Coupled radiation, convection and conduction (CRCC) is a self-consistent heat transfer process by the mechanisms of thermal radiation, convection and thermal conductivity. CRCC occurs between a surface in contact with a moving medium and between various components of dust-loaded flows, the moving medium being considered not only as gas and plasma, but the condensed state as well.
Here, self-consistent heat transfer means, in essence, that each of the above-mentioned mechanisms influences to the same extent the energy balance inside and (or) on the boundary of the domain considered and thereby changes the energy exchange intensity by other mechanisms.
The combined action of radiative, convective and conductive heat transfer must be considered when solving a wide class of heat and mass transfer problems in such fields as power, aerospace and process engineering. These include:
Heat transfer in hypersonic flows around bodies moving in planetary atmospheres;
Heat transfer in arcs, microwaves and optical plasma generators, in combustion chambers and nozzles of rocket engines (gaseous, liquid and solid propelled rocket engines; plasma, nuclear and laser rocket engines), in furnaces of steam boilers and other power facilities;
Heat transfer in various gas discharges and in intensive shock waves, combustion fronts and laser deflagration waves;
Heat transfer in thermal protection materials, in combustion in semi- transparent porous matrices, in laser and plasma engineering, in electronic engineering, etc.
CRCC is the most general type of heat transfer; in this general case of heat transfer in a moving medium, all the three mechanisms act. However (depending on medium temperature, velocity, density, geometry and optical and physical properties), it is possible for one or two mechanisms to dominate. In these limiting cases, heat transfer may be designated as "convective", "conductive," "radiative," "conductive-convective," "radiative-conductive" and "radiative-convective" heat exchanges.
When solving problems of CRCC in whichever mechanisms of the heat transfer are important, a useful first step is to determine the influence of radiation on the moving medium parameters. If this influence is small, one can simplify the problem, treating the medium motion and the radiation heat transfer separately. The problems are then solved successively by the methods for a moving media and for radiation heat transfer — the combined action of the mechanisms being taken into account using the additivity principle. In the case of such weak interaction account may be taken of the interdependency of the processes using perturbation methods. If radiative transfer influences strongly the medium's thermal state and its motion, the additivity principle is inapplicable. This case is the most complicated one in the CRCC problem.
To describe the CRCC processes mathematically and estimate how important is each of the heat transfer mechanisms, the following systems of equations are used [Pai (1966)]; [Bond et al. (1965)]; [Siegel and Howell (1972)]; (Ozisik (1973)]; [Sparrow and Cess (1970)]:
The continuity equation.
(1)where ρ is the medium density, V = {u_{1}, u_{2}, u_{3}} is the flow-velocity vector, u_{i} is x_{i}-velocity component and t is the time. In (1) and thereafter, we use tensor designations, the summation is over repeating indices.
The momentum equation.
(2)where
is the viscous stress tensor, F_{i} is the x_{i}-component of the force F_{i} acting on unit volume (for example, the gravity force F_{i} = ρg_{i}, where g_{i} is the x_{i}-component of the free-fall acceleration vector); p is the gas-dynamic pressure, μ is the dynamic viscosity coefficient; δ_{ik} is the Kronecker delta symbol (δ_{ik} = 1 for i = k and δ_{ik} = 0 for i ≠ k); p_{R, i, j} = is the radiation stress tensor determined when integrating the integral radiation intensity J = over the solid angle Ω = 4π; here, J_{ν} is the spectral radiation intensity, ν is the radiation frequency, ω_{i}and ω_{j} are direction cosines for the vector , which characterizes the radiation propagation direction. In the evaluation of the significance of radiation pressure, a useful dimensionless parameter is R, the ratio of mean radiation pressure of isotropic black-body at a temperature T to gas dynamic pressure
where c = 3 · 10^{8} m/s is the velocity of light, and (Joule/s · m^{2} k^{4}) is the Stefan-Boltzmann constant. At p = 10^{5} Pa, T = 2 · 10^{4} K, R ≈ 1.27-10^{−3} << 1. In most problems of CRCC (at temperatures lower than 20,000 K), one can neglect the radiation stress tensor.
Energy conservation equation.
(3)where
is the viscous energy dissipation function for a Newtonian fluid, S is the energy released or external energy input per unit volume of the gas per unit time; q_{c, i} is the x_{i}-component of the conductive heat flow according to the Fourier law , λ is the heat conductivity coefficient; q_{R, i} is the x_{i}-component of integral radiation flow density
where ni is the unit vector in the x i-direction.
Transport equation for selective radiation.
(4)where J_{ν} = J_{ν} (s, , t); J_{ν}’ = J_{ν} (s, , t ) is the spectral intensity of radiation in the medium; s is the coordinate along the ray in the -direction; k_{ν} = k_{ν}(s), σ_{ν} = σ_{ν}(s) are the volume spectral coefficients of absorption and scattering, p_{ν} ( , ) is the spectral phase function, j_{ν} = j_{ν} (s) is the volume spectral coefficient of the radiation emission. According to Kirchhoff's law, j_{ν} is proportional to the Planck function
(5)where J_{b, ν} is the black-body spectral intensity (Planck's intensity). When solving problems of CRCC concerned with nonrelativistic motion of a medium, the unsteady term on the left-hand side of Eq. (4) can be ignored.
The systems of equations for CRCC is closed by the state equation which, for the perfect medium, is of the form
where M is the molecular weight and R_{0} = 8314 J/K·kmol is the universal gas constant.
