Coupled or combined radiative and convective heat transfer is a particular case of simultaneous radiative, convective and conductive heat transfer which occurs when heat transfer by conduction is negligibly small compared with that by radiation and convection. Combined radiation and convection occurs between the boundary surfaces and the surrounding medium, between the surfaces separated by the moving medium and inside the moving medium. Depending on conditions in the medium, geometrical factors and surface state, regimes of strong and weak interaction between radiative and convective heat transfer are possible. For weak interaction, problems of heat transfer by radiation and convection can be solved successively and independently, while for strong interaction, these processes are essentially interdependent. Thus, energy transfer by one mechanism can influence heat exchange by the other mechanism and vice versa.

Simultaneous radiative and convective heat transfer is taken into account when solving many problems of heat transfer for hypersonic flows around aircraft, for steam boilers, for aircraft and rocket engines for plasma generators, as well as of heat transfer in shocks, deflagration and detonation fronts and laser deflagration waves for various discharges.

Mathematical formulation of combined radiation and convection problems includes the continuity equation, the momentum (in the Euler or Navier-Stokes forms) and energy conservation equations, the state equation and relations for thermodynamic, thermophysical transport and optical properties.

The energy conservation equation has no terms corresponding to conductive heat transfer and is of the dimensionless form

All the letters and symbols in the above equation are defined in the article on Coupled Radiation, Convection and Conduction. That fact that there are no conductive terms in (1) means that heat conductivity is of no importance in the problem, both for the temperature field formation in the moving medium and for its heat transfer with the surface. Thus, heat conductivity can be ignored for small temperature gradients and when the sharp temperature front is replaced by a contact discontinuity. Obviously, in the latter case, the information on temperature distribution at the front is lost and the medium thermal state on both sides of the contact discontinuity should be determined without heat conductivity.

When analyzing combined radiation and convection, the Boltzmann criterion can be represented in the form of the convective to radiative heat flux component ratio

The classical problem of combined radiation and convection is the problem of steady-state gas flow across a flat plate at a constant temperature [Sparrow and Cess (1970)].

The flow scheme is shown in Figure 1 and the mathematical problem is to solve the energy conservation equation

subject to the following boundary conditions: at x = 0, T = T_{∞}, u = u_{∞}; at y = 0, u = u_{∞}, T = T_{w}, v = 0, and the condition for radiation energy intensity (for example, the black-body assumption for the surface). Even such simplified and idealized problem has been addressed by a wide spectrum of methods with a range of results obtained; this is caused by the very complicated radiation problem. When formulating the complete problem for a radiation-scattering medium (for the optical thickness τ_{0} ~ 1), it is simplier to use numerical methods. The assumptions of weak light scattering or implementation of limited optical thickness regimes (τ_{0} >> 1 and τ_{0} << 1) make it possible to obtain analytical solutions. For example, in a "grey" gas without scattering and using the exponential approximation of the kernel of the integro-differential equation obtained from (3) by substituting q_{R, y} for the plane layer, the following solution has been obtained by [Sparrow, E. M. and Cess, R. D. (1970)]:

Here I_{0} and I_{1} are the modified Bessel functions,
, and κ is the volumetric gas absorption coefficient. When gas heat conductivity is excluded, any solution of (3) has a temperature discontinuity.

When solving the heat protection problem for flight vehicles entering the dense atmosphere at superorbital velocities the combined radiation and convection approximation appears to be very useful and fruitful. When a vehicle with bluntness radius ~ 1 m enters the Earth’s is atmosphere at superorbital velocities of v_{∞} = 10 - 16 km/s at an altitude of 75 km, radiation heating of its heating surface exceeds convection heating. Therefore, the thin boundary layer where the convective flow forms can be neglected. The problem where heat protecting material failure is taken into account is the more realistic one. In this case, in a stagnation streamline the shock layer structure can be represented as two oppositely-directed mass flows separated by a thin transient layer in which the viscosity is essential. Replacing the thin transient layer by a contact surface allows the exclusion of viscosity and thermal conductivity. Considering the flow to be steady-state, the governing equation system can be formulated [Olstad (1971)] as:

assuming Hugoniot conditions on the shock and the condition of flow nonpenetration or a given injection rate through the surface. In problems of radiating shock layers, the radiation transfer is considered, as a rule, to be as in a plane-parallel layer, this is justified by a small thickness of the layer. Solving Eqs. (5) makes it possible to obtain flow variable distributions inside the shock layer and the radiation fluxes to the surface and the shock front. In this situation, radiation transfer strongly influences the shock layer parameters, hence, the problem can be attributed to the class of strong interaction of radiation and convection.

Problems on laser wave deflagration applied in laser rocket engines and in the optical plasmatron are also problems involving the strong interaction regime. The plasma cloud, localized in space and with a size of ~ 1 cm and a temperature of ~15,000 K, moves to meet a laser beam (e.g., from a CO_{2} laser) due to heating of the surrounding cold gas by the heat radiation of the plasma itself. The plasma exists due to laser energy absorption. The heating rate of the cold gas by heat radiation essentially exceeds that by heat conductivity, even for extremaly high temperature gradients in the laser deflagration wave. The process runs usually at atmospheric or higher pressures, and this makes it possible to use the equilibrium approximation for the thermodynamic medium state and an equation system similar to (5).

The main difference is that the integral radiation flux vector is represented as a sum

where **q**_{R, T} and **q**_{R, L} are the vectors of integral heat flux density and laser radiation flux density, respectively. To obtain the function **q**_{R, T} the two-dimensional equation for selective radiation transfer must be solved, while for **q**_{R, L} the geometrical optics approximation may be employed.

Combined radiation and convection problems concerned with calculations of heat transfer in devices such as steam boiler furnaces are widely encountered in engineering practice [Siegel and Howell (1972); Ozisik (1973)]. The temperature of the furnace medium (multicomponent disperse system of gas and solid phases) is maintained by combustion, and the general configuration of the temperature field in the furnaces is determined mainly by convective processes formed when the gas-dust component flows in a working space. The surfaces are heated principally by radiation heat transfer from the furnace medium; therefore, the heat conductivity contribution is neglected when computing real furnace steam boilers. Analyzing the combined radiation and convection processes in furnaces, the transfer equation for this complicated geometry should take into account the radiation scattering and absorptive properties of the furnace medium with various additives (the properties of which are hardly known) as well as the optical-physical characteristics of the heat-receptive surfaces.

#### REFERENCES

Olstad, W. B. (1971) Nongrey Radiating Flow about Smooth Symmetric Bodies, *AIAA Journal*, 1, 122-130.

Ozisik, M. N. (1973) *Radiative Transfer and Interaction with Conduction and Convection*, A Wiley-Interscience Publication.

Sparrow, E. M. and Cess, R. D. (1970) *Radiation Heat Transfer*, Brooks/Cole Publishing Company.

Siegel. R. and Howell, J. R. (1972) *Thermal Radiation Heat Transfer*, McGraw-Hill Book Company.

#### References

- Olstad, W. B. (1971) Nongrey Radiating Flow about Smooth Symmetric Bodies,
*AIAA Journal*, 1, 122-130. - Ozisik, M. N. (1973)
*Radiative Transfer and Interaction with Conduction and Convection*, A Wiley-Interscience Publication. DOI: 10.1002/aic.690210139 - Sparrow, E. M. and Cess, R. D. (1970)
*Radiation Heat Transfer*, Brooks/Cole Publishing Company. - Siegel. R. and Howell, J. R. (1972)
*Thermal Radiation Heat Transfer*, McGraw-Hill Book Company.