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[Doc] Introduce some LaTeX macros in Sphinx and fix text subscripts

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Ray Speth 2024-01-02 14:31:01 -05:00 committed by Ray Speth
parent 7aa65e55a0
commit d36a07a2a3
18 changed files with 185 additions and 178 deletions
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9
doc/sphinx/conf.py
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@ -225,6 +225,15 @@ intersphinx_mapping = {
myst_enable_extensions = ["dollarmath", "amsmath", "deflist", "colon_fence"] 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 # Ensure that the primary domain is the Python domain, since we've added the
# MATLAB domain with sphinxcontrib.matlab # MATLAB domain with sphinxcontrib.matlab
primary_domain = 'py' primary_domain = 'py'

2
doc/sphinx/develop/reactor-integration.md
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@ -165,7 +165,7 @@ example, using the {py:class}`ExtensibleReactor` class, the governing equations
reactor are written in the form: 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 where the {ct}`Reactor::eval` method or the `eval()` method of any class derived from

52
doc/sphinx/reference/kinetics/rate-constants.md
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@ -25,22 +25,22 @@ field.
## Falloff Reactions ## Falloff Reactions
A falloff reaction is one that has a rate that is first-order in the total concentration 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 of third-body colliders $[\t{M}]$ at low pressure, like a
[three-body reaction](sec-three-body-reaction), but becomes zero-order in $\MM$ as $\MM$ [three-body reaction](sec-three-body-reaction), but becomes zero-order in $[\t{M}]$ as
increases. Dissociation/association reactions of polyatomic molecules often exhibit this $[\t{M}]$ increases. Dissociation/association reactions of polyatomic molecules often
behavior. exhibit this behavior.
The simplest expression for the rate coefficient for a falloff reaction is the Lindemann The simplest expression for the rate coefficient for a falloff reaction is the Lindemann
form {cite:p}`lindemann1922`: 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 In the low-pressure limit, this approaches $k_0 [\t{M}]$, and in the high-pressure limit
approaches $k_\infty$. it approaches $k_\infty$.
Defining the non-dimensional reduced pressure: 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 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`: A widely-used falloff function is the one proposed by {cite:t}`gilbert1983`:
\begin{gather*} \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)) \\ 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*} \end{gather*}
```{admonition} YAML Usage ```{admonition} YAML Usage
@ -87,13 +87,13 @@ term of the falloff function is not used.
``` ```
(sec-tsang-falloff)= (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 databases of reactions, for example, {cite:t}`tsang1991`. Tsang's approximations are
linear in temperature: 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 $F$ 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. 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 For these reactions, the rate falls off as the pressure increases, due to collisional
stabilization of a reaction intermediate. Example: 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: 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 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 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 $n$ evaluated at $x$, and
$$ $$
\tilde{T} \equiv \frac{2T^{-1} - T_\mathrm{min}^{-1} - T_\mathrm{max}^{-1}} \tilde{T} \equiv \frac{2T^{-1} - T_\t{min}^{-1} - T_\t{max}^{-1}}
{T_\mathrm{max}^{-1} - T_\mathrm{min}^{-1}} {T_\t{max}^{-1} - T_\t{min}^{-1}}
\tilde{P} \equiv \frac{2 \log P - \log P_\mathrm{min} - \log P_\mathrm{max}} \tilde{P} \equiv \frac{2 \log P - \log P_\t{min} - \log P_\t{max}}
{\log P_\mathrm{max} - \log P_\mathrm{min}} {\log P_\t{max} - \log P_\t{min}}
$$ $$
are reduced temperatures and reduced pressures which map the ranges $(T_\mathrm{min}, are reduced temperatures and reduced pressures which map the ranges $(T_\t{min},
T_\mathrm{max})$ and $(P_\mathrm{min}, P_\mathrm{max})$ to $(-1, 1)$. 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 A Chebyshev rate expression is specified in terms of the coefficient matrix $\alpha$ and
the temperature and pressure ranges. 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 Heterogeneous reactions on surfaces are represented by an extended Arrhenius- like rate
expression, which combines the modified Arrhenius rate expression with further 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: 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: 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 surface reactant stoichiometric coefficients, and $W$ is the molecular weight of the gas
phase species. phase species.

