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CoolProp/include/Helmholtz.h
Ian Bell 54d3b45584 Resort Helmholtz.h 
Signed-off-by: Ian Bell <ian.h.bell@gmail.com>
2014-06-30 17:08:39 +02:00

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#ifndef HELMHOLTZ_H
#define HELMHOLTZ_H
#include <vector>
#include "rapidjson/rapidjson_include.h"
namespace CoolProp{
/// The base class class for the Helmholtz energy terms
/**
Residual Helmholtz Energy Terms:
Term | Helmholtz Energy Contribution
---------- | ------------------------------
ResidualHelmholtzPower | \f$ \alpha^r=\left\lbrace\begin{array}{cc}\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} & l_i=0\\ \displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}) & l_i\neq 0\end{array}\right.\f$
ResidualHelmholtzExponential | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\gamma_i\delta^{l_i}) \f$
ResidualHelmholtzLemmon2005 | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}-\tau^{m_i})\f$
ResidualHelmholtzGaussian | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\eta_i(\delta-\epsilon_i)^2-\beta_i(\tau-\gamma_i)^2)\f$
ResidualHelmholtzGERG2008Gaussian | \f$ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\eta_i(\delta-\epsilon_i)^2-\beta_i(\delta-\gamma_i))\f$
ResidualHelmholtzNonAnalytic | \f$ \begin{array}{c}\alpha^r&=&\displaystyle\sum_i n_i \Delta^{b_i}\delta\psi \\ \Delta & = & \theta^2+B_i[(\delta-1)^2]^{a_i}\\ \theta & = & (1-\tau)+A_i[(\delta-1)^2]^{1/(2\beta_i)}\\ \psi & = & \exp(-C_i(\delta-1)^2-D_i(\tau-1)^2) \end{array}\f$
ResidualHelmholtzSAFTAssociating | \f$ \alpha^r = am\left(\ln X-\frac{X}{2}+\frac{1}{2}\right); \f$
Ideal-Gas Helmholtz Energy Terms:
Term | Helmholtz Energy Contribution
---------- | ------------------------------
IdealHelmholtzLead | \f$ \alpha^0 = n_1 + n_2\tau + \ln\delta \f$
IdealHelmholtzEnthalpyEntropyOffset | \f$ \alpha^0 = \displaystyle\frac{\Delta s}{R_u/M}+\frac{\Delta h}{(R_u/M)T}\tau \f$
IdealHelmholtzLogTau | \f$ \alpha^0 = n_1\log\tau \f$
IdealHelmholtzPower | \f$ \alpha^0 = \displaystyle\sum_i n_i\tau^{t_i} \f$
IdealHelmholtzPlanckEinsteinGeneralized | \f$ \alpha^0 = \displaystyle\sum_i n_i\log[c_i+d_i\exp(\theta_i\tau)] \f$
*/
class BaseHelmholtzTerm{
public:
BaseHelmholtzTerm(){};
virtual ~BaseHelmholtzTerm(){};
/// Returns the base, non-dimensional, Helmholtz energy term (no derivatives) [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double base(const long double &tau, const long double &delta) throw() = 0;
/// Returns the first partial derivative of Helmholtz energy term with respect to tau [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dTau(const long double &tau, const long double &delta) throw() = 0;
/// Returns the second partial derivative of Helmholtz energy term with respect to tau [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dTau2(const long double &tau, const long double &delta) throw() = 0;
/// Returns the second mixed partial derivative (delta1,dtau1) of Helmholtz energy term with respect to delta and tau [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dDelta_dTau(const long double &tau, const long double &delta) throw() = 0;
/// Returns the first partial derivative of Helmholtz energy term with respect to delta [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dDelta(const long double &tau, const long double &delta) throw() = 0;
/// Returns the second partial derivative of Helmholtz energy term with respect to delta [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dDelta2(const long double &tau, const long double &delta) throw() = 0;
/// Returns the third mixed partial derivative (delta2,dtau1) of Helmholtz energy term with respect to delta and tau [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dDelta2_dTau(const long double &tau, const long double &delta) throw() = 0;
/// Returns the third mixed partial derivative (delta1,dtau2) of Helmholtz energy term with respect to delta and tau [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dDelta_dTau2(const long double &tau, const long double &delta) throw() = 0;
/// Returns the third partial derivative of Helmholtz energy term with respect to tau [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dTau3(const long double &tau, const long double &delta) throw() = 0;
/// Returns the third partial derivative of Helmholtz energy term with respect to delta [-]
/** @param tau Reciprocal reduced temperature where \f$\tau=T_c / T\f$
* @param delta Reduced density where \f$\delta = \rho / \rho_c \f$
*/
virtual long double dDelta3(const long double &tau, const long double &delta) throw() = 0;
};
// #############################################################################
// #############################################################################
// #############################################################################
// RESIDUAL TERMS
// #############################################################################
// #############################################################################
// #############################################################################
struct ResidualHelmholtzPowerElement
{
long double n,d,t,ld;
int l;
};
/// Power term
/*!
Term of the form
\f[
\alpha^r=\left\lbrace\begin{array}{cc}\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} & l_i=0\\ \displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\delta^{l_i}) & l_i\neq 0\end{array}\right.
