mirror of
https://github.com/Cantera/cantera.git
synced 2025-02-25 18:55:29 -06:00
[Doc] Introduce some LaTeX macros in Sphinx and fix text subscripts
This commit is contained in:
Ray Speth
Ray Speth
parent
7aa65e55a0
commit
d36a07a2a3
@ -225,6 +225,15 @@ intersphinx_mapping = {
|
||||
|
||||
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'
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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}} $$
|
||||
|
||||
where $\Gamma_\mathrm{tot}$ is the total molar site density, $m$ is the sum of all the
|
||||
where $\Gamma_\t{tot}$ is the total molar site density, $m$ is the sum of all the
|
||||
surface reactant stoichiometric coefficients, and $W$ is the molecular weight of the gas
|
||||
phase species.
|
||||
|
||||
|
||||
@ -9,11 +9,11 @@ types.
|
||||
The basic reaction type is a homogeneous reaction with a pressure-independent
|
||||
rate coefficient and mass action kinetics. For example:
|
||||
|
||||
$$ \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)
|
||||
$$
|
||||
|
||||
Special care is required in this case since the units of the pre-exponential factor
|
||||
|
||||
@ -11,30 +11,25 @@ in Section 7.2 of {cite:t}`kee2017` and are implemented by class {ct}`StFlow`.
|
||||
|
||||
*Continuity*:
|
||||
|
||||
$$ \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,
|
||||
|
||||
$$
|
||||
\frac{\partial E}{\partial z} &= \frac{e}{\epsilon_0}\sum_k Z_k n_k ,
|
||||
\pxpy{E}{z} &= \frac{e}{\epsilon_0}\sum_k Z_k n_k ,
|
||||
|
||||
n_k &= N_a \rho Y_k / W_k,
|
||||
|
||||
|
||||
@ -23,8 +23,8 @@ homogeneous phase reactions is $V \dot{\omega}_k$, and the total rate at which m
|
||||
species $k$ changes is:
|
||||
|
||||
$$
|
||||
\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}
|
||||
$$ (const-pressure-mole-reactor-energy)
|
||||
|
||||
@ -22,7 +22,8 @@ reactor's [inlets and outlets](sec-flow-device), and production of homogeneous p
|
||||
species on [surfaces](sec-reactor-surface):
|
||||
|
||||
$$
|
||||
\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)
|
||||
|
||||
@ -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.
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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)
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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
|
||||
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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`.
|
||||
|
||||
|
||||
@ -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:
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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
|
||||
|
||||
Loading…
Reference in New Issue
Block a user