14. Mass Transfer Limitations in Reacting Systems

Giving up is the ultimate tragedy.

—Robert J. Donovan

or

It ain’t over ’til it’s over.

—Yogi Berra NY Yankees

F_Az = A_c W_Az

The Algorithm

1. Mole balance

2. Rate law

3. Stoichiometry

4. Combine

5. Evaluate

In the previous chapters, we have only considered plug flow with no diffusion superimposed, in which case

We now drop the plug-flow assumption and extend our discussion of mass transfer in catalytic and other mass-transfer limited reactions. In Chapter 10 we focused on the middle three steps (3, 4, and 5) in a catalytic reaction and neglected steps (1), (2), (6), and (7) by assuming the reaction was surface-reaction limited. In this chapter we describe the first and last steps (1) and (7), as well as showing other applications in which mass transfer plays a role.

We want to arrive at the mole balance that incorporates both diffusion and reaction effects, such as Equation (14-16) on page 685. I.e.,

We begin with Section 14.1.1 where we write the mole balance on Species A in three dimensions in terms of the molar flux, W_A. In Section 14.1.2 we write W_A in terms of the bulk flow of A in the fluid, B_A and the diffusion flux J_A of A that is superimposed on bulk flow. In Section 14.1.3 we use the previous two subsections as a basis to finally write the molar flux, W_A, in terms of concentration using Fick’s first law, J_A, and the bulk flow, B_A. Next, in Section 14.2 we combine diffusion convective transport and reaction in our mole balance.

We now apply the mole balance to species A, which flows and reacts in an element of volume ΔV = ΔxΔyΔz to obtain the variation of the molar fluxes in three dimensions.

Mole Balance

where r_A is the rate of generation of A by reaction per unit volume (e.g., mol/m³/h).

Dividing by ΔxΔyΔz and taking the limit as they go to zero, we obtain the molar flux balance in rectangular coordinates

The corresponding balance in cylindrical coordinates with no variation in the rotation about the z-axis is

COMSOL

We will now evaluate the flux terms W_A. We have taken the time to derive the molar flux equations in this form because they are now in a form that is consistent with the partial differential equation (PDE) solver COMSOL, which is accessible from the CRE Web site.

Total flux = diffusion + bulk motion

The bulk-flow term for species A is the total flux of all molecules relative to a fixed coordinate times the mole fraction of A, y_A; i.e., B_A = y_A ∑ W_i.

For a two-component system of A diffusing in B, the flux of A is

The diffusional flux, J_A, is the flux of A molecules that is superimposed on the bulk flow. It tells how fast A is moving ahead of the bulk flow velocity, i.e., the molar average velocity.

The flux of species A, W_A, is wrt a fixed coordinate system (e.g., the lab bench) and is just the concentration of A, C_A, times the particle velocity of species A, U_A, at that point

By particle velocities, we mean the vector average of millions of molecules of A at a given point. Similarity for species B: W_B = U_BC_B; substituting into the bulk-flow term

B_A = y_A ∑ W_i = y_A (W_A + W_B) = y_A (C_A U_A + C_BU_B)

Writing the concentration of A and B in the generic form in terms of the mole fraction, y_i, and the total concentration, c, i.e., C_i = y_ic, and then factoring out the total concentration, c, the bulk flow, B_A, is

B_A = (c y_A)(y_A U_A + y_BU_B) = C_A U

Molar average velocity

where U is the molar average velocity: U = ∑ y_i U_i. The molar flux of A can now be written as

We now need to determine the equation for the molar flux of A, J_A, that is superimposed on the molar average velocity.

Experimentation with frog legs led to Fick’s first law.

In rectangular coordinates, the gradient is in the form

Constitutive equations in heat, momentum, and mass transfer

The mass transfer law for the diffusional flux of A resulting from a concentration gradient is analogous to Fourier’s law for heat transfer and is given by Fick’s first law

D_AB is the diffusivity of A in B . Combining Equations (14-7) and (14-6), we obtain an expression for the molar flux of A in terms of concentration for constant total concentration

Molar flux equation

In one dimension, i.e., z, the molar flux term is

Now the task is to evaluate the bulk-flow term.

