NBS TECHNICAL NOTE 1133

NAU- INST. OF STAND & TECH

A111D7 ED^LjflS

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Calculated Operating

Temperatures of Thermally

Insulated Electric Cables

QC

100

U5753

no, 1133

1980

c. 2
NATIONAL BUREAU OF STANDARDS

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Calculated Operating Temperatures

of Thermally Insulated Electric Cables

HiOwil Buret* 01 Stangfll

library, S«01 fctata. Wfe

FEB 1 6 1981

David D. Evans

Center for Fire Research

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Prepared for the Consumer Product Safety Commission

Washington, DC 20207

/

z

*< "'•*>,

% y°fSAU Of *

to J

-fc

/ID. /^3

/W

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

U»6, NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued January 1981
National Bureau of Standards Technical Note 1 133

Nat. Bur. Stand. (U.S.), Tech. Note 11 33, 41 pages (Jan. 1981) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $2.25

(Add 25 percent for other than U.S. mailing)
TABLE OF CONTENTS

Page

LIST OF FIGURES iv

LIST OF TABLES . V

LIST OF NOTATION vi

Abstract 1

1. INTRODUCTION 1

2. MODEL FOR CALCULATION 2

2.1 Representation of Cables 3 2.2 Heat Generated by Current Flow 3

2 .

3

Thermal Insulation Layer 4 2.4 Surface Heat Dissipation 5

3. METHOD FOR CALCULATION AND DISCUSSION 5

3.1 Heat Flow Between Concentric Cylinders 6

3 .

2

Heat Flow in a Cylinder Containing One or Two Heat Sources . 7

3 .

3

Heat Flow in a Cylinder With Temperature Dependent Thermal

Conductivity 10

4. EXPLANATION OF CURVES FOR CABLE JACKET TEMPERATURES 12

5. DISCUSSION OF RESULTS 13

6. CONCLUSIONS 14

7. REFERENCES 14

ill
LIST OF FIGURES

Page

Figure 1. Comparison of actual and model cable 16

Figure 2. Geometry of the insulated cable 16

Figure 3. Thermal conductivity for fibrous insulation 17

Figure 4. Temperature distribution within the insulation layer for a thermally insulated 14 gauge NM cable 18

Figure 5. Dimensional sketch of the insulation cylinder with two heat

sources representing the conductors in a 14 gauge cable . . 19

Figure 6. Normalized dimensional sketch of the insulation cylinder

with two heat sources 19

Figure 7. Transformation of cylinder onto the upper w-plane 19

Figure 8. Steady-state jacket temperature for AWG #14 copper 2-G NM

cable surrounded by insulation A 20

Figure 9. Steady-state jacket temperature for AWG #14 copper 2-G NM

cable surrounded by insulation B 21

Figure 10. Steady- state jacket temperature for AWG #12 copper 2-G NM

cable surrounded by insulation A 22

Figure 11. Steady- state jacket temperature for AWG #12 copper 2-G NW

cable surrounded by insulation B 23

Figure 12. Steady-state jacket temperature for AWG #10 copper 2-G NM

cable surrounded by insulation A 24

Figure 13. Steady-state jacket temperature for AWG #10 copper 2-G NM

cable surrounded by insulation B 25

Figure 14. Steady-state jacket temperature for AWG #8 aluminum 3 CDRS

SE cable surrounded by insulation A 26

Figure 15. Steady-state jacket temperature for AWG #8 aluminum 3 CDRS

SE cable surrounded by insulation B 27

Figure 16. Steady-state jacket temperature for AWG #6 aluminum 3 CDRS

SE cable surrounded by insulation A 28

Figure 17. Steady-state jacket temperature for AWG #6 aluminum 3 CDRS

SE cable surrounded by insulation B 29

Figure 18. Steady-state jacket temperature for AWG #4 aluminum 3 CDRS

SE cable surrounded by insulation A 30

Figure 19. Steady-state jacket temperature for AWG #4 aluminum 3 CDRS

SE cable surrounded by insulation B 31

IV
LIST OF TABLES

Page

Table 1. Effective diameter of cables 32

Table 2. Electrical resistance of metal conductors 32

Table 3. Values of thermal conductivity for air at atmospheric

pressure 33

Table 4. Values of ^ % for air at atmospheric pressure 33
LIST OF NOTATION

p

- specific heat of air [W's/g«°C]

D - diameter of outer surface of a cylinder [cm]

F - analytic function

g

- acceleration of gravity [cm/s 2

] Gr - Grashof number [dimensionless]

k - thermal conductivity of insulation material [W/cm °C]

k - thermal conductivity of insulation A [W/cm °C]

k, - thermal conductivity of insulation B [W/cm °C]

k - effective thermal conductivity [W/cm °C]

L - insulation thickness [cm]

Nu - Nusselt number [dimensionless]

Pr - Prandtl number [dimensionless]

q

- rate of heat production per unit length of conductor [W/cm]

q" - net rate of convective heat transfer per unit surface area [W/cm 2

] q" - net rate of radiative heat transfer per unit surface area [W/cm2

] r - radial distance [cm]

r - dimensionless radial distance r/r

s r. - cable jacket radius [cm]

r. - dimensionless cable jacket radius r./r

3 3 s r

- surface radius [cm]

r - dimensionless outer surface radius r /r

s s s t

- time [s]

