Page 700 - Bird R.B. Transport phenomena

680 Chapter 22 Interphase Transport in Nonisothermal Mixtures
diffusion between 1 and 2 in Fig. 22.3-1 are given by the following expressions, valid for
either laminar or turbulent flow:

f L f 27T I
heat transfer: Q(t) = I | +k XRdQdz (22.3-1)
J о J о
mass transfer: W (t) - x (W (t) + W (t)) = Г f [ +сЯЬ ^ к rf0 rfz (22.3-2)
A0
A0
A0
B0
АВ
J Jo \ dY r=R/
0
Equating the left sides of these equations to /I (7TDL)(T - TJ and k (irDL)(x - x ) re-
1 0 xl A0 M
spectively, we get for the transfer coefficients
heattransfer. Ц 0 - ]R dO dz (22.3-3)
=R/
mass transfer: k At) = AB ЭХА )R dd dz (22.3-4)
x
Jo
TTDL(X A0 - x Al ) J J o o Jo dr
We now introduce the dimensionless variables f = r/D, z = z/D, T = (T - T )/(T^ - T ),
0
o
and x = (x - x )/(x - x ) and rearrange to obtain
A A A0 M A0
U
heat transfer: Nu,(t) = ^- = —f— f ° Г" - § U dz (22.3-5)
(
к 2TTL/D J O Jo \ dr
2тг
к ,D 1 f L/D Г / дхл \
mass transfer: S\\ (f) = -g— = ^ i ^ (22.3-6)
;
x
/ n t
c4t AB lirL/DJo J o V ^ r =J
Here Nu is the Nusselt number for heat transfer without mass transfer, and Sh is the
Sherwood number for isothermal mass transfer at small mass-transfer rates. The Nus-
selt number is a dimensionless temperature gradient integrated over the surface, and
the Sherwood number is a dimensionless concentration gradient integrated over the
surface.
These gradients can, in principle, be evaluated from Eqs. 11.5-7, 8, and 9 (for heat
transfer) and Eqs. 19.5-8, 9, and 11 (for mass transfer), under the following boundary
conditions (with v and & defined as in §14.3 and with time averaging of the solutions if
the flow is turbulent):
velocity and pressure:
at z = 0, v = for 0 < r < \ (22.3-7)
for z > 0 (22.3-8)
at f = 0 and z = 0,Ф = 0 (22.3-9)
temperature:
at z = 0, f = 1 for 0 < r < ^ (22.3-10)
at f = \, f = 0 forO < z < L/D (22.3-11)

concentration:
atz = 0,x == 1 for 0 < r < \ (22.3-12)
A
atr = ix == 0 for 0 < z < L/D (22.3-13)
A
The boundary condition in Eq. 22.3-8, on the velocity at the wall, is accurate for the heat-
transfer system and also for the mass-transfer system provided that x (W + W ) is
A0 A0 B0
small; the latter criterion is discussed in §§22.1 and 8. No boundary conditions are
needed at the outlet plane, z = L/D, when we neglect the d 1'dz 1 terms of the conserva-
2
tion equations in the manner of §4.4 and §14.3.