ABSTRACT
Heat flux ills is evaluated by differentiating Eq. (9.104) with respect to x and evaluating the result at (LUj2, L\yj4). Thus,
(9.106) Substituting Eq. (9.106) into Eq. (9.101), where A = L\yj2, gives
(9.107)
(9.109)
(9.110)
Equations (9.107) and (9.110) specify the two heat fluxes in cell 1. Applying the same procedure to cells 2, 3, and 4 yields the other six heat fluxes at the surface of the control volume. Thus,
(9.111)
(9.112)
(9.113)
(9.114)
(9.115)
(9.116)
Substituting Eqs. (9.107), (9.110), and (9.111) to (9.116) into Eq. (9.100), collecting terms, and simplifying yields the control volume approximation of the heat diffusion equation:
2(3 -IP)(T1 + T3) + 2(3f32 - 1)(T2+ T4) + (f32 + I) X (Ts + T6+ T7 + Tg) - 12(f32 + I)To = 0 (9.117)
(9.118)
where f3 = tll/f..y is the grid aspect ratio. For unity grid aspect ratio (i.e., f3 = I), Eq. (9.117) becomes
1 2(T) +T2+T3+T4)+(Ts+T6+T7+Tg)-12To=0 I The computational stencil corresponding to Eq. (9.118) is illustrated in Figure 9.23.
