Chapter 9. Basics of Convective Mass Transport

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Chapter 9. Basics of Convective Mass Transport

Learning Objectives

After completing this chapter, you will be able to:

Understand the definition of local versus average mass transfer coefficient.
Define the key dimensionless groups in convective transport.
Use common correlations for internal flows and external flows.
Evaluate mass transfer rate into a liquid flowing as a film.
Calculate the mass transfer rate from a solid sphere and from a gas bubble to a fluid.
State and use common correlations for the mass transfer coefficient for some industrial process equipment.

The mass transfer coefficient is an important parameter to calculate the contribution from flow to interfacial mass transfer, and is a key parameter needed to close macroscopic and mesocopic models. We have presented several examples of the use of this parameter in Chapters 3 and 4. The next four chapters deal with evaluation of this parameter in detail. Two common approaches to calculating the mass transfer coefficients are theoretical results obtained from differential models and empirical correlations that are fitted to experimental measurements. Both approaches are obviously complimentary. The theoretical approach is good for simple geometries under laminar flow conditions (since the flow field is now well characterized) while the empirical approach is good for complex geometries and for turbulent flow conditions. This chapter takes more of an empirical approach and provides useful correlations for some common situations. More discussion and analysis of differential models for the convection+diffusion process will be taken up in the next two chapters (10 and 11) where we show how the basic convection-diffusion theory can be applied to predict mass transfer coefficients, at least for cases where the velocity profile is known exactly. Mass transfer under turbulent flow conditions requires additional closure for eddy diffusivity and some semi-theoretical models are then discussed in Chapter 12.

In this chapter we first revisit the definition of mass transfer coefficients introduced in earlier chapters and distinguish between local and averaged values. We then present the common dimensionless groups used to correlate mass transfer data. This is followed by correlations for a number of common standard cases and for some common industrial reactors.

9.1 Definitions for External and Internal Flows

Consider a species A being transported from a solid to a fluid phase. The flux can be represented as a combination of the diffusion and a convection flux:

N_As = J_As + υ_sC_A(s)

where J_As is the diffusion flux from the surface. The second term is the contribution of convection induced by mass transfer. Here υ_s is the normal component of the velocity at the surface and depends on how the other species present in the mixture are being transported from the surface. For example, it is zero for EMD. This term can also be neglected for low flux conditions and for dilute systems.

The diffusive part of flux from a surface is then modeled as a product of the mass transfer coefficient and a driving force:

J_As = k_m ΔC_A

This leads to the following basic definition of the mass transfer coefficient:

k_{m} = \frac{J_{A s}}{Δ C_{A}}

$k_{m} = \frac{J_{A s}}{Δ C_{A}}$

Here ΔC_A is a suitably defined driving force. The choice of the driving force ΔC_A is often not clear. The definition commonly used for external and internal flows is presented in the following.

External flows are flow past solid bodies. Examples are flow over a flat plate, flow around a spherical solid, and so on. Flow past a flat plate with mass transfer from (or to) the plate to the gas shown in Figure 6.3 is an example. The problem is of a boundary layer type and there is a clearly defined external concentration here outside the boundary layer, denoted as C_A∞, which is the concentration far away from the solid. This is also called the bulk gas concentration. The difference, C_As – C_A∞, is used as the driving force. (The C_As is the concentration in the fluid phase adjacent to the solid.)

Internal flows are flows in confined geometries, such as flow in a pipe or flow in channels, thin liquid films, and so on. A pipe with a dissolving wall or a catalytic wall is an example of mass transfer in internal flows. The concentration at the wall itself is C_As, but C_A varies across the pipe as a function of r. The driving force is defined as C_As minus some average concentration in the pipe. This average concentration is taken as the cup mixing concentration, defined as

C_{A b} = \frac{\int_{A} υ_{z} C_{A} d A}{\int_{A} υ_{z} d A}

$C_{A b} = \frac{\int_{A} υ_{z} C_{A} d A}{\int_{A} υ_{z} d A}$

Here υ_z is the local velocity in the flow direction and C_A is the local concentration.

Hence C_Ab – C_As is used as the driving force and the mass transfer coefficient in internal flows is defined as

J_As = k_m(C_Ab – C_As)

Here N_As is the flux from the bulk of the fluid to the wall. Note that C_Ab is an average concentration and not the concentration at any particular radial location. This is in contrast to external flow, where C_A∞ is the actual concentration at the edge of the boundary layer.

Near the entrance to a pipe, the flow is not fully developed and is of a developing internal flow. The mass transfer coefficient here is defined differently from that for a fully developed flow. The reason is that in developing flow there is a mass transfer boundary layer near the wall. Most of the pipe (in the center region) is still at the inlet concentration C_A,i and the concentration varies only in a thin region near the wall and reaches C_As at the wall itself. The driving force used is now defined as C_A,i – C_As. The C_Ab – C_As used for fully developed internal flows is not used in the developing region.

9.2 Relation to Differential Model

How is k_m related to differential description? The mass transfer coefficient is usually defined on the basis of the low flux case where the diffusion at the surface is the only mode of transport. The normal component of the velocity at the wall is taken as zero. Hence locally near the wall the mass transfer rate can be computed using Fick’s law at the wall. The flux to the wall is given by

J_{A w} = - D_{A} (\frac{d C_{A}}{d y})_{y = w}

$J_{A w} = - D_{A} (\frac{d C_{A}}{d y})_{y = w}$

where w is the location of the wall. If one divides the flux by the local driving force one can get the value of the mass transfer coefficient:

k_{m} (Local) = \frac{- D_{A} [(d C_{A} / d y)_{y = w}]}{Δ C_{A}}

$k_{m} (Local) = \frac{- D_{A} [(d C_{A} / d y)_{y = w}]}{Δ C_{A}}$

This relates the mass transfer coefficient to the concentration gradient at the wall. The latter in turn is from a differential model (also called convection-diffusion models) and the above relation is useful to extract the values of mass transfer coefficients from detailed convection-diffusion models. In fact we will use the relation in later chapters to evaluate mass transfer coefficients based on theory.

What if there is high flux, that is, the normal component of the velocity at the wall is significantly different from zero? In this case, a correction factor is applied to the low flux value and the mass transfer rate is calculated as

N_{A} = k_{m} C Δ y_{A} F

$N_{A} = k_{m} C Δ y_{A} F$

For UMD, $F$ $F$ is the drift flux correction, which is equal to the reciprocal of the log mean mole fraction of the non-diffusing component. For EMD the correction factor is not needed and set as one. For the reacting case with reactants diffusing to and products counter-diffusing from the surface, the correction factor will depend on the excess stoichiometric coefficient, ν_E.