T - temperature [°C or K]

T. - cable jacket temperature [°C or K]

T - ambient temperature [°C or K]

T - surface temperature [°C or K] u - variable substitution for T [°C or K] w - image point of z under transformation given by equation 10 x - dimensional coordinate [cm] x - dimensionless coordinate, x/r

s y

- dimensional coordinate [cm]

y

- dimensionless coordinate, y/r

z - dimensional coordinate x + iy [cm]

z - dimensionless coordinate, z/r

VI
LIST OF NOTATION (continued)

a - parameter equal to [ (l-£) / (l+£) ] [dimensionless]

B - coefficient of thermal expansion for air [1/K]

X - thermal conductivity of air [W/cm °C]

y - viscosity of air [g/cm«s]

p - density of air [g/cm 3

]

E, - dimensional distance of the conductor from center of cable [cm]

5 - dimensionless distance of the conductor from center of cable, £/r

Vll

CALCULATED OPERATING TEMPERATURES OF THERMALLY

INSULATED ELECTRIC CABLES

David D. Evans

Steady-state operating temperatures of currentcarrying electric cables buried axially in cylinders

of thermal insulation were calculated. Combinations

of six types of nonmetallic electric cable, two thermal

insulation materials, five thermal insulation thicknesses and a range of currents both greater and less

than the typical rated service currents for the cables were studied.

These calculations show that thermally insulated

electric cables carrying the rated current may exceed

the rated operating temperature limit for common cable

jacket materials.

Key words: Electric cable; electrical fire; models;

thermal insulation; wiring system.

1. INTRODUCTION

Accompanying the present increased use of thermal insulation have been

questions about the potential harmful effects of thermally insulating building electrical wiring systems. Existing electric cables routed through the

wall cavity and attic spaces in a building may be encapsulated by thermal

insulation.

Electric current limits for cables are established by considering the

temperature limits for the electrical insulation material used in construction

of the cable. In establishing the operating temperatures of current-carrying

electric cables, the heat generated by the current flow in the conductors is

considered to be dissipated through the cable jacket directly into air at a temperature of 30°C [1] . The established principle for general wiring is

that no conductor shall be used under such conditions that its temperature, even when carrying current, will exceed the specified maximum operating

temperature for the type of (electrical) insulation involved.

Numbers in brackets refer to the references listed in section 7.
All thermal insulation materials, such as glass fiber, cellulose fiber,

mineral fiber and foam plastics when surrounding electric cables inhibit the

dissipation of the heat generated by the electric current in the conductors.

For thermally insulated cables, the heat that normally would be dissipated

from a current-carrying cable directly from the cable surface to the surrounding air, must overcome the added resistance of the thermal insulation before

dissipation to the air. This additional resistance causes an increase in

the temperature of the cable above that for a cable exposed directly to air

under an identical current load. The increase in cable temperature, when

thermally insulated and carrying the rated current, should not be so large

that the cable operates at temperatures greater than the maximum design

operating temperature.

The calculations presented in this report predict the expected steadystate operating temperature of long electric cables wrapped circumferentially

with thermal insulation forming a cylinder of insulation. All materials are treated as chemically inert so that the only heat generated is by the current

flow through the electrical conductors. The adoption of cylindrical geometry

for the insulation layers around the cable facilitates the calculation of

cable jacket temperatures. Although a cylinder is not an exact representation

of the way insulation may surround electric cables in attics and wall spaces,

the calculated cable jacket temperatures demonstrate the sensitivity of

cable operating temperatures to encapsulation in thermal insulation. These

predicted steady operating temperatures can serve as a basis for discussion

of conditions that may exist in practical building installations.

2. MODEL FOR CALCULATION

The current-carrying electric cable is modeled as a cylindrical heat

source with uniform temperature. The temperatures of materials interior to

the outer cable jacket are assumed to be equal to the cable jacket temperature

For all thermally insulated cases, cylindrical symmetry is maintained by

considering the cable encapsulated in a cylinder of insulation. The thermal

conductivity of the insulation material is temperature dependent. All cable

jacket temperatures are calculated for the steady-state condition, in which

heat generated by the current flow within the conductors per unit length is

conducted radially through the insulation cylinder, and dissipated from the

outer surface by natural convection of air and radiation to the surroundings,

both at the ambient temperature of 30 °C.
2.1 Representation of Cables

For ease of calculation cylindrical symmetry is essential, but none of

the cables of interest were circular in cross section. For all calculations,

the non-circular cable cross section was represented by a circular cross section of equal circumference. This approximation for the cable diameter

is diagramed for a AWG 4 aluminum 3 CDRS SE cable in figure 1. Table 1 lists the effective diameters for all cables studied.

