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Chemical Engineering Module Model Library © COPYRIGHT 1994–2007 by COMSOL AB All rights reserved Patent pending The software described in this document is furnished under a license agreement The software may be used or copied only under the terms of the license agreement No part of this manual may be photocopied or reproduced in any form without prior written consent from COMSOL AB COMSOL, COMSOL Multiphysics, COMSOL Reaction Engineering Lab, and FEMLAB are registered trademarks of COMSOL AB COMSOL Script is a trademark of COMSOL AB Other product or brand names are trademarks or registered trademarks of their respective holders Version: October 2007 COMSOL 3.4 C O N T E N T S Chapter 1: Introduction Model Library Overview Model Library Guide Typographical Conventions C h a p t e r : F l u i d F l ow — Tu t o r i a l s a n d B e n c h m a r k s Pressure Recovery in a Diverging Duct Introduction 10 10 Model Definition 10 Results 11 Modeling Using the Graphical User Interface 13 Flow in a Fuel Cell Stack Introduction 18 18 Model Definition 18 Results 21 Modeling Using the Graphical User Interface 23 Non-Newtonian Flow Introduction 30 30 Model Definition 30 Results 32 Reference 34 Modeling Using the Graphical User Interface 34 Flow Between Two Parallel Plates Introduction 39 39 Model Definition 40 Results 41 Modeling Using the Graphical User Interface 42 CONTENTS |i Variations in Density in Porous Media Flow Introduction 46 46 Model Definition 46 Results 48 Modeling in COMSOL Multiphysics 49 Modeling Using the Graphical User Interface 49 Coupled Free and Porous Media Flow Introduction 57 57 Model Definition 58 Results 60 Modeling Using the Graphical User Interface 61 Turbulent Flow Through a Bending Pipe Introduction 65 65 Model Definition 65 Results and Discussion 67 Reference 69 Modeling Using the Graphical User Interface 69 Oscillating Liquid Cobalt Droplet Introduction 75 75 Model Definition 75 Results and Discussion 78 References 81 Modeling in COMSOL Multiphysics 81 Modeling Using the Graphical User Interface 82 Turbulent Flow over a Backward-Facing Step Introduction 87 87 Model Definition 87 Results 90 Reference 92 Modeling Using the Graphical User Interface 93 ii | C O N T E N T S C p ter 3: Hea t Tra ns fer a nd Non - Is ot her ma l F low — Tut or i a l s a nd B e n c h m a r k s Forced Turbulent Convection Cooling of a Hot Plate Introduction 98 98 Model Definition 98 Results 101 References 106 Modeling in COMSOL Multiphysics 107 Modeling Using the Graphical User Interface 107 Heating of a Finite Slab 114 Introduction 114 Model Definition 114 Results 115 Reference 115 Modeling Using the Graphical User Interface 116 A 3D Model of a MEMS Heat Exchanger 119 Introduction 119 Model Definition 119 Results 121 Reference 123 Modeling Using the Graphical User Interface 123 Non-Isothermal Flow in a Cooling Process 129 Introduction 129 Model Definition 129 Results 131 Modeling Using the Graphical User Interface 133 MEMS Heat Exchanger 142 Introduction 142 Model Definition 142 Results 144 Reference 145 CONTENTS | iii Modeling Using the Graphical User Interface 145 C h a p t e r : M u l t i c o m p o n e n t Tr a n s p o r t — Tu t o r i a l s and Benchmarks Separation through Dialysis 152 Introduction 152 Model Definition 153 Results 158 Modeling in COMSOL Multiphysics 159 References 159 Modeling Using the Graphical User Interface 160 Stefan Tube 164 Introduction 164 Model Definition 164 Results 166 References 168 Modeling Using the Graphical User Interface 169 Maxwell-Stefan Diffusion in a Fuel Cell Unit Cell 173 Introduction 173 Model Definition 173 Results 176 References 179 Modeling Using the Graphical User Interface 180 C h a p t e r : M i xe r s a n d S t i r r e d Ve s se l s iv | C O N T E N T S Laminar Static Mixer 186 Introduction 186 Model Definition 186 Results 187 References 190 Modeling Using the Graphical User Interface 191 Residence Time in a Turbulent Reactor 197 Introduction 197 Model Definition 198 Results 200 Reference 205 Modeling Using the Graphical User Interface—2D Turbulent Reactor 205 Modeling Using the Graphical User Interface—3D