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The Calculation or the Temperature Rise
and Load Capability of Cable Systems
J. H. NEHER M. H. McGRATHMEMBER AIEE MEMBER AIEE
IN 1932 D. M. Simmons' published aseries of articles entitled, "Calculation
of the Electrical Problems of UndergroundCables." Over the intervening 25 yearsthis work has achieved the status of ahandbook on the subject. During thisperiod, however, there have been numer-ous developments in the cable art, andmuch theoretical and experimental workhas been done with a view to obtainingmore accurate methods of evaluating theparameters involved. The advent of thepipe-type cable system has emphasizedthe desirability of a more rational methodof calculating the performance of cablesin duct in order that a realistic comparisonmay be made between the two systems.
In this paper the authors have en-deavored to extend the work of Simmonsby presenting under one cover the basicprinciples involved, together with morerecently developed procedures for han-dling such problems as the effect of theloading cycle and the temperature riseof cables in various types of duct struc-tures. Included as well are expressionsrequired in the evaluation of the basicparameters for certain specialized alliedprocedures. It is thought that a work ofthis type will be useful not only as a guideto engineers entering the field and as areference to the more experienced, butparticularly as a basis for setting up com-putation methods for the preparation ofindustry load capability and a-c/d-c ratiocompilations.The calculation of the temperature rise
of cable systems under essentially steady-state conditions, which includes the effectof operation under a repetitive load cycle,as opposed to transient temperature risesdue to the sudden application of largeamounts of load, is a relatively simpleprocedure and involves only the applica-tion of the thermal equivalents of Ohm'sand Kirchoff's Laws to a relatively simplethermal circuit. Because this circuitusually has a number of parallel pathswith heat flows entering at several points,however, care must be exercised in themethod used of expressing the heat flowsand thermal resistances involved, anddiffering methods are used by various en-gineers. The method employed in thispaper has been selected after careful con-
sideration as being the most consistentand most readily handled over the fullscope of the problem.
All losses will be developed on the basisof watts per conductor foot. The heatflows and temperature rises due to dielec-tric loss and to current-produced losseswillbe treated separately, and, in the lattercase, all heat flows will be expressed interms of the current produced loss originat-ing in one foot of conductor by means ofmultiplying factors which take into ac-count the added losses in the sheath andconduit.
In general, all thermal resistances willbe developed on the basis of the per con-ductor heat flow through them. In thecase of underground cable systems, it isconvenient to utilize an effective thermalresistance for the earth portion of thethermal circuit which includes the effectof the loading cycle and the mutual heat-ing effect of the other cable of the system.All cables in the system will be consideredto carry the same load currents and to beoperating under the same load cycle.The system of nomenclature employed
is in accordance wvith that adopted by theInsulated Conductor Committee as stand-ard, and differs appreciably from that usedin many of the references. This systemrepresents an attempt to utilize in so faras possible the various symbols appearingin the American Standards AssociationStandards for Electrical Quantities, Me-chanics, Heat and Thermo-Dynamics,and Hydraulics, when these symbols canbe used without ambiguity. Certainsymbols which have long been used bycable engineers have been retained, eventhough they are in direct conflict withthe above-mentioned standards.
Nomenclature
(A F) = attainment factor, per unit (pu)A, = cross-section area of a shielding tape
or skid wire, square inchesa = thermal diffusivity, square inches per
hourCI= conductor area, circular inchesd = distance, inchesd,2 etc.= from center of cable no. 1 to center
of cable no. 2 etc.d,2' etc. = from center of cable no. 1 to
image of cable no. 2 etc.d1j etc. =from center of cable no. 1 to a
point of interference
dit' etc. =from image of cable no. 1 to apoint of interference
D = diameter, inchesD= inside of annular conductorDc=outside of conductorD= outside of insulationDs=outside of sheathD.= mean diameter of sheathD= outside of jacketDJ'= effective (circumscribing circle) of
several cables in contactD= inside of duct wall, pipe or conduitDe= diameter at start of the earth portion
of the thermal circuitDz=fictitious diameter at which the effect
of loss factor commencesE=line to neutral voltage, kilovolts (kv)*e=coefficient of surface emissivitye =specific inductive capacitance of insula-
tionf=frequency, cycles per secondF, Fin1 = products of ratios of distancesF(x)=derived Bessel function of x (Table
III and Fig. 1)G =geometric factorGI =applying to insulation resistance (Fig. 2
of reference 1)C2 = applying to dielectric loss (Fig. 2 of
reference 1)Gb =applying to a duct bank (Fig. 2)I=conductor current, kiloamperesk,=skin effect correction factor for annular
and segmental conductorskp=relative transverse conductivity factor
for calculating conductor proximityeffect
I= lay of a shielding tape or skid wire, inchesL=depth of reference cable below earth's
surface, inchesLb = depth to center of a duct bank (or
backfill), inches(If)=load factor, per unit(LF) =loss factor, per unitn=number of conductors per cablen'=number of conductors within a stated
diameterN=number of cables or cable groups in a
systemP=perinmeter of a duct bank or backfill,
inchescos4-=power factor of the insulationq,=ratio of the sum of the losses in the
conductors and sheaths to the lossesin the conductors
q,= ratio of the sum of the losses in theconductors, sheath and conduit tothe losses in the conductors
R =electrical resistance, ohmsRd d-c resistance of conductorRae= total a-c resistance per conductorR, = d-c resistance of sheath or of the
parallel paths in a shield-skid wireassembly
R-thermal resistance (per conductor losses)thermal ohm-feet
= of insulationRI=of jacket
=adbetween cable surface and surroundingenclosure
Paper 57-660, recommended by the AIEE InsulatedConductors Committee and approved by the AIEETechnical Operations Department for presentationat the AIEE Summer General Meeting, Montreal,Que., Canada, June 24-28, 1957. Manuscriptsubmitted March 20, 1957; made available forprinting April 18, 1957.J. H. NisaBR 15 with the Philadelphia ElectricCompany, Philadelphia, Pa.. and M. H. MCGRATHis with the General Cable Corporation, PerthAmboy, N. J.
Neher, McGrath-Temperature and Load Capability of Cable Systems OcToiBE:R 1957752
0.09
04~~~~~ ~ ~~~~~~~~~~~~~~0\0000.25 lit%sl i _ _ _ t ~~~~0.002
0.07ILL ~~~~~~~~~0.06t1 ~~~~~~~~0-.05
0.03
0.2 K m_ t F(°p025
0.10 0.022 2.5 3 4 5 6 768910 15 20 30 40 50 60 o010
R dc/k
Fig. I (above). F(x) and F(xp') as functions oF Rd/k
Fig. 2 (right). Gb For a duct bank
Rd= of duct wall or asphalt mastic covering
R,. = total between sheath and diameterDe including A,, Ad and Rd
R,=between conduit and ambientR'=effective between diameter De and
ambient earth including the effectsof loss factor and mutual heating by W-portioPother cables W,=ortior
Rea effective between conductor and shieldambient for conductor loss W =portioi
Rcg'- effective transient thermal resistance duitof cable system Wd portlol
Rd I=effective between conductor and am- Xm =mutuabient for dielectric loss or sb
An= of the interference effect Y- the inci
Ra= between a steam pipe and ambient YC=due toearth duct
p electrical resistivity, circular mil ohms to siper foot imnit
p-thermal resistivity, degrees centigrade Y7 =due tocentimeters per watt or s
s=distance in a 3-conductor cable between duethe effective current center of the yconductor and the axis of the cable, Yp=due tcinches or cc
S=axial spacing between adjacent cables, Ya = due toinches
1, T=thickness (as indicated), inches General (
T=temperature, degrees centigrade ThermTa of ambient air or earth
T =of conductorTm= mean temperature of medium THE CALciAT=temperature rise, degrees centigrade RISEATC=of conductor due to current produced
losses The temATd = of conductor due to dielectric loss of a cable a
=TnIof a cable due to extraneous heat be considesource
b osd
r= inferred temperature of zero resistance, temperatwdegrees centigrade (C) (used in which maycorrecting Rdc and R, to tempera- current pr(tures other than 20 C) referred to
Vw= wind velocity, miles per hour refer toW=losses developed in a cable, watts per tor, sheat
conductor foot produced
-11.
:, .01)
cr-s
ew
9iN
t. 2~~~~~~~~~~~~~~~~~~.1.2 -i - - - -.e-00 2.5
2.0
<21.0
1.0-_X_L-22r 7 .6
: F k000[ T 1§Mi~~~~o, 01
.5 - -T-
.3 I.7 ._.l .2 . .5 .? .
RATIO Lb/P
a developed in the conductorn developed im the sheath or
dIn developed in the pipe or con-
in developed in the dielectricJ1 reactance, conductor to sheathhield, microhms per footrement of a-c/d-c ratio, pulosses originating in the con-
;or, having components Yc. duekdn effect and Ycp due to prox-
y effectlosses originating in the sheathshield, having components YJCto circulating current effect anddue to eddy current effecto losses originating in the pipeonduitlosses originating in the armor
Considerations of thedI Circuit
ULATION Op TEMPERATURE
perature rise of the conductortbove ambient temperature mayered as being composed of a
re rise due to its own losses,y be divided into a rise due tooduced (I2R) losses (hereinaftermerely as losses) in the conduc-h and conduit AT: and the riseby its dielectric loss ATd.
Thus
TC Ta =AT.+A Td degrees centigrade(1)
Each of these component temperaturerises may be considered as the result of a
rate of heat flow expressed in watts per
foot through a thermal resistance ex-
pressed in thermal ohm feet (degrees centi-grade feet per watt); in other words, theradial rise in degrees centigrade for a heatflow of one watt uniformly distributedover a conductor length of one foot.
Since the losses occur at several posi-tions in the cable system, the heat flow in
the thermal circuit will increase in steps.It is convenient to express all heat flows interms of the loss per foot of conductor, andthus
AT-= Wc(R +qsRs+qAR)degrees centigrade (2)
in which W, represents the losses in one
conductor and R, is the thermal resistanceof the insulation, qs is the ratio of thesum of the losses in the conductors andsheath to the losses in the conductors,
is the total thermal resistance betweensheath and conduit, qo is the ratio of thesum of the losses in conductors, sheath andconduit, to the conductor losses, and A.
Neher, McGrath-Temperature and Load Capability of Cable Systems
la
40
z
i.
753OcToBim 1957
is the thermal resistance between theconduit and ambient.