TABLE 14-1 EVALUATING W_A FOR SPECIES A DIFFUSING IN SPECIES B

F_Az = A_cW_Az

In terms of concentration, the flux is

The molar flow rate is

Similar expressions follow for W_Ax and W_Ay. Substituting for the flux W_Ax, W_Ay, and W_Az into Equation (14-2), we obtain

Flow, diffusion, and reaction

This form is used in COMSOL Multiphysics.

Equation (14-15) is in a user-friendly form to apply to the PDE solver, COMSOL. For one dimension at steady state, Equation (14-15) reduces to

In order to solve Equation (14-16) we need to specify the boundary conditions. In this chapter we will consider some of the simple boundary conditions, and in Chapter 18 we will consider the more complicated boundary conditions, such as the Danckwerts’ boundary conditions.

We will now use this form of the molar flow rate in our mole balance in the z direction of a tubular flow reactor

However, we first have to discuss the boundary conditions in solving this equation.

TABLE 14-2 TYPES OF BOUNDARY CONDITIONS

² For further discussion of mass-transfer fundamentals, see R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed. (New York: Wiley, 2002).

³ E. N. Fuller, P. D. Schettler, and J. C. Giddings, Ind. Eng. Chem., 58(5), 19 (1966). Several other equations for predicting diffusion coefficients can be found in B. E. Polling, J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed. (New York: McGraw-Hill, 2001).

⁴ R. H. Perry and D. W. Green, Chemical Engineer’s Handbook, 7th ed. (New York: McGraw-Hill, 1999).

⁵ To estimate liquid diffusivities for binary systems, see K. A. Reddy and L. K. Doraiswamy, Ind. Eng. Chem. Fund., 6, 77 (1967).

It is important to know the magnitude and the T and P dependence of the diffusivity.

Gas:

Liquid:

Step 1: Perform a differential mole balance on a particular species A or use the general mole balance, to obtain an equation for W_Az, e.g., Equation (14-2).

Step 2: Replace W_Az by the appropriate expression for the concentration gradient.

Step 3: State the boundary conditions.

Step 4: Solve for the concentration profile.

Step 5: Solve for the molar flux.

Steps in modeling mass transfer

In Section 14.3, we are going to apply this algorithm to one of the most important cases, diffusion through a boundary layer. Here, we consider the boundary layer to be a hypothetical “stagnant film” in which all the resistance to mass transfer is lumped.

⁶ E. L. Cussler, Diffusion Mass Transfer in Fluid Systems, 2nd ed. (New York: Cambridge University Press, 1997).

Use Table 14-4 to In Out of the algorithm (Steps 1 → 6) to generate creative solutions.

The purpose of presenting algorithms (e.g., Table 14-4) to solve reaction engineering problems is to give the readers a starting point or framework with which to work if they were to get stuck. It is expected that once readers are familiar and comfortable using the algorithm/framework, they will be able to move in and out of the framework as they develop creative solutions to nonstandard chemical reaction engineering problems.

Expanding the previous six modeling steps just a bit

TABLE 14-4 STEPS IN MODELING CHEMICAL SYSTEMS WITH DIFFUSION AND REACTION

A reasonable representation of the concentration profile for a reactant A diffusing to the external surface is shown in Figure 14-2. As illustrated, the change in concentration of A from C_Ab to C_As takes place in a very narrow fluid layer next to the surface of the sphere. Nearly all of the resistance to mass transfer is found in this layer.

The concept of a hypothetical stagnant film within which all the resistance to external mass transfer exists

A useful way of modeling diffusive transport is to treat the fluid layer next to a solid boundary as a hypothetical stagnant film of thickness δ, which we cannot measure. We say that all the resistance to mass transfer is found (i.e., lumped) within this hypothetical stagnant film of thickness δ, and the properties (i.e., concentration, temperature) of the fluid at the outer edge of the film are identical to those of the bulk fluid. This model can readily be used to solve the differential equation for diffusion through a stagnant film. The dashed line in Figure 14-2b represents the concentration profile predicted by the hypothetical stagnant film model, while the solid line gives the actual profile. If the film thickness is much smaller than the radius of the pellet (which is usually the case), curvature effects can be neglected. As a result, only the one-dimensional diffusion equation must be solved, as was shown in Figure 14-3.