Turbulent Reactor 211 Swirl Flow Around a Rotating Disk 216 Introduction 216 Model Definition 216 Results 218 Reference 221 Modeling Using the Graphical User Interface 222 Modeling of Rotating Parts 227 Introduction 227 Model Definition 228 Results 229 Modeling Using the Graphical User Interface 230 Chapter 6: Multiphase Flow Rigid Body Motion 236 Introduction 236 Model Definition 236 Results 237 Modeling Using the Graphical User Interface 238 Rising Bubble Modeled with the Level Set Method 243 Introduction 243 Model Definition 244 Results and Discussion 246 Modeling in COMSOL Multiphysics 248 CONTENTS |v Modeling Using the Graphical User Interface 249 Boiling Flow 255 Model Definition 255 Results and Discussion 259 Modeling in COMSOL Multiphysics 260 References 261 Modeling Using the Graphical User Interface 261 Droplet Breakup in a T-Junction 270 Introduction 270 Model Definition 270 Results and Discussion 272 Reference 273 Modeling Using the Graphical User Interface 273 Bubble Column Reactor 279 Introduction 279 Model Definition 279 Results and Discussion 283 Modeling in COMSOL Multiphysics 286 Modeling Using the Graphical User Interface 287 Contaminant-Removal from Wastewater in a Secondary Clarifier 294 Introduction vi | C O N T E N T S 294 Model Definition 294 Results and Discussion 298 Modeling in COMSOL Multiphysics 299 Modeling Using the Graphical User Interface 300 Two-Phase Flow Modeling of a Dense Suspension 305 Introduction 305 Model Definition 305 Results 309 References 311 Modeling in COMSOL Multiphysics 312 Modeling Using the Graphical User Interface—Case 312 Modeling Using the Graphical User Interface—Case 316 Chapter 7: Microfluidics Electrokinetic Flow in a DNA Chip 320 Introduction 320 Model Definition 321 Results 324 References 328 Modeling Using the Graphical User Interface 329 Filling of a Capillary Channel 334 Introduction 334 Model Definition 334 Results and Discussion 338 Modeling in COMSOL Multiphysics 342 Modeling Using the Graphical User Interface 342 Transport in an Electrokinetic Valve 348 Introduction 348 Model Definition 349 Results and Discussion 353 Reference 356 Modeling in COMSOL Multiphysics 356 Modeling Using the Graphical User Interface 356 Transport in an Electrokinetic Valve, 3D Model 365 Introduction 365 Model Definition 365 Results and Discussion 370 Modeling in COMSOL Multiphysics 374 Reference 376 Modeling Using the Graphical User Interface 376 Electroosmotic Flow in Porous Media 389 Introduction 389 CONTENTS | vii Model Definition 389 Results 391 References 394 Modeling in COMSOL Multiphysics 395 Modeling Using the Graphical User Interface 395 Microchannel Cell 403 Introduction 403 Model Definition 404 Results 407 Modeling Using the Graphical User Interface 410 C p ter 8: Tra ns p or t, Reacti on s , a n d React ion Engineering viii | C O N T E N T S Fixed-Bed Reactor for Catalytic Hydrocarbon Oxidation 418 Introduction 418 Model Definition 418 Results and Discussion 421 Modeling in COMSOL Multiphysics 424 References 425 Modeling Using the Graphical User Interface 425 Absorption in a Falling Film 433 Introduction 433 Model Definition 434 Results 438 Modeling in COMSOL Multiphysics 439 References 440 Modeling Using the Graphical User Interface 440 Boat Reactor for Low Pressure Chemical Vapor Deposition 447 Introduction 447 Model Definition 448 Results 451 OPTIONS AND SETTINGS In addition to the constants defined in Model 1, add the following ones: NAME EXPRESSION DESCRIPTION A_i 2.01e-6 Time-scaling coefficient, injector A_t 4.91e-8 Time-scaling coefficient, tube A_c 7.54e-6 Time-scaling coefficient, column D1 1.0e-9[m^2] Diffusion coefficient, component D2 5.0e-10[m^2] Diffusion coefficient, component V 1.67e-8[m/s] Flow velocity GEOMETRY MODELING Click the Draw Mode button on the Main toolbar Select the existing line by clicking on it Click the Move button on the Draw toolbar In the Displacement edit field, type 0.11 Click OK Specify three more lines with the following properties: LINE COORDINATES L2 0.01 L3 0.01 0.11 L4 0.23 0.33 PHYSICS SETTINGS Subdomain Settings To simplify the subdomain settings, first define three variables on the subdomain level From the