In practice, the load carried by a cableis rarely constant and varies according toa daily load cycle having a load factor(f). Hence, the losses in the cable willvary according to the correspondingdaily loss cycle having a loss factor (LF).From an examination of a large number ofload cycles and their corresponding loadand loss factors, the following general rela-tionship between load factor and lossfactor has been found to exist.I
(LF)=Q.3 (lf)+0.7 (If)2 per unit (3)
In order to determine the maximumtemperature rise attained by a buriedcable system under a repeated daily loadcycle, the losses and resultant heat flowsare calculated on the basis of the maxi-mum load (usually taken as the averagecurrent for that hour of the daily loadcycle during which the average current isthe highest, i.e. the daily maximum one-hour average load) on which the loss factoris based and the heat flow in the last partof the earth portion of the thermal circuitis reduced by the factor (LF). If thisreduction is considered to start at a pointin the earth corresponding to the diameterDz,s equation 2 becomes
AT- WC[Rj+q,Rse+ qc(Rez+(L F)Rraa)Jdegrees centigrade (4)
In effect this means that the tempera-ture rise from conductor to DA is made todepend on the heat loss corresponding tothe maximum load whereas the tempera-ture rise from diameter D. to ambient ismade to depend on the average loss over a24-hour period. Studies indicate that theprocedure of assuming a fictitious criticaldiameter D, at which an abrupt changeoccurs in loss factor from 100% to actualwill give results which very closelyapproximate those obtained by rigoroustransient analysis. For cables or ductin air where the thermal storage capacityof the system is relatively small, the maxi-mum temperature rise is based upon theheat flow dorresponding to maximum loadwithout reduction of any part of thethermal circuit.When a number of cables are installed
close together in the earth or in a ductbank, each cable will have a heating effectupon all of the others. In calculatingthe temperature rise of any one cable, it isconvenient to handle the heating effects ofthe other cables of the system by suitablymodifying the last term of equation 4.This is permissible since it is assumedthat all the cables are carrying equal cur-rents, and are operating on the same loadcycle. Thus for an N-cable system
ATC Wc(Rg+qsRsa+qe[Rx+(LF)X(fRxaM+N-1)fRPaD (S)
= Wc(Ri+qeRsa+qsR,')degrees centigrade (SA)
where the term in parentheses is indicatedby the effective thermal resistance R/.The temperature rise due to dielectric
loss is a relatively small part of the totaltemperature rise of cable systems op-erating at the lower voltages, but athigher voltages it constitutes an appre-ciable part and must be considered. Al-though the dielectric losses are dis-tributed throughout the insulation, it maybe shown that for single conductor cableand multiconductor shielded cable withround conductors the correct temperaturerise is obtained by considering for tran-sient and steady state that all of thedielectric loss Wd occurs at the middleof the thermal resistance between conduc-tor and sheath or alternately for steady-state conditions alone that the tempera-ture rise between conductor and sheath fora given loss in the dielectric is half asmuch as if that loss were in the conductor.In the case of multiconductor beltedcables, however, the conductors are takenas the source of the dielectric loss.'The resulting temperature rise due to
dielectric loss ATd may be expressedATd = WdRda' degrees centigrade (6)
in which the effective thermal resistanceRdar is based upon Ri, Rk5, and R,'(at unityloss factor) according to the particularcase. The temperature rise at points inthe cable system other than at the con-ductor may be determined readily fromthe foregoing relationships.
THE CALCULATION OF LOAD CAPABILITrIn many cases the permissible maxi-
mum temperature of the conductor isfixed and the magnitude of the conductorcurrent (load capability) required toproduce this temperature is desired.Equation 5(A) may be written in the formATc =I2Rdc(l+ Yc)Aca'
degrees centigrade (7)in which the quantity Rdc (1+ Ye) whichwill be evaluated later represents theeffective electrical resistance of the con-ductor in microhms per foot, and whichwhen multiplied by I2 (I in kiloamperes)will equal the loss Wc in watts per conduc-tor foot actually generated in the conduc-tor; and Rca' is the effective thermalresistance of the thermal circuit.
ca Rt+ q8Rxe+qcR,' thermal ohm-feet(8)
From equation 1 it follows that
J=.i -(Ta+ATd) kiloamperes (9)IRde(l1+ Yc)R?ca
Table 1. Electrical Resistivity oF VariousMaterials
pCircular MilOhms per Foot
Material at 20 C t, C
Copper (100% IACS*)........ 10.371... 234.6Aluminum (61% IACS)....... 17.002... 228.1Commercial Bronze (43.6% ... 23.8 .....564
IACS)(90 Cu-10 Zn)
Brass (27.3% IACS) .......... 38.0 .....912(70 Cu-30 Zn)
Lead (7.84% IACS)........... 132.3 ..... 236
* International Annealed Copper Standard.
Calculation of Losses andAssociated Parameters
CALCULATION OF D-C RESISTANCES
The resistance of the conductor may bedetermined from the following expressionswhich include a lay factor of 2%; seeTable I.
Rdc -C microhms per foot at 20 CC1
(10)12.9 cpe=Cl for 100% IACS copperCI
conductor at 75 C (IOA)21.2
for 61% IACS
aluminum at 75 C (lOB)where CI represents the conductor size incircular inches and where Pr representsthe electrical resistivity in circular milohms per foot. To determine the value ofresistance at temperature T multiply theresistance at 20 C by (r+T)/(r+20)where r is the inferred temperature ofzero resistance.The resistance of the sheath is given
by the expressions
R,= Ps 1Amicrohms per foot at 20 C (II)4Dsmt
Rs =37-9 for lead at 50 CD8aM(IIA)
=4.75 for 61% aluminum at 50 C
(hIB)
where Dsm is the mean diameter of thesheath and I is its thickness, both ininches
D,.m D,-t inches (12)
The resistance of intercalated shieldsor skid wires may be determined from theexpression
Rs (per pathi)- 1h+(TD )m4As Imicrohms per foot at 20 C (13)
where A. is the cross-section area of the
Neher, McGrath-Temperature and Load Capability of Cable Systems OCTOBER 1957754
tape or skid wire and I is its lay. '
over-all resilstance of the shield and c
wire assembly, particularly for nonincalated shields, should be determinedelectrical measurement when possibliCALCULATION o1? Lossus
It is convenient to develop expressfor the losses in the conductor, sheathpipe or condu'it in terms of the componof the a-c/d-c ratio of the cable syl,;which may be expressed as follows4
Rac/Rdcl1+ ye+ y,+ Yv
The a-c/d-c ratio at conductor is 1
and at sheath or shield is 1+ Yc+ Ys
and at pipe or conduit is 1+ Y,+ Ye5The corresponding losses phys'icallyerated in the conductor, sheath, andare
Wc =I'RdC(1 +YC) watts per conductor,
W, =I'Rde Y7 watts per conductor foot
Wp, =IPR& Yp watts per conductor foot
This permits a ready determination oflosses if the segregated a-c/d-c ratiosknown, and conversely, the a-c/d-c t
is readily obtailned after the values olY7, and 17, have been calculated.
It follows from the definitions of q8q. that
qs =1+-
we 1+Y
The factor Y, is the sum of two coinents, Ya, due to skin effect and Yp,proxuimity effect.
watts per conductor foot
The skin effect may be determinedthe skin effect functilon F(x)
YCs= F(xs)Ifk 6.80
x8j=0.8'175.t - at 60 cyclc
in which the factor k, depends upo:conductor construction. For soIliconventional conductors approlvalues of k8 will be found in Table II.function F(x) may be obtained from.III or from the curves of Fig. 1 in Iof the ratio Rd,,/k at 60 cycles.
For annual conductors
k, D..D(lc+2Do)2
in which D, and D0, represent the
Theskiditer-I byle.
'ionsandLents
Table Ii. Recommended Values of kc, and kp
Conductor Construction Coating on Strands Treatment k k
Concentric round........None ..........None........1.0 ....... 1.0Concentric round ........Tin or alloy. None . 1.0........1.0Concentric round ........None..........Yes.........1.0........0.80oCompact round .........None..........Yes.........1.0 ........0.6Compact segmental .......None..........None ........0.435........0.6Compact segmental .......Tin or alloy .......None ........0.5........0.7Compact segmental .......None ..........Yes.........0.435........0.37Compact sector .........None ..........Yes.........1.0........(see note)
;tm NOTBS:1. The term "treated" denotes a completed conductor which has been subjected to a drying and impregnat-Ing process similar to that employed on paper power cable.
(14) 2. Proximity effect on compact sector conductors may be taken as one-halt of that for compact roundhaving the same cross-sectional area and insulation thickness.
+ Ye 3. Proximity effect on annular conductors may be approximated by using the value for a concentricround conductor of the same cross-sectional area and spacing. The Increased diameter of the annulartype and the removal of metal from the center decreases the skin effect but, for a given axial spacing, tendsto result in an increase in Proximity.