We are going to carry out a mole balance on species A diffusing through the fluid between z = z and z = z + Δz at steady state for the unit cross-sectional area, A_c

dividing by Δz and taking the limit as Δz → 0

For diffusion through a stagnant film at dilute concentrations

or for EMCD, we have using Fick’s first law

Substituting for W_Az and dividing by D_AB we have

Integrating twice to get C_A = K₁z + K₂, using the boundary conditions at

we obtain the concentration profile

To find the flux to the surface we substitute Equation (14-25) into Equation (14-24) to obtain

At steady state the flux of A to the surface will be equal to the rate of reaction of A on the surface. We also note that another example of diffusion through a stagnant film as applied to transdermal drug delivery is given in the Chapter 14 Expanded Material on the CRE Web site.

While the boundary-layer thickness will vary around the sphere, we will take it to have a mean film thickness δ. The ratio of the diffusivity D_AB to the film thickness δ is the mass transfer coefficient, k_c, that is,

The mass transfer coefficient

Combining Equations (14-26) and (14-27), we obtain the average molar flux from the bulk fluid to the surface

Molar flux of A to the surface

In this stagnant film model, we consider all the resistance to mass transfer to be lumped into the thickness δ. The reciprocal of the mass transfer coefficient can be thought of as this resistance

For forced convection, the heat transfer coefficient is normally correlated in terms of three dimensionless groups: the Nusselt number, Nu; the Reynolds number, Re; and the Prandtl number, Pr. For the single spherical pellets discussed here, Nu and Re take the following forms

The Prandtl number is not dependent on the geometry of the system

The Nusselt, Prandtl, and Reynolds numbers are used in forced convection heat transfer correlations.

The other symbols are as defined previously.

The heat transfer correlation relating the Nusselt number to the Prandtl and Reynolds numbers for flow around a sphere is⁷

⁷ W. E. Ranz and W. R. Marshall, Jr., Chem. Eng. Prog., 48, 141–146, 173–180 (1952).

Although this correlation can be used over a wide range of Reynolds numbers, it can be shown theoretically that if a sphere is immersed in a stagnant fluid (Re = 0), then

and that at higher Reynolds numbers in which the boundary layer remains laminar, we can neglect the 2 in Equation (14-34), in which case he Nusselt number becomes

Converting a heat transfer correlation to a mass transfer correlation

Although further discussion of heat transfer correlations is no doubt worthwhile, it will not help us to determine the mass transfer coefficient and the mass flux from the bulk fluid to the external pellet surface. However, the preceding discussion on heat transfer was not entirely futile because, for similar geometries, the heat and mass transfer correlations are analogous. If a heat transfer correlation for the Nusselt number exists, the mass transfer coefficient can be estimated by replacing the Nusselt and Prandtl numbers in this correlation by the Sherwood and Schmidt numbers, respectively:

The heat and mass transfer coefficients are analogous.^† The corresponding fluxes are

^† Strictly speaking, replacing the Nusselt number by the Sherwood number is only valid for situations where the Lewis number, Le, is close to 1.

The one-dimensional differential forms of the mass flux for EMCD and the heat flux are, respectively,

For EMCD the heat and molar flux equations are analogous.

If we replace h by k_c and k_t by D_AB in Equation (14-30), i.e.,

we obtain the mass transfer Nusselt number (i.e., the Sherwood number)

Sherwood number

The Prandtl number is the ratio of the kinematic viscosity (i.e., the momentum diffusivity) to the thermal diffusivity. Because the Schmidt number is analogous to the Prandtl number, one would expect that Sc is the ratio of the momentum diffusivity (i.e., the kinematic viscosity), υ, to the mass diffusivity D_AB. Indeed, this is true

The Schmidt number is

Schmidt number

Consequently, the correlation for mass transfer for flow around a spherical pellet is analogous to that given for heat transfer, Equation (14-33); that is,

This relationship is often referred to as the Frössling correlation.⁸

⁸ N. Frössling, Gerlands Beitr. Geophys., 52, 170 (1938).

A few additional correlations for the Sherwood number from which one can determine the mass transfer coefficient are given in Table 14-5.

The Sherwood, Reynolds, and Schmidt numbers are used in forced convection mass transfer correlations.

⁹ A comprehensive list of correlations for mass transfer to particles is given by G. A. Hughmark, Ind. Eng. Chem. Fund., 19(2), 198 (1980).