Options menu, select Expressions>Subdomain Expressions Define the variables A, n1, and n2 in the different subdomains as follows: VARIABLE SUBDOMAIN SUBDOMAINS 2, SUBDOMAIN A A_i A_t A_c n1 0 n01*K1*c1/(1+K1*c1) n2 0 n02*K2*c2/(1+K2*c2) Click OK From the Physics menu, select Subdomain Settings 652 | CHAPTER 10: ELECTROPHORESIS AND CHROMATOGRAPHY Enter subdomain settings for species according to the following table: SETTINGS SUBDOMAINS 1, 2, SUBDOMAIN δts A A*(1+Phi*dn1_dc1) D A*D1 A*D_eff1 u V V Artificial stabilization for species is needed in Subdomains and From the Subdomain selection list, select Subdomains and Click the Artificial Diffusion button Select the Streamline diffusion check box, then select Anisotropic diffusion, and type 0.05 in the Tuning parameter edit field Click OK Enter subdomain settings for species according to the following table: SETTINGS SUBDOMAINS 1, 2, SUBDOMAIN δts A A*(1+Phi*dn2_dc2) D D2*A D_eff2*A u V V Artificial stabilization for species is needed in all subdomains 10 Select all four subdomains and click the Artificial Diffusion button 11 Select the Streamline diffusion check box Select Anisotropic diffusion and type 0.2 in the Tuning parameter edit field Click OK 12 The initial values in Model are zero for both species in Subdomains 2, 3, and In Subdomain 1, set c1(t0) to c01 and c2(t0) to c02 13 Click OK Boundary Conditions Use the same boundary conditions as in Model 1, that is, zero concentrations at the inlet (Boundary 1) and convective flux conditions at the outlet (Boundary 5) for both species Mesh Generation From the Mesh menu, choose Free Mesh Parameters On the Global page click the Custom mesh size button and enter 5e-3 in the Maximum element size edit field Go to the Subdomain page LIQUID CHROMATOGRAPHY | 653 Select Subdomain and type 1e-4 in the Maximum element size edit field Go to the Boundary page Select Boundaries 1, 2, 3, and Type 1e-5 in the Maximum element size edit field Click Remesh When the mesher has finished, click OK COMPUTING THE SOLUTION Keep the solver settings as in Model but change the Times to 0:0.05:1.95 2:450 in the Time stepping frame Compute the solution by clicking the Solve button on the Main toolbar POSTPROCESSING AND VISUALIZATION You create Figure 10-12 on page 644 in the same way as you did Figure 10-10 on page 643, but his time so for output times of s, 80 s, and 160 s and in Subdomain To create Figure 10-13 on page 645, follow these steps: From the Postprocessing menu, select Domain Plot Parameters From the Solutions to use list, select Click the Line/Extrusion tab From the Subdomain selection list, select Subdomains and In the y-axis data area, type c2 in the Expression edit field From the x-axis data list, select x Click the Line Settings button From the Line style list, select Solid line Click OK Click Apply to generate the graph of c2 On the General page, select the Keep current plot check box 10 Return to the Line/Extrusion page In the y-axis data area, type Phi*n2 in the Expression edit field 11 Click the Line settings button Change Line style to Dashed line, then click OK 12 Click Apply to generate the graph of Φn2 13 Finally, type c2+Phi*n2 in the Expression edit field for the y-axis data 14 Click the Line settings button Change Line style to Dash-dot line, then click OK 15 Click OK to plot c2 + Φn2 and close the dialog box 16 In the figure window, use the Zoom Window tool to zoom in near the point connecting Subdomains and To reproduce the plots in Figure 10-14 on page 645, follow these steps: 654 | CHAPTER 10: ELECTROPHORESIS AND CHROMATOGRAPHY From the Postprocessing menu, select Domain Plot Parameters Select all the output times in the Solutions to use list Click the Point tab From the Boundary selection list, select Boundary 5 In the y-axis data area, type c1 in the Expression edit field Click the Line Settings button Click the Color button In the Line color dialog box, select black Click OK From the Line style list, select Solid line Click OK Back in the Domain Plot Parameters dialog box, click Apply 10 On the General page, select the Keep current plot check box 11 Return to the Point page and type c2 in the Expression