Radiative Heat Transfer in Finite Cylindrical Media

Radiative Heat Transfer in Finite Cylindrical Media — P1 Method

This application uses the P1 approximation to solve a 3D radiative transfer problem in an emitting, absorbing, and linearly anisotropic-scattering finite cylindrical medium. The calculated incident radiation and heat fluxes are compared to the results obtained with the highly accurate S6 discrete ordinates method (see Radiative Heat Transfer in Finite Cylindrical Media). The results obtained with this method show fairly good agreement with published results obtained by transformed integral methods (see Ref. 1) for optically thick media.

Introduction

There are numerous engineering applications of radiative transfer in absorbing, emitting, and anisotropically scattering media with variable radiation properties. Examples include, among others, coal-fired combustion systems, light-weight fibrous insulations, and heat transfer systems containing small scattering particles. Furthermore, the efficiency of radiative transfer depends on the boundary conditions, for example, the temperature and the emissivity of the surrounding walls, and the target where heat transfer is desired. Studies have shown that radiative transfer is highly sensitive to the wall emissivity. In this study you build a validation model representing a cylinder with homogeneous walls. Then you go on to consider walls with space-dependent emissivity and investigate the effects of albedo and scattering.

Model Definition

In this tutorial you validate the P1 formulation in COMSOL Multiphysics by examining three benchmark cases. You investigate the method’s efficiency through parametric analyses by changing the single-scattering albedo, wall emissivity, and linear function. In particular, in the final case, you approximate the scattering phase function by a linear function reflecting highly backward, isotropic, and highly forward scattering. The results are compared to the results obtained with the DOM formulation as shown in Radiative Heat Transfer in Finite Cylindrical Media.

The model geometry, shown in Figure 1, is a cylinder of radius R = 0.5 m and height L = 1 m.

Figure 1: Schematic diagram of the physical model.

These examples use the P1 approximation method for predicting the heat flux on the enclosure side walls and the incident radiation distribution inside the domain.

Verification case

For the initial study, assume cold boundaries, that is, T = 0K. Furthermore, assume that the walls diffusively reflect radiation, that is, ε = 0.5. The medium is at a uniform temperature T such that the blackbody radiation intensity Ib(T) = σT4 ⁄ π is equal to 1 W/m2.

Walls With variable Emissivity

Two cases are computed for comparison purposes. These 3D cases represent opaque partial side wall diffuse emission and/or reflection. In both cases, the radiosity on the side walls (Surface 3) varies with the angular coordinate along the full height of the finite cylinder according to ε3 = (1 − y ⁄ R) ⁄ 2. Both cases also have cold walls at the cylinder top (Surface 1) and bottom (Surface 2).

Case 1 has isotropic scattering function and compares results for different albedos. Case 2 has constant albedo and compares results with highly forward, isotropic, and highly backward scattering function parameterized by the Legendre coefficient a1.

Table 1: Nonstandard test cases.
Case	MEDIUM Properties)
1	albedo = 0.1, 0.5, 0.9 a1 = 0
2	albedo = 0.5 a1 = −0.99, 0, 0.99

Thermal Analysis

The P1 approximation method is the simplest form of the method of spherical harmonics (PN) to describe radiative transport in participating media. The radiation transport equation (RTE) for this type of configuration can be written as

where

•

I(Ω, s) is the radiation intensity at a given position s in the direction Ω

•

T is the temperature

•

κ, β, and σs are absorption, extinction, and scattering coefficients, respectively

•

Ib(T) is the blackbody radiation intensity

•

φ(Ω, Ω′) = 1 + a1μ0 where μ0 = Ω ⋅ Ω′ is the cosine of the scattering angle.

The RTE (an integro-partial differential equation) is transformed into a set of partial differential equations. The remaining task is to solve an additional equation for

which adds a heat source/sink to account for radiative heat transfer.

where DP1 is a diffusion coefficient and Qr is the radiative heat source. The boundary condition for opaque surfaces then is

Results and Discussion

The results demonstrate that the P1 method provides a fast alternative to the DOM method. The solution time is only a few seconds whereas it is about 4 hours for the DOM method on the same mesh. The results represent the radiation characteristics well. The numerical values show good agreement for small scattering coefficients (omega) and high absorption coefficients (kappa). The relation between both values is κ = ω ⁄ (ω − 1).