\f]
*/
class ResidualHelmholtzPower : public BaseHelmholtzTerm{
public:
/*std::vector<double> d, ///< The power for the delta terms
t, ///< The powers for the tau terms
l; ///< The powers for delta in the exp terms*/
std::vector<long double> s;
std::size_t N;
std::vector<ResidualHelmholtzPowerElement> elements;
// Default Constructor
ResidualHelmholtzPower(){N = 0;};
// Constructor
ResidualHelmholtzPower(const std::vector<long double> &n, const std::vector<long double> &d,
const std::vector<long double> &t, const std::vector<long double> &l)
{
N = n.size();
s.resize(N);
for (std::size_t i = 0; i < n.size(); ++i)
{
ResidualHelmholtzPowerElement el;
el.n = n[i];
el.d = d[i];
el.t = t[i];
el.ld = l[i];
el.l = (int)el.ld;
elements.push_back(el);
}
};
///< Destructor for the alphar_power class. No implementation
~ResidualHelmholtzPower(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
struct ResidualHelmholtzExponentialElement
{
long double n,d,t,g,l;
};
/**
Term of the form
\f[ \alpha^r=\displaystyle\sum_i n_i \delta^{d_i} \tau^{t_i} \exp(-\gamma_i\delta^{l_i}) \f]
*/
class ResidualHelmholtzExponential : public BaseHelmholtzTerm{
public:
std::vector<long double> s;
std::size_t N;
std::vector<ResidualHelmholtzExponentialElement> elements;
// Default Constructor
ResidualHelmholtzExponential(){N = 0;};
// Constructor
ResidualHelmholtzExponential(const std::vector<long double> &n, const std::vector<long double> &d,
const std::vector<long double> &t, const std::vector<long double> &g,
const std::vector<long double> &l)
{
N = n.size();
s.resize(N);
for (std::size_t i = 0; i < n.size(); ++i)
{
ResidualHelmholtzExponentialElement el;
el.d = d[i];
el.t = t[i];
el.g = g[i];
el.l = l[i];
el.n = n[i];
elements.push_back(el);
}
}
///< Destructor for the alphar_power class. No implementation
~ResidualHelmholtzExponential(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
struct ResidualHelmholtzGaussianElement
{
long double n, d, t, eta, epsilon, beta, gamma;
};
class ResidualHelmholtzGaussian : public BaseHelmholtzTerm{
public:
std::size_t N; ///< The number of terms
std::vector<ResidualHelmholtzGaussianElement> elements;
std::vector<long double> s;
// Default Constructor
ResidualHelmholtzGaussian(){N = 0;};
// Constructor
ResidualHelmholtzGaussian(const std::vector<long double> &n,
const std::vector<long double> &d,
const std::vector<long double> &t,
const std::vector<long double> &eta,
const std::vector<long double> &epsilon,
const std::vector<long double> &beta,
const std::vector<long double> &gamma
)
{
N = n.size();
this->s.resize(N);
for (std::size_t i = 0; i < n.size(); ++i)
{
ResidualHelmholtzGaussianElement el;
el.n = n[i];
el.d = d[i];
el.t = t[i];
el.eta = eta[i];
el.epsilon = epsilon[i];
el.beta = beta[i];
el.gamma = gamma[i];
elements.push_back(el);
}
};
///< Destructor for the alphar_power class. No implementation
~ResidualHelmholtzGaussian(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
class ResidualHelmholtzGERG2008Gaussian : public BaseHelmholtzTerm{
public:
std::vector<long double> s;
std::size_t N;
std::vector<ResidualHelmholtzGaussianElement> elements;
/// Default Constructor
ResidualHelmholtzGERG2008Gaussian(){N = 0;};
/// Constructor
ResidualHelmholtzGERG2008Gaussian(const std::vector<long double> &n,
const std::vector<long double> &d,
const std::vector<long double> &t,
const std::vector<long double> &eta,
const std::vector<long double> &epsilon,
const std::vector<long double> &beta,
const std::vector<long double> &gamma)
{
N = n.size();
s.resize(N);
for (std::size_t i = 0; i < n.size(); ++i)
{
ResidualHelmholtzGaussianElement el;
el.n = n[i];
el.d = d[i];
el.t = t[i];
el.eta = eta[i];
el.epsilon = epsilon[i];
el.beta = beta[i];
el.gamma = gamma[i];
elements.push_back(el);
}
};
///< Destructor. No implementation
~ResidualHelmholtzGERG2008Gaussian(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
struct ResidualHelmholtzLemmon2005Element{
long double n, d, t, ld, md;
int l, m;
};
class ResidualHelmholtzLemmon2005 : public BaseHelmholtzTerm{
public:
std::size_t N;
std::vector<long double> s; ///< Summation container
std::vector<ResidualHelmholtzLemmon2005Element> elements;
// Default Constructor
ResidualHelmholtzLemmon2005(){N = 0;};
// Constructor
ResidualHelmholtzLemmon2005(const std::vector<long double> &n,
const std::vector<long double> &d,
const std::vector<long double> &t,
const std::vector<long double> &l,
const std::vector<long double> &m)
{
N = n.size();
s.resize(N);
for (std::size_t i = 0; i < n.size(); ++i)
{
ResidualHelmholtzLemmon2005Element el;
el.n = n[i];
el.d = d[i];
el.t = t[i];
