RADIATIVE BOUNDARY LAYER
Following from: The radiative transfer equation; Transport approximation; Differential approximations; Two-flux approximation; P_{1} approximation of the spherical harmonics method; The simplest approximations of double spherical harmonics Radiation of the isothermal plane-parallel layer; Radiation of the nonisothermal layer of a scattering medium; Radiative equilibrium in a plane-parallel layer
Leading to: Liquid droplet radiator for space applications
In a moving medium without heat conduction, the temperature field is a result of coupled convective and radiative heat transfer. In the case of a streamlined body, the so-called radiative boundary layer is formed in the flow near the body surface. In this region, the medium temperature differs from that in an undisturbed flow. A calculation of the radiative boundary layer gives the radiation flux on the body surface and, if the model developed by Cess (1964) is applicable, enables us to obtain external boundary conditions for an optically thin viscous and conducting boundary layer.
Let us consider the radiation heat transfer in nonviscous, nonconducting flow over an isothermal flat plate. We assume that the optical properties of the medium and plate are independent of wavelength and temperature. The transport approximation of the medium scattering function is employed. For simplicity, it is also assumed that the undisturbed flow has constant velocity u_{e} and temperature T_{e}. The following 1D energy equation is considered for the model problem:
(1) |
where ρ_{e} and c_{p} are the density and heat capacity of the medium, x and y are the Cartesian coordinates along the plate (in the flow direction) and in the normal direction, and q is the integral radiation flux. From a more general energy equation, one can derive the following condition of the relatively small role of the radiation transfer along the plate:
(2) |
If the radiative conduction approximation is used on the right-hand side of this inequality, the condition of the 1D radiative transfer model applicability is
(3) |
where
(4) |
The Boltzmann number Bo characterizes the relative role of convective and radiative heat fluxes. According to Eq. (3), the radiative transfer along the plate may be important only near the front edge of the plate. In addition to (4), it is convenient to introduce the dimensionless variables
(5) |
In new variables, the energy equation (1) and obvious boundary condition at the front edge of the plate are
(6a) |
(6b) |
There is no Boltzmann number in Eq. (6a), and hence Bo is not a problem parameter. Variations of the Boltzmann number result only in the solution scale along the plate. The greater Bo, the smaller are the relative role of radiation and the thickness of the radiative boundary layer.
The transport RTE and expression for the radiation flux for a gray medium can be written as follows (see article Radiation of an isothermal plane-parallel layer):
(7a) |
(7b) |
In the case of an isotropically emitting and reflecting gray plate with temperature T_{w} and emissivity ε_{w}, the boundary conditions are
(8) |
The problem in Eqs. (6)-(8) is very complicated. It has been solved by Dombrovsky (1978) using the differential approximations for radiative transfer. The other solutions were obtained for a nonscattering medium, when the problem can be transformed to the following nonlinear integro-differential equations (Taitel, 1969; Sparrow and Cess, 1978):
(9) |
where E_{1} and E_{2} are the exponential integrals. An approximate method for the vicinity of the plate front edge has been developed Cess (1964). The case of |1 - T_{w}| << 1 and corresponding linearization has been considered in some detail in the book by Sparrow and Cess (1978). The solution for a blackbody plate (ε_{w} = 1) was obtained by use of three different approximate methods. An exact numerical solution for Eq. (9) at ε_{w} = 1 has been reported earlier by Taitel (1969). A good agreement was found between the exact solution and the results of Cess (1964) obtained by exponential kernel approximation.