Verification case

This case examines the effects of the scattering albedo on the incident radiation and heat fluxes. Figure 2 shows the distribution of the net heat flux qr, net(R, 0, z) versus axial optical thickness. There is good agreement with the results obtained from the DOM model. The results for the net radiative heat flux (Figure 2) show good agreement with distance to the top and bottom boundary. The relative error for small scattering albedo is below 10% and increases with increasing scattering albedo.

Figure 2: The effects of the scattering albedo on the radial heat flux qr, net(R, 0, z); for a hot cylindrical medium enclosed by cold walls, ε1 = ε2 = ε3 = 0.5.

The effects of albedo on the distribution of centerline incident radiation in axial direction are shown in Figure 3. The incident radiation is symmetric with respect to z = L ⁄ 2 plane and decrease with increasing scattering albedo. Furthermore, results become more uniform with larger scattering albedo.

Figure 3: The effects of the scattering albedo on the distribution of centerline incident radiation G(0, 0, z) for a hot cylindrical medium enclosed by cold walls, ε1 = ε2 = ε3 = 0.5.

Figure 4 displays the distributions of the incident radiation across the midplane radius G(x, 0, L ⁄ 2) with respect to normalized optical thickness x ⁄ R.

Figure 4: The effects of the scattering albedo on the distributions of the incident radiation G(x, 0, L ⁄ 2) with respect to normalized optical thickness x ⁄ R for a hot cylindrical medium enclosed by cold walls, ε1 = ε2 = ε3 = 0.5.

For the validation case the relative error compared to the DOM model is below 20%.

Walls With variable Emissivity

The incident radiation at the midplane z = L ⁄ 2 at the radial position R ⁄ 2 is shown in Figure 5. At the side surface R ⁄ 2 distance from the cylinder axis, G(R, 0, L ⁄ 2) changes with the azimuthal angle. The changes in scattering albedo are also illustrated. The smallest albedo makes the biggest change around the azimuthal angle. The numerical error for small scattering albedo again is low (around 10%).

Figure 5: The effects of scattering albedo on the distribution of incident radiation at midplane z = L ⁄ 2 at R ⁄ 2 distance from the cylinder axis with respect to an azimuthal coordinate for Case 1.

Figure 6 shows the effect of a nonzero linear anisotropic scattering coefficient a1. As expected, the differences between isotropic scattering, forward scattering, and backward scattering are most accentuated far from the boundary.

Figure 6: The effects of linear anisotropic scattering coefficient a1 on the distribution of incident radiation G(0, y, L ⁄ 2) with respect to normalized optical thickness − y ⁄ R for Case 2.

Unlike the discrete ordinates method, the P1 method is computationally inexpensive and gives fast results at the expense of accuracy in most cases. For large optical thickness the results show good agreement with the results obtained from the discrete ordinate method and hence with the results presented in the literature.

References

1. X.L. Chen and W.H. Sutton, “Radiative Transfer in Finite Cylindrical Media Using Transformed Integral Equations,” J. Quantitative Spectroscopy and Radiative Transfer, vol. 77, pp. 233–271, 2003.

2. X. Chen, Transformed Integral Equations of Radiative Transfer and Combined Convection-radiation Heat Transfer Enhancement with Porous Insert, Doctoral Thesis, University of Oklahoma, 2003.

3. S.T. Thynell and M.N. Ozisik, “Radiation Transfer in Absorbing, Emitting, Isotropically Scattering, Homogeneous Cylindrical Media,” J. Quantitative Spectroscopy and Radiative Transfer, vol. 38, no. 6, pp. 413–426, 1987.

4. H.Y. Li, M.N. Ozisik, and J.R. Tsai, “Two-dimensional radiation in a cylinder with spatially varying albedo,” J. Thermophysics and Heat Transfer, vol. 6, no. 1, pp. 180–182, 1992.

Heat_Transfer_Module/Verification_Examples/cylinder_participating_media_p1

Modeling Instructions — Validation Case

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
T0	(1[W/m^2]*pi/sigma_const)^(1/4)	86.275 K	Body temperature
Tw	0[K]	0 K	Wall temperature
ew	0.5	0.5	Wall emissivity
omega	0.5	0.5	Single-scattering albedo
sigma_s	omega	0.5	Scattering coefficient
kappa	sigma_s*(1/omega-1)	0.5	Absorption coefficient
R	0.5[m]	0.5 m	Cylinder radius
L	1[m]	1 m	Cylinder length