CEE255A-L3-Diffusion & MassTransfer

CEE255A-L3-Diffusion & MassTransfer - CEE 255A...

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Unformatted text preview: CEE 255A Lecture 3 Mass Transport in Water General Microscopic Mass Balance Convective Diffusion Layer Characteristics Dr. Eric M.V. Hoek Civil & Environmental Engineering Today's Lecture Outline Microscopic Transport in Fluids Phenomenological Transport Models Constant Density, Isothermal Fluid Continuity Equation Viscosity and Momentum Transport Laminar Flow, Newtonian Fluid Navier-Stokes Equation Diffusivity and Mass Transport Multi-component Mass Balance C-D-R Equation Diffusion and Reaction Kinetics Convection & Diffusion Boundary Layers Nernst Layer Approximation & Film Theory Channel Flow with Soluble or Rapidly Reacting Walls Multidimensional Mass Balances Multidimensional Momentum Balances Multi-component Mass Balance Convective Diffusion Layer Characteristics Phenomenological Transport Models Conservation Laws Conservation of momentum, energy, mass, and charge are applied to define the state of a real fluid system quantitatively with the assumption that the fluid is a continuum. Continuum A region of space where characteristic flow scales, L, are large enough that properties such as density and velocity can be assume to vary smoothly and, thus, have point values. If L3 1/N, a continuum exists*. N = characteristic distance between discrete fluid molecules N 3.3 1028 m-3 in water, so for L 3 a continuum exists N 2.5 1025 m-3 in air, so for L 3 nm a continuum exists *note: the symbol "" means "at least two orders of magnitude (~100x) larger than" Phenomenological Transport Models Definition of Flux, J The rate of flow of energy and matter per unit cross-sectional area of a conducting or transporting medium (e.g., water) is proportional to a driving force by some property of the medium Newton's Law rate of momentum flux (stress) is proportional to velocity gradient (strain rate) Key Constitutive Relations = -(du/dy); = viscosity; du/dy = strain rate (velocity gradient) Fick's Law rate of diffusive flux (mass transfer) is proportional to concentration gradient j = - D(dc/dy); D = diffusivity; dc/dy = concentration gradient Darcy's Law rate of convective flux (mass transfer) is proportional to pressure gradient u = -(k/)(dp/dy); k = Darcy permeability; dP/dy = pressure gradient Multidimensional Mass Balances Accumulation z y 1 6 4 2 3 5 = xyz( /t) z Mass Inflow x Face 1 = 1Q1 = 1u1yz Face 3 = 3Q3 = 3v3xz Face 5 = 5Q5 = 5w5xy x y Mass Outflow Face 2 = 2Q2 = 2u2yz Face 4 = 4Q4 = 4v4xz Face 6 = 6Q6 = 6w6xy To find the mass balance equation for an arbitrary point in space, we define the coordinates and the components of the local fluid velocity, as shown above. We may think of it as being a wire frame with flow into and out of all size faces. The coordinate frame is fixed in space and does not move. Considering all the mass flows into and out of the differential volume yields the equations describing rate of mass flow shown at left. (u = Ux; v = Uy; w = Uz) Multidimensional Mass Balances Accumulation = (Mass Inflows Mass Outflows) xyz( /t) = (1u1 - 2u2)yz + (3v3 - 4v4)xz + (5w5 - 6w6)xy Dividing through by - x y z (= -V) yields: - /t = (2u2 - 1u1)/x + (4u4 - 3u3)y + (6u6 - 5u5)z Now let x, y, and z simultaneously approach zero, so that the cube shrinks to a point. Taking the limit of the three ratios on the right hand side, we find: ( u ) ( v) ( w) This is the differential form of the - = + + = ( U ) general statement of "conservation of t x y z u v w 0= + + = U (constant density) x y z mass" (or for constant density the "continuity