In Example 14-1, the surface reaction was extremely rapid and the rate of mass transfer to the surface dictated the overall rate of reaction. We now consider a more general case. The isomerization

is taking place on the surface of a solid sphere (Figure 14-4). The surface reaction follows a Langmuir–Hinshelwood single-site mechanism for which the rate law is

The temperature is sufficiently high that we only need to consider the case of very weak adsorption (i.e., low surface coverage) of A and B; thus

Therefore, the rate law becomes apparent first order

Using boundary conditions 2b and 2c in Table 14-1, we obtain

The concentration C_As is not as easily measured as the bulk concentration. Consequently, we need to eliminate C_As from the equation for the flux and rate of reaction. Solving Equation (14-44) for C_As yields

and the rate of reaction on the surface becomes

Molar flux of A to the surface is equal to the rate of consumption of A on the surface.

One will often find the flux to or from the surface written in terms of an effective transport coefficient k_eff

where

Rapid Reaction. We first consider how the overall rate of reaction may be increased when the rate of mass transfer to the surface limits the overall rate of reaction. Under these circumstances, the specific reaction rate constant is much greater than the mass transfer coefficient

and

To increase the rate of reaction per unit surface area of a solid sphere, one must increase C_A and/or k_c . In this gas-phase catalytic reaction example, and for most liquids, the Schmidt number is sufficiently large that the number 2 in Equation (14-40) is negligible with respect to the second term when the Reynolds number is greater than 25. As a result, Equation (14-40) gives

It is important to know how the mass transfer coefficient varies with fluid velocity, particle size, and physical properties.

Term 1 is a function of the physical properties D_AB and υ, which depend on temperature and pressure only. The diffusivity always increases with increasing temperature for both gas and liquid systems. However, the kinematic viscosity υ increases with temperature (υ ∝ T^3/2) for gases and decreases exponentially with temperature for liquids. Term 2 is a function of flow conditions and particle size. Consequently, to increase k_c and thus the overall rate of reaction per unit surface area, one may either decrease the particle size or increase the velocity of the fluid flowing past the particle. For this particular case of flow past a single sphere, we see that if the velocity is doubled, the mass transfer coefficient and consequently the rate of reaction is increased by a factor of

(U₂/U₁)^0.5 = 2^0.5 = 1.41 or 41%

Mass Transfer Limited

Slow Reaction. Here, the specific reaction rate constant is small with respect to the mass transfer coefficient

Reaction Rate Limited

The specific reaction rate is independent of the velocity of fluid and for the solid sphere considered here, independent of particle size. However, for porous catalyst pellets, k_r may depend on particle size for certain situations, as shown in Chapter 15. We will continue this discussion in Section 14.5.

Mass transfer effects are not important when the reaction rate is limiting.

Figure 14-5 shows the variation in reaction rate with Term 2 in Equation (14-49), the ratio of velocity to particle size. At low velocities, the mass transfer boundary-layer thickness is large and diffusion limits the reaction. As the velocity past the sphere is increased, the boundary-layer thickness decreases, and the mass transfer across the boundary layer no longer limits the rate of reaction. One also notes that for a given (i.e., fixed) velocity, reaction-limiting conditions can be achieved by using very small particles. However, the smaller the particle size, the greater the pressure drop in a packed bed. When one is obtaining reaction-rate data in the laboratory, one must operate at sufficiently high velocities or sufficiently small particle sizes to ensure that the reaction is not mass transfer–limited when collecting data.

When collecting rate-law data, operate in the reaction-limited region.

carried out in a packed-bed reactor (Figure 14-6). A steady-state mole balance on reactant A in the reactor segment between z and z + Δz is

¹⁰

¹⁰ In the nomenclature for Chapter 4, for the Ergun equation for pressure drop.

Dividing Equation (14-51) by A_c Δz and taking the limit as Δz → 0, we have

We now need to express F_Az and in terms of concentration.

The molar flow rate of A in the axial direction is

In almost all situations involving flow in packed-bed reactors, the amount of material transported by diffusion or dispersion in the axial direction is negligible compared with that transported by convection (i.e., bulk flow)

Axial diffusion is neglected.

