Download (1) WWWR 25.6 The moisture in hot, humid, stagnant air

(1) WWWR 25.6
The moisture in hot, humid, stagnant air surrounding a cold-water pipeline continually
diffuses to the cold surface where it condenses. The condensed water form a liquid film
around the pipe, and then continuously drops off the pipe to the ground below. At a
distance of 10 cm from the surface of the pipe, the moisture content of the air is constant.
Close to the pipe, the moisture content approaches vapor pressure evaluated at the
temperature of the pipe.
a. Draw a picture of the physical system, select the coordinate system that best
describe the transfer process and state at least 5 reasonable assumptions of the
mass-transfer aspect of the water condensation process.
b. What is the simplified form of the general differential equation for mass transfer
in terms of flux of water vapor, NA?
c. What is the simplified form of the Fick’s equation for water vapor, N A?
d. What is the simplified form of the general differential equation for mass transfer
in terms of concentration of water vapor, CA?
(2) WWWR 25.11
In the manufacture of semiconducting thin films, a thin film of solid arsenic laid down on
the surface of a silicon wafer by diffusion-limited chemical vapor deposition of arsine,
AsH3.
2 AsH3 (g)  2 As (s) + 3 H2 (g)
The gas head space, 5 cm above the surface of the wafer, is stagnant. Arsenic atoms
deposited on the surface then diffuse into the solid silicon to ‘dope’ the wafer and impart
semiconducting properties to the silicon, as shown in the figure below.
The process temperature is 1050 . The diffusion coefficient of arsenic in silicon is
at this temperature and the maximum solubility of arsenic in silicon is
atoms/cm3. The density of solid silicon is
atoms/cm3. As the diffusion
coefficient is so small, the arsenic atoms do not ‘penetrate’ very far into the solid, usually
less than a few microns. Consequently, a relatively thin silicon wafer can be considered a
“semi-infinite” medium for diffusion.
a. State at least 5 reasonable assumptions for the mass transfer of arsenic in this
doping process.
b. What is the simplified form of the general differential equation for mass transfer
of the arsenic concentration within the silicon? Purpose reasonable boundary
and initial conditions.
(3) WWWR 25.13
One way to deliver a timed dosage within human body is to ingest a capsule and allow it
to settle in the gastrointestinal system. Once inside the body, the capsule will slowly
release the drug to the body by a diffusion-limited process. A suitable drug carrier is a
spherical bead of non-toxic gelatinous material that can pass through the gastrointestinal
system without disintegrating. A water soluble drug (solute A) is uniformly dissolved
within the gel, has an initial concentration, CAO of 50 mg/cm3. The drug loaded within the
spherical gel capsule is the sink for mass transfer. Consider a limiting case where the
drug is immediately consumed or swept away once it reaches the surface, i.e., @ R, C A =
0.
a. In analyzing the process, choose a coordinate system and simplify the general
differential equation for mass transfer of the drug in terms of the flux.
b. What reasonable assumptions were used in your simplifying of the general
differential equation?
c. Simplify Fick’s equation for the drug species and obtain a differential equation in
terms of concentration, CA.
(4) WWWR 26.4
Ethanol is diffusing through a 4-mm stagnant film of water. The ethanol concentration of
the entrance and the exiting planes are maintained at 0.1 and 0.02 mol/m3, respectively. If
the water film temperature is 283 K, determine the steady-state molar flux of ethanol and
the concentration profile as a function of position z within the liquid film. Compare these
results with a 4-mm stagnant film of air at 283 K and 1 atm at the same entrance and exit
ethanol concentrations.
(1) Problem 25.6 (WWWR)
R+10cm
R
(a)
Pipe
R+10cm
Assumptions:
(i)
Pipe is long and diffusion only in r-direction
(ii)
No homogeneous reaction, RA  0
(iii)
Concentration of A @ r = R + 10 is known and constant
P *
(iv)
Concentration of A @ r = R is constant, YA  A
P
(v)
No mixing in gas space (no convection), only diffusion
(vi)
Steady-state because of constant concentrations at R and R+10 (infinite
reservoir and sink)
(b) General differential equation in terms of C A :
dC A
N Ar   DAB
 YA ( N Ar N Br )
dr
dC A
  DAB
 YA N Ar
dr
CDAB dYA
N A, r 
(1  YA ) dr
(c) Simplified differential form:
C
.N A  A  RA
t
1 d

