diameterof the duct, 4 da. The Neher-McGrath equation predicts athermal resistance that is independent of the thickness of theair layer that surrounds the cable, while the new expressionpredicts the more log ical result that the resistance increasesas the thickness of the air layer increases to a maximumresistivity at a ratio of dJ /da of about 0.35, for theinstallation considered. As the thickness of the air layerincreases further, motion of the air surrounding the cableincreases and these convective currents produce a decreasein the thermal resistan ce in the air layer as indicated in Fig.3.

    Table 11THERMA L PROPERTIES FOR THE INTERVENING MEDIA [6,7]

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    In addition to the dependence on the thickness of the airlayer, the thennal resistance is shown to increase as he ratioof / E ~ecreases, while the Neher-McGrath result isindependent of the emissivity ratio. Fig 4. shows the effectof SUrEdce emissivity on the thermal resistance. In manyinstallations, it is not reasonable to assume that theemissivities of both surfaces are equal. Th e emissivity ratio,/E& equal to 1.0 is typical of prop erties of a cable with apolyethylene jacke t in a fiber duct, while a ratio of 2.0 is areasonable ratio for cables in a metallic pipe. Curves areshown in Fig. 4 fo r a ratio of jacket to duct surfaceemissivity of 1 an d 2. As shown in the figure,removal ofthe restriction of equal emissivities can produce thermalresistivities that are higher than those previously calculated,thereby producing a positive percent difference in thethermal resistance.When these improved values for thermal resistances areused in the TOAD mpacity program, significantly differentampacity values can result. The ampacity valuescorrespon ding to the co nditions used i n Fig. 3are shown inFig.4. A program using the Neher-McGrath modelprovides conservative ampacity values over the entire range

    225

    BSbp 2 o s2

    175

    0.25 0.35 0.45 0.55 0.65 0.75 0.85Outer Diameterof Ca bl fi ne r DiameterofDuct

    (dj Id a1Fig. 3. ThermalResistance calculatedby N e h e r - M a t h

    erpres&n and newexpressionforthermal esistance.

    --

    --

    --

    17

    250 T

    1 5 0 4 : : : i : : : ; : : 1 -80.25 0.35 0.45 OS5 0.65 0.75 0.85Outer Diameter of CableAnner Diameterof Duct

    (djIda)

    Fig. 4. Ampacity o f installationcalculated by Neher-McGrathmodel and model using new expression forthennal esistance of fluid layer.

    of cable and duct diam eters, for E ~ / E ~qual o 1.0. For smallair gaps, the percent difference in ampacity between theNeher-McGrath model and the new model can be as arge as11 percent. Therefo re, for this particular cable andinstallation geometry, the ampacity of the circuit aspredicted by the new model is over 11percent greater thanthe value predicted by the Neher-McGrath model when thecable diameter is approximately 85 percent of the ductdiameter.For the same ins tallatio n, but with emissivity ratios,/E& that are not 1.0, the ampacity predicted by Neher-McGrath i s no longer conservative, as also shown in Fig. 4.For an emissivity ratio of 2.0, th e ampacity based on theNeher-McGrath model is higher than the currentcarrying

    capacity of the ins tab tio n as predicted by the new model forcable diameters that are less thanapproximately 0.85 timesthe diameter of the duct. Neglecting the effects of unequalSurface emissivities could result in higher cabletemperatures, therefore shortening the life of the cable.

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    18TV. RESISTANCE OF RECTANGULAR DU CT BANKS

    The final modification to the ampacity model involvesthe calculation of the thermal resistance of a rectangularregion that surrounds the cable. This material can beassumed to be a concrete duct bank or b ac fil l material usedto reduce the thermal resistance of the region adjacent to thecables. For a rectangular region with a shorter dimension ofX b , a longer dimension of yb, and a therm al resistivity of pdb,Neher-McGrath suggest the following expression ((44A) in[3]) to calculate the effective thermal resistance of therectangular region an d soil (modified for metric units).

    +O. 366(p, - p , ) n n f N ( L F ) G ~This expression was derived by replacing the actualrectangular region with a circular region which possesses an

    equivalent thermal resistance. This assumption allows thetwo-dimensional heat transfer problem to be reduced to anapproximate one-dimensional analysis. The radius of theequivalent duct bank can be expressed as

    The geometric factor, G b, in (13) is determined from usingthe Kennelly formula [3].

    The surface of the duct bank is assumed to betransformed into an isotherm of radius Y b and the heat flowthrough the duct bank is assumed to be independent ofangular direction. These assumptions are not particularlyaccurate, because the lower surfaces of the duct bank arehotter than the upper surfaces. Also, more heat flows towardthe surface of the earth, because the thermal resistance of theearth layer over the cables is significantly less than theresistance of the earth un der the cables.The errors introduced by transforming a rectangularregion into one which has a circular cross-section are lesswhen the duct bank or thermal backfill layer are nearlysquare (xb=yb). However when the region is very slender,the results predicted by (13)-(15) are no longer accurate. Inorder to improve on the accuracy of these equations forwidth-to-he ight ratios not close to 1.0, a conformal mapping

    technique was used [9] to predict the value for the thermalresistance of the rectangular region.The conformal mapping technique provides aconduction shape factor, S, and an effective resistivity value,that differs for each cable in the installation. The2iCctive thermal resistance, R not accounting for the

    loss factor canbe expressed as

    Once the value for R,,,,,, is determined , it can be written ina form similar to (13), where the loss factor is 1.0. Thisexpression can then be combined with the Neher-McGrathequation, (13), to arrive at the following expression for theeffective earth resistance, Re , that is valid for any los s factor.

    By comparing the thermal resistances provided by (17)and the corresponding equations suggested by Neher-McGrath, (13)-(15), it is possible to assess the errors in theold method of calculating the ampacity when cables arerouted through rectangular duct banks. The thermalresistances shown in Fig. 5 illustrate the differences in thetwo models for the case of three, horizontally spaced 1500kcmil, single-conductor, 138 kV, aluminum cables with anopen circuited, copper shield that are buried in a duct bankwith a resistivity of 80 "C c d W , and operating at a lossfactor of 1.0. The cables are placed in 1 5.24 cm (6.0 in)ducts at a depth of 91.44 cm (36 inches) below the surface ofthe earth and separated by a distance of 30.48 cm (12.0inches). The operating temperature of the cables was 90 "Cand the qnb ient earth temperature was assumed to be 25 "C.The curves provide results for a range of soil thermalresistivities and different values for x/y.The results in Fig. 5 show that the percentage differencein the effective thermal resistivity of the soiVduct bank layeris small as long as the duct bank is nearly square in cross-section. However, when the duct bank is slender, (x/y)

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    190.2 0.4 0.6 0.8 1 1.2 1.4

    2 i : ; : i ; ; : ' ; i : ; l

    083-2 -2ad

    -12l o 1Fig. 5 . Percent difference in thermal resistance for variousduct bank heights wtha width of 91.44 cm (36 inches).-- e'P&=l--c-- Pe/pdb=2- e l p = 35 db.s 4xF4.e 3

    Pg$ 2c)Q& '

    0

    1 \ - ,IPdb = 4

    - 1 4 I ; i i i 1 i i~0.2 0.4 0.6 0.8 1 1.2 1.4

    Fig. 6. Percent difference in ampacity for various duct bankheights wth a vridth of 91.44 cm (36 inches).

