|Thermodynamics and Propulsion|
The quasi one-dimensional equation that has been developed can also be applied to non-planar geometries, such as cylindrical and spherical shells.
An important case is a cylindrical shell, a geometry often encountered in situations where fluids are pumped and heat is transferred. The configuration is shown in Figure 16.10.
For a steady axisymmetric configuration, the temperature depends only on a single coordinate ( ) and Equation (16.13) can be written as
or, since ,
The steady-flow energy equation (no fluid flow, no work) tells us that , or
The heat transfer rate per unit length is given by
Equation (16.26) is a second order differential equation for . Integrating this equation once gives
where is a constant of integration. Equation (16.29) can be written as
where both sides of Equation (16.30) are exact differentials. It is useful to cast this equation in terms of a dimensionless normalized spatial variable so we can deal with quantities of order unity. To do this, divide through by the inner radius, ,
Integrating (16.31) yields
To find the constants of integration and , boundary conditions are needed. These will be taken to be known temperatures and at and respectively. Applying at gives . Applying at yields
The temperature distribution is thus
As said, it is generally useful to put expressions such as (16.33) into non-dimensional and normalized form so that we can deal with numbers of order unity (this also helps in checking whether results are consistent). If convenient, having an answer that goes to zero at one limit is also useful from the perspective of ensuring the answer makes sense. Equation (16.33) can be put in nondimensional form as
The heat transfer rate, , is given by
per unit length. When the heat flow rate is written so as to incorporate our definition of thermal resistance,
comparison with (16.35) reveals the thermal resistance to be
The cylindrical geometry can be viewed as a limiting case of the planar slab problem. To make the connection, consider the case when . From the series expansion for we recall that
(Look it up, try it numerically, or use the binomial theorem on the series ( ) and integrate term by term.)
The logarithms in Equation (16.34) can thus be written as
in the limit of . Using these expressions in Equation (16.33) gives
With the substitution of , and we obtain
which is the same as Equation (16.19). The plane slab is thus the limiting case of the cylinder if , where the heat transfer can be regarded as taking place in (approximately) a planar slab.
To see when this is appropriate, consider the expansion , which is the ratio of heat flux for a cylinder and a plane slab.
For error, the ratio of thickness to inner radius should be less than 0.2, and for 20% error, the thickness to inner radius should be less than 0.5 (Table 16.2).
A second example is the spherical shell with specified temperatures and , as sketched in Figure 16.11.
The area is now , so the equation for the temperature field is
Integrating Equation (16.42) once yields
Integrating again gives
or, normalizing the spatial variable
where and are constants of integration. As before, we specify the temperatures at and . Use of the first boundary condition gives . Applying the second boundary condition gives
Solving for and ,
In non-dimensional form the temperature distribution is thus