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A ' I H --.. .- _. w ‘- w. ‘- - 7v .Dcv—r fl- .— ....... .0- ”—3-“.o ' '.1 ‘u—OO \. ._... 44‘ p4 - ..--n '0'..- 4:4I :4 II, I ””’I -j IIII' IIILI4I ,III” III THESIS This is to certify that the dissertation entitled THE APPLICATION OF TURBULENT RELAXATION MODELS TO MASS TRANSFER NEAR INTERFACES AT HIGH MOLECULAR SCHMIDT NUMBERS presented by Hsi-Tai Yao has been accepted towards fulfillment " of the requirements for ' Ph.D. degree in Chemical Engineering d¢©.% Major professor I Date April 15, 1982 MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 MSU LlBRARlES .—,‘_. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. (:‘//27J“7/ THE APPLICATION OF TURBULENT RELAXATION MODELS TO MASS TRANSFER NEAR INTERFACES AT HIGH MOLECULAR SCHMIDT NUMBERS By Hsi-Tai Yao A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1982 ABSTRACT THE APPLICATION OF TURBULENT RELAXATION MODELS TO MASS TRANSFER NEAR INTERFACES AT HIGH MOLECULAR SCHMIDT NUMBERS By Hsi-Tai Yao Turbulent mass transfer near interfaces at high Schmidt numbers is important for many industrial operations. An §_priori prediction of the mass transfer rate, however, requires a theory which depends expli- citly on the underlying hydrodynamic structures and the relevant physi- cochemical parameters. A statistical theory for turbulent mass transfer near interfaces at high Schmidt numbers is developed based on an evolution equation for concentration fluctuations and a Green's function technique. The non- linear coupling between fluctuating velocities and fluctuating concen- trations is retained, but similar terms in the equation for the fluctu- ating Green's function are neglected. A set of non-local statistical equations is developed which depends on the relaxation of the mean Green's function and the velocity space-time correlation. For large Schmidt numbers, the kernels of these equations are spatially peaked functions, and the non-local relaxation equations reduce to local turbu- lent models. The first two relaxation equations are analyzed in this research. a» P.’ AP. 7 I c The Hsi-Tai Yao One model relates the gradient of the turbulent flux to the concentra- tion gradient. Another states that the local turbulent flux is the balance of a ”diffusive“ flux and a ”retardation“ flux. Both models are of the non-gradient type. The two relaxation models are used to predict mass transfer rates for three applications: physical absorption, physicochemical absorp- tion, and electrostatic deposition. The theory shows that the turbulent flux near an interface is retarded by a feedback mechanism involving the gradient of the flux itself. This descovery provides a new interpreta- tion for the mechanism of mass transfer at high Schmidt numbers. The theoretical results predict the correct dependence of the mass transfer rate on the Schmidt number for both free and rigid interfaces. The two models are also used with mass transfer data to estimate a self- consistent set of hydrodynamic time scales near a rigid interface, which agrees with previous proposals on the variation of the temporal behavior of turbulent fluctuations within the viscous sublayer. To my parents -and- To my wife, TE-FEN I". hung inee ban 3 ACKNOWLEDGMENTS The author would like to express his deep appreciation to Dr. Charles A. Petty for his guidance and support throughout the course of this work. Gratitude should also be expressed to Drs. Bruce Wilkinson, Dennis Nyquist, Robert Falco, Chang-Yi Wang, and Philip Wood for their concern and interest. The author gratefully acknowledges the National Science Foun- dation (ENG 79-l5256) and the Michigan State University Division of Engineering Research for financial support. The patience and understanding of the author's wife, Te-Fen, is sincerely appreciated. TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES NOTATION . Chapter I INTRODUCTION Motivation for This Research . Objectives of This Research . . Engineering Significance of This Research Methodology for This Research —a—.A_a_a J-‘WNfl 2 THE MEAN FIELD PROBLEM I General Description 2 The Mean Field Equation for Physical Absorption 2.3 The Mean Field Equation for Physicochemical L. NN Absorption . . The Mean Field Equation for Electrostatic Deposition . 3 TURBULENT STRUCTURES NEAR INTERFACES 3.1 Velocity Intensities . . 3.2 Renewal of Interfacial Fluid . 3. 3 Velocity Space- Time Correlation Near an Interface A A BRIEF SURVEY OF PREVIOUS APPROACHES TO TURBULENT MASS TRANSFER NEAR INTERFACES . 4.1 Physical Absorption Across Rigid Interfaces A.2 Physicochemical Absorption Across Free Interfaces A.3 Electrostatic Deposition . 5 CONCENTRATION FLUCTUATIONS NEAR AN INTERFACE 1 Evolution Equation for c (x, t). .2 Evolution Equations for the Mean and Fluctuating Components of the Green's Function . 3 The Generalized Sternberg Approximation .4 Finite Memory and Spatial Smoothing U1U'I U‘lU‘l iv vii viii (13V O‘d IO 10 12 Is 20 25 25 28 32 no no 1.9 53 so so 65 68 69 (haste: 6 (h Chapter 6 A THEORY FOR THE TURBULENT FLUX OF A PASSIVE ADDITIVE . Hierarchy of Relaxation Equations . . Spatial Smoothing of the Relaxation Equations A Self- -Consistent Approach for Estimating the Renewal Rate for Interfacial Fluid . Type II Relaxation Model for Physicochemical Absorption Across a Free Interface . Type I Relaxation Model for Electrostatic Deposition . 0‘ mm O‘C‘O‘ 0‘ U1? DON-i 7 APPLICATION OF THE THEORY TO PHYSICAL ABSORPTION 7.1 Type I Relaxation Model 7.2 Type II Relaxation Model . . 7.3 An Estimate of TNT and TH+ Using the Relaxation Models 7.4 An Analysis of the Mean Fields for Type I and Type II Relaxation Models . . 7.5 A Parametric Study of the Mean Mass Transfer Rate Using Type II Relaxation Near a Rigid Interface . . . 7. 6 A Comparison Between Type I and Type II Relaxation Models for Physical Absorption 8 APPLICATION OF THE THEORY TO PHYSICOCHEMICAL ABSORPTION . 8.1 Type II Relaxation Model 8.2 An Analysis of the Mean Fields Using Type II Relaxation . 8.3 A Parametric Study of the Mean Mass Transfer Rate Using Type II Relaxation . . . 8. 4 A Summary of New Physical Effects 9 APPLICATION OF THE THEORY TO ELECTROSTATIC DEPOSITION . Electrostatic Drift and Brownian Motion Type I Relaxation \DLOKDKD bWN—i Summary of New Physical Effects 10 CONCLUSIONS AND RECOMMENDATION FOR FURTHER RESEARCH . 10.1 Conclusions . . . 10.2 Recommendation for Further Research . APPENDICES Appendix A PROPERTIES OF THE ENSEMBLE AVERAGE OPERATOR . Type II Relaxation Model for Physical Absorption . A Parametric Study of the Mean Mass Transfer Rate 73 73 80 82 84 9O 93 . IOO . IOO . 102 104 . 106 . 114 . 121 124 124 125 132 136 140 140 143 145 153 158 158 165 167 Appendix B PROPERTIES OF GREEN'S FUNCTION B.l Generalized Green's Theorem, Reciprocity Condition and the Green's Function Technique . B.2 The Effect of Boundary Conditions on Green' 5 Function . . . B. 3 Three Examples of Green' 5 Functions . . 8.3.1 Green's Function for the Diffusion Operator . . B. 3. 2 Green's Function for the Convective- Diffusion Equation with First- Order Chemical Reaction . B. 3. 3 Green's Function for One- Dimensional Diffusion with Normal Convection C DERIVATION OF THE GREEN'S FUNCTION FOR ONE- DIMENSIONAL DIFFUSION WITH NORMAL CONVECTION D DERIVATION OF THE RELAXATION KERNEL U.(x,]x,) E DERIVATION OF THE TURBULENT COEFFICIENTS D1 (x ) AND 2 11(x ) FOR PHYSICAL ABSORPTION . . 1 F DERIVATION OF THE TURBULENT COEFFICIENTS O,,(x,) AND £,,(x,) FOR PHYSICOCHEMICAL ABSORPTION G DERIVATION OF THE CONVECTIVE VELOCITY uC FOR ELECTROSTATIC DEPOSITION . . . . . . . . . . . . . . . . H COMPUTER PROGRAM FOR PHYSICAL ABSORPTION USING TYPE II MODEL . I COMPUTER PROGRAM FOR PHYSICOCHEMICAL ABSORPTION USING TYPE II MODEL . J COMPUTER PROGRAM FOR ELECTROSTATIC DEPOSITION USING TYPE I MODEL REFERENCES . vi 169 - 169 176 179 I79 182 183 - 184 . 187 . 189 . 192 . 194 I97 - 202 - 209 - 213 6.1 7.1 8.1 LIST OF TABLES SUMMARY OF DIFFERENT APPLICATIONS ASYMPTOTIC BEHAVIOR OF THE GRADIENT OF TURBULENT FLUX AND THE CONCENTRATION GRADIENT PREDICTED BY THE RELAXATION MODELS EFFECT OF CHEMICAL REACTION ON THE DIFFUSIVE, RETARDATION, AND TOTAL TURBULENT FLUXES, AT x1+= 1.0 AND FOR TH+=TM+= IO . . . vii 98 - 105 - 129 2.2 2-3 IKB N/S 2.1 M I «(II 6.1 5.5 6.3 3.2a 3.2b 7.1 LIST OF FIGURES Geometric parameters characteristic of the mass transfer problems analyzed . Qualitative behavior of physicochemical absorption for slow and fast reactions (OD+EO /(x1) C’(§_.t) cb 011(x1IX1) 011(x1) NOTATION Particle radius. The first non-trivial coefficient in an expansion of (x1) about x1==0 for free interface, defined by Eq.(3.lO). The first non-trivial coefficient in an expansion of (x1) about x1==0 for rigid interface, defined by Eq.(3.8). Instantaneous stochastic concentration; C(éjt) is statis- tically stationary and statistically homogeneous in planes parallel to the mass transfer interface. Mean concentration. Fluctuating stochastic concentration; c’(53t)55c(x,t)- (x1). ‘ Mean concentration at xl-rw. Mean interfacial concentration for physicochemical absorption. Molecular diffusion coefficient. Type II relaxation kernel, defined by Eq.(6.15). Turbulent diffusivity associated with Type II relaxation model, defined by Eq.(6.22). Turbulent eddy diffusivity. Enhancement factor for physicochemical absorption, defined by Eq.(2.21). Magnitude of the charging field. Magnitude of the electric field near a collecting surface. Stochastic Green's function associated with the linear differential operator (-) on the domain 0. Mean Green's function defined by Eq.(5.19). xi G’(§Jtlx,t) G°(§.tI§_.t) Sc T(x1I;1) T(x1) 2(5: 0 {35(x1) Fluctuating Green's function defined by Eq.(5.20). Non-stochastic Grien's function associated with the linear operator ,(°) on the domain 9. First-order reaction rate constant. Mean mass transfer coefficient with chemical reaction, defined by Eq.(2.20). Stochastic differential operator defined by Eq.(5.14). Non-stochastic differential operator defined by Eq.(5.2) for physical absorption; by Eq.(5.3) for physicochemical absorption; and by Eq.(5.4) for electrostatic convection. Non-stochastic differential operator defined by Eq.(5.2) for physical absorption. Type I relaxation kernel defined by Eq.(6.5). Type II relaxation kernel defined by Eq.(6.14). Length scale associated with Type I relaxation model, defined by Eq.(6.19). Memory length associated with Type II relaxation model, defined by Eq.(6.21). Dimensionless group defined by Eq.(6.57). Dimensionless group defined by Eq.(9.8). Ratio defIned as TM/TH. Total charge on a particle. Normal component of the space-time correlation, defined by Eq.(3.lS); in Chapter 8 it denotes the retardation flux. Molecular Schmidt number. Type I relaxation kernel defined by Eq.(6.6). Time scale associated with Type I relaxation model, defined by Eq.(6.20). Instantaneous stochastic velocity; uixgt)is statisti- cally stationary and statistically homogeneous in planes parallel to the mass transfer interface. Mean velocity for fully developed turbulent flow. xii (:1 rn‘ " I g,(5,t) Uc .5 Greek Letters (1 6c Fluctuating stochastic velocity; u?(x,t)EEqu)t)-; also see Figure 2.1. Diffusive length scale defined as b/ug‘; also see Figure 2.1. Electrostatic length scale defined as fl/uD; also see Figure 2.1. Characteristic length scale associated with the turbulent diffusivity 011. Viscous length scale defined as v/u ; also see Figure 2.1. Reactive length scale defined as Mst; also see Figure 2.1. Particle collection efficiency. Characteristic time associated with the mean field, defined by Eq.(6.37). Turbulent relaxation time defined by Eq.(6.36). Viscosity. Kinematic viscosity. Mass density. xiii Characteristic relaxation time for the velocity auto- correlation in a frame of reference moving with the average velocity. Mean period between bursts. Characteristic time for turbulent diffusion, defined by Eq.(6.34). Magnification factor for u by electrostatic augmentation defined by Eq.(6.55). C Augmentation factor for electrostatic deposition, defined by Eq.(2.31). Subscript or Superscript + Used to denote parameters made dimensionless with v and u“ for rigid interface; .3 and 31F for physicochemical absorption; and v and a1F for electrostatic deposition. Used to denote a fluctuating quantity. Used to denote a mean quantity associated with physical absorption. CHAPTER 1 INTRODUCTION 1.1 Motivation for This Research Turbulent mass transfer of a chemical constituent at high molec- ular Schmidt numbers is an important phenomenon with many industrial applications. For example, in combustion processes the transfer of air to the dispersed fuel or solid phase is a key step in sustaining combus- tion and controlling its rate; in fermentation processes the supply of oxygen to micro-organisms in a turbulent liquid phase determines the reaction rate and ultimately the yield of the product. In gas cleaning processes the removal of particles entrained in a turbulent flue gas is essential to the protection of our environment and life. Yet, the problem of turbulent mass transfer near an interface is not fully under- stood. The most current mass transfer data of Shaw and Hanratty [1977a] near a rigid interface shows that the mean mass transfer rate depends on the molecular diffusivity as (‘62) 413'? . (1.1) They pointed out that the conventional eddy diffusivity of the form )1 De = bx] , (102) with the constants b and n depending only on the hydrodynamics, fails IO FEDFOCI ti l'& I actin- 1.... .4 recocnize uh ' HEIGU 8 y I erIyI ‘a: the a’d WOLII E ‘ \ TETE :5 I IV.“ ‘8‘: UT SEI‘IE PE fig». 07 (I. a: "Ire .. O 5 k: to reproduce the experimental data. In general, the physical parameters affecting the fluctuating concentration and velocity fields may poten- tially influence the turbulent flux. For example, Van Driest [1956] recognized the effect of viscous dampening close to a wall on velocity fluctuations, and incorporated a dampening coefficient in the eddy dif- fusivity (see Eq.(4.3)); but, how this coefficient is related to the underlying physicochemical parameters is unknown. Hinze [1975] argued that the eddy diffusivity is related to the Lagrangian auto-correlation and would be affected by the interaction of the eddy with its surround- ing (see Eq.(4.12)). Like Van Driest, Hinze pointed out an important idea, but failed to express this idea in terms of observable physical parameters. A different approach to turbulent mass or momentum transfer problems follows from Sternberg [1961], who suggested that near the wall the non-linear coupling between the fluctuating fields in the equation of change can be neglected. This assumption, which has been extended by several authors to mass transfer problems (see Chapter 4 for a review), leads to a gradient-type model for the turbulent flux. Although a calculation made by Petty [1975] using this assumption shows the same Schmidt number dependence as Eq.(l.1), the neglect of the non- linear terms places too much emphasis on the mean concentration gradient in generating concentration fluctuations (see Chapter 5). This approach suppresses the importance of other dynamic scales on mass transfer. Exactly what role the non-linear coupling terms play near the interface raises an interesting question. Builtjes [1977] pointed out that the gradient-type model has a fundamental difficulty stemming from the fact that the turbulent flux is prOOOTt in the CO" turbulent I've scale tire scale kivs's Of ' oi nerory QTO'Oge—Iec hi the EX TIE'CI'y is COACIIIICF ‘5 h"Ié‘O‘L'TE tain Ipsi I”? USIT‘g a") I HA1,“ 'I' .,, ‘ V‘Ug 9a., . 3 is proportional to the concentration gradient. A sudden spatial change in the concentration gradient will cause an instantaneous change in the turbulent flux. This feature may not be physically realistic if the time scale associated with the turbulence is of the same order as the time scale associated with the mean field. He argued that there are two kinds of “memory” effects associated with turbulence. The first kind of memory is inherent to the turbulence itself, even if turbulence is homogeneous and stationary. An example of this kind of memory is given by the existence of the space-time correlation. The second kind of memory is caused by the reaction of turbulence to the change in external conditions, which he called the ”extra” memory. He viewed turbulence as a hypothetical non-Newtonian fluid with the turbulent stress at a cer- tain instant depending on the whole history of the mean-velocity field. By using a simple expression for the ”extra” memory, he derived a non- gradient turbulent model for momentum transfer in a developing wake and boundary layer: L _____3_mcm mew—coca Emmmcmcu mmmE mzu mo u_um_cmuumcm£u mcmumEmcma u_cuoeooo .p.~ mc:m_u n_m_m cmmE .Auxv>wnuuw u_um:_10cuum_m .cnxwmmmw m>3ummc .Exmxm 633:; .... . :\>Hu @ mo_mum _mu_m>£m ummc_cuc_ mummc0uc_ omen co n_m_m //////r//////V//.x JLJEQIIIIIIIIII o . 0 III“? . _ . _ co>m_n:m " maoom_> _ . _ _ . . _ _ _ I It.“ 12 and,(2) it implies that for a two-component mixture the instantaneous mass flux of the “passive“ additive relative to a fixed frame of refer- ence can be approximated as 23A —"-’- 3A ‘I WA 1‘5- (2.1) In other words, the instantaneous velocity u_is approximately 28/0. The velocity field g(th) and the concentration field c(x,t) of the “passive“ additive are assumed to be statistically homogeneous in planes parallel with the interface and statistically stationary in time (see Chapter 3). Thus, any statistical property of g(xjt) or c(xgt) evaluated at a specific position and time depends only on the coordinate direction normal to the interface (see Figure 2.1). Spatial and tempo- ral statistical correlations of Efént) and c(x)t) depend on x1, as well as [57;] and It-f]. All average operations on g(xjt) and c(x,t) should be interpreted as ensemble averages; Appendix A lists some important properties of this smoothing operation. 2.2 The Mean Field Equation for Physical Absorption The instantaneous concentration field for a ”passive” additive is governed by the usual convective-diffusion equation and continuity equa- tion (see p. 557 in Bird, et al., 1960) 3c 2 3;+§-VC=$VC (2.2) V-H:=o. (2.3) Eq.(2.2) is linear. The stochastic coefficient g(xjt) causes C(ijt) to I3 be a rather complicated function of space and time. Since ReynOIds [1895]. the strategy for studying Eq.(2.2) has been to decompose u_and c into mean and fluctuating components (see Appendix A and Chapter 3), i.e., E {10+ 2’ (2.4) +c’. (2.5) o N Substituting these results into Eq.(2.2) and averaging yields the follow- ing equation for the mean field d .8 Ji JX’ - JX,1 (2.6) where the statistical assumptions discussed previously have been used, viz., <3? = (x1)_§_3 and = (x1). Solutions to Eq.(2.6) will be developed subject to the following two boundary conditions <6) =0, x, =0 (2.7) (c) Cb, 7C.=°°- (2.8) Boundary condition (2.7) occurs when the constituent reacts instan- taneously and irreversibly at the interface. Obviously, Eq.(2.6) is not a closed equation for the mean field. The turbulent flux of the “passive” additive caused by the normal ’ fluctuating velocity u1 must be modeled. This is the main topic of I Chapter 6. An important feature of , which is characteristic of all the problems considered in this research, is that S l ""“"‘I" Thus, C I S in' 14 II .. (u.c.)-o, x,=o. (2.9) This occurs because the normal component of u_is zero at the interface, which also implies that and u’ 1 1 =0 (see are individually zero at x1 Appendix A). Eq.(2.6) can be integrated once without having a model for . Thus, the region near the interface is often referred to as a region of constant total flux since d €553. ’ $777 -_-, ,8 dx, ~(u,’c 2, (2.10) *’=° molecular diffusion--4 turbulent mixing This result partially motivates the introduction of a mean mass transfer coefficient defined by <£¢>OE .810» 3’- (Q) (2.11) X,—90 JX, Cb or, equivalently, the mean field scale introduced in Figure 2.1, viz., 6co 53/°. Thus, once ° has been estinated, the constant total flux due to turbulent mixing and molecular diffusion can be calculated. Because d/dx1 + 0 as x + w, the mass transfer coefficient can also 1 be interpreted as the turbulent flux far from the interface. Thus, we have Eq.(2.11) as well as of tilt ’1. III *0 «C I lb a- PM e 5 fly A n. o s I 4‘ my .9... - F at e L PI. e e ,0, L1 .6 I; b. b F» .) S Ir. t 0 an . .. II D» h“ AIL f I x S c- 15 0 I. I (£2) =- 21‘..<“"°>/Cb- (2.12) A large amount of experimental data has been correlated in terms of the Stanton number * 0 Sta = ('6‘) / u'. (2.13) A summary of this information is given in Chapter 4. Elementary dimen- sional analysis shows that the magnitude of St: depends on the relative magnitudes of the two characteristic length scales OH and OD. Thus, SI+= f(-:-:-). , (2.111) Note that OH/OD = v49 = Sc, the molecular Schmidt number. In Chapter 7 the theory, developed in Chapter 6, for the turbulent flux is used with the mean field equation presented here ‘UD predict the dependence of St+ on Sc. Although Eq.(2.6) applies to a very special class of turbulent flows, it remains valid for free and rigid interfaces, as well as for Newtonian and non-Newtonian fluids; the specific model for will account for these additional physical effects. 2.3 The Mean Field Equation for Physicochemical Absorption The instantaneous concentration field for a “passive” chemical constituent undergoing a first order transformation is ac fi'ffi'VCL'flVzc—ic. (2,15) This tern: :ior. EVEFB 16 This is simply Eq.(2.2) with the addition of an irreversible reaction term; k is an intrinsic reaction rate coefficient. The mean field equa- tion for physicochemical absorption follows directly from Eq.(2.15) by averaging. Therefore, the same procedure which gave Eq.(2.6) also yields d(u.’c’> d1“) alx. =2)? - ‘50:). (2.16) The boundary conditions for are (C) = CO ) XI = O (2.17) =o , x,-.-oo. (2.18) Here the interfacial concentration co arises because of thermodynamic equilibrium with another phase. This implies that the resistance to mass transfer for x1 < O is unimportant. Eq.(2.16), like Eq.(2.6), is not closed. A model for the turbu- lent flux in terms of the mean field is needed before Eq.(2.16) can be integrated and the mass transfer rate calculated. The usual strategy for treating this problem (see the books of Astarita [1967] and of Davies [1972]) is simply to carry over to physicochemical absorption the same turbulent model developed for physical absorption. Thus, if a classical ”eddy” diffusivity model for is used, then the intrinsic rate con- stant k only affects the turbulent flux implicitly through the mean gradient. The ”eddy“ diffusivity is viewed as a hydrodynamic construct independent of the kinetic scale parameter OR. A distinguishing feature of this research is that the turbulent models which enter the mean field equations depend explicitly on all the underlying physical phenomena I7 governing the fluctuating fields u;(x,t) and c’(x,t) (see Chapter 6). An equation analogous to (2.10) follows by integrating Eq.(2.16) to obtain the total flux K, c >+ 5/ [<0 4‘1. (2.19) molecular diffusi:n-//ld x _///// turbulent mixing chemical reaction A mean mass transfer coefficient for physicochemical absorption is salt) defined to recover the total flux analogous to Eq.(2.11), but Eq.(2.12) obviously does not hold. Thus, in the presence of a chemical reaction, «FRI (DON‘T. (2.20) Xi'fl 0 «( , <£¢>Co E ”‘3 dx' x.=o :/ (£5), , (2.21) ° represents the mass transfer rate with no chemical reaction (i.e., k 0). Two asymptotic cases are relatively easy to determine. For very fast reactions, the contribution of the turbulent wsere .: 1. “tn, K. "E'IEC 1‘. 18 flux to the total flux near the interface is small compared with the enhanced flux due to reaction. Thus, according to Eq.(2.16), (6) £900: C.6XP(-x,/ 31: ) (2.22) where 6K E 8/6. (2.23) From Eq.