diff --git a/docs/memo/binary_limit/binary_limit.pdf b/docs/memo/binary_limit/binary_limit.pdf index c9b0981..bc7cc73 100644 Binary files a/docs/memo/binary_limit/binary_limit.pdf and b/docs/memo/binary_limit/binary_limit.pdf differ diff --git a/docs/memo/binary_limit/binary_limit/diffusion.tex b/docs/memo/binary_limit/binary_limit/diffusion.tex index da2407b..f0d3b18 100644 --- a/docs/memo/binary_limit/binary_limit/diffusion.tex +++ b/docs/memo/binary_limit/binary_limit/diffusion.tex @@ -1,7 +1,7 @@ \section{Diffusion} This section discusses the binary limit of the diffusion coefficients in a ternary system. The purpose of the section is to give insight into how the force-flux in a binary system relate to those in a ternary, how to interpret the diffusion coefficients in a ternary system, and raise awareness regarding potential pitfalls when attempting to model a ternary system as a pseudo-binary system. -\subsection{Notation} +\subsection{Notation}\label{sec:diffusion_notation} In the following text, $D_{ii}^{(b)}$ is used to denote the diffusion coefficient of a binary system, where component $i$ is the independent component. $D_{ii}^{(k)}$ denotes the diagonal elements of the Fick diffusion matrix in a ternary system, where component $k$ is the dependent component. $D_{ij}$ denotes the non-diagonal elements of the Fick diffusion matrix in a ternary system, where components $i$ and $j$ are the independent components. diff --git a/docs/memo/binary_limit/binary_limit/main.tex b/docs/memo/binary_limit/binary_limit/main.tex index cc17186..320dd8f 100644 --- a/docs/memo/binary_limit/binary_limit/main.tex +++ b/docs/memo/binary_limit/binary_limit/main.tex @@ -20,6 +20,7 @@ \section{Introduction} \input{diffusion} \clearpage \input{thermal_diffusion} +\input{soret_coefficient} \clearpage diff --git a/docs/memo/binary_limit/binary_limit/soret_coefficient.tex b/docs/memo/binary_limit/binary_limit/soret_coefficient.tex new file mode 100644 index 0000000..1b38912 --- /dev/null +++ b/docs/memo/binary_limit/binary_limit/soret_coefficient.tex @@ -0,0 +1,66 @@ +\section{Soret coefficients} + +The Soret coefficient is a measure of the steady state separation in a mixture induced by a temperature gradient. Ortiz de Zárate \cite{ortiz2019definition} defines the Soret coefficients of a multicomponent mixture ($s$ components) through +\begin{equation} + \begin{bmatrix} + x_1 (1 - x1) & x_1 x_2 & \hdots & x_1 x_{s - 1} \\ + x_2 x_1 & x_2 (1 - x_2) & \hdots & x_2 x_{s - 1} \\ + \vdots & & \ddots & \vdots \\ + x_{s - 1} x_1 & & & x_{s - 1}(1 - x_{s - 1}) + \end{bmatrix} + \begin{pmatrix} + S_{T, 1} \\ + S_{T, 2} \\ + \vdots \\ + S_{T, s - 1} + \end{pmatrix} + = + - \frac{1}{\nabla T} + \begin{pmatrix} + \nabla x_1 \\ + \nabla x_2 \\ + \vdots \\ + \nabla x_{s - 1} + \end{pmatrix}, +\end{equation} +or more compactly, +\begin{equation} + \Mat{X} \Vec{S}_T = - \frac{\nabla \Vec{x}}{\nabla T}. +\end{equation} +Similarly to the thermal diffusion coefficients, this definition carries the advantage that +\begin{equation} + \Mat{X} \Vec{S}_T = - \frac{\nabla \Vec{x}}{\nabla T} \quad \iff \quad \Mat{W} \Vec{S}_T = - \frac{\nabla \Vec{w}}{\nabla T}, +\end{equation} +such that mole- and mass fractions can be used interchangably with the same Soret coefficients. + +In the state with vanishing mass fluxes ($\Vec{J} = \Vec{0}$). From the condition of vanishing mass fluxes, we find that we can compute the Soret coefficients as\footnote{See the memo on definitions of the diffusion coefficient for notes on $\Mat{D}^{(z)}$.