1 | #*--------------------------------------------------------------------- |
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2 | * EMSO Model Library (EML) Copyright (C) 2004 - 2007 ALSOC. |
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3 | * |
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4 | * This LIBRARY is free software; you can distribute it and/or modify |
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5 | * it under the therms of the ALSOC FREE LICENSE as available at |
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6 | * http://www.enq.ufrgs.br/alsoc. |
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7 | * |
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8 | * EMSO Copyright (C) 2004 - 2007 ALSOC, original code |
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9 | * from http://www.rps.eng.br Copyright (C) 2002-2004. |
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10 | * All rights reserved. |
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11 | * |
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12 | * EMSO is distributed under the therms of the ALSOC LICENSE as |
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13 | * available at http://www.enq.ufrgs.br/alsoc. |
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14 | * |
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15 | *---------------------------------------------------------------------- |
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16 | * 7. Diffusion with chemical reaction in a one dimensional slab |
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17 | *---------------------------------------------------------------------- |
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18 | * |
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19 | * Description: |
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20 | * This problem is part of a collection of 10 representative |
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21 | * problems in Chemical Engineering for solution by numerical methods |
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22 | * developed for Cutlip (1998). |
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23 | * |
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24 | * Subject: |
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25 | * * Transport Phenomena |
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26 | * * Reaction Engineering |
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27 | * |
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28 | * Concepts utilized: |
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29 | * Methods for solving second order ODEs with 2 point boundary |
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30 | * values typically used in transport phenomena and reaction kinetics. |
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31 | * |
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32 | * Numerical method: |
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33 | * * Simultaneous ODEs with split boundary conditions |
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34 | * * Resolved by finite difference method |
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35 | * |
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36 | * Reference: |
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37 | * * CUTLIP et al. A collection of 10 numerical problems in |
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38 | * chemical engineering solved by various mathematical software |
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39 | * packages. Comp. Appl. in Eng. Education. v. 6, 169-180, 1998. |
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40 | * * More informations and a detailed description of all problems |
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41 | * is available online in http://www.polymath-software.com/ASEE |
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42 | * |
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43 | *---------------------------------------------------------------------- |
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44 | * Author: Rodolfo Rodrigues |
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45 | * GIMSCOP/UFRGS - Group of Integration, Modeling, Simulation, |
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46 | * Control, and Optimization of Processes |
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47 | * $Id$ |
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48 | *--------------------------------------------------------------------*# |
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49 | using "types"; |
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50 | |
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51 | |
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52 | |
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53 | Model problem |
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54 | PARAMETERS |
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55 | outer N as Integer (Brief="Number of discrete points", Lower=3); |
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56 | |
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57 | Co as conc_mol (Brief="Constant concentration at the surface"); |
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58 | D as diffusivity (Brief="Binary diffusion coefficient"); |
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59 | k as Real (Brief="Homogeneous reaction rate constant", Unit='1/s'); |
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60 | L as length (Brief="Bottom surface"); |
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61 | |
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62 | |
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63 | VARIABLES |
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64 | C(N+2) as conc_mol (Brief="Concentration of reactant"); |
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65 | z(N+2) as length (Brief="Distance", Default=1e-3); |
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66 | dz as length_delta (Brief="Distance increment"); |
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67 | |
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68 | |
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69 | EQUATIONS |
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70 | "Discrete interval" |
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71 | dz = (z(N+2) - z(1))/(N+1); |
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72 | |
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73 | for i in [2:(N+1)] do |
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74 | "Concentration of reactant" |
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75 | (C(i+1) - 2*C(i)+ C(i-1))/(z(i) - z(i-1))^2 |
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76 | = (k/D)*C(i); |
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77 | |
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78 | "Discrete length" |
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79 | z(i) = z(i-1) + dz; |
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80 | end |
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81 | |
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82 | # Boundary conditions |
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83 | "Initial and boundary condition" # z = 0 |
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84 | C(1) = Co; |
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85 | |
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86 | "Upper boundary" # z = L |
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87 | (C(N+2) - C(N+1))/(z(N+2) - z(N+1)) = 0*'kmol/m^4'; |
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88 | |
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89 | |
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90 | SET |
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91 | Co= 0.2*'kmol/m^3'; |
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92 | D = 1.2e-9*'m^2/s'; |