(2.20) it follows that (£2) 5*” r D/BK. (2.211) =(1/ .1). )5; Obviously, for k + O, + °; however, for slow reactions the quan- titative behavior of depends upon the specific model developed for . This region was studied earlier by Petty and Wood [1980b] with the surprising result that E < 1 for k + 0. A more extensive discussion of this and other phenomena related to physicochemical absorption is pre- sented in Chapter 4, but we should underscore here the observation that the slow reaction regime could potentially serve as an unambiguous test for turbulent models. The relative magnitudes of the intrinsic physical scales 6“, OD, and 6k introduced in Figure 2.1 determine the gross behavior of the total flux given by Eq.(2.19). Figure 2.2 illustrates the two limiting regimes of mass transfer for Sc = 103 and 6c: = 1. Here 6c: is the mean field 19 102-- Reaction Turbulent Limiting Limiting '0 Regime Regime + + 60 co J 1 1 4. I 1 1+1» + 10'3 IO-2 10" 1 IO-1 10-2 St -1‘ IO’L- .\ Menez 8 Sandall [1974] (trend only for free interface) -2 \ IO __ \ + \\ Sto \ \ \\ .. \ 103—p— -__——_—— *--- \ \\ IO'“ ‘ ‘” Petty 8 Wood [l980b] \\ (trend only for free interface) Sc=103; St+=10'3 0 Figure 2.2. Qualitative behavior of physicochemical absorption . +': = -1, + 2 = + -1, for slow and fast reactions (OD ..OD/OH Sc , 5co"6co/6H (ScSto) , +_ 6k : Gk/OH) . 20 scale when k E O. In Chapter 8, the quantitative behavior of St+ for + + + . . . D f 6k 3 6c0 15 determIned USIng the theory developed in Chapter 6 and O Eq.(2.16). The two faired lines appearing in Figure 2.2 show the trends of two previous theories. The quantitative behavior of St+ obviously depends on the turbulent model for (uIc’>. 2.4 The Mean Field Equation for Electrostatic Deposition The instantaneous concentration field for submicron charged par- ticles suspended in a turbulent gas phase near an interface is governed by the Smoluchowski equation (see Chandrasekhar, 1943) ac .. 3.9.. .. ‘ at +g-vc up ax, -08v C. (2.25) The charged particles drift with a constant velocity uD toward a collect- ing interface due to an electrostatic force field. The diffusive behav- ior of the small suspended particles is governed by Brownian motion; moreover, the particles are neutrally buoyant and small enough so that an inertial mechanism for deposition is unimportant (cf. Friedlander and Johnstone, 1957). Thus, the interplay between turbulent mixing, electro- static convection, and Brownian diffusion contained in Eq.(2.25) is dev- eloped in Chapter 9. In Chapter 9, the physical origin of the drift velocity and the Brownian diffusion coefficient will be discussed, espe- cially how these parameters depend on particle size. The mean field equation for electrostatic deposition follows from Eq.(2.25) by averaging. As before, the instantaneous solenoidal velocity field is statistically stationary and statistically homogeneous in planes parallel to the collecting surface. Therefore, following the same 21 procedure used to derive Eqs.(2.6) and (2.16) it follows that CHM/d) “ al .3 all“) olx, P dx, = ‘7’“;- (2.26) The boundary conditions for are <¢>=o, x.=0 (2.27) (C) =Cb) x,=oo. (2.28) The interfacial concentration of charged particles is zero because the collector is assumed to be a good conductor. Thus, particles discharge instantaneously once they touch the interface. Once again, the mean field equation is not closed. A model for the turbulent flux is developed in Chapter 6, which depends explicitly on the mean field, hydrodynamic parameters characteristic of u: (see Chapter 3), the Brownian diffusion coefficient, and the drift velocity uD. A novel feature of the turbulent models developed here is that all the underlying physical phenomena influence the fluctuating concentration and, thereby, the turbulent flux. An equation analogous to Eqs.(2.10) and (2.19) follows by integra- ting Eq.(2.26) to obtain the total flux d _ d<¢> __ (u’C’) + u. (c ) 007? ,r,=o ~2 JX, I p . (2.29) molecular diffusion turbulent mixing electrostatic convection 22 As before, a mean mass transfer coefficient for electrostatic deposition can be defined to recover the total flux. Thus, 4(a) I (‘6‘)cb 5.2) 7,:- x.=o = u, C), '< u1c’> x, . (2.30) =3” The presence of electrostatic convection normal to the collection inter- ; face augments the mass transfer rate provided the turbulent flux is not significantly reduced by u The net enhancement of the mass flux 0' due to chemical reaction also depends on the quantitative behavior of according to Eq.(2.19). Thus, one of the goals of this study is to examine theoretically the behavior of for 'slow' reaction rates and for 'weak' electrostatic convection. In Chapter 9, the electrostatic augmentation factor ‘1’; <1€.>/< 1%.; (2.31) is determined by using the theory for developed in Chapter 6 and the mean field equation presented here. Obviously, as uD + 0, w + 1; this corresponds to 6E + w (see Figure 2.1). When OE + O, electrostatic convection dominates transport due to turbulent mixing (see Eq.(2.29)); so, according to Eq.(2.26), (C) “0*" ec[[I—exP(—x,/CSE)] (2.32) where 55 = 23/11., . (2.33) FTC." E » TL W12; and F r 9‘ :r (U H 23 From Eq.(2.30) it follows that (‘82) “0*”: 3/ 55 :(59/85 > ‘6’ :: MD, (2-3'4) which is the well-known Deutsch result (see p. 165 in White, 1963). Eq.(2.34) is a limiting result which holds for 6E << OD, OH. For weak electrostatic convection (OE m 8D N 6“), the total flux given by Eq.(2.29) will depend on the relative magnitudes of all three intrinsic physical scales introduced in Figure 2.1. Figure 2.3 illustrates the two limiting regimes of mass transfer for OH/OD = 103 and 5;; = 1. Here 8;; is the mean field scale when uD = 0. In Chapter 9, the quantitative behavior of w will be examined for 6; f 6; f 63;. Note that Figure 2.2 and Figure 2.3 are similar; however, the intermediate behavior between the two asymptotic limits are different (see Chapter 9). 211 102-- Electrostatic Turbulent 10 Convection Limiting Regime Regime 6 + + D co L I + I I T I I I >65 10’ IO-2 10-1 1 10 102 + -1 ‘ St IO-L' \ 10'2 \\\ Rigid Interface __ \ (trend only) \\ \ \\ -3 \ 10-- _______ _..\_-_ \ \ \x 10‘“ ‘ -r Free Interface ‘\ (trend only) Sc=103; 53:10-3 0 Figure 2.3. weak and strong electric fields (OD+EEOD/cx.t)+g’(s,t) (3-I> where the notation <°> is understood as an ensemble averaged of 3 (see Appendix A). The mean value of the fluctuating component is obviously zero. The average of the product of the velocity field at two differ- space-time positions is A <14(5.t)£62.3)>=<$>cs )(ENL‘ )+_<_,t) g’(x_,t)>#_g, 3’ and 3’ are said to be correlated, and the quantity is called the space-time correlation. When §f=§n it is called the auto-correlation (see Hinze, 1975; and Lumley, 1970). A turbulent field is statistically stationary if its mean com- ponent is independent of time, and homogeneous if its mean component is independent of the spatial coordinates. For a flow field which is statistically homogeneous, the two-point correlation depends only on the distance between the two points, not the 25 26 coordinates of the individual points. If the flow field is also statis- tically stationary, the correlation depends only on the time span lt-ItL not t or t individually (see p. 246 of Monin and Yaglom, 1971, and p. 63 of Lumley, 1970). For turbulent mass transfer near interfaces, the statistical correlation between velocity fluctuations normal to the interface and concentration fluctuations is an important part of the mean flux (see Section 2.2). At high Schmidt numbers, this quantity is controlled by hydrodynamic parameters related to the turbulent intensities, the bursting phenomenon, and the space-time correlation. In this section we briefly examine the behavior of the turbulent intensities near an interface; the other two features are discussed in Sections 3.2 and 3.3. The intensity of a turbulent velocity field is defined as the root mean square of the fluctuating velocity. Because the concentration boundary layer for high Schmidt numbers is deep within the viscous sub- layer, only the first term of a Taylor series expansion of the intensity about the interface is needed. Some information regarding this stems directly from the continuity equation for 27, i.e., 3W fli+1fi- ____ _. (3-3) 3X4 +I 3‘5 3x3 ‘3' and the boundary conditions (see Tennekes and Lumley, 1972). A Taylor series expansion of u; and u; about x1==0 gives 27 3’ U. = Uz(o.x,.. x1+-“ (3.11) 319 I I 130’ U3 = (13(0) X31X31t2+ 3 X, 4“” . (3.5) ax, 5:0 For rigid interfaces, ug=ug=0 at x1=0; however, Bug/Bxllx _0 and 1- I I I o I 3u3/3xllx1==0 are non zero. Thus, uzocx1 and u3°=x1. To fInd U1» Eq.(3.3) is integrated over x1: u"=-J: (313.; +.%_:: ) J1] I z-on’[;"—a( TXIL'.‘ 01:,+...)+ %,(-%‘iz ant-I“- )] J)!" (3'6) which gives ’ 2 ul 0C X, . g (3-7) Eq.(3.7) implies that near the interface the intensity is given by ,3 (ll ‘) + 45! ‘1‘: = a/R X, (3.8) .1. \ where x1+ is x1 made dimensionless with the friction velocity u, and the kinematic viscosity v. 31R+ is a “universal” parameter independent of the Reynolds number. Monin and Yaglom [1971] used the intensity data of Laufer [1953] in the wall region, and determined that alR+==6.4>(10‘5. Recently, Coles [1978] derived a model for the intensity in the sublayer region by matching the velocity profiles in different regions. He found that 28 ,1 V; “3+ 3 (a, > = u‘x 3.44100 x,zexp[~4x:///36J (3.9) 55 + , * - . . . - For x1 +0, =u x3.’-l’-1x10 3xldl-Z, which implles that a1R+=10 5. El Telbany and Reynolds [1981] in their most recent paper reported a1R+=2.5x10'5. ’ and u’ are If the interface is now a so-called free interface, u2 3 not necessarily zero at the interface. Therefore, ugcrxl, which implies ,/<5t=32. (3.12) Here R6 is the Reynolds number based on the momentum boundary layer J. thickness, 6“ is the displacement thickness, and o is the free stream velocity. From Eqs.(3.11) and (3.12), they suggested that TM scales with outer parameters rather than inner parameters. Sideman and Pinczewski [1975] argued that only a small fraction 'b' of the total number of ”eddies“ can penetrate the sublayer fluid. Therefore, the bursting period in the wall layer is much lower than that in the transition region, and is given by TM TM. I = 6 . + . Here TM,1 is the mean period in the wall reglon (x1 ==1), and TM 15 the mean period in the outer region. With b==0.07 and TM==2h3, TM’1==3h70. 3] Davies [1975] modified their formulation by letting b==0.07 x1+, so that + b= 0.07 at x1 =1, and b=1 at x1+=1ll, where intense bursts originate. Thus, + T»: = 3470/x1*. (3.13) In Figure 7.2, a similar trend for TM+ near a rigid interface is devel- oped using different arguments (see Chapter 10 for further discussion). Nakagawa and Nezu [1978] measured the conditionally sampled fluc- 1/ 1 tuating Reynolds stress ufu§/( 2/2) and identified the bursts with the ejection events (uf>>0, u§<<0) and the sweep events (uf<30, u§3>0). Their results show that the bursting period scales with outer parameters as 7Ch‘£° i: Clo A, 7; ‘19 ~ a-r. - fl :Ls+3. (up where h is the flow depth; Tm, Te and T5 are the mean bursting period, ejection period and sweep period, respectively. Their results also confirmed that the probability distribution of the mean bursting period is log-normal. Hatziavramidis and Hanratty [1975] solved the equation of motion numerically for a flow which is homogeneous in x3 and is periodic in time. The period was taken to be the time interval between bursts, and the wavelength of the coherent structure was taken to be the spacing of the Streaky structure near the wall. They chose TM+==100 and calculated velocity profiles which agreed with the measured velocities predicted by Kim et al. [1971]. 32 Berman [1980] deduced TM from the auto-correlation data and noted the results depend on the size of the probe, the flow rate, and even the viscosity of the fluid. The measured TM+ ranged from 200-1000. Berman correctly points out that the wide scatter in the values of TH+ are attributable to the inconsistency in the definition of the bursts, to the different methods of identifying the bursts, and to the technical details employed. Unfortunately, most measurements are made far from the wall compared with the concentration layer thickness (6c+<<5) at high Schmidt numbers. Therefore, the values of TM+ available in the literature are probably lower bounds for the renewal of fluid important for mass transfer at high Schmidt numbers. The TM values near a free interface are sparse. Davies and Lozano [1981] measured the auto-correlation in water near a water-air interface in a stirred tank, and defined the integral of the auto- correlation as the average time for complete replacement of an eddy by another. If this is identified as the mean bursting period, then these experimental results suggest that TM in the liquid phase could be 0.06- 0.3 sec depending on the rate of stirring. 3.3 Velocity Space-Time Correlation Near an Interface The normal component of the two-point space-time correlation for a fully developed flow is defined as (see Favre et al., 1957) A A R 501/09. X» aim/(x1. X» 29.16)) (3.15) The flow has a mean velocity and is statistically homogeneous in Planes parallel with the x2-x3 plane and is statistically stationary. The 33 two points have the same x1 and x2 coordinates, but different x3 coordi- nates. The coordinate system is shown in Figure 3.1. A typical three- dimensional representation of R is also shown in Figure 3.1. Projec- tions of R onto the R-t and R-x planes are shown in Figures 3.2a and 3 3.2b. Other components of the space-time correlation are unimportant for mass transfer at high Schmidt numbers. Some of the characteristics of R are as follows: (1) R is peaked at x3==;3 and t==t. The velocity fluctu- ations are most correlated at the same point and same instant of time. (2) The ridge of the three-dimensional configuration is called the envelope correlation. The projection of the ridge onto the R-t plane maps the locus of the maxima of the correla- tion curves with different 53; its projection ontolthe R-x3 plane maps the locus of the maxima of the correlations with differ- ent T. According to Favre et al. [1957]1, the maxima of the correlations move with an approximate velocity of 0.8. Thus, the velocity fluctuation between two points separated by a distance of 0.8T are highly correlated because informa- A tion is convected to x from x by the movement of the eddies. 3 3 For a fixed £3, the flow remembers its past the most at the time t==t+-€3/0.8. The maxima decreases as time (or distance) increases because the turbulent eddies lose their identities through mixing with the surrounding turbulent 1These comments apply to the longitudinal component of the space- time correlation in a boundary layer flow. We assume that the qualita- tive behavior of the normal component is similar for this and other fully developed flows. 3h A I A ’1 u1(x1,x2,x3,t) u1(x1,x2,x3,t) X. L l A x3-x3=€3 —F x Ill/' 3 Interface ==:;> ’v—’-.'IJ Envelope Correlation \ / 4~—II->X3 / " / t ._.3>(\ \\ \ , A (x3 - x3)/(t - E) = 0.8 Figure 3.1. The normal component of the space-time correlation. 35 >21 auto-correlation T = €3/0.8 envelope correlation E =- t t T 3 cons an 1 .A 53: v v V constant1 £3: -constant2 = is constantz Figure 3.2a. Velocity correlations for fixed distances. T = -constant Figure 3.2b. 36 DO envelope correlation - €3=0.8T T = constant x3 Velocity correlation for fixed times. 37 fluid. The envelope correlation can be visualized as a two- point correlation measured by an observer riding on a frame moving with velocity of 0.8. (3) Because R is homogeneous in the x3 direction, an R-t curve associated with a particular 53 has a mirror image with respect to the plane t==t, corresponding to the correlation curve with separations of -€3. (A) The auto-correlation, obtained by setting €3==0, is of particular importance in turbulent mass transfer theory (see Figure 3.2a). A simple expression for the normal component of the space-time correlation, which reproduces some of the qualitative features men- tioned above is (cf. p. 61 in Yaglom, 1962) R E < Wow) u,’<£.3>> 14 ,-" -" - " - = (u’/1>K(x’) (1319‘? [_ [X XI[_lx1 xtL [X1 X, {U,)7[_ Lit—L]. (3.16) 1m l”; ’ell3 " Eq.(3.l6) has the following properties: (1) When fif‘gy the auto-correlation has a maximum at t==t, and is a symmetrical function of T. The decay of the auto-correlation is characterized with a time constant equal to 1/(1/TH-+/2113). However, the first derivative at A t= t is non-zero. A (2) For x1==x x ==x2, and a fixed T, the maxima always 1’ 2 A occur at x3 = x3 + T. (3) An observer riding on a frame moving with a velocity 38 A A A g=33 w11l have coordinates x =x1, x2=x2, x =x3+T. 1 3 1TH: Eulerian correlation measured by this observer is the envelope correlation if we assume that the maxima of the cor- relation curves move at the speed . The envelope corre- lation has the following simple expression R. n."- ,:.K A [Tl/ = (u. >cx,) «16,8 . (11.1) There are two key ideas in this theory: (1) fluid elements with uniform concentration of passive constituent intermittently penetrate the region near the interface and sweep the stalled elements depleted in passive constituent away from the boundary; and, (2) the mechanism for mass transfer during the stalled stage, in which no concentration fluctua- tions are generated, is unsteady state molecular diffusion from a uni- form initial condition. These assumptionsltmw>allthe turbulent effects into the surface renewal process , thereby decoupling the molecular and turbu- lent effects. Harriott [1962] recognized that the first assumption vio- lated the no-slip condition at rigid interfaces and introduced a vari- able sweeping distance within which the eddies arrived and swept away #0 Al the material near the wall. King [1966] tried to include the turbulent effects in the mass transfer process induced by the small scale fluctu- ation within the eddy by incorporating an eddy diffusion term in the mass transfer equation. Ruckenstein [1971] modeled the mass transfer process to an eddy by treating the eddies as ”roll cells.‘I He simula- ted the micro-structure velocities of the eddies as the superposition of a rotating cell motion on the average bulk liquid velocity, and solved the convective diffusion equation to obtain the mass transfer rate. Despite the difference in the detailed calculations in each model, the overall physical picture of the renewal theory is consistent with the observed coherent motion in the near wall region; that is, the renewing field element may be identified with the wall streaks observed by thu: at al. [1967], and the renewal time may be related to thetime between burst. Thus, according to the renewal theory, mass transfer is generally controlled by the spatial and temporal scales character- izing the Streaky structure near the wall. However, for high Schmidt numbers, these spatial structures (==30 v/u* in diameters) are too big compared with the concentration boundary layer thickness (<15 v/u*) to have a direct effect on the mass transfer. The classical eddy diffusivity model for the turbulent flux assumes that the mean concentration gradient is the driving “force” for turbulent mass transfer (see, e.g., Sideman and Pinczewski [1975] for a lucid review). For a statistically homogeneous concentration field parallel to the mass transfer surface, d (Ll/CI) = -De()‘1) JX, . (4.2) AZ Prandtl [1925] drew an analogy between the kinetic theory of gases and postulated that 08 may be expressed as a product of a characteristic mixing length, and some characteristic velocity associated with the underlying transport process. The analogy, however, neglects the effect of viscous dampening on velocity fluctuations as the wall is approached. Van Driest [1956] took this phenomenon into account by assuming that the dampening coefficient of the form [1-exp(x1/A)]. The parameter A is an empirical spatial scale over which this effect is important (cf. Figure 6.h). The resulting eddy diffusivity is . 1401 > D... =(Km [hum-MA) “7;:— (4.3) where K is the Von Karman constant (cf. p. 89 in Sideman and Pinczewski, 1975)- Lin et al. [1953a] proposed an empirical expression for the momen- tum diffusivity within the viscous sublayer. Because the velocity pro- file calculated using their expression agreed with experimental data, they used this result together with Reynolds' analogy (see p. 99 in Sideman and Pinczewski, 1975) to estimate the mean mass transfer rate and concentration profile. Basically, this approach, which eliminated the idea of a stagnant film close to the wall, assumes that D ac ’ a X, . (14.11) This gives (fig) ac 32/3 according to Eq.(2.6) and Eq.(h.2). + Notter and Sleicher [1971] obtained the diffusivity for x1 +v-[g’gi]— a vow. This should be compared with Eqs.(5.20) and (5.22). Sternberg argued that in some region near the wall the viscous terms in the equation of motion would dominate the inertia terms. Thus, in the limit as the wall is approached, the viscous effects determine the leading terms in an expansion in x1 for the amplitude of the velocity fluctuations. With the Sternberg hypothesis applied to mass transfer, Notter and AA Sleicher [1971] derived an eddy diffusivity in the vicinity of a rigid interface of the form D¢(X,)=GX,3+6X,4, (14.6) They argued that Ia]==|b|. Hence, they predicted that in the limit as the wall was approached the eddy diffusivity would vary as x13. Sirkar and Hanratty [1970] and later Shaw and Hanratty [1977a,b] adopted the same linearized equation for c1 They assumed that this equation was applicable to the high frequency fluctuations, anul derived arl expres- sion for De of the form D I +3 L 0C —-——-5 ”z X, +.... . (11.7) C. Although the justification above for retaining only the lead term in the above series was not given, the linearized equation for c’ was solved to obtain a spectral density function for mass transfer fluctuations which followed the experimental data quite well; however, Eq.(uw7) failed to agree with their most recent data on mass transfer coefficient for high Sc: -.704 St"; (‘59: = 0.0 337 5c . (11.8) u One implication of the x13 dependence of D6 is that the turbulent flux is determined by mass transfer fluctuations at the wall. 1 For instance, because c d3: -———*- consfant , 45 X,'—'" 0 , I AS Eq.(h.2) implies that °c xr". (4.9) With u;crx13 near the wall. Petty [1975, 1978] used the same linearized equation to solve for c’ by the use of a Green's function technique. He found that if the Green's function is spatially peaked compared with the space-time corre- lation for high Sc, then 5' D, 011*): 61,; JTKSc xi’ ,- (4.11) However, if the Green's function is spatially peaked relative to the space-time correlation coefficient, then T" 5 D. and TH is the relaxation time for the envelope correlation (see Chapter 3). As ‘17 Sc-*w, Eq.(4.1l) essentially reduces to Eq.(4.10). Eqs.(h.10) and (h.11) show that the eddy diffusivity is not just a hydrodynamic construct, but depends intrinsically on all the physico- chemical phenomena such as the space-time correlation of velocity fluc- tuations and the dampening of concentration fluctuations due to molecu- lar diffusion. Hinze [1975] used a Lagrangian description of a fluid particle moving through the flow field to derive a similar expression, viz., Q D,=a,,x.