Often the average value over a length of plate or pipe is sufficient for design calculations, especially in the context of mesoscopic models; this is defined as

\begin{matrix} {\bar{k}}_{m} = \frac{1}{L} \int_{0}^{L} k_{m} (local) d x & (9.1) \end{matrix}

$\begin{matrix} {\bar{k}}_{m} = \frac{1}{L} \int_{0}^{L} k_{m} (local) d x & (9.1) \end{matrix}$

For flow over a flat plate, it can be shown that k_m local is proportional to x^–1/2 if the flow is laminar. This is an useful result to remember and we will derive it later based on laminar flow theory in Chapter 11. If k_m local is represented as $A_{1} / \sqrt{x}$ $A_{1} / \sqrt{x}$ , where A₁ is the coefficient of proportionality, we find that the average value is $2 A_{1} / \sqrt{L}$ $2 A_{1} / \sqrt{L}$ .

For laminar flow in a pipe, there is an entry region near the pipe entrance for mass transfer and the local value is a function of axial position, z, in this region. A dependency of z^–1/3 is predicted from theory here. After the entry length the mass transfer coefficient does not change significantly with z and reaches an asymptotic value. For a long pipe the asymptotic value contributes most to the average value. Hence the average value may be taken as a constant equal to the asymptotic value. The following result applies for this parameter as will be proved by detailed analysis in Chapter 10:

\begin{matrix} {\bar{k}}_{m} = 3.6 6 \frac{D_{A}}{d_{t}} & (9.2) \end{matrix}

$\begin{matrix} {\bar{k}}_{m} = 3.6 6 \frac{D_{A}}{d_{t}} & (9.2) \end{matrix}$

This value is applicable to a case where the wall is maintained as a constant concentration (e.g., solid dissolving from a wall or a rapid catalytic reaction at the wall). It is a rather interesting result; the mass transfer coefficient does not depend on the flow velocity for long pipes in laminar flow!

For turbulent flow in a pipe, the mass transfer coefficient is nearly constant along the pipe length. Thus the local value and the average values are nearly the same. Also it is a strong function of the flow velocity, usually to the power of 0.8.

9.3 Key Dimensionless Groups

Mass transfer data can be correlated using three key dimensionless groups: the Sherwood number, the Reynolds number, and the Schmidt number. These groups were introduced earlier in Section 3.4 and we recap them here for ease of reference.

The Sherwood number defined in the following is the dimensionless representation of the mass transfer coefficient:

S h = \frac{k_{m} L r e f}{D_{A}}

$S h = \frac{k_{m} L r e f}{D_{A}}$

Here L_ref is a reference length scale. The pipe diameter is used as the length scale for flow in a pipe while the distance from the leading edge x is used for external flow past a flat plate. For internal flow in non-circular channels, the hydraulic diameter is used as the length scale. This is defined as

hydraulic diameter = 4 \frac{flow area}{wetted perimeter}

$hydraulic diameter = 4 \frac{flow area}{wetted perimeter}$

The Reynolds number is a measure of the effect of flow velocity and is defined in general as

Re = \frac{L_{r e f 〈 υ 〉}}{ν}

$Re = \frac{L_{r e f 〈 υ 〉}}{ν}$

The reference length scale is the pipe diameter for pipe flows. Hence

Re = \frac{d_{p} ρ < υ >}{μ} for pipe flow

$Re = \frac{d_{p} ρ < υ >}{μ} for pipe flow$

The reference length is x, the distance from the leading edge for flow over a flat plate; hence

{Re}_{x} = \frac{x ρ υ \infty}{μ} for flow over a plate

${Re}_{x} = \frac{x ρ υ \infty}{μ} for flow over a plate$

The third group is the Schmidt number:

S c = \frac{μ}{(ρ D_{A - m})}

$S c = \frac{μ}{(ρ D_{A - m})}$

It represents the ratio of momentum diffusivity to mass diffusivity. This group is a physical property and does not depend on the operating parameters. For gases the values are close to one while for liquids the values are high, in the range of 500 to 1500.

9.3.1 Other Derived Dimensionless Groups

The basic groups shown above can be combined to get other groups that can be used in lieu of the Sherwood number. Thus the group Sh/(ReSc) is often used instead of Sh. This group is called the Stanton number:

S t = \frac{S h}{R e S c} = \frac{k_{m}}{υ_{r e f}}

$S t = \frac{S h}{R e S c} = \frac{k_{m}}{υ_{r e f}}$

where υ_ref is the reference velocity, either <υ> or υ_∞.

The j-D factor is also used in correlation. This is defined as

j_{D} = S t (S c)^{2 / 3} = \frac{k_{m}}{υ_{r e f}} (S c)^{2 / 3}

$j_{D} = S t (S c)^{2 / 3} = \frac{k_{m}}{υ_{r e f}} (S c)^{2 / 3}$

where St is the Stanton number for mass transfer. One use of this is that j-factor is closely related to the friction factor in momentum transport, as we will see later.

Another widely used group is the Peclet number, Pe, defined as

Pe = ReSc

In some cases (e.g., laminar flow in a pipe) Sh does not depend on Re and Sc separately and is the function of the product of these quantities. In such cases use of the Peclet number as the correlating group is more convenient.

Now we show several examples of the use of the dimensionless groups for correlation of mass transport data for common flow situations and in process equipment.

9.4 Mass Transfer in Flows in Pipes and Channels

In this section the correlation for mass transfer coefficients for flows in pipe and channels is summarized.

9.4.1 Laminar Flow

The laminar flow is amenable to a theoretical analysis discussed in Section 10.2. The following equation for the average Sherwood number over a length L of pipe can be derived from theory:

\begin{matrix} \bar{S} h_{L} = 3.6 6 + \frac{0.0 668 P e / L *}{1 + 0.0 4 [P e / L *]^{2 / 3}} & (9.3) \end{matrix}

$\begin{matrix} \bar{S} h_{L} = 3.6 6 + \frac{0.0 668 P e / L *}{1 + 0.0 4 [P e / L *]^{2 / 3}} & (9.3) \end{matrix}$

Here L* = L/d_t. As L* becomes large, the average Sherwood number approaches an asymptotic value of 3.66. This equation was also used earlier in Section 4.1 in conjunction with mesoscopic analysis and was used in a model for mass transfer from a dissolving wall. Flow is laminar up to a Reynolds number of 2100.

9.4.2 Turbulent Flow

Gilliland and Sherwood (1934) correlation is commonly used for mass transfer to or from a dissolving wall under turbulent flow conditions:

Sh = 0.023Re^0.83Sc^0.44

For liquid the correlation of Linton and Sherwood (1950) is found to fit the data:

Sh = 0.023Re^0.83Sc^1/3

The equation is very similar to the Dittus-Boelter (1930) equation used for heat transfer, indicating the commonality in the transport of heat and mass. Flow is turbulent when the Reynolds number is greater than 4000.

9.4.3 Channel Flow

For flow between two parallel plates, the Sherwood number is defined using the hydraulic radius as the reference length. This is defined in general as four times the area for flow divided by the wetted perimeter. Using this definition d_h for flow in a channel is equal to twice the spacing between the two plates. Hence the Sherwood number is defined as 2k_mh/D_A.

