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CHAPTER

Introduction to convective 12 mass transfer

12.1 Introduction

Many heat transfer problems are accompanied by mass transfer. For example, water cooling in cooling towers used in thermal power plants involves both heat transfer and mass transfer. A typical example of mass transfer is the immediate spread of a room freshener in an air-conditioned room. Convective mass transfer closely resembles convective heat transfer. The analogy between mass transfer and heat transfer is especially true for low concentrations of the species in the fluid and low mass transfer rates of the species. In this chapter, the laws that govern diffusion and convective mass transfer are discussed. This is followed by a presentation of key equations for the determination of mass transfer coefficients for a gas flow over a volatile liquid or solid surface, based on the convective heat and mass analogy. Simultaneous heat and mass transfer for the case of air blowing over a wet surface is discussed.

12.2 Fick’s law of diffusion

According to Fick’s law of diffusion, the molar flux of a chemical species A in a stationary binary mixture of species A and B in the x direction is proportional to the concentration gradient in the same direction. Mathematically,

JA, x = −DAB	dCA	(12.1)
	dx

where JA,x is the molar flux of the species A in the x direction, DAB is the mass diffusivity or the diffusion coefficient of the species A diffusing through species B; dCdxA is

the concentration gradient in the x direction, with CA being the molar concentration

of species A in the binary mixture. The units of JA,x, DAB, and CA are kmol/m2 s, m2/s, and kmol/m3, respectively.

Fick’s law of diffusion is similar to Fourier’s law of heat conduction, which is expressed as

	dT	(12.2)
q = −k dx		(12.2)
q = −k dx
where k is the thermal conductivity.
Heat Transfer Engineering. http://dx.doi.org/10.1016/B978-0-12-818503-2.00012-5		397

398 CHAPTER 12 Introduction to convective mass transfer

In terms of mole fraction of species, yA, the Fick’s rate equation is

JA, x = −CDAB	dyA	(12.3)
	dx

where C is the total molar concentration equal to the sum of CA and CB for a binary mixture. The Fick’s rate equation for mass flux, jA,x, is

jA, x = −DAB	dρA	= −ρDAB	dwA	(12.4)
			dx
	dx

where ρ, ρA and wA are the total mass density (kg/m3), the mass density (mass concentration) of species A, and the mass fraction of species A, respectively. The unit of

jA,x is kg/m2s. For a binary mixture of A and B, ρ is the sum of ρA and ρB.

For a one-dimensional case, the mass diffusion rate (kg/s) of species A through a nonreacting plane wall (medium B) of thickness L is given by

	(ρA,1 − ρA,2 )	= ρDAB A	(wA,1	− wA,2 )	(12.5)
					(12.5)
mA, plane = DAB A	L			L
	L			L

where A is the area of the plane wall normal to the direction of mass diffusion, and ρA,1, ρA,2 are the mass concentrations of species A on either end of the wall, as shown in Fig. 12.1.

Similarly, the expressions for steady one-dimensional mass diffusion rates through nonreacting cylindrical and spherical walls are given by

	2π LDAB	(ρA,1		− ρA,2 )	(12.6)
mA,cylinder =	2π LDAB		ln(r2 /r1 )
			ln(r2 /r1 )
			(ρA,1	− ρA,2 )	(12.7)
mA,sphere = 4πr1r2 DAB
mA,sphere = 4πr1r2 DAB			(r2	− r1 )
			(r2	− r1 )

FIGURE 12.1 Mass concentration gradient of species A across a plane wall and diffusion of the species.

12.2 Fick’s law of diffusion 399

Note that in the case of a moving medium, JA,x is the diffusive molar flux of species A in the x direction relative to molar average velocity in the x direction, and jA,x is the diffusive mass flux of species A in the x direction relative to mass average velocity in the x direction. Table 12.1 shows some values of the mass diffusion coefficient or diffusivity DA,B for binary mixtures at atmospheric pressure. The kinetic theory of gases shows that, for dilute gases that follow the ideal gas law, the diffusion coefficients approximately vary with pressure and temperature as

DAB	T 3/2	(12.8)
	P

Table 12.1 Binary diffusion coefficients at 1 atm, Barrer (1941), Geankoplis (1972), Mills (1995), Perry (1963), Reid et al. (1977), Thomas (1991), and Black (1980).

