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7 Principles of Mass Transfer 7-1 Introduction to Mass Transfer Concept of Flux Fundamental Equation for Mass Transfer 7-2 Molecular Diffusion Brownian Motion Fick’s First Law Diffusion in the Presence of Fluid Flow Diffusion in Fixed and Relative Frames of Reference Fick’s Second Law Stokes–Einstein Equation 7-3 Sources for Diffusion Coefficients Liquid-Phase Diffusion Coefficients for Large Molecules and Particles Liquid-Phase Diffusion Coefficients for Small Neutral Molecules Liquid-Phase Diffusion Coefficients for Electrolytes Liquid-Phase Diffusion Coefficient for Oxygen Gas-Phase Diffusion Coefficients 7-4 Models for Mass Transfer at an Interface Surface Area Available for Mass Transfer Film Model Penetration and Surface Renewal Models Boundary Layer Models 7-5 Correlations for Mass Transfer Coefficients at an Interface Common Mass Transfer Correlations Relationship between Mass Transfer Coefficients and Diffusing Species 7-6 Design of Treatment Systems Controlled by Mass Transfer 7-7 Evaluating the Concentration Gradient with Operating Diagrams Contact Modes Development of Operating Diagrams Analysis Using Operating Diagrams MWH’s Water Treatment: Principles and Design, Third Edition John C Crittenden, R Rhodes Trussell, David W Hand, Kerry J Howe and George Tchobanoglous Copyright © 2012 John Wiley & Sons, Inc 391 392 Principles of Mass Transfer 7-8 Mass Transfer across a Gas–Liquid Interface Conditions in Bulk Solution Conditions at Interface Overall Mass Transfer Relationship Determining the Phase That Controls Mass Transfer Application of the Two-Film Model Relationship between Overall Mass Transfer Coefficients and Diffusing Species 7-9 Enhancement of Mass Transfer across an Interface by Chemical Reactions Problems and Discussion Topics References Terminology for Mass Transfer Term Definition Absorption Process in which a solute is transferred from one bulk phase and is homogeneously spread throughout another bulk phase (as opposed to collecting at the interface between phases See which is Adsorption) Solid phase onto which a solute accumulates during the process of adsorption Process in which a solute is transferred from one bulk phase and accumulates at the surface of another phase (such as a solid surface), resulting in an increased concentration of molecules in the immediate vicinity of the surface (in contrast to absorption) Transfer of volatile components from water to air System with no flow in or out during the mass transfer operation Typically, two phases are brought together, mass transfer is allowed to proceed until nearly at equilibrium, and then the phases are separated Random motion of solute molecules or particles caused by collisions with solvent molecules Process in which two phases (e.g., liquid and gas, water and powdered activated carbon) contact each other with their mass flow in the same direction Adsorbent Adsorption Air stripping Batch system Brownian motion Co-current flow Principles of Mass Transfer Term Continuous contact operation Definition A process in which two phases are in continuous contact with each other from the inlet to the outlet of the system, with continuously changing concentrations in each phase as a function of position (e.g., a column filled with adsorbent, a countercurrent packed tower, etc.) Countercurrent flow Process in which two phases (e.g., water and air) contact each other with flow in opposite directions, with contact either in stages or continuously Cross flow Process in which two phases (e.g., water and air) contact each other with flows perpendicular to each other Desorption Mass transfer process involving the removal of substances from an adsorbent surface Diffusion Mass transfer process in which solute molecules or small particles are transported from a region of high concentration to a region of lower concentration as a result of Brownian motion Diffusion coefficient Parameter that relates proportionality of the flux of a solute in a solvent to the concentration gradient Frequently used synonymously with diffusivity Diffusivity Used as a synonym for diffusion coefficient Extracting phase Phase to which compounds are transferred in water treatment (e.g., gas phase for stripping, liquid phase for absorption, solid phase for adsorption or ion exchange) Flow-through system System in which one or both phases flow continuously through it during the mass transfer operation Fluid–fluid process Mass transfer process in which fluid is in contact with another fluid (e.g., air and water in a stripping tower) Fluid–solid process Mass transfer process in which fluid is in contact with a solid (e.g., packing), operated as fixed or fluidized beds Mass transfer Transport of components (molecules, particles, etc.) from one location to another (typically from one phase to another) 393 394 Principles of Mass Transfer Term Definition Staged operation Mass transfer system in which two phases contact each other in discrete steps Typically, the two phases are completely mixed with each other in each stage, and then