The numerical model is not intended to calculate the temperatures of

components inside the cable jacket. The temperatures for the conductors in

the cable are necessary only to determine the heat generated by the current

flow. The conductor temperatures were assumed to be equal to the cable

jacket temperature. For thermally insulated cables, the temperature differences between the conductors and the cable jacket are generally a small

fraction of the temperature rise of the cable jacket above ambient temperature. In tests conducted by Beausoliel et al. [2], the temperature difference

between the conductors and the cable jacket was typically only one or two

degrees Celsius.

2.2 Heat Generated by Current Flow

With the temperatures of components internal to the cable jacket set

equal to the jacket temperature, the temperature of the current-carrying

conductors is also equal to the jacket temperature for this calculation.

Knowing the temperature of the conductor, its resistance per unit length can be calculated. For a given electric current, the heat generated per unit

length of conductor can be calculated as the product of the square of the

current and the electrical resistance per unit length (I 2 R). In the calculations, this heat generation is modeled as if it were generated uniformly

along the axis of the cylinder.

Two conductors in each cable were assumed to be carrying equal current

with no current flow in the grounding conductor. It is essential in these

calculations to consider the increase in wire electrical resistance with

increasing temperature. For copper the electrical resistance per unit

length doubles over the range of 20°C to 275°C [3]. Aluminum doubles its

electrical resistance over the range of 20°C to 268°C [4] . Table 2 lists

the equations used to represent the electrical resistance of various size

copper and aluminum conductors as a function of temperature. These equations

are based on data collected over the temperature range of 0°C to 200 °C for

standard annealed copper [3] and 0°C to 100°C for aluminum ECH19 [4]. For

the calculations performed, values of electrical resistance outside these

ranges were generated by straight line extrapolation.
2 . 3 Thermal Insulation Layer

To maintain cylindrical symmetry, the solutions for the jacket temperatures

of the cables were generated by considering a layer of thermal insulation of

a given thickness wrapped circumferentially around the cable. The general

case is represented pictorially in figure 2. Heat generated within the

cable must be conducted through the insulation layer and dissipated from the

outer surface by heat transfer to the surroundings.

The heat conduction calculations take into account the increase in

thermal conductivity of fibrous insulation materials with increasing temperature. The thermal characteristics of two different fibrous insulations were

used in these calculations. Type A (see figure 3 [5]) is a 11 Kg/m 3 bulk

density glass fiber insulation which is representative of residential insulation material. Type B (see figure 3 [6]) is representative of a high density

mineral fiber industrial pipe insulation and would not be expected to be

used as a building insulation. Type B material was included in this study

strictly to demonstrate the sensitivity of predicted cable jacket temperatures

to changes in the thermal conductivity of the thermal insulation. The

thermal conductivity of material A is greater than that of material B over

the temperature range of interest for this study as shown in figure 3. It

is expected that the thermal conductivities of most other common building

insulation materials not included in this study, such as cellulosic fiber

and foam materials, will be at or between the values for the glass fiber and

high density mineral fiber insulations A and B respectively. Therefore the

range of predicted cable jacket temperatures between those for material type

A and B should be representative of the expected variations of results with

many different possible building insulation materials.

The thermal conductivity of each insulation, k and k, for Type A and

a jd Type B, respectively, can be calculated from the following equations representing the data fits shown in figure 3

k = exp [-7.9440 + 0.005194 T (°C)] W/cm°C [5] (1)

3.

k

fa = exp [-8.094 + 0.003834 T (°C)] W/cm°C [6] (2)

Thermal conductivity data outside the range of experimental values were calculated by extrapolation of the data fits using equations 1 and 2.
2.4 Surface Heat Dissipation

Heat generated by the flow of electric current in the conductors is

transferred through the insulation materials and dissipated to the surrounding air at the outer surface. Heat is dissipated at the outer surface by

two mechanisms, convection to the surrounding air at 30 °C and radiation to

the surrounding medium at 30 °C.

Radiative heat transfer was calculated from the familiar Stef an-Boltzmann

law. The net rate of radiative heat transfer per unit area of the cylinder

surface with unit emissivity, q ", at absolute temperature T to the sur- roundings at absolute temperature T is

q " = 5.672 x 10 -12 (T h - T

4

) W/cm2 (3) ^r so

where the units of both temperatures are kelvin.

Convective heat transfer to quiescent air at one atmosphere pressure

and 30°C was calculated from an empirical equation obtained by Bosworth [7]

representing experimental data for natural convection from cylinders over a wide range of variable Gr • Pr. In conventional notation this relationship

is

(Nu) 1 / 2 = 0.62 + 0.35 (Gr • Pr) 1 / 6 (4)

Using this relationship, the rate of convective heat transfer per unit area

of the cylinder surface q

" at temperature T to the air at T is

q^c (|) • 0.62 + 0.35 f!^VV • !^_\ 1/6

1

2

•

( VT

o

j

(5)

All air properties in this expression are evaluated at the film temperature,

the average of surface and ambient temperatures. Values for A and (g3p 2 /y 2

) as a function of film temperature were obtained by interpolation of tabular

values [8] given in tables 3 and 4. The value of Pr = yc /\ was taken as 0.72 for all temperatures.

