APPENDIX.—EMPIRICAL MODEL MENUS

15

A series of menu screens from the interactive computer program is shown, ultimately displaying cable conductor

temperature rise over time with the current load superposed.

CABLE SPECIFICATION

0. #6 AWG, 3/c, G-GC. 5. #1 AWG, 3/c, G-GC.
1. #5 AWG, 3/c, G-GC. 6. 1/0 AWG, 3/c, G-GC.
2. #4 AWG, 3/c, G-GC. 7. 2/0 AWG, 3/c, G-GC.
3. #3 AWG, 3/c, G-GC. 8. 3/0 AWG, 3/c, G-GC.
4. #2 AWG, 3/c, G-GC. 9. 4/0 AWG, 3/c, G-GC.
Current Selection is: 4

Current Shape is : ROUND

Press 'Q' To QUIT Simulation

0-9 To SELECT Cable NUMBER.
'RETURN' To CONTINUE to Duty Screen.
‘r'or'F' To SELECT the Cable SHAPE.



interpolation, the current yielding the 70° C temperature
tise, i¢., rated ampacity, was determined for each cable.
These rated currents for a 20° C ambient were found to be
13% to 30% greater than those specified by ICEA (18).
A plot of these rated currents versus cable size (fig. 7)”
shows that flat cables should have a higher current rating
or ampacity than the same size round cable. This finding
is not consistent with mine trailing cable ampacity tables
(18), in which flat and round unshielded cables are lumped
together. In all likelihood, the flat configurations, with
25% greater surface area for a given wire size, facilitate
heat transfer to air. Interestingly, when the curves show-
ing final temperature rise versus current are normalized
to the rated ampacity for each cable, the data points be-
come tightly clustered. This gives rise to a relationship
where the temperature rise for any cable in the #6 to
#4/0 range can be described as a function of the percent
rated ampacity (fig. 8).”

Next, equation 13 was fitted separately to the heating
and cooling portions of the averaged conductor tempera-
ture curves (fig. 5) to obtain thermal time constants of
heating and cooling. The heating constants were found to
be independent of current load (fig. 9), while the cooling
constants, calculated under no load, varied inversely with
initial temperature (fig. 10). Since the cooling constant
calculations were made with a preheated cable, this slight
inverse relationship was attributed to radiation effects,
which were neglected in equation 13. Multiple regression
techniques were then employed to adapt the model to any
size G-GC cable within the range of #6 to #4/0 AWG.
Comparisons were made between simulated loads and
actual load tests to gauge model accuracy. Figure 11”
depicts model output and an actual load test for the #4/0
AWG cable carrying 300 A, 60% of the time.

This model was then incorporated into an interactive
computer program that displays conductor temperature
given an arbitrary current load determined by the user.
The program consists of a series of menu screens, which
are depicted in the appendix. To begin, the user specifies
the cable to be evaluated. Next, the simulated load is de-
fined, This may be the prior simulation, which is useful in
comparing the thermal performance of differing cable sizes
or shapes. The user is then given the choice of defining
the hypothetical electrical load as a series of current-time
intervals or as a repeatable or cyclic load. Another op-
tion is to read into the program an actual load current
from a data file. The program output displays cable

conductor temperature rise over time with the current load
superposed (appendix),

This program is useful in several respects. It can be
utilized by designers and approvers of new mining ma-
chines to determine the cable temperatures to be antici-
pated given the machine load cycle. Comparisons among
various sizes can facilitate selection of the appropriate
cable and ensure maximum safety and efficiency. It should
be emphasized that the program provides no safety factor;
the temperatures predicted are those that would be ex-
pected from a given input current. Users may want to
incorporate an appropriate safety factor. Further, the pro-
gram could be interfaced with hardware to monitor ma-
chine currents in real time and predict cable conductor
temperatures. This information could be stored to provide
a history of machine and cable usage. Ultimately this pro-
gram could be the basis for a cable protection system. In
its present form the program can provide a measure of
conductor insulation deterioration through integration of
time and temperature data. However, to offer more com-
prehensive protection it should be responsive to transient
overcurrents, Accordingly, laboratory tests were conducted
to evaluate the thermal performance of overloaded trailing
cables.

420 r -
390} , 4

360}

RATED CURRENT, A

30 4

i Ll L L L
80 120 160 200

CABLE SIZE, 10% cmil

240

Figure 7.Rated currents for flat and round configurations.



Temperature Rise (C)

500

450

\ 350

\ 300

\ 250

200

150

aN 100

20
of
0 LI |

| 50

°
0:00 023 046 FO 133 156 2:20 2:43 3:06 3:30 3:53

Commands: (‘T' 'c' 'E' 'S') 'H' is Help

Time (H:M)

it} H 2M 20 SOO

T: 67.00 | I: 209.50

19

Current (Amps)

INT-BU.OF MINES,PGH.,PA 29753


UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT

A ampere e irrational constant approximately
equal to 2.71828...
°c degree Celsius
fps foot per second
cal/(cm+s) calorie per centimeter
per second ft foot
cal/(cm+s*K) calorie per centimeter g/cm? gram per cubic centimeter
per second per kelvin
h hour
cal/(cm*+s+K) calorie per square centimeter
per second per kelvin in inch
cal/(cm*+s+K*) calorie per square centimeter per K kelvin
second per kelvin to the fourth
power min minute
cal/(g*K) calorie per gram per kelvin a ohm
cm centimeter Q-cm —_ ohm-centimeter

mil circular mil s second



16

CURRENT DUTY CYCLE

Please choose method of Duty Cycle Definition
1 REUSE Last Duty Cycle Input.
2. Input Duty Cycle as a Series of INTERVALS.
3. Define SUBINTERVAL to be Used Multiple Times.
4.. Read in Duty Cycle from a FILE.



