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| 1 | // -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- | ||
| 2 | // vi: set et ts=4 sw=4 sts=4: | ||
| 3 | // | ||
| 4 | // SPDX-FileCopyrightInfo: Copyright © DuMux Project contributors, see AUTHORS.md in root folder | ||
| 5 | // SPDX-License-Identifier: GPL-3.0-or-later | ||
| 6 | // | ||
| 7 | /*! | ||
| 8 | * \file | ||
| 9 | * \ingroup Core | ||
| 10 | * \brief Collection of functions, calculating dimensionless numbers. | ||
| 11 | * | ||
| 12 | * All the input to the dimensionless numbers has to be provided as function arguments. | ||
| 13 | * Rendering this collection generic in the sense that it can be used by any model. | ||
| 14 | */ | ||
| 15 | #ifndef DUMUX_COMMON_DIMENSIONLESS_NUMBERS_HH | ||
| 16 | #define DUMUX_COMMON_DIMENSIONLESS_NUMBERS_HH | ||
| 17 | |||
| 18 | #include <cmath> | ||
| 19 | #include <iostream> | ||
| 20 | |||
| 21 | #include <dune/common/exceptions.hh> | ||
| 22 | #include <dune/common/math.hh> | ||
| 23 | |||
| 24 | namespace Dumux { | ||
| 25 | |||
| 26 | /*! | ||
| 27 | * \brief A container for possible values of the property for selecting which Nusselt parametrization to choose. | ||
| 28 | * The actual value is set via the property NusseltFormulation | ||
| 29 | */ | ||
| 30 | enum class NusseltFormulation | ||
| 31 | { | ||
| 32 | dittusBoelter, WakaoKaguei, VDI | ||
| 33 | }; | ||
| 34 | |||
| 35 | /*! | ||
| 36 | * \brief A container for possible values of the property for selecting which Sherwood parametrization to choose. | ||
| 37 | * The actual value is set via the property SherwoodFormulation | ||
| 38 | */ | ||
| 39 | enum class SherwoodFormulation | ||
| 40 | { | ||
| 41 | WakaoKaguei | ||
| 42 | }; | ||
| 43 | |||
| 44 | /*! | ||
| 45 | * \ingroup Core | ||
| 46 | * \brief Collection of functions which calculate dimensionless numbers. | ||
| 47 | * Each number has it's own function. | ||
| 48 | * All the parameters for the calculation have to be handed over. | ||
| 49 | * Rendering this collection generic in the sense that it can be used by any model. | ||
| 50 | */ | ||
| 51 | template <class Scalar> | ||
| 52 | class DimensionlessNumbers | ||
| 53 | { | ||
| 54 | |||
| 55 | public: | ||
| 56 | /*! | ||
| 57 | * \brief Calculate the Reynolds Number [-] (Re). | ||
| 58 | * | ||
| 59 | * The Reynolds number is a measure for the relation of inertial to viscous forces. | ||
| 60 | * The bigger the value, the more important inertial (as compared to viscous) effects become. | ||
| 61 | * According to Bear [Dynamics of fluids in porous media (1972)] Darcy's law is valid for Re<1. | ||
| 62 | * | ||
| 63 | * Source for Reynolds number definition: http://en.wikipedia.org/wiki/Reynolds_number | ||
| 64 | * | ||
| 65 | * \param darcyMagVelocity The absolute value of the darcy velocity. In the context of box models this | ||
| 66 | * leads to a problem: the velocities are defined on the faces while other things (storage, sources, output) | ||
| 67 | * are defined for the volume/vertex. Therefore, some sort of decision needs to be made which velocity to put | ||
| 68 | * into this function (e.g.: face-area weighted average). [m/s] | ||
| 69 | * \param charcteristicLength Typically, in the context of porous media flow, the mean grain size is taken as the characteristic length | ||
| 70 | * for calculation of Re. [m] | ||
| 71 | * \param kinematicViscosity Is defined as the dynamic (or absolute) viscos ity divided by the density. | ||
| 72 | * http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity. [m^2/s] | ||
| 73 | * | ||
| 74 | * \return The Reynolds Number as calculated from the input parameters | ||
| 75 | */ | ||
| 76 | static Scalar reynoldsNumber(const Scalar darcyMagVelocity, | ||
| 77 | const Scalar charcteristicLength, | ||
| 78 | const Scalar kinematicViscosity) | ||
| 79 | { | ||
| 80 |
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35701513 | return darcyMagVelocity * charcteristicLength / kinematicViscosity ; |
| 81 | } | ||
| 82 | |||
| 83 | /*! | ||