To find a self-similar solution of the problem, we consider the flow far from the front edge of the plate, where the thickness of the radiative boundary layer is large, and the following evaluations take place near the plate surface:
(10) |
One can find from Eq. (6a) that
(11) |
This means that heat transfer conditions at sufficiently long distance from the front edge are similar to the radiative equilibrium in an optically thick layer. Therefore, the solution at large τ^{*}_{tr,x} does not depend on optical properties of the medium and plate (see article Radiative equilibrium in a plane-parallel layer). The radiative conduction approximation, which is correct for the radiative equilibrium in an optically thick layer (Dombrovsky, 1974; Sparrow and Cess, 1978), can be used at τ^{*}_{tr,x} >> 1. In this approximation,
(12) |
and the energy equation is
(13) |
We introduce variables
(14) |
assuming the value of to be independent of τ^{*}_{tr,x}. In new variables,
(15) |
Thus, the flow far from the front edge of the plate is self-similar, and the radiative boundary layer optical thickness τ_{tr,δ} ∼ √τ^{*}_{tr,x}. When |1 - T_{w}| << 1, one can obtain from Eq. (15) the following expression for the radiation flux to the wall:
(16) |
and one can see the convenience of new quantities,
(17) |
For an arbitrary temperature of the plate, the boundary value problem has been solved numerically by use of iterations (Dombrovsky, 1978). The solution at |1 - T_{w}| << 1 was used as an initial approximation. Some computational results are presented in Fig. 1.
Figure 1. Self-similar temperature profiles in a radiative boundary layer at various temperatures of the wall: 1, T_{w} = 0; 2, 1.5; 3, 2.0; 4, 3.0.
With the increase of plate temperature, the thickness of the radiative boundary layer increases, and decreases; this is also seen from Eq. (15). The function (T_{w}) can be approximated by the simple formula,
(18) |
It is interesting that derivative ∂T/∂τ^{*}_{tr} at T_{w} > 1 has a maximum at some distance from the plate surface.
In the calculations of a radiative boundary layer at not-so-large values of τ^{*}_{tr,x}, one can use the DP_{0} approximation. This approximation is more accurate in this region than P_{1}, which gives an incorrect result at τ^{*}_{tr,x} < 1 (see article Radiation of isothermal plane-parallel layer). In the self-similar region, DP_{0} underestimates the radiation flux by a factor of 2/√3 compared to the exact solution. The mathematical formulation for radiative transfer in the DP_{0} approximation is
(19a) |
(19b) |
(19c) |
(19d) |
One can be convinced that DP_{0} is equivalent to the exponential kernel approximation used by Sparrow and Cess (1978) at |1 - T_{w}| << 1. The complete mathematical formulation of the problem is as follows:
(20a) |
(20b) |
(20c) |
(20d) |
(20e) |
For a nonscattering medium and blackbody plate, the numerical results can be compared with the exact solution by Taitel (1969). The comparison of calculated dimensionless radiation flux and temperature slip θ_{0} = [T(τ_{tr} = 0) -T_{w}]/(1 -T_{w}) with the exact data presented in Fig. 2 confirms a sufficiently high accuracy of the DP_{0} approximation.
Figure 2. Dimensionless radiation flux (a) and temperature slip (b) in the case of nonscattering medium and black wall: 1 - T_{w} = 0.1; 2, |1 - T_{w}| << 1; T_{w} = 2 [points are from Taitel (1969), solid curves are from author’s data].
The radiation flux decreases with an increase of the distance from the front edge of the plate because of an increasing in the thickness of the radiative boundary layer, which shields the plate from high-temperature region radiation. At large distances from the front edge (τ^{*}_{tr,x} > 5), the radiation flux decreases as 1/√τ^{*}_{tr,x}, i.e., the value of has a limiting constant value corresponding to the self-similar solution. The larger relative temperature of the plate, the smaller is . This is explained by a larger thickness of the boundary layer and corresponding diminishing of the derivative ∂T/∂τ_{tr} near the surface. The value of θ_{0}, which characterizes the “radiation slip” at the plate surface, varies similarly to the radiation flux. Small values of θ_{0} in the case of the hot plate indicate the fact that the value of ∂T/∂τ^{*}_{tr} in a viscous boundary layer may be comparable with the corresponding values outside the boundary layer. Therefore, the division of the flow region in the Cess model may be not correct at T_{w} > T^{*}_{w} > 1. The radiation scattering and the difference of the wall emissivity from unity lead to a decrease in radiation flux, and increase in radiation slip (see Fig. 3). The effect of medium albedo ω_{tr} and wall emissivity ε_{w} on thermal radiation flux decreases with the distance from the front edge of the plate. This effect disappears in the self-similar flow region.
