{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Mass transfer\n", "\n", "* Combustion involves many chemical species.\n", "* A rigorous treatment of mass transfer is often important.\n", "\n", "## Velocity\n", "* If we could see a molecular view of a flow, we would not see a \"velocity,\" we would see many molecules, each with their own velocity. \n", "* If we consider the average velocity of a particular species, we can define average velocities\n", " * Mass average\n", " * Mole average\n", " * Volume average\n", "\n", "### Mass and mole flux\n", "* Total mass flux for a mass average velocity\n", "$$\\dot{m}^{\\prime\\prime} = \\rho v$$\n", "* Species $A$\n", "$$\\dot{m}^{\\prime\\prime}_A = \\rho y_A v_A$$\n", "* Total molar flux for a molar average velocity\n", "$$\\dot{n}^{\\prime\\prime} = c v$$\n", "* Species $A$\n", "$$\\dot{n}^{\\prime\\prime}_A = c x_A v_A$$\n", "\n", "### Diffusion velocity\n", "$$v_A = v_A + v-v$$\n", "$$v_A = v + \\underbrace{(v_A - v)}_{v_A^{D}}$$\n", "* Mass flux\n", "$$\\dot{m}^{\\prime\\prime}_A = \\rho y_Av_A = \\rho y_Av + \\rho y_Av_A^{D}$$\n", "\n", "\n", "$$\\dot{m}^{\\prime\\prime}_A = y_A\\dot{m}^{\\prime\\prime} + j_A$$\n", "\n", "\n", "* This last expression is (Bulk flux of A) + (Diffusion flux of A)\n", "* $j_A$ is the diffusion flux.\n", "* Now, add all species:\n", "$$\\dot{m}^{\\prime\\prime} = \\sum_i\\dot{m}^{\\prime\\prime}_i = \\sum_iy_i\\dot{m}^{\\prime\\prime}+\\sum_ij_i = \\dot{m}^{\\prime\\prime}\\sum_iy_i + \\sum_ij_i = \\dot{m}^{\\prime\\prime} + \\sum_ij_i $$\n", "\n", "\n", "$$\\rightarrow \\sum_ij_i = 0$$\n", "\n", "\n", "* The diffusion flux is a difference from an average flux. So, the sum of all diffusion fluxes should be zero.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Multicomponent mass transfer\n", "\n", "* Model the $j_i$ as gradients of species. \n", "* Using a mole basis is more physical.\n", "* $J_i$ is the molar flux of species $i$\n", "* Fluxes depend on mole fraction gradients.\n", "\n", "#### Two species 1, 2\n", "$$J_1 = -cD\\nabla x_1,$$\n", "$$J_2 = -cD\\nabla x_2.$$\n", "\n", "#### Three species 1, 2, 3\n", "$$J_1 = -cD_{1,1}\\nabla x_1 - cD_{1,2}\\nabla x_2,$$\n", "$$J_2 = -cD_{2,1}\\nabla x_1 - cD_{2,2}\\nabla x_2.$$\n", "\n", "* The flux of one species depends on its own gradient **and** on the gradient of **other** species.\n", " * Other species can *drag* on each other.\n", " * Species can diffuse **up** their gradients!\n", "* **Question** why do we not include $\\nabla x_3$ in $J_1$ above?\n", "\n", "### Matrix form\n", "\n", "\n", "$$\\mathbf{J} = -c[D]\\mathbf{\\nabla x},\\phantom{xxxxxx}\\text{size = }n_s-1$$\n", "\n", "\n", "* $[D]$ is a diffusion matrix defined as follows\n", "\n", "\n", "\n", "$$[D] = [B]^{-1}$$\n", "\n", "$$B_{i,j} = -x_i\\left(\\frac{1}{\\mathcal{D}_{i,j}} - \\frac{1}{\\mathcal{D}_{i,n_s}}\\right),\\phantom{xxx} i\\ne j$$\n", "\n", "$$B_{i,i} = \\frac{x_i}{\\mathcal{D}_{i,n}} + \\sum_{k=1,i\\ne k}^{n_s} \\frac{x_k}{\\mathcal{D}_{i,k}}$$\n", "\n", "\n", "* $\\mathcal{D}_{i,j}$ are binary diffusion coefficients.\n", " * Available in tables. See Turns Appendix D.\n", " * Cantera can provide these also.