Fire Safety Journal 16 (1990) 459-467

Effect of Deposi t ion of Combustible Matter onto Electric Power Cables

B. F. Gray

School of Chemistry, Macquarie University, Sydney NSW 2109, Australia

J. Dewynne

Mathematical Institute, University of Oxford, Oxford, UK

M. Hood

Department of Mathematics, University of Western Australia, Perth, WA 6009, Australia

G. C. Wake

Department of Mathematics and Statistics, Massey University, Palmeston North, New Zealand

&

R. Weber

Department of Mathematics, ADFA, Canberra, ACT, Australia

(Received 29 January 1990; revised version received 9 April 1990; accepted 19 April 1990)

A B S T R A C T

The thermal insulating and self-heating effect of combustible dust deposited on electric power cables is discussed. In many practical situations, such as factories and warehouses, cable runs are exposed to the deposition of many insulating and combustible dusts over long periods. Critical thicknesses for these layers are calculated, i.e. the thicknesses for which cable failure will occur, either due to the onset of runaway combustion reaction in the dust layer, or even before the onset of runaway reaction when temperatures can reach the melting point o f PVC insulation.

N O T A T I O N

a Radius of conductor b Outer radius of PVC insulator

459 Fire Safety Journal 0379-7112/91/$03-50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland

460

b + h E F G I kc ki k' k" l O R R'

Ta x Z

6

0

P Z

B. F. Gray et al.

Radius of dust layer (scaled to be unity) Activation energy of chemical reaction of the dust Integration constant Integration constant Current Thermal conductivity of copper Thermal conductivity of PVC Defined by eqn (7) Defined by eqn (8) Characteristic distance (e.g. outer radius of dust annulus) Heat of reaction/unit mass Conductor resistance/unit length Universal gas constant Ambient temperature Radial distance from cable centre Pre-exponential rate factor

Defined by eqn (10) E

- ( T - - Ta) Dimensionless temperature R' T2a

Thermal conductivity of dust layer Density of reactive dust Defined by eqn (11)

INTRODUCTION

Electric power cables are used in plant installations according to selection criteria set out in various national standards, e.g. AS3000 and AS3008. The bases for selecting and installing the correct size and type of cable are governed by the current flow, the length of installations and type of cable insulation to be used. 'Current rating' of cables is determined by the temperature reached in the conductor under steady-state operation. This temperature is of course determined by the rate of heat conduction through the insulator. Cables placed in a conduit or adjacent to each other on a tray are 'derated' , since the heat generated will be harder to dissipate than that for a single cable. The geometry of the cable arrangements determines the quantitative value of the derating. Standard tables are available for this purpose and are applied to each installation.

A problem not hitherto considered is the collection of combustible (and, at the same time, thermally insulating) dust on the installations.

Combust ib le mat ter on electric p o w e r cables 461

This could achieve considerable thickness over long periods and lead to two possible problems which are intimately related.

Firstly, the layer of dust could ignite if the critical condition were to be reached. This depends, as we shall see later, on the ambient temperature, the thickness and configuration of the dust layer, the electric heating by the current, etc. Secondly, the dust layer, if not extremely reactive, might cause failure of the cable by overheating, since the extra insulation of the dust layer is not allowed for in standard tables, nor is the heat generated by normal decomposition of the dust. One can thus envisage two extreme types of failure: ignition of the dust before cable failure, and cable failure before the dust ignites due to its insulating and thermogenetic properties.

THE PROBLEMS F O R M U L A T E D

The most promising formulation with possibility of mathematical solution is shown in Fig. 1. It is a cable of infinite length inside a conduit with the annular space between insulator and conduit filled with combustible dust of uniform density. The cylindrical symmetry of the problem renders it attractive, as well as the fact that related problems involving ignition of materials by annular sheet sources of energy (representing friction between rotating cylinders) have already been solved. ~'2 It is also well known that the infinite cylinder gives results which are bounds for finite cylinders, erring on the side of safety.

The steady-state energy conservation equations in the conductor, PVC

Fig. 1.

support structure

uniform heat source

electrical insulation

combustible material

conduit

Representation of a cable inside a conduit filled with combustible insulating material.