18

SUBINTERVAL DEFINITION

Enter the SAMPLE Rate (Seconds): 60 _ [Default = 30]

Enter the ON TIME: Hours: OO Min: 45 Sec: 00
Enter On Time CURRENT LEVEL (Amps): 200_

Enter the OFF TIME: Hours:OO Min: 15 Sec: OO

How many times to REPEAT Subinterval 8_ [ Default = 101



IEEE Standard 242 (8) represents the overload region
with the following relationship:

2
T-T, [hi ek
I, |T,-T, [hb 230 +T, a4)
Th 1-¢K B0+T J’
where I, = overload current,

I, = normal current rating,
T = conductor temperature,

T, = normal cable temperature rating,

T, = ambient temperature,

J, = initial current prior to overload,
t = time of overload,
and K = a constant, dependent on cable size and
configuration.

However, the assumptions used to derive this equation
are the same ones used for equation 13, namely that radia-
tion and changes in conductor resistance are neglected.
Consequently, equation 14 cannot be used to accurately
predict conductor temperatures rises over time during
cable overloads.

CONCLUSIONS

The thermal performance of both flat and round type
G-GC portable power cables under normal loading cir-
cumstances has been successfully modeled empirically. An
alternate mathematical model also yielded reasonable pre-
dictions of the stabilized conductor temperature rise for
round cables. The temperature prediction technique of
the empirical model utilized a somewhat simplistic ap-
proach, assuming an exponential rise and fall in conductor
temperatures over time in response to step changes in load
currents. Realistic model parameters were derived from
extensive load tests of cables at or near rated current
carrying capacity. A key finding of this analysis was that
flat cables have a higher current rating or ampacity than
the same size round cable. Additional laboratory tests
were then conducted to measure temperature rises in ca-
bles energized with overcurrents in hopes of incorporating

the thermal response of cables under such overloads into
the thermal model. However, efforts to expand the scope
of the empirical model into the overload range were not
successful.

Nonetheless, the value of the existing thermal model
should not be underestimated. In the form of an inter-
active computer program, it can be used by designers and
approvers of new mining machines to determine the cable
temperatures to be anticipated given the machine load
cycle. Comparisons among various sizes and types can
facilitate selection of the appropriate cable size while
ensuring maximum safety and efficiency. In addition, the
program can be interfaced with hardware to monitor ma-
chine currents in real time and predict cable conductor
temperatures. This information could then be stored to
provide a history of machine-cable usage.

RECOMMENDATIONS FOR FUTURE RESEARCH

As stated previously, the value of the thermal model as
a protection methodology would be augmented if its appli-
cation range were extended beyond normal cable loading.
Since efforts to amplify the empirical model proved unsuc-
cessful, a reevaluation of the mathematical model may be
warranted. Specifically, the assumption of a uniform
temperature across the single lumped metallic conductor
should be investigated further. In addition, adjustments
may be necessary for the assumed jacket emissivity and
convective heat transfer coefficient. The resultant mathe-
matical model would be more complex than the existing
empirical model (a separate theoretical analysis for flat

cable configurations would be still more complicated).
However, these mathematical approaches do have the en-
hanced capability of predicting temperatures under any
loading conditions, including overloads.

Tf conductor temperatures can be successfully pre-
dicted for abnormal as well as normal load conditions, the
next logical step is to incorporate this into a cable pro-
tection system. Overloaded cables may attain high tem-
peratures that can damage cable insulation. Consequently,
the electrical current must be interrupted promptly to
preclude this potential hazard. However, this must not
occur so quickly as to cause nuisance tripping of the cable


The heat transport equations appropriate for the insulation
and cable jacket differ only in the thermal properties of
the materials, Because of the much slower thermal dif-
fusion of the insulation and cable jacket, the thermal
conduction within these materials has to be considered
explicitly. The thermal transport equation for these re-
gions is written in condensed notation in terms of the local
temperature T,(r,t),

aT, at, 10T,
Cc, —< = — +o}, 6)
ee aT (Bs 1)
where 9, = mass density of material k, g/cm?,
Cy, = specific heat of material k, cal/(g+K),

A, = thermal conductivity of material k,
cal/(cms+K),

temperature of material k, K,

a
"

k = 2 for insulation,
and k = 3 for jacket.

The right side of equation 6 is based upon cylindrical sym-
metry. Equations 5 and 6 are written in dimensionless
form as

4 1 1S

du Ty Cp, eR? R2%1Cp,

2 2 | ab
Ty lew 1)

and
a0,
— =P, 8)
ia ®
where u =t/r,
y =1/Ry
% = Ty/To

=R2
Te = Roo Coy/ Ay

and Pp

1/t

Equations 7 and 8 are solvable in principle if specified
initial conditions and boundary conditions are available.
The initial condition is that the cable is at ambient tem-
perature. The boundary conditions are written in the form
of dimensionless material temperature.

@ May = A2-=—@y = 1, @)

ii) Ag—— = A3—— @y = — 10)
@ By say @y R,’ (10)
sae A3 963 3 94
(ii) ER h(6, - 1) + oer of - sey

R,

=f, a)

R,

where h = convective heat transfer coefficient,

cal/(em?+s+K),

Qa
1

Stefan-Boltzmann radiation constant, 1.355
x 10°"? cal/(cm?+sK*),

€ = jacket emissivity (absorptivity),
T, = ambient temperature, K.

Conditions (i) and (ii) state that a continuity of heat
flow occurs where the materials are in contact. Condition
(iii) defines the heat loss at the jacket surface due to free
convection and radiation.