For laminar flow, the average Sherwood number for a channel of length L is given as

\begin{matrix} \bar{S} h_{L} = 7.5 4 + \frac{0.0 3 P e / L *}{1 + 0. 016 [P e / L *]^{2 / 3}} & (9.4) \end{matrix}

$\begin{matrix} \bar{S} h_{L} = 7.5 4 + \frac{0.0 3 P e / L *}{1 + 0. 016 [P e / L *]^{2 / 3}} & (9.4) \end{matrix}$

Here L* is equal to L/d_h. As L or correspondingly L* becomes large, the average Sherwood number approaches an asymptotic value of 7.54. This equation is based on the theoretical result of the convection-diffusion model and is not empirical.

For turbulent flow, the correlations for pipe flow (e.g., the Dittus-Boelter equation) is generally used with the pipe diameter now replaced by the hydraulic diameter. The rationale for using the same correlation is that the viscous sublayer near the wall is very thin and this layer offers most of the resistance to mass transfer. The concentration drop in the core fluid is rather small. Hence the precise shape of the wall region is not very critical.

9.5 Mass Transfer in Flow over a Flat Plate

The dimensionless mass transfer coefficient is the local Sherwood number Sh_x = k_mx/D, and this should be correlated in terms of the local Reynolds number and the Schmidt number Sc. Note that the local distance from the leading edge point is used in the definition of the Reynolds number, Re_x.

A power law correlation type is used. Thus the equation

S h_{x} = {A Re}_{x}^{α} S c^{β}

$S h_{x} = {A Re}_{x}^{α} S c^{β}$

appears to be a good form for correlation based on dimensionless analysis. Dimensional analysis does not resolve what exponents α and β in this equation should be. It is merely a platform for the correlation of the experimental data.

9.5.1 Laminar Flow

The following equation derived from theory shown in Chapter 11 is applicable for laminar flows:

\begin{matrix} S h_{x} = 0.3 32 R e_{x}^{1 / 2} S c^{1 / 3} & (9.5) \end{matrix}

$\begin{matrix} S h_{x} = 0.3 32 R e_{x}^{1 / 2} S c^{1 / 3} & (9.5) \end{matrix}$

Note that the exponent on the Sc is 1/3. This is characteristic of many other problems as well where mass transfer takes place from a no-slip (fluid-solid) boundary.

The previous expression gives the local mass transfer coefficient. An integrated average value is often used in engineering design calculations. Integrating over a length L of plate, an average Sherwood number can be defined and is given by the following expression:

\begin{matrix} S h_{L} = \frac{1}{L} \int_{0}^{L} S h_{x} d x = 0.6 64 R e_{L}^{1 / 2} S c^{1 / 3} & (9.6) \end{matrix}

$\begin{matrix} S h_{L} = \frac{1}{L} \int_{0}^{L} S h_{x} d x = 0.6 64 R e_{L}^{1 / 2} S c^{1 / 3} & (9.6) \end{matrix}$

Note that the above equation should not be used if Re_L > 2 × 10⁵. The flow ceases to be laminar under these conditions. The boundary layer becomes fully turbulent if Re_L > 2 × 10⁶.

9.5.2 Turbulent Flow

For turbulent flow over a plate, the following correlation for local value is commonly used:

\begin{matrix} S h_{x} = 0.0292 {Re}_{x}^{0.8} S c^{1 / 3} & (9.7) \end{matrix}

$\begin{matrix} S h_{x} = 0.0292 {Re}_{x}^{0.8} S c^{1 / 3} & (9.7) \end{matrix}$

Note the change in the dependency to 0.8 on Re, which is now much stronger in turbulent flow. Thus the flow velocity has a much stronger effect on the mass transfer coefficient. The exponent on Sc is still the same as in laminar flow.

The average value is given by integration as

\begin{matrix} S h_{L} = 0.0365 {Re}_{L}^{0.8} S c^{1 / 3} & (9.8) \end{matrix}

$\begin{matrix} S h_{L} = 0.0365 {Re}_{L}^{0.8} S c^{1 / 3} & (9.8) \end{matrix}$

Turbulent mass transfer in both pipes and in flow over flat plates is often analyzed using an analogy with momentum transfer. The analysis is semi-theoretical and discussed in detail in Chapter 12.

9.5.3 The j-Factor

The j-factor form of the correlations for external flows is now shown and the relation to the drag coefficient is indicated.

For laminar flow, Equation 9.6 for Sh_L can be rearranged to

\begin{matrix} j_{D} = \frac{0.6 64}{R e_{L}^{1 / 2}} & (9.9) \end{matrix}

$\begin{matrix} j_{D} = \frac{0.6 64}{R e_{L}^{1 / 2}} & (9.9) \end{matrix}$

The drag coefficient for friction averaged over a length of L has a value of

c_{D L} = \frac{1.2 28}{R e_{L}^{1 / 2}}

$c_{D L} = \frac{1.2 28}{R e_{L}^{1 / 2}}$

Comparing the previous two equations, we find that the j-factor is to the c_D/2. This is one form of the analogy between momentum and mass transfer that is discussed in more detail in Chapter 11.

A similar relation applies for turbulent flow:

\begin{matrix} j_{D} = \frac{0.0 365}{{Re}_{L}^{0.2}} & (9.10) \end{matrix}

$\begin{matrix} j_{D} = \frac{0.0 365}{{Re}_{L}^{0.2}} & (9.10) \end{matrix}$

9.6 Mass Transfer for Film Flow

Film flow over a flat plate under laminar flow conditions is amenable to a theoretical analysis, which is presented in Section 10.4. Although film flow is an idealized case, the results form a basis for correlation of data over more complex equipment such as a packed bed where the liquid flows as thin rivulets over the packing surface. Here we summarize the key correlation and the effect of parameters on the mass transfer coefficient. Two cases are presented: mass transfer from the solid (flat plate surface) to the liquid film and mass transfer from a gas to the film.

9.6.1 Solid to Liquid

An example of this is mass transfer from a dissolving wall over which a thin liquid film is flowing. A second example is a catalytic wall to which mass transfer is taking place from the liquid film. The mass transfer from or to a solid surface is given by the following equation:

\begin{matrix} k_{s l} = 0.5 384 D^{2 / 3} β^{1 / 3} z^{- 1 / 3} & (9.11) \end{matrix}

$\begin{matrix} k_{s l} = 0.5 384 D^{2 / 3} β^{1 / 3} z^{- 1 / 3} & (9.11) \end{matrix}$

Here β is the slope of the velocity profile at the solid–liquid interface. The expression that can be derived for this quantity based on momentum transfer considerations is

\begin{matrix} β = \frac{ρ g}{μ} δ & (9.12) \end{matrix}

$\begin{matrix} β = \frac{ρ g}{μ} δ & (9.12) \end{matrix}$

Here δ is the film thickness, which depends on the flow rate and is given as

\begin{matrix} δ = (\frac{3 Q μ}{ρ g})^{1 / 3} & (9.13) \end{matrix}

$\begin{matrix} δ = (\frac{3 Q μ}{ρ g})^{1 / 3} & (9.13) \end{matrix}$

with Q the flow rate per unit width of the plate. The equation is valid only if the Reynolds number is less than 20. For larger values of Re, the film becomes wavy and eventually turbulent.