3. METHOD FOR CALCULATION AND DISCUSSION

The calculated cable jacket temperatures as a function of electric

current flow in the conductors, and thermal insulation type and thickness were generated by iteration. An assumed jacket temperature was used to
calculate the outer surface temperature at the insulation-air interface.

The heat transfer rate from the outer thermal insulation surface was compared

to the heat generated by the conductors for the given electric current flow

and resistance. The assumed jacket temperature was adjusted until the heat

generated by the conductors equaled the heat loss from the outer surface.

This value of the jacket temperature is the steady-state value for the

conditions of interest. Steady-state jacket temperatures as a function of

current loads for six common electric cables, under various thermal insulation

conditions, were determined in this study.

A basic part of the above iteration method is the calculation of the

outer thermal insulation surface temperature for a given rate of heat production and jacket temperature of the cable. The method for performing this

calculation is discussed below.

3.1 Heat Flow Between Concentric Cylinders

For the calculations performed, the electric cable jacket and surrounding

insulation are modeled as two concentric cylinders. Within the cable jacket

there are two conductors that together produce a known rate of heat per unit

length of cable, 2 q

1 , when there is an electric current flow through the

conductors. The heat flow between the cylindrical outer surface of the

cable jacket and the cylindrical outer surface of the thermal insulation is

assumed to be entirely axisymmetric. The cable is also assumed to be long

so that conduction is radial. If the temperature of either the cable jacket

or the outer surface is known, the other may be calculated.

The temperature distribution between the two concentric cylinders,

representing the outer surfaces of the cable jacket and insulation layer

under steady-state conditions, can be obtained by observing that the heat

flow through any concentric cylindrical surface within the thermal insulation

is constant.

This can be represented by the equation:

dT - 2-rrkr -=— = constant = 2 q

1 (6) dr ^

The solution to equation 6 for the temperature distribution between the

bounding cylindrical surfaces for a given cable jacket temperature and

constant thermal conductivity is:

T = 31 fcn ln £. /A + T. (7)

(

r

TTk jA) *" [] / J
Similarly, for a given outer thermal insulation surface temperature the

temperature distribution is:

T = ¥K * n

( r s/ r

) + T

s

(8)

Temperature dependent thermal conductivity is discussed in section 3.3.

The above solutions assume that the electric cable may be represented

as a circular cylinder in order to simplify the calculations, but actual

cables are not circular in cross section. This approximation will introduce

distortions in the temperature distribution within the thermal insulation.

The greatest distortions will be in the immediate area of the cable jacket.

The amount of variation in predicted temperature between the axisymmetic

solution given by equation 8 and a more refined model of the thermally

insulated electric cable will be examined in the next section.

3.2 Heat Flow in a Cylinder Containing One or Two Heat Sources

In the previous section, a solution for temperature distribution within

the cylindrical layer of thermal insulation surrounding an electric cable

was presented. This solution was based on representing the cable as a single uniform axisymmetric heat source. In an actual electric cable the

centers of the two current-carrying conductors are separated by a distance

of approximately one-half the diameter of the cable. This separation of the

heat sources will produce a temperature distribution within the cylinder of

thermal insulation that is not axisymmetric. Deviations from the axisymmetric temperature distribution will be greatest near the heat sources.

As an illustrative example of the possible differences between the

solution for the single axisymmetric heat source and the solution for two

heat sources representing the separate conductors within the cable a specific

case of a AWG-14 two conductor cable was studied.

Specific parameters for the calculation were: a 30 ampere current

load, a heat generation rate of 0.1265 W/cm per conductor, a 5.91 cm diameter

for the outer surface of the insulation cylinder, a constant thermal con- ductivity for the thermal insulation of 0.000 36 W/cm °C, and a fixed insulation outer surface temperature of 4 3.9°C.

Using equation 8 the axisymmetic temperature distribution within the

insulation can be calculated with T = 43.9°C, r = 5.91/2 = 2.955 cm and

s s q

1 = 0.1265 W/cm and k = 0.00036 W/cm °C. The resulting expression for the

temperature distribution is:
T = 43 - 9 + n (2*!*) , grrogfg

where r is measured in centimeters.

Figure 4 shows the constant temperature lines for the axisymmetic heat

source calculated from equation 9 as the broken lines. Only one quarter of

the insulation cylinder cross section is shown. The size of the cylindrical

model cable for the AWG-14 cable, 0.91 cm diameter (see table 1) is also

shown. Based on the constant temperature lines shown in figure 4 the predicted temperature for the cylindrical model AWG-14 cable jacket is slightly

greater than 250°C.

The solution for the temperature distribution produced by two heat

sources representing the two conductors within the cable was obtained by

conformal transformations [9] and by the method of images. Figure 5 shows a dimensional sketch of two line heat sources representing the separation of

conductors within a AWG-14 cable enclosed by a cylinder of insulation in the

dimensional z-plane.

Figure 6 shows the same configuration in the dimensionless z-plane, in

which all dimensions are normalized by dividing them by the radius of the

outer surface of the insulation cylinder, r = 2.955 cm.

s

Figure 7 shows the mapping of the cylindrical cross section to the

w-plane by the transformation

w = V1 "

Z) (10)

z + 1 The labels in figure 6 help to locate the image points in figure 7.

