Quenching of a billet 32 pts a solid sphere initially

12. Quenching of a billet (32 pts.)
A solid sphere initially at temperature T
0
(radius R , mass m , heat capacity
ˆ
C
p
≈
ˆ
C
V
, thermal conductivity
k ) is totally immersed in a stagnant liquid of mass M greatermuch m and temperature T
∞
. Heat transfer from the surface
of the sphere to the fluid is described in terms of a heat transfer coefficient,
Q = hA ( T |
r = R
- T
∞
) .
For sufficiently small Biot numbers, Bi ≡ hR/k lessmuch 1 , it is reasonable to assume that the temperature within
the sphere is uniform.
Reduce the appropriate macroscopic balances on the following page to determine the temperature of the
sphere as a function of time, T ( t ) , in terms of parameters defined above.
8

Macroscopic Balances
Mass:
dm
tot
dt
= - Δ w
Momentum:
d P
tot
dt
= - Δ
parenleftBigg
( v
2
)
( v )
w + pS
parenrightBigg
u + m
tot
g - F
fs
Energy Balance:
d
dt
( U
tot
+ K
tot
+ Φ
tot
) = - Δ
bracketleftBiggparenleftBigg
ˆ
H +
1
2
( v
3
)
( v )
+ gh
parenrightBigg
w
bracketrightBigg
+ Q + W
m
Mechanical Energy Balance (steady state):
Δ
parenleftBigg
1
2
( v
3
)
( v )
parenrightBigg
+ g Δ h +
integraldisplay
2
1
dp
ρ
=
ˆ
W
m
-
ˆ
E
v
Additional information:
f ( Re ) =
braceleftBigg
16 / Re Re ≤ 2100
0 . 0791 / Re
1 / 4
2100 < Re < 10
5
Δ
ˆ
H =
braceleftBigg
ˆ
C
p
Δ T + Δ p/ρ ≈ Δ
ˆ
U incompr. liquids and solids, const. ρ,
ˆ
C
p
integraltext
2
1
ˆ
C
p
( T ) dT ideal gas
ˆ
E
v
= 2 ( v )
2
L
D
f ( Re ) for a straight pipe with circular cross-section
9

13. Reaction in a fluidized bed reactor (35 pts.)
A tubular fluidized bed reactor (radius R ) contains catalyst particles that cause species A to react via
A → products
as illustrated in the figure below.
z
v
z
= constant
c
A
( z = 0 ) =
c
A0
R
A
=
- ( k a )
1
’’ c
A
Although the reaction is actually heterogeneous, the apparent homogeneous reaction rate (occurring over a
volume containing many particles) is given by
R
A
= -
(
k
primeprime
1
a
)
c
A
,
where a is the surface area per unit volume. Here, R
A
is the rate of appearance of A per unit volume.
The fluid flows through the reactor with velocity v
z
in the z direction. The presence of the particles causes
the velocity profile to be uniform (i.e., v
z
= constant). The average velocity of particles is zero.
Assuming that the inlet concentration of A in inert liquid B is c
A 0
, that diffusion is negligible relative to
convection (in which case the convective molar flux can be written N
A,z
= c
A
v
z
), that the process occurs at
constant temperature, and that mass transfer limitations to reaction are negligible, use a shell mass balance to
determine the steady-state concentration of species A as a function of position, c
A
( z ) .
10