The free convection heat transfer coefficient h is
defined in terms of the temperature difference between
the jacket surface and ambient air, T,(R,) - T,, (15),

b= 2 pnsn6 10° { 2) To] (12)
Fis4 (QR) .


INTRODUCTION

Extreme mechanical abuse of hardware is characteristic
of mobile equipment operation in underground coal mines,
so much so that trailing cables rarely last a year in this
environment (1).4 Abrasive, tensile, and crushing forces
can cause phase-to-phase and phase-to-ground faults and
ultimate cable failure when circuit protective devices are
inadequate. The energy expended in a trailing cable short
circuit is capable of igniting loose coal and coal dust on
a mine floor as well as an explosive mixture of methane
and air. In addition, combustion of the cable insulation
can create smoke and noxious fumes. When sustained,
even overload currents only slightly in excess of the
tated cable ampacity can expose personnel handling the
cable to burn hazards, and enhance thermal degradation
of the insulation and jacket.

Effective cable protection can lessen such potential
dangers in the underground environment by ensuring that
the cable is operated within safe limits that preclude injury
to personnel and damage to the cable itself. This protec-
tion should also ensure that the cable is electrically utilized
in an efficient manner. To accomplish this, the protection
must account for all aspects of cable operation. Device re-
sponse time and sensitivity should be based upon cable
performance, not only during overloads and short circuits,
but also under continuous and intermittent duty. A pre-
requisite for the implementation of such protection is a

comprehensive understanding of the thermal characteristics
of energized cables. In cooperation with the Mine Safety
and Health Administration (MSHA), the American Mining
Congress (AMC), and the Insulated Cable Engineers As-
sociation (ICEA), the U.S. Bureau of Mines conducted re-
search to define the relationship between current load and
resultant temperature rise in trailing cables loaded con-
tinuously and intermittently under normal circumstances
(2-3). Cable characteristics under abnormal or faulted
conditions remained to be investigated, however.

This report documents the accomplishments of a follow-
up Bureau research project supporting the Bureau’s goal
of enhancing the safety of the Nation’s underground min-
ers. The specific objectives of this project were (1) to
define the thermal characteristics of electrically overloaded
trailing cables, and (2) to conceptualize electrical protec-
tion for cables that allows maximum cable efficiency with-
out diminishing electrical safety in underground mines.
The principal product of this work is a computerized
thermal model that can assist designers and approvers of
mining machines in selecting the appropriate size trailing
cable. The authors recognize there are other concerns as-
sociated with mine trailing cable operation that are beyond
the scope of this document. They are noted in the "Rec-
ommendations for Future Research" section.

BACKGROUND

The performance of copper conductor cables under
short circuit has been well established for many years (4)
and may be expressed as

1p Ty + 234
Gi] tF ye = 0.0297logig Teast’ @
where I = short circuit current, A,

A = conductor area, cmil,

short circuit duration, s,
Fi, = skin effect or ac/dc ratio,

T; = final conductor temperature after a
change in current, °C,

‘Italic numbers in parentheses refer to items in the list of references
Preceding the appendix at the end of this report.

and T, = conductor temperature prior to a
change in current, °C.

This equation is the basis of present ICEA specifica
tions for short circuited insulated cables (5-6). This equa-
tion assumes that heat transfer from the conductor is neg-
ligible, a valid assumption for short time periods (7).
However, for fault durations greater than several seconds,
conduction and radiation become significant, increasing the
error in application of this assumption.

The intermediate time zone from 10 s up to 6 h is the
bridge between short circuit performance and continuous
duty, The Institute of Electrical and Electronic Engineers
(IEEE) Buff Book (8) calls this region "the greatest un-
known in the cable thermal characteristic." Emergency
overload ratings have been established for medium-voltage
cables (9), but their applicability to low-voltage portable
power cables is uncertain. In previous Bureau research
(10), the relationship between cable thermal life and tem-
perature was determined through accelerated life testing.
In addition, the impact of current loads on this thermal life
was mathematically derived for normal machine loads (11).


THERMAL MODELING OF PORTABLE POWER CABLES

By Michael R. Yenchek,’ Gregory P. Cole,? and John C. Edwards?

ABSTRACT

The U.S. Bureau of Mines investigated the performance of portable power cables under transient
conditions. This research had a twofold purpose: (1) to define the thermal characteristics of electrically
overloaded trailing cables, and (2) to conceptualize electrical protection for cables that allows maximum
cable efficiency without diminishing electrical safety in underground mines. Several tasks were under-
taken in support of these goals during the 3-year research effort. Overload tests ranging from 2 to 12
times rated ampacity were conducted in the Bureau’s Mine Electrical Laboratory. Two thermal models
of energized type G-GC trailing cables were constructed, one based on thermodynamic theory and the
other using empirical data from previous Bureau load tests. The empirical model was then incorporated
into an interactive computer program that can assist designers and approvers of mining machines in
selecting the appropriate size trailing cable. This program can be the basis for a cable protection system
that ensures that cables are not the source of fires, ignitions, burns, or explosions underground.

'Blectrical engineer.
yputer engineer.
Research physicist.
US. Bureau of Mines, Pittsburgh Research Center, Pittsburgh, PA.


14

protection system or to preclude efficient operation of the
cable. To accurately match protection operating character-
istics to cable thermal performance, the limits of insulation
thermal damage must be defined for overloads. This can
be accomplished by extending the previously derived (10)
Arrhenius plots for EPR insulation in the high tempera-
ture range. Allowable time-current combinations can then
be derived by adding appropriate safety factors,

It is envisioned that any computer-based cable pro-
tection system would augment existing short circuit pro-
tection in the form of fuses and/or circuit breakers. Thus,
at currents above motor or transformer inrush, the present
protective devices would activate with no intentional time
delay.

