# One-dimensional steady-state heat conduction

## Steady-state heat conduction in plane and radial walls

(a)Plane Wall (b)Hollow cylinder or sphere
Heat conduction in different geometric configurations

One-dimensional heat conduction can occur in different geometric configurations. The figure on the right shows heat conduction in a plane wall and in a hollow cylinder or sphere. The energy equationin different geometric configurations can be expressed as:
Plane wall:

$\frac{d}{{dx}}\left( {k\frac{{dT}}{{dx}}} \right) = 0,\qquad {x_1} < x < {x_2} \qquad \qquad(1)$

Hollow cylinder:

$\frac{1}{r}\frac{d}{{dr}}\left( {kr\frac{{dT}}{{dr}}} \right) = 0 \qquad {r_1} < r < {r_2}{\rm{)}} \qquad \qquad(2)$

Hollow sphere:

$\frac{1}{{{r^2}}}\frac{d}{{dr}}\left( {k{r^2}\frac{{dT}}{{dr}}} \right) = 0 \qquad {r_1} < r < {r_2}{\rm{)}} \qquad \qquad(3)$

See Main Article Steady-state heat conduction in plane and radial walls

## Heat Transfer from Extended Surfaces

Figure 1: Fin configurations: (a) straight fin of uniform cross-section on plane wall, (b) straight fin of uniform cross-section on circular tube, (c) annular fin, and (d) straight pin fin

The total thermal resistance includes two convective thermal resistances and one conduction thermal resistance. For the cases that one of the convection thermal resistances is dominant (i.e., significantly greater than the conduction thermal resistance and the other convective thermal resistance), one can increase the heat transfer coefficient (hi or ho) or the heat transfer area (A1 or A2) to enhance the overall heat transfer. Since increasing the heat transfer coefficient is constrained by the type of working fluid and the power required driving the flow, increasing the heat transfer area becomes a natural choice. The increase of surface area can be done by using fins that extend into the fluid. Figure 1 shows some examples of fin configurations. It can be seen that the fins can have either uniform or variable cross-sectional areas. For cases where the fluid is cooling the fin, the fin temperature is the highest at the base (x = 0) and decreases with increasing x as convection takes place on the fin surface. The degree of heat transfer enhancement can be maximized by minimizing the temperature variation, which can be achieved by using fin materials with large thermal conductivity. The solution of heat transfer from extended surfaces (fins) will provide (a) temperature distribution in the fin, and (b) total heat transfer from the finned surface.

See Main Article Heat conduction in extended surface

## Bioheat equation

Heat transfer in living tissue can be considered as heat conduction with internal heat generation from two sources: metabolic heat generation and blood perfusion. The former results from a series of chemical reactions that occur in the living cells, while the latter is due to energy exchange between the living tissue and blood flowing in the small capillaries. If the heat transfer in a tissue can be considered as one-dimensional and the thermal conductivity of the tissue is assumed to be independent from temperature, the energy equation for living tissue can be obtained by

$\frac{1}{{A(s)}}\frac{d}{{ds}}\left( {A(s)\frac{{dT}}{{ds}}} \right) + \frac{{{{q'''}_m} + {{q'''}_b}}}{k} = 0,{\rm{ }}{s_1} < s < {s_2} \qquad \qquad(1)$

where q'''m and q'''b are the rate of metabolic heat generation and blood perfusion, respectively. Equation (1) is generalized for Cartesian (s = x,A(s) = const), cylindrical (s = r,A(s) = r), and spherical (s = r,A(s) = r2) coordinate systems.

See Main Article Bioheat equation

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.