diff --git a/doc/sphinx/conf.py b/doc/sphinx/conf.py index 760679f8ba..2aa47f712a 100644 --- a/doc/sphinx/conf.py +++ b/doc/sphinx/conf.py @@ -225,6 +225,15 @@ def escape_splats(app, what, name, obj, options, lines): myst_enable_extensions = ["dollarmath", "amsmath", "deflist", "colon_fence"] +mathjax3_config = { + 'tex': { + 'macros': { + 't': ['\\mathrm{#1}', 1], + 'pxpy': ['\\frac{\\partial #1}{\\partial #2}', 2] + } + } +} + # Ensure that the primary domain is the Python domain, since we've added the # MATLAB domain with sphinxcontrib.matlab primary_domain = 'py' diff --git a/doc/sphinx/develop/reactor-integration.md b/doc/sphinx/develop/reactor-integration.md index 25898a6204..b04f99fd05 100644 --- a/doc/sphinx/develop/reactor-integration.md +++ b/doc/sphinx/develop/reactor-integration.md @@ -165,7 +165,7 @@ example, using the {py:class}`ExtensibleReactor` class, the governing equations reactor are written in the form: $$ -\mathrm{LHS}_i \frac{dy_i}{dt} = \mathrm{RHS}_i +\t{LHS}_i \frac{dy_i}{dt} = \t{RHS}_i $$ where the {ct}`Reactor::eval` method or the `eval()` method of any class derived from diff --git a/doc/sphinx/reference/kinetics/rate-constants.md b/doc/sphinx/reference/kinetics/rate-constants.md index 6f610f7df2..823dc067a0 100644 --- a/doc/sphinx/reference/kinetics/rate-constants.md +++ b/doc/sphinx/reference/kinetics/rate-constants.md @@ -25,22 +25,22 @@ field. ## Falloff Reactions A falloff reaction is one that has a rate that is first-order in the total concentration -of third-body colliders $\def\MM{[\mathrm{M}]} \MM$ at low pressure, like a -[three-body reaction](sec-three-body-reaction), but becomes zero-order in $\MM$ as $\MM$ -increases. Dissociation/association reactions of polyatomic molecules often exhibit this -behavior. +of third-body colliders $[\t{M}]$ at low pressure, like a +[three-body reaction](sec-three-body-reaction), but becomes zero-order in $[\t{M}]$ as +$[\t{M}]$ increases. Dissociation/association reactions of polyatomic molecules often +exhibit this behavior. The simplest expression for the rate coefficient for a falloff reaction is the Lindemann form {cite:p}`lindemann1922`: -$$ k_f(T, \MM) = \frac{k_0 \MM}{1 + \frac{k_0 \MM}{k_\infty}} $$ +$$ k_f(T, [\t{M}]) = \frac{k_0 [\t{M}]}{1 + \frac{k_0 [\t{M}]}{k_\infty}} $$ -In the low-pressure limit, this approaches $k_0 \MM$, and in the high-pressure limit it -approaches $k_\infty$. +In the low-pressure limit, this approaches $k_0 [\t{M}]$, and in the high-pressure limit +it approaches $k_\infty$. Defining the non-dimensional reduced pressure: -$$ P_r = \frac{k_0 \MM}{k_\infty} $$ +$$ P_r = \frac{k_0 [\t{M}]}{k_\infty} $$ The rate constant may be written as @@ -67,15 +67,15 @@ A falloff reaction may be defined in the YAML format using the A widely-used falloff function is the one proposed by {cite:t}`gilbert1983`: \begin{gather*} -\log_{10} F(T, P_r) = \frac{\log_{10} F_{cent}(T)}{1 + f_1^2} \\ +\log_{10} F(T, P_r) = \frac{\log_{10} F_\t{cent}(T)}{1 + f_1^2} \\ -F_{cent}(T) = (1-A) \exp(-T/T_3) + A \exp (-T/T_1) + \exp(-T_2/T) \\ +F_\t{cent}(T) = (1-A) \exp(-T/T_3) + A \exp (-T/T_1) + \exp(-T_2/T) \\ f_1 = (\log_{10} P_r + C) / (N - 0.14 (\log_{10} P_r + C)) \\ -C = -0.4 - 0.67\; \log_{10} F_{cent} \\ +C = -0.4 - 0.67\; \log_{10} F_\t{cent} \\ -N = 0.75 - 1.27\; \log_{10} F_{cent} +N = 0.75 - 1.27\; \log_{10} F_\t{cent} \end{gather*} ```{admonition} YAML Usage @@ -87,13 +87,13 @@ term of the falloff function is not used. ``` (sec-tsang-falloff)= -### Tsang's Approximation to $F_{cent}$ +### Tsang's Approximation to $F_\t{cent}$ -Wing Tsang presented approximations for the value of $F_{cent}$ for Troe falloff in +Wing Tsang presented approximations for the value of $F_\t{cent}$ for Troe falloff in databases of reactions, for example, {cite:t}`tsang1991`. Tsang's approximations are linear in temperature: -$$ F_{cent} = A + BT $$ +$$ F_\t{cent} = A + BT $$ where $A$ and $B$ are constants. The remaining equations for $C$, $N$, $f_1$, and $F$ from the [Troe](sec-troe-falloff) falloff function are not affected. @@ -132,11 +132,11 @@ An SRI falloff function may be specified in the YAML format using the For these reactions, the rate falls off as the pressure increases, due to collisional stabilization of a reaction intermediate. Example: -$$ \mathrm{Si + SiH_4 (+M) \leftrightarrow Si_2H_2 + H_2 (+M)} $$ +$$ \t{Si + SiH_4 (+M) \leftrightarrow Si_2H_2 + H_2 (+M)} $$ which competes with: -$$ \mathrm{Si + SiH_4 (+M) \leftrightarrow Si_2H_4 (+M)} $$ +$$ \t{Si + SiH_4 (+M) \leftrightarrow Si_2H_4 (+M)} $$ Like falloff reactions, chemically-activated reactions are described by blending between a low-pressure and a high-pressure rate expression. The difference is that the forward @@ -210,15 +210,15 @@ defining the rate, $\phi_n(x)$ is the Chebyshev polynomial of the first kind of $n$ evaluated at $x$, and $$ -\tilde{T} \equiv \frac{2T^{-1} - T_\mathrm{min}^{-1} - T_\mathrm{max}^{-1}} - {T_\mathrm{max}^{-1} - T_\mathrm{min}^{-1}} +\tilde{T} \equiv \frac{2T^{-1} - T_\t{min}^{-1} - T_\t{max}^{-1}} + {T_\t{max}^{-1} - T_\t{min}^{-1}} -\tilde{P} \equiv \frac{2 \log P - \log P_\mathrm{min} - \log P_\mathrm{max}} - {\log P_\mathrm{max} - \log P_\mathrm{min}} +\tilde{P} \equiv \frac{2 \log P - \log P_\t{min} - \log