The average value of the mass transfer coefficient over a length L is obtained by averaging and the final result is

\begin{matrix} {\bar{k}}_{s l} = 0.8 075 D^{2 / 3} β^{1 / 3} L^{- 1 / 3} & (9.14) \end{matrix}

$\begin{matrix} {\bar{k}}_{s l} = 0.8 075 D^{2 / 3} β^{1 / 3} L^{- 1 / 3} & (9.14) \end{matrix}$

The following parametric effects can be deduced by substituting for β from the above equation:

The mass transfer coefficient varies as diffusivity to the power of 2/3. This is characteristic of mass transfer from a solid–fluid boundary where a no-slip boundary condition for velocity is applicable.
It increases with film thickness to the power of 1/3, since β is proportional to δ.
It decreases with length with an exponent of 1/3.
It is proportional to 1/6 exponent of the liquid flow rate per unit width.

9.6.2 Gas to Liquid

The mass transfer from a gas phase to a liquid flowing as a thin film over a solid wall under laminar flow conditions is also amenable to theoretical analysis (Section 10.5). The penetration theory discussed in Section 8.6 is used here and the result is

k_{L} = 2 \sqrt{\frac{D_{A} υ_{max}}{π L}}

$k_{L} = 2 \sqrt{\frac{D_{A} υ_{max}}{π L}}$

Here υ_max is the maximum velocity at the surface, which is given from hydrodynamic considerations as

υ_{max} = \frac{ρ_{L} g δ^{2}}{2 μ}

$υ_{max} = \frac{ρ_{L} g δ^{2}}{2 μ}$

with δ being the film thickness given by Equation 9.13.

Mass transfer in wavy and turbulent film requires somewhat different considerations and is discussed in later chapters. Turbulent flow needs additional correlations for eddy diffusivity.

9.7 Mass Transfer from a Solid Sphere

Mass transfer from a solid sphere to gas is a well studied problem and usually the Froessling-Marshall correlation introduced in Section 3.4 is used. This is reproduced below for completeness:

\begin{matrix} S h = 2 + 0.5 52 R e^{0.5} S c^{1 / 3} & (9.15) \end{matrix}

$\begin{matrix} S h = 2 + 0.5 52 R e^{0.5} S c^{1 / 3} & (9.15) \end{matrix}$

Note the limiting value of 2 for Re → 0, that is, for a stagnant gas. This is a theoretical result that was shown in Section 6.3.

For mass transfer into a liquid stream the correlation of Brian, Hales, and Sherwood (1969) is often used:

Sh = (4 + 1.21Pe^2/3)^1/2

The Peclet number, which is the product of Re and Sc is used in this correlation. The above correlation is suitable for P e < 10⁴. The first term represents the contribution under stagnant conditions: Sh → 2 as Pe → 0. The second term captures the contribution due to flow.

For larger Peclet numbers (> 10⁴) the simpler correlation of Levich is considered suitable since the first term representing the stagnant flow contribution can be neglected:

Sh = 1.01Pe^1/3

Again a 1/3 exponent dependency on the Schimidt number is observed; this is similar to that for flow over a flat plate.

In general, two points are worth noting. First, there is a considerable variation of the local mass transfer coefficient over the angular direction. The mass transfer correlations shown above represent azimuthal averaged values. Second, the natural convection effects often accompany forced convection transfer. The dimensionless group to account for natural convection effects is the Grashof number:

G r = \frac{d^{3} ρ_{L} g Δ ρ}{μ_{L}^{2}}

$G r = \frac{d^{3} ρ_{L} g Δ ρ}{μ_{L}^{2}}$

Natural convection is usually neglected if Re > 0.4Gr^1/2Sc^1/3, as indicated by Steinbeger and Treybal (1960).

Some theoretical analysis is also possible based on the convection-diffusion model. Again hydrodynamics play an important role. For small Re the flow is analyzed using what is known as Stokes’ law. The average mass transfer coefficient can be fitted within 2% accuracy by the following correlation based on theory:

\begin{matrix} S h = 1 + {(1 + P e)}^{1 / 3} & (9.16) \end{matrix}

$\begin{matrix} S h = 1 + {(1 + P e)}^{1 / 3} & (9.16) \end{matrix}$

This correlation is useful if the Reynolds number range is less than one. For a stagnant fluid (Pe = 0), Sh has a value of two, which is also the limiting value obtained from the above equation.

As the Reynolds number increases beyond one, the improvement over the Stokes flow known as the Oseen (1910) model is useful. The convection-diffusion equation can be solved under these conditions and theoretical results for the local mass transfer coefficient can be obtained. The local values are averaged over θ and the results are usually reported in terms of an average Sherwood number. A number of studies have been conducted using this approach and the results are summarized by Leal (2007) and Clift, Grace, and Weber (1978). Interested readers may wish to study these books and the papers cited thereof for further discussion.

9.8 Mass Transfer from a Gas Bubble

It is useful at the outset to point out some differences in mass transfer from a gas bubble and from a solid sphere to a liquid. A gas bubble is a deformable surface and the spherical shape may not hold for all conditions. Even assuming a spherical shape, there may be internal circulation within the bubble itself. In contrast, there is no internal circulation in a solid sphere. Whether the internal circulation is present or not depends on the properties of the liquid, mainly whether there is a surface active agent present in the liquid or not. The internal circulation is reduced in the presence of surface active agents. They adsorb in the rear of the bubble and create a surface tension–driven flow that counters the internal circulation within the bubble. (The surface tension–driven flow is also known as Marangoni flow). The drop or bubble then behaves more like a rigid sphere. For these conditions the rigid sphere correlation given in the earlier section is applicable. The mass transfer coefficient is usually proportional to diffusivity to the 2/3 power in this case.

The regime with significant internal circulation is called the Hadamard-Rybczynski (1911) regime, named after the scientists who proposed this effect. This, applies for clean liquids, that is, in the absence of surface active agents. For these conditions, the following correlation is found to be suitable:

\begin{matrix} S h = 1 + (1 + 0.5 64 P e^{2 / 3})^{3 / 4} & (9.17) \end{matrix}

$\begin{matrix} S h = 1 + (1 + 0.5 64 P e^{2 / 3})^{3 / 4} & (9.17) \end{matrix}$

where Pe = Re Sc with the Reynolds number based on bubble diameter. Hence we need additional correlation to calculate the bubble size. Note that the mass transfer coefficient now becomes proportional to diffusivity to the power of one half.

We now present correlations for industrial contactors where a gas is sparged into a liquid.