	Component A	Component B	T (K)	DAB (m2/s)
Dilute gas mixtures
	Ammonia, NH3	Air	298	2.6 × 10−5
	Benzene	Air	298	0.88 × 10−5
	Carbon dioxide	Air	298	1.6 × 10−5
	Chlorine	Air	298	1.2 × 10−5
	Ethyl alcohol	Air	298	1.2 × 10−5
	Ethyl ether	Air	298	0.93 × 10−5
	Helium, He	Air	298	7.2 × 10−5
	Hydrogen, H2	Air	298	7.2 × 10−5
	Iodine, I2	Air	298	0.83 × 10−5
	Methanol	Air	298	1.6 × 10−5
	Naphthalene	Air	300	0.62 × 10−5
	Oxygen, O2	Air	298	2.1 × 10−5
	Argon, Ar	Nitrogen, N2	293	1.9 × 10−5
	Carbon dioxide, CO2	Nitrogen, N2	293	1.6 × 10−5
	Carbon dioxide, CO2	Water vapor	298	1.6 × 10−5
	Oxygen, O2	Ammonia, NH3	293	2.5 × 10−5
	Oxygen, O2	Benzene	296	0.39 × 10−5
	Oxygen, O2	Water vapor	298	2.5 × 10−5
	Water vapor	Argon, Ar	298	2.4 × 10−5
	Water vapor	Helium, He	298	9.2 × 10−5
	Water vapor	Nitrogen, N2	298	2.5 × 10−5
	Water vapor	Air	273	2.09 × 10−5
	Water vapor	Air	278	2.17 × 10−5
	Water vapor	Air	283	2.25 × 10−5
	Water vapor	Air	288	2.33 × 10−5
	Water vapor	Air	293	2.42 × 10−5
	Water vapor	Air	298	2.5 × 10−5
	Water vapor	Air	303	2.59 × 10−5

(Continued)

400 CHAPTER 12 Introduction to convective mass transfer

Table 12.1 Binary diffusion coefficients at 1 atm (cont.)

	Component A	Component B	T (K)	DAB (m2/s)
	Water vapor	Air	308	2.68 × 10−5
	Water vapor	Air	313	2.77 × 10−5
	Water vapor	Air	323	2.96 × 10−5
	Water vapor	Air	373	3.99 × 10−5
Dilute liquid solutions
	Benzene	Water	293	1 × 10−9
	Carbon dioxide	Water	298	2 × 10−9
	Ethanol	Water	298	1.2	× 10−9
	Glucose	Water	298	0.69 × 10−9
	Hydrogen	Water	298	6.3	× 10−9
	Methane	Water	293	1.5	× 10−9
	Nitrogen	Water	298	2.6	× 10−9
	Oxygen	Water	298	2.4	× 10−9
	Water	Ethanol	298	1.2	× 10−9
	Water	Ethylene glycol	298	0.18 × 10−9
	Water	Methanol	298	1.8	× 10−9
Dilute solid solutions
	Carbon dioxide	Natural rubber	298	1.1	× 10−10
	Nitrogen	Natural rubber	298	1.5	× 10−10
	Oxygen	Natural rubber	298	2.1	× 10−10
	Helium	Pyrex	293	4.5	× 10−15
	Helium	Silicon dioxide	298	4.0	× 10−14
	Hydrogen	Iron	298	2.6	× 10−13
	Cadmium	Copper	293	2.7	× 10−19
	Antimony	Silver	293	3.5	× 10−25
	Bismuth	Lead	293	1.1	× 10−20
	Mercury	Lead	293	2.5	× 10−19

12.3 The convective mass transfer coefficient

The convective mass transfer coefficient, hm, is analogous to the convective heat transfer coefficient, h. The mass transfer by convection will occur as long as the species mass concentration in a fluid (ρA,∞) is different from the species concentration at a surface (ρA,s) over which the fluid flows. The species mass flux (jA) is written as

jA = hm (ρA, s − ρA,∞ )

where jA is in kg/m2 s.

The total mass transfer rate (mA ) on the surface of area As is

mA = hm As (ρA, s − ρA,∞ )

(12.9)

(12.10)

12.4 The velocity, thermal, and concentration boundary layers 401

where mA is in kg/s and hm is the average mass transfer coefficient given by

				=	1
			h		1	h	dA	(12.11)
			h			h	dA	(12.11)
			m		A	m	s
					A	∫As
					s
The unit of hm or		m is m/s.
The unit of hm or	h	m is m/s.
The species molar flux, JA, on the surface is given by
		JA = hm (CA, s − CA,∞ )						(12.12)

where JA is in kmol/m2s. The units of species mass concentration, ρA, and species molar concentration, CA, are kg/m3 and kmol/m3, respectively. The relationship between ρA and CA is ρA/CA = MA, where MA is the molecular weight of species A in kg/kmol.

