Question 5.17: Figure 5.14 is a two-dimensional rectangular enclosure that ...
Figure 5.14 is a two-dimensional rectangular enclosure that is long in the direction normal to the cross section shown. All the surfaces are diffuse-gray and their temperatures and emissivities are given. For simplicity, the four sides are not subdivided.
The configuration factors are obtained by the crossed-string method as F_{1-3}=F_{3-1}= 0.2770,F_{2-4}=F_{4-2}=0.5662,F_{1-2}=F_{1-4}=F_{3-2}=F_{3-4}=0.3615, and F_{4-1}=F_{4-3}=F_{2-1}=0.2169. In what follows an abbreviated notation is used E=(1-\epsilon )/\epsilon ,\vartheta =T^4, and F_{kj}=F_{k-j}. Then the four equations from Equation 5.29 are
\sum\limits_{j=1}^{N}{\left\lgroup \frac{\delta _{kj}}{\epsilon _j} -F_{k-j}\frac{1-\epsilon _j}{\epsilon _j} \right\rgroup } \frac{Q_j}{A_j} =\sum\limits_{j=1}^{N}{\left(\delta _{kj}-F_{k-j}\right)\sigma T_j^4=\sum\limits_{j=1}^{N}{F_{k-j}} \sigma \left(T^4_k-T_j^4\right) } (5.29)
\frac{q_1 }{\epsilon _1}-q_2F_{12}E_2-q_{3}F_{13}E_3 -q_{4}F_{14}E_4=\sigma (\vartheta _1-F_{12}\vartheta _2-F_{13}\vartheta _3-F_{14}\vartheta _4)
-q_1F_{21}E_1+\frac{q_1 }{\epsilon _1}-q_{3}F_{23}E_3 -q_{4}F_{24}E_4=\sigma (-F_{21}\vartheta _1+\vartheta _2-F_{23}\vartheta _3-F_{24}\vartheta _4)
-q_1F_{31}E_1-q_{2}F_{32}E_2+\frac{q_3 }{\epsilon _3} -q_{4}F_{34}E_4=\sigma (-F_{31}\vartheta _1-F_{32}\vartheta _2+\vartheta _3-F_{34}\vartheta _4)
-q_1F_{41}E_1-q_{2}F_{42}E_2 -q_{3}F_{43}E_3+\frac{q_4 }{\epsilon _4}=\sigma (-F_{41}\vartheta _1-F_{42}\vartheta _2-F_{44}T_3+\vartheta _4)
All quantities except the q´s are known. Substitution and solution by a computer software package gives q_1 = –2876.52, q_2 = 1612.53, q_3 = 1508.92, q_4 = –791.97 W/m^2.
Now the method in Section 5.3.2 is used. In Equation 5.37, it is noted that the Γ_{jk} have the same matrix of coefficients for all k. There are four sets of four equations, one set for each k, and within each set j has values 1, 2, 3, and 4. The matrix of the Γ coefficients is (note that the F_{kk} = 0 for this example)
(1-F_{1-1}\rho _1)\Gamma _{1k}-F_{1-2} \rho _2\Gamma _{2k}-F_{1-3} \rho _3\Gamma _{3k}-……-F_{1-N}\rho _N\Gamma _{Nk}=F_{1-k}\epsilon _k
-F_{2-1} \rho _1\Gamma _{1k}+(1-F_{2-2}\rho _2)\Gamma _{2k}-F_{2-3} \rho _3\Gamma _{3k}-……-F_{2-N}\rho _N\Gamma _{Nk}=F_{2-k}\epsilon _k
-F_{3-1} \rho _1\Gamma _{1k}-F_{3-2} \rho _2\Gamma _{2k}+(1-F_{3-3}\rho _3)\Gamma _{3k}-……-F_{3-N}\rho _N\Gamma _{Nk}=F_{3-k}\epsilon _k
. . .
. . . (5.37)
. . .