‘* “f(T)/2. <1). (M3) 0 where RLfr) hsa Lagrangian auto-correlation coefficient and f(T) accounts for the effect of exchange of a fluid particle with its surroundings. The idea that the eddy diffusivity depends on the velocity correlation and the interaction of the eddy with its surroundings is similar to an approach using Green's function advanced in this research. Both theor- ies contain the spatial and the temporal structure of the coherent motions near the wall explicitly. However, for large Schmidt numbers, the spatial structure of the coherent motion is unimportant in deter- mining the mass transfer rate. One physical feature implied by the eddy diffusivity model is that sudden spatial changes in the mean concentration gradient causes an instantaneous change in the turbulent flux. This is fundamentally unrealistic because the time scale associated with the turbulence may not be small compared with the time scale associated with the mean con- centration gradient, so extra time is needed for the turbulent flux to 118 readjust to the new mean concentration field. The failure of gradient- type models for turbulent transport is demonstrated by the momentum transfer experiments of Eskinazi and Yeh [1956] in a concentric annulus with a smooth and a rough wall, and by the experiments of Hanjalic and Launder [1972] for a channel with a smooth and a rough wall. In both experiments the Reynolds stress can be non-zero for zero velocity gradients. Wood and Petty [1981] derived the following non-gradient model from the evolution equation for c’ and the requirement that ==0: (4.1h) This model yields results for the mass transfer coefficient which are quantitatively consistent with the experimental data of Shaw and Hanratty [1977a] for (a range of hydrodynamic parameters estimated using recent data on bursting rates and turbulent intensities near the wall. Brodkey et al. [1978] avoided the use of any turbulent model and tried to solve the simplified instantaneous equation .a_c_ 9a. 2‘2. cit-[m ax, - flax? numerically by simulating the normal velocity ”1' Unfortunately, they assumed that the root-mean-square of ul has the form ’2 Uh 1 (“I > = 4X1+bxl +H° , (“'15) which violates the continuity equation (see Chapter 3). Campbell and 49 Hanratty [1981a,b] also used a numerical method to approach the mass transfer problem. In their first paper, the Fourier transform of the linearized equation for c’ is solved numerically, and the results are fitted to match the high frequency measurements of Sirkar and Hanratty [1970]. The calculations showed that 3 <15.) cc .9 /“ , (11.16) which also follows from Eq.(h.10) when Eqs.(2.6) and (h.2) are used as a model for mass transfer. In a companion paper, Campbell and Hanratty [1981b] studied two different instantaneous mass balance equations which they called “non- linear” models I and II. A record of experimental measurements of the random normal velocity fluctuations were used as input. Model | con- tains only the normal coupling as a source for turbulent mixing; whereas Model II keeps both normal and lateral turbulent mixing terms. Both models predicted the correct dependence of and yE/ on Sc. It is noteworthy that their numerical results showed that a (alRTH)5%, which is closely related to that predicted by Petty [1975] using the Sternberg linearization and a Green's function technique. h.2 Physicochemical Absorption Across Free Interfaces There are two major physical aspects which make a flow field near a free interface qualitatively different from one near a rigid inter- face. First of all, the no-slip condition is no longer valid at the free interface, so the fluid at the surface Eag_have tangential velocity components. This, together with the continuity condition, requires 50 u;o. (11.18) Eq.(h.18) was also tested experimentally by Ueda et al. [1977]. Petty and Wood [1980a] derived an expression for De by solving the evolution equation for c’, and reported that Decrxla. Furthermore, by examining experimental results of gas absorption into falling liquid films and assuming that the eddy diffusivity is related to the small eddies, King [1966] concluded that De (x1) cc x,”', as x,—-0. (£1.19) F01 and analy: integral 1 experimen transfer fixing th transfer . other han ahalys i S , notion of Cient t1“. "if”: e: ‘1 kiqepatli ‘I'Sls the eddies, ted data SCOtt [1 51 Fortescue and Pearson [1967] adopted the surface renewal concept and analyzed the mass transfer into roll cells having size equal to the integral scale of turbulence. Davies and Lozano [1979] compared the experimental data of oxygen absorption into water with various mass transfer theories and found that the Levich approach based on theFWandtl mixing theory gave the best agreement. They thus concluded that mass transfer was controlled by the large scale Prandtl size eddies. 0n the other hand, Lamont and Scott [1970] extended Fortescue and Pearson's analysis, but assumed that the mass transfer rate was controlled by the motion of the viscous small scale motions. The mass transfer coeffi- cient thus calculated has the form (‘5‘ ) = 0 5a.”: (51/ )"4, (11.20) where E is the rate of energy dissipation per unit mass, and v is the kinematic velocity. Prasher [1973] argued using the dimensional anal- ysis that if the eddy diffusivity is solely determined by the small eddies, the resultant is of the same form as Eq.(h.20), and presen- ted data to support this hypothesis. Thus, according to Lamont and Scott [1970], and Prasher [1973], the small scale eddies of Kolmogoroff size control the mass transfer rate; however, Levich [1962], Davies [1972], Fortescue and Pearson [1967] and Davies and Lozano [1979] all support the view that large scale Prandtl size eddies set the mass transfer rate. Petty and Wood [1980a] combined their theory with the data of Davis and Lozano [1979] and deduced hydrodynamic parameters from that. They found that the parameters were independent of Reynolds number if made dimensionless with scales characteristic of small eddies. 52 They therefore concluded that the mass transfer into liquid film was controlled by small eddies. 0n the other hand, Theofanous et al. [1976] presented data which suggested that the energy containing turbulent motions should dominate the mass transfer rate at small Re, while at high Re the energy-dissipating motions controlled the rate. Henstock and Hanratty [1979] argued in their recent paper that an interpretation of gas absorption by liquid layers required a hybrid approach which introduced two length scales, one characteristic of the large scale tur- bulent motions near the mass transfer interface. Their argument tends to imply that both large and small eddies may be responsible for the gas absorption process. When an irreversible chemical reaction is present, the concentra- tion fluctuations are dampened due to the reaction, and this could have a significant affect on the turbulent flux . Most works on turbu- lent absorptions with simultaneous chemical reactions (known as physico- chemical absorption) use an eddy diffusivity obtained from physical absorption data alone (see, for example, Menez and Sandall, 197A; Step- anek and Achwal, 1975a,b; Yeh and Seagrave, 1978; and Gottifredi and Quiroga, 1979). With this type of approach, chemical reaction will affect the turbulent flux only implicitly through the concentration gradient. Petty and Wood [l980a,b], on the other hand, solved the linear- ized evolution equation for c’ and obtained an eddy diffusivity as a function of the reaction rate constant. The turbulent flux is thus affected by the chemical reaction explicitly through the eddy diffusi- vity. An abstract by Corrsin [1972] also mentions this possibility. One consequence of this theory is that a chemical reaction may not in 53 the usual way enhance mass transfer. This could occur because a chem- ical reaction, while steepening the concentration gradient, actually decreases the eddy diffusivity. However, all the experimental work on absorption in the near mass transfer regime (i.e., kinetic limited) show that the enhancement factor is much greater than one. h.3 Electrostatic Deposition The main difference between mass transfer of particles and that of chemical constituents is that particles have finite mass and inertia. If the particles are small enough, Brownian diffusion, turbulent dif- fusion, and inertial impaction may all contribute to the deposition rate. Friedlander and Johnstone [1957], in their pioneering turbulent deposition studies in vertical pipes, argued that the experimental par- ticle deposition rate is considerably higher than can be accounted for by the mechanism of turbulent diffusion alone, if the particle and the fluid eddy diffusivity are assumed equal. They found that the particle must be endowed with a finite momentum to account for the apparent abil- ity to penetrate much of the boundary layer with little or no resistance. They then developed a theory where the particles were assumed to be transported by eddy diffusion to within one “stopping“ distance from the wall, at which point particles made a ”free flight” to the wall, where they arrived with zero velocity. The stopping distance is calcu- lated as the product of vT, where v is the initial free flight velocity and T is the particle relaxation time over which the particle loses its momentum completely to the viscous drag. Friedlander and Johnstone assumed v to be 0.9 u", the normal turbulent intensity in the turbulent 511 core. In the subsequent development of this “diffusion free flight” model by Davies [1966a,b] and by Beal [1970], the same assumption was made regarding the equality of the particle eddy diffusivity and the momentum eddy diffusivity. However, different free flight velocities were used. Beal assumed that the initial free flight velocity was equal to one-half of the axial fluid velocity at the point where the free flight began, plus a velocity due to the Brownian motion of the particle; on the other hand, Davies used the local fluctuating velocity of the fluid as the initial free flight velocity. The theory of Friedlander and Johnstone and that of Beal give results that are in reasonable agreement with experimental data, while Davies' theory gives deposition rates that are too low. Later, Forney and Spielman [197A] modified the earlier model of Friedlander [l95h] for the deposition of coarse particles and found good agreement between the theory and data. Their theory also assumed that v was equal to 0.9 u*. The free flight model may, in a sense, be regarded as the analogy of the “film” theory in turbulent mass transfer. The fluid within the stopping distance has been treated as if it were stagnant, and the only mechanism for mass transfer is the inertial impaction once the particle is transported to within a certain distance. Although the model has provided a simple, straightforward description of the particle depo- sition process, it neglects the important turbulent mixing of the par- ticles within this fictitious and also arbitrary “stopping distance,” especially for small particles. Furthermore, the problem of having to use what appears to be unreasonably high, or arbitrary, free flight velocities to obtain reasonable agreement with experimental data makes the approach fundamentally unsound. 55 Sehmel [1370, 1971] replaced the idea of a free flight mechan- ism with an empirical ”effective” particle diffusivity to incorporate the inertial effect near the deposition surface. The effective diffusi- vity calculated from the experimental deposition data is higher than the fluid eddy diffusivity. Rouhiainen and Stachiewicz [1970] tried to explain the high particle diffusion rate by the shear lift experienced, by a particle moving through the shear field in the boundary layer. Similarly, Hutchinson et al. [1971] sought an explanation on the basis of a statistical, random walk model. Liu and Ilori [197A] adopted the concept of an effective particle diffusivity of Schmel and proposed a model for it. They assumed that the diffusivity was the sum of a fluid eddy diffusivity and a diffusivity related to the inertial effect. The resultant particle diffusivity is much higher than the fluid eddy dif- fusivity, especially in the viscous sublayer. While these theories are able to predict the deposition rate reasonably well, they all need to resort to either some empirical parameters or detailed trajectory cal- culations. When charged particles entrained in a turbulent flow are passed through an imposed electrostatic field, electrostatic convection helps transport the particles towards a deposition surface in addition to the other mechanisms of Brownian motion, turbulent mixing, and inertial impaction. In the classical theory of electrostatic precipitation, Brownian motion is neglected and the turbulent mixing provides a nearly uniform concentration of charged particles in the bulk gas stream. The limiting particle flux is determined by electrostatic convection near the ground plate. With these assumptions the collection efficiency is given by the so-called Deutsch equation (White, 1963) 56 77 = I“ “P ("up 14/62,) , (4.21) where “D is the ”drift” velocity caused by the interaction of an exter- nal electric field with a charged particle; QB and A represent, respec- tively, the volumetric flow rate of the gas phase and the total area available for deposition. Collection efficiencies for large particles are fairly well rep- resented by Eq.(h.21), provided a number of empirical corrections are made to account for imperfect mixing and non-uniformities in electric conditions (see, for example, White, 1963; Gooch and Francis, 1975; Ledbetter, 1978; Reynolds et al., 1975; and McDonald, 1978). However, Leonard et al. [1970] have correctly argued that Eq.(h.21), which con- tains only electrostatic convection in the near wall region as a mechan- ism for particle deposition, does not necessarily represent, as so often assumed, an upper bound to practical collection efficiencies--especially for small submicron sized particles. Other effects, such as Brownian motion and weak turbulent mixing near the collecting surface, can play a significant quantitative role in capturing submicron particles. In the Deutsch model, the turbulent motion provides ample mixing to keep the concentration of particles uniform. Friedlander [1959] con- sidered the dampening of the turbulent mixing in the region close to the collecting wall. Instead of attempting the difficult solution of the convective diffusion equation for this case, the assumption is made that of P09.) = De +UDi (14°22) «IX, 57 where p is the particle fiux normal to the wall, and is a function of the distance in the direction of flow (i.e., parallel with the depo- sition surface). De is the eddy diffusivity. By assuming that c==0 at the wall and that the turbulent Schmidt number equals one, he calcula- ted a collection efficiency which is always greater than that given by the Deutsch equation. This is because the additional turbulent flux near the wall helps the electric convection to bring the particle toward the wall. WilliamsenuIJackson [1962] solved the convective-diffusion equa- tion 3<¢> 1913 a‘(c)_ u 3(6) =' 0‘ a)"; D ax! 3 (4.23) within the viscous sublayer near the collection surface. The eddy dif- fusivity De, the drift velocity1q3,and the gas velocity represent 3 the average values independent of x1. As in the Deutsch equation, all the particles entering the sublayer are rapidly attracted to the collect- ing wall by electric forces; therefore, 3/3x1==0 at the core-sublayer interface. There is no provision for reentrainment. Inyushkin and Averbukh [1962] obtained experimental evidence that turbulent action exhibits both positive and negative effect on particle collection, espe- cially at high Reynolds numbers. They propose that the positive contri- bution of turbulence is due to the inertial impaction of particles through the viscous layer; the negative effect of turbulence arises because turbulent eddies continuously redistribute the particles that the electric field tends to concentrate in the neighborhood of the wall. Cooperman [1960, 1965] incorporated turbulent reentrainment into 58 his model. As in the case of Inyushkin and Averbukh [1962], positive and negative turbulent effects are postulated. However, he argued that the positive effect comes from the turbulent diffusion sweeping parti- cles from the region of higher concentration to the wall for low reen- trainment cases; when the reentrainment is high, a dense cloud of par- ticles is formed near the wall, turbulent diffusion now sweeps the particles away from the wall. For large turbulent diffusion, the effi- ciency calculated by Cooperman obeys an exponential law; the exponent, however, being different from that given by Deutsch. Robinson [1967] assumed that the total particle flux is the sum of the particle flux transported to the wall and a flux of particles eroded away from the wall. Without calculating the detailed concentration profile, he der- ived an expression for the collection efficiency in terms of a: uniform- ity parameter xiicw/c, where cw is the concentration at the wall and E is the concentration averaged over the cross section. Leonard et al. [1979] solved a two-dimensional convective diffu- sion equation for : at.» aka] a , 222 De]: 3,911. + 3):," “up 3X, "' “3);“ . (11.211) The eddy diffusivity D6 is assumed constant. However, to overcome the difficulty of the high deposition rate due to the constant diffusivity, they proposed a boundary condition al Aw+ and the mean concentration . Additional equations relating these variables can be developed from an equation for c’(§,t), which is easily derived by subtracting the mean field equation from the instantaneous mass balance. All three problems formulated earlier give the same type of equation for the fluctuating field, viz., I I J“) I I J'z - (5.6) xi Thus, a characteristic time scale for the production of concentration 1 fluctuations by this mechanism is x1//2 (ETp). From Eq.( 3.8) and Eq.(3.lO), it follows that ,0 ’ free interface (5.7) 7;, = $1067 X; , rigid interface (5.8) 0 Which partly explains the different behavior of St+ with Sc for free and rigid interfaces (see Chapter 7). 62 The term designated (:> in Eq.(5.l) also “produces” concentration fluctuations. The mechanism here involves the coupling of all three components of the fluctuating velocity with the fluctuating concentration gradient. The gradient of the mean turbulent flux is an important com- ponent of (:) inasmuch as it makes <(:)> = O. In the mass transfer studies of Sleicher [1969], Sirkar and Hanratty [1970], and Petty [1975], (:) was neglected by using an argument that near an interface where vis- ’ cous dampening occurs, ”non-linear” fluctuating quantities, such as ulc’, are small compared with uI. The validity of this hypothesis, used earlier by Sternberg [1961] for momentum transfer, has recently been questioned by Shaw and Hanratty [1977a,b] for mass transfer. From the definition of (:) it follows that "@u 5 (128175)»? (5.9) Now, if we let Ac denote a characteristic length scale for concentration fluctuations, then ,, ./ I I 2. V2 (11(3),, (6;) z. °VC )> N . (5.10) A. <(2 1 Furthermore, if /2/)\c W /6C, the production of concentration fluctuations due to the second mechanism is small compared with H<:)H for X, +O; however, for x N 6c the two mechanisms may be comparable. 1 lilthough the above suggestion is speculative, it provides some §_priori motivation for developing a theory which allows for some of the physical effects contained in <:). 63 Random initial and boundary conditions can also be sources of concentration fluctuations, but in this research these are considered unimportant. It is therefore convenient to think of c’(x,t) as the superposition of fluctuations generated by (:) and (:) at other space- time points and propagated to (xgt) by the action of the non-random, AA - linear, integral operator 1,, . The transport phenomena from (fit) to (xjt) accounts for the various ”loss” mechanisms such as molecular dif- fusion and irreversible chemical reaction (cf. p. 384 in Hinze 1975). Formally, C] ==az::' [(::)-I<::) ].. (5.11) Because <®> = 0 and <®> = 0, it follows that = 0 also. This is an important characteristic property of concentration fluctuations which should be retained, if possible, by any approximate model of Eq.(5.11). Petty [1975] and others used Eq.(5.11)‘with (:) = O as a first 0 o I o approx1matlon for c , 1.e., c' :- IJWCD]. (5.12) This representation, which satisfies = 0 independent of the mean field, leads directly to a non-local gradient-type model for the turbu- lent flux (see Chapter 6). The next step would be to construct a repre- sentation of Q), using Eq.(5.12), so that <®>=0. This could then be inserted into Eq.(5.11) to obtain a second approximation. Basically, this approach yields a sequence of successive approximations generated by the coupling of the mean gradient with the normal fluctuating velocity. 64 A characteristic feature of all turbulent flows is that a wide spectrum of space-time scales exists. Which scales are dynamically sig- nificant for a particular transport problem is a central question. 6c I is clearly an important parameter related to the turbulent flux , but the strategy of estimating concentration fluctuations with the mean field as d§g§_§x_machina places undue emphasis on 6c. Which scales cause 6c? The question invites a different formulation of Eq.(5.l) that bridges the mean field to the underlying turbulent field. In this research, Eq.(5.1) is rearranged to give ’_. ______ .- Zc - u. Jx, Ix. (5.13) where OZ : X0 -1_I’- v. (5.111) We are motivated to consider Eq.(5.13); not because Eq.(5.12) is funda- mentally unsound as a first approximation, but rather from the observa- tion that Eq.(5.13) leads to an analysis of mass transfer, which links the statistical properties of the passive additive field more directly to the space-time structure of the turbulence. The ”source” of concentration fluctuations in Eq.(5.13) is no longer an explicit function of c’ (cf. Eq.(5.1)); note also, however, that <(:)> = (:) # O. The random convective mixing of concentration Fluctuations is now contained in the linear, differential operatorwil. A formal representation of c’(x,t), which is the analog of Eq.(5.11), is 65 = eff-1CD - @]- (5.15) Approximate solutions to this equation should be: (I) statistically sta- tionary, (2)5tatistically homogeneous in planes parallel with (O,x2,x3) and, (3) should satisfy =0. This last property leads to some inter- esting and surprising results if we use Eq.(5.15) as a starting point, rather than Eq.(5.11). The physical interpretation of Eq.(5.15) is similar to the one given for Eq.(5.11). Now the distributed “source” of concentration fluc- tuations is G) - 6), instead of G) + Q). The propagation of these fluctuations includes the random mixing effect, which makes the construc- tion of ‘1?! more difficult (see Section 5.2). Figure 5.1 summarizes part of the methodology followed in this research. The remaining sec- tions of this chapter outline the specific mathematical and physical assumptions used to develop a first approximation of the integral opera- tor :fi". 5.2 Evolution Equations for the Mean and Fluctuating Components of the Green's Function Eq.(5.15) can be written in terms of a Green's function associa- ted with the differential operatorx . The result is t I _ _ A A . . ,. A d J C(x’t)— jdtlJflG(51*[§,*)[U/( (_X_ ’t)dxl" T] (516) 1 Eq. for 66 Material Balance for 3 Passive Additive Previous Research This Research STERNBERG HYPOTHESIS Eq' [Or (5) Q’: z;'[®+®] STERNBERG HYPOTHESIS l z§'=1®j Model I 2.536%] / for ... Gradient Model Gradient Model for for I Neglect 2__ Mean Field Memory Relaxation Model Chapter 6 for Figure 5.1. Methodology for a theory of turbulent mass transfer. 67 The lower limit on the temporal integral is -w because c(§,t) is statis- tically stationary. The Green's function G(xjtlgjt) satisfies homoge- neous Dirichlet boundary conditions and the following random differential equation f(G)="§(§-£)c§(f-2)- (5.17) Some useful properties of classical Green's functions are summarized in Appendix B. Physically, G(x,tlxjt) represents a passive additive field caused by an impulse source located at (xjt). Analogous to Eq.