Page 300 - Modelling in Transport Phenomena A Conceptual Approach

280 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

Before solving Eq. (8.2-77), it is convenient to express the governing equation
and the boundary conditions in dimensionless form. The reason for doing this is
the fact that the inventory equations in dimensionless form represent the solution
to the entire class of geometrically similar problems when they are applied to a
particular geometry.
Introduction of the dimensionless variables

(8.2-82)

E
<=- (8.2-83)
L

(8.2-84)

reduces Eqs. (8.2-77)-(8.2-79) to

&e
-- A20 (8.2-85)
dc2 -

at <=0 8=1 (8.2-86)

(8.2-87)

The solution of Q. (8.2-85) is

6 = CI sinh(A<) f C2 cosh(A<) (8.2-88)

where CI and C2 are constants. Application of the boundary conditions, Eqs.
(8.286) and (8.2-87), gives the solution as

coshA cosh(A<) - sinhA sinh(A<)
O= cosh A (8.2-89)

The use of the identity

cosh(z - y) = cosh z cosh y - sinh z sinh y (8.2-90)

reduces the solution to the form
cash [A( 1 - <)I
cosh A (8.2-91)