equation"), which has the same meaning as "Qin = Qout". Multidimensional Mass Balances Explanation of Gradient and Divergence The gradient of a scalar gives a rate of change (i.e., slope) with respect to location, whereas the divergence of a vector gives the net flux of a quantity emanating from a single point... z The Gradient of a Scalar = + + x y z y "slope of in 3-D" The Divergence of a Vector ( U ) = u v w + + x y z x "net rate of mass efflux per unit volume" Note: drawn in 2D Viscosity and Momentum Transport Newton's law of viscosity states there is a linear relation between the shear stresses and rates of strain. (steady state) The "Couette" problem F Vol yx = 0 yx = constant y du U = = ; = dy h x = yx Bird considers yx the viscous flux of momentum in the negative y-direction du du yx = = dy dy Multidimensional Momentum Balances Considering all normal and shear stresses z z 1 3 6 4 2 z x y y x 5 y ji = ij for i j x First subscript indicates the axis to which the face is perpendicular Second subscript indicates the direction to which the shear stress is parallel yx = stress exerted in the x-direction on a fluid surface that is perpendicular to the y-axis Multidimensional Momentum Balances Accumulation z y 1 xyz( U/t) 6 4 2 3 5 z Momentum Inflow x Face 1 = 1u12yz Face 3 = 3v32xz Face 5 = 5w52xy x y Momentum Outflow Face 2 = 2u22yz Face 4 = v xz 2 4 4 To make the working form of the momentum balance for multidimensional flows, we apply the general momentum balance to the flows and forces acting on the small cube in the figure. Considering all the momentum being carried into and out of the system by the mass flowing into and out of the differential volume, yields the equations describing rate of momentum flow into and out of the system. Face 6 = 6w62xy (u = Vx; v = Vy; w = Vz) Multidimensional Momentum Balances Gravity Force (x-dir) z y 1 6 4 2 3 5 mgcos = g cos xyz is angle between x-axis and the gravity vector On faces 1 and 2 = (xx1 - xx2)yz z x Normal Forces (x-dir) y x Considering all forces acting on the body and faces of the cube gives the additional sources of momentum change for the system. xx = -p + 2(du/dx) (this is ~ P = F/A plus a "dilation" term) Shear Force (x-dir) On faces 3 and 4 = (xy4 xy3) xz ; xy = yx = (du/dy+dv/dx) On faces 5 and 6 = (xy6 xy5) xy Multidimensional Momentum Balances Combining all of the above expressions and dividing by x y z, but considering only the x-direction yields 2 2 2 2 2 - 1u12 - 2u 2 3u3 - 4u 4 5u5 - 6u6 - - + g cos + xx1 xx 2 + xy 4 xy 3 + xz 6 xz 5 ( u ) = + + x t x y z y z Now taking the limit as x, y, and z all approach zero gives the "equation of motion" in the x-direction ( u 2 ) ( uv) ( uw) - xx xy xz ( u ) = - + + + g cos + x + y + z t y z x Similar expressions can be written for the y and z directions, or for all three dimensions the equation of motion can be written as ( U ) = -[ U U + U U] + g + terms involving normal t and shear stresses Multidimensional Momentum Balances A special case of the equation of motion can be derived considering a few (very severely) restrictive assumptions. The fluid has constant density (good assumption for water) The flow is laminar (sometimes true for engineered systems) The fluid is Newtonian (good assumption for water) The fluid is behaves as a perfectly elastic, isotropic solid such that the three dimensional stresses obey Hooke's law... (x-direction only) If we make these assumptions, then it can be shown that 2 u 2 u 2u u u u u P + u + v + w = g cos - + 2 + 2 + 2 t x x y z x y z In 3-D vector notation we have the Navier-Stokes eq'n DU = g - P + 2 U t What can you do with N-S eq'n Derive the shape of a laminar flow profile in a tube or thin channel with no-slip b.c. See handout for details. 