To solve CRCC problems for multiphase, niulticomponent and multitemperature media, one should formulate similar equations for each component and include models, where appropriate, for the turbulent motion of the radiating gas. The necessary steps to solve particular problems arc:
Development of relationships for the thermodynamic, thermophysical, transfer and optical properties of the media;
Formulation of initial and boundary conditions, including the specification of thermophysical, physicochemical and optical properties of surfaces;
Solution of the problem for gas flow with internal heat sources;
Solution of selective radiation problems and determination of the radiation flow fields and their divergence.
If the radiation stress tensor is negligibly small compared with the viscous stresses, then the radiational energy influences the medium motion described by Eqs. (1) and (2) only by its influence on the temperature field. The relationship between various heat exchange mechanisms is obtained on the basis of the energy conservation equation (3), written in dimensionless form
where ρ, c_{p}, μ, λ are referred to their values at characteristic temperature T_{0} (accordingly, ρ_{0}, c_{p0}, λ_{0}), and T, p, S, u are referred to T_{0}, ; t and x_{i} are referred to L/u_{0} and L, where L is the typical space length.
Heat transfer mechanisms are characterized by the dimensionless similitude criteria of Prandtl (Pr = c_{p0}μ_{0}/λ_{0}). Reynolds (Re = ρ_{0}u_{0}L/μ_{0}), Boltzmann (Bo = ), Eckert (Ec = ) and the generalized Boltzmann criterion (Bo* = ρ_{0}u_{0}c_{p0}T_{0}/LS_{0}). The Boltzmann criterion gives an estimation of the relationship between convective, conductive and radiation heat transfer
where is the radiation-convective parameter (to compare heat conductivity and radiation contributions into the process); Pe = Re · Pr is the Peclet number (to compare heat conductivity and convection), τ_{0} = L/l_{0} is the typical optical thickness, l_{0} is the mean free path to characterize radiation heat transfer. There are problems in estimating the latter value, since it characterizes the photon mean free path averaged over the entire frequency spectrum, while the spectral mean free paths vary a few orders of magnitude. The approximations of optically thin (τ_{0} << 1) and optically thick (τ_{0} >> 0) media are widely used in CRCC analyses, since they make it possible to avoid the solution of the transport equation (4). In particular, for τ_{0} << 1:
which corresponds to radiation energy losses. In this last equation, k_{p} is the mean Planck absorption coefficient. For τ_{0} >> 1, radiation flow can be represented in the Fourier law form
where is the radiative conduction coefficient, with kp as the mean Rosseland coefficient.
If the medium is optically thin for certain wavelengths and thick for others, the above-mentioned approximations for the radiation flux and its divergence cannot apply. Instead, the transport equation (4) must be solved. For a highly-scattering medium, the integro-differential form of Eq. (4) must be solved and this makes the problem more complicated. The standard way to simplify the problem is to use assumptions on diffusional or δ-like character of the scattering and then to solve the problem employing diffusion approximation. This approximation appears to be even more justified in this case than in the nonscattering case, where it can also be recommended for application.
CRCC problems have been intensively studied theoretically and experimentally since the mid-50s and are central to the achievements in aerodynamics and plasma-dynamics, astrophysics, in nuclear reactor theory, shock waves, and high temperature physics. The classical problems of CRCC are problems of viscous, heat conductive radiating gas flow in shock and boundary layers, and of Couette flow in radiation magnetogasdynamics.
All CRCC regimes appear in the problems of laser wave deflagration, which are analyzed for optical plasmatron and for laser thruster development. For a small laser beam, laser-supported plasma moves to meet the radiation due to the heat conductivity mechanism, giving rise to motion of the surrounding gas. When laser beam size increases, the heat conductivity mechanism for plasma motion is replaced by a radiational one due to the absorption of radiation in the cold gas surrounding the plasma.
REFERENCES
Bond, J. W., Watson, K. M., and Welch, J. A. (1965) Atomic Theory of Gas Dynamics, Addison-Wesley Reading.
Ozisik, M. N. (1973) Radiative Transfer and Interaction with Conduction and Convection, A Wiley-Interscience Publication.
Pai, S. I. (1966) Radiation Gas Dynamics, Springer Veriag. DOI: 10.1016/0375-9474(67)90640-9
Siegel, R. and Howell, J. R. (1972) Thermal Radiation Heat Transfer, Mc. Graw-Hill Book Company.
Sparrow, E. M. and Cess, R. D. (1970) Radiation Heat Transfer, Brooks/Cole Publishing Company.
References
- Bond, J. W., Watson, K. M., and Welch, J. A. (1965) Atomic Theory of Gas Dynamics, Addison-Wesley Reading.
- Ozisik, M. N. (1973) Radiative Transfer and Interaction with Conduction and Convection, A Wiley-Interscience Publication. DOI: 10.1002/aic.690210139
- Pai, S. I. (1966) Radiation Gas Dynamics, Springer Veriag. DOI: 10.1016/0375-9474(67)90640-9
- Siegel, R. and Howell, J. R. (1972) Thermal Radiation Heat Transfer, Mc. Graw-Hill Book Company.
- Sparrow, E. M. and Cess, R. D. (1970) Radiation Heat Transfer, Brooks/Cole Publishing Company.
Heat & Mass Transfer, and Fluids Engineering