(In Chapter 18 we consider the case when dispersive effects (e.g., diffusion) must be taken into account.) Neglecting dispersion, Equation (14-14) becomes

where U is the superficial molar average velocity through the bed (m/s). Substituting for F_Az in Equation (14-52) gives us

For the case of constant superficial velocity U

Differential equation describing flow and reaction in a packed bed

For reactions at steady state, the molar flux of A to the particle surface, W_Ar (mol/m² · s) (see Figure 14-7), is equal to the rate of disappearance of A on the surface – (mol/m² · s); that is

From Section 14.4, the boundary condition at the external surface is

Substituting for in Equation (14-56), we have

In reactions that are completely mass transfer–limited, it is not necessary to know the rate law.

In most mass transfer–limited reactions, the surface concentration is negligible with respect to the bulk concentration (i.e., )

Integrating with the limit, at z = 0, C_A = C_A0

The corresponding variation of reaction rate along the length of the reactor is

The concentration and conversion profiles down a reactor of length L are shown in Figure 14-8.

Reactor concentration profile for a mass transfer–limited reaction

To determine the reactor length L necessary to achieve a conversion X, we combine the definition of conversion

with the evaluation of Equation (14-61) at z = L to obtain

To learn the effect of flow rate on conversion, we need to know how flow rate affects the mass transfer coefficient. That is, we must determine the correlation for the mass transfer coefficient for the particular geometry and flow field. For flow through a packed bed, the correlation given by Thoenes and Kramers for 0.25 < ϕ < 0.5, 40 < Re′ < 4000, and 1 < Sc < 4000 is¹¹

¹¹ D. Thoenes, Jr. and H. Kramers, Chem. Eng. Sci., 8, 271 (1958).

Thoenes–Kramers correlation for flow through packed beds

For constant fluid properties and particle diameter

We see that the mass transfer coefficient increases with the square root of the superficial velocity through the bed. Therefore, for a fixed concentration, C_A, such as that found in a differential reactor, the rate of reaction should vary with _U^1/2

For diffusion-limited reactions, reaction rate depends on particle size and fluid velocity.

However, if the gas velocity is continually increased, a point is reached where the reaction becomes reaction rate–limited and, consequently, is independent of the superficial gas velocity, as shown in Figure 14-5.

Most mass transfer correlations in the literature are reported in terms of the Colburn J factor (i.e., J_D) as a function of the Reynolds number. The relationship between J_D and the numbers we have been discussing is

Colburn J factor

Figure 14-9 shows data from a number of investigations for the J factor as a function of the Reynolds number for a wide range of particle shapes and gas-flow conditions. Note: There are serious deviations from the Colburn analogy when the concentration gradient and temperature gradient are coupled, as shown by Venkatesan and Fogler.¹²

¹² R. Venkatesan and H. S. Fogler, AIChE J., 50, 1623 (July 2004).

Dwidevi and Upadhyay review a number of mass transfer correlations for both fixed and fluidized beds and arrive at the following correlation, which is valid for both gases (Re > 10) and liquids (Re > 0.01) in either fixed or fluidized beds:¹³

¹³ P. N. Dwidevi and S. N. Upadhyay, Ind. Eng. Chem. Process Des. Dev., 16, 157 (1977).

A correlation for flow through packed beds in terms of the Colburn J factor

For nonspherical particles, the equivalent diameter used in the Reynolds and Sherwood numbers is , where A_p is the external surface area of the pellet.

To obtain correlations for mass transfer coefficients for a variety of systems and geometries, see either D. Kunii and O. Levenspiel, Fluidization Engineering, 2nd ed. (Butterworth-Heinemann, 1991), Chap. 7, or W. L. McCabe, J. C. Smith, and P. Harriott, Unit Operations in Chemical Engineering, 6th ed. (New York: McGraw-Hill, 2000). For other correlations for packed beds with different packing arrangements, see I. Colquhoun-Lee and J. Stepanek, Chemical Engineer, 108 (Feb. 1974).

¹⁵ Prof. J. D. Goddard, University of Michigan, 1963–1976. Currently at University of California, San Diego.

J. D. Goddard’s

Back of the Envelope

To help answer these questions, we recall Equation (14-49) and our discussion on page 696. There, we showed the mass transfer coefficient for a packed bed was related to the product of two terms: Term 1 was dependent on the physical properties and Term 2 was dependent on the system properties. Re-writing Equation (14-41) as

one observes from this equation that the mass transfer coefficient increases as the particle size decreases. The use of sufficiently small particles offers another technique to escape from the mass transfer–limited regime into the reaction-rate-limited regime.

Find out how the mass transfer coefficient varies with changes in physical properties and system properties.

