4 Mass transport
4.1 Analogy between mass transport and heat transport
There exist analogies between mass and heat transport.
For the convective transport of both heat ϕq [J s−1 ] and mass ϕm [kg s−1 ]:
ϕq = ϕv · ρcp T, (4.1)
ϕm = ϕv · c. (4.2)
For the convective fluxes of both heat ϕ′′q [J m−2 s−1 ] and mass ϕ′′m [kg m−2 s−1 ]:
ϕ′′q = v · ρcp T, (L14)
ϕ′′m = v · c. (L14)
For the diffusion coefficients of both heat and mass: a [m2 s−1 ], D [m2 s−1 ].
For the diffusive fluxes of both heat ϕ′′q [J m−2 s−1 ] and mass ϕ′′m [kg m−2 s−1 ]:
dT d(ρcp T )
ϕ′′q = −λ = −a (Fourier’s law), (4.3)
dx dx
dc
ϕ′′m = −D (Fick’s law). (4.4)
dx
Fick’s law only applies to binary (so only two substances) systems, and gives a poor
description for polar molecules. The analogy with heat transport therefore does not
hold under every condition.
4.2 Mutual diffusion based on the analogy with heat trans-
port
Mass flow and a driving force can be linked similarly to Newton’s law of cooling.
For a substance A,
ϕm,A = kA∆cA , (4.30)
where k [m s−1 ] is the mass transfer coefficient.
This equation is entirely analogous with heat transport, where 1/k can now be
interpreted as the resistance to mass transport.
Between two flat plates a distance D apart:
D
k= ;
D
for annular space between two cylinders of radii D1 and D2 :
2D
k= ;
D2 ln(D2 /D1 )
and for a sphere in an infinite medium of radius D:
2D
k= .
D
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,In non-dimensional form, the ratio of the convective mass transport and the diffusive
mass transport is known as the Sherwood number (Sh):
kD
Sh = , (4.31)
D
which is analogous to the Nusselt number (Nu).
Similarly for the Prandtl number (Pr), we have the analoglous Schmidt number (Sc):
ν
Sc = . (L14)
D
For mass, we can also analogously to heat work with penetration theory. The
penetration depth is now defined as
√
δ(t) = πDt, (L14)
which again is only valid for a layer/slab of thickness D if δ(t) < 0.6D. With help
of the Fourier number:
Dt
Fo = < 0.1. (4.35)
D2
And, for double-heated layers/slabs or cylindrical/spherical bodies:
Fo < 0.03.
It is then also clear that
r
D D D
ϕ′′m = k∆c = ∆c = √ ∆c = ∆c, (L14)
D πDt πt
which implies that k is dependent on time:
r
D
k(t) = . (4.36)
πt
With the penetration depth δ(t) only having a significantly changed concentration,
it follows that the overall concentration difference is independent on time:
∆c = c1 − c0 ̸= f (t).
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, Similarly, we speak of long-term diffusion into a layer of a stagnant medium or a
solid material if: Fo > 0.1 for diffusion processes from/towards a single boundary
plane; and if Fo > 0.03 for diffusion processes concerning double-sided diffusion
from/towards a slab or a cylindrical/spherical body.
We again work with a mean concentration ⟨c⟩, which yields
D
ϕ′′m = k(c1 − ⟨c⟩) = Sh (c1 − ⟨c⟩),
D
where k is now independent on time and therefore is constant:
D
k = Sh ̸= f (t). (L14)
D
So for long periods of time, we again find
For a flat slab: Sh = 4.93;
For a long cylinder: Sh = 5.8;
For a sphere: Sh = 6.6.
The concentration difference after long times, so Fo > 0, 03, is then also dependent
on time:
∆c = T1 − ⟨T ⟩ = f (t).
Again, the driving force for diffusion can also be described by the concentration at
the centre of a body cc by
ϕ′′m = k(c1 − cc ).
The exact solutions for the total diffusion process for a number of finite-size objects
are shown in TPDC ”Fourier Instationary Heat and Mass Transfer” (p. 90-92) for
the ratios
c1 − cc c1 − ⟨c⟩
and .
c1 − c0 c1 − c0
Note that for smaller values of Fo (e.g. Fo < 0.03), these graphs lead to inaccurate
results and it is therefore better to use penetration theory.