 rN Ar   0 (cylindrical coordiates in r-direction only)
r dr
d
(d)
(rN Ar )  0
dr
 rN Ar  constant
Flux (2r NArL) is constant along diffusion path
boundary conditions: at r = R + 10, YA  YA
at r = R, YA 
PA*
P
note: if gas space is dilute (A) then N Ar  CDAB
dYA
dr
(2) Problem 25.11 (WWWR)
Well-mixed feed gas (constant composition)
…………………………………………………………
diffuser screen
H2 (g)
AsH3 (g)
As, thin film
NA
Si wafer
Assumptions:
(i)
Temperature = constant; DAB and density constant
(ii)
Flux only in the z-direction (one directional diffusion)
(iii)
No homogeneousd reaction
(iv)
Silicon treated as “semi-infinite”
(v)
CA ( z, o)  0
General mass conservation equation:
C
.N A  A  0
t
dC A
N A   DAB
dz
C A
 2C A
 DAB
t
z 2
Boundary conditions:
CA ( z, o)  0
CA (o, t )  CAS
CA(∞,t) = CA  0
z
(3) Problem 25.13 (WWWR)
(a)
R
Spherical coordinates
Assumptions:
(i)
Concentration, C A0 uniform throughout pill at t=0
(ii)
Molecular diffusion only within pill
(iii)
No reaction along diffusion path
(iv)
No bulk contribution term (convection)
(v)
Diffusion only in r-direction
(vi)
Constant DAB
(b) Differential form of Fick’s equation:
dCA
dCA
N Ar   DAB
 y A ( N Ar )   DAB
dr
dr
(c) General differential equation:
C
.N A  A  RA
t
C A
1 
  2  r 2 N Ar 
t
r r
C A 
1  
  2  r 2 DAB

r r 
r 
D   2 C A 
 AB
r

r 2 r 
r 
Boundary conditions:
@ t = 0, CA  CA0 for 0  r  R
@ t > 0, CA  CAS at r = R
C A
 0 at r = 0
r
(4) Problem 26.4
.N A  0 
d
( N Az )  0
dz
N Az =  DAB dCA  y A ( N Az  N Bz )
dz

N Az 
D AB
(C A1  C A2 )
δ
from Hayduk-Laudie
DAB L  13.26  105 μB
1.14
VA
0.589
VC2 H5OH  2(14.8)  6(3.7)  7.4  59.2
VA0.589  0.090
 B 1.14  (1.45cP ) 1.14  0.655
 DABL  7.82 106 cm 2 s 1  7.82 1010 m 2 s 1
7.82 1010 m 2 s 1
mol
NA 
(0.1

0.02)
 1.56 108 mol  m 2 s 1
3
3
4 10 m
m
Conc. Profile:
d dC A
(
)  0  CA  C1 z  C2
dz dz
b.c.
at
z  0, C A  0.1
3
at
z  4 10 , C A  0.02
C A  20 z  0.1
Diffusion through air:
C1  20, C2  0.1
P 1.013 105 Pa
mole

 43.05 3
RT 8.314(283K )
m
CA
CA
0.1
y A1  1 
 2.32 103 ; y A2  2  4.62 104
C
43.05
C
C
Same equations as in water, N Az 
and
 at
DAB  1.32 105 m2 s 1
283K , DAB2
DAB

 (C A1  C A2 )
at 298K ( Appendix, J .1)
2
T2 3 2  D ,T1
5 m
 DAB1 ( )
 1.22 10
T1
 D ,T2
s
1.22 105 m2 s 1
 N Az 
 0.1  0.02  mol  m3
3
4 10 m
 2.44 104 mol  m2 s 1
Concentration profile is the same.