2u 2u 2u P u u u u P 2u + u + v + w = g cos - + 2 + 2 + 2 = 2 t x x y z x y z x y Thin Rectangular Channel y2 uchan = umax - 2 h = channel half height y 1 , h 3u 3 umax ,chan = uchan & 0,chan chan 2 h h h x y = 2h y=0 y = -h Tubular Conduit r2 = umax - 2 1 a , a = tube radius 2u & 0,tube tube a r x r = +a a a r=0 r = -a utube umax ,tube = 2utube Useful to understand mixing and flow in MFRs, PFRs, and PBRs Diffusivity and Mass Transport Transport of mass by diffusion in a fluid mixture of two or more species occurs when there is a spatial gradient in the proportions of the mixture, i.e., a concentration gradient. Mass Concentration i = mi mass of species i kg = = 3 V volume of solution m Molar Concentration ci = ni moles of species i mol = = V volume of solution m3 Mass Fraction i = i mass conc of species i = mass density of solution Mole Fraction xi = ci molar conc of species i = c molar density of solution Mean Molar Mass of a Mixture M = kg = xi M i = ; M i = molecular mass of i c mol Mass Average and Molar Average Velocities 1 i ui ; 1 u * = ci ui ; c u= u = mass flux through a unit area normal to u cu * = molar flux through a unit area normal to u Diffusivity and Mass Transport The terms iui and ciui represent the mass and molar fluxes of species "i", or: ji = iui and ji* = ciui The mass and molar fluxes w.r.t. the mass avg and molar avg velocities are: J i = i (ui - u ) = ji - i u = - Di the flux arises due to a difference in the species "i" and bulk average velocities J i* = ci (ui - u * ) = ji* - ci u = -cDxi If a spatial concentration gradient exists, Fick's first law of diffusion states there is a linear relationship between species flux and the concentration gradient. For infinitely dilute systems: J i = ji - i u = - D i J i* = ji* - ci u * = - Dci D is the "mass diffusivity" or binary diffusion coef. Mass Transfer Fundamentals Convection Convective mass transfer occurs in flowing and mixed fluid systems (i.e., PFR/PBR, and CSTR). In aquatic systems, forced convection (UCA) can transport solutes at rates much higher than diffusion. Schmidt number, Sc = /D Ratio of kinematic viscosity to solute diffusivity Relates the importance of convective and diffusive mass transfer for water is ~10-6 m2/s D for dissolved solute is ~10-9 m2/s Sc ~ 10-6/10-9 ~ 1,000 so convective mass transfer dominates; however, molecular and Brownian diffusion is significant in many W&WW treatment processes Mass Transfer Fundamentals Diffusion Molecules vibrate in a random manner through three dimensions based on their thermal energy (kT) Diffusivity is defined as kT/f, where f (= 6rH) is StokesEinstein friction factor and rH is a hydrodynamic radius. When a concentration gradient exists, molecules at the higher concentration will diffuse to areas of low concentration Consider a stagnant rectangular volume of a mixture of crosssection A and with length L separating concentrations C1 and C2, then at steady state, the rate of mass transfer by diffusion is FA = DA(CA,1 CA,2)/L = [m2/s][m2][mol/m3]/[m] = [mol/s] A,1 A,2 D = solute diffusivity or diffusion coefficient [m2/s] A = cross sectional area through which mass flows [m2] C1/2 = solute concentration at position 1 and 2 [mol/m3] L = distance between concentration 1 and 2 [m] N = F /A = molar flux of solute A due to diffusion [mol/m2s] Mass Transfer Fundamentals Molar Fluxes Mass Transfer usually refers to any process in which diffusion plays a role. Diffusion is the spontaneous intermingling or mixing of atoms or molecules by random thermal motion. It gives rise to motion of