github/CoolProp
1
1
Fork 0
mirror of https://github.com/CoolProp/CoolProp.git synced 2026-02-08 21:05:14 -05:00
Code Issues Packages Projects Releases Wiki Activity

docs for dTau3 derivation from Lemmon2005

Browse Source 
Signed-off-by: Ian Bell <ian.h.bell@gmail.com>
This commit is contained in:
Ian Bell
2014-06-06 18:54:35 +02:00
parent 69c21800bb
commit da7b9c7793
1 changed files with 8 additions and 25 deletions
Download Patch File Download Diff File Expand all files Collapse all files

33
src/Helmholtz.cpp
View File

@@ -879,31 +879,14 @@ long double ResidualHelmholtzLemmon2005::dDelta_dTau2(const long double &tau, co
}
/**
\f[
\frac{{{\partial ^2}{\alpha ^r}}}{{\partial {\tau ^2}}} = {N_k}{\delta ^{{d_k}}}{\tau ^{{t_k} - 2}}\exp \left( { - {\delta ^{{l_k}}}} \right)\exp \left( { - {\tau ^{{m_k}}}} \right)\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right]\\
\f]
\f[
\frac{{{\partial ^2}{\alpha ^r}}}{{\partial {\tau ^2}}} = {N_k}{\delta ^{{d_k}}}\exp \left( { - {\delta ^{{l_k}}}} \right){\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right]\\
\f]
Group all the terms that don't depend on \f$ \tau \f$
\f[
\frac{{{\partial ^2}{\alpha ^r}}}{{\partial {\tau ^2}}} = A{\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right]\\
\f]
\f[
\frac{1}{A}\frac{{{\partial ^3}{\alpha ^r}}}{{\partial {\tau ^3}}} = {\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)\frac{\partial }{{\partial \tau }}\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right] + \frac{\partial }{{\partial \tau }}\left[ {{\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)} \right]\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right]\\
\f]
\f[
\frac{\partial }{{\partial \tau }}\left[ {{\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)} \right] = ({t_k} - 2){\tau ^{{t_k} - 3}}\exp \left( { - {\tau ^{{m_k}}}} \right) + {\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)( - {m_k}{\tau ^{{m_k} - 1}}) = \exp \left( { - {\tau ^{{m_k}}}} \right)\left( {({t_k} - 2){\tau ^{{t_k} - 3}} - {\tau ^{{t_k} - 2}}{m_k}{\tau ^{{m_k} - 1}}} \right)\\
\f]
\f[
\frac{\partial }{{\partial \tau }}\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right] = \left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( { - m_k^2{\tau ^{{m_k} - 1}}} \right) + \left( { - m_k^2{\tau ^{{m_k} - 1}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^3{\tau ^{{m_k} - 1}} = - m_k^2{\tau ^{{m_k} - 1}}\left[ {{t_k} - {m_k}{\tau ^{{m_k}}} + {t_k} - 1 - {m_k}{\tau ^{{m_k}}} + {m_k}} \right] = - m_k^2{\tau ^{{m_k} - 1}}\left[ {2{t_k} - 2{m_k}{\tau ^{{m_k}}} - 1 + {m_k}} \right]\\
\f]
\f[
\frac{1}{A}\frac{{{\partial ^3}{\alpha ^r}}}{{\partial {\tau ^3}}} = {\tau ^{{t_k} - 2}}\exp \left( { - {\tau ^{{m_k}}}} \right)\left( { - m_k^2{\tau ^{{m_k} - 1}}\left[ {2{t_k} - 2{m_k}{\tau ^{{m_k}}} - 1 + {m_k}} \right]} \right) + \exp \left( { - {\tau ^{{m_k}}}} \right)\left( {({t_k} - 2){\tau ^{{t_k} - 3}} - {\tau ^{{t_k} - 2}}{m_k}{\tau ^{{m_k} - 1}}} \right)\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right]\\
\f]
\f[
\frac{1}{A}\frac{{{\partial ^3}{\alpha ^r}}}{{\partial {\tau ^3}}} = \exp \left( { - {\tau ^{{m_k}}}} \right)\left[ { - {\tau ^{{t_k} - 2}}m_k^2{\tau ^{{m_k} - 1}}\left[ {2{t_k} - 2{m_k}{\tau ^{{m_k}}} - 1 + {m_k}} \right] + \left( {({t_k} - 2){\tau ^{{t_k} - 3}} - {\tau ^{{t_k} - 2}}{m_k}{\tau ^{{m_k} - 1}}} \right)\left[ {\left( {{t_k} - {m_k}{\tau ^{{m_k}}}} \right)\left( {{t_k} - 1 - {m_k}{\tau ^{{m_k}}}} \right) - m_k^2{\tau ^{{m_k}}}} \right]} \right]
\f]
Derived using sympy:
from sympy import *
init_printing()
n_i, d_i, t_i, m_i, l_i, tau, delta, eta, gamma, beta, epsilon = symbols('n_i, d_i, t_i, m_i, l_i, tau, delta, eta, gamma, beta, epsilon')
I = n_i*delta**d_i*tau**t_i*exp(-delta**l_i)*exp(-tau**m_i)
ccode(simplify(diff(I,tau,3)))
*/
long double ResidualHelmholtzLemmon2005::dTau3(const long double &tau, const long double &delta) throw()
{