(2.5) the instantaneous Green's function can be decomposed into a mean and fluctuating component, i.e., C} :: < C}‘> .f C}, . (5.18) An equation for follows by simply averaging Eq.(5.17) to obtain ofxm- V~<2’c1'>= wagon—2). (5.191 The following equation for G’ is the result of subtracting Eq.(5.19) from Eq.(5.17) 1261’ = 15’- V + 11’- Vc1’ - (3’. 176'), (5.20) The similarity between Eqs.(5.20) and (5.1) is obvious. Note, however, that G’ is partly generated by a full three-dimensional coupling between 68 the fluctuating velocity and the mean gradient V; in Eq.(5.1), only uId/dx1 appears. Obviously, <<:>> = O and <(:)> = O. A formal solution to Eq.(5.20) can be written in terms of a non- stochastic Green's function associated with the differential operator at, and homogeneous boundary conditions. Thus, GI= «it. [©+®], (5.21) which is Eq.(5.11) again, but with a more general distributed source term. The classical Green's functions associated with the operator for each of the cases defined by Eqs.(5.2)-(5.4) are tabulated in Appen- dix B. 5.3 The Generalized Sternberg Approximation By neglecting the production mechanism described by <:) in Eq.(5.21), the following approximate representation of G’ results G'=$f:[©]. (5.22) This closes the turbulent mass transfer problem. Because of the formal similarity of this result to Eq.(5.12), we refer to Eq.(5.22) as the generalized Sternberg approximation. A second approximation could be developed along the lines suggested for c’ (see comments following Eq.(5.12)). An important feature of Eq.(5.22) is that = 0 for 221. approximate mean Green's function . This stems directly from the linearity of 02;, and the observation that = 0. The foregoing approximation formally belongs to the same class as 69 Eq.(5.12), but the result is fundamentally different. This occurs because , unlike , begins as a spatial delta distribution at t = t and then spreads out over the spatial domain according to the physical phenomena portrayed by Eq.(5.19). The ”production” rate of G’ through the coupling of g: with V will therefore depend on all the velocity scales, not just the ones tuned to BC. Thus, Eq.(5.22) serves as a link between the mean field properties and the intrinsic underlying turbulent scales. Although it may be possible (and interesting) to establish a formal relationship between a theory based on Eq.(5.22) with one which generates an infinite number of terms by iterating Eq.(5.11) using Eq. (5.12), such a question goes far beyond the scope of this research (see, however, the lucid review article of Frisch [1968]). The emphasis here will be limited to the behavior of the mean field using the generalized Sternberg approximation as a working hypothesis. The merits of this suggestion should be judged on the results generated. 5.4 Finite Memory and Spatial Smoothigg Eq.(5.22) can be used to estimate the turbulent flux in Eq.(5.19). Multiplying G’(x,tlx,t) by 37(xjt) and averaging the result gives Gr'(s.+I£.ai1> = t 2 R O a 2 ’ I A a a * A A T!“146m&(§:*I£I1I(t.tlx,t). (5.23) Causality requires that the temporal variables satisfy the following 7O inequality (see Appendix B) t+-> fl\ d~s> H\ "h . (5.24) Eq.(5.23) is a non-local, gradient-type model. A similar result would hold for if Eq.(5.12) were used for c’ rather than Eq.(5.16). The space-time correlation for velocity fluctuations (see Chapter 3) and the Green's function for the operator 20 (see Appendix B) are important elements of the foregoing model for <2fG’>. At high Schmidt numbers the result simplifies considerably. This occurs because the spatial spreading of G°(x,t]§,§) is controlled by molecular diffusion and because the space-time correlation has finite memory. This latter property stems from the well known observation (see Chapter 3) that flow structures near interfaces are characterized by two dynamic stages consisting of a violent burst and renewal of interfacial fluid followed by a long relatively quiescent, but still turbulent, state. Conceptu- ally, the period between burst is controlled by large scale eddies with a mean turnover time TM. The turbulent fluctuations of the smaller entrained eddies near the interface have a mean characteristic time TH. Because is zero for It-fl > IN, the temporal integration in Eq.(5.23) is cut off by TM; the temporal behavior of the mean Green's function (§,§L£,t) is also controlled by these physical considerations. Therefore, lneq.(5.24) is bounded below by t-TM. Now during the interval t-TM 3 § 5 t and in a frame of reference moving with the local mean velocity, the Green's function G°(xjt|3,t) spreads out over a spatial domain with a linear length scale comparable to i/fiTM . 71 For large Schmidt numbers (i.e.,-9 -> O) the hydrodynamic scales charac- teristic of the spatial structure of turbulent fluctuations are orders of magnitude larger than the domain over which G°(° 2) is essentially non-zero. Thus, a spatial smoothing approximation simplifies Eq.(5.23) to the following result A <2’(1§,~t) 635,1] 223)) = -( 14’ch Ig’rx,€-1). 1 4).-31 1P 2 fi 13 ’ Q R 5? 8 A 0" t n. This shows that the temporal properties of the turbulence structure, not the spatial correlation scales, determine the mass transfer rate at high Schmidt numbers. Eq.(5.19) for and Eq.(5.25) for determine the response of the mean Green's function. For this research, only a first order approximation to this equation will be used to determine the statistical behavior of c’(§,t) governed by Eq.(5.16). Thus, in what follows, we neglect all but the first term in an expansion of about the ”unper- turbed'l Green's function, = G° + (higher order terms which depend on (5.26) turbulent mixing). A first order approximation of the turbulent correlation will be calculated according to Eq.(5.25) with replaced by 72 6)) I; . (G)(§,tl5.3)= (5.27) O , otherwise. Eq.(5.27) retains the physical idea that the major resistance to mass transfer near an interface occurs during the ”quiescent” period between burst. The intermittent mixing of interfacial fluid with the bulk fluid has been an important component of mass transfer models for three dec- ades (see Chapter A). CHAPTER 6 A THEORY FOR THE TURBULENT FLUX OF A PASSIVE ADDITIVE 6.1 Hierarchy of Relaxation Equations Two complementary approaches can be followed to obtain an equa- tion for the turbulent flux. The more traditional one is to simply multiply Eq.(5.13) by u;(xjt) and average the result. This gives 4m = (u,’u,’)-77I-, (6.1) which could easily be rearranged and interpreted as a ”transport“ equa- tion for the turbulent flux (see Launder, 1976). Unfortunately, an analysis of this equation quickly reveals that several unknown statis- tical correlations emerge (such as the triple correlation ). Additional equations for these unknowns can easily be derived from Eq.(5.13), but this set of hierarchy equations is unclosed. Although many closure hypotheses for transport-type equations have appeared in the literature'(see Chapter A), this approach introduces many ad_h9£_param- eters which are not only difficult to evaluate, but they often have lit- tle, if any, physical significance. Furthermore, the phenomenological models provide no insight regarding the potential effect of other phen- omena, such as chemical reaction or electrostatic drift, on the turbulent flux. Obviously, a good set of closed ”transport“ equations could pro- vide some understanding of the response of the mean field to various 73 7h boundary conditions, but for the questions posed in this research such an approach is too restrictive. Therefore, another strategy based on Eq. (5.16), rather than Eq.(5.13), will be examined. As previously discussed in Chapter 5, an important property of c’(§,t) is that its average is zero, / _ For Eq.(5.11) this occurs provided the average of the random source is zero; however, for Eq.(5.16), Eq.(6.2) implies a more subtle result, viz., t A . I x A , x a d“) J at L Jn[(6(§,+lg,t)u,(£'+'£'£)J—%] = 0. (6,3) Eq-(6-3) can be rewritten as f Lama #7:}? J13 = J: T(x.l£.)d<:':’>4£., (M) where the two kernels l.(xllx1) and 1r(xllx1) are defined as follows: t A ~ D A A A A LCx.1£.)e-LJ£LJ£] di‘, (G’(z,tlzs.t)U,’tx.t)) (6.5) ,. ‘f A ” A O A TCX.IX.)Z-j 4*} JXaI d£3<6>(l,+(21f) . (6.6) an” -O -0 If and G’ were known, Eq.(6.h) places an important dynamic restriction on the mean fields and . Furthermore, even for an approximate model of the Green's function (see, for example, Chapter 5), Eq.(6.h) and the mean field equation provide a closed model for turbulent mass transfer. A special case of Eq.(6.h) was recently applied to turbu- lent mass transfer near rigid interfaces by Wood and Petty [1981]. The 75 results showed good agreement with the earlier experimental study of Shaw and Hanratty [1977a]. Eq.(6.h) is new and intriguing. It relates the mean fields to the underlying dynamics through the two kernels fl;(x1l;1) and 1“(x1|;,). Figure 6.1 shows the behavior of 1[(x1|x1) for (' A) defined by Eq.(5.27) and Eq.(B.35), i.e., A '- a I. I” _ X-Kt _‘5‘+fi) h 1' 4HBI' T(x.IX.)=J 41 [¢ " 2, fl} (6.7) , J4m- . O + O O D I O The mean field parameter 6C sugnlfles the spatial scale over which ’ I + c I and change. Here 6c IS calculated usung the data of Shaw and Hanratty [1977a]. Obviously, the relaxation of the mean Green's function from its initial delta distribution, together with the finite turnover time of the large scale eddies, plays an important role in determining the spatial extent of the kernel'fl'(xllx1). Therefore, these underlying dynamic processes determine the relative importance of the non-local behavior of the turbulent flux in maintaining = 0. The spatial behavior ofL(x1|;1) is more difficult to assess because it depends on the relaxation of a turbulent space-time corre- lation. According to Eq.(5.22), the space-time correlation appearing in Eq.(6.5) is (cf. Eq.(5.23)) tA [As A . fl = - )3 4* Jfimcr <- M)- WWW). (6.8) In Appendix D, Eq.(6.8) is used to develop an expression for fl;(x1|x1) based on a mean Green's function defined by Eq.(5.27) and Eq.(B.35). The spatial smoothing hypothesis described in Section S.h is also used with 76 + TM = 2 ,OOO Sc=106 TJ=10 6c+ (Data of Shaw 5 Hanratty [1977a]) + TM=lOO A + 1 ’X1 0.5 1.0 1.5 2.0 Figure 6.1. The effect of finite memory on the kernel 1['(X1|;1) for physical absorption at high Schmidt numbers. 77 the result that A I3 LCXJXU=4UI >(‘1)' fl t A t k -L§_$_:Tl o a a o 0 d6] dé f. JXICT.('M)Q§IGH(£1A) (6.9) in“ f O c Eq.(6.9) shows that the relative importance of the non-local behavior of the mean concentration field in keeping = 0 is determined not only by TM and Sc, but also by TH and the intensity of the normal fluctuating velocity. A better understanding of Eq.(6.9) can be obtained by intro- ducing the following approximations: a o 2 R n A ~ 9 A A SEQ(3‘H'£"XH+) —R&°(Xuflfl,,t) (6.10) and, w A A g » j 42 Cr°(x.,+1:?,,t)zl . (6.11) 0 Obviously, Eqs.(6.10) and (6.11) do not apply in general, and should be restricted to: (1)high Schmidt numbers; (2)5nall TM; and, (3) x1>>0. With these reservations, we examine the behavior of flg(x1|;1) at 2: = 1 + . . . . and Sc = 10” for several values of TM In Figure 6.2 usung the approxumate result obtained by combining Eqs.(6.9)-(6.11), i.e., y, 1; ( 37-5,” _ Sc 07- Q3)", 4 T 1" a 5:. I'd". 4’ 4+ 4' '- C ) '2. 4T Lmix.’--——§“—-xi. (Xi-x. “5 1,,“ (6.12) a Figure 6.2 shows that for small T; the spatial domain of &a(X1|X1) is small compared with 6: (see Figure 2.1), and that an additional smooth- ing of Eq.(6.h) may be justified under certain conditions. Also, Figure 78 10 q— 5c+ (Data of Shaw 8 Hanratty [1977a])\ l Sc= 10“ af'R=10'5 + TM = 100 + Figure 6.2. The qualitative effect of finite memory on uglx1121) for physical absorption at high Schmidt numbers. 79 6.2 underscores once again the importance of finite memory (i.e., T; < co) and the underlying turbulent relaxation processes in determin- ing the contribution of the non-local behavior of the mean field on the dynamic constraint = 0. Eq.(6.h) is but one of many relaxation equations derivable from Eq.(5.16). Another, studied in this research, follows by multiplying Eq.(5.16) through by u;(x,t) and averaging. The result is I I a. ¢icx.+lf. X-t) +] (6.16) The spatial extent of these kernels, like [.(xllxl) and 1r(x1|;l), determines the relative importance of the non-local behavior of the mean field on the local turbulent flux. It is noteworthy that Eq.(6.13) would belong to the class of gradient-type models if the term containing L11(x1|;1) were unimportant. However, in general, the non-local behav- ior of the turbulent flux acts through L11(xll;1) to ”retard” itself locally (see Chapter 7). This negative feedback mechanism could provide significant new understanding of turbulent mass transfer. As briefly mentioned in Chapter 1, Builtjes [1977] has noted similar effects for momentum transfer in developing flows. The analysis of a hierarchy of 80 relaxation equations, like Eqs.(6.h) and (6.13), has not been proposed heretofore. In what follows, we refer to results based on Eq.(6.h) as Type I Relaxation and results based on Eq.(6.13) as Type II Relaxation. 6.2 Spatial Smoothing of the Relaxation Equations The first two equations in the hierarchy of relaxation equations and the mean field equation can be used to estimate the mean mass trans- fer rate. Presumably, the exact statistical behavior of the underlying stochasticfieldsis consistent with both non-local constraints, but for approximate relaxation kernels, Eq.(6.A) and Eq.(6.13) may not be indi- vidually satisfied along with the mean field equation. Thus, a multi- plicity of theories can emerge from the formalism which begins with Eq.(5.15). The asymptotic theory examined in this research stems from the observation (see Figures 6.1 and 6.2) that at very high Schmidt numbers and for small TR’ the spatial bandwidth of the relaxation kernels is small compared with the mean field scale 6:. This means that the gradi- ents of the mean fields appear ”smooth” to the relaxation kernels, so Eqs.(6.h) and (6.13) can be reduced to the following approximate equa- tions; d J<“"¢’> thd—d—xl' =- Thu) 4”, (6.17) (Type 1 Relaxation) and d J“) I l (alt )+,€,,(x,) «Ix, ="Du("l)7;‘ . (6.18) (Type II Relaxation) These equations still provide a link between the mean fields and the 81 non-local behavior of the underlying dynamics inasmuch as the relaxa- tion coefficients are simply integrals over the previously defined relaxation kernels, i.e., L00) E],~L(X11£.)J£ (6.19) T (”1’ a I.” T (min d2?) (6-20) (”cm 3]: LucxllvaJdIa (6.21) Du 6x.) 5].” 0,, (26123) J23. (6.22) For physical absorption, a sufficient condition for the I'exis- tence'I of these model equations is J T2175... << 5:. (6.23) Shaw and Hanratty [1977a] and Henstock and Hanratty [1979] have reported experimental data for which 5 "3 + '0 c , (6.211) O «+06 —h5 [30m 5;, ’ (6.25) where m IS the quuId film thickness made d1mensnonless With the wall parameters. Thus, lneq.(6.23) requires the characteristic turnover time .4. TN to be bounded above for Sc + m, i.e., +.4‘ 4, (00 Sc , rigid interface TM << I} 4. lo? X l0 at , free interface. Caution should be exercised in interpreting the above inequality. + O 0 Note that as Sc + w, 5c + 0; therefore, as the Schmidt number Increases, the resistance to mass transfer occurs across smaller and smaller layers of fluid which are not renewed as frequently as fluid in the outer 82 region of the viscous sublayer (see Chapter A). Thus, if T; + m as x: + O, the above inequality may not hold for rigid interfaces even for very large values of Sc. The validity of lneq.(6.23) and its analog for physicochemical absorption (Chapter 8) or electrostatic deposition (Chapter 9) is difficult to assess §_priori; therefore, as a working hypothesis the two asymptotic relaxation models, Eqs.(6.17) and (6.18), are assumed to hold. 6.3 A Self-Consistent Approach for Estimating the Renewal Rate for Interfacial Fluid Eqs.(6.17) and (6.18) do not necessarily yield the same mean field behavior if the relaxation coefficients are estimated using the approximate model developed in Chapter 5. However, it would be inter- esting, but beyond the scope of this research, to determine a sequence of approximate Green's functions which made the first two relaxation equations equivalent. On the other hand, a slightly different and less ambitious question regarding the equivalence of Eqs.(6.17) and (6.18) can be posed which could provide significant new insight regarding the mean renewal frequency for interfacial fluid. Eqs.(2.6) and (6.17) or Eqs.(2.6) and (6.18) can be used to cal- culate the mean fields and 6:(=1/St+Sc). For physical absorption across a rigid interface, four parameters determine the behavior of 6:. These include the Schmidt number Sc, the turbulence intensity coeffi- cient aTR, the mean turnover time I; for I'large“ scale eddies, and a . . + . mean relaxatlon time TH for “small“ scale eddles. Thus, 6: = 5:(Sc,a:R,T;,T;). (6.26) 83 The parametric behavior of 6: predicted by the two reiaxation models is different. This would occur even for an exact model because the physi- cal constraints represented by Eqs.(6.17) and (6.18) are not the same. However, for an ”exact” solution the mean mass transfer rate or, equiva- lently, 6: predicted by both models would agree provided the I'correct" hydrodynamic parameters were available. This qualitative discussion motivates the following Self-Consistent Hypothesis: Eqs.(6.17) and (6.18), together with the appropriate mean field equation, give the same mean mass transfer rate which also agrees with experimental data for some set of hydrodynamic parame- + + + ters (alR,TM,TH). Application of this hypothesis implies equality of the mean field scales predicted by the two models and agreement with experimental data. In . + + . . other words, If 6Cl and 6C2 represent, respectlvely, the mean f1eld scales predicted by Eqs.(6.17) and (6.18), then 6C1 = 6C2 = 10 56°“ . (6.27) For fixed Sc and aTR, Eq.(6.27) gives two relationships for I; and 1;. According to this scheme, the ”hydrodynamic“ parameters depend on the Schmidt number. At first thought, Eq.(6.27) appears ludicrous and nothing more than a fitting algorithm. On the other hand, this hypothesis could be physically meaningful inasmuch as the hydrodynamic parameters T; and I; should be interpreted as properties of the turbulent fluid which offers resistance to mass transfer. Therefore, for low Schmidt number prob- lems, this could be the entire viscous sublayer, but for large Schmidt 8h numbers, only a small portion of the viscous sublayer very near the mass . . . . + + transfer surface IS Important. Intultlvely, we expect TM and TH to . + . . . . Increase as x1 + 0 because of no slip at rIgId Interfaces. In the the- + + . . . ory developed here, TM and TH should be interpreted as characteristics of the temporal behavior of the turbulent fluid which roughly occupies + 1 f 6:). Therefore, the application of the Self- the region (0 f x Consistent Hypothesis does not contradict the physical meaning of these hydrodynamic parameters; on the contrary, it provides a novel means to estimate the renewal rate of interfacial fluid. Obviously, the parame- ters estimated in this way are also subject to the limitations inherent in the approximate relaxation equations. In Chapter 7, this strategy is applied to physical absorption near rigid interfaces, with the inter- esting result that I; and I; increase as 6: + 0. It is noteworthy that solutions to Eq.(6.27) do indeed exist. 6.h Type II Relaxation Model for Physical Absorption The relaxation coefficient 011(x1) defined by Eq.(6.22) contains two contributions: 0 1 Ducx.) = D..(x.)+D.. cm (6.28) where t D O O A n ‘ 0,16,)2 jaij 4;.) 6?.) dx; («TX 5,212,) )(u.’<:.+)u.’cg,$)>] , (6.29) '4” 0 ~99 -09 and a a a’ A I IA A J dfij ”IJ 4;: , which could be estimated by using the generalized Sternberg approximation, Eq.(5.22), and an approximate model for the mean Green's function such as Eq.(5.27). In this way 0:1 could be related to 6° and a triple velocity correlation . However, in what follows the contribution of Dil to D11 is neglected as a first approximation. Hopefully, the assumption that DH 2' D" (6.31) does not upset the relaxation processes important to the turbulent flux. Limiting expressions for 0:1(x1) and 211(x1) near the interface are derived in Appendix E. The results are 4' Du 0“) , / f 0,. (ma "—7 = "fil’"]“1:(‘fu*$6)uxi , (6.32) ' I + I s 1,, (X1): "‘ 113"" = !,(P) a”. 1:56 xi, (6.33) Here p = TM/TH and P - fluvial], ivéUss 1; 7) J’7 where ©(1.5;2;n) is the degenerate hypergeometric function (see p. 1058 in Grodshteyn and Ryzhik, 1965). Figure 6.3 shows the behavior of ffl (p) and erf[/-p-] as well as some other functions of p which arise later. Eqs.(6.32) and (6.33) represent upper bounds to more rigorous expressions which should be used for Sc + w. A criterion for the appli- cability of these equations can be developed from Eq.(E8) and (E10), rewritten as 86 g(p) 5).... w- bjb U1— ox—L \j— m—ii— m_.— 4.. l Figure 6.3. The effect of p on erf/p and the relaxation param- eters fI(p). fn (p). and g(p). 87 T , " (+4 + Dii ‘ am "i 'L'c (xx) (E8) + + 111 = am Xi” F+""+) . (£10) where E 31R x1“ (see Chapter 3). Note that T:(O) = O and O) = O. Eqs.(6.32) and (6.33) follow by retaining only the first non-zero term in a series expansion of T: and F+ about x + = 0. Both 1 + + + . . . Tc and F are bounded for x1 + w, so the approximation used here is + + c’ then St only valid over some region 6; near the wall. If 6; << 6 calculated using Eqs.(6.32) and (6.33) would be too large. From the definition of T: it follows that / , - W." ”625’”; i (75%. i )"1 < ”W |~eXP[-xf'( 31%)“) . <63“) 1 — /E, or 66 = fibTH (see Figure 6.h), This result suggests that 65 = (TE/Sc) which obviously approaches zero as Sc + m. Although retaining only the first term of Eq.(6.3h) in an expansion about x+ = 0 could seriously 1 overestimate T:(x:), this simplification could still provide an accurate estimate of some integral property of the mean field, such as St+ (cf. Section 19.3 in Bird et al., 1960). It is interesting to note that according to Eq.(6.3h) the turbulent diffusivity D11 has the same form as that proposed by Van Driest [1956]. A key feature of Eq.(6.18) is that sudden changes in the mean concentration gradient will not translate into proportional changes in the local flux. Extra time is needed for the turbulent flux to readjust to the new mean concentration field. This translates into a non-zero 88 TH[1-exp(-x1/%5T;)] Figure 6.4. Linearization of the characteristic time for turbulent diffusion. 89 memory length for statistically stationary problems. In Eq.(6.18), both 011(X1) and 211(x1) are positive for x1>:O. Therefore, if the local gradient of steepens (i.e., becomes more negative), turbulent dif- I fusion tends to make more positive, which will cause the local gradient of to increase. According to Eq.(6.18), this will in turn cause the turbulent flux to decrease. This feedback mechanism, which opposes turbulent ”diffusion,” will be referred to as turbulent ”retardation.” The analogous effect for momentum transfer in a develop- ing boundary layer was considered by Builtjes [1977] and was termed “extra” memory. This phenomenon partially motivates the introduction of a finite propagation velocity u _ Du (X!) P _ I" (M) 1 (635) which characterizes the spatial propagation of disturbances in . If the ”memory” length 211330, then up-ran This implies that for a classical eddy diffusivity model, the time scale needed for the turbu- lent flux to adjust to spatial changes is zero. For a Type II relaxation model, a finite relaxation time exists and can be defined as (X ) 10:.) a £21 ' (6.36) LL, . A time scale characteristic of the mean field behavior can also be defined as 56 <£.> 6c :5 (6.37) A ratio of these two characteristic times defines a Deborah number for turbulent mass transfer and provides a measure of the relative impor- tance of the retarding effect in Eq.