| 84 | * \brief Calculate the Prandtl Number [-] (Pr). | ||
| 85 | * | ||
| 86 | * The Prandtl Number is a measure for the relation of viscosity and thermal diffusivity (temperaturleitfaehigkeit). | ||
| 87 | * | ||
| 88 | * It is defined as | ||
| 89 | * \f[ | ||
| 90 | * \textnormal{Pr}= \frac{\nu}{\alpha} = \frac{c_p \mu}{\lambda}\, , | ||
| 91 | * \f] | ||
| 92 | * with kinematic viscosity\f$\nu\f$, thermal diffusivity \f$\alpha\f$, heat capacity \f$c_p\f$, | ||
| 93 | * dynamic viscosity \f$\mu\f$ and thermal conductivity \f$\lambda\f$. | ||
| 94 | * Therefore, Pr is a material specific property (i.e.: not a function of flow directly | ||
| 95 | * but only of temperature, pressure and fluid). | ||
| 96 | * | ||
| 97 | * source for Prandtl number definition: http://en.wikipedia.org/wiki/Prandtl_number | ||
| 98 | * | ||
| 99 | * \param dynamicViscosity Dynamic (absolute) viscosity over density. | ||
| 100 | * http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity [m^2/s] | ||
| 101 | * \param heatCapacity Heat capacity at constant pressure. | ||
| 102 | * Specifies the energy change for a given temperature change [J / (kg K)] | ||
| 103 | * \param thermalConductivity Conductivity to heat. Specifies how well matter transfers energy without moving. [W/(m K)] | ||
| 104 | * \return The Prandtl Number as calculated from the input parameters. | ||
| 105 | */ | ||
| 106 | static Scalar prandtlNumber(const Scalar dynamicViscosity, | ||
| 107 | const Scalar heatCapacity, | ||
| 108 | const Scalar thermalConductivity) | ||
| 109 | { | ||
| 110 | 2964303 | return dynamicViscosity * heatCapacity / thermalConductivity; | |
| 111 | } | ||
| 112 | |||
| 113 | /*! | ||
| 114 | * \brief Calculate the Nusselt Number [-] (Nu). | ||
| 115 | * | ||
| 116 | * The Nusselt Number is a measure for the relation of convective- to conductive heat exchange. | ||
| 117 | * | ||
| 118 | * The Nusselt number is defined as Nu = h d / k, | ||
| 119 | * with h= heat transfer coefficient, d=characteristic length, k=heat conductivity(stagnant). | ||
| 120 | * However, the heat transfer coefficient from one phase to another is typically not known. | ||
| 121 | * Therefore, Nusselt numbers are usually given as *empirical* Nu(Reynolds, Prandtl) for a given flow | ||
| 122 | * field --forced convection-- and *empirical* Nu(Rayleigh, Prandtl) for flow caused by temperature | ||
| 123 | * differences --free convection--. The fluid characteristics enter via the Prandtl number. | ||
| 124 | * | ||
| 125 | * This function implements an *empirical* correlation for the case of porous media flow | ||
| 126 | * (packed bed flow as the chemical engineers call it). | ||
| 127 | * | ||
| 128 | * source for Nusselt number definition: http://en.wikipedia.org/wiki/Nusselt_number | ||
| 129 | * source for further empirical correlations for Nusselt Numbers: | ||
| 130 | * VDI-Gesellschaft, VDI-Waermeatlas, VDI-Verlag Duesseldorf, 2006 | ||
| 131 | * | ||
| 132 | * \param reynoldsNumber Dimensionless number relating inertial and viscous forces [-]. | ||
| 133 | * \param prandtlNumber Dimensionless number relating viscosity and thermal diffusivity (temperaturleitfaehigkeit) [-]. | ||
| 134 | * \param porosity The fraction of the porous medium which is void space. | ||
| 135 | * \param formulation Switch for deciding which parametrization of the Nusselt number is to be used. | ||
| 136 | * Set via the property NusseltFormulation. | ||
| 137 | * \return The Nusselt number as calculated from the input parameters [-]. | ||
| 138 | */ | ||
| 139 | 2964303 | static Scalar nusseltNumberForced(const Scalar reynoldsNumber, | |
| 140 | const Scalar prandtlNumber, | ||
| 141 | const Scalar porosity, | ||
| 142 | NusseltFormulation formulation) | ||
| 143 | { | ||
| 144 |
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2964303 | if (formulation == NusseltFormulation::dittusBoelter){ |
| 145 | /* example: very common and simple case: flow straight circular pipe, only convection (no boiling), | ||
| 146 | * 10000<Re<120000, 0.7<Pr<120, far from pipe entrance, smooth surface of pipe ... | ||
