{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Combustion kinetics I\n", "\n", "* Equilibrium is *approached* in combustion, but often kinetics are limiting and required for an accurate representation of temperature and composition.\n", "* Kinetics are important for soot and NOx\n", "* Flame extinction and ignition phenomena\n", "* **Combustion chemistry is complex:** hundreds (thousands) of species and reactions.\n", "\n", "## Elementary reactions\n", "* Bimolecular reaction\n", "$$A + B \\rightarrow C + D$$\n", "\n", "$$\\frac{d[A]}{dt} = -k[A][B]$$\n", "\n", "* In general:\n", "$$ aA + bB \\rightleftharpoons cC + dD$$\n", "$$\\frac{d[A]}{dt} = -ak_f[A]^a[B]^b + ak_r[C]^c[D]^d$$\n", "* For elementary reactions the powers on [A], [B], etc. are integers, and correspond to the coefficients in the reaction.\n", "\n", "\n", "$$k_f = \\alpha T^{\\beta}\\exp(-E_a/RT)$$\n", "\n", "\n", "* $\\alpha$ is a pre-exponential rate factor.\n", "* $\\beta$ is a temperature coefficient.\n", "* $E_a$ is the activation energy.\n", "\n", "* Unimolecular reaction: $A\\rightarrow B$\n", "* Termolecular reaction: $A+B+M\\rightarrow \\text{products}$\n", "* In combustion, we often see species M, which is some generic species.\n", " * Normally occurs in radical reactions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Global reactions\n", "\n", "* Lump many elementary steps into one (or more) overall reaction.\n", "$$\\text{Fuel} + a\\text{Oxidizer} \\rightarrow b\\text{Products}$$\n", "* For methane\n", "$$\\frac{d[CH_4]}{dt} = -k[CH_4]^n[O_2]^m$$\n", "* Generally $n$, and $m$ are not integers.\n", " * Can be greater than or less than 1.\n", "* The reaction rate is **empirical** and only valid within the range of conditions used to fit the data.\n", "* **Use with caution**\n", "* Can be very convenient.\n", "* See Turns Table 5.1\n", "* See also [Westbrook 1981 simplified reaction mechanisms](https://doi.org/10.1080/00102208108946970)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Compact notation\n", "* To do combustion kinetics requires computer codes.\n", "* Need a *framework* to express reactions and rates on a consistent basis.\n", "\n", "### Reactions\n", "* Consider reaction $i$, made up of **all** species $j$\n", "$$\\sum_{j=1}^{N_{sp}} \\nu_{i,j}^\\prime X_j\\rightleftharpoons \\sum_{j=1}^{N_{sp}} \\nu_{i,j}^{\\prime\\prime}X_j$$\n", "* $\\nu_{i,j}^{\\prime}$ is the **reactant** coefficient of species $j$ in reaction $i$.\n", "* $\\nu_{i,j}^{\\prime\\prime}$ is the **product** coefficient of species $j$ in reaction $i$.\n", "* $X_j$ simplly denotes species $j$ in the reaction.\n", "* Most of the $\\nu$ are zero, since most reactions only involve a very few species.\n", "\n", "#### Matrix form\n", "\n", "$$[V^{\\prime}]\\mathbf{X} \\rightleftharpoons [V^{\\prime\\prime}]\\mathbf{X}$$\n", "\n", "\n", "* $[V^{\\prime}]$ has $N_{rxns}$ rows, and $N_{sp}$ columns. $[V^{\\prime\\prime}]$ also.\n", "* $\\mathbf{X}$ is the vector of all species.