14–41
14-67 Pure N2 gas is flowing through a rubber pipe. The rate at which N2 leaks out by diffusion is to be determined for the
cases of vacuum and atmospheric air outside.
Assumptions 1 Mass diffusion is steady and one-dimensional since the nitrogen concentration in the pipe and thus at the
bar0.79=kPa79)kPa100)(79.0(
22 NN === PyP
bar0.21=kPa21)kPa100)(21.0(
22 OO === PyP
When solubility data is available, the molar flow rate of a gas
N2 diffusion
Vacuum
14–42
14–43
14–69 A balloon is filled with helium gas. A relation for the variation of pressure in the balloon with time as a result of mass
transfer through the balloon material is to be obtained, and the time it takes for the pressure in the balloon to drop from 120 to
100 kPa is to be determined.
Assumptions 1 The pressure of helium inside the balloon remains nearly constant. 2 Mass diffusion is transient since the
conditions inside the balloon change with time. 3 Mass diffusion is one-dimensional since the helium concentration in the
pure helium gas at 120 kPa, the initial partial pressure of helium in the balloon is 120 kPa.
When permeability data is available, the molar flow rate of a gas through a solid wall of thickness L under steady
one-dimensional conditions can be determined from Eq. 14-29,
L
P
A
L
PP
AN AB
A,2A,1
ABwallA,diff, PP =
−
=
(kmol/s)
L
P
AN
dt
dN
ABwallA,diff, P−=−=
(2)
Substituting (2) into (1),
ATR
P
TR
dN
TR
dP uuu
AB
P
He
diffusion
He
25C
120 kPa
Air
Balloon
14–44
14–70C Excess moisture changes the dimensions of wood, and cyclic changes in dimensions weaken the joints, and can
14–71C The condensation or freezing of water vapor in the wall increases the thermal conductivity of the insulation material,
14–72C Vapor barriers are materials that are impermeable to moisture such as sheet metals, heavy metal foils, and thick
14–73C Insulations on chilled water lines are always wrapped with vapor barrier jackets to eliminate the possibility of vapor
14–74C A tank that contains moist air at 3 atm is located in moist air that is at 1 atm. The driving force for moisture transfer
14–75C When the temperature, total pressure, and the relative humidity are given, the vapor pressure can be determined from
14–76C The mass flow rate of water vapor through a wall of thickness L in therms of the partial pressure of water vapor on
both sides of the wall and the permeability of the wall to the water vapor can be expressed as
PP
14–45
14-77 A glass of milk left on top of a counter is tightly sealed by a sheet of 0.009-mm thick aluminum foil. The drop in the
level of the milk in the glass in 12 h due to vapor migration through the foil is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Mass transfer through the foil is one-dimensional. 3 The vapor
permeability of the foil is constant.
Properties The permeance of the foil to water vapor is given to be 2.910-12 kg/s.m2.Pa. The saturation pressure of water at
the two sides of the foil.
The diffusion area of the foil is
222 m 0113.0)m 06.0( ===
rA
.
kg1021.1s) 3600kg/s)(1210(2.79 -611
12,=== −
−tmm vhv
3936 m 1021.1)kg/m 1000/()kg 1021.1(/ −− ===
RH=50%
Moisture
migration
14–46
14–78 The wall of a house is made of a 20-cm thick brick. The amount of moisture flowing through the wall in 24-h is to be
determined.
Assumptions 1 Steady operating conditions exist. 2 Mass transfer through the wall is one-dimensional. 3 The vapor
permeability of the wall is constant.
Properties The permeance of 100 mm thick wall is 4610-12
Pamkg/s 1046
13
=
−
The mass flow rate of water vapor through a plain layer of
thickness L and normal area A is given as (Eq. 14–31)
L
PP
A
L
PP
Am vv
v
sat,221,sat1,21,
−
=
−
=PP
where P is the vapor permeability, is the relative humidity and Psat is the
saturation pressure of water at the specified temperature. Subscripts 1 and 2
denote the states of the air on the two sides of the roof. Substituting, the mass
flow rate of water vapor through the wall is determined to be
2813 mkg/s1088.3
m 20.0
]Pa) 3169(40.0)Pa 7384(40.0[
)kg/s.m.Pa1046(=
−
= −−
v
m
Then the total amount of moisture that flows through the roof during a 24-h period becomes
g 3.35==== −
−kg 0.00335s) 3600)(24mkg/s10(3.88 28
24,tmm vhv
Brick
Vapor
diffusion
Room
25ºC
RH=40%
RH=40%
14–47
14–79 The roof of a house is made of a 30-cm thick concrete layer. The amount of water vapor that will diffuse through a 15
m 8 m section of the roof in 24-h is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Mass transfer through the roof is one-dimensional. 3 The vapor
permeability of the roof is constant.