| 147 | * Dittus, F.W and Boelter, L.M.K, Heat Transfer in Automobile Radiators of the Tubular Type, | ||
| 148 | * Publications in Engineering, Vol. 2, pages 443-461, 1930 | ||
| 149 | */ | ||
| 150 | using std::pow; | ||
| 151 | ✗ | return 0.023 * pow(reynoldsNumber, 0.8) * pow(prandtlNumber,0.33); | |
| 152 | } | ||
| 153 | |||
| 154 |
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2964303 | else if (formulation == NusseltFormulation::WakaoKaguei){ |
| 155 | /* example: flow through porous medium *single phase*, fit to many different data | ||
| 156 | * Wakao and Kaguei, Heat and mass Transfer in Packed Beds, Gordon and Breach Science Publishers, page 293 | ||
| 157 | */ | ||
| 158 | using std::pow; | ||
| 159 | 2964303 | return 2. + 1.1 * pow(prandtlNumber,(1./3.)) * pow(reynoldsNumber, 0.6); | |
| 160 | } | ||
| 161 | |||
| 162 | ✗ | else if (formulation == NusseltFormulation::VDI){ | |
| 163 | /* example: VDI Waermeatlas 10. Auflage 2006, flow in packed beds, page Gj1, see also other sources and limitations therein. | ||
| 164 | * valid for 0.1<Re<10000, 0.6<Pr/Sc<10000, packed beds of perfect spheres. | ||
| 165 | * | ||
| 166 | */ | ||
| 167 | using std::sqrt; | ||
| 168 | using std::pow; | ||
| 169 | using Dune::power; | ||
| 170 | ✗ | Scalar numerator = 0.037 * pow(reynoldsNumber,0.8) * prandtlNumber ; | |
| 171 | ✗ | Scalar reToMin01 = pow(reynoldsNumber,-0.1); | |
| 172 | ✗ | Scalar prTo23 = pow(prandtlNumber, (2./3. ) ) ; // MIND THE pts! :-( otherwise the integer exponent version is chosen | |
| 173 | ✗ | Scalar denominator = 1+ 2.443 * reToMin01 * (prTo23 -1.) ; | |
| 174 | |||
| 175 | ✗ | Scalar nusseltTurbular = numerator / denominator; | |
| 176 | ✗ | Scalar nusseltLaminar = 0.664 * sqrt(reynoldsNumber) * pow(prandtlNumber, (1./3.) ); | |
| 177 | ✗ | Scalar nusseltSingleSphere = 2 + sqrt( power(nusseltLaminar,2) + power(nusseltTurbular,2)); | |
| 178 | |||
| 179 | ✗ | Scalar funckyFactor = 1 + 1.5 * (1.-porosity); // for spheres of same size | |
| 180 | ✗ | Scalar nusseltNumber = funckyFactor * nusseltSingleSphere ; | |
| 181 | |||
| 182 | ✗ | return nusseltNumber; | |
| 183 | } | ||
| 184 | |||
| 185 | else { | ||
| 186 | ✗ | DUNE_THROW(Dune::NotImplemented, "wrong index"); | |
| 187 | } | ||
| 188 | } | ||
| 189 | |||
| 190 | |||
| 191 | /*! | ||
| 192 | * \brief Calculate the Schmidt Number [-] (Sc). | ||
| 193 | * | ||
| 194 | * The Schmidt Number is a measure for the relation of viscosity and mass diffusivity. | ||
| 195 | * | ||
| 196 | * It is defined as | ||
| 197 | * \f[ | ||
| 198 | * \textnormal{Sc}= \frac{\nu}{D} = \frac{\mu}{\rho D}\, , | ||
| 199 | * \f] | ||
| 200 | * with kinematic viscosity\f$\nu\f$, diffusion coefficient \f$D\f$, dynamic viscosity | ||
| 201 | * \f$\mu\f$ and mass density\f$\rho\f$. Therefore, Sc is a material specific property | ||
| 202 | * (i.e.: not a function of flow directly but only of temperature, pressure and fluid). | ||
| 203 | * | ||
| 204 | * source for Schmidt number definition: http://en.wikipedia.org/wiki/Schmidt_number | ||
| 205 | * | ||
| 206 | * \param dynamicViscosity Dynamic (absolute) viscosity over density. | ||
| 207 | * http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity [m^2/s] | ||
| 208 | * \param massDensity Mass density of the considered phase. [kg / m^3] | ||
| 209 | * \param diffusionCoefficient Measure for how well a component can move through a phase due to a concentration gradient. [m^2/s] | ||
| 210 | * \return The Schmidt Number as calculated from the input parameters. | ||
| 211 | */ | ||
| 212 | static Scalar schmidtNumber(const Scalar dynamicViscosity, | ||
| 213 | const Scalar massDensity, | ||
| 214 | const Scalar diffusionCoefficient) | ||
| 215 | { | ||
| 216 | 2361016 | return dynamicViscosity / (massDensity * diffusionCoefficient); | |
| 217 | } | ||
| 218 | |||
| 219 | /*! | ||
| 220 | * \brief Calculate the Sherwood Number [-] (Sh). | ||
| 221 | * | ||
| 222 | * The Sherwood Number is a measure for the relation of convective- to diffusive mass exchange. | ||