9.8.1 Bubble Swarms and Bubble Columns

Mass transfer from a swarm of bubbles to a liquid finds many important applications in industrial processes. For example, bubble column reactors, where a gas is sparged into a tall pool of liquid, find use in a wide variety of processes including oxidation, hydroformylation, Fisher-Tropsh synthesis, and so on (see Ramachandran and Chaudhari, 1983). In this contactor we often have a swarm of bubbles that create the circulation in the liquid; hence the prediction of the mass transfer coefficient can be a difficult task since the velocity field near a bubble cannot be accurately calculated. (Note that CFD simulations can be useful in this context but they involve many closure parameters and provide more of a smeared value based on the local volume averaging.) Further one needs to classify the flow regime since the hydrodynamics is different for small spherical bubbles, large ellipsoidal bubbles, and large spherical cap bubbles. The results also depend on whether there is an internal circulation within the bubbles or not. We can only provide a brief (empirical) discussion on these topics. Clift et al. (1978) provide additional details. The correlations suggested here are for a quick estimation and to get a feel for the numbers. They are also useful to understand the interrelations between various parameters. There is a vast body of literature as well as measurements on these parameters and for more accurate estimation, consult these sources. The paper by Akita and Yoshida (1974) is an useful reference on this topic.

Mass Transfer Coefficient

The correlation of Calderbank and Moo-Young (1961) is widely used in the literature for bubbles of diameter greater than 2.5 mm:

S h = \frac{k_{L} d_{b}}{D_{A}} = 0.4 2 G r^{1 / 3} S c^{1 / 2}

$S h = \frac{k_{L} d_{b}}{D_{A}} = 0.4 2 G r^{1 / 3} S c^{1 / 2}$

where Gr is defined as

G r = \frac{d_{B}^{3} ρ_{L} g Δ ρ}{μ_{L}^{2}}

$G r = \frac{d_{B}^{3} ρ_{L} g Δ ρ}{μ_{L}^{2}}$

A penetration model is used as an alternative with contact time taken as d_B/υ_B, where υ_B is the effective rise velocity of the swarm of bubbles in the liquid:

k_{L} = \sqrt{\frac{4 D_{A} υ_{B}}{π d_{B}}}

$k_{L} = \sqrt{\frac{4 D_{A} υ_{B}}{π d_{B}}}$

For bubbles with diameter less than 2.5 mm the following correlation is commonly used:

Sh = 0.31Gr^1/3Sc^1/3

Small bubbles of this size do not have internal circulation and hence the mass transfer process is closer to that of a solid. Hence the exponent of Sc is 1/3 and correspondingly the dependence on the diffusivity is 2/3. In contrast, for large bubbles the dependency is 1/2.

A knowledge of bubble diameter is needed in order to use these correlations. Also, in order to use the penetration theory, the relative velocity between the gas and liquid is needed. Some correlations for these quantities are summarized in the following.

Bubble Diameter

The maximum bubble size is an important parameter and the average bubble size can be related to this. A force balance model for estimation of this parameter was proposed by Kolmogorov (1949) and Hinze (1955). In a turbulent field, the local fluctuations are responsible for bubble breakup. The surface tension forces oppose this deformation. A balance of these forces can be used to estimate the maximum bubble size. The key dimensionless parameter resulting from these balances is the Weber number, defined below:

Weber number = \frac{τ}{σ / d_{max}} (\frac{ρ_{d}}{ρ_{c}})^{1 / 3}

$Weber number = \frac{τ}{σ / d_{max}} (\frac{ρ_{d}}{ρ_{c}})^{1 / 3}$

where τ is the stress, which can be correlated to the power input per unit mass (defined as ∊ here) by the following equation:

τ = ρ_{c} [2 {(ϵ d_{max})}^{2 / 3}]

$τ = ρ_{c} [2 {(ϵ d_{max})}^{2 / 3}]$

Rearranging we get the maximum bubble diameter in terms of a critical Weber number:

d_{max} = (\frac{W e_{c r i t}}{2})^{3 / 5} (\frac{σ}{ρ_{c}})^{0.6} (\frac{ρ_{d}}{ρ_{c}})^{0.2} ϵ^{- 0.4}

$d_{max} = (\frac{W e_{c r i t}}{2})^{3 / 5} (\frac{σ}{ρ_{c}})^{0.6} (\frac{ρ_{d}}{ρ_{c}})^{0.2} ϵ^{- 0.4}$

The bubble diameter is proportional to the power input to the system by the power of –0.4. This forms the basis for many correlations for the bubble diameter, two of which are summarized below.

Bhavaraju et al. (1978) suggested the following correlation:

d_{B} = 0.5 3 (\frac{μ_{c}}{μ_{d}})^{0.1} (\frac{σ}{ρ_{c}})^{0.6} ϵ^{- 0.4}

$d_{B} = 0.5 3 (\frac{μ_{c}}{μ_{d}})^{0.1} (\frac{σ}{ρ_{c}})^{0.6} ϵ^{- 0.4}$

The following correlation of Calderbank (1958) is also often used:

d_{B} = 4.1 5 ϵ_{G}^{05} (\frac{σ}{ρ_{c}})^{0.6} ϵ^{- 0.4}

$d_{B} = 4.1 5 ϵ_{G}^{05} (\frac{σ}{ρ_{c}})^{0.6} ϵ^{- 0.4}$

This is similar to the Bhavaraju correlation but needs an estimate of the gas holdup ∊_G as an additional input.

Power Input

The power input to the system is needed to find the average bubble diameter. This is taken as the work done in expansion of the gas as it travels from the column bottom to the top; the following simplified formula is often used:

Power input = Q_Gρ_Lgh_L

where Q_G is the gas flow rate and h_L is the clear liquid height. Hence ∊ = u_Gg is used as a simple model for the power input.

Relative Velocity

The terminal velocity of bubble rise is often used for the relative velocity (needed for instance in the penetration model) and the following correlation suggested by Clift et al. (1978) is useful:

υ_B = [2.14σ_ρLd_B] + 0.505(gd_B)]^1/2

This correlation applies if the bubble holdup is small (< 0.1), in which case the bubble motion is not affected by the presence of neighboring bubbles. Note that a correction is needed for a large holdup, which is known as the Richardson-Zaki correction.

Interfacial Area

The interfacial area for mass transfer depends on the bubble diameter and the gas holdup. Assuming an average bubble diameter d_B, the following expression is commonly used:

a_{g l} = \frac{6 ϵ_{g}}{d_{B}}

$a_{g l} = \frac{6 ϵ_{g}}{d_{B}}$

The gas holdup is needed in order to use this equation.

Gas Holdup

A correlation proposed by Yamashita and Inoue (1975) is a commonly used and simple equation suitable for estimation of the gas holdup:

ϵ_{G} = \frac{u_{G}}{2.2 υ G + 0.3 {(g d_{t})}^{0.5}}

$ϵ_{G} = \frac{u_{G}}{2.2 υ G + 0.3 {(g d_{t})}^{0.5}}$

Here u_G is the superficial gas velocity.