    V. CONCLUSIONSThree refinem ents to the Neh er-McG rath ampacity modelare proposed in this paper. The first involves the mutualheating parameter and it removes the assumption that allcables must generate the same amou nt of heat. This

    improvement therefore increases the accuracy of ampacitycalculations when the circulating currents in the metallicshield produce unequal heating among the cables. Forseveral typical cable installations, this refinement to theNeher-McGrath model suggests that the ampacity value canbe increased by up to 7 ercent.The second refinement involves the correlation used tocalculate the thermal resistance of the air or oil layer thatsurround s pipe-type cables or cables in ducts. By replacingold heat transfer correlations by more up-to-date ones an d byexpanding the model to cases not considered by Neher-McGrath, more accurate and more realistic ampacity valuescan be calculated. Applying the new model to typicalinstallations has shown that the ampacity can be increasedup to 11percent in some cases and mu st be reduced by up to5 percent in others. The case for which the new modelrecommends a reduction in the ampacity is of particularimportance, because it suggests that the Neher-McGrathmodel does not provide a conservative ampacity, but onewhich could lead to an unexpected high cable temperature.The final refinement considers the calculation of thethermal resistance of a rectangular duct bank. A proposedexpression for the thermal resistance is based on aconduction shape factor that can be calculated from aconformal mapp ing technique. Unlike the Neher-McGrathtreatment of the type of installation, the new expression isnot limited to duct banks that have a nearly square cross-section. For typical cables installed in rectangular ductbanks, the revised expression for thermal resistance predictsampacities that are up to about 5 percent greater than, orabout 1percent less than, the Neher-McG rath values.

    VI. ACKNOWLEDGMENTSThe work reported in this paper was supported by anEPRI contract (Project 7913-05), Thomas Kendrew andThom as Rodenbaugh project directors.

    VII. NOMENCLATUREA - surface area (cm2)A, B, C, A ', B' - constants for calculation of thermalDIS- effective diameter of multiple cables in contact (cm)0, - diameter at which loss factor becomes significant (cm)d - outer diameter (cm)

    resistance of air layer

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    20dv distance from center of cable i to center of cablej (cm)dvt distance from center of cable i to center of mino r imageF - mutual heating factor (dimensionless )FI-] shape factor from surface i to surfacej (dimensionless)G, - geome tric factor (dimensionless )g - gravitational constant (980 cm/s2)k - therm al conductivity of the ea rth (W/OC cm)LF - loss factor (dimension less)L - depth of shallowest cable below s urface of the earth (cm)N - total number of cables in arrangementn - number of cables per ductn - number of condu ctors per cableP - pressure (atm)Pr - Prandtl number (dimensionless)qr - heat transfer per unit length fo r cable i (W/cm)R - thermal resistance ("C cm/W)Rac* - modified Rayleigh number for concentric cylindersRa, - Rayleigh number based on thickness of fluid layerrb - effective radius of duct bank (cm)S - shape factor for a particular cable (dimensionless)s - spacing between adjacent cables (cm)7' - temperature ("C)x - height of duct bank (cm)x b - length of short side of duct bank (cm)y - width of duct bank (cm)Y b - length of long side of duct bank (cm)Greek Symbolsp - thermal coefficient of expansion ( I P C )S - thickness of fluid layer (cm)E - thermal emissivity (dimensionless)v - kinematic viscosity (cm2/s)p -thermal resistivity ("C cm/W)CJ - Stefan-Boltzniann constant (5.67 x

    of cable j (cm)

    (dimensionless)(dimensionless)

    W/m2K4)Subscriptsa - ai rcond - conductionconv - convectiond - duct or pipedb - duct bank or backfille - earthe f f - effectivej -jacketm - meanrad - radiationw/o-LF - without loss factor

    VIII. REFERENCESKemelly, A. E., "Current Capacity of Electrical CablesSubmerged, Buried, or Suspended in Air," Electric World,Vol. 22, pp. 183,201, 1893.Rosch, S. J , "The Current-Canyiag Capacity of Rubber-Insulated Conductors," AIEE Transactions, Vol. 57, pp. 155-167, March, 1938.Neher J. H. and M. H. McGrath, "The Calculation of theTemperature Rise and Load Capability of Cable Systems,"MEE Transactions,Vol 76, pt. Et, p 752-772, Oct., 1957.Sellers, Sally M., "Calculation of AmpElectrid Cables," M.S. Thesis, TheSchool of Mechanical Engineering, Georgia Institute ofTechnology, Atlanta, GA, February, 1994.Simmons, D. M., "Calculation of the Electrical Problems ofUndergound Cables, Induced Sheath Voltages and Currents,Temperature %se on Brief Overloads, Dielectric Stresses,"The ElectricJournal, Vol. 29, pp. 4764 77, October, 1932.Buller, F. H. and J. H. Neher, "The Thermal ResistanceBetween Cables and a Surrounding Pipe or Duct Wall," AZEETransactions,Vol 69, pt. I, pp. 342-349, 1950.Incropem F. P. and D P. DeWitt, Fundamentals of HeatTransfer, John Wiley& Sons, Inc., 1990.Raithby, 6 . D. and K G T. Hollands, "A General Method ofObtaining Approximate Solutions to Laminar and TurbulentFree Convection Problems," in T. F. Irvine and J. P Hartnett,Eds. ,Advances in Heat Transfer, Vol. 11, Academic Press,New York, pp 265-315, 1975.Wood, Sandra Jean, "Determination of Effective ThermalConductivity of Media Surrounding UndergroundTransmission Cables," M.S. Thesis, The George W. WoodruffSchool of Mechanical Engineering, Georgia Institute ofTechnology, Atlanta, GA, December, 1993.

    [lo] Saieeby, K. E., W. 2. Black, and J. G. Hartley, "EffectiveThermal Resistivity for Power Cables Buried in ThermalBackfill," IEEE Transactions on Power Apparatus andsystems, Vol. PAS-98, pp. 2201-2214, Nov./Dec., 1979.

    lX.BIOGRAPHYS a y M. Sellers received her B.S. from North Carolina State University andher M.S. from Georgia Institute of Technology, both in MechanicalEngineering. She is currently working on her Ph.D. in M.E. at Georgia Tech.W. Z lack received h s B.S. and M.S. in M echanical Engineering from th eUniversity of Illinois and his Pb D . in M.E. from Purdue Universlty He iscurrently Regents Professor and Georgia Power Distinguished Professor ofMechanical Engineering at Georgia Institute of Technology in Atlanta. Hisresearch effort is concentrated in the area of heat transfer from electrical andelectronic equipment. He has been Principal Investigator for several EPRIampacity projects and he is active in IEEE committee work relating to thethermal ratings ofoverhead and underground conductors.

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    21Discussion

    L.H.Fink (ECC,nc., Fairfax, Virginia). The discusser hadthe privilege of sharing an office and working closely withJ.H.Neher during the decade of the 1950s when the Neher-McGrath paper was being written. Neher was painstaking,conscientious, and rigorous in his work, never satisfied withan interim or approximate solution. Accordingly, he neverconsidered "Neher-McGrath" to be the last word, and wouldhave been delighted to know that further work and refine-ments are still being pursued.Having said this, this discusser has serious reservations aboutthe conclusions reached by the authors of the present paper.Neher recogpized the needs, not only to ground his work onthe best available scientific understanding of the phenomenawith which he was dealing, and not only also to makesimplifying assumptions necessary to provide practicable solu-tion techniques, but also, and importantly, to validate the lat-ter by adjusting them to fit laboratory and field test data.This may be seen in his earlier papers on which the Neher-McGrath paper was based, e.g. the authors' reference 6. Theunderlying theory and accompanying linearizations provideconfidence that the relationships may be used for interpola-tion within a range of validity that has been established. Im-proved theoretical relationships may provide a basis of in-creasing the range within which interpolation is valid, butsuch extensions should not be adopted without subjectingthem to validation by test data. The Neher-McGrath resultswill remain valid within their range of validity just becausethey were so validated. The authors' assertion in SectionIDthat some of the Neher-McGrath assumptions "reduce ac-curacy and even can produce erroneous results" can be soonly if the Neher-McGrath equations are used outside thelimits within which they were validated.