el.ld = l[i];
el.md = m[i];
el.l = (int)el.ld;
el.m = (int)el.md;
elements.push_back(el);
}
};
///< Destructor. No implementation
~ResidualHelmholtzLemmon2005(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
struct ResidualHelmholtzNonAnalyticElement
{
long double n, a, b, beta, A, B, C, D;
};
class ResidualHelmholtzNonAnalytic : public BaseHelmholtzTerm{
public:
std::size_t N;
std::vector<long double> s;
std::vector<ResidualHelmholtzNonAnalyticElement> elements;
/// Default Constructor
ResidualHelmholtzNonAnalytic(){N = 0;};
/// Destructor. No implementation
~ResidualHelmholtzNonAnalytic(){};
/// Constructor
ResidualHelmholtzNonAnalytic(const std::vector<long double> &n,
const std::vector<long double> &a,
const std::vector<long double> &b,
const std::vector<long double> &beta,
const std::vector<long double> &A,
const std::vector<long double> &B,
const std::vector<long double> &C,
const std::vector<long double> &D
)
{
N = n.size();
s.resize(N);
for (std::size_t i = 0; i < n.size(); ++i)
{
ResidualHelmholtzNonAnalyticElement el;
el.n = n[i];
el.a = a[i];
el.b = b[i];
el.beta = beta[i];
el.A = A[i];
el.B = B[i];
el.C = C[i];
el.D = D[i];
elements.push_back(el);
}
};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
class ResidualHelmholtzSAFTAssociating : public BaseHelmholtzTerm{
protected:
double a, m,epsilonbar, vbarn, kappabar;
long double Deltabar(const long double &tau, const long double &delta);
long double dDeltabar_ddelta__consttau(const long double &tau, const long double &delta);
long double d2Deltabar_ddelta2__consttau(const long double &tau, const long double &delta);
long double dDeltabar_dtau__constdelta(const long double &tau, const long double &delta);
long double d2Deltabar_dtau2__constdelta(const long double &tau, const long double &delta);
long double d2Deltabar_ddelta_dtau(const long double &tau, const long double &delta);
long double d3Deltabar_dtau3__constdelta(const long double &tau, const long double &delta);
long double d3Deltabar_ddelta_dtau2(const long double &tau, const long double &delta);
long double d3Deltabar_ddelta3__consttau(const long double &tau, const long double &delta);
long double d3Deltabar_ddelta2_dtau(const long double &tau, const long double &delta);
long double X(const long double &delta, const long double &Deltabar);
long double dX_dDeltabar__constdelta(const long double &delta, const long double &Deltabar);
long double dX_ddelta__constDeltabar(const long double &delta, const long double &Deltabar);
long double dX_dtau(const long double &tau, const long double &delta);
long double dX_ddelta(const long double &tau, const long double &delta);
long double d2X_dtau2(const long double &tau, const long double &delta);
long double d2X_ddeltadtau(const long double &tau, const long double &delta);
long double d2X_ddelta2(const long double &tau, const long double &delta);
long double d3X_dtau3(const long double &tau, const long double &delta);
long double d3X_ddelta3(const long double &tau, const long double &delta);
long double d3X_ddeltadtau2(const long double &tau, const long double &delta);
long double d3X_ddelta2dtau(const long double &tau, const long double &delta);
long double g(const long double &eta);
long double dg_deta(const long double &eta);
long double d2g_deta2(const long double &eta);
long double d3g_deta3(const long double &eta);
long double eta(const long double &delta);
public:
/// Default constructor
ResidualHelmholtzSAFTAssociating(){ disabled = true; };
// Constructor
ResidualHelmholtzSAFTAssociating(double a, double m, double epsilonbar, double vbarn, double kappabar)
: a(a), m(m), epsilonbar(epsilonbar), vbarn(vbarn), kappabar(kappabar)
{
disabled = false;
};
bool disabled;
//Destructor. No Implementation
~ResidualHelmholtzSAFTAssociating(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw();
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw();
long double dDelta_dTau(const long double &tau, const long double &delta) throw();
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw();
long double dDelta2_dTau(const long double &tau, const long double &delta) throw();
long double dDelta_dTau2(const long double &tau, const long double &delta) throw();
long double dTau3(const long double &tau, const long double &delta) throw();
};
class ResidualHelmholtzContainer
{
public:
ResidualHelmholtzPower Power;
ResidualHelmholtzExponential Exponential;
ResidualHelmholtzGaussian Gaussian;
ResidualHelmholtzLemmon2005 Lemmon2005;
ResidualHelmholtzNonAnalytic NonAnalytic;
ResidualHelmholtzSAFTAssociating SAFT;
long double base(long double tau, long double delta)