The gas species would remain in equilibrium with the solid or liquid phase. Hence, the species concentration at the surface is determined for the saturated condition corresponding to the surface temperature Ts. For a gas, ρA,s can be obtained using the ideal gas equation as

ρA, s =	Psat (Ts )	=	MA Psat (Ts )	(12.13)
ρA, s =	R T	=	RT	(12.13)
	A s		s

where RA and R are the characteristic gas constant and the universal gas constant, respectively. The challenge in convective mass transfer is to get hm for the engineering problem of interest.

12.4 The velocity, thermal, and concentration boundary layers

Just as a velocity boundary layer develops if the surface velocity and the free stream velocity are different and a thermal boundary layer develops if the surface temperature and the free stream temperature are different, a concentration boundary layer develops if the surface concentration and the free stream concentration of the species are different. Fig. 12.2 A–C show the development of the velocity, thermal, and concentration boundary layers, respectively, for flow over a flat plate. The species molar concentration at the surface, CA,s, is greater than the free stream concentration, CA,∞. The concentration boundary layer thickness δc is normally defined as the value of y, which satisfies

(CA,s − CA )	= 0.99	(12.14)
(CA,s − CA,∞ )

For a stationary surface, the mass transfer at the surface will be by only diffusion and is given by

J	A, s	= −D	AB	∂CA \|	y=0	(12.15)
	A, s		AB	∂y	y=0

402 CHAPTER 12 Introduction to convective mass transfer

FIGURE 12.2 (A) Velocity boundary layer, (B) thermal boundary layer, and (C) concentration boundary layer.

Using Eqs. (12.12) and (12.15), we get

			−D	AB		∂CA \|	y=0
				AB		∂y	y=0		(12.16)
hm = (CA, s − CA,∞ )									(12.16)
hm = (CA, s − CA,∞ )
On mass basis, the equations are
j	A, s	= −D			AB	∂ρA	\|	y=0	(12.17)
	A, s				AB	∂y		y=0

and

12.5 Analogy between momentum, heat transfer, and mass transfer 403

	−D		∂ρA	\|	y=0
		AB
hm =			∂y			(12.18)
	(ρA, s − ρA,∞ )

Note that the convective mass transfer is due to the combined effect of mass diffusion and advection (bulk fluid motion). If there is no flow, then the mass transfer is due to diffusion alone.

12.5 Analogy between momentum, heat transfer, and mass transfer

From the previous section, it is clear that there exists a similarity between the phenomena governing the growth of the velocity, thermal, and concentration boundary layers. The boundary layer equations for momentum, heat, and mass transport processes for a two-dimensional laminar steady, incompressible flow over a flat plate for a constant property fluid are

Momentum transfer

∂u

+ v

∂u

= ν

∂2 u

u ∂x

∂y

∂y2

(12.19)

Heat transfer

∂T

+ v

∂T

= α

∂2 T

(12.20)

∂x

∂y

∂y2

Mass transfer

∂C

+ v

∂C

= D

∂2 C

(12.21)

∂x

∂y

AB ∂y2

It can be seen that Eqs. (12.19)-(12.21) are similar. These equations result in two important nondimensional numbers in addition to the Prandtl number. The Prandtl number, as already discussed, is the ratio of the momentum diffusivity to the thermal diffusivity.

Pr =	ν	(12.22)
	α

The ratio of the momentum diffusivity to the mass diffusivity (diffusion coefficient) is called the Schmidt number,

Sc =	ν	(12.23)
	DAB

The ratio of the thermal diffusivity to the mass diffusivity is called the Lewis number,

Le =	α	(12.24)
	DAB

404 CHAPTER 12 Introduction to convective mass transfer

From Eqs. (12.22), (12.23), and (12.24) it can be seen that,

Pr =	Sc	(12.25)

	Le

The momentum, thermal, and convection boundary layers will be identical when

Pr = Sc = Le = 1

Just as the Nusselt number in convection heat transfer indicates the nondimensional temperature gradient at the surface, we define an analogous nondimensional number called the Sherwood number for convective mass transfer as

Sh =	hm x	(12.26)
	DAB

The Sherwood number indicates the nondimensional concentration gradient at the surface.