The result of the transformation is to map the area inside the cylinder onto

the upper half plane. The circumference of the cylinder becomes the real

axis in the w-plane. To satisfy the boundary conditions for this problem,

the real axis in the w-plane must be a constant potential line. By introducing two heat sources of equal strength along the imaginary axis above the

real axis and as counterparts two identical heat sinks along the imaginary

axis below the real axis, the two heat sources within the cylinder representing the cable conductors may be represented as two corresponding points

on the w-plane above the real axis, see figure 7. The two points representing sinks below the real axis cause the real axis to become a constant

potential line. The complex potential for heat flow between the sources and

sinks is represented by the analytical function, F (w) as:

8
F (w) = „ q, F[£n (w - ia) - In (w + ia) ] + [In (w - i/a) - In (w + i/a)] (11)

where a = (1 - £) / (1 + £) in which 5 is dimensionless distance of the con- ductor from the center of the cable. The real part of F (w) is zero everywhere along the real axis.

Substituting for w in terms of z, yields the form of the solution in

the z -plane.

F <«> = 2?F £n (.

Ml v ~

. ,, ,\ /i(l+-)z - i(l - -) i(l+g)z - xd-cQl + £n I a a i(l-a)z - i(l+a) J lid- i)z - i(l + £)

(12)

Substituting for z = x + iy and extracting the negative of the real

part yields the temperature distribution in the z-plane as a function of

T-T :

Re(F(z)) = T-T

s = ^ -g)y] 2 + [ (l+g) - (l-a)xl 2 a)y] 2 + [ (1-a) - (l+o)x] 2

, n // ta - |)yi 2 I

[(i + ^) - (i - £)x]

l\ [(l + J)yl

2 + [(l - |) - (l + !)£]

(13)

Equation 13 was used to generate the temperature distribution from two

heat sources representing the separate conductors within the electric cable.

This distribution is shown for comparison with the temperature distribution

produced by the single heat source as the solid lines in figure 4. Only one quarter of the cylindrical cross section is shown. For scale, an outline

representing the cable jacket of the 2 conductor AWG-14 cable is shown. It can be seen from figure 4 that the temperature distribution calculated from

the single heat source and cylindrical cable approximation fall between the

minimum and maximum temperatures of the distribution predicted from the two

heat source solution. For this study temperatures at the cable jacket

(250°C contour) are of interest. Even though deviations of the cylindrical

approximation from the better two heat source solution are greatest there,

these deviations are tolerable in this study.
All of the solutions for cable jacket temperatures generated in this

report are calculated on the basis of the cylindrical cable approximation

and a single axial heat source. Even though a solution for two heat sources was available, it is cumbersome and, as just demonstrated, does not provide

much more information than the single source solution. Predicted cable

jacket temperatures, using the above approximations, should be regarded as an average value for the circumference of the cable.

A special case of equation 13 results if both line sources are superimposed

at the origin by setting £ = thus making a = 1. The resulting temperature

distribution in normalized coordinates is

T-T = ^— In

s nk V"x + y (14)

Noting that a radial distance from the origin satisfies r

2 = x

2 + y

2

, equation 14 becomes in dimensional coordinates:

T-T

s = g/_ /

( } irk s /

Equation 15 agrees with the result obtained by other means, equation 8.

The technique used to calculate the temperature distribution produced

by two line sources representing a single cable, may be extended by adding

the additional source-sink pairs to calculate the temperature distribution

produced by several cables.

3.3 Heat Flow in a Cylinder With Temperature

Dependent Thermal Conductivity

As shown in figure 3, the thermal conductivity of fibrous insulation

materials is a strong function of the temperature of the material. For the

two insulation materials, A and B, the thermal conductivity increases exponentially with increasing material temperature. This exponential temperature

dependence for the thermal conductivity can be incorporated into a solution

for the temperature distribution from one axial heat source within a cylinder

of insulation. This modification of the calculation for the single heat

source detailed in section 3.2 was used in the calculation of cable jacket

temperature presented in this report.

10
The case of the constant thermal conductivity was useful in demonstrating

the differences between temperature distribution produced by the single and

dual heat sources. The increase in thermal insulation conductivity with

increasing temperature is significant enough over the temperature range of

interest in this study, that the overall thermal resistance of the insulation

layer surrounding the cable is affected. Better predictions of the operating

temperatures of thermally insulated electric cables are obtained by accounting for thermal conductivity variations.

To incorporate non-constant thermal conductivity in the solution for

the temperature distribution between cylinders, a different method of solution

from that given in section 3.2 for constant thermal conductivity must be

used. For steady-state conduction between cylinders, the rate of heat

conduction through cylindrical shells at all radii must be equal. Using

Fourier's law of heat conduction applied at any radius between the cylinders

representing an electric cable (2 conductors) and the outer surface of the

thermal insulation, the rate of heat conduction per unit length is:

2q'^

- -2^rk ^

(16) dr

Equation 16 is identical to equation 6 , except that the integration of

the equation is complicated by the fact that thermal conductivity, k, is not

assumed constant and is a function of the local temperature, T.

