Supplemental Information
Generalized TEMOM Scheme for Solving the Population
Balance Equation
Mingzhou Yu1,2 , Yueyan Liu1, Jianzhong Lin1*, Martin Seipenbusch3
1 China Jiliang University, Hangzhou 310028, China
2 Key Laboratory of Aerosol Chemistry and Physics, Chinese Academy of Science, Xi’an, China
3Institute for Mechanical processes Engineering and Mechanics, Karlsruhe Institute of Technology,
Germany
Supplementary Information I-Derivation of the generalized TEMOM scheme in the free
molecular regime
At first, it needs to point out the generalized TEMOM equations, including Equation (5) in the
text and in Supplementary Information I and II below, are universal, which have the
uncharged form regardless how order of Taylor expansion series is reserved, thus it is
convenient to be used. We provided its program code as a function along with this paper for
its convenient application. The readers only need to call this function when solving population
balance equation.
Similar to the derivation in the continuum regime shown in the text, we can easily derive the
ODEs for moments in the free molecular regime using the same way. In this regime, the
coagulation kernel is,
1
1
1
2
1
1
2
𝛽(𝑣, 𝑣 ′ ) = 𝐵1 ( + ) (𝑣 3 + 𝑣′3 )
𝑣
𝑣′
3
(S1)
1
6
where 𝐵1 = (4𝜋) (6𝑘𝑏 𝑇/𝜌)1/2,𝑘𝑏 is the Boltzmann constant, 𝑇 is the gas temperature and 𝜌
is the mass density of the particles. In this derivation, Equation (S1) is firstly written as,
𝛽(𝑣, 𝑣
′)
1
1
1
2
1
1
1
1
= 𝐵1 (𝑣 + 𝑣 ′ ) (𝑣 3 + 𝑣′3 )2 = 𝐵1 (𝑣 + 𝑣 ′ )2 (𝑣 6 𝑣 ′
−
1
2
1
1
1
−
1
+ 2𝑣 −6 𝑣 ′ 6 + 𝑣 2 𝑣 ′ 6 )
(S2)
As Equation (S2) is introduced into Equation (2) in the text, then we write κ(𝑣, 𝑣 ′ , 𝑘) in Equation
(2) in the text as follows,
𝑘
𝜅(𝑣, 𝑣 ′ , 𝑘) = 𝐵1 [(𝑣 + 𝑣 ′ )𝑘 − 𝑣 𝑘 − 𝑣 ′ ](𝑣 + 𝑣1 )1⁄2 𝜑
𝑘
= 𝐵1 [(𝑣 + 𝑣 ′ )𝑘+1⁄2 − (𝑣 𝑘 + 𝑣 ′ )(𝑣 + 𝑣1 )]𝜑
𝑘
= 𝐵1 [𝐴1 − (𝑣 𝑘 + 𝑣 ′ )𝐴2]𝜑(S3)
where𝜑 = (𝑣 1⁄6 𝑣1−1
⁄2
+ 2𝑣 −1⁄6 𝑣1−1
⁄6
(S3)
⁄
+ 𝑣 −1⁄2 𝑣11 6), 𝐴1 = (𝑣 + 𝑣′)𝑘+1⁄2 , and 𝐴2 = (𝑣 + 𝑣1 )1⁄2 .
