Page 68 - Introduction to Computational Fluid Dynamics

P1: IWV/ICD
CB908/Date
0521853265c02
EXERCISES
where 0 521 85326 5 May 25, 2005 10:49 47
3 y 2
u fd = u 1 − (2.86)
2 b 2
and y is measured from the symmetry axis. The initial condition is T = T i at z = 0
and the symmetry boundary condition is ∂T /∂y = 0at y = 0. At y = b, however,
T = T w if both walls are at constant wall temperature, or, if constant wall heat
flux is specified, then k ∂T /∂y | b = q w . For this problem, if we set y ≡ x, z ≡ t,
2
2
q = 0, A = 1, and C p u fd ≡ 1.5u (1 − y /b )C p then Equation 2.85 is the same

as Equation 2.5 in which the unsteady term is retained.
Diffusion Mass Transfer
In a binary mixture of species i and j, the equation (in spherical coordinates)
governing radial diffusion of j in a stationary medium i is given by
2
∂ ρ m D 4π r ∂ω j
2 ∂ω j
= ρ m 4πr , (2.87)
∂r (1 − ω j ) ∂r ∂t
where ω j is the mass fraction of j in the mixture and D is the mass diffusivity.
2
Thus, if we set ∂r ≡ ∂x, A = 4π r , k = ρ m D/(1 − ω j ), C p = 1, T = ω j , and,

q = 0 then this equation is the same as Equation 2.5. To solve the equation,
one will need boundary conditions at r = r i and r = r o and the initial condition
at t = 0. Estimation of penetration depth during surface hardening of materials,
estimation of leakage flow of gases from storage vessels, or estimation of burning
rate of volatile fuel in still surroundings are some of the mass transfer problems
of interest. The reader is referred to the unified formulation of the mass transfer
problem by Spalding [72] and to the book by Gupta and Srinivasan [26].

EXERCISES 8
1. Show that the derivative expressions in Equation 2.21 are second-order accurate
if the cell face is midway between adjacent nodes.
2. A slab of thickness 2b is initially at temperature T 0 .At t = 0, the boundary
temperatures at x =−b and +b are raised to T b and maintained there. The
exact solution for evolution of temperature in this case is given by
∞
T − T b sin(λ n b)
2
= 2 cos(λ n x)exp −αλ t ,
n
T 0 − T b λ n b
n=1
where λ n b = (2n − 1)π/2. Hence, considering the data of Problem 1 in the
text, write a computer program to determine the value of t for the centerline
8 All numerical problems given in these exercises can be solved by the generalised computer code
given in Appendix B.