Diffusion to a Cylinder
Consider a uniform flow with velocity U and concentration co approaching a
cylinder of radius a. . For a low Reynolds number condition (with Re << 1), the stream
function near the cylinder is given by
r
r
a
ψ = AUa sin θ [ (2 ln − 1) + ] ,
r
a
a
(1)
where
A=
1
.
2(2 − ln Re )
(2)
The diffusion equation is given by
v θ ∂c
∂c
∂ 2 c 1 ∂c
+ vr
= D( 2 +
).
r ∂θ
∂r
r ∂r
∂r
(3)
a
Figure 1. Schematics of a flow around a circular cylinder.
The boundary conditions for an absorbing surface are given as
d
r=a+ ,
2
r = ∞,
c=0
(4)
c = c∞
where d is the particle diameter.
Using x, y coordinate as shown in Figure 2, we find
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u
∂c
∂c
∂ 2c
+v =D 2 ,
∂x
∂y
∂y
y
(5)
x
a
Figure 2. Surface coordinate system.
The boundary conditions given by (4) for small aerosols reduce to
y = 0,
y = ∞,
c=0
c = c∞
(6)
Introducing the stream function ψ
u=
∂ψ
∂ψ
,
, v=−
∂y
∂x
(7)
and using x and ψ as independent variables, one finds
∂c
∂
∂c
= D [u ] .
∂x
∂ψ ∂ψ
(8)
When the concentration boundary layer is thin, the stream function given by (1) may be
approximated as
ψ ≈ 2AaUy12 sin x 1 ,
(9)
where
y1 =
y
x
, x1 = .
a
a
(10)
Here
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u=
8AU
∂ψ
= (
) sin 1 / 2 x 1ψ1 / 2
a
∂y
(11)
Equation (8) may be restated as
∂c
D ∂
∂c
=
(ψ11 / 2
),
∂χ aAU ∂ψ1
∂ψ1
(12)
where
χ = ∫ sin 1 / 2 x 1dx 1 , ψ1 =
ψ
.
2AaU
(13)
The boundary conditions are now given by
ψ1 = 0,
c=0
(14)
ψ 1 = ∞, c = c ∞
Introducing the similarity variable
ξ=
ψ1
,
χ2/3
(15)
Equation (12) may be restated as
−
APe dc d 1 / 2 dc
).
ξ = (ξ
3 dξ dξ
dξ
(16)
The solution to Equation (16) is given by
c ∞ (APe )1 / 3
c=
1.45
ξ
2
∫ exp{− 9AP z }dz ,
3
(17)
e
0
where
Pe =
2Ua
= R e ⋅ Sc
D
(18)
is the Peclet number.
The averaged Sherwood number is given by
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sh =
h (2a )
= 1.17(APe )1 / 3 ,
D
(19)
where h is the averaged mass transfer coefficient. The collection efficiency is defined as
the fraction of the particles collected from the fluid in the volume swept by the cylinder.
That is
ηR =
hπ(2a )c ∞
= 3.68A1 / 3 Pe−2 / 3 .
(2a ) Uc ∞
(20)
Since Pe = 2aU / D , it follows that η R ~ d −2 / 3 (or d −4 / 3 for free molecular regime).
Variations of sh / A1 / 3 and η R / A1 / 3 with Pe are shown in Figure 3.
100
Sh/A1/3
10
1
0
η/A1/3
0
0
1E-1
1E0
1E1
1E2
1E3
1E4
Pe
Figure 3. Variations of sh / A1 / 3 and η R / A1 / 3 with Pe .
Direct Interception Limit
For Pe → ∞ (absence of diffusion), particles follow the fluid and deposit when a
streamline passes within one radius of the surface. This is referred to as the direct
interception. Schematics of interception process are shown in Figure 4.
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d
U
a
Figure 4. Schematics of interception process.
The capture efficiency then is given by
π/2
∫v|
y =d / 2
ηR = −
dx
= 2AR 2 ,
0
Ua
(21)
where
R=
d
,
2a
(22)
is the interception parameter.
Equations (20) and (21) are for the limiting conditions that the diffusion or the
direct interception are dominant. They show that for fixed U and fiber diameter, the
efficiency decreases with particle diameter d at first and then increases with further
increase in d.
Fiber Efficiency/Filter Efficiency
The efficiency of a fibrous filter is directly related to the single fiber efficiency.
For a regular array shown in Figure 5, the decrease of concentration in a length dz is
given by
υdz
dc = − [ηR ( 2a )c] ( 2 ) ,
1424
3
π2
a3
Collection by one fiber 1
(23)
No. of fibers
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dz
C1
C
C-dC
C2
Figure 5. Schematics of gas flow through a fiber filter.
where υ is the fraction of solid (fibers). Rearranging and integrating (23), it follows that
ηR =
πa
c
ln 1 ,
2υL c 2
(24)
where L is the length of the filter. For ηR the following semi-empirical equation was
suggested (Hinds, 1982):
η r (RPe ) = 1.3RPe1 / 3 + 0.7(RPe1 / 3 ) 3 ,
(25)
which correlates well with the data as shown in Figure 6. Note also that Equation (25)
reduces to (21) as Pe → ∞ and resembles (20) for Pe → 0 . For Re=0.1, the predictions
of Equations (20) and (21) are shown in Figure 6 for comparison. It is seen that the
prediction of Equation (20) for the diffusion limit is in good agreement with the result of
(25) for small Peclet numbers. Similarly the prediction of Equation (21) for the
interception limit asymptotes the empirical Equation (25) at large Peclet numbers.
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1E+06
Eq (25)
1E+04
Eq (21)
1E+02
Eq (20)
1E+00
1E-02
1E-04
0.1
1
10
100
RPe^(1/3)
Figure 6. Variation of filter collection efficiency ηr RPe with RPe1/ 3 .
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