Here, we need to change Equation (S3) to be discrete polynomials for the closure of moment
equations. To achieve this, we need to expand two terms, namely 𝐴1and 𝐴2, around average
volume (𝑢0 ) with three orders, the same as the solution in the continuum regime. Finally we give
the ODEs for both k-th moments in this regime,
𝑑𝑚𝑘
𝑑𝑡
=
1
1
2
3⁄2
𝑚
(𝑚 1 )
0
𝑚1
((
𝑚20
− 4(
𝑚0
𝑚1
𝑚0
3⁄2
)
1
)
1
𝑘+
𝑚20 2 2
+ 2 (𝑚−1 𝑚1 +
2
6
1
𝑚20 (𝑘 − ) (𝑚5 + 𝑚7 𝑚1 + 𝑚11 𝑚−1 + 𝑚13 𝑚−1 + 𝑚3 𝑚1 )2
2
2 6
2 2 6
6
2
6
6
2
6
3
2
1
2
1
3
2
2
1
1
𝑚1
(𝑘 + ) (𝑘 − ) (𝑚5 𝑚−1 + 𝑚7 𝑚−1 + 𝑚1 𝑚1 ) (
𝑚
3
1 2
𝑚2 1 ) ( )
−
𝑚0
6
6
6
1
𝑚20 (𝑘 − ) 2
2
1
𝑘+
2
2
6
2
3
𝑚1
2
𝑚0
(𝑘 − ) (
2
6
2
𝑚0
1
𝑘+
2
1
𝑚1
2
𝑚0
(𝑘 + ) (
3
𝑘−
)
2
1
𝑘−
)
2
1
𝑘+
)
2
1
1
1
1
1
+ √2 ((−3𝑚−1 𝑚𝑘+1 − 6𝑚−1 𝑚𝑘−1 − 3𝑚1 𝑚𝑘−1 ) 𝑚21 − 6𝑚0 (𝑚5 𝑚𝑘−1 + 𝑚7 𝑚𝑘−1 + 𝑚𝑘+5 𝑚−1 + 𝑚𝑘+7 𝑚−1 + 𝑚𝑘+1 𝑚1 + 𝑚1 𝑚𝑘+1 ) 𝑚1
2
2 6
2
2
2 2
2
6
6
6
6
2
6
2
6
6
2
2
6
6
6
6
1
1
1
1
1
1
1
1
+ 𝑚20 (𝑚5 𝑚𝑘+5 + 𝑚 13 𝑚−1 + 𝑚𝑘+2 𝑚1 + 𝑚𝑘+1 𝑚7 + 𝑚11 𝑚𝑘−1 + 𝑚13 𝑚𝑘−1 + 𝑚1 𝑚𝑘+7 + 𝑚𝑘+11 𝑚−1 + 𝑚3 𝑚𝑘+1 )))
4 𝑘+
4
2
2 6
4 6
2 2
2
4 2
2
3
6
2
6
6
2
6
6
6
6
6
6
6
(S4)
Once 𝜙 is specified, we can couple Equation (S4) and the closure model shown in Table 1
for a specific calculation.
In addition, as 𝜙 is 2, we give another way to dispose κ(𝑣, 𝑣 ′ , 𝑘) below in which only
(𝑣 + 𝑣1 )1⁄2 needs to be expanded around 𝑢0 with three orders,
𝜅(0) = −𝐵1 (𝑣 + 𝑣1 )1⁄2 𝜑
1
𝜅 ( ) = B1 [(𝑣 + 𝑣′ ) − (𝑣1⁄2 + 𝑣′
1⁄2
2
)(𝑣 + 𝑣1 )1⁄2 ]𝜑
𝜅(1) = 0
3
2
𝜅 ( ) = 𝐵1 [(𝑣 − 𝑣′) − (𝑣
2
3⁄2
+𝑣
′ 3⁄2
(S5)
)(𝑣 + 𝑣1 )
1⁄2
]𝜑
𝑘
′ 2
𝑘
′
1⁄2
{ 𝜅(2) = 𝐵1 [(𝑣 − 𝑣 ) − 𝑣 − 𝑣 ] (𝑣 + 𝑣1 ) 𝜑
In this case, the specific ODEs for moments was derived as the following form,
𝑑𝑚0
𝑑𝑡
=
1
1
𝑚0
𝑑𝑚1
2
𝑑𝑡
1
8 (𝑚1 )3⁄2
𝑚02
1