P_\t{max}} + {\log P_\t{max} - \log P_\t{min}} $$ -are reduced temperatures and reduced pressures which map the ranges $(T_\mathrm{min}, -T_\mathrm{max})$ and $(P_\mathrm{min}, P_\mathrm{max})$ to $(-1, 1)$. +are reduced temperatures and reduced pressures which map the ranges $(T_\t{min}, +T_\t{max})$ and $(P_\t{min}, P_\t{max})$ to $(-1, 1)$. A Chebyshev rate expression is specified in terms of the coefficient matrix $\alpha$ and the temperature and pressure ranges. @@ -284,7 +284,7 @@ Blowers Masel reactions can be defined in the YAML format using the Heterogeneous reactions on surfaces are represented by an extended Arrhenius- like rate expression, which combines the modified Arrhenius rate expression with further -corrections dependent on the fractional surface coverages $\theta_{k}$ of one or more +corrections dependent on the fractional surface coverages $\theta_k$ of one or more surface species. The forward rate constant for a reaction of this type is: $$ @@ -329,9 +329,9 @@ for all temperatures. The sticking coefficient is related to the forward rate constant by the formula: -$$ k_f = \frac{\gamma}{\Gamma_\mathrm{tot}^m} \sqrt{\frac{RT}{2 \pi W}} $$ +$$ k_f = \frac{\gamma}{\Gamma_\t{tot}^m} \sqrt{\frac{RT}{2 \pi W}} $$ -where $\Gamma_\mathrm{tot}$ is the total molar site density, $m$ is the sum of all the +where $\Gamma_\t{tot}$ is the total molar site density, $m$ is the sum of all the surface reactant stoichiometric coefficients, and $W$ is the molecular weight of the gas phase species. diff --git a/doc/sphinx/reference/kinetics/reaction-rates.md b/doc/sphinx/reference/kinetics/reaction-rates.md index 29b1ffcf92..6f9a06cfe4 100644 --- a/doc/sphinx/reference/kinetics/reaction-rates.md +++ b/doc/sphinx/reference/kinetics/reaction-rates.md @@ -9,11 +9,11 @@ types. The basic reaction type is a homogeneous reaction with a pressure-independent rate coefficient and mass action kinetics. For example: -$$ \mathrm{A + B \rightleftharpoons C + D} $$ +$$ \t{A + B \rightleftharpoons C + D} $$ The forward reaction rate is then calculated as: -$$ R_f = [\mathrm{A}] [\mathrm{B}] k_f $$ +$$ R_f = [\t{A}] [\t{B}] k_f $$ where $k_f$ is the forward rate constant, calculated using one of the available rate parameterizations such as the [modified Arrhenius](sec-arrhenius-rate) form. @@ -32,30 +32,30 @@ reaction type is [`three-body`](sec-yaml-three-body). A three-body reaction is a gas-phase reaction of the form: -$$ \mathrm{A + B + M \rightleftharpoons AB + M} $$ +$$ \t{A + B + M \rightleftharpoons AB + M} $$ -Here $\mathrm{M}$ is an unspecified collision partner that carries away excess energy to -stabilize the $\mathrm{AB}$ molecule (forward direction) or supplies energy to break the -$\mathrm{AB}$ bond (reverse direction). In addition to the generic collision partner -$\mathrm{M}$, it is also possible to explicitly specify a colliding species. In both +Here $\t{M}$ is an unspecified collision partner that carries away excess energy to +stabilize the $\t{AB}$ molecule (forward direction) or supplies energy to break the +$\t{AB}$ bond (reverse direction). In addition to the generic collision partner +$\t{M}$, it is also possible to explicitly specify a colliding species. In both cases, the reaction type can be automatically inferred by Cantera and does not need to be explicitly specified by the user. Different species may be more or less effective in acting as the collision partner. A -species that is much lighter than $\mathrm{A}$ and $\mathrm{B}$ may not be able to +species that is much lighter than $\t{A}$ and $\t{B}$ may not be able to transfer much of its kinetic energy, and so would be inefficient as a collision partner. On the other hand, a species with a transition from its ground state that is nearly -resonant with one in the $\mathrm{AB^*}$ activated complex may be much more effective at +resonant with one in the $\t{AB^*}$ activated complex may be much more effective at exchanging energy than would otherwise be expected. These effects can be accounted for by defining a collision efficiency $\epsilon$ for each species, defined such that the forward reaction rate is -$$ R_f = [\mathrm{A}][\mathrm{B}][\mathrm{M}]k_f(T) $$ +$$ R_f = [\t{A}][\t{B}][\t{M}] k_f(T) $$ where -$$ [\mathrm{M}] = \sum_{k} \epsilon_k C_k $$ +$$ [\t{M}] = \sum_{k} \epsilon_k C_k $$ where $C_k$ is the concentration of species $k$. Since any constant collision efficiency can be absorbed into the rate coefficient $k_f(T)$, the default collision efficiency is @@ -70,9 +70,9 @@ Sometimes, accounting for a particular third body's collision efficiency may req alternate set of rate parameters entirely. In this case, two reactions are written: $$ -\mathrm{A + B + M \rightleftharpoons AB + M \quad (R1)} +\t{A + B + M \rightleftharpoons AB + M \quad (R1)} -\mathrm{A + B + C \rightleftharpoons AB + C \quad (R2)} +\t{A + B + C \rightleftharpoons AB + C \quad (R2)} $$ where the third-body efficiency for C in the first reaction should be explicitly set to @@ -105,14 +105,14 @@ Explicit reaction orders different from the stoichiometric coefficients are some used for non-elementary reactions. For example, consider the global reaction: $$ -\mathrm{C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O} +\t{C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O} $$ the forward rate constant might be given as {cite:p}`westbrook1981`: $$ -k_f = 4.6 \times 10^{11} [\mathrm{C_8H_{18}}]^{0.25} [\mathrm{O_2}]^{1.5} - \exp\left(\frac{30.0\,\mathrm{kcal/mol}}{RT}\right) +k_f = 4.6 \times 10^{11} [\t{C_8H_{18}}]^{0.25} [\t{O_2}]^{1.5} + \exp\left(\frac{30.0\,\t{kcal/mol}}{RT}\right) $$ Special care is required in this case since the units of the pre-exponential factor diff --git a/doc/sphinx/reference/onedim/governing-equations.md b/doc/sphinx/reference/onedim/governing-equations.md index ebbdb6e414..142d64afb0 100644 --- a/doc/sphinx/reference/onedim/governing-equations.md +++ b/doc/sphinx/reference/onedim/governing-equations.md @@ -11,30 +11,25 @@ in Section 7.2 of {cite:t}`kee2017` and are implemented by class {ct}`StFlow`. *Continuity*: -$$ \frac{\partial\rho u}{\partial z} + 2 \rho V = 0 $$ +$$ \pxpy{u}{z} + 2 \rho V = 0 $$ *Radial momentum*: $$ -\rho u \frac{\partial V}{\partial z} + \rho V^2 = - - \Lambda - + \frac{\partial}{\partial z}\left(\mu \frac{\partial V}{\partial z}\right) +\rho u \pxpy{V}{z} + \rho V^2 = - \Lambda + \pxpy{}{z}\left(\mu \pxpy{V}{z}\right) $$ *Energy*: $$ -\rho c_p u \frac{\partial T}{\partial z} = - \frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right) - - \sum_k j_k \frac{\partial h_k}{\partial z} - - \sum_k h_k W_k \dot{\omega}_k +\rho c_p u \pxpy{T}{z} = \pxpy{}{z}\left(\lambda \pxpy{T}{z}\right) + - \sum_k j_k \pxpy{h_k}{z} - \sum_k h_k W_k \dot{\omega}_k $$ *Species*: $$ -\rho u \frac{\partial Y_k}{\partial z} = - \frac{\partial j_k}{\partial z} - + W_k \dot{\omega}_k +\rho u \pxpy{Y_k}{z} = - \pxpy{j_k}{z} + W_k \dot{\omega}_k $$ where the following variables are used: @@ -82,7 +77,7 @@ mixture-averaged or multicomponent formulation. If the mixture-averaged formulat used, the calculation performed is: $$ -j_k^* = - \rho \frac{W_k}{\overline{W}} D_{km}^\prime \frac{\partial X_k}{\partial z} +j_k^* = - \rho \frac{W_k}{\overline{W}} D_{km}^\prime \pxpy{X_k}{z} j_k = j_k^* - Y_k \sum_i j_i^* $$ @@ -97,8 +92,8 @@ guaranteed by the mixture-averaged formulation. When using the multicomponent formulation, the mass fluxes are computed according to: $$ -j_k = \frac{\rho W_k}{\overline{W}^2} \sum_i W_i D_{ki} \frac{\partial X_i}{\partial z} - - \frac{D_k^T}{T} \frac{\partial T}{\partial z} +j_k = \frac{\rho W_k}{\overline{W}^2} \sum_i W_i D_{ki} \pxpy{X_i}{z} + - \frac{D_k^T}{T} \pxpy{T}{z} $$ where $D_{ki}$ is the multicomponent diffusion coefficient and $D_k^T$ is the Soret @@ -118,74 +113,73 @@ freely-propagating flame, the mass flow rate is not an input but is determined indirectly by holding the temperature fixed at an intermediate location within the domain; see [](discretization) for details. -The following equations are solved at the point $z = z_0$: +The following equations are solved at the point $z = z_\t{in}$: $$ -T(z_0) &= T_0 +T(z_\t{in}) &= T_0 -V(z_0) &= V_0 +V(z_\t{in}) &= V_0 -\dot{m}_0 Y_{k,0} - j_k(z_0) - \rho(z_0) u(z_0) Y_k(z_0) &= 0 +\dot{m}_0 Y_{k,\t{in}} - j_k(z_\t{in}) - \rho(z_\t{in}) u(z_\t{in}) Y_k(z_\t{in}) &= 0 $$ If the mass flow rate is specified, we also solve: $$ -\rho(z_0) u(z_0) = \dot{m}_0 +\rho(z_\t{in}) u(z_\t{in}) = \dot{m}_0 $$ Otherwise, we solve: -$$ \Lambda(z_0) = 0 $$ +$$ \Lambda(z_\t{in}) = 0 $$ These equations are implemented by class {ct}`Inlet1D`. ### Outlet boundary -For a boundary located at a point $z_0$ where there is an outflow, we -solve: +For a boundary located at a point $z_\t{out}$ where there is an outflow, we solve: $$ -\Lambda(z_0) = 0 +\Lambda(z_\t{out}) = 0 -\left.\frac{\partial T}{\partial z}\right|_{z_0} = 0 +\left.\pxpy{T}{z}\right|_{z_\t{out}} = 0 -\left.\frac{\partial Y_k}{\partial z}\right|_{z_0} = 0 +\left.\pxpy{Y_k}{z}\right|_{z_\t{out}} = 0 -V(z_0) = 0 +V(z_\t{out}) = 0 $$ These equations are implemented by class {ct}`Outlet1D`. ### Symmetry boundary -For a symmetry boundary located at a point $z_0$, we solve: +For a symmetry boundary located at a point $z_\t{symm}$, we solve: $$ -\rho(z_0) u(z_0) = 0 +\rho(z_\t{symm}) u(z_\t{symm}) = 0 -\left.\frac{\partial V}{\partial z}\right|_{z_0} = 0 +\left.\pxpy{V}{z}\right|_{z_\t{symm}} = 0 -\left.\frac{\partial T}{\partial z}\right|_{z_0} = 0 +\left.\pxpy{T}{z}\right|_{z_\t{symm}} = 0 -j_k(z_0) = 0 +j_k(z_\t{symm}) = 0 $$ These equations are implemented by class {ct}`Symm1D`. ### Reacting surface -For a surface boundary located at a point $z_0$ on which reactions may occur, the -temperature $T_0$ is specified. We solve: +For a surface boundary located at a point $z_\t{surf}$ on which reactions may +occur, the temperature $T_\t{surf}$ is specified. We solve: $$ -\rho(z_0) u(z_0) &= 0 +\rho(z_\t{surf}) u(z_\t{surf}) &= 0 -V(z_0) &= 0 +V(z_\t{surf}) &= 0 -T(z_0) &= T_0 +T(z_\t{surf}) &= T_\t{surf} -j_k(z_0) + \dot{s}_k W_k &= 0 +j_k(z_\t{surf}) + \dot{s}_k W_k &= 0 $$ where $\dot{s}_k$ is the molar production rate of the gas-phase species $k$ on the @@ -201,8 +195,7 @@ drift term to the diffusive fluxes of the mixture-average formulation according {cite:t}`pedersen1993`, $$ -j_k^* = \rho \frac{W_k}{\overline{W}} D_{km}^\prime \frac{\partial X_k}{\partial z} + - s_k \mu_k E Y_k, +j_k^* = \rho \frac{W_k}{\overline{W}} D_{km}^\prime \pxpy{X_k}{z} + s_k \mu_k E Y_k, $$ where $s_k$ is the sign of charge (1,-1, and 0 respectively for positive, negative, and @@ -219,7 +212,7 @@ $$ In addition, Gauss's law is solved simultaneously with the species and energy equations, $$ -\frac{\partial E}{\partial z} &= \frac{e}{\epsilon_0}\sum_k Z_k n_k , +\pxpy{E}{z} &= \frac{e}{\epsilon_0}\sum_k Z_k n_k , n_k &= N_a \rho Y_k / W_k, diff --git a/doc/sphinx/reference/reactors/constant-pressure-mole-reactor.md b/doc/sphinx/reference/reactors/constant-pressure-mole-reactor.md index 440724576b..422ea4f254 100644 --- a/doc/sphinx/reference/reactors/constant-pressure-mole-reactor.md +++ b/doc/sphinx/reference/reactors/constant-pressure-mole-reactor.md @@ -23,8 +23,8 @@ homogeneous phase reactions is $V \dot{\omega}_k$, and the total rate at which m species $k$ changes is: $$ -\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_{in} \dot{n}_{k, in} - - \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} +\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}} + - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}} $$ (const-pressure-mole-reactor-species) Where the subscripts *in* and *out* refer to the sum of the corresponding property over @@ -35,8 +35,8 @@ all inlets and outlets respectively. A dot above a variable signifies a time der Writing the first law for an open system gives: $$ -\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + - \sum_{in} \dot{n}_{in} \hat{h}_{in} - \hat{h} \sum_{out} \dot{n}_{out} +\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} + - \hat{h} \sum_\t{out} \dot{n}_\t{out} $$ where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific @@ -50,6 +50,6 @@ $$ \frac{dH}{dt} = \frac{dU}{dt} + p \frac{dV}{dt} + V \frac{dp}{dt} $$ Noting that $dp/dt = 0$ and substituting into the energy equation yields: $$ -\frac{dH}{dt} = \dot{Q} + \sum_{in} \dot{n}_{in} \hat{h}_{in} - - \hat{h} \sum_{out} \dot{n}_{out} +\frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} + - \hat{h} \sum_\t{out} \dot{n}_\t{out} $$ (const-pressure-mole-reactor-energy) diff --git a/doc/sphinx/reference/reactors/constant-pressure-reactor.md b/doc/sphinx/reference/reactors/constant-pressure-reactor.md index 347065447d..1067249de2 100644 --- a/doc/sphinx/reference/reactors/constant-pressure-reactor.md +++ b/doc/sphinx/reference/reactors/constant-pressure-reactor.md @@ -22,7 +22,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p species on [surfaces](sec-reactor-surface): $$ -\frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall} +\frac{dm}{dt} = \sum_\t{in} \dot{m}_\t{in} - \sum_\t{out} \dot{m}_\t{out} + + \dot{m}_\t{wall} $$ (constpressurereactor-mass) Where the subscripts *in* and *out* refer to the sum of the superscripted property over @@ -33,21 +34,21 @@ all inlets and outlets respectively. A dot above a variable signifies a time der The rate at which species $k$ is generated through homogeneous phase reactions is $V \dot{\omega}_k W_k$, and the total rate at which species $k$ is generated is: -$$ \dot{m}_{k,gen} = V \dot{\omega}_k W_k + \dot{m}_{k,wall} $$ +$$ \dot{m}_{k,\t{gen}} = V \dot{\omega}_k W_k + \dot{m}_{k,\t{wall}} $$ The rate of change in the mass of each species is: $$ -\frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k + - \dot{m}_{k,gen} +\frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}} + - \sum_\t{out} \dot{m}_\t{out} Y_k + \dot{m}_{k,\t{gen}} $$ Expanding the derivative on the left hand side and substituting the equation for $dm/dt$, the equation for each homogeneous phase species is: $$ -m \frac{dY_k}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k) + - \dot{m}_{k,gen} - Y_k \dot{m}_{wall} +m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k) + + \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall} $$ (constpressurereactor-species) ## Energy Equation @@ -55,8 +56,8 @@ $$ (constpressurereactor-species) Writing the first law for an open system gives: $$ -\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + - \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} +\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + + \sum_\t{in} \dot{m}_\t{in} h_\t{in} - h \sum_\t{out} \dot{m}_\t{out} $$ where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific @@ -70,5 +71,6 @@ $$ \frac{dH}{dt} = \frac{dU}{dt} + p \frac{dV}{dt} + V \frac{dp}{dt} $$ Noting that $dp/dt = 0$ and substituting into the energy equation yields: $$ -\frac{dH}{dt} = \dot{Q} + \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} +\frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{m}_\t{in} h_\t{in} + - h \sum_\t{out} \dot{m}_\t{out} $$ (constpressurereactor-energy) diff --git a/doc/sphinx/reference/reactors/controlreactor.md b/doc/sphinx/reference/reactors/controlreactor.md index 6861da799a..d2cd0d2713 100644 --- a/doc/sphinx/reference/reactors/controlreactor.md +++ b/doc/sphinx/reference/reactors/controlreactor.md @@ -21,7 +21,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p species on [surfaces](sec-reactor-surface): $$ -\frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall} +\frac{dm}{dt} = \sum_\t{in} \dot{m}_\t{in} - \sum_\t{out} \dot{m}_\t{out} + + \dot{m}_\t{wall} $$ (mass) Where the subscripts *in* and *out* refer to the sum of the corresponding property over @@ -46,22 +47,22 @@ The rate at which species $k$ is generated through homogeneous phase reactions i $V \dot{\omega}_k W_k$, and the total rate at which species $k$ is generated is: $$ -\dot{m}_{k,gen} = V \dot{\omega}_k W_k + \dot{m}_{k,wall} +\dot{m}_{k,\t{gen}} = V \dot{\omega}_k W_k + \dot{m}_{k,\t{wall}} $$ The rate of change in the mass of each species is: $$ -\frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k + - \dot{m}_{k,gen} +\frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}} + - \sum_\t{out} \dot{m}_\t{out} Y_k + \dot{m}_{k,\t{gen}} $$ Expanding the derivative on the left hand side and substituting the equation for $dm/dt$, the equation for each homogeneous phase species is: $$ -m \frac{dY_k}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k) + - \dot{m}_{k,gen} - Y_k \dot{m}_{wall} +m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k) + + \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall} $$ (species) ## Energy Equation @@ -71,7 +72,7 @@ system: $$ \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + - \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} + \sum_\t{in} \dot{m}_\t{in} h_\t{in} - h \sum_\t{out} \dot{m}_\t{out} $$ (cv-energy) Where $\dot{Q}$ is the net rate of heat addition to the system. diff --git a/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-mole-reactor.md b/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-mole-reactor.md index 7111891b43..a0309a347a 100644 --- a/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-mole-reactor.md +++ b/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-mole-reactor.md @@ -23,8 +23,8 @@ species $k$ is generated through homogeneous phase reactions is $V \dot{\omega}_ the total rate at which moles of species $k$ changes is: $$ -\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_{in} \dot{n}_{k, in} - - \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} +\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}} + - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}} $$ (ig-const-pressure-mole-reactor-species) Where the subscripts *in* and *out* refer to the sum of the corresponding property over @@ -35,8 +35,8 @@ all inlets and outlets respectively. A dot above a variable signifies a time der Writing the first law for an open system gives: $$ -\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + - \sum_{in} \dot{n}_{in} \hat{h}_{in} - \hat{h} \sum_{out} \dot{n}_{out} +\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} + - \hat{h} \sum_\t{out} \dot{n}_\t{out} $$ where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific @@ -50,8 +50,8 @@ $$ \frac{dH}{dt} = \frac{dU}{dt} + p \frac{dV}{dt} + V \frac{dp}{dt} $$ Noting that $dp/dt = 0$ and substituting into the energy equation yields: $$ -\frac{dH}{dt} = \dot{Q} + \sum_{in} \dot{n}_{in} \hat{h}_{in} - - \hat{h} \sum_{out} \dot{n}_{out} +\frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} + - \hat{h} \sum_\t{out} \dot{n}_\t{out} $$ As for the [ideal gas mole reactor](ideal-gas-mole-reactor), we replace the total diff --git a/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-reactor.md b/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-reactor.md index 36ac59f617..96ca482259 100644 --- a/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-reactor.md +++ b/doc/sphinx/reference/reactors/ideal-gas-constant-pressure-reactor.md @@ -21,7 +21,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p species on [surfaces](sec-reactor-surface): $$ -\frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall} +\frac{dm}{dt} = \sum_\t{in} \dot{m}_\t{in} - \sum_\t{out} \dot{m}_\t{out} + + \dot{m}_\t{wall} $$ (igcpr-mass) Where the subscripts *in* and *out* refer to the sum of the corresponding property over @@ -32,21 +33,21 @@ all inlets and outlets respectively. A dot above a variable signifies a time der The rate at which species $k$ is generated through homogeneous phase reactions is $V \dot{\omega}_k W_k$, and the total rate at which species $k$ is generated is: -$$ \dot{m}_{k,gen} = V \dot{\omega}_k W_k + \dot{m}_{k,wall} $$ +$$ \dot{m}_{k,\t{gen}} = V \dot{\omega}_k W_k + \dot{m}_{k,\t{wall}} $$ The rate of change in the mass of each species is: $$ -\frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k - + \dot{m}_{k,gen} +\frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}} + - \sum_\t{out} \dot{m}_\t{out} Y_k + \dot{m}_{k,gen} $$ Expanding the derivative on the left hand side and substituting the equation for $dm/dt$, the equation for each homogeneous phase species is: $$ -m \frac{dY_k}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k) + \dot{m}_{k,gen} - - Y_k \dot{m}_{wall} +m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k) + + \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall} $$ (igcpr-species) ## Energy Equation @@ -66,6 +67,6 @@ Substituting the corresponding derivatives into the constant pressure reactor en equation {eq}`constpressurereactor-energy` yields an equation for the temperature: $$ -m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,gen} - + \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) +m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,\t{gen}} + + \sum_\t{in} \dot{m}_\t{in} \left(h_\t{in} - \sum_k h_k Y_{k,\t{in}} \right) $$ (igcpr-energy) diff --git