edit field 12 Click the Line Settings button 13 From the Line style list, select Dashed line Click OK 14 Click OK to finalize the plot and close the Domain Plot Parameters dialog box Model Library path: Chemical_Engineering_Module/ Electrophoresis_and_Chromatography/liquid_chromatography_3 Modeling Using the Graphical User Interface—Model Start COMSOL Multiphysics In the Model Navigator select Axial symmetry (2D) from the Space dimension list, then click the Multiphysics button From the Application Modes list, select Chemical Engineering Module> Mass Transport>Convection and Diffusion>Transient analysis Click Add From the Application Modes list, select Chemical Engineering Module> Momentum Transport>Laminar Flow>Incompressible Navier-Stokes> Steady-state analysis From the Element list, select P2P-1, then click Add From the Application Modes list, select Chemical Engineering Module> Energy Transport>Convection and Conduction>Steady-state analysis Click Add Click OK LIQUID CHROMATOGRAPHY | 655 GEOMETRY MODELING Press the Shift key and click the Rectangle/Square button on the Draw toolbar In the Size area of the Rectangle dialog box, set the Width to 1.05e-3 and the Height to 5e-2 In the Position area, set r to and z to -2.5e-2 Click OK to create the rectangle R1 Repeat this procedure to create five more rectangles with the following properties: SETTINGS R2 R3 R4 R5 R6 Width 1.05e-3 1.05e-3 1e-4 1e-4 2.5e-3 Height 1e-3 1e-3 2.4e-2 2.4e-2 8e-2 r 0 0 z 2.5e-2 -2.6e-2 2.6e-2 -5e-2 -4e-2 Double-click the EQUAL button in the status bar at the bottom of the user interface Click the Zoom Extents button on the Main toolbar Click in the drawing area and then press Ctrl+A to select all six rectangles Ctrl-click rectangle R6 to remove it from the selection Press Ctrl+C to copy the rectangles R1 through R5, then press Ctrl+V In the Paste dialog box, click OK to paste the copies, R7 through R11, without displacement From the Draw menu, select Create Composite Object In the Object selection list, select R6, R7, R8, R9, R10, and R11 10 In the Set formula edit field, edit the expression to read R6-(R7+R8+R9+R10+R11) 11 Click OK to create the composite object CO1 The geometry-modeling stage is now complete, and the result in the drawing area on your screen should look as in the figure below 656 | CHAPTER 10: ELECTROPHORESIS AND CHROMATOGRAPHY OPTIONS AND SETTINGS From the Options menu, open the Constants dialog box Enter the following constants; when done, click OK NAME EXPRESSION DESCRIPTION rho 1000[kg/m^3] Fluid density dp_bed 1.7[um] Particle size for packed bed dp_frit 3.5[um] Particle size for frits void_bed 0.4 Bed porosity void_frit 0.4 Frit porosity p_in 54.5[MPa] Inlet pressure p_out 0[Pa] Outlet pressure T_in 303[K] Inlet temperature c0 1[mol/m^3] Inlet concentration peak maximum, c0*exp(-a0*(t-t0)^2) t0 2[s] Peak center time a0 5[1/s^2] Peak shape parameter Cp 2.9[kJ/(kg*K)] Heat capacity of fluid k_f 0.25[W/(m*K)] Thermal conductivity of fluid k_e 0.5[W/(m*K)] Thermal conductivity of bed and frit k_s 15[W/(m*K)] Thermal conductivity of column walls eta0 1e-3[Pa*s] Viscosity of fluid at T=293K D_mol0 1e-9[m^2/s] Molecular diffusion coefficient at T=293K z_in -5[cm] Inlet position z_out 5[cm] Outlet position From the Options menu, point to Expressions and select Subdomain Expressions Enter the following expressions; when done, click OK SUBDOMAINS NAME EXPRESSION 1–5 D_mol D_mol0*exp(0.02[1/K]*(T-293[K]))*T/(293[K]) 1–5 eta eta0*exp(-0.02[1/K]*(T-293[K])) 2–4 u_lin U_chns/epsilonp_chns 2–4 H_eff 1.5*dp+D_mol/u_lin+dp^2*u_lin/(6*D_mol) 2–4 D_eff 0.5*H_eff*u_lin LIQUID CHROMATOGRAPHY | 657 SUBDOMAINS NAME EXPRESSION 2–4 epsilon dp^2*epsilonp_chns^3/ (180*(1-epsilonp_chns)^2) 2, dp dp_frit dp dp_bed PHYSICS SETTINGS Subdomain Settings—Convection and Diffusion From the Multiphysics menu, select Convection and Diffusion (chcd) From the Physics menu, select Subdomain Settings Select Subdomain 6, then clear the Active in this domain check box Select Subdomains 1–5 Set the r-velocity to u and the z-velocity to v Select Subdomains and In the D (isotropic) edit field, type D_mol Select Subdomains 2–4 In the D (isotropic) edit