# Multimode heat transfer with radiation

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## Radiation-Conduction Problem

When radiation is combined with conduction and/or convection, the problems are called *conjugate or multi-mode* heat transfer. If temperature is one of the unknowns in the problem being studied, then such problems are highly non-linear, in that the fundamental radiation terms involve temperature to the fourth power, while conduction and convection depend on the first power of temperature or its derivatives. Property variations with temperature may add further nonlinearities.

The mathematics of highly nonlinear equations is not well-developed, and almost all combined mode problems are solved numerically; closed-form analytical solutions seldom exist. As an example of a combined-mode radiation-conduction problem, consider the geometry and conditions shown in Fig. 1. Radiation transfer occurs between the parallel plates and from the plates to the environment. Because the plates are not isothermal, conduction occurs within the plates, and will modify the temperature distributions, which are the unknowns in the problem.

For this case, the usual net radiation equations are as derived as following (Faghri et al., 2010):

Note that the q"_{k,rad} is the heat flux required to balance the radiation at the surface. It can be supplied by conduction, convection, internal generation, or a combination of these. In this case, only conduction is present.
If conduction is assumed to occur only in the x-direction in the thin plates, then two additional energy equations are written for the conducting plates

Equations (3) and (4) contain second-order derivatives, so two boundary conditions are required for each. The appropriate ones are

Equation (5) is a symmetry condition, and eq. (6) equates the conduction into the plate ends with the radiative loss from the plate end.

Equations (3) and (4) can be substituted into eqs. (1) and (2), resulting in two simultaneous integral-differential equations with the unknowns *T*_{1}(*x*_{1}) and *T*_{2}(*x*_{2}). The resulting Equations (7) are highly nonlinear, involving temperatures to the fourth power as well as second derivatives in temperature.

To simplify eqs. (7), define

and eqs. (7) become

with boundary conditions [from eqs. (5) and (6)]

In terms of the nondimensional variables, the required configuration factor is (using *h* = *H* / *W*),

The parameters that appear in the problem are the emissivity ε, the conduction/radiation parameter N_{CR}, and the two geometric parameters δ / *W* and h. The N_{CR} or modifications of it often appear in radiation-conduction problems. Its meaning can be appreciated by multiplying the numerator and denominator by *T*_{ref} to give

The last form in eq. (12) is the ratio of a conduction heat transfer to a radiative emission. The conduction is through the plate thickness σ with a temperature gradient of (*T*_{ref}0) / *W*. The radiation is emission from a black surface of width *W* and temperature *T*_{ref}. The *N*_{CR} is thus a measure of the importance of conduction to radiation in affecting the temperature distributions on the surfaces. For example, if *N*_{CR} = 0, eqs. (9) reduce to the radiation-only solutions. If *N*_{CR} is very large, then the problem becomes a pure conduction problem. (This is shown be reverting to eqs. (3) and (4) and placing them in dimensionless form).
It is unlikely that eqs. (9), subject to boundary conditions eqs. (10), will have a closed-form analytical solution, and a numerical approach must be employed.

## Numerical Method

To solve the radiation-conduction problem defined in eqs. (9-11), the integrals in the equations are replaced by summations over elements of width Δ*x*, i.e.,

Substituting into eq. (9) and using the finite difference form for the second derivatives results in

where, including the configuration factor relations, the Φs are

The boundary conditions become

Because the boundary elements at *n* = 1 and *m* = 1 are finite with width = ΔX, the radiative exchange to the Δ*X* surfaces must be included in addition to the radiation-conduction balance at the end surfaces. The Δ*X* approaches zero in the analytical boundary conditions of eq. (10) so these terms do not appear.
The result is (*M* + *N*) nonlinear algebraic equations with *N* values of θ_{1,n} and *M* values of θ_{2,m}. This set can, in principle, be solved by solvers available for treating a set of non-linear algebraic equations. However, experience indicates that such solvers often fail unless some further steps are taken to rearrange the form of the equation.