separated before being sent to the next stage, where they are remixed (often with the two contacting phases traveling in different directions) Dissolved substance Liquid in which other compounds (solutes) are dissolved General term for the many phenomena commonly included under the terms adsorption and absorption when the nature of the phenomenon involved is unknown or indefinite Removal of a component from one phase by transfer to another (such as air stripping, see above) Solute Solvent Sorption Stripping Several water treatment processes involve the transfer of material from one phase to another (i.e., from liquid to gas, or liquid to solid) Aeration and air stripping (Chap 14), adsorption, (Chap 15), ion exchange (Chap 16), and reverse osmosis (Chap 17) are all processes that involve mass transfer between phases In these processes, the contaminant removal efficiency, the rate of separation, and/or the size of the equipment can be governed by the rate of mass transfer Mass transfer, in the broadest possible definition, is the movement of matter from one location to another, and the rate at which this occurs can be the governing factor in treatment processes Consider a contaminant removal process that relies on an instantaneous reaction at a surface Since the reaction is instantaneous, the rate at which the contaminant is degraded is controlled not by the rate of the reaction but by the rate at which the reactants can be transported to the surface Such a process is called mass transfer limited Mass transfer is a complex topic Books have been written about the topic and the chemical engineering curriculum at many universities includes an entire course in mass transfer This chapter focuses on key principles that are relevant to environmental engineering and water treatment processes Topics discussed in this chapter include an introduction to mass transfer, molecular diffusion and diffusion coefficients, models and correlations for mass transfer coefficients, operating diagrams, and mass transfer across a gas–liquid interface with and without chemical reactions 7-1 Introduction to Mass Transfer 395 7-1 Introduction to Mass Transfer To introduce the subject of mass transfer, the concept of flux and the fundamental equation for mass transfer are introduced in this section In mass transfer operations, the movement of matter is measured as flux Mass flux is defined as the amount of material that flows through a unit area per unit time: m (7-1) JA = At where JA = mass flux of solute A across an interface, mg/m2 ·s m = mass of solute A, mg A = area perpendicular to the direction of flow, m2 t = time, s Concept of Flux Because flux is defined per unit area, it is an intensive property (intensive properties, like concentration or temperature, not depend on the size of the system) Thus, for two systems with the same mass flux, the system with the larger amount of area will have more mass transfer Mass flow is the product of the flux and the area: MA = JA A where (7-2) MA = mass flow of solute A, mg/s As will be seen later in this chapter, increasing the surface area is a key method for increasing the rate of mass transfer (and hence, increasing the efficiency of a separation process that relies on mass transfer) In some cases (principally membrane processes), the material moving across the interface is measured in units of volume instead of mass, and the corresponding flux is called a volumetric flux instead of a mass flux An example of units for a volumetric flux is L/m2 ·s Other situations are best described with molar units, where the units of molar flux are mol/m2 ·s Molar fluxes can be converted to mass fluxes by multiplying by the molecular weight Mass transfer occurs in response to a driving force Forces that can move matter include gravity, magnetism, electrical potential, pressure, and others In each case, the flux of material is proportional to the driving force In environmental engineering, the driving force of interest is a concentration gradient or, in more general terms, a gradient in chemical potential, or Gibbs energy When a concentration gradient is present between two phases in contact with each other or between two locations within a single phase, matter will flow from the region of higher concentration to the Fundamental Equation for Mass Transfer 396 Principles of Mass Transfer region of lower concentration at a rate that is proportional to the difference between the two concentrations, as given by the following equation: JA = kf (CA ) where (7-3) JA = mass flux of component A, g/m2 ·s kf = mass transfer coefficient, m/s CA = difference in concentration of component A, mg/L Equation 7-3 has only two components (the mass transfer coefficient and the concentration gradient), and while this equation seems simple, it has profound implications for many treatment processes The bulk of the rest of this chapter is devoted to the examination of variations of this equation The next four sections are devoted to development of the mass transfer coefficient and models that describe mass transfer Following that, Sect 