(6.18). Eqs.(6.35), (6.36), (6.37), 90 and 6C =$/ can be combined to give A _. ( (ic>)z On 7:- “p 6% ' (6'38) Eqs.(6.32) and (6.33) imply that the propagation velocity up has the following form for p = l + = 3!. = [.87 P u‘ I"??? , (6°39) which suggests that for large Schmidt numbers the retarding effect char- acteristic of Type II relaxation becomes negligible. Also, for p = 1, the local Deborah number evaluated at x1+ = 6C+ is .f + *3/1 .04 Lee). NW T~ 5‘ (6 i6) 9. (31* s." )3 ’ ° which reduces to the following expression if St+ is replaced by the empirical correlation Eq.(6.8) ‘1 .+ +3fi2 —.q 5 .2: 342 a”; T" Sc. (6.111) 6 o 6.5 Type II Relaxation Model for Physicochemical Absorption Across a Free Interface The derivations for D11(x1) and £11(x1) given in Appendix E for physical absorption can easily be extended to include the effect of chemical reaction. The details of the analysis are given in Appendix F, with the result that the relaxation coefficiencs for physicochemical absorption are On 3 D: (X8): 37: 7% er;(f( fi*+ {THE} (6.112) U 4' 4. (I! _ X+3T '* f -(’+Tfli)” 111(i1)=W-'—‘2_L£ e. @053- z;7)J7, (6.113) 91 For free interfaces and for physicochemical absorption, the superscript + signifies that a variable has been made dimensionless with» and 31F- For a rigid interface and for physical absorption, + denotes a dimen- sionless variable using the viscous scales V and u*. As previously discussed in Section 6.A for the no reaction case, Eqs.(6.h2) and (6.A3) were obtained by expanding about x1 = O Eqs.(F3) and (F5), rewritten as 0,, (xi) = an: X.“ 1. (xi) (6.1111) 1,, (X1) a]; 7‘le (XI) . (6.85) Once again, from the definition of Tc it follows that P —(fi+‘b.)t x 1:, =1» I, 6 "14773637631 “'1' .. - HTuiV-D‘t'vyzflj X $1," [ 1 ”(LL16)- Tut ) (6.116) which reduces to Eq.(6.3h) for k = O. This result shows that the char- acteristic scale controlling the turburlent diffusivity depends on the intrinsic rate of reaction and the molecular diffusivity. Thus, _, Tu (Se—I HT”. . (6.117) With 6C =(D/, it follows that 1% IA: if: ‘ <£:’( ac 12.7.0) . (“'8' In Chapter 8, we show that the Type II relaxation model predicts 1 ca co» mo_mum u_um_couumcmzu ogu co co_uumoc m_n_mco>occ_ cmocoiumc_w m mo uuowwo och .m.m oc:m_u +6. Am . m i _ 8o Am couamcu :_ m_m>_mcm co vommmv mE_mom . co_uumom 11w 0. ummu um I '1' I'll: + + oE_mmm 11 co_uummm 30.m 11w.— 93 that 66 is comparable to 6C for T; = T; = 10. This suggests that retaining only the lead term in Eqs.(6.41) and (6.A2) is valid (cf. Figure 6.h). Note, however, that the asymptotic relaxation equation (6.18) may require T; << 10, according to lneq.(6.23); therefore, the same comment which follows Eq.(6.3h) also applies here. The effect of a chemical reaction on the propagation velocity up is shown in Figure 6.6. Although £11 and D11 both decrease as the reac- tion parameter k increases, it is noteworthy that 211 is reduced more than 011, with the result that the ratio increases with k, as illustra- ted by Figure 6.6. This important result will be discussed in more detail in Chapter 8. 6.6 Type I Relaxation Model for Electrostatic Deposition Eq.(6.17) can be rewritten as Jai’c’) d<¢> J'X, zfiu" 47‘1’ (6.19) where the turbulent “convective'I velocity L (X1) 7’ (X5) For uD+ = 0, Petty and Wood [1981] showed that, + +4 + d x! , rigid interface (6.50) u..= + +3 _ 0‘ x! ,freeinter‘ace. (6.51) Note that the only difference in uC for rigid and free interfaces is , . . + . . the lead term of . The dimen510nless parameter a 15 defined as 9A coo.— .>u_uo_m> co_ummma0ca ou~:_w ocu co co_uumoc mo uoomwo 65h oo— — 6.5 .o.o oc:m_u An oE_mom co_uumom ummu co_uommx 02 _.o Illlllli o oE_mmm co_uumom 30_m 0a: 1.0— imp .iON 1mm iom mm 95 + {—4- I {1(f)am TH Sc, , rigid interface (6.52) i I ( J1(f)JTu’5c. , free interface (6.53) where, as before, p = TM/TH. Here the superscript + notation has sev- 9L II eral meanings, depending on the specific physical situation. For a free interface and for electrostatic deposition, + denotes a quantity made dimensionless using the parameters v and 31F (cf. the scaling for physi- cochemical absorption across a free interface); for a rigid interface and for electrostatic deposition, the usual viscous scales V and u* are used to make variables dimensionless. The function fI(p), which is also plotted in Figure 6.3, depends only on p and was derived by Wood and Petty [1981] with the result that I, 15 '2‘! tan-l EL) :4 ftp) W . (6.51) It is noteworthy that the Type I relaxation model defined by the fore- going expressions, like the Type II model, also gives “191/2 for free interfaces; moreover, the model yields the important result that ¢,D'7 for rigid interfaces which has been observed experimentally (see Chapter A). The convective velocity uC with weak electrostatic augmentation, ”D + 0, is derived in Appendix G with the result that + + “c 3 ¢ “CO ’ (6°55) where the magnification factor 6 is defined by 96 .. ”NJ/:1: ¢= T’i-i-M" . (6-56) The dimensionless group M+ can be regarded as the ratio of the flux due to electrostatic drift to the flux due to molecular (or Brownian) dif- fusion. M+ is defined by 4 F «if-‘— M ‘27—'“0 Tu+$c , (6.57) The dimensionless function g(p) appearing in the definition of 6 is “1 given by P 51' __ er}(/—) a [P], e. J; (I) ' (6.58) which is also plotted in Figure 6.3. For electrostatic augmentation of mass transfer, the convective velocity u: depends on “D as well as the molecular diffusivity. Figure 6.7 shows the behavior of ¢(p,M+) for several values of p. One of the important features displayed in Figure 6.7 is that ”D (i.e., M+) causes u: to decrease for p = 1. Thus, electrostatic drift does not necessar- ily augment mass transfer. The efficiency of particle deposition may in fact decline, rather than increase, with the added effect of electro- statics. This important phenomenon will be investigated in Chapter 9. Because Eqs.(6.52) and (6.53) were obtained by expanding the two relaxation coefficients L(x1) and T(x1) about “0 = 0, the quantitative behavior of 6 for large values of M+, which is proportional to uD, may not be represented accurately. Finally, Type II relaxation for electro- static deposition was not examined in this research. Table 6.1 summarizes the type of model studied for each of the 97 ¢ 1.3 ' 1 2'-- p==O 1 1.1 d- 1.0 p=1 Turbulent External Field 0'9 '_ Regime Regime 0.8 ..- O 7 I I L I T 1 3’,” 10'3 10'2 10'1 1 10 102 Figure 6.7. The effect of electrostatic convection on the magnification factor. 8 9 mmo_c0_mcoe_v mco_umnoo .6605 on“ ome On com: mcouoewcma + co_m: _ cm_czocm ”mumwcwwmw up.me.mhm A :.>v U_m_m :o_u.moaoo mcu um mmcmzom_c Amm.ov .Amq.mv .Ao~.~v o “a _ oa>h o_umuwopuum_m ummm ”Ao_+ +3v um_cv :.om A m.>v omen . u_umum0cuoo_o xmmz + . a . I .Z .1...“ co_uummc Am: mv AN: my P e m maoucomoeo; .033. + + + 16.8 09.... __ out. coiacoflz . . . . . . .mu_Eo:uou_m>sm ico>occ_ covcoiumc_m Am— my Ao— NV +x um . mummcouc_ Amm.mv .Amm.wv .Aw—.mv ”H.MP.mHm __ oa>h co_uac0mn< 65 um c0363.. +733 Em; ”3.8:: o_n_mco>mcc_ .ummm Aom.mv .Am:.mv .Ao.~v om e . oa>h . m cmEom mco_umacm u_mmm mcouoemcmm oommcouc_ .060: co.um:u_m x . . . . co_umxm_om mzo_hm=-£ Ll. CXF (‘SCL uzdx,+)dx, 65+ (7.1) _ T". 1:: 9 A (C) - Jo exp C-ScJ 3c+dx,+)dx,+ 6: (7.2) + .. '° '3” ’k 2 3c. ‘Jo exp (“Sc L u: 419*) d£.+ . (7.3) By setting x3 A+ A z: ScJ u. dx, , (7-1') a Eq.(7.1) can easily be integrated to give I I 4' x'+A‘P A + - = [("U'I’ (“SC— 14., 42(1)] 5:. 56.. (7.5) 9 According to Eq.(2.12), - should approach St+ far from the mass transfer surface. Because 6: = 1/St+Sc, Eq.(7.5) clearly has this property. With u: defined by Eqs.(6.50) and (6.51) for rigid and free interfaces, respectively, the above expressions simplify to the follow- ing set of results: 100 lOl + + '4'” _ (are) 6: = I- 6.p(— 5"", if," ) (2.6) ‘l' 4' xi... + A... [+11 5+ .. S (1 (X1 ) (C) Sc. - J; EXP (- c '4. n )JX' (7-7) + i ,, -i/(i-ni) 5.. = F(,,,,,) [(1%) 663.] (7.8) Eq.(7.8) shows how the mean field scale (or, equivalently, the mean mass transfer coefficient) depends on the physical parameters. Because a+ a VSE-for both free and rigid interfaces (see Eqs.(6.52) and (6.53)), we conclude from Eq.(7.8) that _,3 5+ 5c. , rigid interface, n=li cc c (7.9) 5:5 , free interface, n=2. These results have been observed experimentally by Shaw and Hanratty [1977a] for rigid interfaces and by many others for free interfaces (see Chapters 1 and A). Eq.(7.8) also shows that the mass transfer rate has a relatively weak dependence on the hydrodynamic parameters contained in a+. For instance, VI 0 4 , (Th ) rigid interface {2 < ‘ > (7.10) 4 '66 (TH ) ' free interface. The result for rigid interfaces was also observed by Campbell and Hanratty [1981b]in their recent theoretical study and earlier by Petty [1975]. The variable C, defined by Eq.(7.h), can be used to make all the concentration and flux profiles similar. It is obvious from Eq. (7.6) that 6: depends only on the type of interface and 102 + n+| ;: (‘_5i_c_+i’_t_)(x"’) . (7.11) After a few manipulations of Eq.(7.7), it is also clear that C and n determine the behavior of +. The scaling factor in Eq.(7.ll) is pro- portional to 6:— given by (7.8). Based on a numerical simulation of the balance equation, Campbell and Hanratty [1981b] concluded that the above . . . . . + scaling for rigid Interfaces was only valid for x 1 very near the mass transfer interface. The Type | relaxation model for physical absorption is extended in Chapter 9 to include the effect of electrostatic drift. The results presented here are used in Chapter 9 to calculate particle deposition rates in the absence of an electrical field. A computer program with a sample calculation is listed in Appendix J. 7.2 Type II Relaxation Model Eq.(6.18) can be combined with Eq.(2.10), with the result that <“.’¢'>+= —-S‘£+H(§, B) (7.12) ”‘3'8’5[furl-<24)-g<§'*-g4)pg g :: $'fl:P95] " f. m (Sc/Ti)” xi" (7&3) - ‘+ q _ Ba 4[f. , so the feedback mechanism exists implicitly through the effective eddy diffusivity. 6; can be estimated by combining Eqs.(2.10) and (7.15) and using the boundary conditions =0 at x1=0 and =cb for x1+°°. The result is + _ J” g nix,+ (Sc-a (l+%)- (7.17) This integral and the one defined by Eq.(7.l3) are evaluated numeri- cally. A program and sample output are listed in Appendix H for esti- mating the mean mass transfer rate according to the Type II relaxation model. The asymptotic behavior of H(£;B) near the rigid interface and outside the viscous sublayer is given by eg’M . ‘ in ’B = M . i'Z.H“’B) ‘ lam; ) Her/4. (7 ‘8’ If £11(x1)530, the resulting dimensionless flux H(E;B) equals the asymptotic result given by Eq.(7.l8) for 055,500. Note, however, that at the two extremes with and without retardation are not the same 104 + . . because the St In Eq.(7.lZ) would be different for the two cases. Differentiating Eq.(7.lZ) with respect to x1+ yields a’+ + erfCP‘“) . u: 4H fir:- {’(F) (SC/TH) 71—. (7.19) For g + O and w, the asymptotic expression, Eq.(7.lS), shows that d‘U,’c’> 58 £4 81?;— (7 20) ’7}?— 4/dx1 increases with x1 as x1”; for large values -6 1 O 5 of XI, d/dx1 approaches zero as x 1 , the tur- Because £11 a x bulent relaxation term £11d/dxl decays to zero for large x1, and = -Dlld/dx1. Thus, because = constant for large x1, 5 d/dx1 a x; as x1->m. Table 7.1 summarizes the limiting behaviors of the two mean gradients for Type | and Type II relaxation models. 7.3 An Estimate of T; and I; Using the Relaxation Models A self-consistency calculation is made no obtahw a set of hydro- dynamic parameters relevant for turbulent mass transfer at high Schmidt numbers. Note that for high Schmidt numbers, the mass transfer resistance is confined to a thin region close to the wall. The hydro- dynamicparametersshould be characteristic of the temporal behavior of the flow field in this region, and can be obtained from the self- consistent calculations according to the hypothesis made in Section 6.3. To begin the calculation, first fix Sc and let atR equal 10's. St+ is then calculated over a range of T: for a fixed value of p(ETM/TH) for each of the relaxation models. Fortunately, these two calculations cross to give a St+ which is consistent with both Type I and Type II . + models at a particular value of TH. 105 TABLE 7.1 ASYMPTOTIC BEHAVIOR OF THE GRADIENT OF TURBULENT FLUX AND THE CONCENTRATION GRADIENT PREDICTED BY THE RELAXATION MODELS I) + + d/dx1 d/dx1 Relaxation Model + + + 1 + 0 x1 w 1 + 0 x1 + -Sca x + -Sca x Type | +k +u 5 1 ScSt 5 x e e 1 +u - + + - Type II (x1 ) 6 ScSt (X1 ) 5 106 As will be discussed in Section 7.5, the extra memory represented by the turbulent retardation in the Type II model suppresses the mean mass transfer rate for large TH; however, the mean mass transfer rate predicted by the Type 1 model increases with TH, as noted in Section 7.1. This opposite trend in St... predicted by the two models make the crossing possible. A typical example of the calculation is shown in Figure 7.1. If the St+ thus calculated does not agree with the St+ reported by Shaw and Hanratty [1977a], p is changed until St+ agrees with the experimental data as well (see Eq.(6.27)). The set of param- eters obtained by this procedure is shown in Figure 7.2. Figure 7.2 shows that as Sc increases, both T; and T; increase. This behavior, which was anticipated in Section 6.3, is expected because 6: decreases as Sc increases. 7.“ An Analysis of the Mean Fields for Type | and Type II Relaxation Models Figure 7.3 shows the turbulent flux predicted by Type I and Type II models using the parameters obtained from the self-consistency calcu- lations. Because the two equations preserve different dynamic features which determine the turbulent flux, they predict different profiles. The Type I model suggests that the correlation between the fluctuating concentration and velocity fields are weak for small x1, but increases faster than that predicted by the Type II model as x1 gets large. Thus, according to the Type I model the transport mechanism for relatively large x1(x12:2) is predominantly turbulent mixing. The fact that the flux calculated by the Type I model approaches its asymptotic limit faster than that predicted by the Type II model is consistent with the St 16«- Ih._ 12- 10d 107 —x1o-‘* 5c=1o3 I— alR+=1O-5 Type I Model p=1>< P p= IO \ Data of Shaw 8 Hanratty [1977a] Type II Model I 1 1 10 1102 1b3 1?“ Figure 7.1. Self-consistency calculation for physical absorption. 108 150 30— -— lhO __ I30 —— 120 -‘ IOO TH 102.. \\ IO 1 l J O 045 1.0 1.5 2A) .... O 41 5C 70 103 10“ 105 Sc Figure 7.2. The effect of Sc on p and TH for physical absorption obtained from the self-consistency calculations. lO9 I ’ + 8 r-xlO'L' 7 -~ St+=7.7x1o-‘* (Data of Shaw 8 Hanratty [1977a]) 6 ....— 5 __ +._ -5 l. -.. 31R —10 TH =70 Type II Model p=10 3 —— 2 __ Type I Model ==x 1 _- J 0 l l i 1 : ,4)... O S I O l 5 2.0 2 5 3 O 4. Figure 7.3. Turbulent flux profiles for physical absorption using Type I and Type II relaxation models with Sc==900. llO asymptotic behavior discussed in Table 7.1. Because both calculations are made with the parameters obtained from the self-consistent proce- dure, they yield the same asymptotic turbulent flux, which is equal to the St+. Figure 7.4 shows the concentration profiles predicted by the two relaxation equations using the set of parameters obtained from the self- consistency calculations. The self-consistency hypothesis makes the slope of at x1+ = 0 the same for the two models; however, the Type I model predicts a steeper concentration profile in the intermediate region. Note that in the near wall region, the sum of the turbulent flux and the flux due to molecular diffusion is constant for all X}. Recall that the turbulent flux predicted by the Type I model is smaller for the intermediate region, but is larger for large xl than the flux predicted by the Type II model, as seen in Figure 7.3. The flux due to molecular diffusion calculated by the Type I model must be larger than that calculated by the Type II model for intermediate x1, in order for the total flux to remain constant. The concentration profile predicted by the Type I model is thus steeper and approaches unity faster for large x1 than the profile predicted by the Type II model. This observa- tion is also consistent with the asymptotic behavior discussed in Table 7.1. Also shown in Figure 7.h are the experimental data of Lin et al. [l953b] and the numerical calculation of Campbell and Hanratty [l981b]. Both models predict concentration profiles which are higher than the data of Lin et al. because the slopes at x1+==0 were fitted to the more recent mass transfer data of Shaw and Hanratty [1977a], rather than to the Lin et al. data. It is noteworthy that the calculation of Campbell and Hanratty shown in Figure 7.h follows closely the predictions of the lll /Cb l.O-~ / ,< 0.9-#- Type I Model /’/<::' \\§/’/’ / /\\// o 8—— / <\/ /\ /( \\ Data of Lin et al. 0.7—— \ [1953b] for Sc=900 \ and Re=11,OOO- 12,1400 0 6 4:\ Type II Model /\ V ————— Model 0-5-- k calculation of \_ Campbell 8 Hanratty [1981b] 0.11-- \ \/ Sc=9OO \ +_ ..S 0.3-1.— 81R -10 I TH+=7O p==lO 0,2__ 0 1 __ St+Sc=O.69 ' - (Data of Shaw 8 Hanratty [l977a]) I J. 11 i Z»X1 I 3 A Figure 7.4. Concentration profile for physical absorption using Type I and Type II relaxation models with Sc= 900. 112 Type 11 model. As was discussed in Chapter 6, the feedback mechanism in the Type II relaxation model caused by the retardation effect tends to decrease the turbulent flux, as shown in Figure 7.5. Near the wall the turbulent flux with memory increases at a slower rate than the flux without memory, with the result that the retarded flux is smaller every- where. It is noteworthy that for the calculation shown in Figure 7.5, I 1c’> occurs at the edge of the concentration bound- a 59% decrease in (x1) can be approximated by the first term in a Taylor series expansion about x +==O is valid. Eqs.(6.32) and (6.33) were both 1 obtained by assuming that 1 + +4 0,, x, (7.21) I (UK;) 3 u within the viscous sublayer (see p. 279 in Monin and Yaglom, 1971, and Chapter 3). The calculations with uc+==0° were made by neglecting the memory length £11(x1) in Eq.(6.18). If 211(x1) is non-zero, then according to Eq.(6.39), for TH+=7O and Sc=900, u +=0.0015. It is noteworthy that p for TH+=70, up+ changes from 0.0020 for Sc=500 to 0.00045 for Sc= 10,000, which is approximately the same variation as that found in the .1. C O . 1. 4. normal intensnty Vku12>/u% evaluated at x1 =15 The extent of retardation of the turbulent flux depends on the distance from the wall. 'R) see how this retarding effect changes with x the turbulent flux is decomposed into a diffusive flux 1’ 113 I I + ' 12—l-x10’“ + + up =00- St =1.22x103 10"” ==59% decrease in + up++0.0015 8" St+=7.7x10"’ 6..- Sc=900 81R+ 10-5 11 __ TH+=70 5 + p==10 C I 2 -.. t‘.“““Data of Shaw 8 Hanratty | [1977a] I I l I O i 5% i 1 Ii...’+- 0.5 1 5 2 0 2.5 3 0 "1 Figure 7.5. The effect of turbulent retardation on the turbu- lent flux. 114 _ aI D = - DH dx’ (7-22) and a retardation flux d500. This is in sharp con- trast to the behavior of St+ for A/GCEEO. Fron Eq.(7J3),IL+0 as TH-+«5 and hence St+-*0. Although both the turbulent diffusivity and the memory length increase with TH, £11 increases faster than 011 (see Eqs.(6.32) and (6.33)), with the result that the retardation effect overrides turbulent diffusion for large TH. As indicated in Figure 7.8, 118 51:14 Sc=1OL' a1R+= 10-5 p: 3--x10'“ A/9C==0 (Petty [1975]) Type II Model 2-0. 0.78 A/0C=0.027 89 3 Data of Shaw 8 Hanratty 1-- [19778] 0 n1 4| i i i, 10 102 103 10" 105 TH Figure 7.8. The effect of TH+ on the Stanton number for physical absorption. 119 this destroys the turbulent flux. However, if 211130, the turbulent retardation is absent, so the mechanism for retarding the turbulent flux is absent. Under these circumstances, the more the eddy can remember its part (i.e., the larger TH), the larger the flux. Thus, with £11530, St+ increases continuously with TH. The reduction of the flux by TH can also be explained by the fact that u -*0 as TH-tan That is, the turbulent field is totally iso- p lated from its surroundings so the turbulent flux can not sustain itself, with the result ==0. Therefore, with TH==TM-*“5 the mean field equation reduces to the ”film” model and the semi-infinite domain must be replaced by a finite film thickness. Figure 7.9 shows the effect of the ratio TM/TH on St+. As in Figure 7.8, the curve attains a maximum before dropping off at large p. Physically, when p increases, more and more information developed in the past contributes to the relaxation coefficients 011(x1) and £11(x1) at the present time t; when p-Fau the entire history determines the value of the parameters. This results in an increase in both D11 and £11 with p. However, as p increases, the memory length 111 increases at a rate faster than the turbulent diffusivity D11, with the result that the tur- bulent flux decreases with p. If p-+w, then fill->w also and the Type II relaxation model degenerates to the film model previously described. This underscores the physical significance of finite memory for the type of memory function introduced in Chapter 5. However, this result does not exclude the possibility that a class of fading memory models with TM-*«>could be developed, as opposed to the simple one used here (see Hinze, I975)- 120 .co_uac0mnm _mo_m>ca co» Logan: coucmum oz“ :0 oE_u >cosmE ou_:_m mo uommmm och .m.n mc:m_u a mop - Ill-T1 Hmmxmpi >uumccm: w ZmLm mo mama o— _0noz __ mask 8. "+3 80.8 "um I as mno—.1+ m ll+~.— lm.p 1m.P .Io.N IN.N :Io—qu:.N a... 121 7.6 A Comparison Between Type I and Type II Relaxation Models for Physical Absorption The two relaxation equations (6.17) and (6.18) predict different asymptotic behaviors for the mean gradients and different parametric behavior for the mean field scale 6c+‘ For the set of parameters for which the two models yield the same mass transfer rate, Eq.(6.17) pre- dicts a steeper concentration profile and a turbulent flux profile for intermediate values of x1 (see Figures 7.3 and 7.h). However, for the set of hydrodynamic parameters deduced from the literature, Eq.(6.17) predicts a lower Stanton number than Eq.(6.18), and a less steep initial slope in the concentration profile (see Figure 7.7). Thus, the two equations are fundamentally different. Each equation has preserved dif- ferent features of the fluctuating concentration field in determining the turbulent flux. According to the Type II relaxation model, turbulent mixing of a passive additive is reduced by a turbulent retardation effect related to a finite propagation velocity for the turbulent flux . This phenomenon occurs because of the fundamental fact that cannot respond instantaneously to sudden temporal or spatial changes in the mean concentration gradient. As illustrated in Figure 7.5, the effect is significant even fiarthe fully developed flow considered in this research. Previously, Builtjes [1977] studied this phenomenon theoreti- cally and experimentally for momentum transfer in a developing wake and boundary layer. The diffusive and retarding effects of the underlying turbulent mixing process are approximated in the theory presented here by a non- gradient relaxation model for the turbulent flux (see Eq.(6.18)). A 122 time scale (cf. Eq.(6.36)) associated with the retardation effect can be defined in terms of a turbulent diffusivityngl(x1) and a turbulent ’ memory length £11(x1) which appear in the model for . A compari- son of this time scale with one related to the mean concentration field could potentially provide a useful measure of the relative importance of the turbulent retardation. Such a comparison is not uncommon for analo- gous phenomenon characteristic of momentum transfer in viscoelastic fluids (Middleman, 1977) or molecular diffusion in polymers near their glass transition temperature (Vrentas and Duda, 1979). Thus, a charac- teristic Deborah number for turbulent mixing of a passive additive (see Eq.