{
return (Power.base(tau, delta) + Exponential.base(tau, delta)
+Gaussian.base(tau, delta) + Lemmon2005.base(tau, delta)
+NonAnalytic.base(tau, delta) + SAFT.base(tau,delta));
};
long double dDelta(long double tau, long double delta)
{
return (Power.dDelta(tau, delta) + Exponential.dDelta(tau, delta)
+ Gaussian.dDelta(tau, delta) + Lemmon2005.dDelta(tau, delta)
+ NonAnalytic.dDelta(tau, delta) + SAFT.dDelta(tau,delta));
};
long double dTau(long double tau, long double delta)
{
return (Power.dTau(tau, delta) + Exponential.dTau(tau, delta)
+ Gaussian.dTau(tau, delta) + Lemmon2005.dTau(tau, delta)
+ NonAnalytic.dTau(tau, delta) + SAFT.dTau(tau,delta));
};
long double dDelta2(long double tau, long double delta)
{
return (Power.dDelta2(tau, delta) + Exponential.dDelta2(tau, delta)
+Gaussian.dDelta2(tau, delta) + Lemmon2005.dDelta2(tau, delta)
+NonAnalytic.dDelta2(tau, delta) + SAFT.dDelta2(tau,delta));
};
long double dDelta_dTau(long double tau, long double delta)
{
return (Power.dDelta_dTau(tau, delta) + Exponential.dDelta_dTau(tau, delta)
+ Gaussian.dDelta_dTau(tau, delta) + Lemmon2005.dDelta_dTau(tau, delta)
+ NonAnalytic.dDelta_dTau(tau, delta) + SAFT.dDelta_dTau(tau,delta));
};
long double dTau2(long double tau, long double delta)
{
return (Power.dTau2(tau, delta) + Exponential.dTau2(tau, delta)
+ Gaussian.dTau2(tau, delta) + Lemmon2005.dTau2(tau, delta)
+ NonAnalytic.dTau2(tau, delta) + SAFT.dTau2(tau,delta));
};
long double dDelta3(long double tau, long double delta)
{
return (Power.dDelta3(tau, delta) + Exponential.dDelta3(tau, delta)
+Gaussian.dDelta3(tau, delta) + Lemmon2005.dDelta3(tau, delta)
+NonAnalytic.dDelta3(tau, delta) + SAFT.dDelta3(tau,delta));
};
long double dDelta2_dTau(long double tau, long double delta)
{
return (Power.dDelta2_dTau(tau, delta) + Exponential.dDelta2_dTau(tau, delta)
+ Gaussian.dDelta2_dTau(tau, delta) + Lemmon2005.dDelta2_dTau(tau, delta)
+ NonAnalytic.dDelta2_dTau(tau, delta) + SAFT.dDelta2_dTau(tau,delta));
};
long double dDelta_dTau2(long double tau, long double delta)
{
return (Power.dDelta_dTau2(tau, delta) + Exponential.dDelta_dTau2(tau, delta)
+ Gaussian.dDelta_dTau2(tau, delta) + Lemmon2005.dDelta_dTau2(tau, delta)
+ NonAnalytic.dDelta_dTau2(tau, delta) + SAFT.dDelta_dTau2(tau,delta));
};
long double dTau3(long double tau, long double delta)
{
return (Power.dTau3(tau, delta) + Exponential.dTau3(tau, delta)
+Gaussian.dTau3(tau, delta) + Lemmon2005.dTau3(tau, delta)
+NonAnalytic.dTau3(tau, delta) + SAFT.dTau3(tau,delta));
};
};
// #############################################################################
// #############################################################################
// #############################################################################
// IDEAL GAS TERMS
// #############################################################################
// #############################################################################
// #############################################################################
/// The leading term in the EOS used to set the desired reference state
/**
\f[
\alpha^0 = \log(\delta)+a_1+a_2\tau
\f]
*/
class IdealHelmholtzLead : public BaseHelmholtzTerm{
private:
long double a1, a2;
bool enabled;
public:
// Default constructor
IdealHelmholtzLead(){enabled = false;};
// Constructor
IdealHelmholtzLead(const long double a1, const long double a2)
:a1(a1), a2(a2)
{enabled = true;};
//Destructor
~IdealHelmholtzLead(){};
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc){
el.AddMember("type","IdealHelmholtzLead",doc.GetAllocator());
el.AddMember("a1", static_cast<double>(a1), doc.GetAllocator());
el.AddMember("a2", static_cast<double>(a2), doc.GetAllocator());
};
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return log(delta)+a1+a2*tau;
};
long double dDelta(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return 1.0/delta;
};
long double dTau(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return a2;
};
long double dDelta2(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return -1.0/delta/delta;
};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta3(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return 2/delta/delta/delta;
};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau3(const long double &tau, const long double &delta) throw(){return 0.0;};
};
/// The term in the EOS used to shift the reference state of the fluid
/**
\f[
\alpha^0 = a_1+a_2\tau
\f]
*/
class IdealHelmholtzEnthalpyEntropyOffset : public BaseHelmholtzTerm{
private:
long double a1,a2; // Use these variables internally
