Mass Transfer Coefficient
Consider a binary mixture consisting of components A and B. If component A moves with a velocity of vA and the component B with a velocity of vB there is a force against the motion of component A that is proportional to the velocity difference ( v/>. ~ vb) • Th’s is the physical content of Fick’s law in the steadystate condition.
Where jA is the molar flux density (mol/m2 s), DAB is the diffusion factor (m~/s), cA is the concentration of component A (mol/m3), and z is a coordinate parallel to the flux (m). Note that jA = cAwA. On the basis of the force and counterforce, DAB = DBA.
Note: Equation (4.241) characterizes diffusion when the mixture element is in steady state with no turbulence. Diffusion in a pipe can be represented by Eq. (4.241) in convective mass transfer; the flow and turbulence are important.
An important convective flow is created from vaporization alone, if no other component is absorbed from the gas and is replacing the vaporizing component. In drying technology, for example, the diffusion process is considered to be diffusion between water vapor ( A) and dry air (B) (a mixture of nitrogen and oxygen), and only a small amount of dry air replaces the vaporized water, if the volume of the water in the form of liquid is very small. With good accuracy jB = 0, and the diffusion caused by the concentration gradient dcg/dz is fulfilled with convective flow (Stefan flow) according to
• t’, dcB
1b——— Dba gj. + CBV >
Where cBv represents the convective flow that cancels the diffusion. Therefore the Stefan flow is
„ — dЈg v cB a* •
The net flow of component A with Stefan flow taken into consideration is
• i—. acA IA dcA cA _
!A—DABdz +cAv ~Dab^ +Dba 37 ■
With constant temperature (c = cA + cB ■ constant),
+ ^ЈB = 0 and Dab = Dba 
DЈA, dЈB dz dz
Giving
= D — ^ Li ■ <4’2421
Integrating Eq. (4.242) gives
IA = ^~ ln^U. (4.243)
Z c~cA 
Where z is the thickness of the diffusion layer, cA2 = cA(z = 2), and CM = cA(z = 0).
By giving jA a constant value, cA(z) can be calculated from Eq. (4.243) for different z values. The concentration cB can then be calculated as cB(z) = c — cA(z). The result is shown in Fig. 4.34.
Component A diffuses due to the concentration gradient — dcA/dz. Component B diffuses due to the mean molar velocity v, v = (cAvA + cBvB)/c, like a fish swimming upstream with the same velocity as the flowing water, jB = 0, with regard to a fixed point.
In a distillation process the diffusion is nearer to the case jA = ~jB = constant or component B absorbs in place of the vaporizing component A, and now jB 0. If jA = — jB, the concentrations are similar to those presented in Fig. 4.35.
An integral equation consistent for this case is the integrated Eq. (4.21):
1a = — cai) (4.244)
Figures 4.34 and 4.35 represent two extreme cases. Drying processes represent the case shown in Fig. 4.34 and distillation processes represent Fig. 4.35. Neither case represents a convective mass transfer case; while the gas flow is in the boundary layer, other flows are Stefan flow and turbulence. Thus Eqs. (4.243) and (4.244) can seldom be used in practice, but their forms are used in determining the mass transfer factor for different cases.
C = CA + CB Distance from the surface 
1 