Let Δ*X*_{1} = Δ*X*_{2} (equal sized increments on each plate, so that eqs. (14) - (16) become

The parameter *N*_{CR} / (Δ*X*)^{2} now appears in eqs. (12) - (19), and is a guideline for how to approach a numerical solution.

## Conduction Dominated Problems

If conduction dominates (i.e., *N*_{CR}/(Δ*X*)^{2}>>1), then eq. (14) can be rearranged to reflect a conduction problem that is perturbed by the presence of radiation:

and for this case the right-hand-side of eq. (20) will be small and the solution will be close to the conduction-only solution. Equation (20) (including the insertion of the boundary conditions) is a set of equations of the matrix form

where **A** is a tridiagonal matrix of coefficients that need only be inverted once. The solution is then

Equation (22) is solved by assuming the distributions of θ_{1,n} and θ_{2,m}, using these to evaluate Φ_{1,m,n} and Φ_{2,m,n}, which in turn are used to evaluate **C(x)**. The matrix multiplication indicated in eq. (22) gives a new set of θ values; the process is repeated until convergence. Effectively, this method is solving a set of linear equations by interation in the unknown values of θ, and amounts to a series of matrix multiplications.

## Radiation dominated problems

If radiation dominates so that *N*_{CR} / (Δ*X*)^{2}<<1, then eq. (14) can be rearranged to

Again, the set of equations can be arranged as a matrix equation of the form

Matrix inversion of *A*_{1} gives

Again, the *A*_{1} matrix need be inverted only once. Equation (25) is solved by assuming the distributions of θ_{1,n} and θ_{2,m}, using these to evaluate Φ_{1,m,n} and Φ_{2,m,n} and the second derivative terms, which in turn are used to evaluate *C*_{1}(*x*). The matrix multiplication indicated in eq. (25) gives a new set of θ values; the process is repeated until convergence. Effectively, this method is iteravely solving a set of linear equations in the unknown values of θ^{4}.

## Problems with Both Modes Significant

When *N*_{CR} is not near either the large or small limit, then the problem is truly nonlinear, and the solution methods described for small or large *N*_{CR} / (Δ*X*)^{2} values often fail. In that case, eq. (14) can be arranged as

This equation in matrix form becomes

We could define so that eq. (27) becomes

If this approach is followed, an initial guess for the distributions of θ_{1,n} and θ_{2,m} is used to evaluate *A*_{3}(θ) and *C*_{2}(θ), *A*_{3}(θ) is inverted, and a new set of θ values is found. This method has two problems: First, the *A*_{3}(θ) matrix must be inverted at each iteration, and *A*_{3}(θ) is probably a full matrix that can require significant time to invert. Second, this method is often quite unstable, and requires a small relaxation factor to be imposed between iterations to avoid divergence.

To avoid this, a modified Newton-Raphson iteration method can be used. In this case, eq. (27) is rewritten as

The residual is a measure of convergence of the solution, and will approach *g*_{j} = 0 when solution is complete

Next, the function *g*_{jk} is found

and *g*_{jk} is seen to be the gradient in the residual. Now solve the auxiliary equation

The next iterative value for θ_{k} is found from

The values of θ_{k}^{p} are now used to find a new residual, and the process is repeated until convergence. The λ_{k} values are seen to be those that would make the residual equal to zero on the next iteration if the problem were linear. This approach has been found to be quite stable, although sometimes a relaxation factor is needed on θ_{k} between iterations.

Some of the complexities of multi-mode heat transfer when radiation is present have been outlined. A detailed discussion of more advanced numerical techniques is in Hogan and Gartling (2007). There are commercial software packages that can be used for these problems, and they can be particularly useful when convective transfer must be linked to radiation.

## References

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

Hogan, R.E. and Gartling, D.K., 2007, “Solution Strategies for Coupled Conduction/Radiation Problems,” *Commun. Numer. Meth. Engng*,, Vol. 23, pp. 523-542.