9.9 Mass Transfer in Mechanically Agitated Tanks

Another common piece of equipment for gas–liquid contacting is mechanically agitated tanks. These are tanks equipped with an impeller or agitator with gas sparged at the bottom. Mechanical stirring causes the gas to be dispersed into bubbles and also keeps the contents of the tanks well-mixed. Usually the volumetric mass transfer coefficient is measured and correlated. van’t Riet (1979) proposed the following correlation:

k_{L} a_{g l} = 2.6 \times 1 0^{- 2} (P / V)^{0.4} u_{G}^{0.5}

$k_{L} a_{g l} = 2.6 \times 1 0^{- 2} (P / V)^{0.4} u_{G}^{0.5}$

Here P/V is the power input per unit volume of liquid and u_G is the superficial gas velocity. Again a 0.4 dependency on power input is seen, similar to the bubble columns.

The power consumption in an agitated tank in turn is correlated using the power number, which is defined as

P o = Power number = \frac{P}{ρ Ω^{3} d_{i}^{5}}

$P o = Power number = \frac{P}{ρ Ω^{3} d_{i}^{5}}$

where Ω is the speed of agitation in revolutions per second (r.p.s). The power number is usually correlated as a function of the agitator Reynolds number, defined as

R e = \frac{d_{I}^{2} Ω ρ}{μ}

$R e = \frac{d_{I}^{2} Ω ρ}{μ}$

The suggested form of the correlation for power consumption is

Po = f(Re)

Charts of Po versus Re are available in many common reference books (for example, Nagata, 1975). For large Reynolds numbers (Re > 10⁵), the power number becomes independent of the Reynolds number. The explanation is that the viscous effects well captured by the Reynolds number are not so important at these conditions. An asymptotic value of 6.3 is often observed for the power number. Hence the power consumption can be calculated as

P = 6.3 ρ Ω^{3} d_{I}^{5}

$P = 6.3 ρ Ω^{3} d_{I}^{5}$

This provides a simple formula for power consumption and also provides a useful scale-up rule. We find the power consumption is proportional to $d_{I}^{5}$ $d_{I}^{5}$ while the dependence on agitation speed is third power.

For gas–liquid dispersions in agitated vessels, the gas flow rate Q_G is also important. An additional group, the flow number, defined in the following, is also needed to correlate the data:

Flow number = \frac{Q_{G}}{Ω d_{I}^{3}}

$Flow number = \frac{Q_{G}}{Ω d_{I}^{3}}$

Hence the power number is then correlated as a function of the Reynolds number and the flow number.

The power consumption in a gas aerated system is smaller than that for a gas free tank. Calderbank (1958) suggested the following correlation to include the effect of gas sparging:

P = ψP₀

where ψ is a correction factor to include the effect of gas flow. This factor in turn was correlated as

ψ = 1.0 - 1 2.6 \frac{Q_{G}}{Ω d_{I}^{3}} for \frac{Q_{G}}{Ω d_{I}^{3}} < 3.5 \times 10^{- 2}

$ψ = 1.0 - 1 2.6 \frac{Q_{G}}{Ω d_{I}^{3}} for \frac{Q_{G}}{Ω d_{I}^{3}} < 3.5 \times 10^{- 2}$

and

ψ = 0.62 - 1.85 \frac{Q_{G}}{Ω d_{I}^{3}} for \frac{Q_{G}}{Ω d_{I}^{3}} > 3.5 \times 10^{- 2}

$ψ = 0.62 - 1.85 \frac{Q_{G}}{Ω d_{I}^{3}} for \frac{Q_{G}}{Ω d_{I}^{3}} > 3.5 \times 10^{- 2}$

The following correlations for the mass transfer coefficient are commonly used at the first level of analysis:

\begin{matrix} k_{L} = 0.42 [\frac{(ρ_{L} - ρ_{G}) μ_{L} g}{ρ_{L}^{2}}] S c^{- .5} \\ a_{g l} = 1.4 4 σ^{- 0.6} ρ_{L}^{0.2} (u_{G} / υ_{B})^{0.5} (P / V)^{0.4} \end{matrix}

$\begin{matrix} k_{L} = 0.42 [\frac{(ρ_{L} - ρ_{G}) μ_{L} g}{ρ_{L}^{2}}] S c^{- .5} \\ a_{g l} = 1.4 4 σ^{- 0.6} ρ_{L}^{0.2} (u_{G} / υ_{B})^{0.5} (P / V)^{0.4} \end{matrix}$

Note that the intrinsic mass transfer coefficient does not depend on the power input while the interfacial area depends on the power per unit volume (P/V) to an exponent of 0.4. An equal value of P/V is often used as a criteria for scale-up from small vessel to large vessel since it then provides the same value for the volumetric mass transfer coefficient.

9.10 Gas–Liquid Mass Transfer in a Packed Bed Absorber

Gas absorption columns are usually packed with irregularly shaped solid particles whose role is to break up the liquid into flow as rivulets and thereby promote gas–liquid mass transfer. Common packings include Raschig rings, Intalox saddles, Pall rings, and Berl saddles. The mass transfer phenomena is complex and depends on many parameters, for example packing geometry and the resulting flow pattern, wettability of the packing, and so on. Hence the approach is more empirical rather than theoretical and a number of correlations have been proposed. For illustration, we show the equation developed by Onda et al. (1968), which is usually within 20% of the experimental data. The coefficients for the liquid side and the gas side are correlated separately and an overall coefficient is calculated as

\frac{1}{K_{y}} = \frac{1}{k_{y}} + \frac{m}{k_{x}}

$\frac{1}{K_{y}} = \frac{1}{k_{y}} + \frac{m}{k_{x}}$

This is a formula based on two-film theory, discussed in Chapter 6. In addition a correlation for the interfacial transfer area is also needed. These are summarized next.

9.10.1 Liquid Side Coefficient

The parameter k_x is correlated as

\begin{matrix} k_{x} = 0.0 51 C_{L} (ν_{L} g)^{1 / 3} (a_{p} d_{p})^{0.4} R e_{L}^{2 / 3} S c_{L}^{- 1 / 2} & (9.18) \end{matrix}

$\begin{matrix} k_{x} = 0.0 51 C_{L} (ν_{L} g)^{1 / 3} (a_{p} d_{p})^{0.4} R e_{L}^{2 / 3} S c_{L}^{- 1 / 2} & (9.18) \end{matrix}$

The variables used in the above equations are defined in the following. C_L is the total molar concentration of the liquid phase. a_p is the specific surface area of the packing per unit column volume m^–1 and d_p is the nominal size of the packing. Data for common packings are available from the packing manufacturers.

The Reynolds number for the liquid Re_L is based on a_p and is defined as

R e_{L} = \frac{u_{L} ρ_{L}}{a_{p} μ_{L}}

$R e_{L} = \frac{u_{L} ρ_{L}}{a_{p} μ_{L}}$

where u_L is the superficial liquid velocity.