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4.1 Analogy between mass transport and heat transport
There exist analogies between mass and heat transport.
For the convective transport of both heat ϕq [J s−1 ] and mass ϕm [kg s−1 ]:
ϕq = ϕv · ρcp T, (4.1)
ϕm = ϕv · c. (4.2)
For the convective fluxes of both heat ϕ′′q [J m−2 s−1 ] and mass ϕ′′m [kg m−2 s−1 ]:
ϕ′′q = v · ρcp T, (L14)
ϕ′′m = v · c. (L14)
For the diffusion coefficients of both heat and mass: a [m2 s−1 ], D [m2 s−1 ].
For the diffusive fluxes of both heat ϕ′′q [J m−2 s−1 ] and mass ϕ′′m [kg m−2 s−1 ]:
dT d(ρcp T )
ϕ′′q = −λ = −a (Fourier’s law), (4.3)
dx dx
dc
ϕ′′m = −D (Fick’s law). (4.4)
dx
Fick’s law only applies to binary (so only two substances) systems, and gives a poor
description for polar molecules. The analogy with heat transport therefore does not
hold under every condition.
4.2 Mutual diffusion based on the analogy with heat trans-
port
Mass flow and a driving force can be linked similarly to Newton’s law of cooling.
For a substance A,
ϕm,A = kA∆cA , (4.30)
where k [m s−1 ] is the mass transfer coefficient.
This equation is entirely analogous with heat transport, where 1/k can now be
interpreted as the resistance to mass transport.
Between two flat plates a distance D apart:
D
k= ;
D
for annular space between two cylinders of radii D1 and D2 :
2D
k= ;
D2 ln(D2 /D1 )
and for a sphere in an infinite medium of radius D:
2D
k= .
D
23
,In non-dimensional form, the ratio of the convective mass transport and the diffusive
mass transport is known as the Sherwood number (Sh):
kD
Sh = , (4.31)
D
which is analogous to the Nusselt number (Nu).
Similarly for the Prandtl number (Pr), we have the analoglous Schmidt number (Sc):
ν
Sc = . (L14)
D
For mass, we can also analogously to heat work with penetration theory. The
penetration depth is now defined as
√
δ(t) = πDt, (L14)
which again is only valid for a layer/slab of thickness D if δ(t) < 0.6D. With help
of the Fourier number:
Dt
Fo = < 0.1. (4.35)
D2
And, for double-heated layers/slabs or cylindrical/spherical bodies:
Fo < 0.03.
It is then also clear that
r
D D D
ϕ′′m = k∆c = ∆c = √ ∆c = ∆c, (L14)
D πDt πt
which implies that k is dependent on time:
r
D
k(t) = . (4.36)
πt
With the penetration depth δ(t) only having a significantly changed concentration,
it follows that the overall concentration difference is independent on time:
∆c = c1 − c0 ̸= f (t).
24
, Similarly, we speak of long-term diffusion into a layer of a stagnant medium or a
solid material if: Fo > 0.1 for diffusion processes from/towards a single boundary
plane; and if Fo > 0.03 for diffusion processes concerning double-sided diffusion
from/towards a slab or a cylindrical/spherical body.
We again work with a mean concentration ⟨c⟩, which yields
D
ϕ′′m = k(c1 − ⟨c⟩) = Sh (c1 − ⟨c⟩),
D
where k is now independent on time and therefore is constant:
D
k = Sh ̸= f (t). (L14)
D
So for long periods of time, we again find
For a flat slab: Sh = 4.93;
For a long cylinder: Sh = 5.8;
For a sphere: Sh = 6.6.
The concentration difference after long times, so Fo > 0, 03, is then also dependent
on time:
∆c = T1 − ⟨T ⟩ = f (t).
Again, the driving force for diffusion can also be described by the concentration at
the centre of a body cc by
ϕ′′m = k(c1 − cc ).
The exact solutions for the total diffusion process for a number of finite-size objects
are shown in TPDC ”Fourier Instationary Heat and Mass Transfer” (p. 90-92) for
the ratios
c1 − cc c1 − ⟨c⟩
and .
c1 − c0 c1 − c0
Note that for smaller values of Fo (e.g. Fo < 0.03), these graphs lead to inaccurate
results and it is therefore better to use penetration theory.
25