32
doc/sphinx/reference/kinetics/reaction-rates.md
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@ -9,11 +9,11 @@ types.
The basic reaction type is a homogeneous reaction with a pressure-independent The basic reaction type is a homogeneous reaction with a pressure-independent
rate coefficient and mass action kinetics. For example: 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: 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 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. 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: 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 Here $\t{M}$ is an unspecified collision partner that carries away excess energy to
stabilize the $\mathrm{AB}$ molecule (forward direction) or supplies energy to break the stabilize the $\t{AB}$ molecule (forward direction) or supplies energy to break the
$\mathrm{AB}$ bond (reverse direction). In addition to the generic collision partner $\t{AB}$ bond (reverse direction). In addition to the generic collision partner
$\mathrm{M}$, it is also possible to explicitly specify a colliding species. In both $\t{M}$, 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 cases, the reaction type can be automatically inferred by Cantera and does not need to
be explicitly specified by the user. be explicitly specified by the user.
Different species may be more or less effective in acting as the collision partner. A 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. 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 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. exchanging energy than would otherwise be expected.
These effects can be accounted for by defining a collision efficiency $\epsilon$ for These effects can be accounted for by defining a collision efficiency $\epsilon$ for
each species, defined such that the forward reaction rate is 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 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 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 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: 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 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: 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`: 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} k_f = 4.6 \times 10^{11} [\t{C_8H_{18}}]^{0.25} [\t{O_2}]^{1.5}
\exp\left(\frac{30.0\,\mathrm{kcal/mol}}{RT}\right) \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 Special care is required in this case since the units of the pre-exponential factor

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doc/sphinx/reference/onedim/governing-equations.md
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@ -11,30 +11,25 @@ in Section 7.2 of {cite:t}`kee2017` and are implemented by class {ct}`StFlow`.
*Continuity*: *Continuity*:
$$ \frac{\partial\rho u}{\partial z} + 2 \rho V = 0 $$ $$ \pxpy{u}{z} + 2 \rho V = 0 $$
*Radial momentum*: *Radial momentum*:
$$ $$
\rho u \frac{\partial V}{\partial z} + \rho V^2 = \rho u \pxpy{V}{z} + \rho V^2 = - \Lambda + \pxpy{}{z}\left(\mu \pxpy{V}{z}\right)
- \Lambda
+ \frac{\partial}{\partial z}\left(\mu \frac{\partial V}{\partial z}\right)
$$ $$
*Energy*: *Energy*:
$$ $$
\rho c_p u \frac{\partial T}{\partial z} = \rho c_p u \pxpy{T}{z} = \pxpy{}{z}\left(\lambda \pxpy{T}{z}\right)
\frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right) - \sum_k j_k \pxpy{h_k}{z} - \sum_k h_k W_k \dot{\omega}_k
- \sum_k j_k \frac{\partial h_k}{\partial z}
- \sum_k h_k W_k \dot{\omega}_k
$$ $$
*Species*: *Species*:
$$ $$
\rho u \frac{\partial Y_k}{\partial z} = - \frac{\partial j_k}{\partial z} \rho u \pxpy{Y_k}{z} = - \pxpy{j_k}{z} + W_k \dot{\omega}_k
+ W_k \dot{\omega}_k
$$ $$
where the following variables are used: 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: 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^* 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: 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} j_k = \frac{\rho W_k}{\overline{W}^2} \sum_i W_i D_{ki} \pxpy{X_i}{z}
- \frac{D_k^T}{T} \frac{\partial T}{\partial z} - \frac{D_k^T}{T} \pxpy{T}{z}
$$ $$
where $D_{ki}$ is the multicomponent diffusion coefficient and $D_k^T$ is the Soret 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 indirectly by holding the temperature fixed at an intermediate location within the
domain; see [](discretization) for details. 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: 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: Otherwise, we solve:
$$ \Lambda(z_0) = 0 $$ $$ \Lambda(z_\t{in}) = 0 $$
These equations are implemented by class {ct}`Inlet1D`. These equations are implemented by class {ct}`Inlet1D`.
### Outlet boundary ### Outlet boundary
For a boundary located at a point $z_0$ where there is an outflow, we For a boundary located at a point $z_\t{out}$ where there is an outflow, we solve:
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`. These equations are implemented by class {ct}`Outlet1D`.
### Symmetry boundary ### 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`. These equations are implemented by class {ct}`Symm1D`.
### Reacting surface ### Reacting surface
For a surface boundary located at a point $z_0$ on which reactions may occur, the For a surface boundary located at a point $z_\t{surf}$ on which reactions may
temperature $T_0$ is specified. We solve: 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 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`, {cite:t}`pedersen1993`,
$$ $$
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} + s_k \mu_k E Y_k,
s_k \mu_k E Y_k,
$$ $$
where $s_k$ is the sign of charge (1,-1, and 0 respectively for positive, negative, and where $s_k$ is the sign of charge (1,-1, and 0 respectively for positive, negative, and
@ -219,7 +212,7 @@ $$
In addition, Gauss's law is solved simultaneously with the species and energy equations, In addition, Gauss's law is solved simultaneously with the species and energy equations,
$$ $$
\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, n_k &= N_a \rho Y_k / W_k,