(6.38)) was introduced as a possible qualitative measure of the importance of turbulent retardation. However, as noted earlier, the finite propagation velocity uc+(EDII/211u*) may be a more significant . . '1' measure, Inasmuch as numerIcal values of u are related to the turbu- c lent intensity yE/u* evaluated at 8C+ (see the discussion following Eq.(7.21)). At high Schmidt numbers the temporal structure of the turbulence determines the magnitude of turbulent diffusion and retardation. Of course the spatial variation of the turbulent intensity b&/u* also influences the turbulent flux through the parameter a:%, but the spatial scales associated with the space-time correlation of velocity fluctua- tions do not appear in Eqs.(6.32) and (6.33). This follows because we assumed that the spreading rate of the mean Green's function is control- led by molecular diffusion and a finite memory time near the mass trans- fer interface. We need to emphasize, however, that although the approx- imate strategy presented here neglects the effect of <£fG3 on , still influences the mean field through the memory length 123 £11(x1). Finally, an explanation of the mass transfer rates at high Schmidt numbers reported by Shaw and Hanratty[1977a] may require a mechanism for turbulent transport which allows the 'effective' diffusi- vity (see Eq.(7.l6)) to depend on molecular diffusion, turbulent diffu- sion, and turbulent retardation. However, quantitative agreement between the theory and the experimental data is determined by the hydrodynamic parameter characteristic of the region deep within the viscous sublayer. The values of these parameters are not well known at present, but may be estimated by using the Type I and Type II relaxa- tion models simultaneously, as illustrated in Section 7.3. CHAPTER 8 APPLICATION OF THE THEORY TO PHYSICOCHEMICAL ABSORPTION 8.1 Type II Relaxation Model The mean field equation for physicochemical absorption, Eq.(2.16), is made dimensionless with D, 31F, and co (see Section 6.4), with the result that d‘c*_ d(u.’c.’)+ £1? + 4x.” N = c ’ (8'1) cJ’: I , X3: 0 (8.2) 4- C.1|L = 0 "I = °°. (8.3) The mass transfer coefficient , defined by Eq.(2.20), can also be expressed in terms of dimensionless variables: + _ + ('éc:> _-¢JC+' St -<‘£¢) W "' dx.‘ ‘20. (8.4) It is noteworthy that the: coefficients 011+(x1) and 211+(x1) in Eqs.(6.42) and (6.43) are not explicit functions of Sc. The dependent variables c+ and + in Eqs.(8.1) and (6.18) are thereby independent of Sc. It follows directly from Eq.(8.4) that °C.Dl/2, which agrees with the experimental observations on mass transfer near free interface. The boundary value problem, Eq.(8.1)-(8.3), together with the 124 125 second relaxation equation (6.18) was solved numerically using Gear's algorithm together with a standard shooting technique which exploits the linearity of (8.1) and (6.18) (see Appendix I). 8.2 An Analysis of the Mean Fields Using Type II Relaxation The turbulent retardation, which was shown in Chapter 7 to decrease the turbulent flux near a rigid interface, shows the same qual- itative effect when the mass transfer at a free interface is accompanied by a chemical reaction. This is demonstrated in Figure 8.1. Because the memory tends to oppose and retard turbulent I'diffusion" and, conse- quently, the spatial changes in the turbulent flux , the flux calculated with finite up is significantly smaller than that calculated by letting up==W. The rate of change of the flux with distance is also smaller with memory than without memory. The reduction of in- creases as x1 increases and, for the calculation shown in Figure 8.1, the turbulent flux is reduced by almost 50% in the bulk. Note, however, that when chemical reaction is present, the turbulent flux is larger than the flux without reaction if turbulent ”retardation” is also pres- ent. This does not occur when retardation is absent. To understand the underlying mechanisms affected by chemical reaction, the turbulent flux has again been decomposed into a ”diffu- sive” flux __ 4(0) D = - D“ 4x, and a ”retardation” flux 126 up+= 0.0007, k+= 0.1 up+= 0.0006, k+= 0 Figure 8.1. The effect of turbulent retardation on the turbulent flux. 127 R E j“ .d<:(lr’c’> xI Figure 8.2 shows the spatial variation of D and R in the near interface region. Note that all the fluxes have a maximum independent of chemical reaction. Chemical reaction does, however, decrease the magnitude of both the “diffusive” and ”retardation‘l fluxes to different degrees. Table 8.1 shows the effect chemical reaction has on the component fluxes as well as the turbulent flux itself. All values are evaluated at a particular point in the liquid phase, i.e., x+==1. The results in Table 8.1 show that chemical reaction tends to reduce the ”retardation” flux more than the ”diffusive” flux. It is this uneven reduction of the component fluxes which causes the turbulent flux to increase with reaction. Note, however, that the enhancement of the turbulent flux by chemical reaction appears only for intermediate k+. When k+ is very large, the chemical reaction dampening effect becomes so great that it actually destroys the turbulent flux. The mass transfer with large k+ is dominated solely by the chemical reaction. Figure 8.3 shows this effect by plotting the limiting flux versus k+. Figure 8.4 shows the effect of memory on the mean concentration profile. The reduction in the turbulent flux by the "retardation'I effect causes the concentration profile to broaden. This broadening results in an increase in the integral [XI kdx1 in Eq.(2.19), which 0 is the amount that the absorbed A is converted to B by chemical reac- tion within the volume between planes at x1==0 and X}. It is noteworthy 128 12A 11"" 10—_- Dimensionless Flux k+=OJ i I I J. 0 2 4 6 8 Figure 8.2. The effect of chemical flux 0+ and the retardation flux RT. RT 10‘ = Total Flux=D - R \ J . 1 1 l 5"" x1 10 12 14 reaction on the diffusive 129 TABLE 8.1 EFFECT OF CHEMICAL REACTION ON THE DIFFUSIVE RETARDATION, AND TOTAL TURBULENT FLUXES, AT x1+=1.0 AND FOR TH+=TM+=10 9 Flux, F Reduction, % Reduction, _ +_ _ += += k+=0 k+=01 A:F(k -0) F(k 0.1) A/F(k 0) 0+ 0.8769 0.801A 0.0755 8.6 R+ 0.7557 0.6272 0.1175 15.8 + 0.1322 0.1752 -0.0520 -31.8 + Sto, 130 l .".fl-l No Reaction += += TM TH 10 0.01 Figure 8.3. The effect of k+ on the turbulent flux in the bulk. 1y /co 1.0 —- +_ += TH TM 10 k+=0J 0.8-—— u +=O.0007 0.6-— P 0.4 __ + up :00 0.2 -- l I ‘ T l T j 0 2 A 6 8 Figure 8.4. The effect of turbulent retardation on the mean concentration profile. 132 that although the mass transfer rate is reduced by the presence of the memory effect, the theory predicts that more A will be consumed by chemical reaction as it is transferred toward the bulk due to the broad- ening of the concentration profile. 8.3 A Parametric Study of the Mean Mass Transfer Rate Using Type II Relaxation Figure 8.5 shows the effect of the turbulent retardation on the enhancement factor E, defined by Eq.(2.21). One important qualitative effect of memory on the enhancement factor is that instead of exhibit- ing a dip in the intermediate region, as in the no memory case, the enhancement factor with memory is a monotonically decreasing function of 1/k+. This difference in the qualitative behavior of E can be ration- alized by comparing the total flux equations with and without reaction, (2.19) and (2.10). At large x1, -.Dd/dx1 =0; from Figure 8.1, 3o if the retardation mechanism is present. It then follows directly from (2.10) and (2.19) that E3;1. In the case of no memory, <:o for conditions shown in Figure 8.1. If the integral of the rate of reaction does not offset the reduction of , then E<’1 (see Petty and Wood, 1980). In physical terms, the enhancement of mass transfer in the slow reaction regime is caused by the adverse effect of chemical reaction on the turbulent “retardation” relative to l'diffusive” trans- port; this causes an apparent increase in , which makes > °. The curves in Figure 8.5 were again parametrized by the Deborah number X/Oc. If Eq.(8.42) is substituted into Eq.(6.38) and the result \_,( 133 A/OC = 0.0025 6- TH+=TM+= 10 5.. Kinetic Regime (Mass Transfer Limited) 4- .21 3- Mass Transfer Regime (Kinetic Limited) 2—.. 6.2 l.-e 1/k+ Figure 8.5. The effect of turbulent retardation on the enhance- ment factor. 134 evaluated at 5c Efll, then 1(Sc)_ «ARV-i ag-gm‘) — 9. «5% 169‘ sci/Au 71;. (8.5) +__ + + :D /2.11 -1 c 11 ° for k-*0 by the memory effect. Also shown in the figure are the experimental data of Menez and Sandall [1974] for which Sc=:500. It is obvious that the data follows the theory with memory much better tha1 the theory without mem- ory. Although the shape of the curves in Figure 8.6 are independent of the Schmidt number, it does depend on the parameters TM+ and TH+. The values of these parameters used for the specific calculations shown were chosen for illustrative purposes only. Of course, the self-consistency 135 5‘ ..L: .. 0.1., . Halt. .uco_o_wmmoo cmwmcmcu mmmE ecu co co_umccmumc ace—sacs“ mo uommmm och .o.w mc:m_u +x\_ ooo.p oo— o_ _ ..o _ _ _ 1115+ . _ _ 11.o._ AUOH_E_4 u_umc_xv os_mom cmwmcmch mmmz .II +um I..o.~ Aomu_E_4 cmwmcmch mmmzv o mE_mmm o_umc_x LI mmooduemi . 1 :8: :eeemm o m w Noam: mo mama o 136 calculations similar to those used in Chapter 7 can also be applied here to estimate TM+ and TH+- Figure 8.7 shows the effect of TH on . The curve shows the same qualitative behavior as the curve with no chemical reaction (see Figure 7.8). However, when TH-tw, the mass transfer coefficient does not decay to zero; instead, it levels out to a constant value. Note that while TH enhances £11(x1) relative to 011(x1), chemical reaction attenuates £11(x1) more than 011(x1). Thus, the increasing importance of the retardation effect by large TH is buffered by the presence of the chemical reaction. The two adverse effects combined makes both D11 and £11 independent of TH as TH'*“5 with the result that becomes con- stant. The crossing of the curves with A/BCEEO corresponds to the enhancement factor less than unity, as already seen in Figure 8.5. Figure 8.8 shows the effect of p on . As in Figure 8.7, the mass transfer coefficient with reaction approaches a constant as p increases, instead of dropping off continuously for k==0 case. The rea- soning is the same as that for Figure 8.7. 8.4 A Summary of New Physical Effects I Because the diffusion and the retardation components of are affected by chemical reaction differently (see Figure 8.2 and Table 8.1) , the turbulent flux increases with I: >0, for slow reactions. Turbu- lent retardation has the highest effect on + when k+ is small, as shown in Figure 8.6. As k+ gets large, the theories with and without memory tend to predict the same result. The large difference in the + enhancement factor shown in Figure 8.5 for large k stems from the large difference in ° for k+==0. Therefore, the main conclusion developed 137 .p no cu_3 acm_o_mmmou commcmcu mmme come ecu co co_umocmuoL ace—anczu mo uummmm och C 1.4 In. ‘ - nasl W‘lh 2: 2 +5 _ ‘11 “ + $8.055: omN._ co_uommm oz Soduuo: N. _ . I o R o \ \\ New I\ I \ \\ \ NM. Nm.o1\\ \\ C. H+v_v \\\ 8:08”. £5 \\ \\ \\ \\ 2. "+0: co_uummm cu.3 \\ o . \\\\ \\\ \\\ .1. ll 11 \\l/IcOCummm oz \ U ox o} \ anew} .lllll. .m.w mcam_m ..o Im.o +um Io.— 138 .ucm_u_mwmou memcmcu mmmE some :0 >LOEmE mu_c_m wo uommmm och oo— p.o p ...—O O .m.m 0L:m_u co_uomom oz 2. 0+0: eo_eumee £0_2 .IT~.o .me.o ..:.o .lwm.o I..w.o Jfilfigu w.o um 139 herein is that a slow chemical reaction enhances mass transfer by not only steepening the mean conCentration profile, but also by suppressing turbulent retardation. Thus, mass transfer with slow reaction for which 6k+f55co+ could potentially serve as a test for different turbulent models (cf. Figure 8.5). The memory effect causes the concentration boundary layer thick- ness SC to increase relative to OC obtained with up==m. For constant k and TH, 58 is unaffected by memory. Figure 6.5 then suggests that Eqs. (6.42) and (6.43), obtained by expanding Tc(x1) and F(x1) about x ==0, 1 are good approximations with afld_without memory. However, the increase of 5c by memory makes the smoothing assumption (see Section 6.2) better with memory than without. The effect of chemical reaction on the turbulent coefficients D11(x1) and £11(x1) is opposite to the effect of the hydrodynamic param- eters TM and TH. As the result, the mass transfer coefficient with chemical reaction will be independent of the values of the hydrodynamic parameters for large TM and TH. Under these conditions, the mass trans- fer is only a function of molecular diffusivity, chemical kinetics, and alF. This is an important difference between the mass transfer with chemical reaction and that without reaction. CHAPTER 9 APPLICATION OF THE THEORY T0 ELECTROSTATIC DEPOSITION 9.1 Electrostatic Drift and Brownian Motion In this chapter the Type I relaxation model for the turbulent flux is used to investigate the possibility of increasing the rate of l‘l’?‘~’im_fi_rrmu mass transfer by electrostatic augmentation (see Chapter 4 for a dis- cussion of the relevant literature). Eqs.(2.26) and (6.49) are com- bined as a mathematical model for charged particles entrained in a fully developed turbulent flow and attracted by electrostatic forces to a flat surface (see Section 2.4). Turbulent mixing keeps the bulk con- centration uniform over the cross section of the precipitator, so the resistance to mass transfer is confined to a very thin region near the collector. The ground surface at x1==0 is assumed to be a good conduc- tor, so the particle discharges instantaneously. Thus, the mean field is zero at x1==0 and increases rapidly to cb, according to Eq.(2.28) Within this concentration layer, the drift velocity uD(=-uogl) is assumed constant. Other assumptions concerning this problem are summa- rized in Section 2.4. For conductive spherical particles in the size range for which Stokes' law holds, White [1963] has shown that particles of radius 'a' having charge q drift toward the collecting surface with a velocity given by 140 141 U > o. (9.1) D 67/461 Here Ep represents the magnitude of the electric field near the ground plate and u is the viscosity of the gas phase. The limiting (or satu- ration) charge on a particle attainable by a bombardment mechanism is 3 = n5 2 .-. 35., a}, (9.2) where EO represents the magnitude of the charging field, N5 is the num- ber of charges on the particle and e is the elementary charge (see p. 135 of White, 1963). However, when the particles are of submicron size, a diffusive mechanism for charging, which is due to the thermal motion of the gas ions, may also be important. The charge on a particle due to this mechanism is (White, 1951) GH£ T 15‘ [ 7td./&=]~%t¢2' ] 3-:: 41 l 4' 1QLT, *7 . (9.3) Here T is the absolute temperature, “57—is the root-mean-square molecu- lar velocity, N; is the gas ion density, and t is the exposure time. The number of charges on a particle acquired by both mechanisms as a function of particle size is plotted in Figure 9.1. We note that for Eo=50 kv/cm and a charging time of 10 sec., the diffusive charging mechanism is relatively unimportant for particle sizes larger than 10-2 micron. Obviously, for a given particle size the charging field Eo can be increased to a level for which Eq.(9.2) determines q. In what follows, the diffusive mechanism for charging is assumed unimportant. Charge Density (Number of Charges/Particle) 10“- 103-— 102~- 10-- 1J0 10'2 0001 Figure 9.1. 142 Bombardment Mechanism E0 = 50 kv/cm I I ' 0.01 DA 170 Particle Radius, Microns Diffusive Mechanism T=300° K Ni=5x108 ions/cm3 t=10 sec e=4.8x10’10 esu 1? Effect of particle size on equilibrium charge density. 143 According to Eqs.(9.1) and (9.2), the drift velocity “D may be comparable to the friction velocity u“ for very small particles. For weak electrostatic fields with Eo==Ep==150 v/cm, a particle of diameter .L 1‘ = .01 v/u* (==.1 pm for u*==200 cm/s) has a drift velocity uD==10-“u for u==.02 cp and v==.15 cm2/s. For these conditions, turbulent fluc- tuations in the viscous sublayer will play an important role in deter- mining the deposition rate. Moreover, Brownian motion will also be important at this scale. With (see Chandrasekhar, 1943) .8 =m (9.2+) I. the Brownian diffusivity at T=20° C for a z .01 v/u" is z IO-chz/s. Therefore, a characteristic diffusion velocity within the viscous sub- layer could be as large as 10-6u . 9.2 Type I Relaxation Eqs.(2.26) and (6.49) can easily be combined and integrated to give *- 6‘: : J exp [- Sc Jx'(ue++ 01;)dxr] fo. (9,5) 0 O . + .. . This expression reduces to Eq.(7.3) by letting uD::0. Once agaIn the superscript + has a dual meaning, depending on the specific application. For rigid interfaces, the variables have been made dimensionless with J. v and u ; for free interfaces, v and a1; are used. + . . . In Section 6.6, a model for uC for electrostatIc depos1tIon was developed. Inserting (6.55) and (6.50) into Eq.(9.5) yields 144 +,,n+l + no $0.9M x + x - 5+ _. I 3c " so dX; €XP 1"? n - SC “0 xi ] ’ (9.6) where n==4 for a rigid interface and n==2 for a free interface. The other notation was previously defined in Section 6.6. The above inte- gral was evaluated numerically and the Stanton number for a wide range of parameters was calculated using the expression I 3"" Sc 50+ . (9.7) .3 The mean concentration and the turbulent flux were also evaluated using Eqs.(2.26) and (6.49), respectively. The procedure employed for these calculations is described in Appendix J. An important question in this chapter is the effect of particle size on the mass transfer rate. The parameter affects 5c+ through the Schmidt number and the drift velocity. In the previous section, we have already discussed that both Sc and uD+ were proportional to the particle radius. This motivates the introduction of a new dimensionless group N (xi/Sc, (9.8) which is independent of particle size. N can be interpreted as the ratio of a characteristic Reynolds number for electrostatic convection to a characteristic Peclet number. That is, __ (Up/II) N“ (ii/.0) , 1115 where t 0L , rigid interface I} 2104;”, free interface Inserting the expressions for uD and flinto the preceding definition of u: N gives { rE'E’kT ' lzn’fl3u“ , rigid interface (9.9) ,5 _ l N - l I: FV’E,E,/>0, the deposition rate is enhanced and approaches a line with slope +1 on Figure 9.2. This corresponds to the limiting case when electrostatic convection controls the mass transfer rate, so St+-*uD+==NSc. The region over which (x1) changes significantly is very small for St+=uD+=NSc because OC+=1/(ScSt+)=1/(NSc2). Turbulent mixing is rel- 1 I O O O + + atIvely unImportant thIs close to the wall, Inasmuch as uc “aSc/EOC ” -15/2 4. + . + W Sc . Therefore, for large Sc, uc << u and the behaVIor of St D 146 10"+ - A‘-+1 // _ '11 .~~.__"’, 10 10's.- \\~__// -, L N=10’12 St+ Rigid Interface \‘_’/ alR+=10-5 N=10-13 - ‘+= 10 5___ TH 100 + + T /T = 1 M H II=O —-—-EqJ9H) 1°" I I 105 106 107 108 Sc Figure 9.2. The effect of Schmidt number on the particle deposition rate for rigid interface. 147 shown in Figure 9.2 is anticipated. For fixed N, the mass transfer rate passes through a minimum between a regime controlled by turbulent mass transfer and one con- trolled by electrostatic convection. Figure 9.2 shows that St+<=l=Sc$'"7 in the "turbulent'I regime, whereas St+<; 1000-- Rigid Interface a1R+= 10'5 +_ TH -100 + -+_ 100- H)- I 4" N 1.0 ‘ 10L12 10-11 10-10 10‘13 Figure 9.3. The effect of electric field strength on the augmentation factor for rigid interface. . “.11...- _ In» “a... a. 19.. .. -.. I I9 149 different Schmidt numbers. For large N, the curves approach a straight line of slope +1 because St+0=N in this regime. With b==0.1, the aug- mentation factor for relatively large values of N can be written as 7 St I. : -—-—-—‘ z (9 . A}! St. I NSc (9.14) The strong dependence on particle size expressed by Eq.(9.14) suggests that electrostatic augmentation of turbulent deposition rates could form the basis for separating particles of different sizes. With TH+==100 and the range of N shown in Figure 9.3, the electrostatic augmentation will always enhance the particle deposition rate. This qualitative behavior is also found for free interface with TM/TH==1 and T + 10. H 4. Figure 9.4 shows the effect of T on St+ for the particle depo- H sition on a free interface. The Stanton number increases continuously + . with T regardless whether the external field is present. The same H 9 behavior is also found for rigid interface with p= 1. The effect of TH on the augmentation factor for the particle deposition onto a free interface is shown in Figure 9.5. It is inter- esting to note that for the field strength and the Schmidt number shown in the figure, an augmentation factor less than one occurs for p==1 and relatively large T This is due to the reduction of the convective H' velocity uc by the electric field shown in Figure 6.7. The consequence of the reduction of uc by the electric field is that the turbulent flux is reduced relative to the flux without augmentation. This can be seen from Figure 9.6. The reduction in uc gets magnified by large TH. When the reduction of uC by the electric field is too large to be compensated for by increases in uD, an enhancement less than unity can happen. In 150 23..-x10"* 21.__ N==O /— N=10"1° 19..- Free Interface 5 =16 17 -_ c 0 St+ p=1 15.__ 13 a- 11__ N=10'10 9 -— N==O 7 __ 5 41 1 1 I 1 2 I 3 I u iii-" 1.0 10 10 10 10 TH+ Figure 9.4. The effect of TH on the particle deposition rate for free interface. 151 Sc=106 3.0 - Rigid Interface (I) 2.0-—— Free Interface p=1 N=10‘10 1.0 -“ Figure 9.5. The effect of TH+ on the electrostatic augmentation factor for rigid and free interface. -1 __ -8 __x10"‘ 22+ 152 8 10 12 111 16 18 20 22 211 26 28 30x10” l .1 1 l I l J 1 I I l .J l l I I 1* I 1* 1 I ”l T, l .fi'...x Free Interface N=10'11 N=10'10 Figure 9.6. The effect of electric field strength on the turbulent flux for free interface. 153 physical terms, when TH gets large, the turbulence-electric field inter- action reduces the total flux even with an increase in the convective flux caused by the imposed field. The same phenomenon was not observed for a rigid interface due to the different x dependence of “c' 1 Figure 9.7 shows the variation of St versus p for the deposition of particles onto a free interface with and without charges. The aug- mentation enhances the deposition more for small p than for large p. To see why this happens, one is referred to the plot of the magnification factor ¢ versus p shown in Figure 9.8. We note that the reduction of uc by augmentation does not happen for small p. The fact that the turbulence-external field interaction enhances uc for small p then com- bines with the forced convection by external field to increase the depo- sition rate more for small p than for large p. Note that uC becomes negative when p is large (p3:10), which means that the turbulent mixing tends to transport the particles up the concentration field. This is thought to be caused by the expansion about uD==0, among other assump- tions made in the calculations. 