REFERENCES

1. Hansloven, J. J. Trailing Cable Stresses in Underground Coal
Mines. M.S. Thesis, PA State Univ., University Park, PA, 1978, 275 pp.

2. Yenchek, M. R, and P. G. Kovalchik. Thermal Characteristics of
Energized Coal Mine Trailing Cables. IEEE Trans. Ind. Appl, v. 25,
No. 5, Sept/Oct. 1989, pp. 824-829.

3. ___. Thermal Characteristics of Energized Shielded and Reeled
‘Trailing Cables. IEEE Trans. Ind. Appl,, v. 27, No. 4, July/Aug, 1991,
Bp. 791-796,

4. Kaufmann, R.H. Selecting Power Cables To Meet Short Circuit
Requirements. Power Gener. and Power Plant Eng,, v. 53, No. 2, Feb.
1949, pp. 64-67.

5. Insulated Cable Engineers Association (South Yarmouth, MA).
Short-Cireuit Characteristics of Insulated Cable. ICEA Publ. No. P-32-
382, 1987, 22 pp.; available from Natl. Electr. Manuf. Assoc., Washing-
ton, DC.

6. ___. Short-Circuit Performance of Metallic Shielding and
Sheaths of Insulated Cables. ICEA Publ. No. P-45-482, 1987, 11 pp.
available from Natl. Electr. Manuf, Assoc., Washington, DC.

7. Jones, B. W., and J. A. Scott. Short-Time Current Ratings for
Aircraft Wire and Cable. AIEE Trans., v. 65, Oct. 1946, pp. 644-648.

& Institute of Electrical and Electronic Engineers (New York).
IEEE Recommended Practice for Protection and Coordination of Indus-
trial and Commercial Power Systems. ANSI/IEBE Stand. 242, 1986,
p44,

9. Lazar, I. New Insulation Developments for Electrical Wire and
Cable. Specifying Eng., Sept. 1981, pp. 90-93.

10. Yenchek, M. R, and P. G. Kovalchik. Mechanical Performance
of Thermally-Aged Trailing Cable Insulation. IEEE IAS Trans., v. 25,
No, 6, Nov./Dec. 1989, pp. 1000-1005.

IL. Yenchek, M. R,, and P. G. Kovalchik. The Impact of Current
Load on Mine Trailing Cable Thermal Life. Paper in Conference Rec-
ord of the 10th WVU International Mining Electrotechnology Con-
ference, WV Univ., Morgantown, WV, July 1990, pp. 17-22.

12. US. Code of Federal Regulations. Title 30—Mining Resources;
Chapter I—Mine Safety and Health Administration, Department of La-
bor; Subchapter O—Coal Mine Safety and Health; Part 75—Mandatory
Safety Standards—Underground Coal Mines; Section 75.601; July 1,
1989.

13. Fesak, G., W. S. Vilcheck, W. J. Helfrich, and D. C. Deutsch. In-
stantaneous Circuit Breaker Settings for the Short Circuit Protection of
‘Three-Phase 480-, 600-, and 1040-V Trailing Cables. IEEE Trans. Ind.
Appl, v. IA-17, No. 4, July/Aug. 1981, pp. 369-376.

14, Vilcheck, W. S., G. Fesak, and W. J. Helfrich. Instantaneous Cir-
cuit Breaker Settings for the Short-Circuit Protection of Direct Current
300- and 600-V ‘Trailing Cables. IEEE Trans. Ind. Appl. v. IA-17, No. 4,
July/Aug. 1981, pp. 362-368.

15. Luxbacher, G. W. Evaluation of the Effectiveness of Molded-Case
Circuit Breakers for Trailing Cable Protection. Ph.D. Thesis, PA State
Univ., University Park, PA, Nov. 1980, p. 128.

16. Ames, W. F. Numerical Method for Partial Differential Equa-
tions. Barnes and Noble, 1969, pp. 52-53.

17, Graham, RC. Overload Classification for Secondary Network
Cables. AIEE Trans,, v. 74, Pt. III-PAS, Oct, 1955, pp. 916-919.

18, Insulated Cable Engineers Association (South Yarmouth, MA).
Ethylene-Propylene-Rubber-Insulated Wire and Cable for the Trans-
mission and Distribution of Electrical Energy. Pt. 8 (appendices).
ICEA Publ. $-68-516, 1985, p. 8; available from Natl. Electr. Manuf.
Assoc. Washington, DC.



Thermal production of heat occurs in the conductor as
the result of electrical current through a small, nonzero
resistance, Symmetry with respect to a longitudinal axis,
at the cable center and parallel to the cable, results in the
radial transport of thermal energy to the outer surface of
the cable jacket, where the energy is lost to the external
environment (air) through free convection and radiation.
The proper description of heat generation and heat dissi-
pation results in a thermal transport equation that can be
solved for the temperature over the cross section of the
cable at any time. The presence of several conductors
within the cable creates a complex geometry for the exact
analytic representation of the current and heat flow within
the cable. However, the near symmetry of the conductors
about the cable longitudinal axis permits the represen-
tation of the n conductors by a single conductor of equiv-
alent cross-sectional area in order to define a boundary
surface of radius R, for the heat generation source of
nwires. A simple geometric equivalence of wire cross-
sectional area defines R, as

R, =n xR,, 2)
where R, = radius of boundary surface of conduc-
tors, cm,

n = number of conductors,
and R, = conductor radius, cm.

Surrounding the equivalent-heat-generating cylindrical
element of radius R, is an insulation of radius R, with
respect to the central axis. Concentric with the equivalent-
heat-generating element is a cable jacket of radius R,. A
schematic of the equivalent cable cross section is shown in
figure 1.

Each wire of electrical resistivity p (ohm-centimeters)
contributes to the total heat conduction rate S$ for uniform
current I (amperes).