12
doc/sphinx/reference/reactors/constant-pressure-mole-reactor.md
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@ -23,8 +23,8 @@ homogeneous phase reactions is $V \dot{\omega}_k$, and the total rate at which m
species $k$ changes is: species $k$ changes is:
$$ $$
\frac{dn_k}{dt} = V \dot{\omega}_k + \sum_{in} \dot{n}_{k, in} \frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}}
- \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}}
$$ (const-pressure-mole-reactor-species) $$ (const-pressure-mole-reactor-species)
Where the subscripts *in* and *out* refer to the sum of the corresponding property over 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: Writing the first law for an open system gives:
$$ $$
\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in}
\sum_{in} \dot{n}_{in} \hat{h}_{in} - \hat{h} \sum_{out} \dot{n}_{out} - \hat{h} \sum_\t{out} \dot{n}_\t{out}
$$ $$
where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific 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: 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} \frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in}
- \hat{h} \sum_{out} \dot{n}_{out} - \hat{h} \sum_\t{out} \dot{n}_\t{out}
$$ (const-pressure-mole-reactor-energy) $$ (const-pressure-mole-reactor-energy)

20
doc/sphinx/reference/reactors/constant-pressure-reactor.md
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@ -22,7 +22,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p
species on [surfaces](sec-reactor-surface): 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}
$$ (constpressurereactor-mass) $$ (constpressurereactor-mass)
Where the subscripts *in* and *out* refer to the sum of the superscripted property over 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 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{\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: 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 + \frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}}
\dot{m}_{k,gen} - \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 Expanding the derivative on the left hand side and substituting the equation for
$dm/dt$, the equation for each homogeneous phase species is: $dm/dt$, the equation for each homogeneous phase species is:
$$ $$
m \frac{dY_k}{dt} = \sum_{in} \dot{m}_{in} (Y_{k,in} - Y_k) + m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k)
\dot{m}_{k,gen} - Y_k \dot{m}_{wall} + \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall}
$$ (constpressurereactor-species) $$ (constpressurereactor-species)
## Energy Equation ## Energy Equation
@ -55,8 +56,8 @@ $$ (constpressurereactor-species)
Writing the first law for an open system gives: Writing the first law for an open system gives:
$$ $$
\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q}
\sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} + \sum_\t{in} \dot{m}_\t{in} h_\t{in} - h \sum_\t{out} \dot{m}_\t{out}
$$ $$
where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific
@ -70,5 +71,6 @@ $$ \frac{dH}{dt} = \frac{dU}{dt} + p \frac{dV}{dt} + V \frac{dp}{dt} $$
Noting that $dp/dt = 0$ and substituting into the energy equation yields: Noting that $dp/dt = 0$ and substituting into the energy equation yields:
$$ $$
\frac{dH}{dt} = \dot{Q} + \sum_{in} \dot{m}_{in} h_{in} - h \sum_{out} \dot{m}_{out} \frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{m}_\t{in} h_\t{in}
- h \sum_\t{out} \dot{m}_\t{out}
$$ (constpressurereactor-energy) $$ (constpressurereactor-energy)