} +\begin{equation} + \begin{split} + \Vec{J} = - c \left( \Mat{X} \Vec{D}_{T}^{(z)} \nabla T + \Mat{X} \Mat{D}^{(z)} \Mat{X}^{-1} \nabla \Vec{x} \right) &= \Vec{0} \\ + - \frac{\nabla \Vec{x}}{\nabla T} &= \Mat{X} \left(\Mat{D}^{(z)}\right)^{-1} \Vec{D}_{T}^{(z)} \\ + \Vec{S}_T &= \left(\Mat{D}^{(z)}\right)^{-1} \Vec{D}_{T}^{(z)}. + \end{split} +\end{equation} + +For a binary system (1, 2), this definition thus yields +\begin{equation} + S_{T, 1}^{(b)} = \frac{D_{T, 1}^{(z, b)}}{D_{11}^{(z, b)}}. +\end{equation} + +From the preceding relations it is possible to show that when using this definition of the Soret coefficient \cite{ortiz2019definition}, +\begin{equation} + \begin{split} + \lim_{x_1 \to 0} S_{T, 2} = S_{T, 2}^{(b, 3)} \\ + \lim_{x_2 \to 0} S_{T, 1} = S_{T, 1}^{(b, 3)} \\ + \lim_{x_3 \to 0} S_{T, 1} - S_{T, 2} = S_{T, 1}^{(b, 2)}, + \end{split} + \label{eq:soret_binary_limit} +\end{equation} +where $S_{T, i}^{b, j}$ denotes the Soret coefficient of component $i$ in a binary mixture with $j$, with species $j$ being the dependent species. Figure \ref{fig:soret_binary_limit} shows how this convergence behaviour is obeyed. + +\begin{figure} + \centering + \includegraphics[width=.85\textwidth]{soret_limit.pdf} + \caption{The convergence of the ternary Soret coefficient to the corresponding binaries as indicated in Eq. \eqref{eq:soret_binary_limit}.} + \label{fig:soret_binary_limit} +\end{figure} \ No newline at end of file diff --git a/docs/memo/binary_limit/binary_limit/soret_limit.pdf b/docs/memo/binary_limit/binary_limit/soret_limit.pdf new file mode 100644 index 0000000..c585cdd Binary files /dev/null and b/docs/memo/binary_limit/binary_limit/soret_limit.pdf differ diff --git a/docs/memo/binary_limit/binary_limit/thermal_diffusion.tex b/docs/memo/binary_limit/binary_limit/thermal_diffusion.tex index 3eac91b..6841968 100644 --- a/docs/memo/binary_limit/binary_limit/thermal_diffusion.tex +++ b/docs/memo/binary_limit/binary_limit/thermal_diffusion.tex @@ -10,6 +10,8 @@ \subsection{Notation} The notation $D_{T,i}^{(z,tj)}$ denotes the thermal diffusion coefficient of species $i$ in a ternary mixture with species $j$ taken to be the dependent species, as defined in ref. \cite{ortiz2019definition} while $D_{T,i}^{(z,bj)}$ denotes the thermal diffusion coefficient of species $i$ in a binary mixture, with species $j$ taken to be the dependent species. +Boldface roman font ($\Vec{v}$) is used to indicate vectors, and slanted underlined boldface figures ($\Mat{D}$) are used to indicate matrices. The notation $\nabla \Vec{v}$, should be understood not as a dot product, but as a vector of gradients (i.e. $\nabla \Vec{v} \equiv (\nabla v_1, \nabla v_2, ...)