SME
Transport Phenomena · Problem type
Evaporation through a stagnant gas
Use this for “Water at 30 °C (4.25 kPa) evaporates up a 15 cm tube into dry air at 101.3 kPa, D = 2.7 × 10⁻⁵ m²/s. What is the flux?”
Diagram
A diffuses up through gas that stays put, so it is carried by its own flow too: the curve bends from the straight dashed line.
NA
=
(
cD
L
)
ln(
1 − xA2
1 − xA1
)
NA
=
0.0003378
mol/(
m
2
·s)
This is an example. Type your own values to start a new problem.
c (mol/m³)
=
1,000
×
P (kPa)
8.314
×
T (K)
xA1
=
Pₛₐₜ (kPa)
P (kPa)
NA (mol/(m²·s))
=
c (mol/m³)
×
DAB (m²/s)
L (m)
×
ln
1
−
1
−
Formulas
c
=
1,000
×
P
8.314 ×
T
40.88
=
1,000
×
101.325
8.314 × 298.15
xA1
=
Pₛₐₜ
P
0.03129
=
3.17
101.325
NA
=
(
c
×
DAB
L
)
×
ln(
1 − xA2
1 − xA1
)
3.3781 ×
10
-4
=
(40.88
×
2.6 ×
10
-5
0.1
)
×
ln(
1 − 0
1 − 0.03129
)
Assumptions
•
Steady; the air doesn’t dissolve in the liquid, so it stands still in the tube.
•
Ideal gas, c = P ÷ (RT) with R = 8.314 J/(mol·K); P in kPa, so × 1000 for pascals.
•
The vapour at the surface is at its vapour pressure; T and P are uniform.
Step-by-step
WE KNOW
P = 101.325 kPa, T = 298.15 K, Pₛₐₜ = 3.17 kPa, xA2 = 0, DAB = 2.6 × 10⁻⁵ m²/s, L = 0.1 m
FIND
total concentration (c), mole fraction at the surface (xA1), molar flux (NA)
Step 1 · Find total concentration (c)
Use c = 1,000 × P ÷ (8.314 × T)
The
ideal
gas
law,
n
V
=
P
RT
,
with
P
in
pascals.
c
=
1,000
×
101.325
8.314 × 298.15
c
=
101,325
2,478.8
c
=
40.88
mol/
m
3
Step 2 · Find mole fraction at the surface (xA1)
Use xA1 = Pₛₐₜ ÷ P
The vapour’s share of the pressure just above the liquid.
xA1
=
3.17
101.325
xA1 = 0.03129
Step 3 · Find molar flux (NA)
Use NA = (c × DAB ÷ L) × ln((1 − xA2) ÷ (1 − xA1))
Fick’s law with the vapour carrying itself along: the still air turns the drop into a log.
NA
=
(40.88
×
2.6 ×
10
-5
0.1
)
×
ln(
1 − 0
1 − 0.03129
)
NA
=
0.010628
×
ln(
1
0.96871
)
NA = 0.010628 × 0.031785
NA
=
3.3781 ×
10
-4
mol/(
m
2
·s)
Check
Put the numbers back in. Both sides should match.
40.88
=
1,000
×
101.325
8.314 × 298.15
✓
0.03129
=
3.17
101.325
✓
3.3781 ×
10
-4
=
(40.88
×
2.6 ×
10
-5
0.1
)
×
ln(
1 − 0
1 − 0.03129
)
✓
Topic check
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