S= fax p x) , Q)
(4184 ® xR?)

where S = heat conduction rate, cal/(cmss),

"

p = electrical resistivity, Q-cm,
I = electrical current, A,

and mn = 3,14159265.

For a cable length, 1, the appropriate heat transport
equation for radial heat flow defines a partial differential

equation for the temperature T,(t) of the region r < R, at
time t,

a, aT,
2
IeRG71Cp, «Is - 2AR Doz" In,» 4

where 1 = cable length, cm,
0, = conductor mass density, g/cm?,
C, = conductor specific heat, cal/(g+K),

T, = conductor temperature, K,

T, = insulation temperature, K,
2 = insulation thermal conductivity,
cal/(cm+s¢K),
t = time, s,
and r = radial distance from longitudinal axis of
cable, cm.

The second term on the right side of equation 4 represents
the conductive loss of thermal energy into the insulation
surrounding the conductor. Equation 4 is based upon the
model assumption that the conductor’s high thermal con-
ductivity rapidly establishes a uniform temperature over
the conductor. Equation 4 is simplified to read

a) os 2, oh
TRE Rae Ror
mR o (OF

©

Conductor

Figure 1.Equivalent cable cross section for mathematical
model.



ee

EES

h REPORT OF ESSA LONS! /1993
af a ERED SECTE ING }

Thermal Modeling of Portable Power Cables

By Michael R. Yenchek, Gregory P. Cole, and John C. Edwards

| UNITED STATES DEPARTMENT OF THE INTERIOR

BUREAU OF MINES | : z


the jacket surface yielded a rise time of 4.6 min for wire
currents of 100, 200, 400, and 800 A. For the #2/0 AWG,
3/C case reported in table 3, the rise time is 30.4 min for
currents of 100 and 200 A. For a 400-A current, the rise
time is 26.6 min, The temperature dependence in the
jacket boundary condition causes the rise time to (1) differ
from the isothermal boundary condition case, and (2) vary
with the heat production rate (wire current).

Table 3.—Measured and predicted wire temperature rises

Cable, 3/C, Current, Stabilized wire temperature rise, “C
found G-GC A Measured (2) Model prediction
#2/0AWG.. 100 8 10
200 28 37
400 128 132
#4/0AWG.. 100 8 G
200 20 22
400 68 79

The FORTRAN program developed has the capability
to impose a duty cycle on the current. An example is pre-
sented in figure 4. Figure 4 represents the evolution
of the temperature rise above ambient for a #4/0 AWG,
3/C, round cable subjected to a maximum current of
800 A with a 25% duty cycle for a 50.4-min period.

ee
% 60 1
@& 650 4
g =| 1
- nN AKAL 4
gM ANW

= 20 i’ KEY 4
# i — Conductor-insulation

- Insulation- jacket
—— Cable surface

10) fl L 1 n 1 1
200 300 400 500 600 700

TIME, min

Figure 4—Mathematical model predicted temperature rise in
#4/0 AWG, 3/C, round, G-GC cable loaded at 800 A, 25% of the
time.

EMPIRICAL MODEL

Concurrent with the theoretical analysis, a thermal
model empirically based on laboratory test results was
constructed for 3/C, G-GC trailing cables suspended in
open air, The purpose of this model was to predict cable
conductor temperature fluctuations over time given current
loading.

Previously, the Bureau had conducted extensive re-
search to determine the relationship between current load
and temperature rises in coal mine trailing cables (2).
Portable power cables were continuously and periodically
loaded with direct current of various magnitudes. Tem-
perature rises within and on the cables were measured
with thermocouples and the data were recorded by com-
puter. A typical steady state test result is shown in fig-
ure 5.

A literature search (17) revealed that, neglecting ra-
diation and changes in conductor resistance, cable temper-
atures should rise and fall exponentially according to the
following relationship:

T=(T,-T) ae +T,, (13)

where T = cable temperature rise above ambient at

the end of time interval t, °C,

T, = initial temperature rise at the start of
that interval, °C,
T; = final, stabilized temperature rise, °C,
t = time, s,
and + = thermal time constant, s.

The thermal time constant represents a ratio of system
heat capacity to the rate of heat transfer from the system.
During heating, for example, the constant is equal to the
time required to reach 63% of the final temperature rise.
Thermal time constants are influenced by three factors:
(1) the volume-to-surface area ratio or cable geometry;
(2) the product of the density and specific heat of the
cable, thermodynamic properties; and (3) the convective
heat transfer coefficient, or the rate at which heat is re-
moved from the cable surface.

The empirical model incorporated the steady state load
test results for three round and three flat G-GC cables
ranging from a fraction to up to 150% of rated ampacity
(18). In constructing the model, the relationship be-
tween stabilized conductor temperature rise and constant
load current was initially established for cach cable by
fitting functions of the form y = ax® to the test data us-
ing commercial data analysis software (fig. 6). Through


The partial differential equations 7 and 8 and associated
boundary conditions must be solved with a numerical pro-
cedure. A finite difference method that is implicit in time
and central in space was selected to reduce the partial dif-
ferential equations to a set of coupled algebraic equations
that could be solved iteratively at each step with a Gaus-
sian elimination method (16). A FORTRAN computer
program was developed to solve these algebraic equations.
The program has the flexibility to solve each particular
application for a known set of physical properties charac-
teristic of the cable.