    In Section IV , the authors undertake to correct the Neher-McGrath factor Gb for duct banks. Neher was well aware ofthe limited validity of that factor as given in the paper. In1958, Fink and Smerke [A] published an analysis of the ef-fect of (backfilled) trench dimensions on the effective earththermal resistance around pipe type cables, using conformalmapping. In an appendix, this analysis was extended (atNeher's request) to duct configurations, and was later usedinternally to correct Gb for various duct bank configurations.The authors should provide the conformal mapping relation-ships that were used in obtaining their results, in order toenable independent assessment.REFERENCEA L.H.Fink, J.J.Smerke: Control of the Thermal En-

    vironment of Cable Systems - Part II; Truns. AZEEPart IJI, PAS, Vol. 77,1958, pp.161-68.

    Manuscript received February 8, 1995.

    George J. Anders. OntarioHydro. Most utilities in the USA used the Neher-McGrath (NM) aper as a basis for computation of am pacities of electric powercables. The paper has been published in 1957 and is based on the analytical andexperimental work performed in the preceding 30 years. As the authors of thediscussed paper rightly pointed out, the NM paper is based on number ofsimplification s and uses several correlations which are now dated. T he authorsof the discussed paper are to be congratulated on attempting to update some ofthe compu tational procedure s used in North Am erican standards which a rebased on the NM paper.The purpose of this discussion is to review the contents of the paper in thecontext of published and stand ardized information and to supply additionalinformation on the topics addressed in the paper. The follow ing is a discussionof the three contributions ntroduced.by the authors.

    Just one reservation will be mentioned. In Section III of thepaper, the authors (as did Neher-McGrath) ground theiranalysis in concentric cylinder geometry. It is evident,however, that this is rarely if ever true in practice: individualcables in ducts will be resting on the bottom of the interiorduct wall; individual phase cables in pipe will either be(randomly) cradled or triangular at low loads, or birdcaged athigh loads. Whether we are dealing with convection andradiation (for cables in duct), or with convection and conduc-tion (for high-pressure Oil pipe-type cable, calculations based

    Mutual Heating FactorT he computaaon of the external thermal resistance discussed under the headingof Mutual Heating Factor is one of the topics dealt with in an internationalstandard. The analysis of unequally loaded cables leading to Equation (2) in thepaper 1s described in the E C Standard 287 (1982) and the approach ~roposdthere is identical to the one proposed to the authors. Tosee that, we first observethat for several loaded cables placed underground, we must deal withsupemposed heat fields. The principle of superposition IS applicable if weassume that each cable acts as a line source and does not distort the heat fielddu e to the other cables. Therefore, in the NM and the discussed paper, it isassumed that the cables are spaced sufficiently apart so that this assumption isapproxim ately valid. The axial seDaration of the cables should be at least twoon concentric configuration must be treated with the greatest

    caution. In either case, the effective distance between thecable surface and the duct or pipe wall will not vary directlywith the respective diameters. Caveat: It cannot be unfair toecho the authors' own words: such assumptions, ifthe resultsare not validated against test data, can 'reduce accuracy andeven can produce erroneous results.''In short, until the results of this paper have been validatedand, if necessary, corrected by reference to appropriatelaboratory or field test data, they should be considered only 'as hypotheses.

    cable diameters. The case when &e superposition principle is not applicable isalso dealt with in IEC Publication287 (1982)..The method suggested in the IEC Publication 287 for the calculation of ratingsof a group of cables set apart is to calculate the temperature rise at the surface ofthe cable under consideration by the other cables of the group, and to subtractthis rise from the value of AT used in the equation for the rated current. Anestimate of the power dissipated per unit length of each cable must be madebeforehand, and this can be subsequently amended as a result of he calculationwhere this becomes necessary, which also applies to equation (2) of thediscussed paper.Section 9.3.1 of IEC 287gives the following equation for the temperature r ise atthe surface of the cable p produced by the power qk watt per unit lengthdissipated in cable k (a unity load factor is considered below, but the reasoning

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    22applies also in the case of the nonunity load factor):

    The distan ces of d, , and d, , are measured from the centre of the pI h cable to thecentre of cable k. and the centre of the reflecuon of cable k in the ground-msurface, respecuvely. p, is the thermal resishvity of the soil Equauon D1 isa l l y developed based on the Kemelly's theorem. All the vanables areexpressed in metric unitsThus, the temperature nse AT, above ambient at the surface of the p th cable,whose ratlng is being d et em ne d, caused by the pow er dissipated by tbe other(q-1) cables in the group, is given by:AT, =AI; ,+AT, ,+ ...+A T p + ..+A$,

    temperatures, and the second is to wnte full set of energy balance equations.The last approach is more accurate and is formulated for a general case of cablesenclosed in walls in the paper presented at the Same Winter Meetlng as thediscussed paper (Anders, 1995). Since the authors of the paper face the sameproblem of defining the value of T o, it will be appreciated if they could revealbow they defme this temperature.Thesolutlon of equatlons D4 nd D 5 yields the conductor rated current I and thejacket temperatureWith regards to the value of T , , on e more point may be of interest Thetemperature T, appearing in equations 9 and 10 should actually be thetemperature of the inner wall of the duct or pipe while the value in equatlon 12is actually the temperature of the a r. Could the authors comment on this?Conunuing with the problem of cables in ducts or pipes, equatlon 9 in the paperonginates from the following standard defmitlon of the radiahve heat transfer:

    (assuming that T , is defined in terms of T, ).

    with the term ATppexcluded from the summauon. grad = F ~ , d G ( T J 4 - (D6)The value of AT in equatlon 5 in the NM paper for the rated current IS thenreduced by the amou nt of ATP and the rating of pt hcable is determined using avalue of Re corresponding to an isolatedcable at position p . Tius calculmon 1sperformed for all cables in the group and is repeated where neceSSary to avoidpossibility of overheaung any cable.Substituting equation D1 into the right-hand-side of equation D2 an d taking thedefinitlon of tbe extemal thermal resistance of a cable (the ratio of the cablesurface temperature nse to the total cabIe losses), the fdlowing generalexpression for the extemal fhermal resistance of cablep isobtained:

    where:=Bolrvnann

    T i =the average temperature of the wall inner surface, K, (astensk denotesabsolute temperatures)F,,d =thermal radiation shape factor, its value. depends on the ge ometry ofhesystem,A, =is the area of the cable surface effectwe for heat radiation, (d);orunit length of the cable.

    equal to 5.667x10-8 w,(mZ.K4);

    where u = 4 L l d e .

    Two points are worth mentioning in the context of the discussed paper: (1) b evalue of he radlation shape factor, and (2) the value of thevariable A, .The radiation shape factor is obtained considering two long mncentnccylinders. In the caseof a single cable in the riser, we have:

    It is evident that the second term in the summa tion in equation D3 is equivalentto equauon 2 in the paper. Moreover, it is more practlcal in computerapplicatlon to use duectly equatlon D2 as described above than to use equationD3 . A =area per unit length

    5dw (I+c,E , + A p d A ~ E ~ ) - 'where:

    (D7)

    d =the reflectivity of the cable outside surface;= he emissivity of the cable outside surface?esistanceof Fluid Layer for Cables in Ducts and PipesThe second refinement introduced in the paper deals with the calculation of theresistance of fluid layer for cables in ducts and pipes. In the opinion of thisdiscusser, the updating of the computation of the thermal resistance of air forcables in ducts is an im portant contribution of the paper. However, one shouldbear in mind tha t the method for the computation of the thermal resistance of thefluid in a duct or pipe introduced by Neher and M cGrath and adopted in the LECStandard 287 is very useful precisely because it is simple and does not dep end onthe unknown cable surface temperature. If the method proposed by the authorsis adopted, than there is no need to compute the thermal resistance of the fluid,but it is sufficient to solve the energy balance equa tion 4 in the paper which canbe stated as (the notation is the Same as n the NM paper):

    ci,,E~

    It is evident that equation D7 is equivalent to the fomu lauon in equatlon 10 ofthe paper where a group of cables is replaced by an effechve diameter d, of asingle cable and gray, diffuse surfaces are assumed. However, it is msufficien tto just use an e f f a v e d ia me te r in equation D7 when several cables are ininstalled in one duct. For the case of several cables in a duct, equatlon D7 takesthe form:

    = he reflec tivity of rbe wall inner surface:=the emissivity of the wall inne r surface;

    The conductive, radiauve and conv ectlve heat tran sfer rates are obmned fromequatlons 8, 10 an d 12 in the paper, respechvely. Equatlon D4 ha s threetemperature of the fluid in the duct. The second equatlon relates the current inthe conductor with the jacket temperature:T, -T ] = I z ~R , , + Y , + Y , + Y , ) - R , + A T , )where AT , is the cable surface temperatureR, s the equivalent intem d thermal resistance of the cable.As for the third unknown vanable, there are two ways to approach the problem.One, is to assume that is equal to the mean value of jacket and ambient

    For the cdse of threecables in a duct, we have:

    5 = 6rul E 0 (D9)unknown parame ters: the conductor current, temperature of the cable jacket and 3

    1 - L A - KAd E d

    Thesecond point mnc erns computaaon of A, When several cables are locatedin one duct, the m & a l radian area between them must be subtracted from thearea radiatlng to the duct inner surface. The most common installaoons haveeither one or three cables inside the guard. For example, for three cables intouching trefoil formation, we have (Wee dy, 1988):A, =%de -3.0.618de

    (D5)( (due to dlelectnc h s e s and

    (D10)

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    23Resistanceof Rectangular Duct BanksThe third topic introduced in the paper deals with computation of the externalthermal resistance of cables in duct banks or backfills. Th e NM method assumesthat the duct bank surface s isothermal and as pointed out in the Nh4 paper andin IEC 287, it is valid for the ratio of duct bank dim ensions in the range of 1/3 to3. In the studies performed by the authors, the results of the method presentedin the paper are com pared with the NM results for this ratio equal to 0.2.The approach proposed by the authors is not well explained and the crucialinformation how to obtain the conduction shape factor S and the effectiveresistivity value ps is withheld. In this context, the following comm ents are inorder. There are several drawbacks to the conformal transformation m ethod usedby the authors (see reference CIGRE, 1985 for the full description of thistechnique). The major one is that the equations describing the transformednetwork are equivalent to finite difference equations obtained by discretizing theheat equation in the transformed plane and, hence, the complexity of anumeric al solution of the heat conduction problem is not avoided. Anothe rdrawback is that both the earth and cable surfaces are assumed to be isothermal.In addiuon, transformation point by point of the boundaries between regionswith different resistivities is very laborious and the resulting computer softwarecannot handle effic iently more than four cables in one in stallation. All theselimitations can easily be overcom e by the applica tion of the finite eleme ntmethod. The nonuniform soil conditions and n o n i so t h e d boundaries arehandled naturally by this method. The com putational efficiencyof this approachis also quite satisfying. With the presently available personal compu ters,calculations involving networks with several thousand nodes can be performedin a matter of minutes.' The finite element method is described in (A nders andCoates, 1995).The limitation of the NM method for the calculation of the extemal thermalresistance of cables in duct banks or backfills has been recognized by severalauthors. El-Kady and H orrocks (1985) used the finite element method to developthe values of the geometric factor for thermal envelopes with ratios beyond therange specified above. Th en approach ha s an advantage of providing a table ofgeom etric factors for a very wide range of cable installations. Thi s table is useddirectly in the cable ampacity program written by the discusser (Anders et al.,1991) and the results are very satisfactory. In later works of El-Kady and others(El-Kad y et al., 1988, Tarasiew icz et al., 1987) the basic method of Neher andMcGrath was extended to remove the assumption that the external perimeter ofthe rectangle is isothermal.

    REFERENCESAnde rs, G.J. Moshref, A., and R oiz, J., (1991) "Advanced Computer Programsfor Power Cable Ampacity Calculations",IEEE Computer Applicationsin Power, Vo1.3, No.3, July 1990, pp.42-45.Anders, G.J.1995) "Ratmg of Cables on Riser Poles, In Trays, in Tunnels andShafts -- A Review", to be published m IEEE Tra ns. on Power Delivery.Anders, G.J. and Coates, M., (1995) Rating of Electric Pow er Cables. Am pacifyCalculations of Transmission, Distribution and Industrial Cables", tobe published by IEEE Press.CIGRE (1985), "The Calculation of the Effective ThermalResistance of CablesLaid in Materials Havmg Different Thermal Resistivities", Electra,NO.98, pp. 19- 42 .El-Kady, M.A., Hom cks, D.J., (19 83, "Extended Values of Geometric Factorof Extemal Thermal Resistance of Cables in Duct Banks", EEE Trans.on Power Apparatus and Systems, Vol. PAS-104, pp. 1958 - 1962.El-K ady , An ders, G.J., M.A., H o m k s , D.J., Mot lis, J., (1988) "ModifiedValues for Geometric Factor of External Th ermal Resistance of C ablesin Ducts", IEEE Tran sactions on Power Delivery, Vol. 3, No.4, Ocmber

    1988, pp 1307-1309.IEC Standard 287 (1982). "Calculation of the Con tinuous Current Ratin g ofCables (100% load factor)", 2nd edition..' The discusser performed computations on the network of 7000 nodes in lessthan 5 minutes using 66 MHz, 486 PC.

    Tarasiew icz, E., El-Kady, M.A., Anders, G.J., (1987) "Gene ralized Coefficien tsof External Thermal Resistance for Ampacity Evaluation ofUnderground Multiple Cable Systems", IEEE Transactions on PowerDelivery, Vol. PWRD-2, NO. 1, January 1987, pp.15-20.Weedy, B.M., (1988) "ThermalDesign of Underground Systems", John Wileyand S ons, Chichester, U.K.Manuscript received February 13, 1995.

    Vincent Morgan (CSIRO Division of Applied Physics, Sydney, Australia): It is appropiatethat the often-used heat transfer correla tions for cables, given by Neher and McG rath [3]nearly forty years ago, should be re-assessed in the light of recent research. However, thetransition from a set of mixed units to metric units runs the risk of generating errors. Onesuch error appears to have occurred in (3,hich is converted from (19) in reference [6].The coefficient in the numerator of the first term within the square brackets on the right-hand side should read 0.092 / (2.54)3'4=0.0457, instead of 0.092 / (2.54)=0.0362.The correlation (11) given by Raithby and Hollands 181 for the natural convective heattransfer from isothermal concentric horizontal cylinders agrees closely with the numericalresults obtained by Farouk and Guceri [A]. The range of the Rayleigh number in (11) islb - 10'. The type of flow in the annulus depends on the ratio of the inner diameter to thethickness of the gap and the Grasbof number [B].It is stated that Fig. 3 (not Fig. 4) shows that the calculated thermal resistance to theannular air layer increases as the ratio E, I E, decreases. The labeling of the curves indicatesthat this is true for the Neher-McGrath results, but not for the authors' results.The authors have noted that the surface of the duct bank is not isothermal, but they haveassume d the, surface to be transformed into a circular isotherm. Other researchers [Cl haveassumed the more realistic condition of uniform flux, and have used a numerical method toderive modified geometric factors for various configurations of the duct bank.Neher and&cGrath [3] mentioned the effect of solar heating at the surface of the soil, butthey did not elaborate on this, instead preferring to take the ambient temperature at thedepth of the cable. Would the authors care to comment?REFERENCES[A ] B. Farouk and S.I. Guceri, "Laminar and Turbulent Convection in the Annulusbetween Horizontal Concentric Cylinders", 3. Heat Tranfer, Vol. 104, pp. 631-636,November 1982.R.E. Power, C.T. Carley and E.H. Bishop, "Free Convection Flow Patterns inCylindrical Annuli", J. Heat Transfer, Vol. 91, pp. 310-314, August 1969.M.A. El-Kady, G.A. Anders, I. M o t h a n d D.J. Horrocks, "Modified Values fo rGeometric Factor of Extemal Thermal Resistance of Cables in Duct Banks", IEEE

    Trans. on Power Delivery, Vol. 3, pp. 1303-1309, October 1988.