9.10.2 Gas Side Coefficient

The parameter k_y is correlated as

\begin{matrix} k_{y} = 5.2 3 C_{G} (ν_{G} a_{p}) (a_{p} d_{p})^{- 2} {Re}_{G}^{0.7} {S c}_{G}^{- 2 / 3} & (9.19) \end{matrix}

$\begin{matrix} k_{y} = 5.2 3 C_{G} (ν_{G} a_{p}) (a_{p} d_{p})^{- 2} {Re}_{G}^{0.7} {S c}_{G}^{- 2 / 3} & (9.19) \end{matrix}$

C_G is the total molar concentration of the gas phase. The Reynolds number for the gas phase Re_G is also based on a_p and is defined as

{Re}_{G} = \frac{u_{G ρ G}}{a_{p} μ_{G}}

${Re}_{G} = \frac{u_{G ρ G}}{a_{p} μ_{G}}$

where u_G is the superficial liquid velocity.

9.10.3 Transfer Area

Onda et al. (1967) proposed the following correlation for the gas–liquid interfacial area:

\frac{a_{g l}}{a_{p}} = 1 - exp [- 1.45 {Re}_{L}^{0.1} {F r}_{L}^{- 0.05} W e_{L}^{0.2} {(σ_{c} / σ)}^{0.75}]

$\frac{a_{g l}}{a_{p}} = 1 - exp [- 1.45 {Re}_{L}^{0.1} {F r}_{L}^{- 0.05} W e_{L}^{0.2} {(σ_{c} / σ)}^{0.75}]$

Note that the left-hand side represents the fraction of the surface area of the packing that is wetted by the liquid. The additional parameters needed in the preceding equation are defined as follows:

Froude number = F r_{L} = \frac{α_{p} u_{L}^{2}}{g}

$Froude number = F r_{L} = \frac{α_{p} u_{L}^{2}}{g}$

weber number = W e_{L} = \frac{u_{L}^{2} ρ L}{σ a_{p}}

$weber number = W e_{L} = \frac{u_{L}^{2} ρ L}{σ a_{p}}$

where σ is the surface tension of the liquid and σ_c is a critical surface tension of the packing material for which values are available. Typical values for illustration are 0.061 N/m for ceramic, 0.033 for polyethylene and 0.075 for steel.

Summary

The mass transfer coefficient is defined as flux (under low flux conditions) divided by a suitably defined concentration difference. The difference between the surface concentration and the bulk concentration is used for external flow while the difference in surface concentration and cup mixing concentration is used for internal flows.
The mass transfer coefficient may vary along the surface and a local value can be ascribed as a function of the length along the surface. In practice however it is convenient to use an average value for a given length or geometry in conjunction with mesoscopic or macroscopic balances. The variation along the length is strong in laminar flow but very small in turbulent flows.
In laminar pipe flows an asymptotic value of the mass transfer coefficient is reached for long pipes and the value does not depend on the flow velocity.
Key dimensionless parameters to correlate mass transfer data are the Sherwood number, the Reynolds number, and the Schmidt number. Data can also be correlated using the Stanton number or the j-factor.
The convective mass transfer in boundary layers in laminar flow over a flat plate is a well studied problem and is amenable to a complete theoretical analysis, which is presented in Chapter 11. In this chapter the correlations are shown for laminar and turbulent flows for both local and average values.
Mass transfer into a falling film is an important problem in convective mass transfer. The solid dissolution at the walls and the gas absorption at the interface are the two important prototype problems. The length of the wall (contactor) is usually small and hence an asymptotic entry region model type is generally used for these cases. The local and average value of the mass transfer coefficient can be predicted from such a model as shown in the next chapter.
It is important to note the dependency on the diffusion coefficient in film flow. In the solid dissolution (no slip boundary) case the dependency is to the power of 2/3 while in the gas absorption (no shear boundary) case it is 1/2.
Mass transfer from a solid sphere is a complex problem to compute theoretically but in practice many empirically based correlations (e.g., Brian, Hales, and Sherwood 1969; Levich, 1962) are available; these can be used to estimate this parameter.
Gas–liquid mass transfer from a single bubble is complex as the bubble surface is a deformable surface, in contrast to a solid surface. Further, there may be internal circulation flow within the bubble. The correlation of Calderbank and Moo-Young (1961) is commonly used which is presented in this chapter.
For gas–liquid mass transfer in process equipment, a volumetric mass transfer value is often used. This is a product of the intrinsic mass transfer coefficient and the gas–liquid interfacial area per unit volume of the reactor. Illustrative correlations for bubble column reactors and stirred tank reactors are presented in this chapter.
For the gas–liquid mass transfer in a packed column, separate correlations are proposed for the intrinsic liquid side and gas side coefficient and for the transfer area. The correlation of Onda et al. (1967, 1968) shown in this chapter is useful to get an estimate of these quantities and the overall volumetric coefficient can then be calculated by combining these.

Review Questions

9.1 What representative concentration difference is used in pipe flow mass transfer?

9.2 Define the cup mixing concentration and indicate its use.

9.3 How does the local coefficient vary with the distance along a plate for external flow over a flat plate? Assume laminar conditions.

9.4 How does the local coefficient vary with the axial distance in a long pipe? State the answer for both laminar flow and turbulent flow.

9.5 Given an expression for the local mass transfer coefficient, how can the average mass transfer coefficient be calculated?

9.6 Name and define the various groups used in convective mass transfer.

9.7 What is the relation of the hydraulic diameter to the length or width in a square channel?

9.8 What is the j-factor for a mass transfer coefficient? Why is it commonly used?

9.9 What is the asymptotic value for the Sherwood number for solid dissolution from a long pipe?

9.10 At what point does the flow become unstable in an external flow over a plate?

9.11 How does the mass transfer coefficient vary with local Reynolds number for laminar flow and for turbulent flow over a plate?

9.12 How does the mass transfer coefficient vary with the diffusion coefficient for solid–fluid mass transfer and for fluid–fluid mass transfer?

9.13 When does a gas bubble show strong internal circulation?

9.14 What is Marangoni flow?

9.15 What is the Hadamard-Rybcznski regime?

9.16 For mass transfer from a gas bubble to a liquid, what is the dependency of the mass transfer coefficient on the diffusion coefficient?

9.17 What is the dependency in general of the mass transfer coefficient on power input per unit volume?

9.18 Which parameter changes strongly on power input, k_L or a_gl?

9.19 An agitated vessel is scaled up to ten times the lab scale size. By what factor does power consumption increase? By what factor does power consumption per unit volume increase? Assume that geometric similarity is maintained.

9.20 What is the physical interpretation of the Froude and Weber numbers?

Problems

9.1 Mass transfer cofficients for a pipe. Calculate the mass transfer coefficient form a bulk liquid to pipe wall for the following conditions: pipe diameter = 1 cm; length of the pipe = 100 cm; D_A = 2 × 10^–9 m²/s. The physical properties are similar to water and the range of velocity from 1 to 20 cm/sec is to be examined. Repeat if the fluid is a gas with properties similar to air with D_A = 2 × 10^–5 m²/s.

