The following identities and definitions are necessary for further development of the problem

Where In(x) and K,(x) are modified Bessel functions of order n, and of the first and second kind, respectively.

Proceeding as in section 2.1.a, and using (17), the solutions for the two finite regions of

Mt, Pt, and Qt must satisfy the boundary and continuity conditions (7). The exterior boundary condition, that 0-0 at the surfaces z=0, z=l, and r=b, is satisfied by the above equations. For region 1, c<r<r'

Τ

b. Multiple Line Heat Sources

Consider, as in section 2.1.b, m equally spaced equal sources at r=r'. Equation (19)

where q=mi and j=2n-1.

Σ

n=1

j

{ro

Yo(r',r) +2 — Y,(r', r) cos qu

•}

It is of particular interest to evaluate the mean temperature at r=c and z=1/2,

(20)

to determine the effects of longitudinal heat flow by the ratio, 7, of this temperature to the average temperature for the infinitely long cylinder at r=c as given by (13):

Numerical solutions for 7 versus the dimension ratio /b for various values of σ, are shown in figure 4 for fixed radius dimension ratios b/a=5.0, r'/a=0.7, c/a=0.2.

An interesting special case of multiple line sources in an infinitely long cylinder is the case where k1-k2. Substitution of o=1 in (11) gives

For r=c and for c<<r'<b, only the first term in (22) need be considered. Physically, this special case represents a single cylinder of homogeneous material, in which m heater wires parallel to the axis are embedded at a radius r' with equal angular spacing. The temperature is measured at the axis of the cylinder.

For the special case in a finite cylinder where c=0 and k1=k2 or o=1, the y, of eq (19) becomes

3. Discussion

The variations in temperature at a radius c (c<r') with respect to angle are illustrated in figures 2 and 3, using fixed values of the radius dimension ratios b/a, r'a and c/a. Figures 2 and 3, for the infinitely long cylinder, show that the angular variation decreases with increase in the number of line sources (m), and also with increase in the value of σ= (k/k2). The latter is intuitively evident since the magnitudes of all temperature differences in the inner cylinder obviously are reduced as its conductivity increases relative to that of the outer cylinder.

Figure 4 shows the above-datum temperature at the center and midpoint of a cylinder of finite length as a fraction of the corresponding temperature attained in an infinitely long cylinder (eq (1)), for variations of σ and of l/b, the ratio of the cylinder length to the outside radius of the outer cylinder. Since this analysis assumes that the ends of the two concentric cylinders are maintained at the same temperature (0-0) as the outer surface at r=b, the l/b ratio necessary to maintain a specified 7-value from figure 4 may be somewhat different from that encountered in most applications. With an increasing value of o, longitudinal heat flow in the inner cylinder increases relative to the radial heat flow, hence it is necessary to increase the cylinder length to avoid an increase in the error. For values of a much less than unity, the length of cylinder necessary to restrict the error is controlled chiefly by the effect of longitudinal heat flow to the ends of the outer cylinder.

This analysis confirms that for steady radial heat flow in angularly isotropic infinitely long circular cylinders, the temperature at the axis is equal to the average temperature over any coaxial cylindrical surface which does not envelop a heat source. Thus, in such a cylindrical system having steady line heat sources parallel to the axis and all at a radius r', the temperature at the axis is equal to the average temperature at the cylindrical surface of radius r'. If the line sources were uniformly spaced angularly, there would be no difference in average temperature or heat flux at a radius greater than r', whether the heat was supplied by the distributed line sources and the average temperature at r' was determined by a measurement at the axis, or whether the same aggregate heat was supplied at the axis, and the temperature at was determined by averaging the temperatures measured at the several angular positions at r'. In short, the axial position and the equiangular positions at r', respectively, can be used interchangeably as sites for heat supply or temperature measurement, as convenient, without affecting the average heat flow pattern in the cylinder beyond r'.

The use of equiangularly distributed line sources for heat supply has advantages in some applications. For example, the apparatus described by Flynn [2] used a cylindrical ceramic core as a central heat source in testing a hollow cylinder of fine granular material. If the granular material were very coarse, the thermal contact between the granular particles and the smooth convex ceramic surface might be enough different from that between particles to cause an unwanted and not easily ascertainable temperature drop in the critical region near the ceramic core. If, on the other hand, such a coarse granular material were tested as a complete cylinder, with temperature measured at the axis and line heat sources at radius r' in the material (i.e., the case of σ=1, eq (22)), the oddities of thermal contact of particles with the line heat source would be inappreciable at short distances from the line sources, since the heat flow there would be that occurring in the particulate region.

Another example, of possible experimental interest, is that of conducting a measurement of the thermal conductivity of a solid by the Powell "stacked-disk" method [3], but using the axial hole for the temperature measurement and the inner circle of holes (preferably three or more) for heat supply, in reverse of the arrangement used by Powell and by others. In this way, uncertainties arising from measurements of temperature in regions of large temperature gradient, or due to disturbances of the heat flow pattern by the thermocouples, would be largely avoided.

The author is indebted to J. C. Jaeger, Professor of Geophysics in the Australian National University, for his comments on an earlier draft of this paper.

4. References

[1] H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, 2d ed., pp. 385, 423 (Oxford University Press, London, 1959).

[2] D. R. Flynn, A radial flow apparatus for determining the thermal conductivity of loose-fill insulations to high temperatures, J. Research NBS 67C (Eng. and Instr.) No. 2, 129-137 (Apr.-June 1963). [3] R. W. Powell, Further measurements of the thermal and electrical conductivity of iron at high temperatures, Proc. Phys. Soc. 51, p. 411 (London, 1939).

(Paper 67C2-125)