-F_{N-1} \rho _1\Gamma _{1k}-F_{N-2} \rho _2\Gamma _{2k}-F_{N-3} \rho _3\Gamma _{3k}-……+(1-F_{N-N}\rho _N)\Gamma _{Nk}=F_{N-K}\epsilon _k
m=\left[\begin{matrix}1 & \rho _{2}F_{12} & \rho _{3}F_{13}&-\rho _{4}F_{14} \\ -\rho _{1}F_{21} & 1 & -\rho _{3}F_{23} & -\rho _{4}F_{24} \\ -\rho _{1}F_{31} & -\rho _{2}F_{32} &1 & -\rho _{4}F_{34} \\ -\rho _{1}F_{41} & -\rho _{2}F_{42} &-\rho _{2}F_{24} &1 \end{matrix}\right]
When the values are substituted into this matrix it becomes
m=\left[\begin{matrix}1 &− 0.25306 & − 0. 04155 &−0.19883 \\ − 0 .06507 & 1 & −0.03254 & -0.31140 \\ -0.08310 & -0.25306 & 1 & -0.19883 \\ -0.06507 & -0.39633 &-0.03254 &1 \end{matrix}\right]
The matrix of the four columns of values on the right-hand sides of the four sets of equations is, in symbolic and numerical form,
f=\left[\begin{matrix}F_{11}\epsilon _1 &F_{12}\epsilon _2 &F_{13}\epsilon _3&F_{14}\epsilon _4 \\F_{21}\epsilon _1 & F_{22}\epsilon _2 & F_{23}\epsilon _3 & F_{24}\epsilon _4 \\ F_{31}\epsilon _1 & F_{32}\epsilon _2 & F_{33}\epsilon _3 & F_{34}\epsilon _4\\ F_{41}\epsilon _1 & F_{42}\epsilon _2 & F_{43}\epsilon _3 & F_{44}\epsilon _4 \end{matrix}\right]
f=\left[\begin{matrix}0 &0.10845 &0.23544 & 0.16268\\0.15183 & 0 & 0. 18437 & 0.25479 \\0.19389 & 0.10845 & 0& 0.16268\\ 0.15183 & 0.16986 &0.18437 & 0 \end{matrix}\right]
The matrix of Γ_{kj} factors is obtained by matrix inversion as [Γ_{kj}] = m^{−1}f,
\Gamma =\left[\begin{matrix}0.13233 &0.18271 &0.39315 & 0.29181\\0.25579 & 0.08738 & 0.32299 & 0.33384 \\0.32377 & 0.19000 & 0.18278 & 0.30345\\ 0.27236 & 0.22256 &0.34391 & 0.16117 \end{matrix}\right]
By summing values in each row, it is evident that \sum\limits_{j=1}^{4}{\Gamma _{kj}}=1. The q are then found from Equation 5.39,
Q_k=A_k\epsilon _k\sigma T^4_k\sum\limits_{j=1}^{N}{\Gamma _{kj}}-\sum\limits_{j=1}^{N}{A_k\epsilon _k \Gamma_{kj} \sigma T^4_j}=A_k\epsilon _k\sum\limits_{j=1}^{N}{\Gamma _{kj}\sigma \left(T^4_k-T^4_j\right) } (5.39)
and they agree with the values given previously.
The solution is now obtained by the method in Section 5.3.3. The coefficient matrix for the set of Equations 5.45a,b is
\sum\limits_{j=1}^{N}{\left(\frac{\delta _{kj}}{\epsilon _j}-F_{k-j}\frac{1-\epsilon _j}{\epsilon _j} \right)q_j=-F_{k-n } } (k\neq n) (5.45a)
\sum\limits_{j=1}^{N}{\left(\frac{\delta _{nj}}{\epsilon _j}-F_{n-n}\frac{1-\epsilon _j}{\epsilon _j} \right)q_j=-F_{k-n } } (5.45b)
D=\begin{bmatrix} 1/\epsilon _1 & -\rho _2(F_{12}/\epsilon _2)& -\rho _3(F_{13}/\epsilon _3)& -\rho _4(F_{14}/\epsilon _4) \\ -\rho _1(F_{21}/\epsilon _1) & 1/\epsilon _2 &-\rho _3(F_{23}/\epsilon _3)&-\rho _4(F_{24}/\epsilon _4)\\-\rho _1(F_{31}/\epsilon _1) &-\rho _2(F_{32}/\epsilon _2)&1/\epsilon _3&-\rho _4(F_{34}/\epsilon _4)\\ -\rho _1(F_{41}/\epsilon _1) &-\rho _2(F_{24}/\epsilon _2)&-\rho _3(F_{43}/\epsilon _3)&1/\epsilon _4\end{bmatrix}
and this gives in numerical form
D=\begin{bmatrix}1 .42857 & -0.84352& -0.04888& -0.44184 \\ -0.09296 & 3.33333 &-0.03828&-0.69201\\-0.11871 &-0.84352& 1.17647 &-0.44184\\ -0.09296&-1.32111&-0.03828&2.22222\end{bmatrix}
The matrix for the right-side coefficients of Equation 5.45a,b to determine –q is (note that F_{nm} = 0 for this example)
M=\begin{bmatrix}-1 & F_{12}& F_{13}& F_{14} \\ F_{21} & -1 &F_{23}&F_{24}\\F_{31} &F_{32}& -1 &F_{34}\\ F_{41} &F_{42}&F_{43}&-1\end{bmatrix}
To relate the – q and \mathscr{F} from Equation 5.44, the emissivity matrix is
F_{kn}=-q_k (k\neq n) (5.44)
F_{nn}=-q_n + \epsilon _n
\epsilon =\begin{bmatrix} \epsilon _{1} & 0 & 0 & 0\\ 0 & \epsilon _2 & 0& 0\\0& 0&\epsilon _3& 0& \\ 0& 0& 0& \epsilon _4\end{bmatrix}
Then, according to Equation 5.44, the F factors are obtained from the matrix operations
\mathscr{F} =D^{−1}M+ a
This yields the matrix of \mathscr{F} values:
\mathscr{F} =\begin{bmatrix} 0.09263 & 0.12790 & 0.27520 & 0.20427\\ 0.07674 & 0.02621 & 0.09690& 0.10015\\0.27520& 0.16150 &0.15537& 0.25793 \\0.12256&0.10015 &0.15476 &0.07253 \end{bmatrix}
From these values, it is evident that there is the relation \mathscr{F}_{kj} = ϵ_kG_{kj} as in Equation 5.41. The q are computed from Equation 5.40, and the values are the same as before.
Q_k=A_k\sum\limits_{j=1}^{N}{\mathscr{F}_{kj}\sigma \left(T^4_k-T^4_j\right) } (1\leq k\leq N) (5.40)
\mathscr{F}_{kj} =\epsilon _kG_{kj} (5.41)