bool enabled;
std::string reference;
public:
IdealHelmholtzEnthalpyEntropyOffset(){enabled = false;};
// Constructor
IdealHelmholtzEnthalpyEntropyOffset(long double a1, long double a2, std::string reference):a1(a1), a2(a2){this->reference = reference; enabled = true;};
// Set the values in the class
void set(long double a1, long double a2, std::string reference){this->a1 = a1; this->a2 = a2; this->reference = reference; enabled = true;}
//Destructor
~IdealHelmholtzEnthalpyEntropyOffset(){};
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc){
el.AddMember("type","IdealHelmholtzEnthalpyEntropyOffset",doc.GetAllocator());
el.AddMember("a1", static_cast<double>(a1), doc.GetAllocator());
el.AddMember("a2", static_cast<double>(a2), doc.GetAllocator());
};
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return a1+a2*tau;
};
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return a2;
};
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau3(const long double &tau, const long double &delta) throw(){return 0.0;};
};
/**
\f[
\alpha^0 = a_1\ln\tau
\f]
*/
class IdealHelmholtzLogTau : public BaseHelmholtzTerm
{
private:
long double a1;
bool enabled;
public:
/// Default constructor
IdealHelmholtzLogTau(){enabled = false;};
// Constructor
IdealHelmholtzLogTau(long double a1){this->a1=a1; enabled = true;};
bool is_enabled(){return enabled;};
//Destructor
~IdealHelmholtzLogTau(){};
void to_json(rapidjson::Value &el, rapidjson::Document &doc){
el.AddMember("type", "IdealHelmholtzLogTau", doc.GetAllocator());
el.AddMember("a1", static_cast<double>(a1), doc.GetAllocator());
};
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return a1*log(tau);
};
long double dTau(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return a1/tau;
};
long double dTau2(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return -a1/tau/tau;
};
long double dTau3(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return 2*a1/tau/tau/tau;
};
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0.0;};
};
/**
\f[
\alpha^0 = \displaystyle\sum_i n_i\tau^{t_i}
\f]
*/
class IdealHelmholtzPower : public BaseHelmholtzTerm{
private:
std::vector<long double> n, t; // Use these variables internally
std::size_t N;
bool enabled;
public:
IdealHelmholtzPower(){enabled = false;};
// Constructor
IdealHelmholtzPower(const std::vector<long double> &n, const std::vector<long double> &t)
:n(n), t(t)
{
this->N = n.size();
enabled = true;
};
//Destructor
~IdealHelmholtzPower(){};
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc)
{
el.AddMember("type","IdealHelmholtzPower",doc.GetAllocator());
cpjson::set_long_double_array("n",n,el,doc);
cpjson::set_long_double_array("t",t,el,doc);
};
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i = 0; i<N; ++i){s += n[i]*pow(tau, t[i]);} return s;
};
long double dTau(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i = 0; i<N; ++i){s += n[i]*t[i]*pow(tau, t[i]-1);} return s;
};
long double dTau2(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i = 0; i<N; ++i){s += n[i]*t[i]*(t[i]-1)*pow(tau, t[i]-2);} return s;
};
long double dTau3(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i = 0; i<N; ++i){s += n[i]*t[i]*(t[i]-1)*(t[i]-2)*pow(tau, t[i]-3);} return s;
};
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0.0;};
};
/**
\f[
\alpha^0 = \displaystyle\sum_i n_i\log[c_i+d_i\exp(\theta_i\tau)]
\f]
To convert conventional Plank-Einstein forms, given by
\f$
\frac{c_p^0}{R} = a_k\displaystyle\frac{\left( b_k/T \right)^2\exp \left( b_k/T \right)}{\left(\exp \left(b_k/T\right) - 1 \right)^2}
\f$
and
\f$
\alpha^0 = a_k\ln \left[1 - \exp \left( \frac{-b_k\tau}{T_c} \right) \right]
\f$
use \f$c = 1\f$, \f$d = -1\f$, \f$n = a\f$, \f$\theta = -\displaystyle\frac{b_k}{T_c}\f$
To convert the second form of Plank-Einstein terms, given by
\f$
\frac{c_p^0}{R} = a_k\displaystyle\frac{\left( -b_k/T \right)^2\exp \left( b_k/T \right)}{c\left(\exp \left(-b_k/T\right) + 1 \right)^2}
\f$
and
\f$
\alpha^0 = a_k\ln \left[c + \exp \left( \frac{b_k\tau}{T_c} \right) \right]
\f$
use \f$c = 1\f$, \f$d = 1\f$, \f$n = -a\f$, \f$\theta = \displaystyle\frac{b_k}{T_c}\f$
Converting Aly-Lee tems is a bit more complex
Aly-Lee starts as
\f[\frac{c_p^0}{R_u} = A + B\left(\frac{C/T}{\sinh(C/T)}\right)^2 + D\left(\frac{E/T}{\cosh(E/T)}\right)^2\f]
Constant is separated out, and handled separately. sinh part can be expanded as
\f[B\left(\frac{C/T}{\sinh(C/T)}\right)^2 = \frac{B(-2C/T)^2\exp(-2C/T)}{(1-\exp(-2C/T))^2}\f]
where
\f[n_k = B\f]
\f[\theta_k = -\frac{2C}{T_c}\f]
\f[c_k = 1\f]
\f[d_k = -1\f]
cosh part can be expanded as