    [B][C]

    Manuscript received March 2, 1995.

    A. Ernst and D. W. Purnhagen (Underground Systems,Inc., Armonk, NY): Our comments on this paper can besummarized as follows:

    1)Equation 2 for the mutual heating factor F in thispaper is mathematically correct and calculations using thisequation d o agree with calculations based on the Neherand McGrath paper for cables with unequal losses. Theauthors have mis-applied equation 44 of the Neher andMcGrath paper which leads them to the incorrect conclu-sion that a discrepancy exists between their equation andthe Neher and McGrath method.

    2) The Neher and McGrath equations for thermal resis-tance from cable surface to duct or pipe wall were derivedby fitting equations to data. An y alternative equationshould fit the same data but the authors offer no experi-mental substantiation for their formulation. We have re-viewed available data and reach the following conclusions:

    The equations in this paper do not agree with available

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    24data for cables installed in conduits where air is theintervening medium or for high pressure gas filled pipetype cables.Use of these equations for cable in conduit or highpressure gas installations can lead to substantial errorsin cable ampacity or temperature computations.However the equations do agree with the data for pipetype cables where dielectric liquid is the interveningmedium.In the first part of this paper the authors state that the

    Neher and McGrath paper is incorrect for cases wherethe cables have unequal losses such as occurs when singleconductor cables are installed in separate ducts and circu-lating currents flow in the metallic shields or sheaths. Thisstatement is not correct.Equation 44 of the Neher and McGrath paper (equa-tion l of the Black and Sellers paper) applies only for thecase of a single cable or a group of cables with equallosses. Apparently the authors have used this equation tocompute what they refer to as the Neher-McGrathcomputations in making the comparisons of Figures 1 and2 of their paper.

    For the case of unequal losses equations lA, 9A, 48, 49and 50 of the Neher and McGrath paper apply. Eachcable or group of cables with equal losses is consideredseparately. The mutual heating due to the remainingcables is added to the earth temperature at the location ofthe cable being rated (or subtracted from the allowabletemperature rise to compute ampacity). When cables havedifferent losses an iterative or simultaneous solution withan equation for each cable may be required. However ifthe ratio of losses is known a single equation can bederived from the Neher and McGrath equations. In thiscase the earth thermal resistance is separated into selfand mutual heating terms. Each term is multiplied bythe appropriate losses. By combining these terms theformulation of Equations 1 and 2 of the paper can bederived. The process is repeated for each cable to deter-mine which is the hottest. The ampacity should be thesame as derived from the Neher and McGrath equationsenumerated above although a small difference will occurif the temperatures of all cables are assumed to be thesame.In applying the authors equations 1 and 2 it should berealized that the earth thermal resistance IS not actually afunction of heat. Although the equivalent mutual heatingfactor of equation 2 is a function of heat flows the earththermal resistance is assumed constant. The temperaturerise produced by each heat source is directly proportionalto the heat generation. This is significant since in practicethe earth thermal resistivity may be variable with heatflow but the equations are not meant to account for thisphenomenon.

    The next topic in this paper is a a so-called improvedmethod for calculating Rsd, the thermal resistance fromcable to duct or pipe.

    The Neher &McGrath paper describes in some detailhow data from heat transfer tests on cables installed inducts or steel pipes were used to compute the constants inEquations 41, 53 and 54. In these tests (references [2] and[3]) the cables were resting as they do in an actualinstallation in contact with the conduits. The tests coveredvarious duct materials having different conductivity and arange of cable sizes and duct diameters were covered. Thefill ratios ranged from about 0.2 to 0.9. (Higher fill ratiosare impracticaI since cables must be pulled into ducts anda minimum clearance or % fill is normally recommended.)

    The test results clearly showed that where gas in theintervening medium the conductivity of the duct materialstrongly influences the heat transfer as does the diameterof the cable. The alternate equations (8-12) in this paperdo not fit available data for cables installed in ducts orpipes where air or compressed nitrogen is the interveningmedium. Errors as large as -50% in Rsd and +12.5% inampacity or +25% in temperature rise of conductor overambient will occur using these equations for non metallicducts. Errors on the conservative side of as much as +70to +100% in Rsd occur for cables in steel pipe with air orhigh pressure nitrogen as the medium.TabZe A-1 summarizes the results of tests for cables infiber or Transite ducts and Tuble A-2 for cables in steelpipes and compares the measured thermal resistance withthat computed by Equations 41 and 53 of Neher andMcGrath and equations 8-12 of the Black and Sellerspaper.

    For the case of 3 cables in a pipe we have tabulated thetotal loss and computed Rsd for an equivalent singleconductor cable so that the thermal resistance has unitsof degree C per total watts. Equations 8, 10 and 12 of thepaper for thermal conductance have the factor nn (num-ber of conductors) in the numerator rather than thedenominator but in this case nn =1 so the effect of thiserror is nil. Application of equations as shown results inan additional error of l/n2 in Rsd for multiple conduc-tors in a duct (1/9 for 3 conductors per duct).As can be seen from the table the Neher and McGrath

    equations fit the data quite well (within 20% for equation41 and within 15% for equation 53).The Black and Sellers method results in substantialdeviations from the data (typically -20% to -50% fornon-metallic ducts and +50% to +70% for cables in steelpipes).The above discrepancies result at least in part sinceequations 8-12 are based on correlations for concentriccylinders rather than for cylinders which are thermally incontact.TableB compares thermal resistance derived from tests

    on high pressure pipe cables with various dielectric liq-uids. These are compared with Equations 41 and 54 ofNeher and McGrath and also with correlations derivedfrom tests at the EPRI Waltz Mill test site conducted byUnderground Systems, Inc. The Black and Sellers equa-tions (with the nn factor moved to the denominator) wereevaluated for the same data and are also tabulated. HOW-

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    ever, the radiation term which is does not apply, wasommitted.Thc Neher and McGrath equations agree with the datafor high viscosity mineral oil since this data was used toderive the equations. However the Neher and McGrathequations do not agree as well with the data for lowerviscosity lqiuids. For these, either the Waltz Mill correla-tion or the equations of the present paper are moreaccurate.

    For the case of cables installed in steel pipes filled withdielectric liquids, equations 8-12 agree quite well withavailable data. In this case the conductance through theliquid is large compared with that directly from cable topipe which the equations ignore.

    Typically the accuracy of the Neher and McGrath equa-tions for naturally cooled pipe cables is sufficient since thetemperature rise from cable to pipe is small comparedwith the total temperature rise from conductor to ambi-ent. However the Waltz Mill tests were undertaken todetermine more accurate formulations for the tempera-ture rise of the cable to the fluid in forced-coozed modewhere this temperature rise can be a significant parame-ter.

    The third topic of the paper concerns heat conductionthrough earth of non-homogeneous conductivity.

    The conformal mapping technique is not new. It is astandard method for solving two dimensional field prob-lems. The method was applied to cable problems at leastas far back as 1958 (L. Fink and J. Smerke, reference [4]).Other papers on the same subject appearedin the AIEETransactions and in the IEE in England (Luoni [51).

    12

    2 5 Finally the technique was the subject of a CIGRE Work-ing Group report (reference [6]).

    Properly applied, this technique is very useful. Whenthe backfill thermal resistivity differs from that of thesummnding native earth, the position of cables within thebackfill as well as the backfill shape can effect the result-ing temperature rise and can be evaluated by this tech-nique. These effects are not adequately treated by theNeher and McGrath paper, although for the majority ofcases encountered in practice the errors involved are notsignificant compared to the errors involved in estimating

    Type F F F F F F F F T T TD s 3.5 3.5 3.5 0.69 0.69 0.69 3.13 3.13 3.5 3.5 3.5

    the1.2.3.4.