a/doc/sphinx/reference/reactors/ideal-gas-mole-reactor.md b/doc/sphinx/reference/reactors/ideal-gas-mole-reactor.md index d12ddd7c0d..fb26f00dca 100644 --- a/doc/sphinx/reference/reactors/ideal-gas-mole-reactor.md +++ b/doc/sphinx/reference/reactors/ideal-gas-mole-reactor.md @@ -35,8 +35,8 @@ species $k$ is generated through homogeneous phase reactions is $V \dot{\omega}_ the total rate at which moles of species $k$ changes is: $$ -\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_{in} \dot{n}_{k, in} - - \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} +\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}} + - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}} $$ (ig-mole-reactor-species) ## Energy Equation diff --git a/doc/sphinx/reference/reactors/ideal-gas-reactor.md b/doc/sphinx/reference/reactors/ideal-gas-reactor.md index 8801f46cd5..26cdf24d1c 100644 --- a/doc/sphinx/reference/reactors/ideal-gas-reactor.md +++ b/doc/sphinx/reference/reactors/ideal-gas-reactor.md @@ -21,7 +21,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of gas phase spe [surfaces](sec-reactor-surface): $$ -\frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall} +\frac{dm}{dt} = \sum_\t{in} \dot{m}_\t{in} - \sum_\t{out} \dot{m}_\t{out} + + \dot{m}_\t{wall} $$ (igr-mass) where the subscripts *in* and *out* refer to the sum of the corresponding property over @@ -45,21 +46,21 @@ $v_w(t)$ is the velocity of the wall as a function of time. The rate at which species $k$ is generated through homogeneous phase reactions is $V \dot{\omega}_k W_k$, and the total rate at which species $k$ is generated is: -$$ \dot{m}_{k,gen} = V \dot{\omega}_k W_k + \dot{m}_{k,wall} $$ +$$ \dot{m}_{k,\t{gen}} = V \dot{\omega}_k W_k + \dot{m}_{k,\t{wall}} $$ The rate of change in the mass of each species is: $$ -\frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k + - \dot{m}_{k,gen} +\frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}} - \sum_\t{out} \dot{m}_\t{out} Y_k + + \dot{m}_{k,\t{gen}} $$ Expanding the derivative on the left hand side and substituting the equation for $dm/dt$, the equation for each homogeneous phase species is: $$ -m \frac{dY_k}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k)+ \dot{m}_{k,gen} - - Y_k \dot{m}_{wall} +m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k)+ \dot{m}_{k,\t{gen}} + - Y_k \dot{m}_\t{wall} $$ (igr-species) ## Energy Equation @@ -80,8 +81,8 @@ Substituting this into the energy equation for the control volume reactor {eq}`cv-energy` yields an equation for the temperature: $$ -m c_v \frac{dT}{dt} =& - p \frac{dV}{dt} + \dot{Q} + \sum_{in} \dot{m}_{in} \left( h_{in} - \sum_k u_k Y_{k,in} \right) \\ - &- \frac{p V}{m} \sum_{out} \dot{m}_{out} - \sum_k \dot{m}_{k,gen} u_k +m c_v \frac{dT}{dt} =& - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{m}_\t{in} \left( h_\t{in} - \sum_k u_k Y_{k,\t{in}} \right) \\ + &- \frac{p V}{m} \sum_\t{out} \dot{m}_\t{out} - \sum_k \dot{m}_{k,\t{gen}} u_k $$ (igr-energy) While this form of the energy equation is somewhat more complicated, it significantly diff --git a/doc/sphinx/reference/reactors/interactions.md b/doc/sphinx/reference/reactors/interactions.md index eee5eefec9..9075e6f649 100644 --- a/doc/sphinx/reference/reactors/interactions.md +++ b/doc/sphinx/reference/reactors/interactions.md @@ -102,7 +102,7 @@ on outlet of the reactor. The mass flow rate of the pressure controller is equal of the primary mass flow rate, plus a small correction dependent on the pressure difference: -$$ \dot m = \dot m_{\text{primary}} + K_v f(P_1 - P_2) $$ +$$ \dot m = \dot m_\t{primary} + K_v f(P_1 - P_2) $$ where $K_v$ is a proportionality constant and $f$ is a function of the pressure drop that defaults to $f(P_1 - P_2) = P_1 - P_2$. If $\dot m < 0$, the mass flow rate will be @@ -130,7 +130,7 @@ integrated to determine any wall property. Since it is the wall, or piston, velo that enters the energy equation, this means that it is the velocity, not the acceleration or displacement, that is specified. The wall velocity is computed from -$$ v = K(P_{\mathrm{left}} - P_{\mathrm{right}}) + v_0(t) $$ +$$ v = K(P_\t{left} - P_\t{right}) + v_0(t) $$ where $K$ is a non-negative constant, and $v_0(t)$ is a specified function of time. The velocity is positive if the wall is moving to the right. @@ -144,8 +144,8 @@ for the reactor on the right). The heat flux $\dot{Q}_w$ through a wall $w$ conn reactors *left* and *right* is computed as: $$ -\dot{Q}_w = U A (T_{\mathrm{left}} - T_{\mathrm{right}}) - + \epsilon\sigma A (T_{\mathrm{left}}^4 - T_{\mathrm{right}}^4) + A q_0(t) +\dot{Q}_w = U A (T_\t{left} - T_\t{right}) + + \epsilon\sigma A (T_\t{left}^4 - T_\t{right}^4) + A q_0(t) $$ where $U$ is a user-specified heat transfer coefficient (W/m{sup}`2`-K), $A$ is the wall @@ -172,12 +172,12 @@ species $k$ on surface $w$ is $\dot{s}_{k,w}$ (in kmol/s/m{sup}`2`). The total mass production rate for homogeneous phase species $k$ on all surfaces is: -$$ \dot{m}_{k,surf} = W_k \sum_w A_w \dot{s}_{k,w} $$ +$$ \dot{m}_{k,\t{surf}} = W_k \sum_w A_w \dot{s}_{k,w} $$ where $W_k$ is the molecular weight of species $k$ and $A_w$ is the area of each surface. The net mass flux from all reacting surfaces is then: -$$ \dot{m}_{surf} = \sum_k \dot{m}_{k,surf} $$ +$$ \dot{m}_\t{surf} = \sum_k \dot{m}_{k,\t{surf}} $$ For each surface species $i$, the rate of change of the site fraction $\theta_{i,w}$ on each surface $w$ is integrated with time: @@ -191,10 +191,10 @@ additional ODEs appended to the state vector for the corresponding reactor. ### Mole-based reactors -In the case of mole based reactors, $\dot{n}_{surf}$ is used instead, and is calculated -as: +In the case of mole based reactors, $\dot{n}_\t{surf}$ is used instead, and is +calculated as: -$$ \dot{n}_{k,surf} = A_{w}\sum_{w}\dot{s}_{w, k} $$ +$$ \dot{n}_{k,\t{surf}} = A_{w}\sum_{w}\dot{s}_{w, k} $$ and the conservation equation for each surface species $i$ is diff --git a/doc/sphinx/reference/reactors/mole-reactor.md b/doc/sphinx/reference/reactors/mole-reactor.md index 76a7073e74..f7ba0ef7cb 100644 --- a/doc/sphinx/reference/reactors/mole-reactor.md +++ b/doc/sphinx/reference/reactors/mole-reactor.md @@ -35,8 +35,8 @@ species $k$ is generated through homogeneous phase reactions is $V \dot{\omega}_ the total rate at which moles of species $k$ changes is: $$ -\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_{in} \dot{n}_{k, in} - - \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} +\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}} + - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}} $$ (molereactor-species) where the subscripts *in* and *out* refer to the sum of the corresponding property over @@ -48,8 +48,8 @@ The equation for the total internal energy is found by writing the first law for system: $$ -\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_{in} \dot{n}_{in} \hat{h}_{in} - - \hat{h} \sum_{out} \dot{n}_{out} +\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in} + - \hat{h} \sum_\t{out} \dot{n}_\t{out} $$ (molereactor-energy) where $\dot{Q}$ is the net rate of heat addition to the system and $\hat{h}$ is the diff --git a/doc/sphinx/reference/thermo/species-thermo.md b/doc/sphinx/reference/thermo/species-thermo.md index ee555d5e6f..5aa78345e7 100644 --- a/doc/sphinx/reference/thermo/species-thermo.md +++ b/doc/sphinx/reference/thermo/species-thermo.md @@ -42,18 +42,18 @@ While each of these functions is implemented explicitly in Cantera for computati efficiency, $\hat{h}^\circ(T)$ and $\hat{s}^\circ(T)$ can be expressed in terms of $\hat{c}^\circ_p(T)$ using the relations -$$ \hat{h}^\circ(T) = \hat{h}^\circ(T_\mathrm{ref}) + - \int_{T_\mathrm{ref}}^T \hat{c}^\circ_p(T) \; dT $$ +$$ \hat{h}^\circ(T) = \hat{h}^\circ(T_\t{ref}) + + \int_{T_\t{ref}}^T \hat{c}^\circ_p(T) \; dT $$ and -$$ \hat{s}^\circ(T) = \hat{s}^\circ(T_\mathrm{ref}) + - \int_{T_\mathrm{ref}}^T \frac{\hat{c}^\circ_p(T)}{T} \; dT $$ +$$ \hat{s}^\circ(T) = \hat{s}^\circ(T_\t{ref}) + + \int_{T_\t{ref}}^T \frac{\hat{c}^\circ_p(T)}{T} \; dT $$ respectively. This means that a parameterization of $\hat{c}_p^\circ(T)$ plus the -constants $\hat{h}^\circ(T_\mathrm{ref})$ and $\hat{s}^\circ(T_\mathrm{ref})$ at a -reference temperature $T_\mathrm{ref}$ is sufficient to define the standard state -properties for a species. +constants $\hat{h}^\circ(T_\t{ref})$ and $\hat{s}^\circ(T_\t{ref})$ at a reference +temperature $T_\t{ref}$ is sufficient to define the standard state properties for a +species. The models described in this section can be used to provide standard state thermodynamic data for each species in a phase. They are implemented by classes deriving from @@ -169,16 +169,16 @@ narrow temperature range. In such cases, the heat capacity can be approximated a constant, and simple expressions can be used for the thermodynamic properties: $$ -\hat{c}_p^\circ(T) &= \hat{c}_p^\circ(T_\mathrm{ref}) +\hat{c}_p^\circ(T) &= \hat{c}_p^\circ(T_\t{ref}) -\hat{h}^\circ(T) &= \hat{h}^\circ\left(T_\mathrm{ref}\right) + \hat{c}_p^\circ \left(T-T_\mathrm{ref}\right) +\hat{h}^\circ(T) &= \hat{h}^\circ\left(T_\t{ref}\right) + \hat{c}_p^\circ \left(T-T_\t{ref}\right) -\hat{s}^\circ(T) &= \hat{s}^\circ(T_\mathrm{ref}) + \hat{c}_p^\circ \ln{\left(\frac{T}{T_\mathrm{ref}}\right)} +\hat{s}^\circ(T) &= \hat{s}^\circ(T_\t{ref}) + \hat{c}_p^\circ \ln{\left(\frac{T}{T_\t{ref}}\right)} $$ -The parameterization uses four constants: $T_\mathrm{ref}$, -$\hat{c}_p^\circ(T_\mathrm{ref})$, $\hat{h}^\circ(T_\mathrm{ref})$, and -$\hat{s}^\circ(T)$. The default value of $T_\mathrm{ref}$ is 298.15 K; the default value +The parameterization uses four constants: $T_\t{ref}$, +$\hat{c}_p^\circ(T_\t{ref})$, $\hat{h}^\circ(T_\t{ref})$, and +$\hat{s}^\circ(T)$. The default value of $T_\t{ref}$ is 298.15 K; the default value for the other parameters is 0.0. This model is implemented by the C++ class {ct}`ConstCpPoly`. diff --git a/doc/sphinx/userguide/creating-mechanisms.md b/doc/sphinx/userguide/creating-mechanisms.md index 426284cb41..95b4c6932f 100644 --- a/doc/sphinx/userguide/creating-mechanisms.md +++ b/doc/sphinx/userguide/creating-mechanisms.md @@ -562,10 +562,10 @@ it is usually best not to specify units for $A$, in which case they will be comp taking all of these factors into account. ```{note} -If $b \ne 0$, then the term $T^b$ should have units of $\mathrm{K}^b$, which would +If $b \ne 0$, then the term $T^b$ should have units of $\t{K}^b$, which would change the units of $A$. This is not done, however, so the units associated