more in model

Author: xavierr

docs/src/manuals/user_guide/models.md

@@ -3,43 +3,43 @@ In the following, all the symbols that are not introduced directly in the text a
 The mass and charge conservation in the electrolyte are given by
 ```math
 \begin{aligned}
-  \fracpar{}{t}(\epsi_\elyte c_\elyte) + \dive \Nvec_{\elyte} &=  R_{\elyte},\\
-  \dive \jvec_{\elyte} &= FR_{\elyte},
+  \frac{\partial }{\partial t}(\varepsilon_\text{elyte} c_\text{elyte}) + \nabla\cdot \textbf{N}_{\text{elyte}} &=  R_{\text{elyte}},\\
+  \nabla\cdot \textbf{j}_{\text{elyte}} &= FR_{\text{elyte}},
 \end{aligned}
 ```
 where the fluxes are given by
 ```math
 \begin{aligned}
-  \jvec_\elyte   &= -\kappa_{\elyte,\eff}\grad\phi_\elyte - \kappa_{\elyte,\eff}\frac{1 - t_+}{z_+F}\left(\fracpar{\mu}{c}\right)\grad c_\elyte,\\
-  \Nvec_{\elyte} &= - D_{\elyte,\eff}\grad c_{\elyte} + \frac{t_+}{z_+F}\jvec_\elyte.
+  \textbf{j}_\text{elyte}   &= -\kappa_{\text{elyte},\text{eff}}\nabla\phi_\text{elyte} - \kappa_{\text{elyte},\text{eff}}\frac{1 - t_+}{z_+F}\left(\frac{\partial \mu}{\partial c}\right)\nabla c_\text{elyte},\\
+  \textbf{N}_{\text{elyte}} &= - D_{\text{elyte},\text{eff}}\nabla c_{\text{elyte}} + \frac{t_+}{z_+F}\textbf{j}_\text{elyte}.
 \end{aligned}
 ```
-The volumetric reaction rate is given as ``R_\elyte = -\sum_\elde \gamma_\elde R_\elde`` where ``\gamma_{\elde}`` is the volumetric surface area and the expression for ``R_\elde`` is given below. Note that the reaction rates depends on the spatial variable ``x``. For the chemical potential, we use ``\mu = 2RT\log(c_\elyte)``. The effective quantities are computed from the intrinsic properties and the volume fraction using a Bruggemann coefficient, denoted ``b``, which yields ``\kappa_{\elyte,\eff} = \epsi_\elyte^{b}\kappa_{\elyte}`` and ``D_{\elyte,\eff} = \epsi_\elyte^{b}D_{\elyte}``. For the electrolyte, we have a spatially dependent Bruggeman coefficient.
+The volumetric reaction rate is given as ``R_\text{elyte} = -\sum_\text{elde} \gamma_\text{elde} R_\text{elde}`` where ``\gamma_{\text{elde}}`` is the volumetric surface area and the expression for ``R_\text{elde}`` is given below. Note that the reaction rates depends on the spatial variable ``x``. For the chemical potential, we use ``\mu = 2RT\log(c_\text{elyte})``. The effective quantities are computed from the intrinsic properties and the volume fraction using a Bruggemann coefficient, denoted ``b``, which yields ``\kappa_{\text{elyte},\text{eff}} = \varepsilon_\text{elyte}^{b}\kappa_{\text{elyte}}`` and ``D_{\text{elyte},\text{eff}} = \varepsilon_\text{elyte}^{b}D_{\text{elyte}}``. For the electrolyte, we have a spatially dependent Bruggeman coefficient.
 
 In the electrode, the charge conservation equation is given by
 ```math
-  -\dive (\kappa_{\elde, \eff} \grad \phi_\elde) = F\gamma_\elde R_{\elde}.
+  -\nabla\cdot (\kappa_{\text{elde}, \text{eff}} \nabla \phi_\text{elde}) = F\gamma_\text{elde} R_{\text{elde}}.
 ```
-We use a pseudo particle model for ``c_\elde(t, x, r)``
+We use a pseudo particle model for ``c_\text{elde}(t, x, r)``
 ```math
-\fracpar{}{t}c_\elde - \frac1{r^2}\fracpar{}{r}(r^2D_\elde\fracpar{}{r} c_\elde) = 0.
+\frac{\partial }{\partial t}c_\text{elde} - \frac1{r^2}\frac{\partial }{\partial r}(r^2D_\text{elde}\frac{\partial }{\partial r} c_\text{elde}) = 0.
 ```
 with boundary condition
 ```math
-- 4\pi r_p^2 D_\elde \fracpar{c_\elde}{r}(t, x, r_p) = \frac{\gamma_\elde R_\elde}{\epsi_\elde}\frac{4\pi r_p^3}{3}.
+- 4\pi r_p^2 D_\text{elde} \frac{\partial c_\text{elde}}{\partial r}(t, x, r_p) = \frac{\gamma_\text{elde} R_\text{elde}}{\varepsilon_\text{elde}}\frac{4\pi r_p^3}{3}.
 ```
 
-Reaction kinetics. The reaction rate ``R_\elde`` at each electrode is given
+Reaction kinetics. The reaction rate ``R_\text{elde}`` at each electrode is given
 ```math
-R_\elde = j_{\elde}(c_\elde, c_\elyte, T)(e^{\alpha F\frac{\eta_\elde}{RT}} - e^{-(1 - \alpha) F\frac{\eta_\elde}{RT}} ) .
+R_\text{elde} = j_{\text{elde}}(c_\text{elde}, c_\text{elyte}, T)(e^{\alpha F\frac{\eta_\text{elde}}{RT}} - e^{-(1 - \alpha) F\frac{\eta_\text{elde}}{RT}} ) .
 ```
-where ``\eta_\elde`` and ``j_\elde`` denote the overpotential and the reaction exchange current density. The overpotential
-``\eta_\elde`` is given by
+where ``\eta_\text{elde}`` and ``j_\text{elde}`` denote the overpotential and the reaction exchange current density. The overpotential
+``\eta_\text{elde}`` is given by
 ```math
-\eta_\elde = \phi_\elde - \phi_\elyte - U_\elde(c_\elde, T).
+\eta_\text{elde} = \phi_\text{elde} - \phi_\text{elyte} - U_\text{elde}(c_\text{elde}, T).
 ```
-where ``U_\elde`` denotes the open circuit potential, given as a function of the Lithium concentration in the electrode
+where ``U_\text{elde}`` denotes the open circuit potential, given as a function of the Lithium concentration in the electrode
 and the temperature.  The exchange current density is given by
 ```math
-  j_\elde = k_{\elde,0} e^{-\frac{E_a}{R}(1/T - 1/T_{\text{ref}})}\left(c_\elyte(c_{\elde,\max} - c_\elde)c_\elde\right)^{\frac12}.
+  j_\text{elde} = k_{\text{elde},0} e^{-\frac{E_a}{R}(1/T - 1/T_{\text{ref}})}\left(c_\text{elyte}(c_{\text{elde},\max} - c_\text{elde})c_\text{elde}\right)^{\frac12}.
 ```