field type D_eff*epsilonp_chns, and in the δts edit field type epsilonp_chns Click OK Subdomain Settings—Incompressible Navier Stokes From the Multiphysics menu, select Incompressible Navier-Stokes (chns) From the Physics menu, select Subdomain Settings Select Subdomain 6, then clear the Active in this domain check box Select Subdomains 1–5 In the ρ edit field type rho, and in the η edit field type eta Click the Artificial Diffusion button Clear the Streamline diffusion check box, then click OK Select Subdomains 2–4 Select the Flow in porous media (Brinkman equations) check box, then type epsilon in the κ edit field Select Subdomains and In the εp edit field, type void_frit Select Subdomain In the εp edit field, type void_bed Click OK Subdomain Settings—Convection and Conduction From the Multiphysics menu, select Convection and Conduction (chcc) From the Physics menu, select Subdomain Settings On the Physics page, select Subdomains 1–5 658 | CHAPTER 10: ELECTROPHORESIS AND CHROMATOGRAPHY In the ρ edit field, type rho; in the Cp edit field, type Cp; and in the u edit fields, type u and v, respectively Select Subdomains and In the k (isotropic) edit field, type k_f Select Subdomains 2–4 In the k (isotropic) edit field, type k_e Select Subdomain In the k (isotropic) edit field, type k_s Select Subdomain In the Q edit field, type -v*pz Click the Init tab 10 Select all subdomains In the Initial value edit field, type T_in-(z-z_in)*(p_out-p_in)/(z_out-z_in)/(rho*Cp) 11 Click OK Boundary Conditions—Convection and Diffusion From the Multiphysics menu, select Convection and Diffusion (chcd) From the Physics menu, select Boundary Settings Enter the following boundary settings: SETTINGS BOUNDARIES 1, 3, 5, 7, BOUNDARY BOUNDARY 11 Boundary condition Axial symmetry Concentration Convective flux c0*exp(-a0*(t-t0)^2) c0 For all other boundaries, the default boundary condition (insulation/symmetry) applies Click OK Boundary Conditions—Incompressible Navier Stokes From the Multiphysics menu, select Incompressible Navier-Stokes (chns) From the Physics menu, select Boundary Settings Enter the following boundary settings: SETTINGS BOUNDARIES 1, 3, 5, 7, BOUNDARY BOUNDARY 11 Boundary type Symmetry boundary Stress Stress Boundary condition Axial symmetry Normal stress, normal flow Normal stress p_in p_out f0 For all other boundaries the default settings apply (wall/no slip) Click OK LIQUID CHROMATOGRAPHY | 659 Boundary Conditions—Convection and Conduction From the Multiphysics menu, select Convection and Conduction (chcc) From the Physics menu, select Boundary Settings Enter the following boundary settings: SETTINGS BOUNDARIES 1, 3, 5, 7, BOUNDARY BOUNDARY 11 Boundary condition Axial symmetry Temperature Convective flux T_in T0 For all other boundaries, the default boundary condition (thermal insulation) applies Click OK MESH GENERATION From the Mesh menu, choose Free Mesh Parameters On the Subdomain page, enter the following settings: SETTING SUBDOMAINS 1, SUBDOMAINS 2, SUBDOMAIN SUBDOMAIN Maximum element size 2.5e-5 1e-4 2e-4 3e-3 Click the Boundary page Select Boundaries and 10, then set the Maximum element size to 5e-6 Click the Advanced tab In the z-direction scale factor edit field, type 0.5 Click the Remesh button Click OK COMPUTING THE SOLUTION First, compute the steady-state velocity and temperature fields Then, using these fields as an initial value, compute the transient solution for the analyte concentration Record the entire procedure in a Solver Manager script Click the Solver Parameters button on the Main toolbar From the Analysis list, select Stationary Click OK to close the Solver Parameters dialog box Click the Solver Manager button on the Main toolbar On the Initial Value page, go to the Initial value area and click the Initial value expression option button 660 | CHAPTER 10: ELECTROPHORESIS AND CHROMATOGRAPHY On the Solve For page, select Incompressible Navier-Stokes (chns) only On the Output page, select Incompressible Navier-Stokes (chns) and Convection and Conduction (chcc) Click Apply On the Script page, click the Add Current Solver Settings button 10 On the Initial Value page, click the Current