    5.6.

    7.

    thermal resistivities.

    Neher and McGrath, AIEE Transactions on PowerApparatus and Systems, October 1957.Buller and Neher, AIEE Trans. on P.A.S. v. 69, 1950.Greebler and Barnett, AIEE Trans. on PAS v. 69,1950, pp. 357-367.Fink and Smerke, Control of the Thermal Environ-ment of Buried Cable Systems - Pt. 11, AIEE Trans.on PAS June 1958.Luoni et al., I.E.E. Proceedings, Paper 6596 P, Octo-ber 1971.C.I.G.R.E. Working Group 21-02, Calculation of theeffective external thermal resistance of cables laid inmaterials having different thermal resistivities, Elec-tra No. 98, Paris, France.D. W . Purnhagen, Designers Handbook for Forced-Cooled High Pressure Oil Filled Pipe Type CableSystems, EPRI EL-3624, Palo Alto, CA, July 1984.

    References

    Table A-1Rsd Com parison for Cables in Conduit

    I I I I I I I I I I3 I D o I 4.0 I 4.0 I 4.0 I 3.5 1 3.5 I 3.5 I 3.5 I 3.5 I 4.0 I 4.0 I 4.0

    7 Watts/ft 12.48 d T

    Rs d =8i

    1.Type: F =Fiber, T =Transite duct, 1 cable with air at 1 atm.2. Ds =0.d. of cable in inches5. 6 =air gap dimension (D p-Ds)/27. Measured h eat flow in watts per ftSource: References 2 and 33. D p = id of conduit in inches6. Tm =mean temperature of air space8. d T =measured tem perature rise of cablesurface over du ct i.d.

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    ---i 26Table A-2Rsd Comparison for Cables in Steel Pipe

    Steel- HPGF HPGF HPGF HPGF

    3 Dp 6.07 6.07 6.07 6.07

    1.Type: Steel - air 3 cables in steel pipe with air at 1 atm.HPGF =3 cables in steel pipe with Nitrogen at14.7 atm2. Ds =2.155 X od of cable in inches3. Dp =id of conduit in inches5. 6 - air gap dimension (Dp-Ds)/26. Tm =mean temperature of air space7. Watts = measured heat flow in watts per ft8 dT =average measured temperature rise of cable surface9. Rsd =measured thermal resistance in C per total wattsSource: Reference 2

    over duct id

    Table BRsd ComparisonHigh Pressure Fluid Filled Pipe-Type Cable

    8.125 8.125 10.25 10.25

    4 AT 5 c 1oc 1oc 1ocEq. 41Eq. 54

    Black &Sellers

    1.Cable diameter (3 phases/pipe)2. I.D. of pipe3. Mean Fluid Temperature4. Temperature rise of cable surface over pipe5. HVP =High Viscosity Polybutene; LVAB low viscosity alkyl-6. Neher B McGrath Equation 41, Th Ohm-ft7. Neher & McGrath Equation 54, Th Ohm-ft8. Equations 8, 12 (revised with nn in denominator), Th Ohm-ft9. Waltz Mill Forced-Cooled Test data correlation, fully developed

    benzene

    laminar flow, ref 7Manuscript received March 13, 1995.

    Sally M . Sellers and W . Z . Black (GeorgiaInstitute of Technology, Atlanta, GA) We wouldlike to thank the reviewers for their comments and questionsconcerning our paper. Their input will help clarify areas in thepaper that were not thoroughly explained.One common thread that appears in a majority of the reviewerscomments involves our intent in writing this paper. Ourprimary goal was to present refinements to the Neher andMcGrath paper. We had neither the resources nor thelaboratory facilities to verify thd proposed refinements to theirthermal model. Obviously an experimental study of the extentnecessary to verify, for example, the resistance of a series ofduct banks, ranging from square to oblong would require asubstantial investment of time, labor and capitol. However,we are completeIy confident that the refinements to the Neher-McGrath model that are proposed in the paper are accurate,even though they have not been verified in laboratory tests.The refinements that we recommend are based on sound heattransfer principles and founded on heat transfer correlations thathave been veS ie d in laboratory settings. We are confident thatwhen the accuracy of our refiyments is checked, they willprove to be sound.We also wish to reiterate our purpose in proposing the threerefinements that are addressed in the paper. We were carefulnot to refer to these as corrections of errors or extensions tothe equations that appeared in the model. In the title andthroughout the text, we pointed out that we were proposingrefinements to the Neher-McGrath method. We were alsocarefnl to Iabel the curves that compared our proposed thermalresistance and ampacity values with those given in the Neher-McGrath paper as percent differences, not percent errors.Obviously the thermal model proposed by Neher and McGrathhas withstood the test of time and it is, and will remain, thesingle most important paper dealing with the problem ofampacity in underground cables. Considering the assumptionsstated in the paper and using the most up-to-date heat transfercorrelations available in the 1950s, t represents a correct anderror-free model. However, the sophistication of heat transferknowledge has progressed significantly in the nearly fortyyears since this paper has been published. More refinedconvective heat transfer correlations covering a far broaderrange of geometrics and flow conditions are available todaythan existed in the 1950s.This factor alone should encouragea periodic reexamination of a thermal model such as the oneproposed by Neher and McGrath. A reexamination in no wayimplies that the method is incorrect, simply that its accuracycould be enhanced through the incorporation of improved heattransfer correlations.Furthermore, we feel the calculational procedure that wasappropriate for the slide rule era of the 1950s should beperiodically revisited in a rapidly evolving age of personalcomputers. Assumptions made in the original model tosimplify the calculations and to avoid an iterative solution cannow be relaxed. Using todays computers, the formidable taskof repeating the calculations until the correct temperaturesthroughout the thermal circuit are known is no longer as timeconsuming as it was in the 1950s. Therefore, some of theassumptions that Neher and McGrath made in theirformulation to simplify the calculations no longer need to beused and the model can be made more flexible and accurate.

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    27Another aspect of the Neher-McGrath model should also beaddressed. Their model is based on the electric analog of heatconduction through a series of solid media. In order for theanalog method to successfully predict the ampacity of anunderground cable installation, all thermal resistances in thecircuit, the maximum conductor operating temperature, and theambient soil temperature must be known. Furthermore, allsources of heat that are generated in the circuit such as thoseresulting from circulating currents in the shield, current in theconductor, and dielectric losses must be either known orexpressed in terms of the conductor current. In this way, thethermal circuit can be reduced to a single equation in terms ofthe single unknown which is the current in the conductor. Ifany of the heat generation terms or the thermal resistanceterms are temperature dependent, then the analog modelrequires that the local temperatures be approximatedso that theterms can be accurately quantified and the model will retain itssimple nature. This is the difficulty Neher and McGrath facedin several areas when they wrote their paper. They were wellaware that the convective and radiative heat transfer resistancesacross air or liquid layers that exist in pipe-type cableinstallations are strongly temperature dependent. Therefore,they assumed the temperature differences and meantemperatures that existed in the air and fluid layers in thecircuit. Furthermore the radiative resistances across gas layersare non-linear with temperature and Neher-McGrath chose tosimplify the resistance by linearizing this term. Unless thesurface temperatures of the two opposing, radiating surfaces areapproximated, the analog method requires iteration on thevarious component temperatures until they converge to valuesthat conserve energy for the entire installation. Obviously amathematical model that requires repeated applications of theequations before converging to a correct set of temperatures isnot a very popular approach unless a personal computer is usedto solve the set of equations. Given today's sophisticated levelof computing capabilities, it is no longer necessary to assumethe local temperatures in the thermal network in order toestimate the temperature dependent resistance terms. Iterativealgorithms are common-place and for this reason, it is nolonger necessary to use some of the assumptions that wereused by Neher and McGrath to mathematically simplify theirmodel. This philosophy motivated us to suggest some of therefinements that we proposed in the paper.In order to consolidate our comments to the questions posed bythe reviewers, we will organize our responses in the same orderas the threesrefinements hat appear in the paper.