+ Y, 4. The values listed above for compact segmental refer to four segment constructions. The "uncosted-treated" values may also be taken as applicable to four segm'ent comnpact segmental with hollow core (ap-proximately 0.75 inch clear). For "uncoated-treated" six segment hollow core compact segmental limited
gen- test data indicates ks and kV vialues of 0.39 and 0.33 respectively.pipe-
foot(15)
(16)
(17)
Table Ill. Skin Effect in %/ in Solid Round Conductor and in Conventional Round ConcentricStrand Conductors
100 F(x), Skin Effect %7r 0 1 2 3 4 5 6 7 8 9
~tnte 0.3... 0.00... 0.00o... 0.01... 0.01 ... 0.01.. 0.01 ... 0.01... 0.01 ... 0.01... 0.01
ae 0.4... 0.01 ... 0.01... 0.02... 0.02... 0.02... 0.02... 0.02... 0.03... 0.03... 0.03ae 0.5... 0.03 ... 0.04... 0.04... 0.04... 0.05... 0.05... 0.05... 0.06... 0.08... 0.08
matio 0.8... 0.07... 0.07... 0.08... 0.08... 0.09 ... 0.10... 0.10... 0.11 ... 0.11 ... 0.12f C
0.7... 0.12... 0.13... 0.14... 0.15... 0.18... 0.17... 0.18... 0.19... 0.19... 0.20C, 0.8... 0.21 ... 0.22... 0.24... 0.25... 0.26... 0.28... 0.29... 0.30... 0.31... 0.33
0.9... 0.34... 0.38 ... 0.38... 0.39... 0.41 ... 0.43... 0.45... 0.47... 0.48... 0.501n.0... 0.52... 0.54... 0.58... 0.58... 0.81... 0.83... 0.65... 0.68... 0.70... 0.73ad 1.1... 0.76... 0.79... 0.81... 0.84... 0.87... 0.90... 0.94... 0.97... 1.00... 1.03
1.2... 1 07... 11.. 1.14... 1.18... 1.22... 1.26... 1.30:.. 1.34... 1.38... 1.421.3... 1.47... 1.52... 1.56..... 1... .6... 1.71... 1.76... 1.81... 1.86... 1.921.4... 1.97... 2.02... 2.08... 2.14... 2.20... 2.26... 2.32... 2.39... 2.45... 2.52
(18) 1.5... 2.58... 2.65... 2.72... 2.79... 2.88... 2.93... 3.01... 3.08... 3.16... 3.241... 3.32... 3.40... 3.49... 3.57... 3.66... 3.75... 3.83... 3.92... 4.02... 4.111... 4.21... 4.30... 4.40... 4.50... 4.80... 4.70... 4.81 ... 4.91... 5.02... 5.13
1.8... 5.24... 5.35... 5.47... 5.58... 5.70... 5.82... 5.94... 6.08.. 61... 6.31(19) 1.9... 6.44... 8.57... 6.70... 6.83... 8.97... 7.11... 7.24... 7.38... 7.53... 7.67
2.0... 7.82... 7.98... 8.11 ... 8.26... 8.42... 8 57... 8.73... 8 89... 9.05... 9.212.1... 9.38... 9.54... 9.71... 9.88... 10.05... 10.22... 10.40... 10.58... 10.76... 10.942.2... 11.13... 11.31... 11.50... 11.69... 11.88... 12.07... 12.27... 12.47... 12.67... 12.87
inpo- 2.3... 13.07... 13.27... 13.48... 13.68... 13.90... 14.11... 14.33... 14.54... 14.76... 14.98due 2.4... 15.21 ... 15.43... 15.66... 15.89... 16.12... 16.35... 18.58... 16.82... 17.16... 17.30
2.5.., 17.54... 17.78... 18.03... 18.27... 18.52... 18.78... 19.03... 19.28... 19.54... 19.802.6... 20.08... 20.32... 20.58... 20.85... 21.12... 21.38... 21.65... 21.93... 22.20... 22.482.7... 22.75... 23.03... 23.31 ... 23.60... 23.88... 24.17... 24.45... 24.74... 25.03... 25.332.8... 25.62... 25.92... 26.21 ... 26.51... 26.81 ... 27.11... 27.42... 27.72... 28.03... 28.34
(2) 2.9... 28.65... 28.96... 29.27... 29.58... 29.90... 30.21 ... 30.53... 30.85... 31.17... 31.49(2) 3.0... 31.81... 32.13... 32.45... 32.78... 33.11 ... 33.44... 33.77... 34.10... 34.43... 34.77
3.1... 35.10... 35.44... 35.78... 36.11 .. 36.45... 36.79... 37.13... 37.47... 37.82... 38.16from 3.2... 38.50... 38.85... 39.20... 39.55... 39.89... 40.24... 40.59... 40.94... 41.29... 41.65
3.3... 42.00... 42.35... 42.71... 43.08... 43.42... 43.78... 44.14... 44.49... 44.85... 45.213.4... 45.57... 45.93... 46.29... 46.66... 47.02... 47.38... 47.74... 48.11... 48.47... 48.843.5... 49.20... 49.57... 49.94... 50.30... 50.67.. 51.04... 51.40... 51.77... 52.14... 52.51
(21) 3.8... 52..88... 53.25... 53.62... 53.99... 54.36... 54.73... 55.10... 55.48... 55.85... 58.223.7... 56.59... 56.96... 57.33... 57.71 .. 58.08... 58.45... 58.82.... 59.20... 59.57... 59.943.8... 60.31... 60.69... 81.08... 61.44... 61.81 ... 82.18... 82.56... 62.93... 63.30... 63.68
es 3.9... 84.05... 64.42... 64.80... 65.17... 65.55... 65.92... 88.29... 66.67... 67.04... 67.414.0... 67.79... 68.18... 68.53.;.. 68.91... 69.28... 69.65... 70.03... 70.40... 70.77... 71.14
(22) 4.1... 71.52... 71.89... 72.26... 72.63... 73.00... 73.38... 73.75... 74.12... 74.49... 74.864.2... 75.23... 75.60. 75.97... 76.34... 76.71 ... 77.08... 77.45... 77.82... 78.19... 78.56
nte 4.3... 78.93... 79.30... '79.67... 80.04... 80.41 ... 80.78... 81.14... 81.51 ... 81.88... 82.254.the .. 82.61... 82.98... 83.35... 83.61... 84.08... 84.45... 84.81 ... 85.18... 85.55... 85.91
id or 4.5.-. 86.28... 86.64... 87.01... 87.37... 87.73... 88.10... 88.46... 88.82... 89.19... 89.554.6 .. 89.91 ... 90.28... 90.64... 91.00... 91.37... 91.73... 92.09 ... 92.45 ... 92.81... 93.17
priate 4.7... 93.53... 93.89... 94.25... 94.61... 94.97... 95.33... 95.69... 98.05... 96.41... 96.77The 4.8... 97.13... 97.49... 97.85... 98.21... 98.57... 98.92... 99.28... 99.64... .100.00... .100.35
4.9.. .100.71.. .101.07... .101.42... .101.78... .102.14... .102.49... 102.85....103.21.. .103.56.. .103.92iFableterms
(23)
outer
and inner diameters of the annular con-ductor. In comparison wi'th the rigorousBessel function solution for the skin effectin an isolated tubular conductor, it hasbeen found that the 60-cycle ski'n effect of
annular conductor when computed byequation 23 will not be in error by morethan 0.01 in absolute magnitude forcopper or aluminum IPCEA (InsulatedPower Cable Engineers Association) filled
OCTOBER 1957Neher, McGrath-TemPerature and Load Capability of Cable Systems75 755C)CTLOBBR1957
Table IV. Mutual Reactance at 60 Cycles, Conductor to Sheath (or Shield)
Dam/2S 0 1 2 3 4 5 6 7 8 9
0.4. 21.1... 20.5.... 19.9. 10 18.3. 17.8. 17.4.16.9. 16.40.3. 27.7... 26.9... 26.2. 25.5. 24.8. 24.1. 23.5. 22.9.£..22.2. 21.60.2. 37 0... 359.. ..34.8..33.8..32.8..31.9. 31.0.....30.1. 29.3. 28.4O.1. 52.9... 50.7.. .48.7. 46.9. 45.2. 43.6..4'.42.1. 40.7.. 39.4. 38.2
core conductors up through 5.0 CI and forhollow core concentrically stranded copperor alumninum oil-filled cable conductorsup through 4.0 CI.
For values of xp below 3.5, a rangewhich appear to cover most cases of prac-tical interest at power frequencies, theconductor proximity effect for cables inequilateral triangular formation in thesame or in separate ducts may be cal-culated from the following equation basedon an approximate expression given byArnold' (equation 7) for a system ofthree homogeneous, straight, parallel,solid conductors of circular cross sectionarranged in equilateral formation andcarrying balanced 3-phase current remotefrom all other conductors or conductingmaterial. The empirical transverse con-ductance factor k, is introduced to makethe expression applicable to strandedconductors. Experimental results sug-gest the values of kp shown in Table IT.
2~Ycp F(xp)(DC )X
[F(x:; +.27 +0.312(S ) ]2 (24)6.80
XV ~~at 60 cycles (25)V\Rdc/kpWhen the second term in the brackets
is small with respect to the first tern as itusually is, equation 24 may be written
Y=p-4F'x [0.295(Dc/S)S1CY,4F(Xt)F(x,) +0.27.]=4( S F(xp') (24A)
where the function F(xp) is shown inFig. 1.The average proximity effect for con-
ductors in cradle configuration in thesame duct or in separate ducts in a forma-tion approximating a regular polygon may
Table V. SpeciRc Inductive Capacitance ofInsulations
Matera e
Polyethylene .................2.3Paper insulation (solid type) ... 3.7 (IPCEA value)Paper Insulation (other types).. 3.3-4.2Rubber and rubber-like com-pounds ....................5 (IPCXA value)
Varnished cambric............5 (IPCEA value)
also be estimated from equation 24 and24(A). In such cases, S should be takenas the axial spacing between adjacentconductors.The factor Ys is the sum of two factors,
Ys, due to circulating current effect andY. due to eddy current effects.
WS=I'Rdc( Yu + Y8,)watts per conductor foot (26)
Because of the large sheath losses whichresult from short-circuited sheath opera-tion with appreciable separation betweenmetallic sheathed single conductor cables,this mode of operation is usually restrictedto triplex cable or three single-conductorcables contained in the same duct. Thecirculating current effect in three metallicsheathed single-conductor cables arrangedin equilateral configuration is given by
y R,JRdc1 +(R8/Xm)' (27)
When (Rs/Xm)2 is large with respect tounity as usually is the case of shielded non-leaded cables, equation 27 reduces to
Yc=R approximately (27A)
Xm 0.882f log 2S/Dsmmicrohms per foot (28)
=52.9 log 2S/Dsmmicrohms per foot at 60 cycles (28A)
where S is the axial spacing of adjacentcables. For a cradled configuration Xmmay be approximated from
2.52SI IS\Xm=52.9 log -D 1-I I
microhms per foot at 60 cycles (29)= 52.9 log 2.3 S/Dsm
approximately (29A)
Table IV provides a convenient means fordetermining Xm for cables in equilateralconfiguration.The eddy-current effect for single-
conductor cables in equilateral configura-tion with open-circuited sheaths is
y _- 3Rs/Rdc X
so 5 (2Rs 2 ( 2S
2S )2125 DS) (30)
when (5.2 Rl/f)2 is large in respect to 1/5
(2 SiDsm) as in the case of lead sheaths.