solution option button in the Initial value area 11 On the Solve For page, select Incompressible Navier-Stokes (chns) and Convection and Conduction (chcc) 12 Click Apply 13 On the Script page, click the Add Current Solver Settings button 14 Click OK Before completing the solver-script generation, set up the visualization of the time-dependent solution From the Postprocessing menu, choose Probe Plot Parameters Click New For the Plot type choose Integration, and for the Domain type choose Boundary In the Plot name edit field, type Inlet, then click OK to close the New Probe Plot dialog box From the Boundary selection list, select In the Expression edit field, type 2*pi*(v*c-Dzz_c_chcd*cz)*r Click the Title/Axis button Click the option button next to the Title edit field, then enter the title Axial flux of analyte Click OK to close the Title/Axis Settings dialog box Select the Plot all plots in the same axis check box 10 Click Apply 11 Repeat Steps 2–10 to generate a second plot with the Plot name set to Outlet and with Boundary 11 selected from the Boundary selection list Continue with the transient solution set-up Click the Solver Parameters button on the Main toolbar Change the Analysis back to Transient LIQUID CHROMATOGRAPHY | 661 In the Time stepping area, type 0:1:9 in the Times edit field On the Time Stepping page, select the Manual tuning of step size check box Set the Initial time step to 1e-4 and the Maximum time step to 1e-2 Click OK to close the Solver Parameters dialog box Click the Solver Manager button on the Main toolbar On the Solve For page, select Convection and Diffusion (chcd) only On Output page, select all three application modes Click Apply 10 On the Script page, click the Add Current Solver Settings button 11 Select the Solve using a script check box The solver script is now complete and ready to run 12 Click OK to close the Solver Manager, then click the Solve button on the Main toolbar to compute the solution POSTPROCESSING AND VISUALIZATION The default plot is the concentration-peak evolution plot shown in Figure 10-16 To plot the temperature field, follow these steps: Click the Plot Parameters button on the Main toolbar Click the Surface page On the Surface Data page, type T in the Expression edit field Click OK Double-click the EQUAL button on the status bar at the bottom of the user interface, then click the Zoom Extents button on the Main toolbar to toggle the axis-scale status To reproduce Figure 10-15, follow these steps: From the Postprocessing menu, select Domain Plot Parameters From the Solutions to use list, select Click the Surface tab Select all subdomains In the Expression edit field, type T Click OK to generate the plot and close the dialog box 662 | CHAPTER 10: ELECTROPHORESIS AND CHROMATOGRAPHY I N D E X A absorption process 433 chemical activation energy 463 reactions 435 active layer 581 species 435 agglomerate model 582 chemical vapor deposition 447 anisotropic porous medium 448 CHEMKIN transport data file 475 annular flow 434 chlor-alkali membrane cell 530 anode 433 concentration application mode B overvoltage 590 convection and diffusion 560 conductivity 533, 544 Darcy’s law 46, 585 conjugate heat transfer 98 Navier-Stokes 40, 58, 145, 454 conservation of mass, checking 84 Arrhenius law 419, 463 contact angle 334, 337 boat reactor 447 convection and conduction 143 boundary condition convection and diffusion application mode 560 no-slip 39, 144, 449 convective flow 461, 462 boundary conditions slip/symmetry 39 conversions 422 straight-out 39 corner smoothing 67 breakup of droplets 270 current density 530, 553 Brinkman equations 57 current distribution 531 CVD 447 C capillary channel 334 capillary forces 334, 541 D Darcy’s law application mode 46, 585 carbonate 433 dialysis 152 Carreau viscosity model 30 diaphragm process 530 catalyst diffuse double layer 320 agglomerate 543 diffusion boundary layer 433 layer 461 diffusivity 450 Dirac delta function 76 catalytic combustion 461 discontinuities 159 converter 461 DNA Chip 320 purification of emissions 461 droplet 75 catalytic process 418 droplet breakup 270 cathode 433, 541 dynamic viscosity 11, 40, 58 caustic solution 