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@ -21,7 +21,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p
species on [surfaces](sec-reactor-surface): 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}
$$ (mass) $$ (mass)
Where the subscripts *in* and *out* refer to the sum of the corresponding property over 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: $V \dot{\omega}_k W_k$, 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: 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 + \frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}}
\dot{m}_{k,gen} - \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 Expanding the derivative on the left hand side and substituting the equation
for $dm/dt$, the equation for each homogeneous phase species is: 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) + m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k) +
\dot{m}_{k,gen} - Y_k \dot{m}_{wall} \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall}
$$ (species) $$ (species)
## Energy Equation ## Energy Equation
@ -71,7 +72,7 @@ system:
$$ $$
\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \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}
$$ (cv-energy) $$ (cv-energy)
Where $\dot{Q}$ is the net rate of heat addition to the system. Where $\dot{Q}$ is the net rate of heat addition to the system.

12
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@ -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: 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} \frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}}
- \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}}
$$ (ig-const-pressure-mole-reactor-species) $$ (ig-const-pressure-mole-reactor-species)
Where the subscripts *in* and *out* refer to the sum of the corresponding property over 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: Writing the first law for an open system gives:
$$ $$
\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in}
\sum_{in} \dot{n}_{in} \hat{h}_{in} - \hat{h} \sum_{out} \dot{n}_{out} - \hat{h} \sum_\t{out} \dot{n}_\t{out}
$$ $$
where positive $\dot{Q}$ represents heat addition to the system and $h$ is the specific 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: 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} \frac{dH}{dt} = \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in}
- \hat{h} \sum_{out} \dot{n}_{out} - \hat{h} \sum_\t{out} \dot{n}_\t{out}
$$ $$
As for the [ideal gas mole reactor](ideal-gas-mole-reactor), we replace the total As for the [ideal gas mole reactor](ideal-gas-mole-reactor), we replace the total

17
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@ -21,7 +21,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p
species on [surfaces](sec-reactor-surface): 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}
$$ (igcpr-mass) $$ (igcpr-mass)
Where the subscripts *in* and *out* refer to the sum of the corresponding property over 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 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: $V \dot{\omega}_k W_k$, 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: 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 \frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}}
+ \dot{m}_{k,gen} - \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 Expanding the derivative on the left hand side and substituting the equation
for $dm/dt$, the equation for each homogeneous phase species is: 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} m \frac{dY_k}{dt} = \sum_\t{in} \dot{m}_\t{in} (Y_{k,\t{in}} - Y_k)
- Y_k \dot{m}_{wall} + \dot{m}_{k,\t{gen}} - Y_k \dot{m}_\t{wall}
$$ (igcpr-species) $$ (igcpr-species)
## Energy Equation ## 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: 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} m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,\t{gen}}
+ \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) + \sum_\t{in} \dot{m}_\t{in} \left(h_\t{in} - \sum_k h_k Y_{k,\t{in}} \right)
$$ (igcpr-energy) $$ (igcpr-energy)

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@ -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: 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} \frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}}
- \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}}
$$ (ig-mole-reactor-species) $$ (ig-mole-reactor-species)
## Energy Equation ## Energy Equation