9.4 Summary of New Physical Effects A non-gradient turbulent model which relates the gradient of the turbulent particle flux to the concentration gradient by a I'convective'I velocity u is used to determine the particle deposition rate. Both the c convective velocity and the drift velocity ”D are functions of Sc. As Sc increases, uc and ”0 increase, but the Brownian diffusivity decrea- ses. Thus, depending on the values of Sc, the mass transfer rate can be characterized by two extreme regimes: a turbulent regime or an elec- trostatic regime. In the turbulent regime, mass transfer is controlled 154 10'3—- N=10’1° I N==0 St+ Free Interface Sc=106 +_. TH --10 10-“ I l > T T 0.01 0.1 1.0 p Figure 9.7. The effect of the finite memory time on the Stanton number for free interface. £1.04 155 3.0- 2.0- 1.0- + f» 0.01 0.1 1.0 p Figure 9.8. tion factor. The effect of finite memory time on the magnifica- 156 primarily by Brownian diffusion and turbulent mixing near the interface. Therefore, increasing Sc causes a decrease in . In the electro- static regime, however, mass transfer is controlled by electrostatic convection. Here increasing Sc causes to increase. Figures 9.2 and 9.3 illustrate these features. An important result predicted by the theory is that the turbulent particle flux is reduced by the presence of an electrostatic field for certain p. This occurs because the magnification factor decreases with N as shown in Figure 6.7. This phenomenon has not been reported hereto- fore. It is noteworthy that the calculations presented were made by expanding ”D about uD==0. It can be shown that the actual magnification factor ¢ approaches zero for large N, rather than leveling out to a finite value as shown in Figure 6.7. Thus, the potential effect of electrostatic convection on the turbulent flux may be more pronounced for larger field strengths than indicated here. We have assumed in this work that inertial impaction of particles onto the collecting surface is negligible. The validity of this assump- tion requires that the particle ”stopping” distance be much smaller than the thickness of the concentration layer. Under normal conditions, this requirement can be met for small particles (821 um or smaller) or for particles with densities smaller than the fluid density. Thus, in the collection of submicron-sized particles for which Brownian diffusion and turbulent mixing deep within the concentration boundary layer are impor- tant, theories such as those developed by Friedlander [1959] and applied by Forney and Spielman [1974] may not be appropriate. However, for large particles, the theory developed here would grossly underestimate the mass transfer rate by orders of magnitudes. 157 The mass transfer coefficient with augmentation is bounded below by the mass transfer coefficient with NEED for the parameters shown in Figure 9.2. With free interface and high TH+ this may not be the case, and the curve with augmentation dips below the curve with NEEO for some Sc before it increases with +1 slope at high Sc. This is because the reduction of uC by N causes the augmentation factor to be less than . . + . . . unIty. For a fixed TH , however, the reductIon of u wIll not occur If c p is small. Thus, the critical parameters which determine whether or not the enhancement factor is less than unity are the relaxation time scales TM and TH. CHAPTER 10 CONCLUSIONS AND RECOMMENDATION FOR FURTHER RESEARCH 10.1 Conclusions The basic strategy for this research is based on the evolution equation (5.13), which links the concentration fluctuation c’ to the underlying turbulent field. This strategy makes the concentration fluc- tuations depend not only on the mean field scale 6c, but also on the smaller scales characteristic of the fluctuating velocity field. To close the turbulent mass transfer problem, we apply the Sternberg hypo- thesis to the level of fluctuating Green's function, thereby neglecting the coupling between the fluctuating fields in generating 0’ (see Eqs. (5.20) and (5.22)). Because is a fully three-dimensional function which evolves temporally, the coupling of u: with V depends on a_l_ the velocity scales, not just the ones related to 8:. Another important closure assumption made in this research is the replacement of by the "unperturbed'l Green's function G° with a fin- ite cut-off time TM. This approximation makes the evolution of depend only on molecular diffusion. However, the important physical idea that the major resistance to mass transfer near an interface occurs during the time between bursts is retained by the inclusion of the finite cut-off time. Of course, the next level of the research would be to construct a containing more hydrodynamic information by including the turbulent mixing in Eq.(5.19). 158 159 A sufficient condition for the existence of the local turbulent models (6.17) and (6.18) is JTM/Se << 3:, (10.1) which may hold for all Sc provided TM+ does not increase too rapidly with Sc. From Figures 6.1 and 6.2, the spatial spreading of the kernels usand [I can be made small for large Schmidt numbers and/or small TM+. Under these conditions, the two local relaxation models hold, and the spatial structures of the turbulent field are unimportant for mass trans- fer. Note, however, that the strategy does not necessarily limit itself to the condition (10.1). Proper modifications in and the spatial l'smoothing" hypothesis should facilitate the use of the theory for low Schmidt number problems. The two relaxation equations are not necessarily equivalent if the relaxation coefficients are estimated using the approximate model developed in Chapter 5. The two models represent different physical constraints for the fluctuating fields. Eq.(6.17) tries to preserve the important constraint that ED. Eq.(6.18), on the other hand, tries to preserve the detailed underlying mechanisms in balancing the local fluxes. Because the physical effects retained in the two models are different, they predict different parametric behavior of . However, for an ”exact“ solution the mean mass transfer rate predicted by both models should agree, provided the "correct'I hydrodynamic parameters are used. This argument forms the basis for the Self-Consistent Hypothesis discussed in Section 6.3. The values of the hydrodynamic parameters TM+ and TH+ obtained from the self-consistent calculations increase with Schmidt numbers. 160 Note that the parameters are characteristics of the temporal behavior of the fluid which offers resistance to mass transfer. As Schmidt num- ber increases, the resistance is confined to a smaller and smaller region near the interface. The fluid is renewed less frequently due to the no-slip condition, and TM+ and TH+ increase with Schmidt number. This result is consistent with the idea proposed by Harriott [1962] and Davies [1975]. The calculations shown in Figure 7.2 may be correlated as «f ~ [077 TM - xl-I 1.3 (10.2) which should be compared with the (X1+)-1 dependence in Eq.(3.l3) assumed by Davies [1975]. The feedback mechanism acting through the ”memory” length 211 in the Type II relaxation model opposes turbulent ”diffusion.” Conse- quently, the turbulent flux with retardation is smaller than the flux without retardation (see Figures 7.5 and 8.1). Thus, turbulent retarda- tion (or the “extra” memory by Builtjes, 1977) has an important effect on the turbulent flux even for fully developed flows. This conclusion complements the results of Builtjes [1977], who assumed that the ”extra” memory is only important in developing flows. Futhermore, the Type II model predicts that the turbulent flux decreases as the memory time of the eddies TM and TH increases; the model degenerates to the film theory when TM and TH-tw. Thus, the turbulent memory effects have a general feature to retard the flux, as predicted by the Type II model. Another important result predicted by the Type II model is that chemical reaction always enhances mass transfer (see Figure 8.5). This 161 is because chemical reaction reduces the retardation flux more than the diffusive flux. The result reported by Petty and Wood [1980b] that the enhancement factor could be less than one for relatively weak chemical reactions can be recovered by the same model if turbulent retardation is absent. This again demonstrates the important role turbulent retarda- tion plays in turbulent mass transfer. The augmentation factor predicted by the Type I model may be less than one for weak electrostatic fields and free interfaces (see Figure 9.5). This surprising result comes about because the convection velo- city is decreased by the presence of the external field. Of course, this qualitative feature may be model sensitive and needs to be veri- fied experimentally. When a strong external field is applied, the elec- trostatic convection will be the dominating transport mechanism. The turbulent mixing becomes unimportant and the reduction of the augmenta- tion factor does not occur. Thus, the weak_reaction or external field regime in which 5k and 5E are small compared to 5c could serve as an unambiguous test for turbulent models. The Type I model predicts that the particle size (whichis.propor- tional UOSc) has an opposite effect on the mass transfer rateirithe tur- bulent regime and the electrostatic regime (see Figures 9.2 and 9.3). In the turbulent regime, mass transfer is controlled primarily by Brownian diffusion and turbulent mixing near the interface. Therefore, increasing particle size causes a decrease in . In the electro- static regime, however, mass transfer is controlled by electrostatic convection. Because “D is also proportional to particle size, increa- ing particle size in this region causes to increase. From Figure 9.3 and Eq.(9.14), the augmentation factor has a strong particle size 162 dependence for relatively large values of N. Thus, Type I model pre- dicts that the electrostatic augmentation of turbulent deposition rates could be used to separate particles of different sizes. This possibil- ity invites experimental verification. Recall, however, that the results shown in Figures 9.2 and 9.3 were obtained by assuming that the diffusive mechanism is unimportant and the limiting charge on a particle is determined by a bombardment mechanism. The qualitative feature shown in Figures 9.2 and 9.3 may be changed if the diffusive mechanism is also included. One of the physical assumptions made throughout the research is that the turbulent mixing in the bulk is so effective that the resis- tance to mass transfer is concentrated in a region close to the inter- face. In reality, the bulk flow may also offer finite resistance to mass transfer. In such a case, the theory derived in this research can provide a sound boundary condition for the large scale calculation. For example, Eq.(2.30) for electrostatic deposition can be expressed as ::<:c') (10.3) X,=O 9 for a region very close to the wall. Note that this equation has the same form as that proposed by Leonard et al. [1979], with A==-OC (see Eq.(4.25)). Here Sc==OC(Sc, N; hydrodynamic parameter) is a function of all the physical parameters relevant to mass transfer in the near wall region. Eq.(10.3) can now serve as a boundary condi- tion for a turbulent model (e.g., a transport model) which appropriately 163 describes the transport phenomena over a larger scale. Thus, one of the most important contributions of this research is that it provides a link between a small scale theory and a large scale theory, which illustrates the complementary nature of the relaxation approach and the transport approach to turbulent modeling. Some of the key conclusions developed as a result of this research are summarized as follows: (I) The theory applies a Green's function technique to turbulent mass transfer near an interface. This methodology has been used by many authors to solve transport problems in isotropic, homogeneous turbulent flows, as exemplified by the work of Kraichnan [1959] and of Hill [1979]. This research, however, employs the technique to interfacial mass transfer in the region near the interface, and incorporates the "wall'I effect in the theory. The Green's function for electrostatic deposition, Eq.(B.42), is new with this research. (2) The theory retains the non-linear coupling in the equation for c’, but neglects this phenomenon in the equation for G’. The coupling between the fluctuating velocity field and the mean gradient is one-dimensional in Eq.(5.1), but fully three-dimensional in Eq.(5.20). Neglecting the non- linear coupling in Eq.(5.1) places too much emphasis on 6c in determining c’. However, because is a fully three- dimensional function which evolves with time, the coupling u’°V also evolves and selects the turbulent scales which are correlated with at different times. The production 164 of G’ by this coupling mechanism will therefore depend on all the velocity scales, not just the one related to 0c. Further- more, starts out as a sharp peak at t==t, the spreading rate being controlled by molecular diffusion according to Eq. (5.27). 0’, however, is zero at t==;, and gradually develops into a random field for t>*t. Thus, for large Schmidt num- bers and short time behavior, the magnitude of the linear coupling in Eq.(5.20) may be larger than that of the non- linear coupling. This heuristic argument partially justifies the closure assumption used in this research. Of course the results calculated using this theory would provide additional justifications for this assumption. (3) The methodology developed in this research, summar- ized by Figure 5.1, yields non-gradient type turbulent relax- ation models. The relaxation parameters in Eqs.(6.17) and (6.18) depend not only on the underlying flow structures, but also on the relevant physicochemical parameters. (4) The Type I relaxation model, Eq.(6.17), predicts analytically that G=Sc"7 for rigid interfaces and a=THya° for rigid interfaces, which also agrees with a numerical simulation of mass trans- fer developed by Campbell and Hanratty [1981b]. The applica- tion of Eq.(6.17) to the deposition of charged particles on a free interface predicts that the deposition rate may not_ be enhanced by electrostatic augmentation. This phenomenon 165 has not been reported heretofore. (5) The Type II relaxation model, Eq.(6.18), predicts that the turbulent flux near an interface is retarded by a feedback mechanism involving the gradient of the flux itself. This mechanism is important even in fully developed flow. When applied to physical absorption near a rigid interface, the model predicts a Schmidt number dependence of which is very close to -.7; the results agree with experimental data better than those predicted by the theory which neglects the turbulent retardation. When applied to physicochemical absorption near a free interface, the theory predicts that the enhancement factor is always greater than one, which would not be the case if the turbulent retardation were neglected. (6) The two models can be used with mass transfer data to estimate the hydrodynamic time scales characteristic of the region close to a rigid interface which are consistent with previous proposals on the variation of the temporal behavior of turbulent fluctuations within the viscous sublayer. 10.2 Recommendation for Future Research The theory developed in this research provides a clear and sys- tematic way of approaching the turbulent mass transfer problem. The strategy used in this thesis is a first-order approximation and can be readily improved and extended for further research. A few suggestions are listed below: (1) The effect of the turbulent mixing term (255’) on should be studied using Eq.(6.8) as the starting point. The 166 evolution of is therefore determined not only by molecular diffusion, but also by the underlying turbulent structures. A more realistic memory time than the cut-off time TM should also be developed to describe the bursting phenomenon near interfaces. (2) D31 should be included in the expression for 011. A self-consistent concept similar to the one developed in Sec- tion 6.3 may possibly be implemented to back calculate 0:1. (3) The analysis should be extended to study the mass transfer problems at low Schmidt numbers and large TM“ The relaxation kernels under these conditions are no longer spatially peaked functions, so the non-local, non-gradient type relaxation equations, (6.4) and (6.13), must be used. (4) This research could easily be extended to describe mass transfer in developing flows where the concentration field is no longer statistically homogeneous in planes par- allel with the mass transfer interface. The effect of turbu- lent retardation on the mass transfer rate may be much larger for this class of problem (cf. Builtjes, 1977). (5) A careful analysis of the fundamental closure assump- tion used in this research should be developed by calculating the non-linear coupling terms in Eq.(5.20) using G’ developed in this thesis. (6) Mass transfer experiments for the w§§k_chemical reaction regime and the weak electrostatic convection regime could provide useful information for evaluating the theories applied in Chapters 7, 8, and 9. APPENDICES APPENDIX A PROPERTIES OF THE ENSEMBLE AVERAGE OPERATOR APPENDIX A PROPERTIES OF THE ENSEMBLE AVERAGE OPERATOR The ensemble average of a quantity is defined as an average taken over a large number of experiments that have the same initial and boun- dary conditions. Thus, the ensemble average of a velocity field u_of N identical experiments is given by II = Z ng,(§,tl (A.1) l=l N , where <°> is the ensemble average operator. By introducing a probability density function B(u) (see Lumley, 1970), which has the property J B(5)d£(2) = I, (M) E then the ensemble average velocity is formally defined as <2=] EB +'§é . (A-h) 167 168 A similar expression holds for v, I y=+\_{. (A.5) Some properties of the ensemble average operator used in this thesis are as follows: U)> = <><> = (3) ((2)13 = <<_u_>> =g (4) <_Li+_\_/_> = <_u_>+ (l) (5) <2)? = + (6) 8/3t=; V= (7) Jd§2=; fdt= $2 $2 Thus, <°> is linear and commutes with space and time differentiation. Property (7) states that <°> also commutes with space and time integra- tion. APPENDIX 8 PROPERTIES OF GREEN'S FUNCTION APPENDIX 8 PROPERTIES OF GREEN'S FUNCTION B.1 Generalized Green's Theorem, Reciprocity Condition and the Green's Function Technique Let L be a linear differential operator and w a solution to the equation Lfil’(x,.xs.x,.t)=0. (3.1) An adjoint operator L of L is defined as uLu-vLu. = V-E(u.\r) (0.2) ) where E-is called the bilinear concomitant vector. Eq.(B.2) states that for a given operator L, there exists a corresponding operator L such that uLv-vLu can be expressed as a divergence of a vector P, (This discussion parallels the treatment of Green's functions in Chapter 7 of Morse and Feshbach, 1953.) For example, if L is the diffusion operator, 3’ L L E a axia- - at ’ then N at a ._ 3 -——+— L " ax," at . so that 169 170 " -3_[ fl. 3.: -2. uLv-ULu-ax’ u&X)_-J3X, at[14‘1"]. If L is an integral Operator, the adjoint operator is defined as [uLvdfi-J Imam/5:0. 1; 11 Eq.(B.2) is the generalization of the classical Green's theorem, which reads uv‘v-vv‘azv-(uvv-vvu), (8.3) An operator is self-adjoint if ULV-VL“=V°pCu:U’). (8.4) but this may not be satisfied by all operators. For instance, if Eq.(B.4) does not hold. The Green's function G associated with the operator L in Eq.(B.1) satisfies the following equation LG(L.+I”,€)= -J<.§-§)J(t-€) (8.5) where 5 is the Delta distribution. Physically, the Green's function 171 G(x,tlx,t) measures the effect at the location x_(observation point) and time t of an instantaneous point source located at x_(the source point) and at time t. The relaxation of the Green's function over the domain after t is determined by the operator L. The adjoint Green's function G is governed by [Giulifin-5cs-£)Jce-?). (3.6) To see how the Green's function is used to solve the inhomoge- neous problem, consider the equation Lil'crnt) =-{’c.<..t) (0.7) with inhomogeneous boundary conditions. The inhomogeneity p(x) is called the source function. Multiply (8.6) by w(x,t) and (8.7) by G(x,tl2,£). Subtracting the two equations and integrating the result over the physical domain and over time yields 8 ... ‘l’cin’e‘ 2 = Loft [nJEffL'UG'CJS.+I£,€) + sztjdé [5(L.t12,3)L1}I-1/’Zé(£.tli)3’] . (3.8) .n. -0 With Eq.(B.2) and the divergence theorem, Eq.(B.8) can be reduced to 3 ., , fl ‘. . . \f(£.f)= J 4*[J£f(§t)6(£tli,€)+ fJS‘ ~f[&(£,tl§.t), 7(3)]. (3.9) -w n The surface §:==E:S’ in the second term on the right-hand side of Eq. (8.9) is now the generalized surface which includes the physical boundary 172 surface and a surface associated with time. 2:, defined as 8'5 ”'5‘ ’ 1.21 is the unit normal to the generalized surface. n_is the unit normal to the physical boundary surface; St is a unit vector for the t-axis. The vector E_is E=Ps X, I +'F; 'e Eq.(B.9) can be rewritten as follows: t 4 [i " ~ A A 778th!” 4t (6 J5f(£)*)6(é.t In) (I) +If «2 Iain-2Iéiisiisnmaifi (II) A {let + JJS. d2? §£° E[G(i,%l$,f)’+(gvf)l:=‘ (8.10) (111) °° Symbolically, O can be expressed as 4 _, _, he V’zL (f(1))+B(EI£.§‘)+l(81“,w) (I) (I) (III) ’ where L (., 1:.Iéjfiaé‘é; -I B (-) 51* 4'2 §d§§;and, 173 Eq.(B.10) states that w is made up of three components (I), (II), and (III): (I) represents the contribution due to the inhomogeneity 9(fint); (II) represents the contribution due to the boundary conditions; and, (III) represents the contribution from the initial condition. In order to express the solution 0 in terms of G and flg£_G, it is necessary for a simple algebraic relation between G and G to exist. This leads to the generalized reciprocity condition. To this end, we consider the equations for G and G LG(XI3)=-5(l-£) (0.11.) ”N i a LGIXI§)=-5(X-X). (B.llb) Here the Green's functions are expressed in terms of the generalized coordinates X_and g_representing all the independent variables which are relevant. Thus, in a problem where time is also a variable, X_consists of x_and t, and (8.11) are equivalent to (8.5) and (8.6). Multiply (B.lla) by G and (B.llb) by G, subtract, and integrate over the relevant volume in X space. Employing the generalized Green's theorem (8.2), we obtain G(>§I§)-0(§It)=f 2-B[é(2i‘li).cf< \a H C) G and G are so correlated that no ‘5 5 a E-E[G(XIX).G(XIX)]=0, 13.211 (ii) With Eq.(B.21), the reciprocity condition requires 176 IX) )=Ef( IX) CT(ZSI I!) (3.22) (iii) For the inhomogeneous equation L7=~fitx the solution is |))( (3)]43 (3.23) I><> 1M.) =jAIto. If the boundary is not present, the Green's function is symmetric with respect to §f=23 When the boundary is present, how- ever, the symmetry property of the Green's function is upset. If G==O at the boundary, the problem can be regarded as if an infinite chemical 178 reaction took place at the boundary. Thus, the boundary acts as a sink for the diffusing material making the Green's function (i.e., concentra- tion) different from the Green's function for free space. The Green's function for the diffusion operator will be discussed in more detail in Section 8.4. Another effect the boundary conditions have on the Green's func- tion is the irreversibility in the coordinate. The operator for the wave equation is self-adjoint (L==L), so that a directionality, for example in the time coordinates, cannot arise from it. However, if we require that G satisfies the initial condition (which is a boundary condition in the general sense) ¢==80/8t==0 for tt, The diffusion operator is not self-adjoint. L describes the forward diffusion from a point source to the final distribution as time increases; L describes the same process in reverse time order, beginning with the final distribution and going backward in time to the original point source. It is interesting to note that the reciprocity condition for the diffusion Green's function (8.28) is the same as the reciprocity 180 condition for the Green's function for the wave equation (8.25). How- ever, the directionality in time in the two cases arises for different reasons: the cause in the diffusion case is the asymmetry in the oper- ator, whereas the cause in the wave case is the asymmetry in the initial condition. By combining Eq.