C = c A + CB 
О ‘лі 
CB2 
CA1 
0 
Distance from the surface FIGURE 4.35 Diffusion with equal measures, jA = js : fully permeable surface. 
C U 
129 
T.3 HEAT AND MASS TRANSFER 
Considering the case of Eq. (4.244), it is normal to describe a real mass transfer case by taking into consideration the boundary layer flows and the turbulence by using a mass transfer factor k’c, which is defined by
(4.245) 
J. A ~ k’c (CA1 ~ CA2) •
For Eq. (4.243), which can be written similar to Eq. (4.244) as
SHAPE \* MERGEFORMAT
(CA1 _ cAl) 
Ja 
Г)Сг ^авс _ £лів£п£в2 СВі
Z Cg 1 Z(CA1~CA2)
In 
CB2
CB1
_ QabЈ ——— zm—^ — cAo),
Z (CA icA2)KAl A~”
It is seen that by defining a logarithmic concentration difference
(4.246) (4.247) 
CB2~CB1
CBM —
In
CB
Eq. (4.243) can be written in an identical form:
D
%CBM
Ja
Based on this, it is normal to define a mass transfer factor consistent with this case, analogous with Eq. (4.245):
(4.248) 
І A = MCA1ЈA2)
Assuming that the relation of Eqs. (4.243) and (4.244) represents correctly the ratio of the real mass transfer flows, if it is valid that
IaUb = °) = /a(Eq — (4.243))
Ja a a = 7b) 7 A (Eq • (4.244)) ’
With Eq. (4.237) and the equations defining the mass transfer factors, Eqs.
(4.245) and (4.248) give
Kc = ~k’c. (4.249)
CBM
If the ideal gas law is used for the gases the concentrations can be shown by using partial pressures:
CA = ^ = yA, (4.250)
Where p — pA + pB is the total pressure and yA is the molar fraction of component A in the gas. The total concentration c = cA + cB can be expressed in
Terms of pressure:
C = (4.251)
Where R = 8.314 J/kmol and T is the temperature (K). Partial pressure pA can be calculated from
Pa = = MAcA, (4.252)
Where MA and MB are the molar masses of components A and B. Equation
(4.246) can be expressed in a form using partial pressures:
PBM = Pllz2M = UCBu_ (4.253,
LnPsi c
Pb i
Equation (4.247) can then be written as
By using different potential differences,
CaCaz = ^PaPai) = c(yAiyAi) = jAAC(.PAPA2) > (4.255)
Singlematerial flow can be written in various ways:
Ja = kc(cA1 — cA1) = kG(pA1 — pA2) = ky(yM — yA1) . (4.256a)
Instead of the molar flow, the mass flow can be used:
= ma1a = kp(pAi — pA2). (4.256b)
With the use of Eqs. (4.255) and (4.256a), the following relationships between the mass transfer factors are obtained:
Kc = ky=^ky = kG = RTkc. (4.257a)
Correspondingly, using Eqs. (4.255) and (4.256b) the following is obtained:
The total mass balance is the sum of Eqs. (4.260) and (4.261):
D(MAlAx + MBlBx) + d(MAIAy + MBIBy) dx dy
(4,262)
+ + MBIBz) dЈ _ Q
Dz dt 5
Where p = pA + pB. For a twocomponent or binary mixture, MArA + MBrB = 0 is valid.
The mass flow density of the mixture in the x direction is defined by
Pux = MAIAx + MBIBx, (4.263)
With corresponding equations for other directions or velocities uy and u… Using these notations, the mass balance of the mixture (Eq. (4.262)) is written as
+ + + ^ = 0 . (4.264)
Dx dy dz dt ‘
For the mass balance of component A, diffusion velocity and the corresponding diffusion factor are defined with regard to the mean molar velocity v, defined by the equation
VX = ^Ax + ^Bx, (4.265)
Where Ax is the velocity of component A in the x direction, assuming a stable coordinate system. According to this the mass flow density is
Ma! ax = PavAx = MAcAvAx. (4.266)
The velocity of the mass center of the system, ux (Eq. (4.263)), can be written as
PuAx = Pa^ax + PbVbx (4.267)
Using velocities vAx and vBx. The mass flow density may be written in the fol
Lowing ways:
Pvax = Pa(vax ~ vj + pAvx (4.268a)
= PA(vAxux) + PAux (4.268b)
= P. iv’V~vV(v’v — ux) + Paux • (4.268c)
V Ax yx
In a boundary layer equation the mass center is considered with the help of the velocity (ux, uz) and therefore a distribution of the velocity of the mass center is desirable. The diffusion velocity and diffusion factor are determined with regard to velocity vx, giving a formula for vAx — vx, but not for
VAx ~ ux ■ A useful approach is offered by Eq. (4.268c), using the artificial
Multiplication factor (vAx — ux)/(wAx — vx).
From Eq. (4.267) and p = pA + pB, it is seen that the following equation is valid:
Pa(vAxux) = — Pb(VbxUx) = ^^(VaxVbx) ■ (4.269)
The last term is best understood by noting that
(vAx ~ vSx) = (v.4* “ «,) — (vBx — «*)
And then using the first part of Eq. (4.269).
Using Eq. (4.265) and c = cA + cB, the following connections result:
(4.270) 
CaCd
Ca(vAx~’vx) ~ ^cb(‘vEx~vx) ~ ~~(vAx~ vBx) •
According to Eqs. (4.269) and (4.270),
Pb
(V/u ~ vBx) _ Psf
VAxvx (vA*“vB*) pcЯ
C
Because pA = MAcA, pB = MBcB, and p = Me, whereM = (ci4/dMA + (cB/c)MB,
(4.271)
Vyt. r ~ ux Mb
VAx — yx M ’ Substituting Eq. (4.271) in Eq. (4.268c) gives
_ MBpA
PAvAx — 
M 
<vAx — Vx) + Pa“x.
For diffusion flow, Pa(^ax~ V. v) > Fick’s model (Eq. (4.241)) gives
(4.273) (4.274) (4.275) (4.276) 
Pa(vAx~vx) = MAjAx = — MaDab^ .
Using important connections,
m f, M R,. . M A M D 3 t d
MaIax ~ PAvAx = ~J/[MAiA + pAux = — Dab^ + p^w.,. .
To shorten the notations, the following definition is used:
Ry = n
■AB M AB’
Using this, the mass flow density is represented as
MAIAx = — MA D’ab ^r + pux,
And correspondingly for directions y and z. Substituting Eq. (4.276) into Eq. (4.260) and keeping D’AB constant gives
Dx 
Dt 
D{pAux) , d(pAuv) , d(pAuz) + —3^ + °ab dpA = MAr 
A’a 
32Ca, d"cA d^cA^ _ dx2 dy2 dz2 


