Question 7.14: A wedge consists of two very long parallel sides of equal wi...
A wedge consists of two very long parallel sides of equal width joined at 90°, Figure 7.30. The surface temperatures are T_1 = 1000 K and T_2 = 2000 Κ. The effects of the ends may be neglected. Surface 1 is diffuse-gray with ϵ_1 = 0.5, and surface 2 is directional-gray with directional total emissivity and absorptivity,
\epsilon_2(θ2) = α_2(θ_2) = 0.5cosθ_2 (7.101)
Assume for simplicity that surface 2 reflects diffusely. Set up a Monte Carlo flowchart for determining the energy to be added to each surface to maintain its temperature. Assume that the environment is at T_e = 0 K.
The energy flux emitted by surface 1 is \epsilon _1\sigma T_1^4. If N_1 emitted energy bundles are followed per unit time and area from surface 1, the energy per bundle is w=\epsilon _1\sigma T_1^4/N_1. The energy flux emitted from surface 2 is
2\sigma T_2^4\int_{\theta =0}^{\pi /2 }{\epsilon _2(\theta )\cos \theta \sin \theta d\theta }=\sigma T_2^4\int_{\theta =0}^{\pi /2}{\cos ^2\theta \sin \theta d\theta =\frac{\sigma T_2^4}{3} }
If the same amount of energy w is assigned to each bundle emitted by wall 2 as for wall 1, then wN_2=σT_2^{4}/3 Substituting for w, ϵ_1 T_1, and T_2 gives
N_2=\frac{\sigma T_2^4}{3} \frac{N_1}{\epsilon _1\sigma T_1^4}=\frac{32 }{3}N_1 (7.102)
Because all bundles have equal energy and 32/3 as many bundles are emitted from surface 2 as from surface 1, it is evident that surface 2 will make the major contribution to the energy transfer. Now, the distributions of directions for emitted bundles from the two surfaces will be derived. Surface 1 emits diffusely, so Equation 7.97b
\sin \theta _1=\sqrt{R_{\theta _1, diffuse-gray}} (7.97b)
applies. For the directional-gray surface 2, the second line in Table 7.2 is used with ϵ_2(θ_2) from Equation 7.101,
| TABLE 7.2 Convenient Functions Relating Random Numbers to Variables for Emission (Assume No Dependence on Circumferential Angle \phi ) |
||
| Variable | Type of Emission | Relation |
| Cone angle θ | Diffuse | \sin \theta =R_{\theta }^{1/2} |
| Directional-gray | R_\theta =\frac{2\int_{\theta ^*=0}^{\theta }{\epsilon (\theta ^*)}\sin \theta ^*\cos \theta ^*d\theta ^* }{\epsilon } | |
| Directional-nongray | R_\theta =\frac{2\pi \int_{\theta ^*=0}^{\theta }\int_{\lambda =0}^{\infty }{} {\epsilon_\lambda (\theta ^*)}I_{\lambda b}\sin \theta ^*\cos \theta ^*d\lambda d\theta ^* }{\epsilon \sigma T^4} | |
| Circumferential angle ϕ | Diffuse | \phi =2\pi R_\phi |
| Wavelength λ | Black or gray | F_{0-\lambda T}=R_{\lambda T} |
| Diffuse-nongray | R_\lambda =\frac{\int_{\lambda ^*=0}^{\lambda }{\epsilon _{\lambda ^*}E_{\lambda ^*b}d\lambda ^*} }{\epsilon \sigma T^4} | |
| Directional-nongray | R_\lambda =\frac{2\pi \int_{\lambda ^*=0}^{\lambda }\int_{\theta =0}^{\pi /2 }{} {\epsilon_{\lambda ^*} (\theta ^*)}I_{\lambda^* b}\sin \theta \cos \theta d\theta d\theta ^* }{\epsilon \sigma T^4} | |
R_{\theta _2} =\frac{2}{\epsilon _2}\int_{\theta _2^*=0}^{\theta _2}{(0.5\cos \theta _2^*)\sin{\theta _2^*}d\theta _2^* }
The ϵ_2(θ_2) is substituted from Equation 2.6b to give
Hemispherical total emissivity (in terms of directional total emissivity) \equiv \epsilon (T)=\frac{1}{\pi }\int_{\smallfrown }^{}{\epsilon (\theta ,\phi ,T)\cos \theta d\Omega } (2.6b)
R_{\theta _2}=\frac{\int_{0}^{\theta _2}{\cos ^2\theta _2^*}\sin\theta _2^*d\theta _2^* }{\cos ^2\theta _2\sin \theta _2d\theta _2} =1-\cos ^3\theta_2
Because R and 1 − R are both uniform random distributions in the range 0 ≤ R ≤ 1, this result can be conveniently written as \cos \theta _2=R_{\theta _2} ^{1/3} By similar reasoning, Equation 7.97b can be written as \cos \theta _1=R_{\theta _1} ^{1/2}.Since there is no dependence on angle \pi for either surface, Equation 7.98
\phi_1=2\pi R_{\phi _1} (7.98)
applies for both surfaces.
Next, the position must be determined on each surface from which each bundle will be emitted. Because the wedge sides are isothermal, emission is uniform from each side. In such a case, random positions x (Figure 7.30) could be picked on each side as points of emission. This requires generation of a random number. The computer time required to generate a random number can be eliminated by noting that bundle emission is the initial process in each Monte Carlo history; hence, there is no prior history to be removed by using a random number. In this case initial x positions along L can be sequentially chosen as x = (n/N)L, where n is the sample-history index for the history being begun, 1 ≤ n ≤ N. The remaining calculations are to determine whether each emitted bundle will strike the adjacent wall or will leave the cavity. From Figure 7.30, for either surface, when π ≤ ϕ ≤ 2π the bundles will leave the cavity for any θ, and when 0 < ϕ < π they will leave if \sin \theta \lt (x/\sin \phi )/[(x/\sin \phi )^2+L^2]^{1/2}=1/[1+(L\sin \phi /x)^2]^{1/2}. The angle of incidence θ_i on a surface is given in terms of the angles θ_δ and ϕ_δ at which the bundle leaves the other surface, by \cos \theta _i=\sin \theta _\delta \sin \phi _\delta . All the necessary relations are now available. A flow diagram is constructed to combine these relations in the correct sequence. Diffuse reflection is assumed from both surfaces. The resulting diagram is in Figure 7.31, showing one way of constructing the flow of events. The indices δ, δ′, and δ″ are used to reduce the size of the chart. The index δ always refers to the wall from which the original emission of the bundle occurred, and δ′ refers to the wall from which emission or reflection is presently occurring. The δ″ is used to make the emitted distribution of θ angles correspond to either R_{\theta_1}^{1/2} or R_{\theta_2}^{1/2} and have all the reflected bundles correspond to a diffuse distribution.