    Mutual Heating FactorThe mutual heating parameter that we present in Eqn. 1 isequivalent to the way in which mutual heating is treated in theIEC Standards Publication 287 as pointed out by Dr. Anders.Since the factor F is a function of not only the geometry butalso the rate at which heat is generated in every cable in thesystem, the temperature rise at any cable can only bedetermined once the ampacity in every cable is known. Thisfact forces an iterative solution in which the ampacities areassumed and the temperature rise resulting from each cable iscalculated. The process continues until ampacity valuesconverge to ones that provide the given conductortemperatures. If one assumes all cables produce the same

    ' amount of heat, the factor F becomes only a function of

    geometry and an iterative procedure is not necessary. As weshow in Fig. 2, in those situations for which the heatgenerated in the cables is not equal, the assumption of equalheat generation will cause ampacity errors less than 8 percent.Our purpose in presenting a different form of the mutualheating parameter in Eqn. 2 of the paper was to point out howunequal heating influences the value of F. We are notimplying that Neher and McGrath made an error in theirconsideration of mutual heating, only that unequal heating canbe factored into the expression for F and unequal heating doesnot need to be handled by calculating interference temperaturesas described in their paper. We feel that the simplemodification to the mutual heating parameter as described inEqn. 2 is less complex than the interference temperaturetechnique described in the Neher and McGrath paper. The useof equations 48, 49, and 50 is not necessary if the expressionfor F is used to account for the effects of uneven heating. Aswe pointed out in the our paper, the expression for F given byEqn. 2 reduces to Eqn. 46 in Neher-McGrath when all cablesdissipate heat equally. The results in Figs. 1 and 2 wereprovided to show the percent differences in earth resistance andampacity that will arise due to the assumption of equal heatingwhen cables actually generate heat at different rates.Resistance of Fluid Layers for Cables in Ducts and PiDesThe numerical error identified in Eqn. 5 by Dr. Morgan wasdiscovered shortly after we submitted our paper for review andit has been corrected. This equation was provided in the paperfor reference purposes only. It was not used to generate any ofthe curves in the paper.Our comments in reference to Fig. 3 of the paper and ourdiscussion of the thermal resistance of the fluid layer for cablesin ducts and pipes attempted to point out the fact that Eqns. 41and 41a in the Neher-McGrath paper were limited by severalassumptions. Both of these equations do not express thethermal resistance in terms of the thickness of the fluid layersurrounding the cables. Also both of these equations do notinclude the influence of the emissivity of either the jacket orpipe material. Presumably both of these parameters areaccounted for by the values of A, B and C (for Eqn. 41) or Aand B' (for Eqn. 41a). Unfortunately the values for theseconstants are provided only for a metallic conduit, only twotypes of duct (fiber and transite), and only three types of fluids(air at 1 atm, gas at 200 psi and oil). The user is unable tocalculate the thermal resistance for other, more modern cableinstallations and more up-to-date duct materials. An equationsuch as Eqn. 10 that is expressed in terms of the emissivity ofboth the duct and jacket material and Eqn. 12 that is written interms of the geometry and properties of the intervening fluidare more general and can be used for any emissivity, any cableand duct size, and any cable and duct material. Furthermore,they are not limited to any value for the mean temperature.Even though there are no experimental resistancemeasurements for cables in PVC ducts, for example, Eqns. 8-12 could be at least used to provide guidance for ampacitycalculations The user is therefore left with a calculationaltool rather than forced to choose from a limited set ofgeometrics and system materials.Mr. Fink points out that the expressions developed by Neherand McGrath for the air and oil thermal resistance were

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    28validated in a series of experimental tests. In fact theseexperiments were used to determine the constants in Eqns. 41and 41a. Chronologically the theoretical expression for theresistance of the convective, radiative and conductive heattransfer were formulated first and presented in the 1950Bullerand Neher paper. In order to derive these resistances, a numberof assumptions were made, as pointed out in the paper. Whenthe experimental work was completed, the theoreticalexpressions for the thermal resistances were corrected oradjusted to bring them into line with the experimental data.As Mr. Fink points out, some of the differences between thetheoretical expressions and the experimental measurementsmay have resulted from the assumptions of concentricgeometry and heat flow which is independent of azimuthalangle. The Buller and Neher expressions, as well as ours, areboth based on these assumptions.We feel that our expressions for the thermal resistances are'more accurate than those provided in the paper by Buller andNeher, simply because they account for more the variables thatare known to affect the heat transfer and they are not restrictedby some of the limiting assumptions applied by Buller andNeher. We show a comparison between the two models laterin this discussion to support this conclusion. Nonetheless, werecognize that our thermal resistance equations remainunverified by experimental measurements. We would welcomea series of tests that involved modern installation materials andpractices. These tests are long overdue and would provide avaluable service to the power industry.Regarding the value of Ta that we used when evaluating thethermal resistance of the fluid layer, we use the temperatureequal to the inner surface of the duct or pipe. We determinethis value by taking an energy balance on each layer of theinstallation geometry. In that way the temperature profilethroughout the entire geometry is known and any quantity thatis temperature dependent can be accurately evaluated. Thereforeis it not necessary to assume any temperature to evaluate atemperature dependent thermal resistance. This process alsoresults in a more general thermal model whose accuracy isimproved because we do not have to assume any intermediatetemperatures The driving potential for heat transfer across thefluid layer is Tj-Ta, where Ta is the temperature of the innersurface of the duct, which is also equal to the air temperaturein contact with the duct. In the paper we define AT as thetemperature difference between the outer surface of the cableand the inside surface of the duct.Regarding the thermal resistance equations for cables inconduit, we provided equations that are similar to the onespresented by Buller and Neher in 1950. Like Buller andNeher's equations, they have not been adjusted to bring theminto line with experimental data. If we had adjusted them,obviously they would have come closer to matching theavailable experimental data. Since our equations have not beenadjusted to fit experimental data, it seems more appropriate'tocompare our equations with those of Buller and Neher. Wehave carried out a comparison between the two and the resultsare shown in Table C-1 (we will use similar organization asErnst and Purnhagen in their tables so that values can be easilycompared). For the eleven cases compared in Table C-1 theresistances predicted by Eqns. 8-12 give results that are ingeneral closer to the experimental results than those given in

    the paper by Buller and Neher. Also, we have compared ourresults as a percent difference from the experimental results andnot as a ratio as calculated by Ernst and Purnhagen. Thevalues for percentage differences are provided in Table C- 1.We have carefully examined the data in Table A-2provided byErnst and Purnhagen and we are unable confirm theirinterpretation of the results from Eqns. 8-12. The values thatwe calculate for the resistance of the fluid layer in the pipe aregiven in Table C-2 along with the percent difference betweenthese values and the measured values. The percentagedifferences that appear in Table C-2 are significantly less thanportrayed by the ratio values in row 15of Table A-2. We areuncertain of the exact values that Ernst and Purhhagen used forthermo-physical properties of the fluids so this may be thesource of the differences. We have included the properties thatwe used in Table C- 2 so there can be no misunderstanding asto how we calculated the resistance values. The values for theproperties are from reference [B]. Our resistance values areslightly greater than those presented in the Buller-Neher paper.Finally we have verified all of the values that appear in Ernstand Purnhagen's Table B. For the conditions of oil in a pipe-type cable, Eqns. 8-12 predict fluid layer thermal resistancesthat are quite close to the Waltz Mill test data. The thermalresistance values predicted by Eqns. 41 and 41a in the Neher-McGrath paper and the correlations in the Buller and Neherpaper show much larger differences. The flujd properties usedto calculate the values in the table appear at the bottom of thetable. These properties help explain an important point.Equations 41 and 41a are limited to the types of installationslisted in Table VII of the Neher-McGrath paper. The constantsin this table were determined after curve-fitting Eqn. 41 toexperimental data. The experimental data was collected for apipe-type cable that utilized an insulating oil with theproperties shown in Table 11. When the oil properties changeto those of a high viscosity polybutene (HVP) or a lowviscosity alkyl benzene (LVAB), the convective flow of the oilis changed and the ampacity can be significantly altered.Since the constants in Eqns. 41 and 41a are matched to a pipe-type cable surrounded by a generic oil, it should not besurprising that the thermal resistances of an oil with differentproperties will produce large differences with measured values.These equations are being used in an application in which theywere not intended to be used. On the other hand, Eqns. 8-12have retained a general form that includes the properties of theoil and they would be expected to provide a better estimate forthe oil layer when they are provided with accurate thermo-physical properties of the oil. In other words it should beexpected that a general expression would be an improvementover an expression that is limited to a single fluid.

