Page 614 - Bird R.B. Transport phenomena

594 Chapter 19 Equations of Change for Multicomponent Systems

To get the temperature profile, we use the energy flux from Eq. 19.3-6 for an ideal gas along
with Eq. 9.8-8:

= —к— + NA С (Т — Тп) (19 4-7)
Л
л. I iv Ay \^ pA \ i I Q/ VA^«" ' /
Here we have chosen T as the reference temperature for the enthalpy. Insertion of this ex-
o
pression for e into Eq. 19.4-2 and integration between the limits T = T at у = 0, and T = T at
y o x
у = 8 gives
T - T 1 - exp[(N C^/%]
o Av
(19.4-8)
- exp[(N C /k)8]
Ay pA
It can be seen that the temperature profile is not linear for this system except in the limit as
N C /k -» 0. Note the similarity between Eqs. 19.4-6 and 8.
Ay pA
The conduction energy flux at the wall is greater here than in the absence of mass trans-
fer. Thus, using a superscript zero to indicate the conditions in the absence of mass transfer,
we may write
-k(dT/dy)\ = 0 _ ~(N C /k)8
y
pA
Ay
-KdT/dy)% l - ехр[(ДГ С Д)5]
0 Лу рЛ
We see then that the rate of heat transfer is directly affected by simultaneous mass transfer,
whereas the mass flux is not directly affected by simultaneous heat transfer. In applications at
temperatures below the normal boiling point of species A, the quantity N C /k is small, and
pA
Ay
the right side of Eq. 19.4-9 is very nearly unity (see Problem 19A.1). The interaction between
heat and mass transfer is further discussed in Chapter 22.
(b) If both A and В are condensing at the wall, then Eqs. 19.4-1 and 2, when integrated, lead
to N Ay = N A0 and e y = e , where the subscript "0" quantities are evaluated at у = 0. We also in-
0
tegrate the analog of Eq. 19.4-1 for В to get N By = N B0 and obtain
-сЯЬ АВ ^ + x (N A0 + N ) = N A0 (19.4-10)
B0
A
-kj^+ (N H A + N H ) = e 0 (19.4-11)
A0
B
B0
In the second of these equations, we replace H A by C (T — T ) and H by C pB (T - T ), and
o
pA
B
o
since the reference temperature is T , we may replace e by q , the conductive heat flux at the
o
Q
0
wall. In the first equation, we subtract x (N + N ) from both sides to make the equation
AQ A0 B0
similar in form to the temperature equation just obtained. Thus
-сЯЬ АВ ^ + (AU + N )(x A - x ) = N A0 - x (N AQ + N ) (19.4-12)
A0
B0
B0
AQ
-k ^ + (N C + N C )(T - T ) - q (19.4-13)
A0 pA B0 pB o 0
Integration with respect to у and application of the boundary conditions at у = 0 gives
п9л и)
-
(Ы С + N C,, )(T - Го) у]
А0
p рЛ B0 B
^ 1 exp^NC + N C ) | J (19.4-15)
B0 pB