9.2 Mass transfer over the surface of a pond. An open tank is rectangular with a length of 100 m and long in the other direction. Air at 300 K and 1 atm flows parallel to the surface of the pond with a bulk velocity of 6 m/sec. Determine the position in the tank at which the flow is no longer laminar. For further calculations. assume that the flow is fully turbulent beyond this point although there is a transition flow region in between. Calculate and plot the local and average value of the gas side mass transfer coefficient from the leading edge to the edge of the plate. Use ν = 0.15 cm²/s and D = 0.085 cm²/s.

9.3 Mass transfer for flow over a plate. A thin 1.0 mm coat of fresh paint has just been sprayed over a 1.5 m by 1.5 m square steel body part that can be modeled as a flat surface. The paint contains benzene as the solvent and exerts a partial pressure of 0.137 atm at the process temperature of 300 K. Air at 1 atm pressure at a velocity of 1 m/s is blown over the surface. Determine the average mass transfer convective coefficient and the solvent evaporation rate from the surface in g/min.

9.4 Mass transfer from a solid and from a gas in film flow. Consider a liquid flowing down a vertical wall at a rate of 1 × 10^–5 m²/s per meter unit width. The wall is coated with benzoic acid. Find the local mass transfer coefficient as a function of distance from the wall. Find the average mass transfer coefficient for a wall of 50 cm. Use D = 2 × 10^–9 m²/s, C_As = 20 mol/m³, and the physical properties of water.

Repeat the analysis of the this problem if instead a gas such as oxygen is being absorbed into the liquid film.

9.5 Film thickness and mass transfer. Water trickling down a thin vertical wall is used in pollution treatment (an example of a trickle bed filter). Consider water flowing at a rate of 1. × 10^–6 m²/s per unit meter width of vertical wall. Water is exposed to air and there is mass transfer of oxygen into the liquid from the interface. Find the thickness of the film. Find the value of the local mass transfer coefficient at a point 20 cm from the entrance. Based on this value find the hypothetical thickness of the mass transfer film thickness. Compare this to the hydrodynamic film thickness and explain the difference. Use the following values for the physical properties: μ = 0.001 P a.s; D = 2 × 10^–9 m²/s.

9.6 Intrinsic mass transfer coefficient in a bubble column. Consider a bubble column with gas and liquid properties similar to that for air and water. The diffusion coefficient of the gas in the liquid is 2 × 10^–9 m²/s. The gas superficial velocity is 5 cm/sec. Estimate the following parameters: the bubble size d_B; the bubble rise velocity υ_B; exposure or contact time d_B/υ_B; mass transfer coefficient based on the penetration model; and mass transfer coefficient using the equation of Calderbank and Moo-Young.

9.7 Volumetric mass transfer coefficient in a bubble column. For a system with properties similar to air and water and D = 2 × 10^–9 m²/s calculate and plot the volumetric mass transfer coefficient k_La_gl as a function of superficial gas velocity in the range of 5 to 15 cm/s.

9.8 Power consumption and mass transfer in an agitated tank. Calculate the power consumption and the gas–liquid mass transfer coefficient as a function of speed of agitation for the following conditions: tank diameter = 10 cm, impeller diameter = 5 cm, and superficial gas velocity = 0.2 cm/s. Physical properties may be assumed to be the same as that for an air-water system.

9.9 Scale-up of an agitated tank. The mass transfer coefficient in an agitated vessel of 10 cm diameter was measured as 0.2 s^–1 at a speed of revolution of 10 rps. The impeller diameter is usually half the tank diameter and the height of clear liquid is 10 cm. What should be the speed of agitation to have the same mass transfer coefficient in a larger vessel of 100 cm diameter? Assume geometric similarity.

9.10 Mass transfer in a packed bed reactor. The correlation of Gupta and Thodos (1963) is suitable for mass transfer to a solid in a packed bed:

ϵ j_{D} = \frac{2.06}{{Re}^{0.505}}

$ϵ j_{D} = \frac{2.06}{{Re}^{0.505}}$

Here ∊ is the void fraction or the bed porosity. The Reynolds number Re is evaluated using the packing diameter and superficial velocity. Consider a tube of 2.5 cm internal diameter packed with alumina pellets of 3 mm diameter where the void fraction of the bed is 0.38. Air flows through the tube with a superficial mass velocity of 1.3 kg/m²s at 600 K. Find the j_D factor and the mass transfer coefficient if the diffusivity of the reacting gas is D_A = 2 × 10^–5 m²/s.

9.11 Mass transfer in a fluid bed reactor. Mass transfer data from a solid to a gas in a fluidized bed is given by the following correlation:

ϵ j_{D} = 0.010 + \frac{0.863}{{Re}^{0.585} - 0.483}

$ϵ j_{D} = 0.010 + \frac{0.863}{{Re}^{0.585} - 0.483}$

Based on this equation, calculate the gas to solid mass transfer coefficient in a fluidized bed reactor operating with 1 mm catalyst particles at a gas velocity of 15 cm/sec. The bed porosity ∊ is 0.42.

9.12 Mass transfer coefficient in a packed absorber. Evaluate k_x, k_y, and the transfer area a_gl in a packed column of 1.2 m² cross-sectional area. The column is packed with ceramic Berl saddles of 0.038 m nominal diameter and provides a specific surface area of 125 m^–1. The flow rate of the gas and the liquid are 0.35 kg/s and 1.90 kg/s with properties similar to air and water. The tower operates at 1 atm pressure and 300 K temperature. The critical surface tension of ceramic is 0.061 N/m.

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Table of Contents for Chapter 9. Basics of Convective Mass Transport

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Chapter 9. Basics of Convective Mass Transport

9.1 Definitions for External and Internal Flows

9.2 Relation to Differential Model

9.3 Key Dimensionless Groups

9.3.1 Other Derived Dimensionless Groups

9.4 Mass Transfer in Flows in Pipes and Channels

9.4.1 Laminar Flow

9.4.2 Turbulent Flow

9.4.3 Channel Flow

9.5 Mass Transfer in Flow over a Flat Plate

9.5.1 Laminar Flow

9.5.2 Turbulent Flow

9.5.3 The j-Factor

9.6 Mass Transfer for Film Flow

9.6.1 Solid to Liquid

9.6.2 Gas to Liquid

9.7 Mass Transfer from a Solid Sphere

9.8 Mass Transfer from a Gas Bubble

9.8.1 Bubble Swarms and Bubble Columns

Mass Transfer Coefficient

Bubble Diameter

Power Input

Relative Velocity

Interfacial Area

Gas Holdup

9.9 Mass Transfer in Mechanically Agitated Tanks

9.10 Gas–Liquid Mass Transfer in a Packed Bed Absorber

9.10.1 Liquid Side Coefficient

9.10.2 Gas Side Coefficient

9.10.3 Transfer Area

Summary

Review Questions

Problems

Table of Contents for
Chapter 9. Basics of Convective Mass Transport