Physical specifications for EPR insulation and neoprene
jackets were obtained from cable manufacturers and used
in the model calculations (table 1):

Table 1.—Physical data
Property Con. «EPR Neoprene
ductor
Mass density .... g/om®.. 8.96 1.35 15
Specific heat (g*K).. 0.092 0.393 0.335
Thermal conductivity
cal/(omeseK).. 0.96 6.62x 10% —6.62x 10*

Calculations were made for two sizes of round, 3/C,
G-GC cables, #2/0 and #4/0 AWG. Dimensions and
wire resistances are listed in table 2. ~

Table 2.—Cable dimensions and wire resistance

Property #2/0 AWG, #4/0 AWG,
round G-GC round G-GC
Wire diameter ...... in. 0.419 0.528
Insulation thickness .. in. 0.08 0.08
dacket diameter ..... in. 1.65 2.04
Wire resistance
‘ohm per 1,000 ft. . 0.101 0.0626

AWG == American wire gauge.

Figures 2 and 3 show the calculated temperatures above
ambient (20° C) at y = 1, R,/R,, and R,/R,, of the #2/0
and #4/0 cables for a constant wire current of 400 A.
The temperatures of the wire (conductor), insulation-
jacket interface, and jacket external surface all reach a
steady state value within a 2-h time period.

Table 3 shows a comparison of measured and predicted
steady state wire temperature rises above ambient for cur-
rents of 100, 200, and 400 A in each cable. There is a rea-
sonable agreement between measured and predicted val-
ues. Discrepancies are quite possibly due to the inexact
definition of the ‘free convective cable-air heat transfer
coefficient. This, and the radiation boundary condition,
are temperature dependent. Their influence is seen from

200 T T T T

TEMPERATURE RISE (AT), °C

sor; KEY 7
/ —— Conductor-insulation

20 j~77 Insulation-jacket
. —'— Cable surface
|

10 Nl J | |

° 30 60 90 120 150
ee TIME, min

Figure 2.-Mathematical model predicted temperature rise In
#2/0 AWG, 3/C, round, G-GC cable loaded at 400 A.

TEMPERATURE RISE (AT), °C

207, i V4 — Conductor- insulation
f - == Insulation-jacket
i —-— Cable surface
10 L 1 L 1 L
° 30 60 90 120 150 180
TIME, min

Figure 3.—Mathematical model predicted temperature rise in
#4/0 AWG, 3/C, round, G-GC cable loaded at 400 A.

a comparison of the rise time for the conductor temper-
ature rise above ambient to reach e of the maximum tem-
perature rise above ambient. For a #2/0 AWG, 3/C ca-
ble, calculations with an isothermal boundary condition at



Further laboratory investigations of cable transient per-
formance were necessary to quantify the thermal effects of
fault currents and to incorporate those effects into the
time-current characteristics of effective cable protection.

Short circuit protection of trailing cables in coal mines
is presently required by Federal regulation (12). This pro-
tection typically involves the activation of a dual-element
fuse or molded-case circuit breaker with no intentional
time delay. The trip settings specified by Federal regula-
tions are based upon available fault currents and cable
size. Higher trip levels may be permitted when nuisance
tripping results from motor starting or transformer mag-
netizing currents (13-14).

However, protection of cables against overloads and
lower current, higher resistance arcing faults is not man-
dated by Federal law. The absence of this regulatory im-
petus means that many cables can be vulnerable to severe
damage over a significant portion of their operating range.
Indeed, Luxbacher (15), in discussions with mine opera-
tors, found that even circuit breakers equipped with ther-
mal overload protection rarely tripped during these faults,
Yet it is not uncommon to see cables that have been dam-
aged by relatively low-level overcurrents. A protection
scheme accurately tuned to cable damage limits can
obviate such occurrences.

ACKNOWLEDGMENT

The authors acknowledge with thanks the assistance
provided by Dr. William Z. Black of Technology, Inc.,

Atlanta, GA, who helped interpret the laboratory test
results from a thermodynamic standpoint.

CABLE PROTECTION CONCEPTS

An effective cable protection system allows the cable to
be operated within the limits of its thermal capability while
precluding conduction of unsafe load currents. Hazardous
combinations of current and time, in the long term, can
thermally degrade the cable insulation and jacket. In the
short term, they can ignite loose coal dust or methane ac-
cumulations. Several approaches to cable protection had
been envisioned at the outset of the project, each having
advantages and drawbacks. The primary consideration was
identification of the appropriate parameter to be sensed,
ie. cable temperature or load current.

Originally, temperature monitoring was favored as it
provides a direct measure of the cable’s thermal history.
Thermocouples, thermistors, and resistance temperature
detectors were under consideration as temperature sensors
that could be embedded within the metallic conductor.
Ideally, this cable temperature monitor should be installed
in the same environment as the cable. The main drawback
in having the sensor in open air, underground, is its
susceptibility to physical damage. The load center from
which the trailing cable originates can provide the nec-
essary mechanical protection. Also, in the load center the

recommended cable protection system would logically be
near other electrical protective devices such as ground
monitors. Ambient compensation could even be incorpo-
rated to account for the elevated temperatures within this
enclosure, However, no practical means could be found to
compensate for the thermal influence of nearby cable ter-
minations or end effects that would distort cable temper-
ature data (2). Consequently, development of the cable
Protection concept using temperature monitoring was
abandoned in favor of the traditional approach of load
current sensing,

Load currents can be measured with a current trans-
former (CT) that encircles a power conductor. Cable tem-
peratures can then be inferred from the output signal of
the CT secondary. The success of this methodology de-
pends upon how accurately temperature fluctuations can
be predicted or modeled. In the course of this research
two thermal models of energized type G-GC trailing cables
were constructed, one based on thermodynamic theory and
the other using empirical data from previous Bureau load
tests (2).