\n", " * $\\mathcal{D}_{i,j} \\propto T^{3/2}/P$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Fick's law\n", "\n", "* Beware, there are various forms of \"Fick's\" law.\n", "* We will use the \"best\" one\n", "\n", "\n", "$$J_i = -cD_{i,e}\\nabla x_i$$\n", "\n", "\n", "* $D_{i,e}$ is an *effective diffusivity*\n", "* To get mass flux, multiply by $M_i$\n", "\n", "$$j_i = M_iJ_i = -M_icD_{i,e}\\nabla x_i$$\n", "\n", "* Use $c=\\rho/M$, and $x_i = y_iM/M_i$\n", "\n", "$$ j_i = -\\frac{M_i\\rho}{M}D_{i,e}\\frac{M}{M_i}\\nabla y_i - \\frac{M_i\\rho}{M}D_{i,e}\\frac{y_i}{M_i}\\nabla M$$\n", "\n", "\n", "$$j_i = -\\rho D_{i,e}\\nabla y_i - \\left(\\rho D_{i,e}y_i\\frac{\\nabla M}{M}\\right)$$\n", "\n", "\n", "* The second term, in parentheses is often ignored. \n", " * Not a great assumption for combustion.\n", " * Often used in combustion models for turbulent flow.\n", " \n", "**Note**: $J_i$ is relative to a *molar* average velocity. So, our conversion to $j_i$ and the use of a *mass* average velocity is not fully consistent.\n", "\n", "**Note**: In combustion, we often have lots of $N_2$ and using Fick's law instead of a full multicomponent treatment is not that bad.\n", "\n", "\n", "$$D_{i,e} = \\left[\\frac{(1-x_i)}{\\sum_{j=1,j\\ne i}^n \\frac{x_j}{\\mathcal{D}_{i,j}}}\\right]$$\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following illustrates the importance of using the \"full\" Fick's law form given above.\n", "This is from [Pitsch and Peters (1998)](https://www.sciencedirect.com/science/article/pii/S0010218097002782).\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Heat, Mass, Momentum\n", "\n", "* Mass transfer: \n", "$$\\rho y_i \\rightarrow j_i = -\\rho D\\nabla y_i$$\n", "* Heat transfer: \n", "$$\\rho h \\rightarrow q = -\\lambda \\nabla T$$\n", "* Momentum transfer: \n", "$$\\rho u \\rightarrow \\tau = -\\mu \\nabla v$$\n", "\n", "Also,\n", "$$ y_i \\rightarrow -D\\nabla y_i$$\n", "$$ T \\rightarrow -\\alpha\\nabla T$$\n", "$$ v \\rightarrow -\\nu\\nabla v$$\n", "\n", "* $\\alpha = \\lambda/(\\rho c_p)$\n", "* $\\nu = \\mu/\\rho$\n", "* $D$, $\\alpha$, $\\nu$ all have units of $m/s^2$\n", "\n", "* For constant properties, the unsteady diffusion equation for some scalar $\\eta$, with diffusivity $\\Gamma$ is \n", "$$\\frac{\\partial \\eta}{\\partial t} = \\Gamma \\frac{\\partial^2\\eta}{\\partial x^2}$$\n", "\n", "### Dimensionless numbers\n", "\n", "* Lewis number\n", "\n", "$$ Le = \\frac{\\alpha}{D}$$\n", "\n", "* Schmidt number\n", "\n", "$$ Sc = \\frac{\\nu}{D}$$\n", "\n", "* Pr number\n", "\n", "$$ Pr = \\frac{\\nu}{\\alpha}$$\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## One-dimensional species transport equation\n", "\n", "$$\\frac{\\partial \\rho y_i}{\\partial t} + \\frac{\\partial}{\\partial x}(\\rho y_i v_i) = S_i$$\n", "\n", "$$v_i = v + v_i^D$$\n", "$$\\rho y_i v_i^D = j_i$$\n", "\n", "\n", "$$\\frac{\\partial \\rho y_i}{\\partial t} + \\frac{\\partial}{\\partial x}(\\rho y_i v) + \\frac{\\partial j_i}{\\partial x} = S_i$$\n", "\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "hide_input": false, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 2 }