chemical species relative to the motion of the mixture. In absence of other gradients (temperature, electrical potential, or gravitational potential), solutes diffuse from regions of higher concentration to regions of lower concentration This gradient results in a molar flux of a solute in the direction of the concentration gradient, regardless of fluid flow. NA = iNAx + jNAy + kNAz [= moles/m2-s] ... a vector quantity Total Molar Flux of A relative to a fixed coordinate, Mass Transfer Fundamentals Molar Fluxes Total Molar Flux of A, NA = JA + UCA Assuming a binary system; A = solute, B = solvent Diffusive Flux, JA = -DAB CA = -CDAB yA C = total concentration (mol/m3); CA = concentration of solute A DAB = diffusivity of solute A in solvent B (m2/s) yA = mole fraction of A (mol A/mol A+B) CA = concentration of solute A (mol/m3) U = yiUi = molar average velocity (m/s) yi = mole fraction of A (mol A/mol A+B) For a binary mixture of solute A in solvent B, we let UA and UB be the velocities of species A and B, respectively, thus U = yAUA + yBUB. When the concentration of A is dilute, yA yB and yB 1 and the molar Convective Flux, UCA = (Q/A)CA Mass Transfer Fundamentals Molar Fluxes Forced Axial Convection with Diffusion to a Surface Assuming a binary system; A = solute, B = solvent NAx = JAx + UxCA UxCA (axial diffusion neglected) NAy = JAy + UyCA JAy = -DABdCA /dy (normal convection neglected) Impermeable Surface FAx = NAxAx = UxCAAx = QCA For very low flow rates ("creeping flow"), axial diffusion may be non-negligible, and thus: FAx = NAxAx = Ax(JAx + UxCA) = Ax(-DABdCA /dx + UxCA) For permeable surfaces ("membranes"), normal convection may be non-negligible, and thus: F = N A = A (J + U C ) = A (-D dC /dy + U C ) Mass Transfer Fundamentals Boundary Conditions Specify a concentration at the boundary At y = 0 (the surface), CA = CA0 no reaction At y = 0 (the surface), CA = 0 instantaneous reaction R r Specify a flux at the boundary No mass transfer, NAy = 0 For example, at the wall of a non-reacting pipe, d CA/dr = 0 at r = R Since the diffusivity is finite, the only way the flux can be zero is if the concentration gradient is zero... Molar flux equals the rate of reaction on the surface NA (surface) = - rA' (surface) NA (boundary) = kc (CAb CAs) ; k is mass transfer coefficient; b = bulk, s = surface Molar flux equals convective transport across boundary layer Specify a plane of symmetry Concentration gradient is zero at a plane of symmetry Mass Transfer Fundamentals Film Theory Diffusion through a Stagnant Film to a Surface Hypothesis: A thin boundary layer ("film") exists near the surface through which convective mass transfer is negligible...a reaction occurs at the surface such that CAb > CAs bulk fluid flow Solid Surface CAb y=0 Solution: Perform Mole Balance Put FA in terms of NA Replace NA with dCA/dy State boundary conditions Solve for concentration profile Solve for molar flux +y Solid Surface y y + y CAs y= Mass Transfer Fundamentals Film Theory Step 1: The general mole balance equation [ flow in ] - [ flow out ] + [ reaction ] = [ accumulation ] FAy y - FAy y + y +0=0 Dividing by -y gives: FAy y + y - FAy y y dy =0 =0 Taking the limit as -y 0 gives: dFAy Step 2: Substitute for FA in terms of NA and Ax FA = NAAx or dNA/dy = 0 Step 3: Replace NA with expression for dCA/dy NA = -DAB dCA/dy Mass Transfer Fundamentals Film Theory Step 4: State the boundary conditions When y = 0, ... CA = Cab When y = , ... CA = CAs ( is the film layer thickness) (d2CA/dy2 = 0) dCA/dy = K1 (constant of integration) (dCA/dy = K1) CA = K1y + K2 ... (find K1 & K2 from b.c.'s) When y = 0, ... CA = Cab ... CAb = 0 + K2 When y = , ... CA = CAs ... CAs = K1 + K2 = K1 + CAb Therefore, K1 = (CAs - CAb) / , and finally: Step 5: Solve for the concentration profile (integrate) C A - C Ab y = C As - C Ab Mass Transfer Fundamentals Film Theory 1.0 C As = 0) 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 y / 0.6 0.8 1.0 C A - C Ab y = C As - C Ab y C A = C Ab + ( C As - C Ab ) C A /C Ab (assumes Mass Transfer Fundamentals Film Theory Step 6: Solve for the molar flux of A through the film NA = -DAB (dCA/dy) where CA = CAb + (CAs CAb)y/ NA = -DAB d[CAb + (CAs CAb)y/ ]/dy NA = -DAB (CAs CAb) / Note: DAB/ = kc = solute mass transfer coefficient CAb ; y = 0 Solid Surface CAs ; y = Effect of Temperature on Solute Diffusivity/Mass Transfer 1 T2 DAB (T2 ) = DAB (T1 ) 2 T1 (What if the surface is permeable?) Mass Transfer Coefficient Spherical Particle NA = -DAB(CAsCAb)/ DAB/' = kc' kc' = local MTC kc = ( kc'dA) / A NA = kc (CAb CAs) Average molar flux to the surface is: Recall that the Sherwood number represents a dimensionless mass transfer coefficient, Sh = kcdp/DAB = 2.0 + 0.6Re1/2Sc1/3 Re = Udp/ Sc = /DAB 1/2 1/3 2/3 1/2 1/2 1 /6 Mass Transfer Coefficients Tube & Channel Reynolds number: Re = u0dH/ ; u0 = Q/A; dH = 4A/Pw; = / Re < 2000 (LAMINAR) 2000 < Re < 4000 (TRANSITIONAL) Re > 4000 (TURBULENT) utube r2 = umax 1 - 2 , r = tube radius; h uchan y2 = umax 1 - 2 , h = channel half height h 2 umax ,tube = 2u0 ; & 0,tube umax ,chan & D 2u = 0 ; ktube ( x ) = 0.678 0 x r ; 3 1 ktube k chan & D = 1.02 0 L 2 2 1 3 & D 3u 3 = u0 ; & 0,chan = 0 ; kchan ( x) = 0.538 0 x 2 h LAMINAR TURBULENT 1 3 2 ; 3 1 & D = 0.807 0 L 1 3 CHARACTERISTIC LENGTH d H = 4(r 2 ) / 2r = 2r dH = 4WH 2H (2W + 2 H ) Shtube Shchan k d d = tube H = 1.62 Re Sc H D L = 0.04 Re 4 Sc 3 3 3 5 k d d = chan H = 1.85 Re Sc H D L 1 = 0.04 Re 4 Sc 3 3 5 Heterogeneous Reaction Kinetics Steps Involved in Heterogeneous Reactions 1. Diffusion of target solute from bulk to interface Most often the rate limiting step in all heterogeneous reactors Not an important step in aeration and stripping In the case of aeration the surface reaction is a phase change In the case of GAC adsorption there is no surface reaction Not an important step in aeration and stripping More often the rate limiting step in heterogeneous catalysis 2. Adsorption of target solute to the surface 3. Solute surface reaction/phase transfer occurs 4. Desorption of product/by-product from the surface 5. Diffusion product/by-product from interface to bulk Heterogeneous Kinetics and Reactor Models Mass Transfer-Limited Reactions in Packed Beds Fi0 = Q0Ci0 Fi = QCi A+B Products + un-reacted A and B Fi0 (x) x Fi (x + x) Ri = -ri'dV = -ri"aAxdx Ai = dNi/dt = 0 ri" = reaction rate per unit surface area of packing material [mol/s m2] a = specific surface area of packing material [m2/m3] = 6(1 )/dp for spherical particles; dp = particle diameter [m] = porosity of the packed bed [m3-void/m3-reactor] Ax = cross-sectional area of reactor [m2] Neglecting axial diffusion/dispersion gives (see Fogler): Ci kc a 1 kc a = exp - x ... locally, or over the whole reactor ... ln L = Ci 0 1- X U U Heterogeneous Kinetics and Reactor Models Mass Transfer-Limited Reactions in Packed Beds Bed average mass transfer coefficient: Sh' = 1.0( Re') 1/ 2 Sc1/ 3 Sh' = Sh 1- kc d p U d p = = DAB 1 - (1 - ) DAB Re Re' = 1- 12 3 Axial pressure drop in a packed bed reactor (cost energy, $$$) p (1 - ) 2 (1 - ) 2 = 1.75 3 U + 150 U 3 2 x dp dp Ergun equation valid for all Reynolds number flow Dispersion Coefficient versus Peclet No. Dispersion of a Tracer in a Capillary Tube (PFR) ...
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