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93 | k = 1e-3/'s'; |
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94 | L = 1e-3*'m'; |
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95 | end |
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96 | |
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97 | |
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98 | |
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99 | FlowSheet numerical_solution |
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100 | PARAMETERS |
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101 | N as Integer; |
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102 | |
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103 | |
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104 | DEVICES |
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105 | reac as problem; |
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106 | |
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107 | |
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108 | SET |
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109 | N = 10; # Number of discrete points |
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110 | |
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111 | |
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112 | SPECIFY |
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113 | reac.z(1) = 0*'m'; |
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114 | |
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115 | reac.z(N+2) = reac.L; |
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116 | |
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117 | OPTIONS |
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118 | Dynamic = false; |
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119 | end |
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120 | |
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121 | |
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122 | |
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123 | FlowSheet comparative |
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124 | PARAMETERS |
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125 | N as Integer; |
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126 | |
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127 | |
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128 | VARIABLES |
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129 | C_(N+2) as conc_mol (Brief="Concentration of reactant by analytical solution"); |
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130 | |
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131 | r_ as Real (Brief="Pearson product-moment correlation coefficient"); |
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132 | |
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133 | Cm as conc_mol (Brief="Arithmetic mean of calculated C"); |
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134 | C_m as conc_mol (Brief="Arithmetic mean of analytical C"); |
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135 | |
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136 | |
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137 | DEVICES |
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138 | reac as problem; |
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139 | |
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140 | |
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141 | SET |
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142 | N = 10; # Number of discrete points |
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143 | |
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144 | |
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145 | EQUATIONS |
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146 | "Analytical solution" |
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147 | C_ = reac.Co*cosh(reac.L*sqrt(reac.k/reac.D)*(1 - reac.z/reac.L)) |
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148 | /cosh(reac.L*sqrt(reac.k/reac.D)); |
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149 | |
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150 | "Pearson correlation coefficient" # used by softwares like MS Excel |
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151 | r_ = (sum((reac.C - Cm)*(C_ - C_m)))/sqrt(sum((reac.C - Cm)^2)*sum((C_ - C_m)^2)); |
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152 | |
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153 | "Arithmetic mean of C" |
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154 | Cm = sum(reac.C)/(N+1); |
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155 | |
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156 | "Arithmetic mean of C_" |
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157 | C_m = sum(C_)/(N+1); |
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158 | |
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159 | |
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160 | SPECIFY |
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161 | reac.z(1) = 0*'m'; |
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162 | |
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163 | reac.z(N+2) = reac.L; |
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164 | |
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165 | OPTIONS |
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166 | Dynamic = false; |
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167 | end |
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168 | |
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169 | |
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170 | FlowSheet analytical_solution |
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171 | PARAMETERS |
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172 | Co as conc_mol; |
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173 | L as length; |
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174 | k as Real(Unit='1/s'); |
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175 | D as diffusivity; |
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176 | |
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177 | |
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178 | VARIABLES |
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179 | C as conc_mol (Default=0.2); |
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180 | z as length (Default=1e-3); |
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181 | |
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182 | |
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183 | EQUATIONS |
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184 | "Change time in z" |
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185 | z = time*'m/s'; |
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186 | |
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187 | "Analytical solution" |
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188 | C = Co*cosh(L*sqrt(k/D)*(1 - z/L))/cosh(L*sqrt(k/D)); |
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189 | |
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190 | |
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191 | SET |
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192 | Co= 0.2*'kmol/m^3'; |
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193 | D = 1.2e-9*'m^2/s'; |
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194 | k = 1e-3/'s'; |
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195 | L = 1e-3*'m'; |
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196 | |
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197 | |
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198 | OPTIONS |
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199 | TimeStart = 0; |
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200 | TimeStep = 1e-6; |
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201 | TimeEnd = 1e-3; |
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202 | end |
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