Equation 16 may be integrated by introducing a new variable u for T and

that,

a constant thermal conductivity k . The new variable, u is defined such

k

T

( d7} = k( dT} (17)

with the boundary conditions,

= T . at r = r

.

(18)

3 J

T at r = r (19)

s s

Integrating equation 17 using boundary conditions given in equations If 9, the ef

determined as:

and 19, the effective value of constant thermal conductivity k can be

11
T

k

T

= ^^-

T . :

Thus k^ is the average thermal conductivity between the temperature

extremes at the boundaries of the insulation layer. It is interesting to

note that by introducing the temperature dependent thermal conductivity both

boundary conditions at r . and r must be used in the integrate process for

equation 16. In integrating equation 6, only one was needed.

Substituting u for T and k for k, equation 16 may be integrated using

either boundary condition equation 18 or 19. Using equation 18, the temperature distribution within the insulation layer is:

T = 2^ £n (r j/r) + T

j (21 >

Equation 21 is identical to equation 7 except for the effective

substitution of k for k.

The thermal conductivity of both the glass fiber insulation material A

and mineral fiber material B may be expressed in the form k = exp (a + bT) as given in equations 1 and 2. Using equation 20 the effective or average

thermal conductivity is

k = 1 [exp (a+bT ) - exp (a+bT.)l (22)

(T-T.)b L

: J s j

In this study, a surface temperature at the outer surface of the

insulation layer, T , was found by iteration using equations 21 and 22 for

each assumed value of the cable jacket temperature, T..

4. EXPLANATION OF CURVES FOR CABLE JACKET TEMPERATURES

The results of the calculations performed are presented in figures 8 through 19. In each of these figures, predicted cable jacket temperatures

are plotted as a function of the current for a specific cable and insulation

material. Results are shown for a range of currents, above and below the

rated current. The typical rated service current for each cable is recorded

in each figure. For ease of comparison, predicted temperatures for each

12
cable buried in insulation B are immediately following those for insulation

A. Predictions are presented for 14, 12, 10 AWG copper and 8, 6, 4 AWG

aluminum cables. An ambient temperature of 30°C is assumed for all calculations

To demonstrate the effect of thermal insulation on the cable jacket

temperature, predicted values for the cable alone are given in each plot

labeled, "no insulation". In each figure 8 through 19, curves for cable

jacket temperature are shown for 2.5 cm, 5 cm, 10 cm, and 20 cm thick layers

of insulation surrounding the cable as diagramed in figure 2.

5. DISCUSSION OF RESULTS

Comparing the results for insulations A and B for like conditions, the

cable jacket temperatures for the cable buried in insulation B are in all cases substantially greater than for the cable in insulation A. The cable

jacket temperature is quite sensitive to the thermal conductivity of the

insulation material.

All cables can exceed the common temperature limits for the jacket

materials when the cable is sufficiently insulated and carrying the typical

rated current. In all cases, these calculations show that a 60°C temperature limit common for cable jacket materials is exceeded at typical rated

current for all cables studied, when wrapped with a 5 cm thick layer of

either insulation A or B. For many cases this temperature limit was exceeded

with only a 2.5 cm thick layer. As wiring systems in buildings may be

buried in insulation layers that are substantially thicker and provide more

resistance to heat dissipation, cable jacket temperatures may routinely

exceed recommended limits in normal service.

Cables that were loaded beyond the recommended current limits can become hot enough to ignite combustibles. Ignition of cellulosic insulation

by buried electric cables has been reported by Gross [10] for a 14 gauge

cable carrying 28.5 amperes, which was 19 percent of the typical rated 15

ampere current load.

It should be reemphasized that the predictions shown in figures 8 through 19 are for steady -state conditions and a 30 °C ambient temperature.

Normally, a few hours would be required for the cable and insulation material

initially at ambient conditions to reach steady-state temperatures, but more

than 60 percent of the final temperature rise can be expected to occur within the first hour. Ambient temperatures above 30 °C will elevate the

13
cable temperature above the predicted values. At low final cable temperatures,

the change in predicted cable jacket temperature will be almost identical to

the change in ambient temperature. At high final cable temperatures, the

change in cable temperatures will be less than the change in ambient temperature,

6. CONCLUSIONS

Predictions of the steady-state operating temperatures of thermally

insulated current-carrying electric cables have been presented for the

special case of a cable buried axially in a cylinder of thermal insulation.

These predictions show that electric cables carrying the typical rated

current can exceed the common 60 °C operating temperature limit for common

jacket materials with a 5 cm thick layer and often just a 2.5 cm thick layer

of thermal insulation wrapped circumferentially around the cable. This

suggests that the jacket materials of electric cables that are frequently

encapsulated in layers of building insulation could be exceeding recommended

upper temperature limits for the materials, if the typical rated current

load is maintained long enough to establish nearly steady conditions. For

cables operated for an extended period of time above typical rated current,

sufficiently high temperatures may be reached to induce ignition of

combustible insulation materials.