396 D,m 5 Ds", 2
RsRds 2S 22approximately at 60 cycles (30A)
When the sheaths are short-circuited, thesheath eddy loss will be reduced and maybe approximated by multiplying equations30 or 30(A) by the ratio
R82/(Ra2+Xm2)
In computing average eddy current forcradled configuration, S should be takenequal to the axial spacing and not to thegeometric-mean spacing. Equations 30and 30(A) may be used to compute theeddy-current effect for single-conductorcables installed in separate ducts.Strictly speaking, these equations applyonly to three cables in eqwulateral con-figuration but can be used to estimatelosses in large cable groups when latter areso oriented as to approximate a regularpolygon.The eddy-current effect for a 3-conduc-
tor cable is given by Arnold.6
3RM (2s/Dsm)2 (2s/Dam)4Yae8 + +Rdc 5.2R8\s 52,
f +1 41-1+1
(2s/Dsm )"
(5.2Rf) +1 I(31)
When (5.2R,/f)2 is large with respect tounity,
396 ( 2s\RsRdc (5Dam)
approximately at 60 cycles (31A)
s=1.155T+0.6OXthe V gauge depth forcompact sectors
1.155T+0.58 D, for round conductors(32)
and T is the insulation thickness, includ-ing thickness of shielding tapes, if any.While equation 31(A) will suffice for leadsheath cables, equation 31 should be usedfor aluminum sheaths.On 3-conductor shielded paper lead
cable it is customary to employ a 3- or 5-mil copper tape or bronze tape inter-calated with a paper tape for shielding andbinder purposes. The lineal d-c resist-ance of a copper tape 5 mils by 0.75 inchis about 2,200 microhms per foot of tapeat 20 C. The d-c resistance per footof cable will be equal to the lineal resist-ance of the tape multiplied by the laycorrection factor as given by the expres-sion under the square-root sign in equation13. In practice the lay correction factormay vary from 4 to 12 or more resultingin shielding and binder assembly resist-
Neher, McGrath-Temperature a-nd Load Capability of Cable Systems756 OcrOBER 1957
ances of approximately 10,000 or moremicrohms per foot of cable. Even onthe assumption that the assembly resist-ance is halved because of contact with ad-jacent conductors and the lead sheathcomputations made using equations 27and 30 show that the resulting circulatingand eddy current losses are a fraction of1% on sizes of practical interest. For thisreason it is customary to assume that thelosses in the shielding and binder tapesof 3-conductor shielded paper lead cableare negligible. In cases of nonleaded rub-ber power cables where lapped metallictapes are frequently employed, tubeeffects may be present and may materiallylower the resistance of the shielding assem-bly and hence increase the losses to apoint where they are of practical signifi-cance.An exact determination of the pipe loss
effect Yp in the case of single-conductorcables installed in nonmagnetic conduitor pipe is a rather involved procedureas indicated in reference 7. Equation 31may be used to obtain a rough estimateof Yp for cables in cradled formation onthe bottom of a nonmagnetic pipe, how-ever by taking the average of the resultsobtained for wide triangular spacingwith s=(Dp-D,)/2 and for close tri-angle spacing at the center of the pipewith s=0.578 D. The mean diameter ofthe pipe and its resistance per foot shouldbe substituted for Dam and R8 respectively.
For magnetic pipes or conduit thefollowing empirical relationships8 may beemployed
1.54s-0.115D, (3-conductor cable)Rdc
(33)
Y= 0.89S-0.11D(single-conductor,Rdc
close triangular) (34)
Y=0.34S+0.175D (single-conductor,Rdc
cradled) (35)
These expressions apply to steel pipe8and should be multiplied by 0.8 for ironconduit."The expressions given for Y, and Y.
above should be multiplied by 1.7 to findthe corresponding in-pipe effects for mag-netic pipe or conduit for both triangularand cradled configurations.
CALCuLATION OF DIELECTic Loss
The dielectric loss Wd for 3-conductorshieldedsdand single-conductor cable isgiven by the expression
= 0.00276E'er cos qbwatts per
log (2T+Dc)/Dcconductor foot at 60 cycles (36)
and for 3-conductor belted cable by'
0.Ol9E2er cos 0'Wd= watts per
G2conductor foot at 60 cycles (37)
where E is the phase to neutral voltagein kilovolts, er is the specific inductivecapacitance of the insulation (Table V) Tis its thickness and cos 4 is its power factor.The geometric factor G2 may be foundfrom Fig. 2 of reference 1.
For compact sector conductors the di-electric loss may be taken equal to that fora concentric round conductor having thesame cross-sectional area and insulationthickness.
Calculation of Thermal Resistance
THERMAL RESISTANCE OF THE INSULATION
For a single conductor cable,
Ri= 0.012pi log D,/D, thermal ohm-feet(38)
where At is the thermal resistivity of theinsulation (Table VI) and Di is itsdiameter. In multiconductor cablesthere is a multipath heat flow between theconductor and sheath. The following ex-pression1 represents an equivalent valuewhich, when multiplied by the heat flowfrom one conductor, will produce theactual temperature elevation of theconductor above the sheath.
Rj =0.00522A,G, thermal ohm-feet (39)
Values of the geometric factor G, for 3-conductor belted and shielded cables aregiven in Fig. 2 and Table VIII respec-tively of reference 1. On large size sec-tor conductors with relatively thin in-sulation walls (i.e. ratios of insulationthickness to conductor diameter of theorder of 0.2 or less); values of GI for 3-conductor shielded cable as determinedby back calculation, on the basis of anassumed insulation resistivity, from lab-oratory heat-run temperature-rise data,have not always confirmed theoreticalvalues, and, in some cases, have yieldedGI values which approach those for anonshielded, nonbelted construction.
Table VI. Thermal Resistivity of VeriounsMaterils
Material ,o, C Cm/Watt
Paper insulation (solid type). ..700 (IPCEA value)Varnished cambric............ 600 (IPCEA value)Paper insulation (other types).. 500-550Rubber and rubber-like........ 500 (IPCEA value)Jute and textile protective
covering ................... 50Fiber duct ................... 480Polyethylene ................. 450Transite duct ................ 200Somastic................. 100Concrete ..................... 85
THERMAL RESISTANCE OF JACKETS, DucTWALLS, AND SOMASTIC COATINGS
The equivalent thermal resistance ofrelatively thin cylindrical sections such asjackets and fiber duct walls may bedetermined from the expression
R=0.0104fin'(D ) thermal ohm-feet
(40)
with appropriate subscripts applied toA, pi, and D in which D represents theoutside diameter of the section and t itsthickness. n' is the number of conductorscontained with the section contributingto the heat flow through it.
THERMAL RESISTANCE BETWEEN CABLESURFACE AND SURROUNDING PIPE,CONDUIT, OR DUCT WALL
Theoretical expressions for the thermalresistance between a cable surface and asurrounding enclosure are given in refer-ence 10. As indicated in Appendix I,these have been simplified to the generalform
Rd =I +(B+ CTm)Ds' thermal ohm-feet
(41)
in which A, B, and C are constants, D,'represents the equivalent diameter of thecable or group of cables and n' the numberof conductors contained within D/'. Tmis the mean temperature of the interven-ing medium. The constants A, B, and C
Table VIL Constants for Use in Equations 41 and 41(A)
Condition A B C A' B'
In metallic conduit ................... 17 ... ...... 3.6 ........ 0.029 .... ............... 0.19In fiber duct in air ................... 17 ......... 2.1 .........0.016 .........5.6 . 0.33In fiber duct in concrete .............. 17 ... ...... 2.3 ........ 0.024 .........4.6 . 0.27In transite duct in air ................ 17 .........3.0 ......... 0.014 .........4.4 . 0.26In transite duct in concrete ........... 17 ......... 2.9 .........0.029 .........3.7 . 0.22Gas-filled pipe cable at 200 psi ......... 3.1 ......... 0...... ..0.0053. ..2.1. 0.68Oil-filled pipe cable. 0.84......... . ......... 0.0065...... 2.1 . 2.45
D-'1 .00 Xdiameter of cable for one cable1 .65 X diameter of cable for two cables2.15 Xdiameter of cable for three cables2.50 X diameter of cable for four cables
Neher, McGrath-Temperature and Load Capability of Cable SystemsOMrOBriR 1957 757
given in Table VII have been determinedfrom the experimental data given in refer-ences 10 and 11.
If representative values of Tm =60 Care assumed, equation 41 reduces to
A,u - "'Athermal ohm-feet (41A)
It should be noted that in the case ofducts, A. is calculated to the inside of theduct wall and the thermal resistance ofthe duct wall should be added to obtainA,..
THERMAL RESISTANCE FROM CABLES,CONDUITS, OR Duc'rs SUSPENDED INAmRThe thermal resistance R. between
cables, conduits, or ducts suspended in stillair may be determined from the followingexpression which is developed in Ap-pendix I.
15.6n'Ds' [(AT/D.')""4+1.6f(1 +0.0167Tm)I
thermal ohm-feet (42)
In this equation AT represents the differ-ence between the cable surface tempera-ture T, and ambient air temperature Ta indegrees centigrade, Tm the average ofthese temperatures and e the coefficient ofemissivity of the cable surface. Assum-ing representative values of TJ=60 andTa=30 C, and a range in D.' of from 2to 10 inches, equation 42 may be simplifiedto
9.5n'R.5=' thermal ohm-feet1+1.7D.'(e+0.41)(42A)
The value of e may be taken as equalto 0.95 for pipes, conduits or ducts, andpainted or braided surfaces, and from 0.2to 0.5 for lead and aluminum sheaths,depending upon whether the surface isbright or corroded. It is interesting tonote that equation 42(A) checks theIPCEA method of determining R. veryclosely with e=0.41 for diameters up to3.5 inches. In the IPCEA method R,=0.00411 n'B/Ds' where B=650+314 Ds'for
Ds'=0-1.75 inches and B 1,200 for largervalues of Ds'
EFFECTIVE THERMAL RESISTANCEBETWEEN CABLES, DUCTS, OR PIPES,AND AMBIENT EARTH
As previously indicated, an effectivethermal resistance A,' may be employed torepresent the earth portion of the thermalcircui't in the case of buried cable systems.This effective thermal resistance includesthe effect of loss factor and, in the case ofa multicable installation, also the mutual
heating effects of the other cables of thesystem. In the case of cables in a con-crete duct bank, it is desirable to furtherrecognize a difference between the thermalresistivity of the concrete Pc and thethermal resistivity of the surroundingearth Ae.The thermal resistance between any
point in the earth surrounding a buriedcable and ambient earth is given by theexpression1lRpa =0.012p. log d'/d thermal ohm-feet
(43)in which As is the thermal resistivity of theearth, d' is the distance from the imageof the cable to the point P, and d is thedistance from the cable center to P.From this equation and the principlesdiscussed in references 3, 12, and 13, thefollowing expressions may be developed,applicable to directly buried cables andto pipe-type cables.
