# Topic 3.0 Unsteady State Diffusion ECE 2102

```Topic 3.0 Unsteady State Diffusion
by Engr. Mudono
3.1 Introduction
3.2 Unsteady State Molecular Diffusion and Fick’s
Second Law
3.3 Transient Diffusion in a Semi-infinite Medium
3.4 Diffusion through a varying cross-sectional areas
3.5 Diffusion through a spherical body
3.1 Introduction to Unsteady State Molecular Diffusion
• Unsteady-state molecular diffusion or transient diffusion
describes processes where the diffusion flux and the
concentrations change with time.
• Since solids are not easily transported through equipment
as fluids, the application of batch and semi batch
conditions arise much more frequently than with fluids.
• Even in continuous operation, e.g., a continuous drier, the
history of each solid piece as it passes through equipment
is representative of the unsteady state.
• These generally fall into two categories:
1. a process that is in an unsteady state only during its
initial startup, and
2. a process in which the concentration is continually
changing throughout its duration.
• These cases are therefore of considerable importance.
• The time-dependent differential equations are simple
to derive from the general differential equation of mass
transfer.
• The equation of continuity for component A in terms of mass:
πππ΄
π» β ππ΄ +
− ππ΄ = 0 −− −1
ππ‘
• The equation of continuity for component A in terms of moles:
ππΆπ΄
π» β π΅π΄ +
− ππ΄ = 0 −− −2
ππ‘
• Where there is no bulk flow, and in the absence of chemical
reaction, Fick's second law, can be used to solve problems of
unsteady-state diffusion by integration with appropriate
boundary conditions.
Mass transfer chart for solid objects
Dimensionless number in mass transfer chart
3.2 Unsteady State Molecular Diffusion and Fick’s
Second Law
• All mass transfer processes will have an initial period
of time with unsteady – state conditions where the
concentration at certain point varies with time until
Fig 3.1 Unsteady state molecular diffusion mass transfer
πππ‘ πππ‘π ππ πππππ ππππ€ πππ‘π
πππ‘ πππ‘π ππ πππππ ππππ€ πππ‘π
•
−
ππ’π‘ ππ π‘βπ ππππ‘πππ π£πππ’ππ
πππ‘π π‘βπ ππππ‘πππ π£πππ’ππ
=
πππ‘π ππ ππππ’πππ’πππ‘πππ ππ π ππππππ  π΄
ππ π‘βπ ππππ‘πππ π£πππ’ππ
• The molar flow rate of species A by diffusion at the plane X=X
is given by Fick's law:
ππΆπ΄
ππ΄π₯ = −π·π΄π΅ π
−− −1
ππ₯ π₯
• The molar flow rate of species A by diffusion at the plane x =
x+Δx, is:
ππΆπ΄
ππ΄π₯ = −π·π΄π΅ π
−− −2
ππ₯ π₯+βπ₯
• The accumulation of species A in the control volume is:
ππΆπ΄
π
βπ₯ −− −3
ππ₯
• Combining eqns. 1, 2 &amp; 3:
ππΆπ΄
ππΆπ΄
−π·π΄π΅ π
+ π·π΄π΅ π
ππ₯ π₯
ππ₯
π₯+βπ₯
ππΆπ΄
=π
βπ₯ −− −4
ππ₯
• Rearranging and simplifying:
π·π΄π΅
ππΆπ΄ ππ₯
− ππΆπ΄ ππ₯
βπ₯
π₯+βπ₯
π₯
ππΆπ΄
=
−− −5
ππ‘
• In the limit, as Δx→0
ππΆπ΄
π 2 πΆπ΄
= π·π΄π΅
−− −6
2
ππ‘
ππ₯
πΈππ. 6 πΉπππ ′ π  π πππππ πππ€ πππ πππ
− ππππππ πππππ ππππππ’πππ πππππ’π πππ
• For the more general three-dimensional case where
concentration gradients are changing in the x, y and z
directions, these changes must be added to give:
ππΆπ΄
=
ππ‘
2
π πΆπ΄
π·π΄π΅
2
ππ₯
+
2
π πΆπ΄
2
ππ¦
+
2
π πΆπ΄
2
ππ§
−− −7
πΉπππ ′ π  π πππππ πππ€ πππ π’ππ π‘ππππ¦ π π‘ππ‘π πππππ’π πππ
ππ π‘βπππ ππππππ‘πππ
• Examining eqn.6:
ππΆπ΄
π 2 πΆπ΄
= π·π΄π΅
−− −6
2
ππ‘
ππ₯
• πΆπ΄ : Concentration of component A (kg/m3, kmol/m3)
• t: time (s)
• DAB: mass diffusivity (m2/s)
• x: distance (m)
• We need to employ the following boundary conditions:
πΉππ π‘ = 0, πΆπ΄ = πΆπ΄0 ππ‘ 0 ≤ π₯ ≤ ∞
πΉππ π‘ = 0, πΆπ΄ = πΆπ΄π  , π‘βπ ππππ π‘πππ‘ πππππππ‘πππ‘πππ ππ‘ π₯ = 0
πΆπ΄ = πΆπ΄0 ππ‘ π₯ = ∞
• Including these three boundary conditions Fick's second law
can be solved to yield:
πΆπ΄π₯ −πΆπ΄0
πΆπ΄π  −πΆπ΄0
= 1 − πππ
π₯
2 π«π‘
−− −8
• The error function(erf) is tabulated and it is just a
mathematical function that can only be represented by an
integral, you can use it just by looking up values in a table
and interpolating.