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@ -21,7 +21,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of gas phase spe
[surfaces](sec-reactor-surface): [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}
$$ (igr-mass) $$ (igr-mass)
where the subscripts *in* and *out* refer to the sum of the corresponding property over 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 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: $V \dot{\omega}_k W_k$, 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: 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 + \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} \dot{m}_{k,\t{gen}}
$$ $$
Expanding the derivative on the left hand side and substituting the equation Expanding the derivative on the left hand side and substituting the equation
for $dm/dt$, the equation for each homogeneous phase species is: 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} 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}_{wall} - Y_k \dot{m}_\t{wall}
$$ (igr-species) $$ (igr-species)
## Energy Equation ## 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: {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) \\ 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_{out} \dot{m}_{out} - \sum_k \dot{m}_{k,gen} u_k &- \frac{p V}{m} \sum_\t{out} \dot{m}_\t{out} - \sum_k \dot{m}_{k,\t{gen}} u_k
$$ (igr-energy) $$ (igr-energy)
While this form of the energy equation is somewhat more complicated, it significantly While this form of the energy equation is somewhat more complicated, it significantly

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@ -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 of the primary mass flow rate, plus a small correction dependent on the pressure
difference: 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 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 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 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 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 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. 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: reactors *left* and *right* is computed as:
$$ $$
\dot{Q}_w = U A (T_{\mathrm{left}} - T_{\mathrm{right}}) \dot{Q}_w = U A (T_\t{left} - T_\t{right})
+ \epsilon\sigma A (T_{\mathrm{left}}^4 - T_{\mathrm{right}}^4) + A q_0(t) + \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 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: 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 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: 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 For each surface species $i$, the rate of change of the site fraction $\theta_{i,w}$ on
each surface $w$ is integrated with time: 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 ### Mole-based reactors
In the case of mole based reactors, $\dot{n}_{surf}$ is used instead, and is calculated In the case of mole based reactors, $\dot{n}_\t{surf}$ is used instead, and is
as: 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 and the conservation equation for each surface species $i$ is

8
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@ -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: 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} \frac{dn_k}{dt} = V \dot{\omega}_k + \sum_\t{in} \dot{n}_{k, \t{in}}
- \sum_{out} \dot{n}_{k, out} + \dot{n}_{k, wall} - \sum_\t{out} \dot{n}_{k, \t{out}} + \dot{n}_{k, \t{wall}}
$$ (molereactor-species) $$ (molereactor-species)
where the subscripts *in* and *out* refer to the sum of the corresponding property over 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: system:
$$ $$
\frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_{in} \dot{n}_{in} \hat{h}_{in} \frac{dU}{dt} = - p \frac{dV}{dt} + \dot{Q} + \sum_\t{in} \dot{n}_\t{in} \hat{h}_\t{in}
- \hat{h} \sum_{out} \dot{n}_{out} - \hat{h} \sum_\t{out} \dot{n}_\t{out}
$$ (molereactor-energy) $$ (molereactor-energy)
where $\dot{Q}$ is the net rate of heat addition to the system and $\hat{h}$ is the where $\dot{Q}$ is the net rate of heat addition to the system and $\hat{h}$ is the

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@ -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 efficiency, $\hat{h}^\circ(T)$ and $\hat{s}^\circ(T)$ can be expressed in terms of
$\hat{c}^\circ_p(T)$ using the relations $\hat{c}^\circ_p(T)$ using the relations
$$ \hat{h}^\circ(T) = \hat{h}^\circ(T_\mathrm{ref}) + $$ \hat{h}^\circ(T) = \hat{h}^\circ(T_\t{ref}) +
\int_{T_\mathrm{ref}}^T \hat{c}^\circ_p(T) \; dT $$ \int_{T_\t{ref}}^T \hat{c}^\circ_p(T) \; dT $$
and and
$$ \hat{s}^\circ(T) = \hat{s}^\circ(T_\mathrm{ref}) + $$ \hat{s}^\circ(T) = \hat{s}^\circ(T_\t{ref}) +
\int_{T_\mathrm{ref}}^T \frac{\hat{c}^\circ_p(T)}{T} \; dT $$ \int_{T_\t{ref}}^T \frac{\hat{c}^\circ_p(T)}{T} \; dT $$
respectively. This means that a parameterization of $\hat{c}_p^\circ(T)$ plus the 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 constants $\hat{h}^\circ(T_\t{ref})$ and $\hat{s}^\circ(T_\t{ref})$ at a reference
reference temperature $T_\mathrm{ref}$ is sufficient to define the standard state temperature $T_\t{ref}$ is sufficient to define the standard state properties for a
properties for a species. species.
The models described in this section can be used to provide standard state thermodynamic 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 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: 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: $T_\mathrm{ref}$, The parameterization uses four constants: $T_\t{ref}$,
$\hat{c}_p^\circ(T_\mathrm{ref})$, $\hat{h}^\circ(T_\mathrm{ref})$, and $\hat{c}_p^\circ(T_\t{ref})$, $\hat{h}^\circ(T_\t{ref})$, and
$\hat{s}^\circ(T)$. The default value of $T_\mathrm{ref}$ is 298.15 K; the default value $\hat{s}^\circ(T)$. 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 for the other parameters is 0.0. This model is implemented by the C++ class
{ct}`ConstCpPoly`. {ct}`ConstCpPoly`.