MATHEMATICAL MODEL

A mathematical model was developed that predicts the
transient heating and cooling of round, type G-GC trailing
cables suspended in open air, The cable features n copper
conductors (n = 2 or 3, typically) of radius R,. Each con-
ductor is covered by insulation, a dielectric material such

as ethylene-propylene-rubber (EPR) or polychloroprene
(neoprene). The insulated conductors are embedded in a
solid cable jacket material such as chlorosulfonated poly-
ethylene or neoprene for mechanical protection.


INTERVAL DEFINITION

Enter the SAMPLE Rate (Seconds): 60 [Default = 30]
Enter the number of INTERVALS : 4 — [Default=1]

DURATION of Interval #2 ?

Hours + 2_ Min: 30 Sec: OO
CURRENT Level (Amps): 130_

7



CONTENTS

Page
Abstract .... 1
Introduction 2
Background . 2
Acknowledgment . 3
Cable protection concepts 3
Mathematical model . 3
Empirical model . . 7
Overload tests . a
Conclusions ... B
Recommendations for future research . B
References ........0.0e eee eee 14
Appendix—Empirical model menus . . 15
ILLUSTRATIONS
1, Equivalent cable cross section for mathematical model ......... 0.0. c eee ce eee eeeeteeaeaees 4
2, Mathematical model predicted temperature rise in #2/0 AWG, 3/C, round, G-GC cable loaded at 400A .. 6
3. Mathematical model predicted temperature rise in #4/0 AWG, 3/C, round, G-GC cable loaded at 400A .. 6
4. Mathematical model predicted temperature rise in #4/0 AWG, 3/C, round, G-GC cable loaded at 800 A,
25% of the time 7
5. Measured temperature rise in #2/0 AWG, 3/C, round, G-GC cable loaded at 400 A 8
6. Temperature rise versus current in #6 AWG, 3/C, round, G-GC cable . 8
7. Rated currents for flat and round configurations .... 9
8. Conductor temperature rises and percent rated load ... 10
9. Heating time constant versus current for #6 AWG, 3/C, round, G-GC cable ... 10
10. Cooling time constant versus initial temperature in #6 AWG, 3/C, round, G-GC cable . oe 10
11, Comparisons of actual load test results with empirical model predictions for #4/0 AWG, 3/C, round,
G-GC cable ......... . 11
12. Overload test setup ..... 2
13. Light (200% rating) overload on #2/0 AWG, 3/C, round, G-GC cable . 2
14. Heavy (1,100% rating) overload on #2/0 AWG, 3/C, round, G-GC cable . na
15. Heavy (1,100% rating) overload on #2/0 AWG, 3/C, round, G-GC cable with time scale expanded na
TABLES
1, Physical data 6
2. Cable dimensions and wire resistance : 6
3. Measured and predicted wire temperature rises .... 7


10

210 T T T T T T T
°

180 4
oO
$

= = (oy, [2-32
Z sob Ty = 70 (%1) J
3
~ Oo
a
z 120 ° 4
Ww
°
= 90- 4
é v°
& 60F 9 4 4
9°
2
30 4
0 i L 1 1 L L
02 04 0.6 08 1.0 1.2 1.4 1.6
RATED CURRENT LOAD, %
Figure 8.—Conductor temperature rises and percent rated load.

30 T T T T T T
sg
Ee
5 25 ° °
é 2 -
2 ° ° i
8 20F ° ° 4
Ww
=
Fos 1 ! ! ! | !

40 60 80 100 120 140 160 180
CURRENT, A
Figure 9.—Heating time constant versus current for #6 AWG, 3/C, round, G-GC cable.

£
€
g
rd
a
z
6
oO
w
=
F 10 1 1 1 i l i l 1 i i

fe} 20 40 60 80 100 120 140 160 180 200 220
INITIAL TEMPERATURE, °C

Figure 10.—Cooling time constant versus initial temperature in #6 AWG, 3/C, round, G-GC cable.



Power source

fA»
YI Alternating current

Tat cable Saker
zs ald |

iS

Computer terminal Datalogger

Figure 12—Overload test setup.

160
a”
140 Key aN
Conductor insulation NS
120 Insulation jacket os NY

Cable surface

TEMPERATURE RISE (AT), °C

10 12 4 6 18 20

Figure 13.—Light (200% rating) overload on #2/0 AWG, 3/C,
round, G-GC cable.

150 5 T T T
iss
PoOSAS
gio} sy a
= ' oe
5 ! ae
wg 90ft 4
2 '
= 1 ——
id 17
Li
g sort /
'
&
z H ! KEY
@ sot! = Conductor- insulation 71
Hl — Insulation- jacket
H Cable surface
ot 3 5

200 300 400
TIME, 5

TINE

Figure 14.Heavy (1,100% rating) overload on #2/0 AWG,
3/C, round, G-GC cable.

conductor-insulation interface. However, with heavier
overloads, when the conductor temperature peaked, the
temperature of the surface was barely elevated above the
lab ambient. In fact, as shown in figure 14, temperatures
at the surface reached a maximum 6 min after power was
deenergized, and this maximum was only one-third the rise
at the conductor-insulation interface. This delay in heat
transfer from the conductor to the insulation, jacket, and
eventually air means that cable jackets are stressed ther-
mally to a relatively lesser degree during transient events
than during loading near the cable’s rated ampacity. Con-
sequently, it is believed that the key to effective cable
overload protection is the maintenance of conductor-
insulation temperatures below the EPR damage threshold.