If the mixture density p = pA + pB is held constant, Eq. (4.264) gives
Dx + dz ~ °AB 
Dx dzz 
A Pcp 
D~T d^T dzT? 1 1 dx^ dy“ dz*, 
Dlc 
+ u ^Јa+ ^ca — n’ Dx +uy dy z dz ~ST AB 
A + CA. u VA 2 
Dx~ dy 

















D2u^ 
D mv 











FIGURE 4.37 Boundary layer flow. 
Where uG is the velocity outside the boundary layer or formally, cAs = cA(z — 0); cAo is the concentration of component A outside the boundary layer, Ts = T(z = 0), and Ta is the temperature outside the boundary layer. The dimension L is the characteristic length. All dimensionless variables range between 0 and 1. For example, ux(x, z) = u0wx(Ј,(x), r}(z)). Using the chain rule,
DuY 
A Dx 
A Dx 
Dx 
3 d^ dx / Un d2w 
V 
Dwx Dx 
«o A L 3З 
L 

D2wr d2w 
Dwr WxT^ + w 
Dw ■■~dt 
XA Re 
(4.285! 
Ar 3rf Where Re = (u0L)/v is the Reynolds number. The dimensionless form of the continuity equation (4.278) (uv = 0) in twodimensional boundary layer flow is 
Аu)x 
Mo A L dx 




0. 
(4.286) 
Dwx dw.
X I i.
3З DV
For the two equations (4.285) and (4.286) and two unknown variables wx(t„ rrj), boundary conditions are 17 = 0; wx = 0, 17 = wx = 1, = 0. The boundary condition wz{17 = 0) is not given in a mass
Transfer case, as it depends on the vaporization.
The dimensionless form of Eq. (4.282) is