462 B.F. Gray et al.

insulator and outer dust annulus take, respectively, the following forms:

d20 1 dO E F R ~- --7-~- + R'T2kc = 0 x u x ( 0 < x < a ) , (1) dx 2

d20 1 dO f - - = 0 (a < x < b ) , ( 2 )

dx 2 x d x

d20 1 dO dx---- S + ~ + 6e ° = 0 (b < x < 1), (3)

where 0 is a dimensionless tempera ture rise in the assembly, refer red to ambient tempera ture (0 = 0), x is a dimensionless radial distance f rom the centre of the cable and 6 is a dimensionless measure of the heat production rate by the decomposit ion and /o r oxidation of the dust. At the interfaces between the three layers, energy flux and tempera ture continuity conditions are applied. Analytical solutions of eqns (1)-(3) are available, and the continuity conditions plus the boundary condition 0 = 0 along with the symmetry condition d 0 / d x = 0 at x = 0 serve to determine all the integration constants.

The form of the solution in the reactive layer is

2F2Gx F-2 0 = In (4)

6(1 + GxF) 2

where F and G are integration constants. If 00 is the central (maximum) tempera ture in the assembly,

application of the boundary and continuity conditions gives us an exact parametric relationship be tween 6 and 0o:

eO ° = [(F + k' + 2) + b-V(F - k' - 2)12b v-2

4FZk,, , (5)

2Fe(F - k' - 2)(F + k' + 2)b -F 6 = [(F + k' + 2) + (F + k' - 2)b-F]2' IF[ > [k' + 21 (6)

The constants k' and k" are defined as

E F R k ' = , - ~ < k ' < - 0 (7)

2R' T2ki

[ eFR EFR k" = exp - - In ( b / a )

2R, T2k i 4 ~ ~-kc/ (8) t

Both a and b are scaled with respect to the radius of the annulus of dust (b + h).

Combustible matter on electric power cables 463

O o

cr

~ 0 . . . . . .

I

Schematic of the relationship between the heat production rate 6 and the central temperature 0o.

A typical graph of 6 versus 0 is shown in Fig. 2. This curve would be calculated for a particular pair of values of k' and k" as well as given radii for conductor, insulator and dust. It is seen that for 6 < 6= (the critical value of 6) there are two solutions for the temperature profile. Of course, for 6 > 6~ there is no steady-state solution for this problem.

The critical values of 6 and 00 are calculated for a given set of parameters k ' , k" and cable dimension, and this enables the construc- tion of a table such as Table 1. Table 1 is calculated for k" = 0-5. Other values of k" affect 0~ r through the formula

0~ ~= 0~(0-5) - l n ( k " / 0 . 5 ) (9)

with 6= independent of k". The thickness of the layers is expressed in units of the radius of the dust layer.

To use this table we have to calculate 6 for a given material from the formula

6 = P Q Z I 2 E e - ~ / R ' r "

~ T 2 R ' (lo)

Experiments carried out using various dusts in an equicylinder of gauze of radius 7.5 cm determined the critical ambient temperature for each (Gray, B. F. unpublished; and Ref. 3). For an equicylinder, di=--2-78.

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3-5

Combust ib le matter on electric p o w e r cables

For l, Ta and E known we thus have

2.78 - pa____Z 12E e_E/R,r~ Z R'T~

12E = X R--~a 2 e-E/RT~

where

SO

pQZ x= Z

465

giving X = 5.3 x 1013 K / c m 2

We substitute this figure in eqn (12) and obtain a value of 6 for a given thickness of dust on the cable; for example, for a dust layer nine times thicker than the PVC insulation (still only 1-17-cm radius), then 6 = 1.7 × 10 -8. This is considerably less than the critical value for this case (h = 9b in Table 1) of 0or = 6.0 × 10 -4. We can increase the dust layer until the calculated value of 6 computed from eqn (12) equals the critical value of 6 computed f rom eqns (5) and (6). In this case, this occurs around 15 cm sawdust a round the cable, an amount which could easily occur in practice.