Figure 3. Effect of medium albedo and wall emissivity on radiation flux (a) and temperature slip (b): I, ω = 0, ε_{w} = 1; II, ω = 0.5, ε_{w} = 1; III, ω = 0, ε_{w} = 0.5 (1, T_{w} = 0.1; 2, |1 - T_{w}| << 1; 3, T_{w} = 2).
There are some practical problems dealing with the radiative cooling of a disperse system in vacuum, when the main characteristics of the process can be obtained by use of the methods discussed. One example is a high-altitude fly of a rocket with a metalized-propellant engine. The combustion products leaving the nozzle form a disperse system of metal oxide particles in the highly rarefied atmosphere. The temperature field and thermal radiation of the exhaust jet can be calculated without taking into account any thermal interaction of gases and particles (Cohen, 1988). On this basis, the mathematical formulation of the problem includes coupled RTE and the energy equation with terms corresponding to the thermal radiation and convective heat transfer in a moving medium. Another problem related to the same type of combined heat transfer problems is the calculation of a liquid droplet radiator (LDR) for space applications. In an LDR, the generated sheet of fluid droplets is cooled due to thermal radiation in vacuum. The droplets are then collected and return into the thermal control system. The LDR calculation is simpler than that for a high-altitude rocket jet because of the integral character of the problem: it is not necessary to determine the angular dependences of the radiation intensity. In addition to this fact, the simple shape of the droplet sheet makes possible a 1D solution, and the composition and initial temperature of the disperse system are also known better. At the same time, the LDR calculation can be considered as an illustration of the methods described, and is the natural continuation of the radiative boundary layer analysis.
REFERENCES
Cess, R. D., Radiation Effects Upon Boundary-Layer Flow of an Absorbing Gas, ASME J. Heat Transfer, vol. 86, no. 4, pp. 469-475, 1964.
Cohen, D. L., Simple Model for Particle Radiative Transfer in Vacuum Particle Plumes, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 365-367, 1988.
Dombrovsky, L. A., Radiative Equilibrium in a Plane-Parallel Layer of Absorbing and Scattering Medium, Fluid Dyn., vol. 9, no. 4, pp. 663-666, 1974.
Dombrovsky, L. A., Heat Transfer When a Non-Heat-Conducting Radiating Medium Flows Around a Flat Plate, High Temp., vol. 16, no. 5, pp. 859-864, 1978.
Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, McGraw-Hill, New York, 1978.
Taitel, Y., Exact Solution for the “Radiation Layer” Over a Flat Plate, ASME J. Heat Transfer, vol. 91, no. 1, pp. 188-189, 1969.
Les références
- Cess, R. D., Radiation Effects Upon Boundary-Layer Flow of an Absorbing Gas, ASME J. Heat Transfer, vol. 86, no. 4, pp. 469-475, 1964.
- Cohen, D. L., Simple Model for Particle Radiative Transfer in Vacuum Particle Plumes, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 365-367, 1988.
- Dombrovsky, L. A., Radiative Equilibrium in a Plane-Parallel Layer of Absorbing and Scattering Medium, Fluid Dyn., vol. 9, no. 4, pp. 663-666, 1974.
- Dombrovsky, L. A., Heat Transfer When a Non-Heat-Conducting Radiating Medium Flows Around a Flat Plate, High Temp., vol. 16, no. 5, pp. 859-864, 1978.
- Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, McGraw-Hill, New York, 1978.
- Taitel, Y., Exact Solution for the “Radiation Layer” Over a Flat Plate, ASME J. Heat Transfer, vol. 91, no. 1, pp. 188-189, 1969.