Question 7.10: Using the ring-to-ring configuration factor, evaluate the co...
Using the ring-to-ring configuration factor, evaluate the configuration factor from a ring element on the interior of a right circular cylinder to the cylinder base for the geometry in Figure 7.19, when x = r. Use the trapezoidal rule and compare the result with the analytical solution.
The factor from dA_1 to a ring dA_2 on the base surface is
dF_{d1-d2}(X,R)=\frac{2XR(1+X^2-R^2)dR}{[(1+X^2+R^2)^2-4R^2]^{3/2}} (7.32)
where X = x/r and R = ρ/r. For this example X = 1, so f_j(X = 1, R_j) in Figure 7.18 is given by
f_j(X=1,R_j)\equiv f_j(1)=\frac{2R_j(2-R_j^2)}{(4+R_j^4)^{3/2}}
where R_j = jΔR and ΔR = 1/N. In particular, letting f_j (X = 1, jΔR) ≡ f_j (1),
f_0(1)=0; f_j(1)=\frac{2j\Delta R[2-(j\Delta R)^2]}{[4+(j\Delta R)^4]^{3/2}} ; f_N=\frac{2}{5^{3/2}}
These terms are substituted into Equation 7.31, and for N = 5 yields F_{d1-2}(X=1)=(1/5)[(1/2 ×0+0.09794+0.18225+0.23451+0.23500 +(1/2)×0.17889]=0.16783
The exact configuration factor is in Appendix C as
F_{d1-2}(X=1)=\frac{X^{*2}+\frac{1}{2} }{(X^{*2}+1)^{{1}/{2}}}-X^{*}; X^*=\frac{x}{2r}=\frac{X}{2}=\frac{1}{2}
which gives F_{d1-2}(X=1)=F_{d1-2}(X^*=0.5)=0.17082 Larger numbers of increments improve the accuracy as follows:
| % Error | F_{d1-2}(X=1) | N |
| −1.75 | 0.16783 | 5 |
| −0.44 | 0.17007 | 10 |
| −0.02 | 0.17079 | 50 |
| −0.006 | 0.17081 | 100 |
| 0 | 0.17082 | 200 |