R,' =0.012Asn'X
[log Dx+(LF) log [() F]]
thermal ohm-feet (44)
iti which Ds is the diameter at which theearth portion of the thermal circuit com-mences and n' is the number of conduc-tors contained within De. The fictitiousdiameter D. at which the effect of lossfactor commences is a function of thediffusivity of the medium a and the lengthof the loss cycle.3
D,= 1.02-\/a(length of cycle in hours)inches (45)
The empirical development of this equa-tion is discussed in Appendix III. For adaily loss cycle and a representative valueof a = 2.75 square inches per hour forearth, Dx is equal to 8.3 inches. It shouldbe noted that the value of Dx obtainedfrom equation 45 is applicable for pipediameters exceeding D-, in which case thefirst term of equation 44 is negative.The factor F accounts for the mutual
heating effect of the other cables of thecable system, and consists of the productof the ratios of the distance from thereference cable to the image of eachof the other cables to the distance to thatcable. Thus,
F-d12 (d13'\(dIN'\terms(d12 )ld13 ) d,NI)
(46)
It will be noted that the value of F willvary depending upon which cable isselected as the reference, and the maxi-mum conductor temperature will occurin the cable for which 4LF/D. is a maxi-
mum. N refers to the number of cables orpipes, and F is equal to unity when N= 1.When the cable system is contained
within a concrete envelope such as aduct bank, the effect of the differingthermal resistivity of the concrete en-velope is conveniently handled by first as-suming that the thermal resistivity of themedium is that of concrete pc through-out and then correcting that portion ly-ing beyond the concrete envelope to thethermal resistivity of the earth A.. Thus
R0 = 0.012pcnIX
-D+(LF) log [(D F +
0.012(A,6 - Ac)n'N(LF)Gbthermal ohm-feet (44A)
The geometric factor Gb, as developedin Appendix II is a function of the depthto the center of the concrete enclosureLb and its perimeter P, and may be foundconveniently from Fig. 2 in terms of theratio Lb/P and the ratio of the longest toshort dimension of the enclosure.For buried cable systems Ta should be
taken as the ambient temperature at thedepth of the hottest cable. As indicatedin reference 12, the expressions usedthroughout this paper for the thermalresistance and temperature rise of buriedcable systems are based on the hypothe-sis suggested by Kennelly applied inaccordance with the principle of super-position. According to this hypothesis,the isothermal-heat flow field and tem-perature rise at any point in the soil sur-rounding a buried cable can be representedby the steady-state solution for the heatflow between two parallel cylinders(constituting a heat source and sink)located in a vertical plane in an infinitemedium of uniform temperature andthermal resistivity with an axial separa-tion between cylinders of twice the actualdepth of burial and with source and sinkrespectively generating and absorbingheat at identical rates, thereby resultingin the temperature of the horizontal mid-plane between cylinders (i.e., correspond-ing to the surface of the earth) remaining,by symmetry, undisturbed.The principle of superposition, as
applied to the case at hand, can be statedin thermal terms as follows: If the ther-mal network has more than one source oftemperature rise, the heat that flows atany point, or the temperature drop be-tween any two points, is the sum of theheat flows and temperature drops atthese points which would exist if eachsource of temperature rise were consideredseparately. In the case at hand, thesources of heat flow and temperature riseto be superimposed are, namely, the heat
Neher, McGrath-Temperature and Load Capability of Cable Systems OCTOIBER 1957758
from the cable, the outward flow of heatfrom the core of the earth, and the in-ward heat flow solar radiation, and, whenpresent, the heat flow from interferingsources. By employing as the ambienttemperature in the calculations the tem-perature at the depth of burial of thehottest cable, the combined heat flowfrom earth core and solarradiation sourcesis superimposed upon that produced atthe surface of the hottest cable by theheat flow from that cable and interferingsources which are calculated separatelywith all other heat flows absent. Thecombined heat flow from earth core andsolar sources results in an earth tempera-ture which decreaseswithdepth insummer;increases with depth in winter; remainsabout constant at any given depth on theaverage over a year; approximates con-stancy at all depths at midseason, andin turn results in flow of heat from cablesources to earth's surface, directly to sur-face in midseason and winter and in-directly to surface in summer.
Factors which tend to invalidate thecombined Kennelly-superposition princi-ple method are departure of the tempera-ture of the surface of earth from a trueisothermal (as evidenced by melting ofsnow in winter directly over a buriedsteam main) and nonuniformity ofthermal resistivity (due to such phe-nomena as radial and vertical migrationof moisture). The extent to which theKennelly-superposition principle methodis invalidated, however, is not of practicalimportance provided that an over-all oreffective thermal resistivity is employed inthe Kennelly equation.
Special Conditions
Although the majority of cable tem-perature calculations may be made bythe foregoing procedure, conditions fre-quently arise which require somewhatspecialized treatment. Some of theseare covered herein.
EMERGENCY RATINGS
Under emergency conditions it is fre-quently necessary to exceed the statednormal temperature limit of the conductorTC and to set an emergency temperaturelimit T6'. If the duration of the emer-gency is long enough for steady-state con-ditions to obtain, then the emergencyrating I' may be found by equation 9substituting Tc' for TC and correcting Rd,for the increased conductor temperature.
If the duration of the emergency is lessthan that required for steady-state con-ditions to obtain, the emergency ratingof the line may be determined from
I iTC'- RdRdc(l+Yc)(1c+'-Y-)-(Ta+' Td) kiloamperes (47)z ~~~RdA8l Yc)Rcs'
in which R,c' is the effective transientthermal resistance of the cable system forthe stated period of time. Proceduresfor calculating RAt' for times up to severalhours are given in reference 14, and forlonger times in references 15-17.
THE EFFECT OF EXTRANEOUS HEATSOURCESIn the case of multicable installations
the assumption has been made that allcables are of the same size and are sim-ilarly loaded. When this is not the casethe temperature rise or load capabilityof one particular equal cable group may bedetermined by treating the heating effectof other cable groups separately, intro-ducing an interference temperature riseATint in equations I and 9. Thus
T- Ta=, ATc+Ta+AThntdegrees centigrade (1A)
= Tc-(Ta+ATd+ATfnt)Rdc(l + Yc)R?ca'
kiloamperes (9A)
in which ATin, represents the sum of anumber of interference effects, for eachof which
ATn [Wcq.(LF)+WdJR in
degrees centigrade (48)fn '=0.012p,n' log Fj.g thermal ohm-feet
(49)
Fn =(d1j')(d2i')(daj1). .. .dNi' (N term)(dF(d (d2i)(d3j) ... d (N
(50)
where the parameters apply to each sys-tem which may be considered as a unit.For cables in duct
Rint = 0.012n'I[c log Fint+N(f3-AC)GX]thermal ohm-feet (49A)
Because of the mutual heating betweencable groups, the temperature rise of theinterfering groups should be rechecked.If all the cable groups are to be givenmutually compatible ratings, it is neces-sary to evaluate Wc for each group bysuccessive approximations, or by settingup a system of simultaneous equations,substituting for W, its value by equation15 and solving for I.In case AT,nt or a component of it is
produced by an adjacent steam main, thetemperature of the steam T, rather thanthe heat flow from it is usually given.Thus
ATtnt[Ty-Ta }
e c. ti d (degrees centigrade (51 )
where R4. is the thermal resistance be-tween the steam pipe and ambient earth.
AERIL CABLESIn the case of aerial cables it may be
desirable to consider both the effects ofsolar radiation which increases the tem-perature rise and the effect of the windwhich decreases it.24 Under maximumsunlight conditions, a lead-sheathed cablewill absorb about 4.3 watts per foot perinch of profile"8 which must be returnedto the atmosphere through the thermalresistance Rc/n'. This effect is con-veniently treated as an interferencetemperature rise according to the rela-tionshipATing =4.3DJ'Re/n'
degrees centigrade (47A)For black surfaces this value should beincreased about 75%.As indicated in Appendix II, the follow-
ing expression for R. may be used whereV,O is the velocity of the wind in miles perhour
3.5n'
D,'( V/ V, /DK' +0.62e)thermal ohm-feet (42B)
USE OF Low-REsisTIVTy BACKFLLIn cases where the thermal resistivity
of the earth is excessively high, the valueof R.' may be reduced by backfilling thetrench with soil or sand having a lowervalue of thermal resistivity. Equation44(A) may be used for this case if of, thethermal resistivity of the backfill is sub-stituted for fi. and Gb applies to thezone having the backfill in place of thezone occupied by the concrete.
SINGLE-CONDUCrOR CABLES IN DUcrwITH SOLIDLY BONDED SHEATHSThe relatively large and unequal sheath
losses in the three phases which may resultfrom this type of operation may be deter-mined from Table VI of reference 1. Itwill be noted that
DC; = (1R )RdcJ121 Rft 12/
Ysa (;) I (52)
where expressions for I,,2/I2 etc., appearin the table. The resulting unequal valuesof YC in the three phases will yield unequalvalues of q., and equation 5 becomes forphase no. 1, the instance given as equa-tion 5(A) on the following page.
Neher, McGrath-Temperature and Load Capability of Cable SystemsOCTOBER 1957 759
ATec Wc[RA+qa,IRas+R,,+(LF)RspI +Nq,.(LF)Rp.] thermal ohm-feet (SA)
where qsa is the average of q,,, q4s, and q,3.
ARNOREI) CABLES
In multiconductor armored cables aloss occurs in the armor which may beconsidered as an alternate to the conduitor pipe loss. If the armor is nonmag-netic, the component of armor loss Y.to be used instead of Y, in equations 14and 19 may be calculated by the equa-tions for sheath loss substituting theresistance and mean diameter of thearmor for those of the sheath. In cal-culating the armor resistance, accountshould be taken of the spiralling effect forwhich equation 13 suitably modifiedmay be used. If the armor is mag-netic, one would expect an increase inthe factors Y, and Y. in equation 14since this occurs in the case of magneticconduit. Unfortunately, no simple pro-cedure is available for calculating theseeffects. A rough estimate of the induc-tive effects may be made by using the pro-cedure given above for magnetic conduit.A simple method of approximating the
losses in single conductor cables with steel-wire armor at spacings ordinarily em-ployed in submarine installations is to as-sume that the combined sheath and armorcurrent is equal to the conductor current.1The effective a-c resistance of the armormay be taken as 30 to 60% greater thanits d-c resistance corrected for lay as in-dicated above. If more accurate calcula-tions are desired references 19 and 20will be found useful.
EFFECT OF FORCED COOLING
The temperature rise of cables in pipesor tunnels may be reduced by forcing airaxially along the system. Similarly, inthe case of oil-filled pipe cable, oil maybe circulated through the pipe. Underthese conditions, the temperature rise isnot uniform along the cable and increasesin the direction of flow of the coolingmedium. The solution of this problem isdiscussed in reference 21.