For coal dust, where E/R ' -5000 , similar calculations give X = 3.37 x l 0 s and k' = 1.63. The critical depth of dust in this case, for the same cable, is of the same order of magnitude.

nT a Z = 2 .78 / -~ - x e e'R'r~ (11)

enabling effective experimental measurement of the group X. Equat ion (10) thus becomes

xI2Ee-E/R'r~ 6 - (12)

R'T

where l and Ta are now appropriate for the configuration in question, i.e. an annulus at an ambient tempera ture of 30 or 40 °C, for example, as possible realistic temperatures .

As an example, let us consider bagasse or sawdust, for which E = 125 kJ /mol and T~ r = 470 K for a 7-5-cm-radius equicylinder.

For this material , then, E/R ' -15000K, so eqn (12) gives us, at criticality,

X(7.5)z15000 exp ( - 15000/470) 2.78 =

(470) 2

466 B. F. Gray et al.

Even in the subcritical cases, however, where neither dust is close to ignition, we would expect the insulating effect of the dust to cause melting of the PVC insulation; for example, for the cable carrying a current of 14 amps with a layer of dust nine times thicker than the PVC, the temperature at the conductor centre can easily be calculated to be around 300 °C for an ambient temperature of 20 °C. This assumes a thermal conductivity for the dust of 5 x 10 -4 W/cm ~ K ~ and that of the PVC to be 1.67 x 10-3W/cm 1 K ~. With no dust, the corresponding calculation gives a temperature of 36 °C above ambient for this case. With a dust layer of equal thickness to that of the PVC the temperature rise is 77 °C.

Clearly, even for ignitable materials, well before ignition takes place, the insulating effect of the dust will cause the PVC to melt unless very considerable derating is done.

POSSIBLE SOURCES OF E R R O R

A possibly important source is the density of the dust layer. In the above calculations it is implicitly assumed to be the same as that of the samples tested for criticality. The density achieved by natural settling over a period is unknown, but is likely to be less than that used in the experiments. If it were 10% of this, both X and 6 would be reduced by this factor.

We do not know how the conductivity of the dust layers depends on its density, and the above calculation assumes it is independent of density. The thermal conductivity of air is approximately 1-5 x 10 -5 W/cm ~ K 1, which is somewhat smaller than that for either sawdust or coal at the packing density used. Thus, we might expect a very porous layer of dust or still air to make the system even less stable than calculated above.

A closely related problem, which has not been addressed here, is that of transient energy deposition in similar circumstances arising from switching and current surges. It is very likely that supercritical conditions can be set up in this way also.

CONCLUSIONS

1. Practically likely thicknesses of dust deposited on power cables can lead to two kinds of failure.

Combustible matter on electric power cables 467

2. The first kind arises purely from the thermal insulating properties of the dust; it need not necessarily be combustible. Temperatures high enough to melt the PVC insulator can easily be attained with normal currents, largely due to the fact that those dusts most likely have a thermal conductivity an order of magnitude lower than PVC. 3. The second kind arises when, in addition to being a good thermal insulator, the dust is combustible. This results in the melting point of PVC being attained at lower current densities than in the previous case, due to the extra heat from the combustion reaction. 4. In the case of combustible dusts, it is also possible that a critical thickness of dust can be exceeded which will lead to runaway reaction and ignition of the dust layer. For example, for sawdust, at an ambient (air) temperature of 30°C, a layer 15-cm thick would lead to runaway reaction if deposited around a cable carrying 14amps. Of course the PVC insulator will also have melted, but even if this did not lead to short circuit in this case, fire is guaranteed from self-ignition of the dust.

REFERENCES

1. Gray, B. F. & Wake, G. C., Criticality in the infinite state and cylinder with surface heat sources. Combustion and Flame, 55 (1984) 23-30.

2. Gomez, A., Wake, G. C. & Gray, B. F., Friction and localised heat initiation of ignition. Combustion and Flame, 61 (1985), 177-87.

3. Gray, B. F., Griffiths, J. F. & Hasko, S. M., Spontaneous ignition hazards in stockpiles of cellulosic materials. J. Chem. Tech. Biotech., 34A (1984) 453-63.