7

.

REFERENCES

[1] Allowable ampacities of insulated conductors rated 0-2000 volts, 60° to

90°C, Table 310-316, National Electric Code, 1978, National Fire Protection Association, Boston, MA, p. 132.

[2] Beausoliel, R. W. , Meese, W. J., and Galowin, L. S., Exploratory study

of temperatures produced by self-heating of residential branch circuit

wiring when surrounded by thermal insulation, Nat. Bur. Stand. (U.S.),

NBSIR 78-1477, July 1978.

[3] Copper wire tables, National Bureau of Standards Handbook 100, February

1966.

[4] Peterson, C. and Thomas, J. L., Aluminum wire tables, National Bureau

of Standards Handbook 109, February 1972.

[5] Pellane, C. M. and Neal, J. E., Heat flow principles: heat flow practice,

Johns-Manville Refractory Products Engineering Information.

14
[6] Standard Recommended Practice for Determination of Heat Gain or Loss;

and Surface Temperatures of Insulated Pipe and Equipment Systems by the

Use of a Computer Program, 1978 Annual Book of ASTM Standards, Part 18,

C680, p. 430.

[7] Bosworth, R. C. L. , Heat transfer phenomena, the flow of heat in physical

systems, John Wiley and Sons, New York, 1952, p. 101.

[8] Kreith, F. , Principles of heat transfer, International Textbook Company,

Scranton, 1965, p. 595.

[9] Churchill, R. V., Complex variables and applications, Second Edition,

McGraw-Hill Book Company, New York, 1960.

[10] Gross, D., A preliminary study of the fire safety of thermal insulation

for use in attics or enclosed spaces in residential housing, Nat. Bur.

Stand. (U.S.), NBSIR 78-1497, July 1978.

15
#4 AWG 3 CDRS

Aluminum SE Cable

Model Cable

1.3cm

Effective diameter

Figure 1. Comparison of actual and model cable

Natural convection

30°C ambient

Radiation

Figure 2. Geometry of the insulated cable

16
oE

u

>-

D

Z

o

u

<

40

30

20

15

10

9

8

7

6

100 200 300

TEMPERATURE °C

400

Figure 3. Thermal conductivity for fibrous insulation

17
S-l m

u

Q)>

(0 -Hc

o

•HP

n3 rH3

encH QJ rHX

s: (0 P U c S

-H s

x:

4-" aj H tn 5 a

rd C tn H •<* P H

3

ja T3 H QJ

S-l P

JJ t0

0} rH •H 3

13 W

c

H U

3 >i

-P .H

<0 H

S-l (0 Q) e

Oj S-l = a)

0) r^ H P

QJ

S-l

tn
X—LOCATION OF

HEAT SOURCE

^X'

3 mxlO" 2

Figure 5. Dimensional sketch of the insulation cylinder with two heat sources representing the conductors in a 14 gauge cable

yj \d^

E [ Y

h^0.0846 |C ^

A I A

r

X

J 1 X

Figure 6. Normalized dimensional sketch of the insulation cylinder

with two heat sources

Fiqure 7. Transformation of cylinder onto the upper w-plane

19
300

280

260 -

240 -

220 -

200 -

180 -

160 -

140 - g 120 -

100

80

60

40

20

Typical rated service: 15 amperes

oo

2

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

10 15 20

CURRENT AMPERES

25 30

Figure 8. Steady-state jacket temperature for AWG #14 copper 2-G NM

cable surrounded by insulation A

20
oo

2

E

300

280

260 -

240

220

200

180

160

140

120

100

80

60

40

20

Typical rated service: 15 amperes

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

10 15 20

CURRENT AMPERES

25 30

Figure 9. Steady-state jacket temperature for AWG #14 copper 2-G NM

cable surrounded by insulation B

21
oo

2

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

Typical

! rated service: 20

I amperes

I l _

- Curve

no.

Insulation

thickness -

"

1 20cm "