Bad idea!! Robert was right to worry.

For a packed catalyst bed, the temperature-dependence part of the mass transfer coefficient for a gas-phase reaction can be written as

Important concept

Depending on how one fixes or changes the molar feed rate, F_T0, U may also depend on the feed temperature. As an engineer, it is extremely important that you reason out the effects of changing conditions, as illustrated in the preceding two examples.

a. For equimolar counterdiffusion (EMCD) or for dilute concentration of the solute

b. For diffusion through a stagnant gas

c. For negligible diffusion

2. The rate of mass transfer from the bulk fluid to a boundary at concentration C_As is

where k_c is the mass transfer coefficient.

3. The Sherwood and Schmidt numbers are, respectively,

4. If a heat transfer correlation exists for a given system and geometry, the mass transfer correlation may be found by replacing the Nusselt number by the Sherwood number and the Prandtl number by the Schmidt number in the existing heat transfer correlation.

5. Increasing the gas-phase velocity and decreasing the particle size will increase the overall rate of reaction for reactions that are externally mass transfer–limited.

6. The conversion for externally mass transfer–limited reactions can be found from the equation

7. Back-of-the-envelope calculations should be carried out to determine the magnitude and direction that changes in process variables will have on conversion. What if . . .?

1. Transdermal Drug Delivery

2. Shrinking Core Model

3. Diffusion through Film to a Catalyst Particle

4. WP14-(a) Revisit Transdermal Drug Delivery

5. Additional Homework Problems

• Learning Resources

1. Summary Notes

Diffusion through a Stagnant Film

4. Solved Problems

Example CD14-1   Calculating Steady State Diffusion

Example CD14-2   Relative Fluxes W_A, B_A, and J_A

Example CD14-3   Diffusion through a Stagnant Gas

Example CD14-4   Measuring Gas-Phase Diffusivities

Example CD14-5   Diffusion through a Film to a Catalyst Particle

Example CD14-6   Measuring Liquid-Phase Diffusivities

• Professional Reference Shelf

R14.1. Mass Transfer-Limited Reactions on Metallic Surfaces

A. Catalyst Monoliths

B. Wire Gauze Reactors

R14.2. Methods to Experimentally Measure Diffusivities

R14.3. Facilitated Heat Transfer

R14.4. Shrinking Core Model

R14.5. Dissolution of Monodisperse Particles

R14.6. Dissolution of Polydisperse Solids (e.g., pills in the stomach)

(a) Make up your data and reaction.

(b) Use a real reaction and real data.

The journals listed at the end of Chapter 1 may be useful for part (b).

Q14-2 (Seargeant Amberrcromby). Capt. Apollo is piloting a shuttlecraft on his way to space station Klingon. Just as he is about to maneuver to dock his craft using the hydrazine system discussed in Example 14-2, the shuttle craft’s thrusters do not respond properly and it crashes into the station, killing Capt. Apollo (Star Wars 7 (fall 2015)). An investigation reveals that Lt. Darkside prepared the packed beds used to maneuver the shuttle and Lt. Data prepared the hydrazine-helium gas mixture. Foul play is suspected and Sgt. Ambercromby arrives on the scene to investigate.

(a) What are the first three questions he asks?

(b) Make a list of possible explanations for the crash, supporting each one by an equation or reason.

(a) Example 14-1. How would your answers change if the temperature was increased by 50°C, the particle diameter was doubled, and fluid velocity was cut in half? Assume properties of water can be used for this system.

(b) Example 14-2. How would your answers change if you had a 50–50 mixture of hydrazine and helium? If you increase d_p by a factor of 5?

(c) Example 14-3. What if you were asked for representative values for Re, Sc, Sh, and k_c for both liquid- and gas-phase systems for a velocity of 10 cm/s and a pipe diameter of 5 cm (or a packed-bed diameter of 0.2 cm)? What numbers would you give?

(d) Example 14-4. How would your answers change if the reaction were carried out in the liquid phase where kinematic viscosity varied as ?

P14-2_B Assume the minimum respiration rate of a chipmunk is 1.5 micromoles of O₂/min. The corresponding volumetric rate of gas intake is 0.05 dm³/min at STP.

(a) What is the deepest a chipmunk can burrow a 3-cm diameter hole beneath the surface in Ann Arbor, Michigan? D_AB = 1.8 × 10^–5 m²/s

(b) In Boulder, Colorado?