Appendix I
Development of Equations 41, 42,and Table VII
Theoretical and semiempirical expressionsfor the thermal resistance between cablesand an enclosing pipe or duct wall are
given in reference 10. Further data on thethermal resistance between cables andfiber and transite ducts are given in ref-erence 11. For purposes of cable rating,it is desirable to develop standardizedexpressions for these thermal resistances
Table VIII. Constants For Use in Equation 53
AverageCondition a b c AT
Cable in metallic conduit ...................0.07 .. 0.121 ... 0.0017 ......,.20Cable in fiber duct In air. ................. 0.07 .. 0.036 .. 0,.,.O..0009. 20Cable In fiber duct in concrete ........ .....0.0 7 ..0.043 . 0..0.0014. , 20Cable in transte duct in air ................0.07... ,0.086.0.0008.. .. 20Cable in transite duct In concrete........... 0.07.. 0.079.0.0016.... 20Gas-filled pipe-type cable at 200 psi......... 0.07.. 0.121.0.0017.... 10
based upon all of the data available andincluding the effect of the temperature ofthe intervening medium.The theoretical expression for the case
where the intervening medium is air or gasas presented in reference 10 may be general-ized in the following form:
Rsd r ATPS 1/ (53)DJ' [a( D,7) +b+cTm]
where
A.d = the effective thermal resistance be-tween cable and enclosure in thermalohm-feet
D,J'=the cable diameter or equivalentdiameter of three cables in inches
AT= the temperature differential in degreescentigrade
P=the pressure in atmospheresTm=mean temperature of the medium in
degrees centigraden'=number of conductors involved
The constants a, b, and c in this equationhave been established empirically as follows:Considering b+cT, as a constant for themoment, the analysis given in reference10 results in a value of a = 0.07. With athus established, the data given in reference10 for cable in pipe, and in reference 11for cable in fiber and transite ducts wereanalyzed in similar manner to give thevalues of b and c which are shown in TableVIII.
In order to avoid a reiterative calculationprocedure, it is desirable to assume a valuefor AT since its actual value will dependupon A.d and the heat flow. Fortunately,as AT occurs to the 1/4 power in equation53, the use of an average value as indicatedin Table VIII will not introduce a seriouserror.By further restricting the range of
D,' to 1-4 inches for cable in duct orconduit and to 3-5 inches for pipe-typecables, equation 53 is reduced to equation41.
1_8_+(B;thermal ohm-feet1+(B+CTm)Da'
(41)
in which the values of the constants A,B, and C appear in Table VII.
In the case of oil-filled pipe cable, theanalysis given in reference 10 gives thefollowing expression
n'0.d0.60+0.025(Da'?TmSAT)114
thermal ohm-feet (54)
Assuming an average value of AT= 7 C
and a range of 150-350 for D,'Tm, equation54 reduces to equation 41 with the valuesof A, B, and C given in Table VII .
In the case of cables or pipes suspendedin still air, the heat loss by radiation maybe determined by the Stefan-Bolzmannformula
n'W (radiation)0.139Ds'e[( T +273)4'-( Ta +273)4 10'6
watts per foot (55)
where e is the coefficient of emissivityof the cable or pipe surface. Over thelimited temperature. range in whch we areinterested, equation 55 may be simplifiedto'0n'W (radiation) -0.102D,'ATeX
(1 +0.0167Tm) watts per foot (55A)Over the same temperature range the
heat loss by conveetion from hoizontalcables or pipes is given with sufficientaccuracy by the expression
n'W(convection) -0.064 Ds'AT(AT/ID').14watts per foot (56)
in which the numerical constant 0.064has been selected for the best fit with thecarefully determined test results reportedby Heilman2' on 1.3, 3.5 and 10.8-inchdiameter black pipes ( = 0.95). Inci-dentally, this value also represents thebest fit with the test data on 1.9-4.5 inchdiameter black pipes reported by Rosch."For vertical cables or pipes the value ofthis numerical constant may be increasedby 22%."2Combining equations 55(A) and 56 we
obtain the relationship
ATn'W (total)
15.6n'Ds'1(AT/Ds')"/'+1.6e(l +0.0167Tm)]
thermal ohm-feet (42)
If the cable is subjected to wind havinga velocity of V. miles per hour, the follow-ing expression derived from the work ofSchurig and Frick'4 should be substitutedfor the convection component.
n'W (convection) =0.286Ds'AT-1Vw1/D8'watts per foot (56A)
Combining equations 55(A) and 56(A)with Tm=45 C
AT 3.5n'6tn'W(total) Dhe(maVlohDmt+4.62B)
thermal ohm-feet (42B)
Neher, McGrath-Temperature and Load Capability of Cable Systems OCTOBER 1957760
Appendix 11Determination of the Geometric
Factor Gb for Duct BankConsiderimg the surface of the duct
bank to act as an isothermal circle ofradius ra, the thermal resistance betweenthe duct bank and the earth's surface willbe a logarithmic function of rb and Lb thedistance of the center of the bank belowthe surface. Using the long form of theKennelly Formula's we may define thegeometric factor Gb as
Ga- log La'+VLbS- arb
-log [La/ra+N/(Lb/rt)2-11 (57)
In order to evaluate rb in terms of thedimensions of a rectangular duct bank, letthe smaller dimension of the bank be xand the larger dimension by y. The radiusof a circle inscribed within the duct banktouching the sides is
r -x/2 (58)
and the radius of a larger circle embracingthe four cowers is
r2=-\/X2+y2 (59)2
Let us assume that the circle of radius rblies between these circles and the magnitudeof ra is such that it divides the thermalresistance between ri and rs in directrelation to the portions of the heat fieldbetween ri and r2 occupied and unoccupiedby the duct bank. Thus
rb xy-7rr,2/ 2log = (r- log or
r2 Trrf5X7 / tlog-m lowg-ta w(t,'-71ri) \ 71from which
log rb - _-) log (1+ ) +log
(60)It is desirable to derive ra in terms of theperimeter P of the duct bank. Thus
P=2(x+y) 4x
( +y/x)2
and therefore
log - -log p(12 4(1+y/x) (61)
The curves of Fig. 2 have been developedfrom equations 57, 60, and 61 for severalvalues of the ratio y/x. It should benoted in passing that the value of rb -0.112P used in reference 13 applies to ay/x ratio of about 2/1 only.
Appendix III
Empirical Evaluation of D2
In order to evaluate the effect of a cyclicload upon the maximum temperature riseof a cable system simply, it is customary toassume that the heat flow in the final
Table IX. Comparison of Values of % (AF)for Sinusokial Los Cycles at 30%
Los Factor
% (AM)Description,
System Inches Neher Shankln Wiseman
I. 4.5 pipe.....63/63 ...61/62... 63/65II. 6.0 pipe.....56/56 ... 60/57....53/60III. 8.6 pipe.....56/56... 59/58 .. 54/63IV. 10.6 pipe.....58/58 ... 61/59... 65/53V. . 5 cable..... 80/80VI. 1.5 cable.... 77/76.. .77/76 ... 77/77Vii. 1. 9 cable....71/71VIII.... 2.0 cable..... 63/62IX. 2.0 cable..... 75/74X. 3.4 cable..... 77/78Xa* . 3.4 cable... .83/80.. 83/81XI. 3.7 cable....76/74.. 74/73XII . 4.2 cable....70/66.. .70/67XIII.... 4.5 cable.... 69/64.. .65/64....61/63* Diffuslvity -4.7 square Inches per hour.
portion of the thermal circuit is reducedby a factor equal to the loss factor of thecyclic load. The point at which thisreduction commences may be convenientlyexpressed in terms of a fictitious diameterDr. Thus
Artea =A+(LF)Ara thermal ohm-feet (62)For greater accuracy, it is desirable toestablish the value of Dz empirically ratherthan to assume that Ds is equal to thediameter D, at which the earth portion ofthe thermal circuit commences.
Equation 62 may be written in the form
Aa'-Re.+A6x+(LF)(42 -R&2)thermal ohm-feet (62A)
In terms of the attainment factor (A F), onemay write
RcA< (A F)Rca-(A F)(ce+Rea)thermal ohm-feet (63)
Equating equations 62(A) and 63 obtainsthe relationship
A.u (1-x),R.-xAc, thermal ohm-feet (64)
where
I1-(AF)1-(LF)
Since
(65)
A=0.012n', log D2/D.thermal ohm-feet (66)
log Dz/D - , [(1-x)*a.-xR cS] (67)
The first paper of reference 3 presentsthe results of a study in which a numberof typical daily loss cycles and also sinu-soidal loss cycles of the same loss factorwere applied to a number of typical buriedcable systems. The results indicated thatin all cases the sinusoidal loss cycle of thesame loss factor adequately expressed themaximum temperature rise which wasobtained with any of the actual loss cyclesconsidered.An analysis by equations 65 and 67 of
the calculated values of attainment factorsfor sinusoidal loss cycles given in Table IIand the corresponding cable system param-eters given in Table I of the first paper ofreference 3 yields a most probable value of
D =-8.3 inches. As indicated in the thirdpaper of reference 3, however, theoreticallyDr should vary as the square root of theproduct of the diffusivity and the timelength of the loading cycle. Hence as thediffusivity was taken as 2.76 square inchesper hour in the above,
Ds i.02XVa#Xlength of cycle in hours inches
(45)Table IX presents a comparison of the
values of per cent attainment factor forsinusoidal loss cycles at 30% loss factor ascalculated by equations 45,8,62(A), and 63and as they appear in Table II of the firstpaper of reference 3.