7-6 will explore how operating diagrams can be used to describe the concentration gradient, and the last two sections describe mass transfer across a gas–liquid interface 7-2 Molecular Diffusion In the previous section, it was noted that mass flux is the product of a mass transfer coefficient and a driving force (see Eq 7-3) A special case of mass transfer is molecular diffusion, in which solute molecules or particles flow from a region of higher concentration to a region of lower concentration solely due to kinetic energy of the solution molecules, that is, when no external forces are present to cause fluid movement Molecular diffusion is a fundamental concept in many mass transfer problems Although mass transfer coefficients are often determined using empirical correlations, the correlations are based on models of mass transfer that in turn depend on molecular diffusion at some level, and the diffusion coefficient will be a required parameter As a result, an understanding of molecular diffusion is a necessary part of an understanding of mass transfer Several important concepts related to molecular diffusion, including Brownian motion, Fick’s first and second laws, and the Stokes–Einstein equation, are described in this section Brownian Motion Brownian motion is the random motion of a particle or solute molecule due to the internal energy of the molecules in the fluid As a result of this internal thermal energy, all molecules are in constant motion A solute molecule or small particle suspended in a gas or liquid phase will be bombarded on all sides by the movement of the surrounding gas or liquid molecules The random collisions cause unequal forces that cause the solute molecule to move in random directions The random motion 7-2 Molecular Diffusion 397 Dye–water bulk interface Water molecule Dye added to water Figure 7-1 Mechanism by which Brownian motion leads to diffusion The left side has about times as many dye molecules, consequently about times as many pass the interface from left to right compared to the number passing in the other direction Dye molecule caused by these collisions is called Brownian motion after Robert Brown, who described it (Brown, 1827) In a completely quiescent fluid, molecular diffusion by Brownian motion will cause matter to flow from regions of high concentration to regions of low concentration If Brownian motion is strictly random, how does it result in the movement of matter in a specific direction defined by the concentration gradient? That question can be answered by considering the probability associated with the movement of groups of molecules Consider a beaker containing water in which one drop of a blue dye has been placed Molecules, both water molecules and dye molecules, are randomly moving in all directions An imaginary boundary in the solution, as shown on Fig 7-1, has a greater concentration of dye molecules on one side than the other In response to completely random movement, the rate at which dye molecules cross the boundary in each direction is proportional to the number of dye molecules on each side; that is, the more dye molecules present, the more that can randomly cross the boundary from that direction The net result is a bulk movement from concentrated regions to dilute ones Net movement of dye molecules across any particular interface ceases when the concentration is the same on both sides In this way, molecular diffusion stops (although Brownian motion continues) when the dye is uniformly distributed throughout the beaker; that is, the concentration is the same everywhere When the concentration is the same everywhere, the solution in the beaker has reached equilibrium With Brownian motion as a foundation, molecular diffusion can be described by Fick’s first law (Fick, 1855): JA = −DAB dCA dz (7-4) Fick’s First Law 398 Principles of Mass Transfer where JA = mass flux of component A due to diffusion, mg/m2 · s DAB = diffusion coefficient of component A in solvent B, m2 /s CA = concentration of component A, mg/L z = distance in direction of concentration gradient, m The term dCA /dz is the concentration gradient, that is, the change in concentration per unit change in distance The negative sign in Fick’s first law arises because material flows from regions of high concentration to low concentration; thus, positive flux is in the direction of a negative concentration gradient The diffusion coefficient describes the proportionality between a measured concentration gradient and the measured flux of material Typical values of diffusion coefficients for solutes in gases and liquids are as follows: Liquids: ∼10−10 to 10−9 m2 /s (10−6 to 10−5 cm2 /s) Gases: ∼10−6 to 10−5 m2 /s (10−2 to 10−1 cm2 /s) Diffusion in the Presence of Fluid Flow Strictly speaking, Fick’s first law describes the flux of component A with respect to the centroid of the diffusing mass of solute In other words, Fick’s first law describes the rate of diffusion from a relative point of view; if the fluid is moving, the mass transfer due to diffusion is superimposed on top of, or in addition to, mass