# 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 steady-state condition.

M = — Dab^, (4.241)

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 i-cA2)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 = MCA1-ЈA2)

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:

C-A = ^ = 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

= ST<4’254>

By using different potential differences,

Ca-Caz = ^Pa-Pai) = c(yAi-yAi) = jAAC(.PA-PA2) > (4.255)

Single-material 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 two-component 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(vAx-ux) + 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(vAx-ux) = — Pb(Vbx-Ux) = ^^(Vax-Vbx) ■ (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

VAx-vx (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

M-aIax ~ 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
 (4.277)

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
 Dividing Eq. (4.277) by MA and using Eq. (4.278) gives

 Because cA = (pA/MA). In a similar manner, the energy balance equation can be determined:

 DT, dT, 0T 07 dx + Uy dy z 3z 37

 (4.280)

 Where a — /pCp and Q is the heat generation per unit volume due to the chemical reactions (W m-3), or Q= rAAH, where AH is the reaction heat (J/mol). The thermal conductivity of the mixture is A (W itT1 K_1),and cp is the specific heat (J kg“1 K’1), or pcp = pAcpA + pBcpB. Equations (4.279) and (4.280) are similar. Figure 4.37 shows a two­dimensional boundary layer flow over a plane. Ignoring any chemical re­actions and considering steady-state conditions, Eqs. (4.279) and (4.280) give

 (4.281)

 D-Ј+u ^ dx z dz

 D_T. d_T dx2 dz2

 (4.282)

 D2u^
 D mv
 Assuming laminar flow for a linear momentum equation in the x direction (an approximation from the Navier-Stokes equations) gives

 D«a Dz

 (4.283)

 Dx

 Where v is the kinematic viscosity (m’/s). Equations (4.281)-(4.283) have to be solved at the same time as the continuity equation (4.278). The following dimensionless variables are used:

 Ux WY = — «o (4.284a) Uz W, = —2 ‘ uo (4.284b) „ _ CA — cAs A r r CAo cAs (4.284c) T — T 0 — s T ~T 1 o 1 s (4.284d) KM II US (4.284e)

 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
 X-A 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 two-dimensional boundary layer flow is
 Аu)x
 Mo A L dx
 D2ux Dx2

 Treating the other terms in a similar manner, the linear momentum equation in a dimensionless form is obtained:

 0

 (4.286)

Dwx dw.

X I i.

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

 De ^ de W*d\$ + w*dTi

 A Lu,

 A/PC» Lun

 A Lu„

 Re Pr’

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
 (4.290)

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
 K (^As ^Ao)

 (4.293)

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 =
 DZA Dr)

 (4.294)

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

M.„’^-0-Re’Pr, = G(Re’Pr)’

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
 Sh

 Nu = Re = Pr = and Sc Dab a v a

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).