This "thermal inertia" is also manifested in the shape
of the conductor temperature rise curve. At light over-
loads, this curve is an ascending exponential typical of
those at or below rated ampacity (equation 13). At high
overloads, a double exponential characteristic shape is
apparent (fig. 15). This is important from a thermal mod-
eling standpoint, as equation 13 can no longer be used to
accurately predict temperature rise over time. Rather, the
overload thermal rise curve seems to be a combination of
equations 1 and 13, the transition point occurring when
heat buildup in the conductor begins to be transferred to
the insulation. In fact, this delay in heat transfer probably
happened in all cable load tests, but was not readily ob-
served in tests at or near rated ampacities because of
1-min sampling rates (fig. 5). An attempt was made to
mathematically account for this phenomenon by calculating
the slope of both curves for a given current and time and
executing a transition from equation 1 to equation 13 when
the slopes equated. However, these calculations proved
too unwieldy for inclusion in the thermal model.

150

ROT T
1 ssl
1 ae
9 120h / KEY 4
*. ! — == Conductor-insulation
5 1 += Insulation- jacket
~ ! ssoeeee Cable surface
wy 90F 1 4
« ! ——|
g | i aa
Beor 7 a“ 4
rd 1 a
& :
a I /
if a;
F 30h / 4
,
so]
yy .
° Z Jeaeepeeeeeeeeye }
20 40 60 80 100
TIME, s

TIME

Figure 15.—Heavy (1,100% rating) overload on #2/0 AWG,
3/C, round, G-GC cable with time scale expanded.



Library of Congress Cataloging in Publication Data:

Yenchek, M. R, (Michael R.)

‘Thermal modeling of portable power cables / by Michael R. Yenchek, Gregory P.
Cole, and John C, Edwards.

Pp. _ cm. — (Report of investigations / United States Department of the
Interior, Bureau of Mines; 9463)

Includes bibliographical references (p. 14).
1. Coal mining machinery—Electric equipment—Protection. 2. Electric ca-

bles—Thermal properties—Mathematical models. 3, Electric currents—Heating ef-
fects—Mathematical models. I. Cole, Gregory P. Il. Edwards, John C. Ill. Title.

IV. Series: Report of investigations (United States. Bureau of Mines); 9463.

TN23.U43  [TN813] 622 s—dc20 —_[622’.334] 92-44394 CIP



Report of Investigations 9463
Thermal Modeling of Portable Power Cables

By Michael R. Yenchek, Gregory P. Cole, and John C. Edwards

UNITED STATES DEPARTMENT OF THE INTERIOR
Bruce Babbitt, Secretary

BUREAU OF MINES


140 r : ,
enema!
go 1207 “a KEY 4
° fo - Conductor - insulation
z 100 F Z nsulation- jacket
5 / pentane CANN ~ Cable surface
w fo
Bef
PA paneer
= sob i/ oo
z iy
B aol H/
40h fj
= It
F oli
20h!
id
ie
I 1
0
TIME, h

Figure 5.—Measured temperature rise in #2/0 AWG, 3/C, round, G-GC cable loaded at 400 A.

y

°

ro}
T

160

Oy
°o
T

TEMPERATURE RISE (AT), °C
@
is}
T

s
ts)
T

i it 1 i i 1
20 40 60 80 100 120 140 160 180

CURRENT, A

Figure 6.—Temperature rise versus current in #6 AWG, 3/C, round, G-GC cable.



KEY

—— Current

1

= Load test data
—-— Empirical model predictions

25

700
P 207 600
é
3 500
y isp «
4 400 E
Ww Ww
& c
p 10 300 &
& is)
& 200
& 5H
- U

100

ol °
100 150 200 250 300
TIME, min

Figure 11.-Comparisons of actual load test results with empirical model predictions for #4/0 AWG, 3/C,

round, G-GC cable.

OVERLOAD TESTS

In laboratory load tests, drag cable samples were sub-
jected to relatively long-duration overcurrents ranging from.
2 to 12 times the cable’s rated 90° C ampacity. The test
setup, essentially identical to that used previously in steady
state and periodic load tests (2), will be summarized here.

The cables were suspended horizontally by rope 3 ft
above and parallel to the lab floor. The test site ambient
was maintained at 20°+2° C. Air currents in the vicinity
of the cable from forced room ventilation ranged from 0
to 1 fps. All load tests were conducted using single-phase
alternating current. Test currents less than 2,250 A were
obtained from the three-phase supply of a motor test sta-
tion in the Mine Electrical Laboratory at the Bureau.
Above this level, a high-current step-down welding trans-
former was utilized as the power source. Type T thermo-
couples (#24 AWG, insulated with Teflon fluorocarbon
polymer) with accuracies of +0.5° C or 0.4% of reading,
whichever was greater, were employed to measure temper-
atures within and on the cable. The thermocouples were
in turn connected to a 32-channel datalogger that was
programmed to read and transmit data to a personal com-
puter. Commercial data acquisition software was used to
collect the data, display them in real time, and store them
on a disk (fig. 12).

The overload test currents were automatically inter-
rupted when the temperature at any of the 30 monitored
cable locations approached the range where the EPR insu-
lation could be thermally damaged (170° to 210° C). This
was accomplished by connecting the datalogger relay cir-
cuit to either the trip contacts of a molded-case circuit
breaker inserted in series with the cable conductors or the
emergency stop contacts of the high-current welding trans-
former. A constant current controller automatically ad-
justed the power source output voltage to compensate for
changes in conductor resistance during conductor heating.
Sampling rates ranged from 1 min for light overloads to
2s at 1,200% ampacity.

Typical load test results are shown in figures 13 and
14 for the #2/0 AWG round cable. Test duration varied
inversely with current. For example, with a 200% overload
(fig. 13) it took about 27 min for the conductor to achieve
a 150° C temperature rise, while at 11 times ampacity
(fig. 14) the conductor reached this level in 25 s. An
interesting phenomenon was observed at the conductor-
insulation, insulation-jacket, and cable surface-air inter-
faces. At light overloads, these locations. appeared to
reach a maximum temperature nearly simultaneously, tem-
perature rise at the surface being roughly half that at the