Appendix IV. Calculations ForRepresentative Cable Systems
15-Kv 350-MCM-3-ConductorShielded Compact Sector Paper andLead Cable Suspended in Air
D O0.616 (equivalent round); V-gaugedepth =0.539 inch
D. 2.129; T=0.175 inch; t=0.120 inch
12.9 1234.5+81~Tc~81 C; R1c 0.350 234.5+75,
-37.8 microhms per foot (Eq. IOA)
D,,,2.129-0.1202.009 inches (Eq. 12)
37.9R. _ =7 157 m*crhmR,2.009(0.120) -17mcom
per foot at 50 C (Eq. 11A)
k, 1.0; k,=0.6 (equivalent round)(Table II)
Rdc/ke =37.8; Ycs =0.008(Eq. 21 and Fig. 1)
S-0.616 +2(0.175+0.008) =0.982 inches
Rd,/kp-62.6; F(x,')=0.003 (Fig. 1)
YCP 2[14(0o192 0]00)30.002(Eq. 24A, and note to Table II)
1+ Yc-1 +0.008+0.002 - 1.010
s= 1.155(0.175+0.008)+0.60(0.539)-0.534 inch (Eq. 32)
396 2(0.534))2= -157(37.6)1 2.009 -.1
Rac/R410 1.010+0.019 = 1.029
0.019q,2=qg =1+j--1 19
(Eq. 31A)
(Eq. 14)
(Eqs. 18-19)
, = 3.7 (Table V); E=15// -8.7;cos -0.022
Wd 0.00276 (8.7)'[3.7(0.022)]d g2(0.175)+0.681g 0.681
-0.094 watt per conductor foot(Eq. 36 and text)
(Note: In computing dielectric loss on
Neher, McGrath-Temperature and Load Capability of Cabk SystemsOCTrOBE,R1957 761
sector conductors, the equivalent diameterof the conductor is taken equal to that of aconcentric round conductor, i.e., 0.681inch for 350 MCM.)
pA=700 (Table VI); GI=0.45(Table VIII of reference 1)
/i =0.00522{700(0.45)}= 1.64thermal ohm-feet (Eq. 39)
n'-=3; e-0.41 (assumed)
A.= 9.5(3)1+1.7(2.129(0.41+0.41)]
=7.18 thermal ohm-feet (Eq. 42A)
fica = 1.64+1.019(7.18) = 8.96thermal ohm-feet (Eq. 8)
ATd-0.094(0.82 +7.18) = 0.75 C (Eq. 6)
Ta =40 C (assumed)
I = 81-(40+0.8)37.6[1.010(8.96)]= 0.344 kiloampere (Eq. 9)
If the cable is outdoors in sunlight andsubjected to an 0.84 mile per hour wind
3.5(3)2.129 -VO.84/2.129+0.62(0.41)1
=5.59 thermal ohm-feet (Eq. 42B)
Rca'= 1.64+1.019(5.59) = 7.34thermal ohm-feet (Eq. 8)
ATtn g = (4.3)(2.129) - = 17.1 C
(Eq. 47A)Ta - 30 C (assumed)
I-181 -(30+0.6+17.1)(37.6)(1.010)(7.34)=0.346 kiloampere (Eq. 9)
In this particular case the net effect ofsolar radiation and an 0.84 mile per hourwind is to effectively raise the ambienttemperature by 10 degrees, which is arough estimating value commonly used.It should be noted, however, that thiswill not always be true, and the procedureoutlined above is preferable.
69-Kv 1,500-MCM-Single-'Conductor Oil"Filled Cable in DuctTwo identical cable circuits will be
considered in a 2 by 3 fiber and concreteduct structure having the dimensionsshown in Fig. 3.
Do=0.600; Dc=1.543; Di=2.113;T=0.285; D8 =2.373; t =0.130 inches
T0=75C; 12.9Tc= 75 C; Rdc15= 8.60
1.50microhms per foot (Eq. 1OA)
DJ, =2.373 -0.130=2.243 inches (Eq. 12)
Ro= 37.9 1 -= 130 microhms(2.243)(0.130)
per foot at 50 C (Eq. 11A)
1.543-0.600 1.543+1.200\'k
1.543 +0.600 1.543 +0.600/=0.72; kp=0.8 (Eq. 23 and Table II)
%0- --.
- I%
d131% 96' K'-
dg3 -96"
(I\
/.. I.49 Images\7 -.(13 \
'
d42d2z 96.5"
d :87.5"
I d---- cl'3,- 78.5"
Lb ' 43.5"
Fig. 3. Assumed duct bank confAguration for typical calculations on 69-kv 1,500-MCMoil-Ailed cable (Appendix IV)
Rde/k = 11.9; Yc1= 0.075(Eq. 21 and Fig. 1)
S=9.0 (Fig. 3); Rdc/kp=10.75;F(xp')=0.075 (Fig. 1)
Ycp=4(92 0.075=0.007 (Eq. 24A)9.0/
1+ Ye= 1+0.075+0.007= 1.082
Assuming the sheaths to be open-circuited,
396 12.243 2
"~=130(8.60) \2(9.0)/
5/2.243\'01 0 (q 3A
[l122(9.0))Rac/Rde 1.082+0.006 = 1.088 (Eq. 14)
0.006
q=q,s1=+ 8=1.006 (Eqs. 18-19)1.082
er= (Table V); E 69/"3--40;cos #=0.005
0.00276(40)2(3.5)(0.005)2.113
log -~1.543
=0.57 watt per conductor foot (Eq. 36)
Neher, McGrath-Temperature and Load Capability of Cable Systems
18"
762 OC,rOBIER 1957
p=550 (Table VI)
R2=0.012 (550 log 1-543=0.90 thermal ohm-foot (Eq. 38)
n'=1; R =2.370.2 =174
thermal ohm-feet (Eq. 41A)
Pd = 480 (Table VI); t= 0.25;De=5.0+0.5=5.50 for fiber duct
- 0.0104(480)(0.25) = 0.245.50-0.25
thermal ohm-foot (Eq. 40)
p.= 120 (asumed); 1.=85 (Table VI);L=L&=43.5 inches (Fig. 3)
N=6; (LF) 0.80 (assumed);
( 96.5 (87.5)(78.5)\9,I9 12.7\ 9 /12.7)=42,200 (Fig. 3 and Eq. 46)
Lbp43.5 048;27 =15
2(18+27) 18
Gb=0.87 (Fig. 2)
Re' (at 80% loss factor) =(0.012)(85)(1)X
(log83+0.80 log (42,200)]) +5.5 ~~~8.3
0.012(120 -85)(1)(6)(0.80)(0.87)=6.79 thermal ohm-feet (Eq. 44A)
R8 (at unity loss factor)=8.44thermal ohm-feet (Eq. 44A)
Rca' =0.90+1.006 (1.74+0.24+6.79)=9.72 thermal ohm-feet (Eq. 8)
ATd=0.57( +1.74+0.24+8.44)
=6.2 C (Eq. 6)
Ta=25 C (assumed);1 75-(25+6.2)8.60(1.082)(9.72)= 0.696 kiloampere (Eq. 9)
To illustrate the case where the cablecircuits are not identical, consider thesecond circuit to have 750-MCM con-ductors. For the first circuit,
N=3; (LF)=0.80 (assumed);
F (6)(78) =92.4 (Eq. 46)
Rc' =0.012(85)(1)Xlo
8.3+0 o (-4(43.5) 92.4\1J
[o5.5 o 8.30.012(120- 85)(1)(3)(0.80)(0.87)
3.74 thermal ohm-feet (Eq. 44A)F~(96.4V87.5\ (78.5 4
9 /(12.7)(Eq. 50)
Ring 0.012(1)X[85 log 456 +3(120 -85)(0.87)]= 3.81 thermal ohm-feet (Eq. 49)
Ra' = 0.90 +1.006(l.74+0.24 +3.74)=6.65 thermal ohm-feet (Eq. 8)
ATd=0.57(0.45+1.75+0.24+4.63)=4.0 C(Eq. 6)
Wc,-(II2)(8.60)( 1.082) =9.3 1 I, 2
watts per conductor foot (Eq. 15)
A Tint= (9.31121( 1.006)(0.80) +0.571)3.81=2.17+28.5j12 degrees centigrade in
circuit no. 2 (Eq. 48)
Similar calculations for the second circuityield the following values.
Rca'= 7.18; ATd =3.4; Wc2 = 17.44122;ATint=1.71+53.2122 in circuit no. 1
2 75 -(25 +4.0 + 1.71 +53.2I22)(9.31)(6.65)=0.715-0.85912' (Eq. 9A)
I275 -(25+3.4+2.17+28.5I, 2)X(17.44)(7.18)
=0.355-0.228I12 (Eq. 9A)
Solving simultaneously I,=0.714; I2=0.487 kiloampere.
138-Kv 2,000-MCM High-PressureOil-Filled Pipe-Type Cable 8.625-Inch-Outside-Diameter Pipe
The cable shielding will consist of anintercalated 7/8(0.003)-inch bronze tape-1-inch lay, and a single 0.1(0.2)-inch D-shaped brass skid wire-1.5-inch lay. Thecables will lie in cradled configuration.
DC=1.632; Dj=2.642; T=0.505;DJ=2.661; Dp=8.125
T12.9\1234.5+70T7C ; RdC 2.00,0\k234.5+75 /
6.35 microhms per foot (Eq. IOA)
For shielding tape As =7/8(0.003) - 0.00263;1=1.0; p=23.8; r=564 (Table I)
23.8w /.6i\4(0.00263) ( 1)
(S64+50)=62,900 microhmsr564 +2per foot at 50 C (Eq. 13)
For skid wire AJ =2 ir(0.1)' = 0.0157;21= 1.5; p=38; T 912 (Table I)
4(0j0157) I(1 65r)(912 +50\912+20/ =11,100 microhms
per foot at 50 C (Eq. 13)
R8(net)[(62.9)(111 1]1,000=9,435 microhms per foot at 50 C
k,=0.435; kp=0.37 (Table II)
Rd,ks= 14.6; Ycs =0.052(1.7) = 0.088(Eq. 21, Fig. 1, and text)
S=2.66+0.10=2.76; Rde/kp=17.2;F(xp')=0.035 (Fig. 1)
( 1.632\'2YCP = 4( 2 ) (0.035)(1.7)=0.0832.76/
(Eq. 24A and text)
1+ YC = 1+0.088+0.083 = 1.171
Xm = 52.9 log(2.3)(2.76)Xm=529 log 2.66
=20.0 microhms per foot (Eq. 29A)
YS = YSC (20.0)2(1.7) =0.011(9.435)(6.35)(Eq. 27A and text)
(0.34)(2.76)+(0.175)(8. 13)0.372Vp 6.35
(Eq. 35)
Rac/Rdc 1.171+0.011+0.372 = 1.554(Eq. 14)
0.011 0.011+0.372qs = 1+ =1.009; q 1.171
1.171117= 1.327 (Eqs. 18-19)
, =3.5 (Table V); E= 138/F3=80;cos +=0.005
0.00276(80)2(3.5)(0.005)2.642
log
= 1.48 watts per conductor foot (Eq. 36)
P3t=550 (Table VI); Rj=0.012X
(550 log 1.38 thermal
ohm-feet (Eq. 38)
n'=3; DJ'=2.15(2.66)-5.72;3(2.1)
Rad = = 0.77 thermal5.72 +2.45
ohm-foot (Eq. 41A)
pd=100 (Table VI); t=0.50;De = 8.63+ 1.0 =9.63 for 1/2-inch
wall of asphalt mastic
0.0104(100)(3)(0.50)9.63-0.50= 0.17 thermal ohm-foot (Eq. 40)
Assume p.=80, L=36 inches, (LF)=0.85;N=1, F=1
R,' (at 85% loss factor) =0.012(80)(3)X[ 8.3 1~~4(36)log -983 +0.85 log ( 1)19.63 8.3=2.85 thermal ohm-feet (Eq. 44)
1' (at unity loss factor) =3.38thermal ohm-feet (Eq. 44)
Rea' = 1.38+1.009(0.77)+1.327(0.17+2.85)
=6.17 thermal ohm-feet (Eq. 8)
ATd= 1.48(0.69+0.77+0.17+3.38) =7.4 C(Eq. 6)
Ta=25 C (assumed);
I_ | 70-(25+7.4)(6.35)(1.171)(6.17)
-0.905 kiloampere (Eq. 9)
Neher, McGrath-Temperature and Load Capability of Cable SystemsOCToBm 1957 763
References1. CALCULATION OP TEE ELECTRICAL PROBLEMSOJ UNDEROROUND CABLES, D. M. Simmons. ThwElectric Journal, East Pittsburgh, Pa., May-Nov. 1932.