\f[D\left(\frac{E/T}{\cosh(E/T)}\right)^2 = \frac{D(-2E/T)^2\exp(-2E/T)}{(1+\exp(-2E/T))^2}\f]
where
\f[n_k = -D\f]
\f[\theta_k = -\frac{2E}{T_c}\f]
\f[c_k = 1\f]
\f[d_k = 1\f]
*/
class IdealHelmholtzPlanckEinsteinGeneralized : public BaseHelmholtzTerm{
private:
std::vector<long double> n,theta,c,d; // Use these variables internally
std::size_t N;
bool enabled;
public:
IdealHelmholtzPlanckEinsteinGeneralized(){N = 0; enabled = false;}
// Constructor with std::vector instances
IdealHelmholtzPlanckEinsteinGeneralized(std::vector<long double> n, std::vector<long double> theta, std::vector<long double> c, std::vector<long double> d)
:n(n), theta(theta), c(c), d(d)
{
N = n.size();
enabled = true;
};
// Destructor
~IdealHelmholtzPlanckEinsteinGeneralized(){};
// Extend the vectors to allow for multiple instances feeding values to this function
void extend(std::vector<long double> n, std::vector<long double> theta, std::vector<long double> c, std::vector<long double> d)
{
this->n.insert(this->n.end(), n.begin(), n.end());
this->theta.insert(this->theta.end(), theta.begin(), theta.end());
this->c.insert(this->c.end(), c.begin(), c.end());
this->d.insert(this->d.end(), d.begin(), d.end());
N += n.size();
}
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc)
{
el.AddMember("type","IdealHelmholtzPlanckEinsteinGeneralized",doc.GetAllocator());
cpjson::set_long_double_array("n",n,el,doc);
cpjson::set_long_double_array("theta",theta,el,doc);
};
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i=0; i < N; ++i){
s += n[i]*log(c[i]+d[i]*exp(theta[i]*tau));
}
return s;
};
long double dTau(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i=0; i < N; ++i){s += n[i]*theta[i]*d[i]*exp(theta[i]*tau)/(c[i]+d[i]*exp(theta[i]*tau));}
return s;
};
long double dTau2(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i=0; i < N; ++i){s += n[i]*pow(theta[i],2)*c[i]*d[i]*exp(theta[i]*tau)/pow(c[i]+d[i]*exp(theta[i]*tau),2);} return s;
};
long double dTau3(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
long double s=0; for (std::size_t i=0; i < N; ++i){s += n[i]*pow(theta[i],3)*c[i]*d[i]*(c[i]-d[i]*exp(theta[i]*tau))*exp(theta[i]*tau)/pow(c[i]+d[i]*exp(theta[i]*tau),3);} return s;
};
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0;};
};
class IdealHelmholtzCP0Constant : public BaseHelmholtzTerm{
private:
double cp_over_R,Tc,T0,tau0; // Use these variables internally
bool enabled;
public:
/// Default constructor
IdealHelmholtzCP0Constant(){enabled = false;};
/// Constructor with just a single double value
IdealHelmholtzCP0Constant(long double cp_over_R, long double Tc, long double T0)
: cp_over_R(cp_over_R), Tc(Tc), T0(T0)
{
enabled = true; tau0 = Tc/T0;
};
/// Destructor
~IdealHelmholtzCP0Constant(){};
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc)
{
el.AddMember("type","IdealGasHelmholtzCP0Constant", doc.GetAllocator());
el.AddMember("cp_over_R", cp_over_R, doc.GetAllocator());
el.AddMember("Tc", Tc, doc.GetAllocator());
el.AddMember("T0", T0, doc.GetAllocator());
};
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return cp_over_R-cp_over_R*tau/tau0+cp_over_R*log(tau/tau0);
};
long double dTau(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return cp_over_R/tau-cp_over_R/tau0;
};
long double dTau2(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return -cp_over_R/(tau*tau);
};
long double dTau3(const long double &tau, const long double &delta) throw(){
if (!enabled){return 0.0;}
return 2*cp_over_R/(tau*tau*tau);
};
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0.0;};
};
class IdealHelmholtzCP0PolyT : public BaseHelmholtzTerm{
private:
std::vector<long double> c, t;
long double Tc, T0, tau0; // Use these variables internally
std::size_t N;
bool enabled;
public:
/// Destructor
IdealHelmholtzCP0PolyT(){N = 0; enabled = false;};
/// Constructor with std::vectors
IdealHelmholtzCP0PolyT(const std::vector<long double> &c, const std::vector<long double> &t, double Tc, double T0)
: c(c), t(t), Tc(Tc), T0(T0)
{
assert(c.size() == t.size());
tau0 = Tc/T0;
enabled = true;
N = c.size();
};
void extend(const std::vector<long double> &c, const std::vector<long double> &t)
{
this->c.insert(this->c.end(), c.begin(), c.end());
this->t.insert(this->t.end(), t.begin(), t.end());
N += c.size();
}
/// Destructor
~IdealHelmholtzCP0PolyT(){};
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
// Term and its derivatives
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau3(const long double &tau, const long double &delta) throw();
};
/// Term in the ideal-gas specific heat equation that is based on Aly-Lee formulation