433 charge balance 543 E effective diffusion coefficient 463, 543 effective exchange current density 543 INDEX| 663 electric energy 580 hydroxide ion 437 electrode reactions 554 hypochlorite 438 electrolysis 530, 552 hypochlorous acid 438 electron transfer reaction 532, 541, 543 I electroneutrality 533 ideal gas 46 immiscible fluids 243 electroosmotic flow 320 incompressible Navier-Stokes 143 electrophoretic flow 321 interdigitated flow 580 element internal convection 530 lagrange-cubic 441 ion conducting electrolyte 541 equilibrium constant 435 exchange current density 534, 554 K kinetic energy 10 explicit streamline diffusion 149 parameters 435 F falling film 433 kinetic gas theory 174 Faraday’s constant 533 Faraday’s law 533 L laminar flow 39 Fick’s law 405 laminar static mixer 186 fixed bed reactor 418 level set method 243 flux 164 mass-conserving 75 continuous 159 liquid phase sintering 75 flux of species 436 low pressure CVD 447 free convection 434 frictional losses 13 G laminar film 434 M mass balance 435 fuel cell 173, 541, 602 mass transport 164, 447 fuel cell, 3D 580 Maxwell-Stefan diffusion 173, 174 gas absorption 433 gas constant 463 gas diffusion electrodes 581 gas evolution 532 MCFC 541 mean concentration 466 medium water 143 membrane cell 530 H Hagen-Poiseuille law 462 profile 10 heat capacity 120, 143 source 120 Henry constant 437 hydrogen 433 hydrogen ion 437 664 | I N D E X meniscus 338 mercury cell process 530 microchips 447 migration 366, 532 molecular diffusion 461 molten carbonate fuel cell 541 momentum transport 164, 447 monolith channels 462 multitube fixed-bed reactor 418 N Navier-Stokes R cylindrical coordinates 10 reaction kinetics 418 Navier-Stokes application mode 40, 58, reaction rate 463 realizability constraint 101 145, 454 O Navier-Stokes equations 10, 186 reattachment 87 with surface tension 76, 335 recirculation 87 Nernst-Planck 351, 368 residence time 418 Nernst-Planck equation 368 Reynolds number 10, 39, 66 no-slip boundary condition 39, 144, 449 Reynolds-averaged Navier-Stokes 98 rising bubble 243 Ohm’s law for electrolytes 533 ohmic losses 535 P RANS 98 S scaled problem 154 oil bubble 243 scaled variables 463 overpotential 545 secondary current distribution 531 overvoltage 541 selective catalytic reduction 461 model 470 parabolic velocity profile 143, 462 separation 87 partial pressure 437 separation process 152 PDE coefficients 116 shear rate 30 PEMFC 580, 602 shear-thinning fluid 33 permeability of silane 448 porous media 47, 58 silicon 447 permeate 152 slip/symmetry boundary condition 39 phthalic anhydride reactor 418 standard electrode potential 554 plug flow 418 static mixer 186 polystyrene 30 stiff-spring method 159 porosity 582, 605 straight-out boundary condition 39 porous electrode 541 streamline diffusion porous media 46 explicit 149 permeability of 47 subdomain integration 469 porous media flow 57 supporting electrolyte 367 potential distribution 531, 546 surface tension 334 pressure drop 13 T Tafel equation 175 energy 10 thermal conductivity 120 recovery 10 transfer coefficient 534 proton exchange membrane fuel cell 580, 602 tubular reactor 418 turbulent flow 87 proton transport 581 turbulent viscosity 99 PVC 530 twisted blade mixer 186 INDEX| 665 two-phase flow 75 typographical conventions U unit cell 531 V variables scaled 463 velocity distribution 11 viscosity 143 turbulent 99 viscous layer 41 viscous losses 13 void fraction 510 volatile organic compound 461 W wafer bundle 450 wall adhesion 334 washcoat 461, 462 wetted wall 337 Y 666 | I N D E X Young’s law 334 ... flow and reaction -engineering applications, including coupling the Chemical Engineering Module to the COMSOL Reaction Engineering Lab® These models illustrate various chemical -engineering- specific... the Chemical Engineering Module | CHAPTER 1: INTRODUCTION documentation set Titled the Chemical Engineering Module User’s Guide, it introduces you to the basic functionality in the module, covers... summarizes key information about the entries in the Chemical Engineering Module Model Library as well as the Chemical Engineering Module User’s Guide The solution time is the elapsed time measured