\n", "\n", "#### Example: Hydrogen:\n", "* Species: $O_2$, $H_2$, $H_2O$, $HO_2$, $O$, $H$, $OH$, $M$\n", "* Reactions:\n", "$$H_2 + O_2 \\rightleftharpoons HO_2 + H$$\n", "$$H + O_2 \\rightleftharpoons OH + O$$\n", "$$OH + H_2 \\rightleftharpoons H_2O + H$$\n", "$$H + O_2 + M \\rightleftharpoons HO_2 + M$$\n", "\n", "$[V^{\\prime}] = $\n", "\n", "\n", "| Rxn | $O_2$| $H_2$| $H_2O$| $HO_2$| $O$| $H$| $OH$| $M$ |\n", "|-----|------|------|-------|-------|----|----|-----|-----|\n", "| 1 |1 |1 | 0 | 0 | 0 |0 |0 |0 |\n", "| 2 |1 | 0 | 0 | 0 | 0 |1 |0 |0 |\n", "| 3 | 0 |1 | 0 | 0 | 0 |0 |1 |0 |\n", "| 4 |1 | 0 | 0 | 0 | 0 |1 |0 |1 |\n", "\n", "$[V^{\\prime\\prime}] = $\n", "\n", "| Rxn | $O_2$| $H_2$| $H_2O$| $HO_2$| $O$| $H$| $OH$| $M$ |\n", "|-----|------|------|-------|-------|----|----|-----|-----|\n", "| 1 | 0 | 0 | 0 |1 | 0 |1 | 0 | 0 |\n", "| 2 | 0 | 0 | 0 | 0 |1 | 0 |1 | 0 |\n", "| 3 | 0 | 0 |1 | 0 | 0 |1 | 0 | 0 |\n", "| 4 | 0 | 0 | 0 |1 | 0 | 0 | 0 |1 |" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Reaction rates\n", "\n", "* Reaction rates are (forward rate) - (reverse rate)\n", "\n", "\n", "$$R_{f,i} = k_{f,i}\\prod_{j=1}^{N_{sp}}[X_j]^{\\nu_{i,j}^{\\prime}}$$\n", "$$R_{r,i} = k_{r,i}\\prod_{j=1}^{N_{sp}}[X_j]^{\\nu_{i,j}^{\\prime\\prime}}$$\n", "\n", "\n", "* For the reaction $$ aA + bB \\rightleftharpoons cC + dD,$$ we have\n", "$$R_f = k_f[A]^a[B]^b[C]^0[D]^0 = k_f[A]^a[B]^b,$$\n", "$$R_r = k_r[A]^0[B]^0[C]^c[D]^d = k_r[C]^c[D]^d.$$\n", "\n", "* The net rate for each reaction $i$ is called the **rate of progress variable**, and is given by\n", "\n", "$$q_i = R_{f,i} - R_{r,i}$$\n", "\n", "\n", "* The net reaction rate for species $j$ is then \n", "$$\\dot{\\omega}_j = \\sum_{i=1}^{N_{rxns}}\\underbrace{(\\nu_{i,j}^{\\prime\\prime}-\\nu_{i,j}^{\\prime})}_{\\nu_{i,j}}q_i$$\n", "Or,\n", "\n", "\n", "$$\\boldsymbol{\\omega} = [V]^T\\mathbf{q},$$\n", "\n", "\n", "where the elements of $[V]$ are $\\nu_{i,j}^{\\prime\\prime}-\\nu_{i,j}^{\\prime}$.\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Reverse reaction rate constants\n", "\n", "* Reaction rate constants are independent of composition.\n", "* Hence, for a reaction at equilibrium, $q_i=R_{f,i}-R_{r,i}=0$. \n", "* This gives $R_{f,i}=R_{r,i}$, or\n", "$$k_{f,i}\\prod_j[X_j]^{\\nu_{i,j}^\\prime} = k_{r,i}\\prod_j[X_j]^{\\nu_{i,j}^{\\prime\\prime}}$$\n", "$$\\frac{k_{f,i}}{k_{r,i}} = \\prod_j[X_j]^{\\nu_{i,j}} = \\prod_j\\left[\\frac{P_i}{RT}\\right]^{\\nu_{i,j}}\n", "= \\prod_j\\left[\\frac{P_i}{P_o}\\right]^{\\nu_{i,j}}\\left[\\frac{P_o}{RT}\\right]^{\\nu_{i,j}}\n", "= K_{eq}\\prod_j\\left[\\frac{P_o}{RT}\\right]^{\\nu_{i,j}} = K_c$$\n", "\n", "\n", "$$\\frac{k_{f,i}}{k_{r,i}} = K_c = K_{eq}\\prod_j\\left[\\frac{P_o}{RT}\\right]^{\\nu_{i,j}}$$\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Quasi-steady-state approximation (QSSA)\n", "\n", "* Fast reacting species, like radical intermediates may be formed and destroyed very fast relative to other species.\n", "* Assume that the fast species are in steady state relative to the slower species.\n", "\n", "### Zeldovich Mechanism\n", "\n", "$$O + N_2 \\rightarrow NO + N\\phantom{xxxxx}\\text{slow}$$\n", "$$N + O_2 \\rightarrow NO + O\\phantom{xxxxx}\\text{fast}$$\n", "\n", "$$\\frac{d[N]}{dt} = k_1[O][N_2] - k_2[N][O_2] = 0$$\n", "Solve this for $[N]$\n", "\n", "\n", "$$[N]_{ss} = \\frac{k_1}{k_2}\\frac{[O][N_2]}{[O_2]}.$$\n", "\n", "\n", "* We have replaced an ODE with an algebraic expression for $[N]$ in terms of other species.\n", "* $[N]_{ss}$ is a function of time though, through the time dependence of $[O]$, $[N_2]$, and $[O_2]$.\n", "\n", "## Partial equilibrium\n", "\n", "* We can also treat fast reactions as if they were in equilibrium (see Turns).\n", "* This again leads to algebraic expressions." ] }, { "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 }