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@ -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. taking all of these factors into account.
```{note} ```{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 $A$ 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 $T^b$ 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 $E_a$. 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 Explicit reaction orders different from the stoichiometric coefficients are sometimes
used for non-elementary reactions. For example, consider the global reaction: 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`: 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} k_f = 4.6 \times 10^{11} [\t{C_8H_{18}}]^{0.25} [\t{O_2}]^{1.5}
\exp\left(\frac{30.0\,\mathrm{kcal/mol}}{RT}\right) \exp\left(\frac{30.0\,\t{kcal/mol}}{RT}\right)
$$ $$
This reaction could be defined as: This reaction could be defined as:

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@ -39,17 +39,18 @@ As shown in the derivations of the governing equations, the equations implemente
the {ct}`IdealGasConstPressureReactor` class are: 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} m c_p \frac{dT}{dt} = \dot{Q} - \sum_k h_k \dot{m}_{k,\t{gen}}
+ \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) + \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 \frac{d(mY_k)}{dt} = \sum_\t{in} \dot{m}_\t{in} Y_{k,\t{in}}
+ \dot{m}_{k,gen} - \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) 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: the thermal mass, the energy equation would become:
$$ $$
\left(m c_p + m_{rock} c_{p,rock}\right) \frac{dT}{dt} = \dot{Q} \left(m c_p + m_\t{rock} c_{p,\t{rock}}\right) \frac{dT}{dt} = \dot{Q}
- \sum_k h_k \dot{m}_{k,gen} - \sum_k h_k \dot{m}_{k,\t{gen}}
+ \sum_{in} \dot{m}_{in} \left(h_{in} - \sum_k h_k Y_{k,in} \right) + \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 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 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 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 pressure reactor can be left unmodified. To implement this change, we define a new class

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@ -65,18 +65,17 @@ three methods:
determines the step size by estimating the local error, which must satisfy tolerance 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. 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 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 $\Delta t_{\mathrm{max}}$. step must not be larger than a predefined maximum time step $\Delta t_\t{max}$. The
The new time $t_{\mathrm{new}}$ at the end of the single step is returned by this new time $t_\t{new}$ at the end of the single step is returned by this function. This
function. This method produces the highest time resolution in the output data of the method produces the highest time resolution in the output data of the methods
methods implemented in Cantera. implemented in Cantera.
- `advance(t_new)`: This method computes the state of the system at the user-provided - `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 time $t_\t{new}$. $t_\t{new}$ is the absolute time from the initial time of the
of the system. Although the user specifies the time when integration should stop, system. Although the user specifies the time when integration should stop, SUNDIALS
SUNDIALS chooses the time step size as the network is integrated. Many internal chooses the time step size as the network is integrated. Many internal SUNDIALS time
SUNDIALS time steps are usually required to reach $t_{\mathrm{new}}$. As such, steps are usually required to reach $t_\t{new}$. As such, `advance(t_new)` preserves
`advance(t_new)` preserves the accuracy of using `step()` but allows consistent the accuracy of using `step()` but allows consistent spacing in the output data.
spacing in the output data.
- `advance_to_steady_state(max_steps, residual_threshold, atol, write_residuals)` - `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 *Python interface only*: If the steady state solution of a reactor network is of