Page 318 - Bird R.B. Transport phenomena

302 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

significantly in the axial direction. Then in the second term we replace v z by the superfi-
cial velocity v , because the latter is the effective fluid velocity in the reactor. Then Eq.
0
10.5-4 becomes
pC v Щ- = K ^ + S (10.5-5)
p 0 eff zz c
dz ' dz 1
This is the differential equation for the temperature in zone II. The same equation ap-
plies in zones I and III with the source term set equal to zero. The differential equations
for the temperature are then

Zone I (z < 0) pC v — = K eiizz — - (10.5-6)
0
p
dz ' dz 1
Zone II ( 0 < z < L ) pC^ ^r- = «eff ^r + S F(0) (10.5-7)
dz ' dz
и o 22 1 cl
Zone III (z > L) C v Ц- = ic ^— (10.5-8)
P p 0 eff zz
dz ' dz 2
Here we have assumed that we can use the same value of the effective thermal conduc-
tivity in all three zones. These three second-order differential equations are subject to the
following six boundary conditions:

B.C.I: atz = — с T 1 = T, (10.5-9)
B.C. 2: at z = 0, T 1 = T n (10.5-10)
dT l dT n
B.C. 3: at z = 0,
^ггЧ~=^2гЦ~ (10.5-11)
B.C. 4: at z = L, ' dz ' dz
T u = T m (10.5-12)
B.C. 5: at z = L,
' dz ' dz
B.C. 6: atz = oo, T m = finite (10.5-14)
Equations 10.5-10 to 13 express the continuity of temperature and heat flux at the bound-
aries between the zones. Equations 10.5-9 and 14 specify requirements at the two ends of
the system.
The solution of Eqs. 10.5-6 to 14 is considered here for arbitrary F(@). In many cases
of practical interest, the convective heat transport is far more important than the axial
conductive heat transport. Therefore, here we drop the conductive terms entirely (those
containing /c cff/Z2 ). This treatment of the problem still contains the salient features of the
solution in the limit of large Pe = RePr (see Problem 10B.18 for a fuller treatment).
If we introduce a dimensiqnless axial coordinate Z = z/L and a dimensionless
chemical heat source N = S L/pC v (T } — T ), then Eqs. 10.5-6 to 8 become
p 0
c]
o
Zone I (z<0) ^ = 0 (10.5-15)
dZ
Zone II (0<z<L) ^ - = NF(®) (10.5-16)
dZ
Zone III (z>L) ^ - = 0 (10.5-17)

dZ
for which we need three boundary conditions:
B.C. 1: at Z = -oo, 0 1 = l (10.5-18)
B.C. 2: atZ = 0, & = в и (10.5-19)
B.C. 3: at Z = 1, в 11 = в 111 (10.5-20)