• You will not need to calculate error functions numerically, but
π₯
2
−π¦ 2
πππ π₯ =
π
ππ¦ −− −9
∏ 0
• The error function erf(x) can also be calculated from the
infinite series:
erf π₯ = π₯ −
π₯3
3
+
1 π₯5
2! 5
−
1 π₯7
3! 7
+ β― −− −10
• However, many problems in unsteady-state diffusion can be
solved without the complication of error function calculation.
• For certain problems, one can employ a simple
relationship between the time and distance at which a
certain concentration will occur.
π₯2
1 − πππ
= ππππ π‘πππ‘ ππ
= ππππ π‘πππ‘ −− −11
π«π‘
2 π«π‘
π₯
• Factors that Influence Diffusion Rate:
1. Both the diffusing species and the host material affect D.
2. Temperature
π
− π ππ
π« = π«π π
−− −12
ππ
πππ« = πππ«π −
ππ
ππ
ππ
ππππ« = ππππ«π −
−− −13
2.303 ππ
• Therefore, a plot of lnD versus 1/T should yield a straight line with
slope -Qd/R and intercept lnD0.
• For one-dimensional diffusion in the radial direction only for
cylindrical coordinates, Fick's second law becomes:
ππΆπ΄ π·π΄π΅ π
ππΆπ΄
=
π
−− −14 (ππ¦πππππππππ)
ππ‘
π ππ
ππ
• For one-dimensional diffusion in the radial direction only for
spherical coordinates, Fick's second law becomes:
ππΆπ΄ π·π΄π΅ π
ππΆπ΄
2
= 2
π
−− −15(π πβππππππ)
ππ‘
π ππ
ππ
3.3 Transient Diffusion in a Semi-infinite Medium
• The boundary conditions for this case to solve (eqn.6) are:
• At t = 0 0 &lt; x &lt; ∞ πΆπ΄ = πΆπ΄0
t&gt;0
x = 0 πΆπ΄ = πΆπ΄π πΆπ΄π = ππππ‘πππ πππππππ‘πππ‘πππ
t&gt;0
x = ∞ πΆπ΄ = πΆπ΄0
ππΆπ΄
π 2 πΆπ΄
= π·π΄π΅
−− −6
2
ππ‘
ππ₯
• To solve the above partial differential equation, either the
method of combination of variables or the Laplace method is
applicable.
• The result, in terms of the fractional accomplished
concentration change (θ), is:
πΆπ΄ − πΆπ΄0
π₯
π=
= ππππ
−− −10
πΆπ΄1 − πΆπ΄0
2 π·π΄π΅ π‘
• Equation 10 is used to compute the concentration in the semiinfinite medium, as a function of time and distance from the
surface, assuming no bulk flow.
• Thus, it applies most rigorously to diffusion in solids, and also
to stagnant liquid and gases when the medium is dilute in the
diffusing solute.
• The instantaneous rate of mass transfer across the surface of
the medium at X = 0 can be obtained by taking the derivative
of (eqn. 10) with respect to distance and substituting it into
Fick's first law applied at the surface of the medium:
ππΆπ΄
ππ΄ = −π·π΄π΅ π
ππ§
= π·π΄π΅ π
π₯=0
πΆπ΄π − πΆπ΄0
ππ·π΄π΅ π‘
π₯2
ππ₯π −
4π·π΄π΅ π‘
− −11
π₯=0
• Thus:
ππ΄
π₯=0
=
π·π΄π΅
π πΆπ΄π  − πΆπ΄0 −− −16
ππ‘
• We can determine the total number of moles of solute, NA,
transferred into the semi-infinite medium by integrating eqn. 16
with respect to time:
π‘
ππ΄ =
ππ΄
0
π₯=0
ππ‘ =
π·π΄π΅
π πΆπ΄π  − πΆπ΄0
ππ‘
π‘
0
ππ‘
π‘
= 2π πΆπ΄π  − πΆπ΄0
π·π΄π΅ π‘
−− −17
π
Table 3.1 The Error Function
Mass transfer chart
3.3.1 Semi-infinite Medium correlations
3.4 Diffusion through a varying cross-section area
• The mole rate (π΅A,kmol/s ) through a system of a varying cross
sectional area is constant, while the mole flux (NA,kmol/m2.s ) is
variable.
• The mass transfer through a cone and sphere can be consider as a
mass transfer through a system of varying cross sectional area.
