14–61
14–62
14–63
14–64
14–65
14–98 A steel component is to be surface hardened by packing it in a carbonaceous material in a furnace at 500 K. The length
of time the component should be kept in the furnace is to be determined.
Assumptions 1 Carbon penetrates into a very thin layer beneath the surface of the component, and thus the component can be
modeled as a semi-infinite medium regardless of its thickness or shape. 2 The initial carbon concentration in the steel
500 K
==
−
−
tD
x
AB
2
erfc22.0
0010.0011.0
0010.00032.0
8674.0
2
=
tD
x
AB
( )
( )
220
2
8674.0/s)m101.2(4
8674.04 AB
D
Therefore, the steel component must be held in the furnace forever to achieve the desired level of hardening.
Discussion The diffusion coefficient of carbon in steel increases exponentially with temperature, and thus this process is
commonly done at high temperatures to keep the diffusion time to a reasonable level.
Carbon
Steel part
14–66
14–67
14-100 A piece of steel was exposed to a carburizing atmosphere for an hour, and the percentage of mass concentration of
carbon at 0.2 mm and 0.4 mm below the surface are to be determined.
Assumptions Carbon penetrates into a very thin layer beneath the surface of the component, and thus the component can be
modeled as a semi-infinite medium regardless of its thickness or shape.
−
−)s 3600)(/sm 101(2
002.0007.0
211
002.0007.0
002.0),( =
−
−txwA
Thus, mass concentration of carbon at x = 0.2 mm and t = 1 h is
( )
054.1erfc
002.0007.0
002.0),( =
−
−txwA
Thus, mass concentration of carbon at x = 0.4 mm and t = 1 h is
14–68
14–69
Diffusion in a Moving Medium
14-102C The diffusion velocity at a location is the average velocity of a group of molecules at that location moving under
the influence of concentration gradient. The average velocity of a species in a moving medium is equal to the sum of the bulk
14-103C The mass-average velocity of a medium at some location is the average velocity of the mass at that location
relative to an external reference point. The molar-average velocity of a medium at some location is the average velocity of
14-104C The mass-average velocity of a medium at some location is the average velocity of the mass at that location
relative to an external reference point. It is the velocity that would be measured by a velocity sensor such as a pitot tube, a
14-106C The diffusion of a vapor through a stationary gas column is called the Stefan flow. The Stefan’s law can be
expressed as
LA
AB
y
DC
,
1
−
14–70
14–107 The amount of chloroform that diffuses from a Stefan tube at a specified temperature and pressure over a specified
time period is measured. The mass diffusivity of chloroform in air is to be determined.
263.0
atm 1
atm 263.0
0,
0, === P
P
yA
A
3
3kmol/m 0409.0
K)25273)(K/kmolmkPa 314.8(
kPa 325.101 =
+
== RT
P
C
14–71
14-108E The amount of water that evaporates from a Stefan tube at a specified temperature and pressure over a specified
time period is measured. The diffusion coefficient of water vapor in air is to be determined.
Assumptions 1 Water vapor and atmospheric air are ideal gases. 2 The amount of air dissolved in liquid water is negligible.
14–72
14–73
14–74
14-111 Methanol undergoes evaporation in a vertical tube. (a) The evaporation rate of methanol is to be determined,
and (b) the mole fraction of methanol vapor as a function of the tube height is to be plotted.
DAB = 1.62 × 10−5 m2/s.
14–75
14–76
14-112 A hydrogen tank is maintained at atmospheric temperature and pressure by venting to the atmosphere through the
charging valve. The initial mass flow rate of hydrogen out of the tank is to be determined.
Assumptions 1 Steady operating conditions at initial conditions exist. 2 Hydrogen and atmospheric air are ideal gases. 3 No
/sm1018.8
88.0
102.7
atm)(in
25
5
atm 1, −
−
=
== P
D
DAB
AB
( )
( )
( )
kg/s104.2 8−− === kg/kmol2kmol/s10081.2 8
H
H2
2MNm
14–77
14–78
14–79
14–80
14-116EE The pressure in a helium pipeline is maintained constant by venting to the atmosphere through a long tube. The
mass flow rates of helium and air, and the net flow velocity at the bottom of the tube are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Helium and atmospheric air are ideal gases. 3 No chemical reactions
occur in the tube. 4 Air concentration in the pipeline and helium concentration in the atmosphere are negligible so that the
Noting that the pressure of helium is 14.5 psia at the bottom of the tube
(x = 0) and 0 at the top (x = L), its molar flow rate is
lbmol/s 10642.5
ft 30
psia )05.14(
R) 540/lbmol.R)(psia.ft (10.73
)ft 10727.8(/s)ft 1075.7(
11
3
2424
,0,
Adiff,helium
−
−−
=
−
=
−
== L
PP
TR
AD
NN LAA
u
AB
Therefore, the mass flow rate of helium through the tube is
lbm/s 102.26 10−− === lbm/lbmol) lbmol/s)(4 10642.5()( 11
heliumhelium MNm
which corresponds to 0.00712 lbm per year.
01034.2
lbm/s )1064.11026.27(
lbm/s 101.64 10
910
9
total
air
air =
+−
== −
−−
−
m
m
w
3
3
helium lbm/ft 00201.0
R) 0/lbm.R)(54psia.ft (2.681
psia 5.14 === RT
P
He
Air
Helium, 80F
14.5 psia
7 lbm/s
30 ft
0.4 in.