-

2 10cm -

3

4

5cm

2.5cm

X -

— 5 No insulation s ~

- yT -

\^z>^^

-

5 ^^____=i

I i

1

1

—

10 15 20 25

CURRENT AMPERES

30 35

Figure 10. Steady-state jacket temperature for AWG #12 copper 2-G NM

cable surrounded by insulation A

22
oo

XL

300

280

260

240

220

200

180

160

140

120

100 -

80

60

40

20

_ ! j __—

Typical rated service: 20 amperes

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

1 i

10 15 20 25

CURRENT AMPERES

30 35

Figure 11. Steady-state jacket temperature for AWG #12 copper 2-G NM

cable surrounded by insulation B

23
300

280

260

240

220

200

P

18° | 160 h

g 140

|

120

100

80

60

40

20

Typical rated service: 30 amperes

Curve

na

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

10 20 30 40

CURRENT AMPERES

50 60

Figure 12. Steady-state jacket temperature for AWG #10 copper 2-G NM

cable surrounded by insulation A

24
oo

300 -

280 -

260 -

240 -

220 -

200 -

180 -

160 -

140 -

120 -

100 -

80 -

60 -

40 -

20 -_

1 r~

Typical rated service: 30 amperes

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

1

10 20 30

CURRENT AMPERES

40 50 60

Figure 13. Steady-state jacket temperature for AWG #10 copper 2-G NM

cable surrounded by insulation B

25
o

2UJQ_

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

- Typical rated service: 30 amperes

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

10 20 30 40

CURRENT AMPERES

50 60

Figure 14. Steady-state jacket temperature for AWG #8 aluminum 3 CDRS

SE cable surrounded by insulation A

26
300

280

260 -

240 -

220 -

200 -

180 -

160 - 2 140

120 -

100

80

60

40

20

1 r~

Typical rated service: 30 amperes

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

10 20 30 40

CURRENT AMPERES

50 60

Figure 15. Steady-state jacket temperature for AWG #8 aluminum 3 CDRS

SE cable surrounded by insulation B

27
300

oo

t—

280 -

260 -

240 -

220 -

200 -

180 -

160 -

140

120

100

80

60

40

20

10

1 1 r~~

Typical rated service: 40 amperes

Curve

no.

12

3

4

5

Insulation

thickness

20cm

10cm

5cm

2.5cm

No insulation

l 1

20 30 40 50

CURRENT AMPERES

60 70

Figure 16. Steady-state jacket temperature for AWG #6 aluminum 3 CDRS

SE cable surrounded by insulation A

28
oo

2

10 20 30 40 50

CURRENT AMPERES

60 70

Figure 17. Steady-state jacket temperature for AWG #6 aluminum 3 CDRS

SE cable surrounded by insulation B

29
oo

300

280

260

240

220

200

180

| 160

|

140

| 120

100

80

60

40

20

I I I - Typical rated service: 55 amperes

i I _

_ Curve Insulation

no. thickness

1 20cm

2 10cm —

3 5cm

4 2.5cm

^r —

_ 5 No insulation

/2 yS

- y/ *>^ -

- ^^ 4^^

-

- ^^

5 -

_j I

i

i i

-

20 30 40 50 60

CURRENT AMPERES

70 80

Figure 18. Steady-state jacket temperature for AWG #4 aluminum 3 CDRS

SE cable surrounded by insulation A

30
300

280

I I

Typical rated service: 55

i amperes

l I

260 Curve

no.

Insulation

thickness

/ -

240

1 20cm

' —

220 2 10cm

200

3

4

5cm

2.5cm / - o

180

o

5 No insulation / ^ / / - £ 160

£ 140

UJ -

|

120 -

100 -

80 -

60 -

5 —

40

"""I 20 -

l I

i

i i

20 30 40 50 60

CURRENT AMPERES

70 80

Figure 19. Steady-state jacket temperature for AWG #4 aluminum 3 CDRS

SE cable surrounded by insulation B

31
Table 1. Effective diameter of cables

2-G AWG Gauge/Metal Effective Diameter (cm)

14/Cu 0.91

12/Cu 1.02

10/Cu 1.15

4/A1 1.85

6/A1 1.54

8/A1 1.46

Table 2. Electrical resistance of metal conductors

1 General Form:

R = A [1 + BT(°C)] x 10~ 5 ohms/cm

Copper B = 0.00427 [1/°C]

Aluminum B = 0.00438 [1/°C]

AWG Gauge/Metal A* [ohms/cm]

14/Cu 7.63

12/Cu 4.80

10/Cu 3.02

4/A1 1.229

6/A1 1.954

8/A1 3.106

*For stranded aluminum conductors, increase table value of A by 2%.

32
Table 3. Values of thermal conductivity for air at

atmospheric pressure

Temperature Thermal Conductivity

°C W/cm°C

0.000242

38 0.000266

148 0.000334

260 0.000400

371 0.000464

482 0.000524

815 0.000692

Table 4. Values of 2-SP f or a ^ r at atmospheric pressure

Temperature

°C

ggp

2 V

1°/C cm3

201

38 112

148 28

260 10.1

371 4.48

482 2.29

815 0.447

33 o U. S. GOVERNMENT PRINTING OFFICE : 1980 340-997/329
NBS-114A irev, 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions,)

1. PUBLICATION OR

REPORT NO. NBS TN 1133

2. Performing Organ. Report No. 3. Publ ication Date

January 1981

4. TITLE AND SUBTITLE

Calculated Operating Temperatures of Thermally Insulated Electric Cables

5. AUTHOR(S)

David D . Evan

s

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

U.S. Consumer Product Safety Commission

Washington, DC 20207

10. SUPPLEMENTARY NOTES

2] Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Steady-state operating temperatures of current-carrying electric cables

buried axially in cylinders of thermal insulation were calculated.

Combinations of six types of nonmetallic electric cable, two thermal

insulation materials, five thermal insulation thicknesses and a range

of currents both greater and less than the typical rated service currents

for the cables were studied.

These calculations show that thermally insulated electric cables carrying

the rated current may exceed the rated operating temperature limit for common cable jacket materials.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Electric cable; electrical fire; models; thermal insulation; wiring system.

13. AVAILABILITY

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