(√2 ((−6𝑚−1 𝑚1 − 6𝑚−2 1 ) 𝑚12 − 6𝑚0 (𝑚−1 𝑚7 + 2𝑚−1 𝑚5 + 𝑚1 𝑚1 ) 𝑚1 + (𝑚7 𝑚1 + 𝑚11 𝑚−1 + 𝑚13 𝑚−1 + 𝑚3 𝑚1 + 𝑚52 ) 𝑚02 ))
2
6
6
3
6
2
6
𝑚
3
= − √2 (𝑚−1 𝑚2 + 2𝑚−1 𝑚1 + 𝑚1 ) √ 1 + 2𝑚1 𝑚1 + 4𝑚5 𝑚−1 −
4
𝑚
4
2
3
6
3
0
6
2
6
2
6
2
6
2
6
√2(𝑚1 𝑚1 +𝑚
1 𝑚5 +2𝑚 1 𝑚4 +𝑚1 𝑚2 +2𝑚1 𝑚5 +𝑚7 )
−
−
2 3
6 3
2 3
3 6
6
𝑚1
√𝑚
0
6
6
6
6
6
+
6
2
1
1
2
16 (𝑚1 )3⁄2
2
6
6
(√2 (2𝑚1 𝑚7 + 𝑚2 𝑚1 + 𝑚−1 𝑚8 + 2𝑚−1 𝑚7 + 2𝑚1 𝑚5 +
6
𝑚0
6
2
3
6
3
2
3
2𝑚1 𝑚11 + 𝑚2 𝑚3 + 4𝑚4 𝑚5 + 𝑚13 ))
3
𝑑𝑚1
𝑑𝑡
𝑑𝑚3
2
𝑑𝑡
6
3
3
2
6
6
=0
3
𝑚
3
= − 4 √2 (𝑚1 𝑚1 + 𝑚−1 𝑚5 + 2𝑚−1 𝑚4 ) √𝑚1 + 2𝑚3 𝑚1 + 4𝑚7 𝑚1 + 4𝑚52 + 4𝑚11 𝑚−1 + 2𝑚13 𝑚−1 − 4
6
2
6
3
0
3
2
6
6
2
6
6
6
6
√2(𝑚1 𝑚7 +𝑚2 𝑚1 +𝑚
6
6
2
1 𝑚8 +2𝑚 1 𝑚7 +𝑚1 𝑚5 +2𝑚4 𝑚5 )
−2
−6 3
2 3
3 6
3
𝑚1
√𝑚
0
1
+ 16
1
𝑚 3⁄2
( 1)
𝑚0
(√2 (𝑚1 𝑚13 + 2𝑚2 𝑚7 +
6
6
𝑚3 𝑚1 + 𝑚−1 𝑚11 + 2𝑚−1 𝑚10 + 2𝑚1 𝑚8 + 𝑚3 𝑚5 + 2𝑚4 𝑚11 + 4𝑚7 𝑚5 ))
6
𝑑𝑚2
𝑑𝑡
1
= −4
2
1
𝑚 3⁄2
( 1 ) 𝑚02
𝑚0
3
6
3
2
3
2
3
3
3
6
6
1
1
2
(((−6𝑚1 𝑚7 − 6𝑚52) 𝑚12 − 6𝑚0 (𝑚1 𝑚13 + 𝑚3 𝑚7 + 2𝑚5 𝑚11 ) 𝑚1 + 𝑚02 (𝑚3 𝑚13 + 𝑚5 𝑚17 + 2 𝑚1 𝑚19 + 2 𝑚7 𝑚5 + 𝑚11
)) √2)
2
6
6
2
6
2
6
6
6
2
6
6
6
2
6
6
2
6
(S6)
Supplementary Information II-Derivation of the generalized TEMOM in the
continuum-slip regime for agglomerate
In the continuum-slip regime, the coagulation kernel for agglomerate is shown as,
𝛽(𝑣, 𝑣 ′ ) = 𝐵2 {(
1
+
𝑣𝑓
1
𝑓
𝑣′
𝑓
1
1
) (𝑣 𝑓 + 𝑣 ′ ) + 𝜙𝑣𝑝0 𝑓−3 (𝑣2𝑓 +
1
𝑣′
2𝑓
𝑓
)(𝑣 𝑓 + 𝑣 ′ )}
(S7)
where 𝑣 is the collision volume of agglomerates with 𝑣𝑐 = 𝑣𝑝01−3/𝐷𝑓 𝑣 3/𝐷𝑓 , 𝜙 = 𝜆𝐴/
(3/4𝜋)1/3,𝐴 = 1.591, 𝑓 = 1/𝐷𝑓 , and 𝑣 is the real volume of the agglomerate. Here, 𝑣𝑝0 is the
volume of primary particles whose diameter is usually specified to be a specific value, such as 1
nm, and 𝐷𝑓 is the fractal dimension whose value is theoretically from 1 to 3. 𝐵2 = 2𝑘𝐵 𝑇/3𝜇,
where 𝑘𝐵 is the Boltzmann constant, T denotes the air temperature, and 𝜇 denotes the gas