transfer due to the movement of the fluid The mass flow of component A due strictly to advection (in the absence of diffusion) may be written as MA = QCA where (7-5) MA = mass flow of solute A due to advection, mg/s Q = flow rate of fluid, m3 /s In terms of flux, the mass flow is divided by the perpendicular area: JA = where QCA = v (CA ) A (7-6) JA = mass flux of component A due to advection, mg/m2 ·s A = cross-sectional area perpendicular to direction of flow, m2 v = fluid velocity in direction of concentration gradient, where v = Q /A Consequently, when matter is being transported by both fluid flow and diffusion, Eqs 7-2, 7-4, 7-5, and 7-6 can be combined to define the net mass flow and mass flux as follows: MA = QCA − DAB dCA A dz (7-7) 7-2 Molecular Diffusion 399 and JA = v (CA ) − DAB dCA dz (7-8) The governing equations for unit processes are often developed by writing mass balance expressions around a control volume using a fixed point of view (stationary frame of reference) It is useful, therefore, to examine the difference between the mathematical expression for diffusion that is defined from a relative reference frame (flux = J ) and from a stationary reference frame (flux = N ) The expression for the molar flux of component A in solvent B can be written as a fraction of total molar flux, where the total molar flux is the sum of the fluxes of components A and B: NA = xA NTOT = xA (NA + NB ) where (7-9) NA = molar flux of component A relative to stationary frame of reference, mol/m2 ·s NB = molar flux of solvent B relative to stationary frame of reference, mol/m2 ·s N TOT = total molar flux (NA + NB ), mol/m2 · s xA = mole fraction of A in solution, mol/mol When matter is transported by both fluid flow and diffusion, the overall flux from a stationary reference frame is the sum of fluxes described in Eqs 7-4 and 7-9: dCA (7-10) NA = xA (NA + NB ) − DAB dz where DAB = diffusion coefficient of component A in solvent B, m2 /s CA = molar concentration of component A, mol/L z = position in direction of flow and diffusion flux (or in direction of concentration gradient), m In Eq 7-10, the first term on the right side describes the molar flux of A due to the movement of the fluid, and the second term describes the molar flux of A due to diffusion, superimposed on the movement of the fluid For the case of no advective flow of the solvent (NB = 0), Eq 7-10 can be algebraically rearranged to yield the expression   dCA −DAB NA = (7-11) − xA dz where NA = molar flux of component A relative to stationary frame of reference, mol/m2 ·s An example of a situation where there is advective flow of the solution but no advective flow of the solvent is when a solute evaporates from a surface Diffusion in Fixed and Relative Frames of Reference 400 Principles of Mass Transfer The solute evaporates and moves away from the surface, and because the solute is moving and the solute is a component of the solution, the solution can be seen as moving However, the solvent (in this case, the air) is not moving toward the surface For many environmental applications, particularly in aqueous solutions, xA is very small For example, the aqueous solubilities of chloroform and oxygen are about 9.3 g/L and 9.3 mg/L at 20◦ C, respectively; consequently, the largest mole fractions that can be found in water are 0.0014 and 5.23 × 10−6 , respectively In these cases, the 1/(1 − xA ) is negligible and JA = NA , that is, molar flux due to diffusion is the same regardless of whether a stationary or relative frame of reference is used Fick’s first law (Eq 7-4) is also valid when the sum of the fluxes NA and NB are equal to zero, as in a case where the diffusion of species A is countered by the diffusion of B (equal molar counterdiffusion) For highly miscible solvents in water or VOCs in gases, however, it is advisable to examine whether the 1/(1 − xA ) factor is important These cases are rare, and for most applications throughout the remainder of the book, Fick’s first law (Eq 7-4) is applied directly even though a stationary frame of reference is being used Fick’s Second Law Fick’s first law describes diffusion when the concentration gradient is constant Fick’s second law describes the rate of change of concentration when the diffusion into a control volume is different from the diffusion leaving a control volume Fick’s second law can be derived by a mass balance on a differential element with volume az, in which the only mass transport is due to diffusion: [accum] = [mass in] − [mass out] (7-12) dC = JA,z A − JA,z+z A (7-13) dt V = volume, m3 CA = concentration of component A, mg/L t = time, s JA,z = flux of component A entering the control volume, mg/m2 ·s JA,z+z = flux of component A leaving the control volume, mg/m2 ·s A = cross-sectional area of control volume, m2 V where Substituting Eq 7-4 and replacing the volume of the control volume with the differential element volume Az results in ∂CA ∂CA,z ∂CA,z + z = −DAB A + DAB A (7-14) Az ∂t ∂z ∂z where z = length of differential element, m3 DAB = diffusion coefficient of component A in solvent B, m2 /s