Properties The permeability of the roof to water vapor is given to be 24.710-12
14–48
14–80 Prob. 14-79 is reconsidered. The effects of temperature and relative humidity of air inside the house on the
amount of water vapor that will migrate through the roof are to be investigated.
L=0.30 [m]
T_1=25 [C]
phi_1=0.50
P_atm=100 [kPa]
T1
[C]
mv
[kg]
10
12
14
16
18
20
22
24
26
28
30
0.33
0.4046
0.4883
0.5821
0.6871
0.8043
0.9349
1.08
1.242
1.421
1.619
10 14 18 22 26 30
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
T1 [C]
mv [kg]
14–49
1
mv
[kg]
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
0.6176
0.6717
0.7259
0.78
0.8341
0.8882
0.9423
0.9965
1.051
1.105
1.159
1.213
1.267
1.321
1.375
1.429
1.484
1.538
1.592
1.646
1.7
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
0.6
0.8
1
1.2
1.4
1.6
1.8
1
mv [kg]
14–50
14–51
14–82 The inside wall of a house is finished with 9.5-mm thick gypsum wallboard. The maximum amount of water vapor that
will diffuse through a 3 m 8 m section of the wall in 24-h is to be determined.
14-10). The saturation pressure of water at 20ºC is 2339 Pa (Table 14-9).
Analysis The mass flow rate of water vapor through a plain
layer of thickness L and normal area A is given as (Eq. 14-31)
,21,
L
PP
Am vv
v
−
=
P
Plaster
board
14–52
14–83 The inside wall of a house is finished with 9.5-mm thick gypsum wallboard with a 0.051-mm thick polyethylene film
on one side. The maximum amount of water vapor that will diffuse through a 3 m 8 m section of the wall in 24-h is to be
14-9).
Analysis The mass flow rate of water vapor through a two–layer plain wall
of normal area A is given as (Eqs. 14-33 and 14-35)
sat,221,sat1
,21,
vv
PP
PP
−
−
humidity and Psat is the saturation pressure of water at the specified
temperature. Subscripts 1 and 2 denote the air on the two sides of the wall.
g 26.4=kg 0.0264=== −
−s) 3600kg/s)(241006.(3 7
24,tmm vhv
Plaster
board
Room
Outdoors
9.5 mm
14–53
Transient Mass Diffusion
14–84C The diffusion of a solid species into another solid of finite thickness can usually be treated as a diffusion process in a
14–85C When the density of a species A in a semi-infinite medium is known initially and at the surface, the concentration of
the species A at a specified location and time can be determined from
−
x
CtxC
iAA
),(
,
14–86C The penetration depth is defined as the location where the tangent to the concentration profile at the surface (x = 0)
intercepts the
iAA CC ,
=
line, and it represents the depth of diffusion at a given time. The penetration depth can be
diff
14–54
14–55
14–56
14–89 A piece of steel undergoing a decarburization process, the depth below the surface of the steel at which the
concentration of carbon is reduced to 40% from its initial value as a result of the decarburization process for (a) an hour and
(b) ten hours are to be determined.
Assumptions Carbon penetrates into a very thin layer beneath the surface of the component, and thus the component can be
14–57
14–58
14–91 A thick nickel wall is exposed to pure hydrogen gas on one side of the surface. The hydrogen concentrations
varying with time are to be evaluated at different distances from the surface.
D_AB=1.0e-11 [m^2/s]
c_2=SemiInf1(c_i,c_s,D_AB,x_2,time)
150000 0.7728 0.5637 0.2482
Discussion Hydrogen concentration increases with time as it diffuses through the nickel wall.
0.2
0.4
0.6
CA(x,t) / CA,s
x = 2 mm
14–59
14–60
14–93 Chemical resistant gloves of 0.67 mm thickness are used for handling tetrachloroethylene solution. The time it
will take for the solution to penetrate to the inner glove surface at 1% of the concentration at the outer surface is to be
determined.
Assumptions 1 The glove wall can be modeled as a semi-infinite medium. 2 The solution concentration in the glove is
=
=
tD
x
tD
x
C
txC
AB
AB
sA
A
2
erfc01.0
2
erfc
),(
,
82.1
2
=
tD
x
AB
s 1.13=
=
=
−/sm 100.3
1
2)82.1(
m 00067.0
1
2)82.1(
28
2
2
AB
D
x
t
Discussion It only takes the tetrachloroethylene solution about 1 second for 1% of the concentration at the exposed outer