    I

    Resistance of Rectangular Duct BanksSeveral questions were asked about the conformal mappingtechnique. We made no claims in the p'aper about thistechnique being new. It has obviously been used by othersdating back many years. The conformal mapping techniquethat we used to calculate the thermal resistance of the ductbank does not assume the outer surface of the duct is anisothermal surface. AJso it does not assume that the inner ductsurface is uniformly heated. The thermal boundary conditions

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    29that were assumed were an isothermal surface at the insidesurface of the duct bank and an isothermal surface at thehorizontal earth/air interface. These boundary conditions donot result in an isothermal duct bank surface as was assumedby Neher and McGrath. The details of the conformal mappingtechnique are given in reference 9 in the paper. They may alsobe found in an IEEE paper [D] that will be submitted forreview in the next few months.As pointed out in the paper, we have used both a finiteelement program (see Ref. 10 in the paper) as well as aconformal mapping model to evaluate the influence of non-square duct banks on the ampacity of cables. We ultimatelyselected the conformal mapping method for reasons ofsimplicity. We were con cerned about having a relatively smallampacity program being burdened by a much larger and morecomplex finite element program. We chose to use theconformal mapping program for a wide variety of ductgeometries and calculate the shape factors defined in Eqn. 16.The conformal mapping model calculates a ratio of theeffective resistivity to the conduction shape factor. In doingso , the com putational time is m inimized and the mode l retainsa reasonable degree of simplicity. Further, calculating aneffective thermal resistance of $e duct bank and earth allowsus to keep the sam e form for the duct bank thermal resistanceproposed by Ne her and McGrath (see Eqn. 17). W e have foundthat use of a finite element program to achieve the same goalrequires far greater computational time with no improvementin accuracy.Regarding Dr. Morgan's question about the influence of solarheating, the thermal model for ampacity assumes that theambient soil temperature and the aidearth interface ismaintained at a constant temperature. Rarely is the soiltemperature consta nt. In most locations the surfacetemperature more c losely follows seasonal variations in the airtemperature, while the temperature variations at greater depths

    are dampen ed and lag behind swings in the surface temperature.For any period in which the temperature of the surface of theearth is wanner than the subsurface temperature, the ampacityshould be reduced if it is ca lculated on the basis of the ambientearth temperature at the cable depth. If the surface temperatureis less than the soil temperature at the cable depth, theampac ity could be increased. Determining the precise amountthe ampa city could be increased or decreased depends upon thesoil resistivity a nd the tem perature distribution in the soil andis not a simple matter to quantify. From a practicalstandpoint, the soil cond itions in the immed iate vicinity of thecable have a far greater influence on cable ampacity than theconditions of the soil near the earth/air interface. It isprobably far more important to be concerned about the localvariations in the soil resistivity at the cable depth thanvariations of the soil temperature. After all, the effect ofvariations in the local so il resistivy caused by local chang es insoil composition or moisture content, for example, areprobably far greater on the ampacity than the influence ofvariations in local soil temperature with depth. Therefore athorough know ledge of the therm al resistivity of the soil at thecable depth a long the entire route of the circuit is more criticalto a calculation of the ampacity than a knowledge of thetemperature profile of the soil as a function of depth.

    References[AI Greebler, Paul, and Guy F. Barnett, "Heat TransferStudy on Power Cable Ducts and Duct Assemblies,"AIEE Transactions on Power Apparatus and Systems,Vol. 69, pp. 357-367, 1950.[B] CRC Handbook of Tables for Apulied EngineeringScience, 2nd. ed., Ray E. Bolz and George L. Tuve,1973.

    TABLE C-1COMPARISON OF THERMAL RESISTANCE FOR CABLE IN CONDUIT

    References [6 ]and [A]1. Type: F=fiber, T =transite duct2. D, =0. . of cable in inches, 1 cable with air at I atm3. Da =I. D. of conduit in inches7. experimental thermal res istance (516)8. thermal resistance as predicted by Buller and Neher (eqn. 5)9. percent differenc e in thermal resistance between lines 7 and 810. thermal resistance as predicted by eqn s. 8 - 121I . percent difference in thermal resistance between lines 7 and 10

    Radiative Properties:emissivity of jacket: 0.95emissivity of inner surface of duct: 0.80

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    30

    1011

    [C] Purnhagen, D. W., "Designers Handbook for Forced -Cooled High Pressure Oil Pipe Type Cable Systems,"EPRI EL - 3624, Palo Alto, CA, July 1984.

    R, 1.167 0.665 0.656 0.763 0.6659% diff. 36.6 45.2 31.6 40.3 32.1

    PI Hartley, J. G., and Sandra J. Woods, "A Method forDetermining the Effective Thermal Resistivity ofRectangular Duct Banks," to be submitted for review toIEEE Transactions on Power Apparatus and Systems,1995.

    TABLE C-2COMPARISON OF THERMAL RESISTANCE FOR

    CABLES IN STEELPIPEI[1 IType I steel 1 HPGF I HPGF I HPGF I HPGF 11

    16 1 Wlft 28.6 I23.4 1 28.9 1 6.8 1 15.917 R e x . 0.855 0.458 0.498 0.544 0.5038 R 1.037 0.641 0.633 0.724 0.6309 % diff. 21.4 39.9 26.9 33.0 25.2Reference [6]1. Type: steellair=3 cables in steel pipe with ai r at 1 atm2. D,' =2.155 x0.D. of cable in inches3. D, =I. D. of conduit in inches7. experimental thermal resistance (5+6)8. thermal resistance as predicted by Buller and Neher (eq n. 5)9. percent difference in thermal resistance between lines 7 and 810. thermal resistance as predicted by eqns. 8 - 1211. percent difference in thermal resistance between lines 7 and 10Properties used in eqns. 8 - 12 for Nitrogen Gas [B]:

    HPGF=3 cables in steel pipe with N2 at 14.6atm

    conductivity: 0.000261 W/cm"Clanematic viscosity: 0.01074 cm2/sPrandtl number: 0.7327emissivity of jacket: 0.95emissivity of inner surface of pipe: 0.70

    Radiative Properties:

    -2.01 2.35 I0.968 I 0.843 I-0.85 I 2.59 1Reference [C ]2. D, =0.D. of cable in inches (3 phases/pipe)3. D, =I. D. of conduit in inches6. experimental thermal esistance7. thermalresistance as predicted by B uller and8. percent difference in thermal resistance betwe9 thermal resistance as predicted by eqns. 8 - 1 210. p ercent difference in therm d resistance betwProperties used in eqns. 8 - 12 [C] :HVP =High Viscosity Polybutefieconductivity: 0.001092 W/cmCkinematic viscosity: 1.2181 cm2/sPrandtl number: 1915.0coefficient of thermal expansion: 0.0007

    conduchvity: 0.001 102 W/cmCkinematic viscosity. 0.1928 cm2/sPrandtl number: 276.8coefficient of therma l expansion: 0.0007

    LVAB =JAW Viscosity Alkylbenzene

    Manuscript received May 26,1995.

    ..31 1 16.511

    :r (eqn. 2 in [6])nes 6 and 7lines 6 and 9

    'C

    'C