(B.28) with the generalized reciprocity condition, N A A G(£,-€I§,i)=6’<§,f15'5) (3.22) it follows that G(§,—tI§,-€)=G(§.flx.t). (3.29) The general solution of the inhomogeneous equation Llf'z-o‘DVW-ajlfl-z-fcsd) (3.30) T (tdgfdé' 66+"/’6&]+L ng‘l’GrL . (3.31) The Green's function for the unbounded domain can be constructed by the use ofeaFouriertransformtechnique(see Morse and Feshbach, 1953) It has the following form 181 I 3 ~15—2I‘/4.0(f—€) ) e (3.32) G(51+I£1€)=(,j4w(f—t)£ for t> t. It is interesting to note that IJ86I(3,+I£.€)=I .0. for t>’t. This is a statement of conservation of mass, for the material introduced at the point source is not consumed in free space. The one-dimensional Green's function in the semi-infinite domain with G==O at x1==0 can be constructed by the method of images. The result is 6%.: I [ -(X1-Ia)‘ --'(xl""?l)1 J anus—ff) 6 {4.0” 43 ”up , (3.33) An extension of Eq.(B.26) is to include a constant convective velocity in the x3 direction (see p. 647 in Monin and Yaglom, 1971). The governing equation is a a A " $VzG-§;3—;Z=’5(!-£)5U'*). (3.311) The solution is Guiliiel=G,(Xutlia:€IG'sChi/gibéfixsil’zsfz). (3.3s) 182 where S_“' _(X+£I)1] “PL x :0) ]-.exp[ 449th?) 4.8(t—t) G" (Xilt'£"€)= A 7373(3-3) ‘ z - 93-2,; exP[ 43(f-t)] G1 (x1)t[;11€ ) = 143.3 (16:1?) (x,—£,-(1t-é‘))’L A A ‘_ e>( [' 3 e—— &3(X;,f[x3i‘£)’ P . 4.805-!) 11(413H-37 Furthermore, the following integrals hold: (3 dx: G, (x,,tI>?1.€)= €'/["I/II/4-8(*‘75] (3.36) / obi,- G;(x.-,tI)?c.€)=/. 7i" (=24. (3.37) 0 8.3.2 Green's Function for the Convective-Diffusion Equation with First-Order Chemical Reaction The operator is (K’_ 1 a 2.- L =03V'<“3>;;,’k’at. (3.33) The Green's function associated with the convective-diffusion operator defined by Eq.(B.34) is related to L(k) by 183 (K) -t(£—1?) -ktt-t‘) )1 L (Ere. ):”€ 5(5-8)J(1€-f‘). (8.39) Eq.(B.39) implies that the Green's function associated with the operator (8.38) is w) .-£(f'€) C} :r C; {1 (8.40) where G is the Green's function defined by Eq.(B.35). 8.3.3 Green's Function for One- Dimensional Diffusion with Normal Convection The operator describing this process is Lsfiig+up%;--§i (3.111) The Green's function satisfying the homogeneous boundary condi- tions is derived in Appendix C. The result is I [ (..Lxl-§V+Up(t'€))z) Innis-e") ex? 4-9 (8‘1?) G(x,,tl§,.€)= fiat/.0 ,. - " .. 2.. 9 9X? (-(XI+XI'+U9 (t 16)) )} 43(13— 1?) (3.112) The method of images cannot be used to solve for G1. If we introduce an image source at -x1 which is to be convected toward the interface by -uD, the two streams will mix, rendering the net uD==O. APPENDIX C DERIVATION OF THE GREEN'S FUNCTION FOR ONE-DIMENSIONAL DIFFUSION WITH NORMAL CONVECTION APPENDIX C DERIVATION OF THE GREEN'S FUNCTION FOR ONE-DIMENSIONAL DIFFUSION WITH NORMAL CONVECTION The Green's function to be solved satisfies the following equa- tion 3 a 3 A (“83“;-TUDfi’W)q=-J(Xl-£)J(f-t), (C.l) with the boundary condition 5:0, X,=0. (C°2) The Green's function can be decomposed into two parts, namely; G=u+w (0.3) where u is the Green's function associated with Eq.(C.1) for the unbounded domain. u has the form (see Appendix 8) 2. -(X,—)?I-Uplf’€)) ] ex ,. ‘7' ,. 0.11 [47’39hé-f’ , ( ) The subsidiary function w satisfies the equation 184 185 31w 342 9w 093,914+ up”, - t, -—o (0.5) with w=0 at x1=0. We now take Laplace transform of Eqs.(C.3) and (C.5). with the result that G = (I '1' (B (C.6) 4‘13 J83 —- and .8 “f + Ddx. [9 . ( 7) Here the, overbar stands for the Laplace transform, and p is the param- eter in the Laplace transform. The Laplace transform of u can be obtained by using the standard table of Laplace transforms (see Abramowitz and Stegun, 1964) with the result that (XL- £1) Up G: e ’2.” fexpl-IX,-x,|fl(p’+4ebf ] ((2.8) [up +4.0? .23 The solution to (C.7) is £5 = A8XP[’(U°+IL:;4M)XI ] (0.9) 186 where A is the coefficient to be determined. Combining Eqs.(C.8) and (C.9) and letting G==0 at x1==0 yields allow/2.9 6XP["XAI u;++.9P/3-9] A = (0.10) fu01+43f - With (C.8), (C.9), and (C.IO), it follows that 1 - ' -[fi-fidllp A G: . e. 2.0 u (~IxI-x1I/uo‘+4.8P ((4, +4910 1’ JD Aide - 5, U A __ :0 472%) 0 ix) 1H1)! “fwd? 6 fl ex!) 2'” fl) (C.11) The Inverse Laplace transform of Eq.(C.ll) yields Eq.(B.42). The fore- going solution strategy is similar to the one presented in Carslaw and Jaeger's [1959] book; however, Eq.(C.11) is new with this research. APPENDIX D DERIVATION OF THE RELAXATION KERNEL E(x1[x1) APPENDIX D DERIVATION OF THE RELAXATION KERNELIL (x1 lxl) Because of the approximation made in Eq.(5.27), that (xjt123t) = 0 if It‘s t-TM, G’(_x,tl2<_,lt) is also zero for Its t-TM. The lower limit of E in Eq.(6.5) is thus t-TM, instead of -w. By combining Eqs.(6.5) and (6.8), it follows that f a, ,, L(X.I£1)=/ .0?ng dxfijtdfj 4.519%. I 1;) {”15” -co at» 4‘} :1 -[< u,’(i,€)2’<£,§)>- 38°“ 13)]. (0.1) If $G°(2I“) is spatially peaked with respect to the correlation , then the smoothing approximation leading to Eq.(5.23) can also be used to replace by in Eq.(D.1), with the result that i- , t L.y£jdx:ja’£6’ (8"). (0.2) 0 IR A ] dtggffgfie ” t .0. a 0 °G°('I8)a‘; fr) (X‘IA). (0.3) Here the one-dimensional Green's function is defined as J43; 60(2)») (0.1») fl 0 k 9 O __ “ G" (xlitl£,t)= J- JX3 «a Eq.(6.9) follows from the definition of Gl° in Eq.(D.4). APPENDIX E DERIVATION OF THE TURBULENT COEFFICIENTS 011(x1) AND 211(x1) FOR PHYSICAL ABSORPTION APPENDIX E DERIVATION OF THE TURBULENT COEFFICIENTS 011(X1) AND 111(X1) FOR PHYSICAL ABSORPTION With the assumption Eq.(5.27) and the generalized Sternberg approximation Eq.(5.22), Eq.(6.22) reduces to Du (X)? it]: 4" fixj dxzfjg G ch,1u£i.t)a=x1 on the semi-infinite domain is unavailable. However, if for short times the mean velocity is replaced by its spatial average over the vis- cous sublayer (i.e., uA), then an approximate G° can be constructed (see Appendix 8). Obviously, this strategy suppresses the interaction bet- ween vertical mixing and the shear field, but at high Schmidt numbers this should only be important for t-‘E >> TM, the finite memory time of the velocity auto-correlation. Therefore, we assume that the 'unper- turbed' Green's function applicable for large Sc near a rigid_interface can be represented by Eq.(B.35) with replaced by “A- The Green's function defined by Eq.(B.35) is sharply peaked about (x1,x2,x3)= (x1,;2,;3+-uAT) for small values of T. G° is a spatial delta distribu- tion for T= 0. Because the hydrodynamic space-time correlation appearing in Eq. (E.1) quickly relaxes to zero, the spatial scale over which Go varies is 189 190 small compared with the size of the viscous sublayer. Therefore, Eq. (E.l) can be simplified by using a smoothing approximation (see discus- sion preceeding Eq.(5.26» withthe result that .. D"(m=(it’J*d€-‘exp['%]¢'{ [$3] (5.2) t'Iw Eq.(E.2) was obtained by assuming that the velocity auto-correlation in a frame of reference with velocity uA can be described as a simple exponential function with a relaxation time TH and a finite cut- off TM. Eq.(6.32) follows from Eq.(E.2) by expanding erf[x1/Vfiifitfrf7] about x1==0 and integrating over time. To derive Eq.(6.33), Eqs.(5.25) and (6.1h) are substituted into Eq.(6.21) with the result that t f k Inna): (uf(£.+)g''f d1?!" Jf£da {ddfi + e-‘W fl t-f - o R o R ‘ A E ' [6. T” GCx.t|£,$)VG (Ltlxflc’) (5.3) Integrating over 9 reduces (E.3) to 1 -£ t t a. A ‘51 I I. or.) . ’j e. ,, I“ .....-) é‘nn t A t A b {I -Xiwaf't) w k D A 2 ° j M a I‘m] ”"3 Giulia). (5.1.) A To calculate the integral over 2, in Eq.(E.4), expand the inte- gral about x1==0 and retain only the first non-trivial term. This pro- cedure yields l9l j “W" 3.3/25 “P331434 ( ’=—_ A “ x’ ”-0 .n-r. 1: fie—J— (£ -4) (5'5) The integral over t can be related to the degenerate hypergeometric function @(l.5;2;n) (see Gradshteyn and Ryzhik, 1965). Eq.(E.S) is equivalent to Eq.(6.33) with Th I Ib/fi: f(TI-‘T Ei’J e-7?(/.53237)J7 (E.6) 0 where n==(t-t)/TH. For TM/TH==1, f(l)==0.h5. Also see Figure 6.3. Extent.- nannz- ‘n‘n‘u . . APPENDIX F DERIVATION OF THE TURBULENT COEFFICIENTS 011(x1) AND 211(x1) FOR PHYSICOCHEMICAL ABSORPTION APPENDIX F DERIVATION OF THE TURBULENT COEFFICIENTS 011(x1) AND £11(x1) FOR PHYSICOCHEMICAL ABSORPTION With Eq.(B.40), the derivation leading to Eq.(6.32) can be repeated to yield - (Lt 13.3) < Watt) u,’(£. {2}] (m) With Eq.(5.27) and the smoothing approximation, Eq.(F.1) simplifies to Dung) = . V [G-‘(L-tlbt) e. 192 193 A Integrating over 9 and using the smoothing approximation, (F.3) reduces to R f t A —kc{-£)—% 2110(07-(‘1/5] “(ff d; ed; a M. .2 Ins—um?) Q 4“ 0 fl .1fo 4096' (Lillgtt). (FA) -d -. z 5 ‘° 3: ~42 Ana-2) .J X: 'e. o The integral over 2, is calculated by first expanding the integrand about x1==0 and retaining only the first non-trivial term. Eq.(6.h3) is obtained by performing the spatial integration over £1 and then the time A A integration over t. 1 -. {{n‘ufi m‘i alas-‘- .... ' q APPENDIX G DERIVATION OF THE CONVECTIVE VELOCITY uC FOR ELECTROSTATIC DEPOSITION APPENDIX G DERIVATION OF THE CONVECTIVE VELOCITY Uc FOR ELECTROSTATIC DEPOSITION From Eqs.(6.9) and (6.19), it follows that L0.) = (“(1)00 ff diftdf: 6-37—j0 dx'C-bop (A) a? 4:4 a In ' 33;] CiCHdei, (6.1) D where l‘ 0 Cf, cxutlfiut) )) w w 0 A A A A I dX;J dx‘sq' ( X], X], X;-X;,X3-X3I+'f) -w -a, D U I 4'le d£G°cn.£,x.-£.,x3-£,t-€). (m) -.. -.. The two integrals in Eq.(6.2) are equivalent because G° is homogeneous in x2 and x3. From Eqs.(5.27), (6.5), and (6.20) it follows that 194 195 f TCx,) =1 at I dfi e°c 23) *‘Wn Ji 61* I I t A v A =I 1* J 4:3 mocxuflfi's), (3,3) é’tn a To further evaluate L(x1) and T(x1), we need to know Gf’and GodA . I0 1 x1 With BG°/3x2=BG°/8x3=6°=0 as x2 and x3-> 1'00, an equation for Gf’ follows by integrating Eq.(G.3) over x2 and x3: 5%;: ll )6}, i965. —— -' -' l”: -2). ax'1-I' ”ax, at 50: x)5(f (6.1-I) .8 Eq.(G.h) can be solved by the use of the standard Laplace trans- form technique; the result is given in Appendix C. Also, a, A a f A J Jan Gr, CXlt‘flxll'C) o I x +u (£45) “WM” x_,-____u, f- -3) =5I'”'J‘(f’4e;-€J)'e " 404m: 5 D]. (“'5) Eq.(G.l) is now expanded first about x1==0 and then about uD==O. With Eqs.(B.h2) and (F.5), it follows _ x,cx.2Tu jf’ -5 ./ P-5 £,(XI) - 'n'JB [ 2 ‘, C: 'thh 4/ I ‘1; _ P z +-’%-”./%("f/F us:- . ’1 e‘ 45)] M 196 Likewise, a limiting expression for T(x1) follows by substituting Eq. ((3.5) into Eq.(G.3), expanding the integrand first about x1=O and then about u0 = O: a ’CM T” --°—-] (6.7) T"”="'[l W T 2.9 Dividing Eq (6.6) by Eq (G 7) and rearranging Y‘e'ds 5q5'(6'55) and (6.56). APPENDIX H COMPUTER PROGRAM FOR PHYSICAL ABSORPTION USING TYPE ll MODEL APPENDIX H COMPUTER PROGRAM FOR PHYSICAL ABSORPTION USING TYPE II MODEL ’ The integration in Eq.(7.l3) for + is done using an IMXL subroutine DCADRE with Romberg extrapolation technique. Because the argument of the exponential function has certain lower and upper limits, the program is designed so that when the values of the argument is smal- ler than the lower limit or larger than the upper limit, the asymptotic expression Eq.(7.18) is used. The integral in Eq.(7.l7) for 6C+ is done using the standard Simpson's rule. The value of H in Eq.(7.l3) depends on the function (see Figure 6.3) Lip) = ‘21 Ire-7 é (1.552,o7)/’7 a . An analytical expression for the degenerate hypergeometric func- tion is not available, so a series representation of ¢ (I.S;2;n) is used in the integration: (aéh)($6 d'I) .L. 1 2.3 2!? 4,... éC/Jsli’l): I +%_/-’- 7+ +12/2) (3/2. +I)--- (3/2 +M-I) , _ ” ... (2)(2+I>~- (2+n-—-l) II!” + The term-by-term integration is carried out until the inclusion I97 198 of the (n+-l)th term does not change the partial sum of the first n terms by a preset tolerance. This calculation is done in the subroutine V. 199 FTN 5.10552 TAIITS OPT=1 PROGRAM STANTON .. . fil- .I.x....\<..r.1-.z.; ; . in...» E 0 o A 0 or E S T R. E U U T. H E H T HS N T! O o 0 0 U 6 I N EH 0 . T H A o R T. 9 R N S 9.. u. TO A A N T H 0 NF H A ‘1 O I E N T R R 1 T HN I: T. .l \- T o H C U0 0 o R o a O N G... 1. H I. E 1 E A T U RS 0 E O G O H o R F AS .... H C. RI. I. T N E E |i T C! ‘2 T C T. I; ER 3 3 L O 3 O I O A HP 0 T o 0 o N o R In TX 0 N 0 S: o A 1 0 T E U I. U 1.7. U . R N E A R A A 0 A R E RC T P T LT T 6 C E N .1 El. 0 o AA 0 . P O 0 A HT 5 N 5 ‘1 I. 5 X D. D a U0 0 E o .l T o a N E R X a T o H a 3 NE 0 A o E E ‘1 NP I. T T H E o .l H 1. R OHCZ z . NT. .1. T H O F. p ITZ. D O o 5 0C 0 6 R H C TS U N U 1. 1 90 U R o .1 R T F AAOA A A C X: A E0 R E R U TT T O 0 E2 T GR E A F. E TE F. s F \a T F PE T 9 0 o a THOC a C 2 n. ERNC A7. 9 B ‘1 N .1 STG. H o H A 0 HEE. L R 9 T0 TO I. 1. I . G TTH... 0 0 A T] CT. C EDT... 0 1. o 0 r T1. ST R 0 UC UT C HS A 1. A 1 T T FA A I. R F PE 0C 3 T 0C C C o 0 TC L F. C TV PON C .16, .L I o O 0 Tu.’ TA 9 E UA ORU I R6 . T R .1 o G 0.. THO.) NU R R 05 TPF 3 0 . T. 0 I. 0 1.. NC . T E0 R’D 0 I S H t F 0C T C O ‘- r.-.1EC vir ETA TI: U E T 0 "TC S C I! a? 0 HR 0C U TRAC U1 A H!- "U U SAGS Q. 1 c T U 12.5 CT \I O. a PC T’TR TA A CH 0. 0 o H E 6 GE 0 R0 AR. 0 NT '3 T T T T Q T M A H . o 0 RD 04 AN oRZX H .15 ).F.E UF . H .1. U 0 U o 1. AAT. 05...... U C 9 Trfiuu- A O 1. CSR. SN.) 0 NS 1 \n "L. E.) AAN 0 NS N.) C. 0. T) 1 N50). IE! st E! C 0 Q r. 0’ HT. t 9C8 E9 01R CAI-L2 T A AE. HE HE C T 0 HS T N8, 0 HE TCF. .3 AC, «AC 0 RLZP PTV P TV 0 l. o TTTP ac C 9 0A TV NCT. 9 ORR,,1C 5 B a C 0 A C A I: o 7. 2C 9 0 ’Al O A AS 0 IHTGEHHAS a SNF 0,5 F )3 0T 0 HN 0F HO 0 0. Z )5 T T F5 'TLSTT '0 O 1 GTCRZLO O RAKU 9 SN" N COOR c. 1F? 9 20 O SNN . 51M oTYHMUOOU H N [EH 10 EH10 [TU C NT. . r N7. . CCD H10. OINLE . :T NHTTSOSA : IL oC:E 00 C: .0 205 IOATZC A AEPT .. .0 HT 0 A 0 OCT. OHT CC122T T ZHAIC’LQB ZCTQS 0H] 2 OARICC RE ZCRXZ )9.) ASNNNln: 0 0T TVT F. o o Iv 03.... .IJTF. F o/JE. FOUT 01 853/ BR 090E FJE F RN 0R..:1n1 WEN :RTTT. 01 0C11LTZ oCT 0T 11.0 0C 0T ITCSNtuO . 0 N3 0 F. OXA/TTC 0T 1!. GELWr.RR ..53T4r.U3NVH o o : : 03.... o LVCIETJV E0 01....JN r; : ..IHOZTSAA1.LSH ....CHuv [Jun .10 OHRIHTRQ‘I ACTUASTTCROIH T3:VJZO—L 2VLCTISJDZ:VCI‘ASJDTHC:ZC CRC...)CHZ.H :3:T.~JVCCA~)J0 RIEOXEEIOEF LT:RRR:....UuV:1A::FNF2A FR:L:N...JAFR:L..VVUF1:FOHTF2LH ....X:2RVAFR:L:N PORCEARAQRISFFPPPPVZJSIHHSJZ JIHSSIPJEJEZHSIPJEJEISIHZIGT IH...TZBAEHHDESIPJEJE I1 a..p. 'A .0. 0 HXOS H NHO NT 9 VC TEII T IT? IA 3 . . . . . . . . . . . . . . . . . . CC CC C CC CC 14‘3‘9367890123“5679.Q 91523qu6790,0123R5679u0,310.3R567990123‘5é00780 .I 1214R‘a‘478 gflIQAZQDRSGT 11161.11111122226uzz29.2‘.33331u‘~333a RRR“RR“R‘rJSC.55p—_5555L_6666 15666 1077777777 200 FTN 5.10552 :1 OPT TAIITS PROGRAM STANTON BoAEPRgRERRQERROPQTERTOH , F A o 0 Z M- a U N. ”no C E9 SI HE A. TV 01 A tr. TS C230 0 CHlo O uCHEJN E6 2233:VC119J3TWC:ETCIZ : .. : ...DAFR :L: NNUFTHPFR ..OV .. : : ..ZORTORROOOOHO OZOEOOUON ZQAHHSIPJEJEISCHT‘IPZGEZQAHHGPSCPPFFFF Y 60 101.214q56700 TTHHBHeeace TOCClo-HIT‘Qo'Clo-HZTOClo-HBTT loE'ST GO TO 40 E H B F RGUMENT OF THE EXPONENTIAL IS TOO 816 SO THE S USED AGAIN- T THEN vaAERRoRERRQERRORoIERToH 101T ' F L G F 2 ATOA RR RERRQERPORoTERTOH 0' TT ICFTAU'CTAU'SCCITT0'noS'SUHT C9 : 3| N:o K 9:01.22 H 1C 0 O 1 91 TR E T QR ENE 00R 9‘ K IETHOO HFTD C 985T!- c FJA T 2ND. T: NNoC E2 5 oUTSIE A0 C A. 0 TNFTSR RIT 0 ON T E N0 8549. BC K OTENAA Si O I, M C I THHRC A 0 U NR2. 01 ST 0T IP02 F. P COTE-0T0 3N" "KN. CZ 0C OBAS.CAX IIU OECn C OF OOFIOO 3 26$ L [R 5 EPUSREUSZIIuTIQ 0W, LC0221 2 2RX33EUQRCCCCSCQ FOUT 011 o F. 09E :N TTTTAT. 3 ..IH 07““? 9 02260: NS 5 O C COD“ 0 U 0000 TA 3 Q 5 0505 7 3 0505 .. 1122 2 2 334° . . CC .34567890133 Gr. .67 590123§567899123459 gnu9n99000036000911111111112222222 11‘11‘111 11111111111111.1111! S: 09.5? NHTGIF A. FE 9 9 TrTLITT TTCSVAFTcTUOTa UA/OTTITTAAAALAZTA'AAAA .CHTNITNNHWHW WTQHIHWHC TCNCIRRRRERE 0R : TRRRDJ TFICFRFFFFE O 11 1 FTN 5.1955? =1 OPT 7Q/175 FUNCTION V voZIOSAV[81/(A0o(361.l) O 6 T 0 T T o 1 0 O G P C F H O NC 3 T F. 0 EP T . H H A HX T E T TA C TE 0 o H I I O 1 1 T TI: T It TT 9 C o) n. P 0.... N o v on C . 08 C TT 0. X N 06 O LN N 5A T I C 5A 0 CO 0 . D F o I A CT I CC A T O T 0T C 3~2N T TD 0 OTZ P TN I SE]. C LX 2 ILOT. Lo 1 0’0. N QE 0 A :ClBNEo. B 333? D U . A 2 C LT9:CU Z E ABCC N2 Fun Hn.o IF. I .I A..1.N77| Vrloo.n 0. _V T01 .1 .01 0302311.. AINSIZCTO R LLoHACEl 0132A VOiLgEBTACES CAAAOATVU AAITC..S:DI.: .... :8: : F :V:NC:S:D..C: :N: CTD Er... :FNLqu:319.3:120..2A10F3L3VNF12:00:EN RRAQICECENAAAAZQJBDPBSBCTBEBECTAANAGVRE o 0 0 2 R 5 123.56789012!OSST890123.567890125RSHTR 1111T111112222222222333333333 501.55? FTN =1 OPT 7&1175 FUNCTION F PCX-lo/(NOXOOQTT TIDN FCXT H V 3N 123N56 201 kmiancum. --u waves-m nanohom u¢~a~.m mp—csom ©®C$Oom cmwroom onccoom ommnoom zmht¢om mn—smom «vhmnom mwmhsot nmnucon «fromoa n: «cownomwo whomouooHZozmoco—u~ p1 cuqooo. nomooc. rupees. ~o~mmo. aomooo. acmmwo. -noco. uswooo. ccmaco. osscoo. nnosov. ocmsoc. pwomoo. «osmco. «coacvo masoco. s~cw~o. mmcskc. rqqmov. mmmong. nwmmcwo oceans. meadow. wwcmmm. ~c.c¢~. cupwmu. mnemmo. «mecca. wouwwmcc. u": 1 no J<¢cubz~ weiwno—uo hm mvsooOo moormo. mewowo o¢~0n00 ochmoOo hmoFOOo omnmcwo accomoo cummFOo nmownOo ~oco~mo mnwoumo smoQN—o «acme:- hmiwncwmo MI» w— ztnaou pm<4 wt» ~¢sm00. «momma. comooo. oe~oooo caoooo. manhoo- awomom. ouooxo. oqgwso. mammmo. soprano wauhsc. soo~o~. uzO¢uo uao : wxb mud wzzaqou cu hm¢~u uzhlil n xumaaz h~tIum I» .chuc—ouuuq APPENDIX I COMPUTER PROGRAM FOR PHYSICOCHEMICAL ABSORPTION USING TYPE ll MODEL APPENDIX I COMPUTER PROGRAM FOR PHYSICOCHEMICAL ABSORPTION USING TYPE II MODEL The boundary value problem (8.1), (8.2), and (8.3), along with the model equation (6.18), is solved using an IMXL subroutine DGEAR with an implicit multistep Gear's algorithm. However, the model equation (6.18), when expressed in the form d =f<< 7; ) is singular at x1==0. To avoid the singularity problem, the numerical integration begins at x1==Ax13>O, an arbitrarily small distance away from the interface with the new boundary conditions al .- 15X C A + de x'ga I) die) __ J“) g dxi T 77 I“? AX, . Two initial concentration gradients at x1==0 are guessed: one using data of Shaw and Hanratty [l977a] with no chemical reaction, another using one-half the first value. The boundary value problem is then solved to a preset value of x1==DELTA. The third initial concen- tration gradient is calculated using the linear combination of the first 202 203 two guesses, so that the concentration at x1==DELTA differs from zero by a preset number. The concentration gradient at x1==DELTA is now checked to see if its value differs from zero by a preset number. If it is larger than that number, DELTA is doubled and the same procedure is repeated over again until both concentration and its gradients are smal- ler than the preset tolerances. The coefficient £11 depends on an integral similar to fn(p) UT Eq.(7.l3). The same algorithm used in Appendix H is also used to calcu- late £11. o12.53.28 OQIIQIBZ FTN 5.16552 :1 OPT TQIITS PROGRAM STANTON NCINPUToOUTPUTT UKCQPTOKC'KCO TAUHOTAUHQSCOKQKFLAG 5 THAT MEMORY LENGTH IS NONZEPOo o O I! U UN H lbw DIG-III 2 ‘32 up cm: 2 did "-1 6> U (m U" .10 0 IL! 2 XI.) 3 52‘ U h."- Cbzo I514»: 21 30—A< '0- >10 0 ..JU 0': Juan (ILUII o-u-ooo 2 I: (we: r; ¢u>zz u0c3nmm4 co «0 0d ¢e~ up uUJut uuohc oz7uz OhlhthdCVquvHM~OOJ uzomuzzwhomc4n2~~n~44~hbo~ «ZUXOHUflUkk‘OKKJCZHu0h~0~ u—~¢ u.uNueuucuonuuzzoucmxoum<~mu~-~¢0u¢>~— 0 Nu Cnrfl'? c DX>>>>Q~hItt~nDxu—Quaum>ux>>>n w—mm>>mo~°o~r .11 F. u I-CIIHCU- hat—~20 m IIhCC‘AADUu. O. o o ficxHflOO— on. I (V) coo-we (“JO-c 3 Jun02>>ou1<wzco 0 rue 0 cm at 5:0. hat-(t- v-C_I Z OZN I IO" N «I utUIt—Kd om «cm—oomuuuv and!» uteri-(CD ZNmCIII>Od> IIOINIIII ...... omziicv- a:uv—o>uc¢wu II II III—(squIIarxb—cu 04—09 a o C O 0 0 026: 0.] 2‘" 0‘ N CD 00-. '1‘ .1 (042-1 .1 0 Ch- 0- any, ‘2 C tit->— UO .0- K u-n-I CV) 04$. x» cm zpoz NU 00-! (m D 1‘ CLU 2H DU 0 I 2090-— KACUF- OZv-o‘ I o. ugh: udOHAUI-JD ZZVO wuzurt— Om. («UDMDOM H: C 02> ”DICK .Jl-‘C X 00" 12'. .12 OOH. beach-Duo 2t- ~10200¢0¢2 lnF-ICUKZO-N I9 w(:o~:o_lo.l~ H.1(ODO-UUUCM mat-cc Dun—- ~UOO¢UUII 1— zen-czzxzo-II 04 o. t—I— ¢ 9 nor”: o:o—< m \COmMO-O-md 0.00.0000........OOOOIOQOQI....QOOOOOOOOOQOOOO 204 OYQXENDQTOLOMETHOMITERQINDEXQIUKoUKoIERT a- D 2 H X 9 .- d J O .0 O 2 IOU O '0‘! In X U .0“ ‘0 5b.: DD- 7 C) c. UO. UO ILNO _l 0.42 \ o 2 U ‘0 v 0:: cu EZICSAVEI-SAVCZTT'YINIOCSAVEI/CSAVFI-SAVEflTTOYTN? O \D O p. O U Q N- I U 0 CI. ‘5 . 2|- inc-j CX> o Noon- ZXXO N C9 NDK 2'14" ’- II TOO-0% U, H II um Cl oE-ZT GO TO 70 2) GO TO 80 "A55 TRANSFER COEFFICIENT WITHOUT CHEMICAL REACTION. “bus-alga." 19“ II II II b06161.- " H «om-sou I .- UFF-F-Z—d-l—CO— out-n-2(lt quIu2¢QJ~2hO—ath¢ .---7h‘h'szdLIb-‘bfi-5" ZJUHZHV>=JdHNW~ ~>~=z> vJ vow u..J 0272w‘hjz’oc xu. ILU Ohfl-‘OICO-u-IHCOC‘ O a In D O O '31: o>>0hu~>mcnw n c c In 0 N II II OOIIIIUIIC ll 0| CIIOIIIIIC UU UU UUUUUUUUUU C"KCO “NntIDCDNCU‘O—ONIOCWOKDU‘DHNWOU’I{NTU'Ov‘NFIcl! \I‘NQU‘OF‘NI’)‘(“DP-QU‘O-ON'OCIPCFCDO‘CH-‘N'TQIDONCO'DNN'TQID‘DN c-I“anoint—"II.-NOTNNNNNNNNnnnnnnannCCOCOQC 'CClfilfilI-mlfiifid‘IDIDWQOOQOOI‘QQGFNNP‘FNNP“ 012.53.28 OIIIQIOZ FIN 501.552 :1 191175 OPT PROGRAH STANTON NOU SET TO 2 AND THE CALCULATION UITH CHEMICAL REACTION IS STARTED II THEN KoKCoUCoE ASS TRANSFER COEFFICIENT UITH CHEMICAL REACTION. E0. 0 00 II‘ 0".- I U22“? U~M~OUXJO GRADIENT AT THE UALL3'0EIIo Y 10 AND REPEAT THE CALCULATION. oGTo I0.) GO TO I D I ~1UQUJI§¥U~¥U~QUQUD¥N0QK IL.- D QU‘OWINWQIDQNQO‘ DF‘NVICII'HDNQU‘OI‘NWCWVDNQU‘O-INI'). FNQQQDQCGIWEU‘U‘U‘U‘ U‘U‘U‘C‘U‘ U‘DOOOOOOOOOn—Iddfi dundnudfidHH—ddd If I I U 0 2055 U U m m I I a- o o .x K Inln In an I O O m- «In In on on o o u 0 mo 0 now. on .- u.— XU H OH «5 o o I o vI I III 0)" II UN 01: I U m: D 0 Id 1 I I Oh-I- o» C‘I I IDI- I.) I I 00-. ox a: D2 KW In nu"- ID 0 O I um. Om ID A> oz 0 I I IDI-I u-I I00 0 Grill II on on OZUU u-uu Id can-1:: .4 o o IIJIL HI I .2 o 0 III II I HI I XI I II II o o In: D UUIDID UC ( D: I 0 NO- I— FIFO 0. an. O on“ In OI o I I UU oxox I o I o «Imam IDIDI I u .0 o 0 all II thDIn d—OUU o o_J o woo-IX! UOUO NU 00.42.. I o o o hu—LJAI I I I O I II II II 0 o NI mt ouumm III—IIOXX O o 00-! all Inc-1 Iguana—nom— oddQ—IOOUU 0') .I .J o .0. N I IDOII I-I OI on H II II I\X\X¥¥x¥ I I In. In d O". I I IDI IDI I I I 1' OVZUtL-JII'YI CNCC’ 'C OGCZCD “8412‘ IL IICHDU IIILCJZZUHILOKOQOGIOOI OI COCO? “OIL-IOU.“ "I‘LIIILILILILU O 6 6 I.) DO 0 0° C OIDOID c-n ON If) to on Nmnn u-n a «no N NNNN .12.53o28 OAIIQ/82 FTN 5.16552 :1 TAIITS OPT SUDPOUTINE FCN A U m C A It a .— u-a C. O I n v \ a A If) n U x >- V I on U CI. :0 ..J I \ A U U x 0‘ (h v 0.1 I I UH. 0‘ I! I-x '0 Q (0 V II ..I! )- u :0 O O UU 0 0 «All; x A Id" U N tut III I! a; ..- a... c-a > N (rut—1 o a - QQI—J I a > > O. a x I 00182)! - x b»: cum: on 9 O CAI—n d a xmr-Zh-u-n o u o. O- J I U ZUHUA\ x U UCdZ—A c I ZO‘I-I out U aha- U \z DU m '63 LUIQO» U 30. IhD-UI I >>- UI-I—IOII II II" 2 n.- A A AA HM\20c-n u-I Our) I-u g(u U we =22>JU uuwwz OHO II. ’HIZK «mt—IriflU—v I-II-t: OLI’ECVKWKDZCFD SOOUILQ JQ 2mau7 V9 UU~>U>U>>¢ZU C-FCN “Nine WOFQOO'ONVO won—"4 04/14/82 .12.53.2a rrn 5.1.552 OPI:I 70/175 SUBROUTINE FCNJ FCN. HOSCIKOKFLAG w‘o-n Oahu fifinxuu 2 ‘0 o-a UU\- 0.! IL-IIIIL :E’C!‘- t-I.J C-IIJ 0 In t- .— EIEI/IO-‘JE: DIIIXI‘SC/LIIIXI IDIIXI ~U\VD II II II II II II C-FCN «NntlnsnbOOOuNncmOFCU‘o-awmc «uduuuu—uuNNNNN a an -0" O- aux X)! -—o (DC DC .0 CU UU um VII/I mo 90 XX ox ow av A: 3"! xx ex um an O I ID! I II II II II II II ------ --- F‘NnleflkC-‘Nlflz o o o o o oo— o o ’a' cuthuuunnnU—a—unnn NNFJD 001(vavvvquwvvvvouva-C‘ 3 OUILDDODDDJDODOOD?DDOUZ mace—aaaaanumaamaauamazm O O 0 DC)" II II II 206 04/14/82 }I2053o28 FTN 5.10552 =1 OPT 74/175 FUNCTION DIl IOTAUHQTAUHQSCQKQKFLAG I'IX'03I0ERFISORTIIKOIo/TAUHIOTAUHII/(SORTIKOIo/TAUHIOSCI out x< v\ at and Dr- III ?\ O ~28“? I-O (a UZJ II D ’t‘dI-C DOM—«.12 ILUKDKU «aw-acme .12o53o28 04/14/82 FIN 501.552 :1 OPT 74/175 207 FUNCTION LII ¢ N o n m o N a o N C \ c I" \ Q '-v c A I D 4 .- x o M N 9 m x In v 9 h " K 0 c If) If) U 2 \ I- A I-L . ‘ x “ D ' d "‘ o— . o z A ' x ‘ a C o ‘ ‘ r- 2 ~ . \ 4 ‘ a 0 IL “ II. u-I x 0 x o 0 U D . 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