2. LOAD FACTOR AND EQUIVALENT HoURSCOMPARED, P. H. Buller, C. A. Woodrow. Elec-trical World, New York, N. Y., vol. 92, no. 2, 1928,pp. 59-60.
3. SYMPOSIUm ON TBMPERATURE RISE OP CABLES,AIRE Committee Report. AIEE Traxsactions,vol. 72, pt. III, June 1953, pp. 530-62.
4. A-C RESISTANCE OF SSGMBNTAL CABLBS IN
STEEL PIPE, L. Meyerhoff, G. S. Eager, Jr. Ibid.,vol. 68, pt. II, 1949, pp. 816-34.
S. PROXIMITY EFFBCT IN SOLID AND HOLLOWROUND CONDUCTORS, A. R. M. Arnold. Journal,Institution of Electrical Engineers, London.England, vol. 88, pt. It, Aug. 1941, pp. 349-59.
6. EDDY-CURRENT LOSsRS IN MULTI-CORB PAPER-INSULATBD LEAD-COVBRBD CABLES, ARMORBDAND UNARMORBD, CARRYING BALANCBD 3-PlASECURRENT, A. H. M. Arnold. Ibid., pt. I, Feb.1941, pp. 52-63.
7. PIPE LOSsES IN NONmAGNRTIC PIPE, L.Meyerhoff. AIEE Transactions, vol. 72, pt. III,Dec. 1953, pp. 1260-75.
8. A-C RESISTANCE OP PIPE-CABLE SYSTEMS
WITU SBEMENTAL CONDUCTORS, AIRE CommitteeReport. Ibid., vol. 71, pt. III, Jan. 1952, pp. 393-414.9. A-C RESISTANCS OF CONVBNTIONAL STRANDPOWER CABLBS IN NONMBTALLIC DUCT AND INIRON CONDUIT, R. W. Burrell, M. Morris. Ibid.,vol. 74, pt. III, Oct. 1955, pp. 1014-23.
10. THE THERMAL RHSISTANCH BETWEEN CABLESAND A SURROUNDING PIPE OR DUCT WALL, F. H.Buller, J. H. Neher. Ibid., vol. 69, pt. 1, 1950,pp. 342-49.11. HEAT TRANSFER STUDY ON POWER CABLSDUCTS AND DUCT AssBMBLIES, Paul Greebler, GuyF. Barnett. Ibid., vol. 69, pt. I, 1950, pp. 357-67.
12. THE TEMPBRATURB RISE OF BURIBD CABLESAND PIPES, J. H. Meher. Ibid., vol. 68, pt. I, 1949,pp. 9-21.13. THE TEMPERATURB RISE OF CABLES IN ADucT BANIC, J. H. Neher. Ibid., pp. 540-49.14. OI. FLOW AND PRESSURE CALCULATIONS FORSELF-CONTAINBD OIL-FILLBD CABLB SYSTEMS,F. H. Buller, J. H. Neber, F. 0. Wollaston. Ibid.,vol. 75, pt. III, Apr. 1956, pp. 180-94.15. THERMAL TRANSIENTS ON BURIBD CABLES,F. H. Buller. Ibid., vol. 70, pt. I, 1951, pp. 45-55.16. TEE DBTERMINATION OF TEMPERATURETRANSIENTS IN CABLB SYSTEMS BY MEANS OF ANANALOGUE COMPUTER, J. H. Neher. Ibid., pt. II,1951, pp. 1361-71.
17. A SIMPLIFIxD MATHEMATICAL PROCBDURBPOR DETERUINING THE TRANSIENT TEMPERATURERISEsOF CABLE SYSTEMS, J. H. Neher. Ibid., vol.72, pt. III, Aug. 1953, pp. 712-18.
18. THs HBATING OP CABLES EXPOSED TO THESUN IN RACIKS, B. B. Wedmore. Jourxal, Institu-tion of Electrical Engineers, vol. 75, 1934, pp.737-48.19. LossE IN ARMORED SINGLE-CONDUCTOR,LEAD-COVERED A-C CABLBS, 0. R. Schurig, H. P.Kuehni, F. H. Buller. AIRE Transaction:, vol.48, Apr. 1929, pp. 417-35.20. CONTRIBUTION TO THE STUDY OF Lossss ANDOF SELP-INDUCTION OF SINGLE'CONDUCTOR AR-MORBD CABLECs, L. Bosone. Elettrotecxica, Milan,Italy, 1931, p. 2.
21. ARTIFICIAL COOLING OP POWER CABLB, F. H.BuUer. AIEE Transactions, vol. 71, pt. III, Aug.1952, pp. 634-41.22. SURFACE HEAT TRANSMISSON, R. H. Hettman.Transactions, American Society of MechanicalEngineers, New York, N. Y., vol. 51, pt. 1, 1929,pp. 287-302.
23. THEB CURRENT-CARRYING CAPACITY OF RUB-BER-INSULATED CONDUCrORS, S. J. Rosch. AIEETransactions, vol. 57, Apr. 1938, pp. 155-07.24. HEATINo AND CURRENT-CARRYING CAPACITYOF BARB CONDUCTORS FOR OUTDOOR SBRVICB,0. R. Schurig, G. W. Frick. General ElectricReview, Schenectady, N. Y., vol. 33, 1930, p. 141.
DiscussionC. C. Barnes (Central Electricity Authority,London, England): This paper is an excel-lent and up-to-date study of a most impor-tant subject. For 25 years D. M. Simmons'articles have been used for fundamentalstudy on current rating problems, but thenumerous cable developments and changesin installation techniques introduced inrecent years have made a modern assess-
ment of this subject very necessary. Theessential duty of a power cable is that itshould transmit the maximum current (orpower) for specified installation conditions.There are three main factors which deter-mine the safe continuous current that a
cable will carry.
1. The maximum permissible temperatureat which its components may be operatedwith a reasonable factor of safety.
2. The heat-dissipating properties of thecable.
3. The installation conditions and ambientconditions obtaining.
In Great Britain the basic referencedocument is ERA (The British Electricaland Allied Industries Research Association)report F/T1311 published in 1939, and in1955 revised current rating tables forsolid-type cables up to and including 33 kvwere published in ERA report F/T183.A more detailed report summarizing themethod of computing current ratings forsolid-type, oil-filled, and gas-pressure cablesis now being finalized and will be publishedas ERA report FIT187 some time in 1958.
Until recent years current ratings inGreat Britain have usually been consideredon a continuous basis, but the importanceof taking into consideration cyclic ratingshas now been carefully studied, since con-
tinued high metal prices have forced cableusers to review carefully the effects ofcyclic loadings. A report has recently been
issued in which a simple method is pre-sented for the rapid calculation of cyclicratings.'
Table V gives specific inductive capaci-tance values for paper as: paper insulation(solid type), 3.7 (IPCEA value); paperinsulation (other type), 3.3-4.2. Is it pos-sible to list the other types and theirappropriate specific inductive capacitancevalues or alternatively simply use anaverage specific inductive capacitance valueof 3.7, for example, for all types of paperinsulation?
Reference is made to the adoption of thehypothesis suggested by Kennelly as the
basis of the paper-this is a logical approachbut it appears to differ from the basis ofcomputing ratings hitherto adopted in theUnited States. An amplification of theauthors' viewpoint on this important issuewill be welcomed.With reference to the use of low-resistivity
backfill, recent studies in Great Britainhave shown that the method of backfillingcable trenches deserves careful considera-tion as attention to this point can resultin increases up to 20% in load currents.
Equation 43 gives the thermal resistancebetween any point in the earth surroundinga buried cable and ambient earth. It is
Table X. Temperature Limits for Belted-, Saeened- and HSLt-Type Cables
Laid Direct or In Air In Ducts
Aluminum AluminumSheathed Sheathed
Lead Sheathed Lead SheathedArmoured Armored
System Voltage and Type Un- or Un- Un- or Un-of Cable Armored armored armored Armored armored armored
1.1 kvSingle-core . . 80. 80. .............. 60. 80Twin and multicore belted...... 80. 80. 80. 80.60 . 80
3.3 kv and 6.6 kvSingle-core . .................. 80 . 80. .60 . 80Three-core belted-type..... 80. 80. 80.8060 80
11 kvSingle-core . .................. 70 . 70. .50 . 70Three-core belted-type.......... 65. 65 . 65. 65.50. 65Three-core screened-type........ 70. 70. 70.70.50 . 70
22 kvSingle-core ..................... 65. 50 65
Three-core belted-type.......... 55. 55..55 . 50Three-core screened-type........ 65 . 65. 65.65.50 . 65Three-core (SL$or SAC)........ 65 ..65.65 . .65
33 kv (screened)Single-core . ..... 65 ... 50Three-core ...... 65. 65. .65 . 50Three-core HSL ...... 65. . . 65
* Measured In degrees centigrade.t Hochstater separate lead.Separate lead sheathed.
5 Separate aluminum sheathed.
Neher, McGrath-Temperature and Load Capability of Cable Systems764 OCTOBER 1957