(c) How would your answers to (a) and (b) change in the dead of winter when T = 0°F?

(d) Critique and extend this problem (e.g., CO₂ poisoning). Thanks to Professor Robert Kabel at Pennsylvania State University.

Hint: Review derivations and equations for W_A and W_B to see how they can be applied to this problem.

P14-3_B Pure oxygen is being absorbed by xylene in a catalyzed reaction in the experimental apparatus sketched in Figure P14-3_B. Under constant conditions of temperature and liquid composition, the following data were obtained:

No gaseous products were formed by the chemical reaction. What would you conclude about the relative importance of liquid-phase diffusion and about the order of the kinetics of this reaction? (California Professional Engineers Exam)

P14-4_C In a diving-chamber experiment, a human subject breathed a mixture of O₂ and He while small areas of his skin were exposed to nitrogen gas. After some time, the exposed areas became blotchy, with small blisters forming on the skin. Model the skin as consisting of two adjacent layers, one of thickness δ₁ and the other of thickness δ₂. If counterdiffusion of He out through the skin occurs at the same time as N₂ diffuses into the skin, at what point in the skin layers is the sum of the partial pressures a maximum? If the saturation partial pressure for the sum of the gases is 101 kPa, can the blisters be a result of the sum of the gas partial pressures exceeding the saturation partial pressure and the gas coming out of the solution (i.e., the skin)?

Before answering any of these questions, derive the concentration profiles for N₂ and He in the skin layers.

Diffusivity of He and N₂ in the inner skin layer = 5 × 10^–7 cm²/s and 1.5 × 10^–7 cm²/s, respectively

Diffusivity of He and N₂ in the outer skin layer = 10^–5 cm²/s and 3.3 × 10^–4 cm²/s, respectively

Hint: See Transdermal Drug Delivery in Expanded Material on the CRE Web site.

P14-5_B The decomposition of cyclohexane to benzene and hydrogen is mass transfer–limited at high temperatures. The reaction is carried out in a 5-cm-ID pipe 20 m in length packed with cylindrical pellets 0.5 cm in diameter and 0.5 cm in length. The pellets are coated with the catalyst only on the outside. The bed porosity is 40%. The entering volumetric flow rate is 60 dm³/min.

(a) Calculate the number of pipes necessary to achieve 99.9% conversion of cyclohexane from an entering gas stream of 5% cyclohexane and 95% H₂ at 2 atm and 500°C.

(b) Plot conversion as a function of pipe length.

(c) How much would your answer change if the pellet diameter and length were each cut in half?

(d) How would your answer to part (a) change if the feed were pure cyclohexane?

(e) What do you believe is the point of this problem? Is the focus really green CRE? How so?

Green engineering

P14-6_C Lead titanate, PbTiO₃, is a material having remarkable ferroelectric, pyroelectric, and piezoelectric properties [J. Elec. Chem. Soc., 135, 3137 (1988)]. A thin film of PbTiO₃ was deposited in a CVD reactor. The deposition rate is given below as a function of a temperature and flow rate over the film.

What are all the things, qualitative and quantative, that you can learn from these data?

P14-7_B A plant is removing a trace of Cl₂ from a waste-gas stream by passing it over a solid granulm absorbent in a tubular packed bed (Figure P14-7). At present, 63.2% removal is being acomplished, but it is believed that greater removal could be achieved if the flow rate were increased by a factor of 4, the particle diameter were decreased by a factor of 3, and the packed tube length increased by 50%. What percentage of chlorine would be removed under the proposed scheme? (The chlorine transferring to the absorbent is removed completely by a virtually instantaneous chemical reaction.) (Ans.: 98%)

P14-8_B In a certain chemical plant, a reversible fluid-phase isomerization

is carried out over a solid catalyst in a tubular packed-bed reactor. If the reaction is so rapid that mass transfer between the catalyst surface and the bulk fluid is rate-limiting, show that the kinetics are described in terms of the bulk concentrations C_A and C_B by

It is desired to double the capacity of the existing plant by processing twice the feed of reactant A while maintaining the same fractional conversion of A to B in the reactor. How much larger a reactor, in terms of catalyst weight, would be required if all other operating variables are held constant? You may use the Thoenes–Kramers correlation for mass transfer coefficients in a packed bed.

