User talk:Sidharth

FPROPS uses function pointers from the underlying correlation (helmholtz or Pengrob) to calculate fundamental quantities with Temperature and <math> \rho</math> as the input. Following are those functions of fprops :-

fprops_p() fprops_u() fprops_h() fprops_s() fprops_g()
fprops_dpdrho_T() fprops_alphap() fprops_betaP() fprops_a()
fprops_cp() fprops_cv() fprops_w()


Both the underlying correlations provide these 13 functions calculated from first principles, with T and <math>\rho</math> as inputs. So the TTSE implementation for a specific liquid and a specific correlation (H or P) should eventually generate tables for each of the above 13 entries. For the saturation function fprops_sat(), takes only 1 input (temperature) and returns saturated liquid density, saturated vapour density and the saturation pressure. This function involves solving equations iteratively and tabulation should speed up calculation on saturation curve.

To begin with we can tabulate the following functions:

• helmholtz_p
• helmholtz_s
• helmholtz_u
• helmholtz_g
• helmholtz_h

In order to use look up tables for TTSE method inside these routines we need to complete the set of all partial derivatives we need :-

For each of the above variable X (where X is either of P,s,u,g and h) we need <math> \frac{\partial X}{\partial T}\bigg|_{\rho} ,

\frac{\partial^2 X}{\partial T^2}\bigg|_{\rho} ,
   \frac{\partial X}{\partial \rho}\bigg|_{T} , 
\frac{\partial^2 X}{\partial \rho^2}\bigg|_{T}</math>   and <math>
  \frac{\partial^2 X}{\partial \rho \partial T} </math>

Many of the partial derivatives are already listed in helmholtz.c. Specifically the new 18 ones we need to implement are:

Variable	Partial derivative with T constant	Partial derivative with <math>\rho</math> constant	Partial second derivative with <math>\rho</math> constant	Partial second derivative with <math>T</math> constant	Partial mixed second derivative
Pressure, P	<already implemented>	<already implemented>	<already implemented>	helmholtz_d2pdT2_rho() <math>\frac{\partial^2 p}{\partial (T^2)} </math>	helmholtz_d2pdrhodT() <math>\frac{\partial^2 p}{\partial (T \rho)} </math>
Enthalpy, h	<already implemented>	<already implemented>	helmholtz_d2hdrho2_T() <math>\frac{\partial^2 h}{\partial \rho^2}\bigg|_{T}</math>	helmholtz_d2hdT2_rho() <math>\frac{\partial^2 h}{\partial T ^2}\bigg|_{\rho}</math>	helmholtz_d2hdrhodT() <math>\frac{\partial^2 h}{\partial (T \rho)} </math>
Internal Energy, u	<already implemented>	<already implemented>	helmholtz_d2udrho2_T() <math>\frac{\partial^2 u}{\partial \rho^2}\bigg|_{T}</math>	helmholtz_d2udT2_rho() <math>\frac{\partial^2 u}{\partial T^2}\bigg|_{\rho}</math>	helmholtz_d2udrhodT() <math>\frac{\partial^2 u}{\partial (T \rho)}</math>
Entropy, s	helmholtz_dsdT_rho() <math>\frac{\partial s}{\partial T}\bigg|_{\rho}</math>	helmholtz_dsdrho_T() <math>\frac{\partial s}{\partial \rho}\bigg|_{T}</math>	helmholtz_d2sdrho2_T() <math>\frac{\partial^2 s}{\partial \rho^2}\bigg|_{T}</math>	helmholtz_d2sdT2_rho() <math>\frac{\partial^2 s}{\partial T^2}\bigg|_{\rho}</math>	helmholtz_d2sdrhodT() <math>\frac{\partial^2 s}{\partial (T \rho)}</math>
Gibbs Free Energy, g	helmholtz_dgdT_rho() <math>\frac{\partial g}{\partial T}\bigg|_{\rho}</math>	helmholtz_dgdrho_T() <math>\frac{\partial g}{\partial \rho}\bigg|_{T}</math>	helmholtz_d2gdrho2_T() <math>\frac{\partial^2 g}{\partial \rho^2}\bigg|_{T}</math>	helmholtz_d2gdT2_rho() <math>\frac{\partial^2 g}{\partial T^2}\bigg|_{\rho}</math>	helmholtz_d2gdrhodT() <math>\frac{\partial^2 g}{\partial (T \rho)}</math>

To evaluate these drivatives we need the underlying partial derivatives of the residual and the ideal part of helmholtz function. So the three new functions needed in helmholtz.c and cp0.c

Third partial derivative of ideal part of helmholtz (all other partial derivatives invloving delta are zero) in file cp0.c

• ideal_phi_tautautau()

Third derivative of helmholtz residual function, with respect to delta once and tau twice

• helm_resid_deltautau()

Third derivative of helmholtz residual function, with respect to delta twice and tau once

• helm_resid_deldeltau()

Third derivative of helmholtz residual function, with respect to tau thrice

• helm_resid_tautautau()

Once we have the above four routines then we can compute all the functions in the table. The detailed expressions to be evaluated can be found in this reference - [1]