Page 464 - Modelling in Transport Phenomena A Conceptual Approach

444 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

(10.2-49)

Substitution of Eqs. (10.2-48) and (10.2-49) into Eq. (10.2-44) gives

@f
- df
d772 + 277 - = 0 (10.2-50)
d77
The boundary conditions associated with Eq. (10.2-50) are

at q=O f=1 (10.2-51)
at q=m f=O
The integrating factor for Eq. (10.2-50) is exp($). Multiplication of Eq. (10.2-50)
by the integrating factor yields

f (e+ $-) = o (10.2-52)

which implies that
(10.2-53)

Integration of Eq. (10.2-53) gives

f = CIA" e-U2du + C2 (10.254)

where u is a dummy variable of integration. Application of the boundary condition
at 11 = 0 gives CZ = 1. On the other hand, application of the boundary condition
atq=lgives
1 n
(10.2-55)
Jo
Therefore, the solution becomes
2 q
f = 1 - Jd e-"'du = 1 - erf(q) (10.2-56)

-- - 1 - erf (-) S
T - To
Tl -To &Kt (10.2-57)
where erf(z) is the error function defmed by