/** Specific heat is of the form:
\f[
\frac{c_p^0}{R_u} = A + B\left(\frac{C/T}{\sinh(C/T)}\right)^2 + D\left(\frac{E/T}{\cosh(E/T)}\right)^2
\f]
Second partial of ideal-gas Helmholtz energy given directly by specific heat (\f$\displaystyle\alpha_{\tau\tau}^0=-\frac{1}{\tau^2}\frac{c_p^0}{R_u} \f$) - this is obtained by real gas \f$c_p\f$ relationship, and killing off residual Helmholtz terms
\f[
\alpha^0_{\tau\tau} = -\frac{A}{\tau^2} - \frac{B}{\tau^2}\left(\frac{C/T}{\sinh(C/T)}\right)^2 - \frac{D}{\tau^2}\left(\frac{E/T}{\cosh(E/T)}\right)^2
\f]
or in terms of \f$ \tau \f$:
\f[
\alpha^0_{\tau\tau} = -\frac{A}{\tau^2} - \frac{BC^2}{T_c^2}\left(\frac{1}{\sinh(C\tau/T_c)}\right)^2 - \frac{DE^2}{T_c^2}\left(\frac{1}{\cosh(E\tau/T_c)}\right)^2
\f]
Third partial:
\f[
\alpha^0_{\tau\tau\tau} = 2\frac{A}{\tau^3} + 2\frac{BC^3}{T_c^3}\frac{\cosh(C\tau/T_c)}{\sinh^3(C\tau/T_c)} +2 \frac{DE^3}{T_c^3}\frac{\sinh(E\tau/T_c)}{\cosh^3(E\tau/T_c)}
\f]
Now coming back to the ideal gas Helmholtz energy definition:
\f[
\alpha^0 = -\tau\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau+\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau}\frac{c_p^0}{R_u}d\tau
\f]
Applying derivative
\f[
\alpha^0_{\tau} = -\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau-\tau\frac{\partial}{\partial \tau}\left[\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau \right]+\frac{\partial}{\partial \tau}\left[\displaystyle\int_{\tau_0}^{\tau} \frac{1}{\tau}\frac{c_p^0}{R_u}d\tau \right]
\f]
Fundamental theorem of calculus
\f[
\alpha^0_{\tau} = -\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau-\tau \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau+\frac{1}{\tau}\frac{c_p^0}{R_u}
\f]
Last two terms cancel, leaving
\f[
\alpha^0_{\tau} = -\int_{\tau_0}^{\tau} \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau
\f]
Another derivative yields (from fundamental theorem of calculus)
\f[
\alpha^0_{\tau\tau} = - \frac{1}{\tau^2}\frac{c_p^0}{R_u}
\f]
see also Jaeschke and Schley, 1995, (http://link.springer.com/article/10.1007%2FBF02083547#page-1)
*/
class IdealHelmholtzCP0AlyLee : public BaseHelmholtzTerm{
private:
std::vector<long double> c;
long double Tc, tau0, T0; // Use these variables internally
bool enabled;
public:
IdealHelmholtzCP0AlyLee(){enabled = false;};
/// Constructor with std::vectors
IdealHelmholtzCP0AlyLee(std::vector<long double> c, double Tc, double T0)
:c(c), Tc(Tc), T0(T0)
{
tau0=Tc/T0;
enabled = true;
};
/// Destructor
~IdealHelmholtzCP0AlyLee(){};
bool is_enabled(){return enabled;};
void to_json(rapidjson::Value &el, rapidjson::Document &doc);
/// The antiderivative given by \f$ \displaystyle\int \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau \f$
/**
sympy code for this derivative:
from sympy import *
a1,a2,a3,a4,a5,Tc,tau = symbols('a1,a2,a3,a4,a5,Tc,tau', real = True)
integrand = a1 + a2*(a3/Tc/sinh(a3*tau/Tc))**2 + a4*(a5/Tc/cosh(a5*tau/Tc))**2
integrand = integrand.rewrite(exp)
antideriv = trigsimp(integrate(integrand,tau))
display(antideriv)
print latex(antideriv)
print ccode(antideriv)
\f[
\displaystyle\int \frac{1}{\tau^2}\frac{c_p^0}{R_u}d\tau = -\frac{a_0}{\tau}+\frac{2a_1a_2}{T_c\left[\exp\left(-\frac{2a_2\tau}{T_c}\right)-1\right]}+\frac{2a_3a_4}{T_c\left[\exp\left(-\frac{2a_4\tau}{T_c}\right)+1\right]}
\f]
*/
long double anti_deriv_cp0_tau2(const long double &tau);
/// The antiderivative given by \f$ \displaystyle\int \frac{1}{\tau}\frac{c_p^0}{R_u}d\tau \f$
/**
sympy code for this derivative:
a_0,a_1,a_2,a_3,a_4,Tc,tau = symbols('a_0,a_1,a_2,a_3,a_4,Tc,tau', real = True)
integrand = a_0/tau + a_1/tau*(a_2*tau/Tc/sinh(a_2*tau/Tc))**2 + a_3/tau*(a_4*tau/Tc/cosh(a_4*tau/Tc))**2
term2 = a_1/tau*(a_2*tau/Tc/sinh(a_2*tau/Tc))**2
term2 = term2.rewrite(exp) # Unpack the sinh to exp functions
antideriv2 = trigsimp(integrate(term2,tau))
display(antideriv2)
print latex(antideriv2)
print ccode(antideriv2)
term3 = a_3/tau*(a_4*tau/Tc/cosh(a_4*tau/Tc))**2
term3 = term3.rewrite(exp) # Unpack the cosh to exp functions
antideriv3 = factor(trigsimp(integrate(term3,tau).rewrite(exp)))
display(antideriv3)
print latex(antideriv3)
print ccode(antideriv3)
Can be broken into three parts (trick is to express \f$sinh\f$ and \f$cosh\f$ in terms of \f$exp\f$ function)
Term 2:
\f[
\displaystyle\int \frac{a_1a_2^2}{T_c^2}\frac{\tau}{\sinh\left(\displaystyle\frac{a_2\tau}{T_c}\right)^2} d\tau = \frac{2 a_{1} a_{2} \tau}{- Tc + Tc e^{- \frac{2 a_{2}}{Tc} \tau}} + a_{1} \log{\left (-1 + e^{- \frac{2 a_{2}}{Tc} \tau} \right )} + \frac{2 a_{1}}{Tc} a_{2} \tau
\f]
Term 3:
\f[