P14-9_B The irreversible gas-phase reaction

is carried out adiabatically over a packed bed of solid catalyst particles. The reaction is first order in the concentration of A on the catalyst surface

The feed consists of 50% (mole) A and 50% inerts, and enters the bed at a temperature of 300 K. The entering volumetric flow rate is 10 dm³/s (i.e., 10,000 cm³/s). The relationship between the Sherwood number and the Reynolds number is

Sh = 100 Re^1/2

As a first approximation, one may neglect pressure drop. The entering concentration of A is 1.0 M. Calculate the catalyst weight necessary to achieve 60% conversion of A for

(a) isothermal operation.

(b) adiabatic operation.

Additional information:

Kinematic viscosity: μ/ρ = 0.02 cm²/s

Particle diameter: d_p = 0.1 cm

Superficial velocity: U = 10 cm/s

Catalyst surface area/mass of catalyst bed: a = 60 cm²/g-cat

Diffusivity of A: D_e = 10^–2 cm²/s

Heat of reaction: = –10,000 cal/g mol A

Heat capacities: C_pA = C_pB = 25 cal/g mol · K, C _pS (solvent) = 75 cal/g mol · K

k′ (300 K) 0.01 cm³/s ·g-cat with E 4000 cal/mol

P14-10_B Transdermal Drug Delivery. See photo on page 713. The principles of steady-state diffusion have been used in a number of drug-delivery systems. Specifically, medicated patches are commonly attached to the skin to deliver drugs for nicotine withdrawal, birth control, and motion sickness, to name a few. The U.S. transdermal drug-delivery is a multi-billion dollar market. Equations similar to Equation (14-24) have been used to model the release, diffusion, and absorption of the drug from the patch into the body. The figure shown in the Expanded Material on page 713 shows a drug-delivery vehicle (patch) along with the concentration gradient in the epidermis and dermis skin layers.

(a) Use a shell balance to show

(b) Show the concentration profile in the epidermis layer

(c) Show the concentration profile in the dermis layer

(d) Equate the fluxes using and at z = δ₁ to show

(e) What are the concentration profiles in the dermis and epidermis layers?

(f) Show the flux in the dermis layer is

(g) What is the flux in the epidermis layer?

P14-11_D (Estimating glacial ages) The following oxygen-18 data were obtained from soil samples taken at different depths in Ontario, Canada. Assuming that all the ¹⁸O was laid down during the last glacial age and that the transport of ¹⁸O to the surface takes place by molecular diffusion, estimate the number of years since the last glacial age from the following data. Independent measurements give the diffusivity of ¹⁸O in soil as 2.64 × 10^–10 m²/s.

C₀ is the concentration of ¹⁸O at 25 m. Hint: A knowledge of error function solutions may or may not be helpful. (Ans.: t = 5,616 years)

BIRD, R. B., W. E. STEWART, and E. N. LIGHTFOOT, Transport Phenomena, 2nd ed. New York: Wiley, 2002, Chaps. 17 and 18.

Collins, S., Mockingjay (The Final Book of the Hunger Games). New York: Scholastic, 2014.

CUSSLER, E. L., Diffusion Mass Transfer in Fluid Systems, 3rd ed. New York: Cambridge University Press, 2009.

GEANKOPLIS, C. J., Transport Processes and Unit Operations. Upper Saddle River, NJ: Prentice Hall, 2003.

LEVICH, V. G., Physiochemical Hydrodynamics. Upper Saddle River, NJ: Prentice Hall, 1962, Chaps. 1 and 4.

2. Experimental values of the diffusivity can be found in a number of sources, two of which are

PERRY, R. H., D. W. GREEN, and J. O. MALONEY, Chemical Engineers’ Handbook, 8th ed. New York: McGraw-Hill, 2007.

SHERWOOD, T. K., R. L. PIGFORD, and C. R. WILKE, Mass Transfer. New York: McGraw-Hill, 1975.

3. A number of correlations for the mass transfer coefficient can be found in

LYDERSEN, A. L., Mass Transfer in Engineering Practice. New York: Wiley-Interscience, 1983, Chap. 1.

MCCABE, W. L., J. C. SMITH, and P. HARRIOTT, Unit Operations of Chemical Engineering, 6th ed. New York: McGraw-Hill, 2000, Chap. 17.