Where Pr = {jxCp)/A is the Prandtl number and p. is the dynamic viscosity, giving
Boundary conditions for the dimensionless temperature are
(4,288) 
0 = 0 at tj = 0 ■>7 = at 0 — 1 From Eqs. (4.285)(4.286) and (4.287)(4.288),
0 — F(^, tj, Re, Pr). (4.289)
Equation (4.289) is an approximation in the mass transfer case, as the boundary conditions cannot always set w.(z = 0) = 0. For the case jA ~ —/B, we nearly have wz(z = 0) = 0, and the analogy equation is based on this situation.
The dimensionless form of Eq. (4.281) is
Dab
Lur
D’ab = D’ab /v =___________ 1
Re Sc 
Lu., Lu./v
Where Sc = ’/D’ab is the Schmidt number. We thus have
1 
D zA d zA d? dv2 
DzA dzA Wxd^+Wzdv Re Sc 


Boundary conditions for the dimensionless concentration are
(4.291) 
At 17 = 0, zA = 0 at 17 = zA = 1.
Equation (4.287) is in exactly the same form as Eq. (4.290), and the boundary conditions (4.288) and (4.291) are also similar.
If the solution to Eq. (4.289) is known, it is also valid for (4.290)(4.291); hence
17, Re, Sc). (4.292)
The function F is then the same in Eqs. (4.289) and (4.292). This is not strictly correct, however; see the comments after Eq. (4.289).
We can apply this result to determine the analogy between mass and heat transfer factors. Mass flow density jA (mol/m2 s) can be given as
DcA Dz 
/a = ~D 
AB 




The mass transfer factor k’ is used because Eqs. (4.289) and (4.292) demand the boundary condition wz( 17 = 0) = 0, which represents the case jA — — jB. Strictly speaking, a new mass transfer factor should be defined that represents the situation MAjA = — MBfB or wz — 0.
Using the dimensionless quantities zA and 77, Eqs. (4.284c) and (4.284f), Eq. (4.293) can be written as
Drj 
Rf —■ 0 
/a— DAB(cAo — cAs)j^
In which
KL Dab 
Sh = 




The dimensionless quantity Sh is called the Sherwood number. The heat transfer factor a is defined bv
(4.295) 
Q = — A 
.
Where q is the heat flow density from the surface to the surroundings. Using dimensionless variables © and 17 from Eqs. (4.284d) and (4.284f), Eq. (4.295) gives
(4.296) 
Where the dimensionless quantity Nu is the Nusselt number.
According to Eqs. (4.289) and (4.292) it is seen that with constant < or x
Dr) 
= 0, Re, Sc) = G(Re, Sc)
7) = 0 arl
Which leads to the important results
(4.297) (4.298) 
Sh = G(Re, Sc)
Nu = G(Re, Pr)
The above shows how the dimensionless numbers are used to provide the most accurate solution. Collecting these definitions together,
D’a 
AB 



Note the diffusion factor appearing in the Schmidt number, =
(Mb/M)Dab (Eq. (4.275)).
The preceding discussion has attempted to formulate the situation for laminar boundary layer flow as accurately as possible and to obtain precise correlation between the heat transfer and mass transfer factors.
It is not possible to translate the above reasoning to turbulent flow, as turbulent flow equations are not reliable. However, in practice it is typical to assume that the same analogy is also valid for turbulent flow. Because of this hypothesis level, it is quite futile to use the diffusion factor D’AB in the Schmidt number; instead we will directly use the number DAB as in the Sherwood number. Hence in practical calculations Sc = v/DAB.
Heat transfer is defined by Nu = ARmPrK. The function G( ■) is given as G(Re, Pr) = ARewPrK.
According to the analogy model, it is valid that
Sh = G(Re, Sc) = AReK’Sc" (4.299)
( Sc Pr 
This allows the mass transfer factor to be calculated. The above equation can be refined to
Sh — G(Re, Sc) = ARemPr’
It follows that
V/D
AB 
PCp/X 
K’c L _ aL Dab ^
Simplifying to
(4.300) 
^Le1""
F>cp
Where Le = (DAsPcp)/& = Pr/Sc is the Lewis number (or Luikov’s number in the Russian literature).
Posted in INDUSTRIAL VENTILATION DESIGN GUIDEBOOK