• On the other hand, the transfer through a cylinder can be consider as a
mass transfer through a system of constant cross section area.
ππππ πππ‘π
ππ΄ ππππ
ππ΄ =
=
−− −1
2
π π’πππππ ππππ
π π .π
Fig 3.2 Variable cross sectional area
3.5 Diffusion through a spherical body
ππΆπ΄ πΆπ΄
ππ΄ = −π·π΄π΅
+
ππ΄ + ππ΅ −− −2
ππ πΆπ
ππ΄
ππΆπ΄ πΆπ΄ ππ΄ ππ΅
= −π·π΄π΅
+
+
−− −3
π
ππ πΆπ π
π
ππΆπ΄ πΆπ΄
ππ΄ = −π·π΄π΅ π
+
ππ΄ + ππ΅ −− −4
ππ πΆπ
• But: The surface area of sphere: S=4πr2
ππΆπ΄ πΆπ΄
ππ΄ = −4ππ π·π΄π΅
+
ππ΄ + ππ΅ −− −5
ππ πΆπ
2
Case 1: Diffusion through a stagnant layer ππ΅ = 0
ππΆ
πΆ
π΄
π΄
2
ππ΄ = −4ππ π·π΄π΅
+
ππ΄ + 0
ππ πΆπ
ππ΄ 1 − πΆπ΄
π1
ππ΄
π0
ππΆ
π΄
2
= −4ππ π·π΄π΅ πΆπ
ππ
ππ
πΆπ − πΆπ΄2
= −4ππ·π΄π΅ πΆπ ππ
2
π
πΆπ − πΆπ΄1
4ππ·π΄π΅ πΆπ
πΆπ − πΆπ΄2
ππ΄ =
ππ
−− −6
1 1
πΆπ − πΆπ΄1
−
• The most important things is to calculate the mass transfer rate for the
π
sphere surface where the surface area is constant ππππ :
4ππ·π΄π΅ πΆπ
πΆπ − πΆπ΄2
ππ΄ π =
ππ
1 1
πΆπ − πΆπ΄1
−
π0 π1
ππ΄ πππππ
4ππ·π΄π΅ πΆπ
πΆπ − πΆπ΄2
=
ππ
1 1
πΆπ − πΆπ΄1
−
π0 π1
π·π΄π΅ πΆπ
πΆπ − πΆπ΄2
ππ΄ =
ππ
−− −7
1
πΆπ − πΆπ΄1
2 1
π0
−
π0 π1
• Mole flux from the sphere surface (eqn.7)
• When the mass transfer from surface to a large distance
compare to the sphere surface (π«π): π1 → ∞ πππ πΆπ΄2 = 0
π·π΄π΅ πΆπ
πΆπ − πΆπ΄2
ππ΄ =
ππ
1
πΆ
−
πΆ
2 1
π
π΄1
π0
−
π0 ∞
π·π΄π΅ πΆπ
πΆπ − πΆπ΄2
ππ΄ =
ππ
−− −8
π0
πΆπ − πΆπ΄1
• In partial pressure form:
ππ΄ =
π·π΄π΅ ππ
ππ −ππ΄2
ππ
π0 ππ
ππ −ππ΄1
−− −9
Case II: Equimolecular Counter Diffusion (ππ΄ = −ππ΅ ):
ππΆπ΄ πΆπ΄
ππ΄ = −π·π΄π΅
+
ππ΄ + ππ΅ −− −10
ππ πΆπ
ππ΄
ππΆπ΄ πΆπ΄ ππ΄ ππ΅
= −π·π΄π΅
+
+
−− −11
π
ππ πΆπ π
π
ππΆπ΄
ππ΄ = −4ππ π·π΄π΅
−− −12
ππ
2
π1
ππ΄
π0
ππ
=
−4ππ·
π΄π΅
2
π
πΆπ΄2
ππΆπ΄ −− −13
πΆπ΄1
1 1
ππ΄
−
= 4ππ·π΄π΅ πΆπ΄1 − πΆπ΄2 −− −14
π0 π1
4ππ·π΄π΅
ππ΄ =
πΆπ΄1 − πΆπ΄2
1 1
−
π0 π1
• For the mass transfer from surface (S=4ππ02 ):
π·π΄π΅ πΆπ
ππ΄ =
1
2 1
π0
−
π0 π1
• In the case of π1 is very large ⇒
1
π1
πΆπ΄1 − πΆπ΄2
=0
π·π΄π΅
ππ΄ =
πΆπ΄1 − πΆπ΄2
π0
• In the form of partial pressure:
π·π΄π΅
ππ΄ =
ππ΄1 − ππ΄2 −− −15
π0 ππ
Case III: Unequimolecular Counter Diffusion (ππ΄ = −πππ΅ ):
π·π΄π΅ ππ
1
ππ − (1 − π)ππ΄2
ππ΄ =
ππ
−− −16
ππ π0 (1 − π)
ππ − (1 − π)ππ΄1
```