^\top$, in which case the equations should be understood to hold componentwise for these gradients. + \subsection{Definitions of the thermal diffusion coefficient} The default definition of the thermal diffusion coefficient in a multicomponent mixture in the KineticGas package is @@ -62,7 +64,7 @@ \subsection{Definitions of the thermal diffusion coefficient} where $D_{T,1}^{(z,t)}$ is the thermal diffusion coefficient of species 1 in a ternary mixture, and $D_{T,1}^{(z,b3)}$ is the thermal diffusion coefficient of species 1 in a binary mixture with species 3, both as defined by Eq. \eqref{eq:zarate_def}. that is: For a ternary system (1, 2, 3), when the mole fraction of species 2 tends to zero, the thermal diffusion coefficient of species 1 approaches the thermal diffusion coefficient of species 1 in the binary mixture (1, 3). This relation follows from an argument analogous to that in section \ref{sec:diff_indep}. -We can relate the thermal diffusion coefficients $\Vec{D}_T^{(n)}$ to the $\Vec{D}_T^{(z)}$, by rewriting Eq. \eqref{eq:DTn_def} as +For an ideal gas can relate the thermal diffusion coefficients $\Vec{D}_T^{(n)}$ to the $\Vec{D}_T^{(z)}$, by rewriting Eq. \eqref{eq:DTn_def} as \begin{equation} \begin{split} \Vec{J}^{(n, n)} &= - \Mat{D}^{(n)} ( c \nabla \Vec{x} + \Vec{x} \nabla c) + \Vec{D}_T^{(n)} \nabla \ln T \\ @@ -72,11 +74,31 @@ \subsection{Definitions of the thermal diffusion coefficient} \end{equation} such that $\Vec{D}_T^{(z)}$ is given by the solution to \begin{equation} - - c \Mat{X} \Vec{D}_T^{(z)} = \frac{1}{T} \left( \Vec{D}_T^{(n)} + c \Mat{D}^{(n)}\Vec{x}\right). + - c \Mat{X} \Vec{D}_T^{(z)} = \frac{1}{T} \left( \Vec{D}_T^{(n)} + c \Mat{D}^{(n)}\Vec{x}\right), \quad \text{Ideal gas} \label{eq:DTz_DTn_relate} \end{equation} +where we have made use of $\nabla c = c \nabla \ln T$ for an ideal gas. For a non-ideal gas the corresponding expression is +\begin{equation} + \begin{split} + \Vec{J}^{(n, n)} &= - \Mat{D}^{(n)} \left[\ppder{\vec{c}}{T}_{\vec{x}, p} \nabla T + \Mat{\Gamma}_c \nabla \vec{x}\right] + \frac{1}{T}\Vec{D}_T^{(n)} \nabla T \\ + &= - \Mat{D}^{(n)} \Mat{\Gamma}_c \nabla \vec{x} + \left[\frac{1}{T}\Vec{D}_T^{(n)} - \Mat{D}^{(n)} \ppder{\vec{c}}{T}_{\vec{x}, p}\right] \nabla T, + \end{split} +\end{equation} +where +\begin{equation} + \left[\Mat{\Gamma}_c\right]_{ij} = \ppder{c_i}{x_j}_{T, p, x_{k \neq j}}, +\end{equation} +and it can be useful to note that +\begin{equation} + \begin{split} + c_i &= \frac{x_i}{v} \implies \\ + \ppder{c_i}{T}_{p, \vec{x}} &= - \frac{x_i}{v^2} \ppder{v}{T}_{p, \vec{x}}, \\ + \ppder{c_i}{x_j}_{T, p, x_{k \neq j}} &= \frac{\delta_{ij}}{v} - \frac{c_i v_j}{v}, + \end{split} +\end{equation} +and to keep in mind that these matrices and vectors are in $\mathbb{R}^{s - 1}$, because the dependent component is excluded. \textbf{Note:} At the time of writing, the expressions for non-ideal gas have not been implemented. -By the same manipulation, and noting that we can write $\nabla \vec{x} = \Mat{T}^{x \leftmapsto w} \nabla \Vec{w}$, we can rewrite equation $\eqref{eq:Dtm_def}$ as +Returning to an