with $A$ are really the units for $k_f$. One way to formally express this is to replace $T^b$ -by the non-dimensional quantity $[T/(1\;\mathrm{K})]^b$. +by the non-dimensional quantity $[T/(1\;\t{K})]^b$. ``` The key `E` is used to specify $E_a$. @@ -611,13 +611,13 @@ negative-A: true Explicit reaction orders different from the stoichiometric coefficients are sometimes used for non-elementary reactions. For example, consider the global reaction: -$$ \mathrm{C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O} $$ +$$ \t{C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O} $$ the forward rate constant might be given as {cite:p}`westbrook1981`: $$ -k_f = 4.6 \times 10^{11} [\mathrm{C_8H_{18}}]^{0.25} [\mathrm{O_2}]^{1.5} - \exp\left(\frac{30.0\,\mathrm{kcal/mol}}{RT}\right) +k_f = 4.6 \times 10^{11} [\t{C_8H_{18}}]^{0.25} [\t{O_2}]^{1.5} + \exp\left(\frac{30.0\,\t{kcal/mol}}{RT}\right) $$ This reaction could be defined as: diff --git a/doc/sphinx/userguide/extensible-reactor.md b/doc/sphinx/userguide/extensible-reactor.md index a2120290f9..e17ea82d22 100644 --- a/doc/sphinx/userguide/extensible-reactor.md +++ b/doc/sphinx/userguide/extensible-reactor.md @@ -39,17 +39,18 @@ As shown in the derivations of the governing equations, the equations implemente the {ct}`IdealGasConstPressureReactor` class are: $$ -\frac{dm}{dt} = \sum_{in} \dot{m}_{in} - \sum_{out} \dot{m}_{out} + \dot{m}_{wall} +\frac{dm}{dt} = \sum_\t{in} \dot{m}_\t{in} - \sum_\t{out} \dot{m}_\t{out} + + \dot{m}_\t{wall} $$ $$ -m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,gen} - + \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) +m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,\t{gen}} + + \sum_\t{in} \dot{m}_\t{in} \left(h_\t{in} - \sum_k h_k Y_{k,\t{in}} \right) $$ $$ -\frac{d(mY_k)}{dt} = \sum_{in} \dot{m}_{in} Y_{k,in} - \sum_{out} \dot{m}_{out} Y_k - + \dot{m}_{k,gen} +\frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}} + - \sum_\t{out} \dot{m}_\t{out} Y_k + \dot{m}_{k,gen} $$ Each of these equations is written with an expression on the left-hand side (LHS) @@ -63,13 +64,13 @@ For example, to add a term for a large mass, say a rock, inside the reactor that the thermal mass, the energy equation would become: $$ -\left(m c_p + m_{rock} c_{p,rock}\right) \frac{dT}{dt} = \dot{Q} - - \sum_k h_k \dot{m}_{k,gen} - + \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) +\left(m c_p + m_\t{rock} c_{p,\t{rock}}\right) \frac{dT}{dt} = \dot{Q} + - \sum_k h_k \dot{m}_{k,\t{gen}} + + \sum_\t{in} \dot{m}_\t{in} \left(h_\t{in} - \sum_k h_k Y_{k,\t{in}} \right) $$ Here, the LHS coefficient has changed from $m c_p$ to -$m c_p + m_{\mathrm{rock}} c_{p,\mathrm{rock}}$. Since the rock does not change the +$m c_p + m_{\t{rock}} c_{p,\t{rock}}$. Since the rock does not change the composition of the species in the reactor and does not change the mass flow rate of any inlets or outlets, the other governing equations defining the ideal gas constant pressure reactor can be left unmodified. To implement this change, we define a new class diff --git a/doc/sphinx/userguide/reactor-tutorial.md b/doc/sphinx/userguide/reactor-tutorial.md index 545d341672..155a609167 100644 --- a/doc/sphinx/userguide/reactor-tutorial.md +++ b/doc/sphinx/userguide/reactor-tutorial.md @@ -65,18 +65,17 @@ three methods: determines the step size by estimating the local error, which must satisfy tolerance conditions. The step is redone with reduced step size whenever that error test fails. SUNDIALS also periodically checks if the maximum step size is being used. The time - step must not be larger than a predefined maximum time step $\Delta t_{\mathrm{max}}$. - The new time $t_{\mathrm{new}}$ at the end of the single step is returned by this - function. This method produces the highest time resolution in the output data of the - methods implemented in Cantera. + step must not be larger than a predefined maximum time step $\Delta t_\t{max}$. The + new time $t_\t{new}$ at the end of the single step is returned by this function. This + method produces the highest time resolution in the output data of the methods + implemented in Cantera. - `advance(t_new)`: This method computes the state of the system at the user-provided - time $t_{\mathrm{new}}$. $t_{\mathrm{new}}$ is the absolute time from the initial time - of the system. Although the user specifies the time when integration should stop, - SUNDIALS chooses the time step size as the network is integrated. Many internal - SUNDIALS time steps are usually required to reach $t_{\mathrm{new}}$. As such, - `advance(t_new)` preserves the accuracy of using `step()` but allows consistent - spacing in the output data. + time $t_\t{new}$. $t_\t{new}$ is the absolute time from the initial time of the + system. Although the user specifies the time when integration should stop, SUNDIALS + chooses the time step size as the network is integrated. Many internal SUNDIALS time + steps are usually required to reach $t_\t{new}$. As such, `advance(t_new)` preserves + the accuracy of using `step()` but allows consistent spacing in the output data. - `advance_to_steady_state(max_steps, residual_threshold, atol, write_residuals)` *Python interface only*: If the steady state solution of a reactor network is of