For convective heat transfer, the Stanton number was defined earlier as

				hx
St =	Nu	=		k			=	h	(12.27)
St =	Re.Pr	=	ρu∞ x		.	µcp	=	ρu∞ cp	(12.27)
			µ		.	k

Similarly, for convective mass transfer, the mass transfer Stanton number can now be defined as

				hm x
Stm =	Sh	=		DAB				=	hm	(12.28)
Stm =	Re.Sc	=	ρu∞ x		.		µ	=	u	(12.28)
	Re.Sc		ρu∞ x				µ		u
			µ			ρDAB			∞
			µ			ρDAB

In forced convective heat transfer, the Nusselt number depends on the Reynolds number and the Prandtl number. Similarly, in forced convective mass transfer, the Sherwood number depends on the Reynolds number and the Schmidt number.

Mathematically

Nu = f1 ( Re, Pr)

Sh = f2 ( Re, Sc)

In free convective heat transfer, the Nusselt number depends on the Grashof number and the Prandtl number. Similarly, in free convective mass transfer, the Sherwood number depends on the Grashof number and the Schmidt number.

Nu = f3 (Gr, Pr)

Sh = f4 (Gr, Sc)

12.5 Analogy between momentum, heat transfer, and mass transfer 405

Note that the Grashof number, in this case, is obtained from

Gr =	g( ρ/ρ)L3	(12.29)
	ν 2

Eq. (12.29) can be used for both temperature-driven and or concentration-driven free convection flows. If no concentration gradients exist in fluids, the density difference, ∆ρ, results from the temperature difference alone. In fluids with concentration gradients, the density difference results from the combined effect of the temperature difference and the concentration difference.

12.5.1 The Reynolds analogy

When the three	boundary layers are identical, that is, ν = α = DAB and hence
Pr = Sc = Le = 1,	the relationship between the skin friction coefficient, Cf; heat

transfer Stanton number, St; and the mass transfer Stanton number, Stm, can be expressed as

C f	= St = Stm	(12.30)
2

This relationship is referred to as the Reynolds analogy and is used to determine skin friction coefficient, heat transfer coefficient, and mass transfer coefficient, if one of them is known. This holds the key to solving many heat transfer problems as a mass transfer study would suffice in order to obtain the heat transfer. A heat transfer study, in general, is more cumbersome than a mass transfer study, and the above analogy is handy for several engineering problems.

12.5.2 The Chilton-Colburn analogy

When Pr ≠ Sc ≠ 1, the analogy that needs to be used is the Chilton-Colburn analogy, which is of the form

C2f = St Pr2 /3 = Stm Sc2/3

The above equation is valid for 0.6 < Pr < 60				and 0.6 < Sc < 3000.
From Eq. (12.31)
	St	Sc 2/3		2/3
		=	= Le
		=	= Le
	Stm	Pr
	h	= ρcp Le2/3
	h	= ρcp Le2/3
	h
	m

(12.31)

(12.32)

406 CHAPTER 12 Introduction to convective mass transfer

12.6 Convective mass transfer relations

The relations for the mass transfer coefficient can be defined using either the ChiltonColburn analogy or the suitable Nusselt number correlations, with the Nusselt number replaced by the Sherwood number and the Prandtl number by the Schmidt number.

12.6.1 Flow over a flat plate

For laminar flow on a flat plate, the local Nusselt number Nux is given by

Nux = 0.332Re1/2x Pr1/3

(12.33)

The ratio of the hydrodynamic boundary layer thickness (δ) to the thermal boundary layer thickness (δt) is given by

δ	= Pr1/3	(12.34)

δt

Similarly, for mass transfer, the local Sherwood number, Shx, is given by

Shx = 0.332 Re1/2x Sc1/3

(12.35)

The ratio of the hydrodynamic boundary layer thickness (δ) to the concentration boundary layer thickness (δc) is given by

δ	= Sc	1/3	(12.36)
δc

The expressions for the mean Nusselt number and the mean Sherwood number

are


				= 0.664ReL1/2 Pr1/3		(12.37)
	NuL			= 0.664ReL1/2 Pr1/3		(12.37)
			= 0.664ReL1/2Sc1/3			(12.38)
		ShL	= 0.664ReL1/2Sc1/3			(12.38)
Eqs. (12.37) and (12.38) are valid for					Pr ≥ 0.6 and Sc ≥ 0.6 , respectively.
If the turbulent boundary layer exists					(Rex,critical = 5 × 105 )	after the initial laminar

boundary layer, the correlations for the mean Nusselt number and Sherwood number are


NuL			= (0.037 ReL4 /5	− 870)Pr1/3	(12.39)
		= (0.037 ReL4 /5		− 870)Sc1/3
	ShL	= (0.037 ReL4 /5		− 870)Sc1/3	(12.40)

From the Chilton-Colburn analogy, the appropriate relations for mass transfer are as follows.

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