\displaystyle\int \frac{a_1a_2^2}{T_c^2}\frac{\tau}{\cosh\left(\displaystyle\frac{a_2\tau}{T_c}\right)^2} d\tau = - \frac{a_{3}}{Tc \left(e^{\frac{2 a_{4}}{Tc} \tau} + 1\right)} \left(Tc e^{\frac{2 a_{4}}{Tc} \tau} \log{\left (e^{\frac{2 a_{4}}{Tc} \tau} + 1 \right )} + Tc \log{\left (e^{\frac{2 a_{4}}{Tc} \tau} + 1 \right )} - 2 a_{4} \tau e^{\frac{2 a_{4}}{Tc} \tau}\right)
\f]
*/
long double anti_deriv_cp0_tau(const long double &tau);
long double base(const long double &tau, const long double &delta) throw();
long double dDelta(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau(const long double &tau, const long double &delta) throw();
long double dDelta2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau2(const long double &tau, const long double &delta) throw();
long double dDelta3(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta2_dTau(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dDelta_dTau2(const long double &tau, const long double &delta) throw(){return 0.0;};
long double dTau3(const long double &tau, const long double &delta) throw();
};
class IdealHelmholtzContainer
{
public:
IdealHelmholtzLead Lead;
IdealHelmholtzEnthalpyEntropyOffset EnthalpyEntropyOffset;
IdealHelmholtzLogTau LogTau;
IdealHelmholtzPower Power;
IdealHelmholtzPlanckEinsteinGeneralized PlanckEinstein;
IdealHelmholtzCP0Constant CP0Constant;
IdealHelmholtzCP0PolyT CP0PolyT;
long double base(const long double &tau, const long double &delta)
{
return (Lead.base(tau, delta) + EnthalpyEntropyOffset.base(tau, delta)
+ LogTau.base(tau, delta) + Power.base(tau, delta)
+ PlanckEinstein.base(tau, delta)
+ CP0Constant.base(tau, delta) + CP0PolyT.base(tau, delta)
);
};
long double dDelta(const long double &tau, const long double &delta)
{
return (Lead.dDelta(tau, delta) + EnthalpyEntropyOffset.dDelta(tau, delta)
+ LogTau.dDelta(tau, delta) + Power.dDelta(tau, delta)
+ PlanckEinstein.dDelta(tau, delta)
+ CP0Constant.dDelta(tau, delta) + CP0PolyT.dDelta(tau, delta)
);
};
long double dTau(const long double &tau, const long double &delta)
{
return (Lead.dTau(tau, delta) + EnthalpyEntropyOffset.dTau(tau, delta)
+ LogTau.dTau(tau, delta) + Power.dTau(tau, delta)
+ PlanckEinstein.dTau(tau, delta)
+ CP0Constant.dTau(tau, delta) + CP0PolyT.dTau(tau, delta)
);
};
long double dDelta2(const long double &tau, const long double &delta)
{
return (Lead.dDelta2(tau, delta) + EnthalpyEntropyOffset.dDelta2(tau, delta)
+ LogTau.dDelta2(tau, delta) + Power.dDelta2(tau, delta)
+ PlanckEinstein.dDelta2(tau, delta)
+ CP0Constant.dDelta2(tau, delta) + CP0PolyT.dDelta2(tau, delta)
);
};
long double dDelta_dTau(const long double &tau, const long double &delta)
{
return (Lead.dDelta_dTau(tau, delta) + EnthalpyEntropyOffset.dDelta_dTau(tau, delta)
+ LogTau.dDelta_dTau(tau, delta) + Power.dDelta_dTau(tau, delta)
+ PlanckEinstein.dDelta_dTau(tau, delta)
+ CP0Constant.dDelta_dTau(tau, delta) + CP0PolyT.dDelta_dTau(tau, delta)
);
};
long double dTau2(const long double &tau, const long double &delta)
{
return (Lead.dTau2(tau, delta) + EnthalpyEntropyOffset.dTau2(tau, delta)
+ LogTau.dTau2(tau, delta) + Power.dTau2(tau, delta)
+ PlanckEinstein.dTau2(tau, delta)
+ CP0Constant.dTau2(tau, delta) + CP0PolyT.dTau2(tau, delta)
);
};
long double dDelta3(const long double &tau, const long double &delta)
{
return (Lead.dDelta3(tau, delta) + EnthalpyEntropyOffset.dDelta3(tau, delta)
+ LogTau.dDelta3(tau, delta) + Power.dDelta3(tau, delta)
+ PlanckEinstein.dDelta3(tau, delta)
+ CP0Constant.dDelta3(tau, delta) + CP0PolyT.dDelta3(tau, delta)
);
};
long double dDelta2_dTau(const long double &tau, const long double &delta)
{
return (Lead.dDelta2_dTau(tau, delta) + EnthalpyEntropyOffset.dDelta2_dTau(tau, delta)
+ LogTau.dDelta2_dTau(tau, delta) + Power.dDelta2_dTau(tau, delta)
+ PlanckEinstein.dDelta2_dTau(tau, delta)
+ CP0Constant.dDelta2_dTau(tau, delta) + CP0PolyT.dDelta2_dTau(tau, delta)
);
};
long double dDelta_dTau2(const long double &tau, const long double &delta)
{
return (Lead.dDelta_dTau2(tau, delta) + EnthalpyEntropyOffset.dDelta_dTau2(tau, delta)
+ LogTau.dDelta_dTau2(tau, delta) + Power.dDelta_dTau2(tau, delta)
+ PlanckEinstein.dDelta_dTau2(tau, delta)
+ CP0Constant.dDelta_dTau2(tau, delta) + CP0PolyT.dDelta_dTau2(tau, delta)
);
};
long double dTau3(const long double &tau, const long double &delta)
{
return (Lead.dTau3(tau, delta) + EnthalpyEntropyOffset.dTau3(tau, delta)
+ LogTau.dTau3(tau, delta) + Power.dTau3(tau, delta)
+ PlanckEinstein.dTau3(tau, delta)
+ CP0Constant.dTau3(tau, delta) + CP0PolyT.dTau3(tau, delta)
);
};
};
};
#endif
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