ideal gas, using $\nabla c = c \nabla \ln T$, and noting that we can write $\nabla \vec{x} = \Mat{T}^{x \leftmapsto w} \nabla \Vec{w}$, we can rewrite equation $\eqref{eq:Dtm_def}$ as \begin{equation} \begin{split} \Vec{J}^{(n, m)} &= - \Mat{D}^{(m)} \nabla \Vec{c} + \Vec{D}_T^{(m)} \nabla \ln T \\ @@ -96,24 +118,30 @@ \subsection{Definitions of the thermal diffusion coefficient} \begin{split} \frac{1}{c} \Mat{X}^{-1}\left( \Vec{D}_T^{(n)} + c \Mat{D}^{(n)}\Vec{x}\right) &= \frac{1}{\rho} \Mat{W}^{-1} \diag(\Vec{M}) \left( \Vec{D}_T^{(m)} + c \Mat{D}^{(m)}\Vec{x}\right) \\ &= \frac{1}{\rho} \Mat{W}^{-1} \diag(\Vec{M}) \Mat{\Psi}^{m \leftmapsto n} \left( \Vec{D}_T^{(n)} + c \Mat{D}^{(n)}\Vec{x}\right) \\ - \frac{1}{c} \Mat{X}^{-1} \Mat{\Psi}^{n \leftmapsto m} \left( \Vec{D}_T^{(m)} + c \Mat{D}^{(m)}\Vec{x}\right) &= \frac{1}{\rho} \Mat{W}^{-1} \diag(\Vec{M}) \left( \Vec{D}_T^{(m)} + c \Mat{D}^{(m)}\Vec{x}\right) + \Mat{I} &= \frac{c}{\rho} \Mat{X} \Mat{W}^{-1} \diag(\Vec{M}) \Mat{\Psi}^{m \leftmapsto n}, \quad \text{and} \\ + \frac{1}{c} \Mat{X}^{-1} \Mat{\Psi}^{n \leftmapsto m} \left( \Vec{D}_T^{(m)} + c \Mat{D}^{(m)}\Vec{x}\right) &= \frac{1}{\rho} \Mat{W}^{-1} \diag(\Vec{M}) \left( \Vec{D}_T^{(m)} + c \Mat{D}^{(m)}\Vec{x}\right) \\ + \frac{\rho}{c} \diag(\Vec{M}^{-1}) \Mat{W} \Mat{X}^{-1} \Mat{\Psi}^{n \leftmapsto m} &= \Mat{I} \end{split} \end{equation} -As a second side-effect of this procedure, we can relate the diffusion matrices, as defined by Eqs. \eqref{eq:zarate_def} to $\Mat{D}^{(m)}$ and $\Mat{D}^{(n)}$ as +As a second side-effect of this procedure, we can relate the diffusion matrices (for an ideal gas), as defined by Eqs. \eqref{eq:zarate_def} to $\Mat{D}^{(m)}$ and $\Mat{D}^{(n)}$ as \begin{equation} \begin{split} \Mat{D}^{(x)} &= \Mat{D}^{(n)} \\ \Mat{D}^{(w)} &= \frac{c}{\rho} \diag(\Vec{M}) \Mat{D}^{(m)} \Mat{T}^{x \leftmapsto w} \end{split} + \label{eq:zarate_D_impl} \end{equation} and remark that Ortiz de Zárate gives the relation \begin{equation} - \Mat{W}^{-1} \Mat{D}^{(w)} \Mat{W} = \Mat{X}^{-1} \Mat{D}^{(x)} \Mat{X}, + \Mat{W}^{-1} \Mat{D}^{(w)} \Mat{W} = \Mat{X}^{-1} \Mat{D}^{(x)} \Mat{X} \equiv \Mat{D}^{(z)}, + \label{eq:zarate_D_equiv} \end{equation} which can serve as a second consistency check. -All the consistency checks mentioned above have been carried out numerically and been found to pass. +All the consistency checks mentioned above have been carried out numerically and been found to pass. + +In light of Eq. \eqref{eq:zarate_D_impl}, the simplest way to implement these diffusion matrices is likely to implement $\Mat{D}^{(n)}$, and to compute $\Mat{D}^{(w)}$ and $\Mat{D}^{(z)}$ using the relations in Eq. \eqref{eq:zarate_D_equiv}. This is the approach taken in the KineticGas package. \subsubsection{The binary case} @@ -147,4 +175,4 @@ \subsection{The binary limit} \includegraphics[width=.85\textwidth]{ternary_DT_large.pdf} \caption{The same thermal diffusion coefficients as in Fig. \ref{fig:ternary_DT}, with deviations and for a larger composition span.} \label{fig:ternary_DT_large} -\end{figure} +\end{figure} \ No newline at end of file