| 223 | * | ||
| 224 | * The Sherwood number is defined as Sh = K L/D, | ||
| 225 | * with K= mass transfer coefficient, L=characteristic length, D=mass diffusivity (stagnant). | ||
| 226 | * | ||
| 227 | * However, the mass transfer coefficient from one phase to another is typically not known. | ||
| 228 | * Therefore, Sherwood numbers are usually given as *empirical* Sh(Reynolds, Schmidt) for a given flow | ||
| 229 | * field (and fluid). | ||
| 230 | * | ||
| 231 | * Often, even the Sherwood number is not known. By means of the Chilton-Colburn analogy it can be deduced | ||
| 232 | * from the Nusselt number. According to the Chilton-Colburn analogy in a known Nusselt correltion one | ||
| 233 | * basically replaces Pr with Sc and Nu with Sh. For some very special cases this is actually accurate. | ||
| 234 | * (Source: Course Notes, Waerme- und Stoffuebertragung, Prof. Hans Hasse, Uni Stuttgart) | ||
| 235 | * | ||
| 236 | * This function implements an *empirical* correlation for the case of porous media flow | ||
| 237 | * (packed bed flow as the chemical engineers call it). | ||
| 238 | * | ||
| 239 | * source for Sherwood number definition: http://en.wikipedia.org/wiki/Sherwood_number | ||
| 240 | * | ||
| 241 | * \param schmidtNumber Dimensionless number relating viscosity and mass diffusivity [-]. | ||
| 242 | * \param reynoldsNumber Dimensionless number relating inertial and viscous forces [-]. | ||
| 243 | * \param formulation Switch for deciding which parametrization of the Sherwood number is to be used. | ||
| 244 | * Set via the property SherwoodFormulation. | ||
| 245 | * \return The Nusselt number as calculated from the input parameters [-]. | ||
| 246 | */ | ||
| 247 | |||
| 248 | 2361016 | static Scalar sherwoodNumber(const Scalar reynoldsNumber, | |
| 249 | const Scalar schmidtNumber, | ||
| 250 | SherwoodFormulation formulation) | ||
| 251 | { | ||
| 252 |
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2361016 | if (formulation == SherwoodFormulation::WakaoKaguei){ |
| 253 | /* example: flow through porous medium *single phase* | ||
| 254 | * Wakao and Kaguei, Heat and mass Transfer in Packed Beds, Gordon and Breach Science Publishers, page 156 | ||
| 255 | */ | ||
| 256 | using std::cbrt; | ||
| 257 | using std::pow; | ||
| 258 | 2361016 | return 2. + 1.1 * cbrt(schmidtNumber) * pow(reynoldsNumber, 0.6); | |
| 259 | } | ||
| 260 | |||
| 261 | else { | ||
| 262 | ✗ | DUNE_THROW(Dune::NotImplemented, "wrong index"); | |
| 263 | } | ||
| 264 | } | ||
| 265 | |||
| 266 | |||
| 267 | /*! | ||
| 268 | * \brief Calculate the thermal diffusivity alpha [m^2/s]. | ||
| 269 | * | ||
| 270 | * The thermal diffusivity is a measure for how fast "temperature (not heat!) spreads". | ||
| 271 | * It is defined as \f$\alpha = \frac{k}{\rho c_p}\f$ | ||
| 272 | * with \f$\alpha\f$: \f$k\f$: thermal conductivity [W/mK], \f$\rho\f$: density [kg/m^3], | ||
| 273 | * \f$c_p\f$: cpecific heat capacity at constant pressure [J/kgK]. | ||
| 274 | * | ||
| 275 | * Source for thermal diffusivity definition: http://en.wikipedia.org/wiki/Thermal_diffusivity | ||
| 276 | * | ||
| 277 | * \param thermalConductivity A material property defining how well heat is transported via conduction [W/(mK)]. | ||
| 278 | * \param phaseDensity The density of the phase for which the thermal diffusivity is to be calculated [kg/m^3]. | ||
| 279 | * \param heatCapacity A measure for how a much a material changes temperature for a given change of energy (at p=const.) [J/(kgm^3)]. | ||
| 280 | * \return The thermal diffusivity as calculated from the input parameters [m^2/s]. | ||
| 281 | */ | ||
| 282 | static Scalar thermalDiffusivity(const Scalar & thermalConductivity , | ||
| 283 | const Scalar & phaseDensity , | ||
| 284 | const Scalar & heatCapacity) | ||
| 285 | { | ||
| 286 | return thermalConductivity / (phaseDensity * heatCapacity); | ||
| 287 | } | ||
| 288 | |||
| 289 | }; | ||
| 290 | |||
| 291 | } // end namespace Dumux | ||
| 292 | |||
| 293 | #endif | ||
| 294 |