viscosity. As Equation (S7) is introduced into Equation (2) in the text, then we can obtain the
following ODEs with respect to k-th moment,
𝑑𝑚𝑘
𝑑𝑡
=
𝑑𝑚𝑘
𝑑𝑡
|
𝑚𝑎𝑖𝑛
+
𝑑𝑚𝑘
𝑑𝑡
|
𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
(S8)
where,
𝑑𝑚𝑘
= 2𝑘−1 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚𝑓+1 𝑚−𝑓+1 − 2𝑘 𝑢0 𝑘−1 𝑘(𝑘 − 2)𝑚𝑓 𝑚−𝑓+1
|
𝑑𝑡 𝑚𝑎𝑖𝑛
+ 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚𝑓 𝑚−𝑓+2 − 2𝑚𝑓 𝑚−𝑓+𝑘 + 2𝑘−1 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚0 𝑚2
+ 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚1 𝑚1 + 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚−𝑓 𝑚𝑓+2
− 2𝑘+1 𝑢0 𝑘−1 𝑘(𝑘 − 2)𝑚0𝑚1 − 2𝑘 𝑢0 𝑘−1 𝑘(𝑘 − 2)𝑚−𝑓 𝑚𝑓+1
+ 2𝑘 𝑢0 𝑘 (𝑘 − 1)(𝑘 − 2)𝑚0 𝑚0 + 2𝑘 𝑢0 𝑘 (𝑘 − 1)(𝑘 − 2)𝑚𝑓 𝑚−𝑓 − 2𝑚𝑓+𝑘 𝑚−𝑓
− 4𝑚0 𝑚𝑘
𝑑𝑚𝑘
= 2𝑘−1 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚𝑓+1 𝑚1−2𝑓 − 2𝑘 𝑢0 𝑘−1 𝑘(𝑘 − 2)𝑚𝑓 𝑚1−2𝑓
|
𝑑𝑡 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
+ 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚1 𝑚−𝑓+1 − 2𝑘−1 𝑢0 𝑘−1 𝑘(𝑘 − 2)𝑚0 𝑚−𝑓+1
+ 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚1 𝑚−𝑓+1 − 2𝑘−1 𝑢0 𝑘−1 𝑘(𝑘 − 2)𝑚0 𝑚−𝑓+1
+ 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚𝑓 𝑚2−2𝑓 + 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚0𝑚2−𝑓 − 2𝑚0 𝑚−𝑓+𝑘
+ 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚−2𝑓 𝑚𝑓+2 + 2𝑘−2 𝑢0 𝑘−2 𝑘(𝑘 − 1)𝑚2 𝑚−𝑓 − 2𝑚𝑓 𝑚−2𝑓+𝑘
+ 2𝑘 𝑢0 𝑘−2 𝑘(2 − 𝑘)𝑚−2𝑓 𝑚𝑓+1 + 2𝑘 𝑢0 𝑘−2 𝑘(2 − 𝑘)𝑚1 𝑚−𝑓 + 2𝑘 𝑢0 𝑘 (𝑘 − 1)(𝑘
− 2)𝑚−2𝑓 𝑚𝑓 + 2𝑘 𝑢0 𝑘 (𝑘 − 1)(𝑘 − 2)𝑚0 𝑚−𝑓 − 2𝑚−2𝑓 𝑚𝑓+𝑘 − 2𝑚𝑘 𝑚−𝑓
It should be noted here the generalized TEMOM for aerosols over the entire size regime can be
achieved through harmonic mean solution or Dahneke’s solution (Otto, Fissan, & Park, 1999)
based on the above works.