Peng-Robinson equation of state: p = RT/(1/rho - b) - a*Alpha(T/Tc) / (1/rho^2 + 2*b/rho - b^2) with a = 0.45724 * (R*Tc)^2 / Pc b = 0.0778 * (R*Tc) / Pc Alpha(T/Tc) = [1 + fw * (1 - sqrt(T/ Tc))]^2 fw = 0.37464 + 1.54226*omega - 0.2699*omega*omega. More...

#include <PengRobinsonEoS.h>

Inheritance diagram for Nektar::PengRobinsonEoS:

[legend]

Static Public Member Functions
static EquationOfStateSharedPtr	create (const LibUtilities::SessionReaderSharedPtr &pSession)
	Creates an instance of this class. More...

Static Public Attributes
static std::string	className
	Name of the class. More...

Protected Member Functions
NekDouble	v_GetTemperature (const NekDouble &rho, const NekDouble &e) final

vec_t	v_GetTemperature (const vec_t &rho, const vec_t &e) final

NekDouble	v_GetPressure (const NekDouble &rho, const NekDouble &e) final

vec_t	v_GetPressure (const vec_t &rho, const vec_t &e) final

NekDouble	v_GetEntropy (const NekDouble &rho, const NekDouble &e) final

NekDouble	v_GetDPDrho_e (const NekDouble &rho, const NekDouble &e) final

NekDouble	v_GetDPDe_rho (const NekDouble &rho, const NekDouble &e) final

NekDouble	v_GetEFromRhoP (const NekDouble &rho, const NekDouble &p) final

NekDouble	v_GetRhoFromPT (const NekDouble &rho, const NekDouble &p) final

Protected Member Functions inherited from Nektar::EquationOfState
	EquationOfState (const LibUtilities::SessionReaderSharedPtr &pSession)
	Constructor. More...

	EquationOfState (const NekDouble &gamma, const NekDouble &gasConstant)
	Programmatic Constructor. More...

virtual NekDouble	v_GetTemperature (const NekDouble &rho, const NekDouble &e)=0

virtual vec_t	v_GetTemperature (const vec_t &rho, const vec_t &e)=0

virtual NekDouble	v_GetPressure (const NekDouble &rho, const NekDouble &e)=0

virtual vec_t	v_GetPressure (const vec_t &rho, const vec_t &e)=0

virtual NekDouble	v_GetSoundSpeed (const NekDouble &rho, const NekDouble &e)

virtual NekDouble	v_GetEntropy (const NekDouble &rho, const NekDouble &e)=0

virtual NekDouble	v_GetDPDrho_e (const NekDouble &rho, const NekDouble &e)=0

virtual NekDouble	v_GetDPDe_rho (const NekDouble &rho, const NekDouble &e)=0

virtual NekDouble	v_GetEFromRhoP (const NekDouble &rho, const NekDouble &p)=0

virtual NekDouble	v_GetRhoFromPT (const NekDouble &rho, const NekDouble &p)=0

Protected Attributes
NekDouble	m_a

NekDouble	m_b

NekDouble	m_Tc

NekDouble	m_Pc

NekDouble	m_omega

NekDouble	m_fw

Protected Attributes inherited from Nektar::EquationOfState
NekDouble	m_gamma

NekDouble	m_gasConstant

NekDouble	m_gammaMone

NekDouble	m_gammaMoneOgasConst

Private Member Functions
	PengRobinsonEoS (const LibUtilities::SessionReaderSharedPtr &pSession)

	~PengRobinsonEoS (void) override

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value \|\| tinysimd::is_vector_floating_point<T>::value>::type>
T	Alpha (const T &temp)

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value \|\| tinysimd::is_vector_floating_point<T>::value>::type>
T	LogTerm (const T &rho)

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value \|\| tinysimd::is_vector_floating_point<T>::value>::type>
T	GetTemperatureKernel (const T &rho, const T &e)

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value \|\| tinysimd::is_vector_floating_point<T>::value>::type>
T	GetPressureKernel (const T &rho, const T &e)

Friends
class	MemoryManager< PengRobinsonEoS >

Additional Inherited Members
Public Member Functions inherited from Nektar::EquationOfState
virtual	~EquationOfState ()

NekDouble	GetTemperature (const NekDouble &rho, const NekDouble &e)
	Calculate the temperature. More...

vec_t	GetTemperature (const vec_t &rho, const vec_t &e)

NekDouble	GetPressure (const NekDouble &rho, const NekDouble &e)
	Calculate the pressure. More...

vec_t	GetPressure (const vec_t &rho, const vec_t &e)

NekDouble	GetSoundSpeed (const NekDouble &rho, const NekDouble &e)
	Calculate the sound speed. More...

NekDouble	GetEntropy (const NekDouble &rho, const NekDouble &e)
	Calculate the entropy. More...

NekDouble	GetDPDrho_e (const NekDouble &rho, const NekDouble &e)
	Calculate the partial derivative of P(rho,e) with respect to rho. More...

NekDouble	GetDPDe_rho (const NekDouble &rho, const NekDouble &e)
	Calculate the partial derivative of P(rho,e) with respect to e. More...

NekDouble	GetEFromRhoP (const NekDouble &rho, const NekDouble &p)
	Obtain the internal energy from rho and P. More...

NekDouble	GetRhoFromPT (const NekDouble &p, const NekDouble &T)
	Obtain the density from P and T. More...

Detailed Description

Peng-Robinson equation of state: p = RT/(1/rho - b) - a*Alpha(T/Tc) / (1/rho^2 + 2*b/rho - b^2) with a = 0.45724 * (R*Tc)^2 / Pc b = 0.0778 * (R*Tc) / Pc Alpha(T/Tc) = [1 + fw * (1 - sqrt(T/ Tc))]^2 fw = 0.37464 + 1.54226*omega - 0.2699*omega*omega.

Definition at line 51 of file PengRobinsonEoS.h.

Constructor & Destructor Documentation

◆ PengRobinsonEoS()

Nektar::PengRobinsonEoS::PengRobinsonEoS ( const LibUtilities::SessionReaderSharedPtr & pSession )

private

Definition at line 47 of file PengRobinsonEoS.cpp.

    : EquationOfState(pSession)
{
    pSession->LoadParameter("Tcrit", m_Tc);
    pSession->LoadParameter("Pcrit", m_Pc);
    pSession->LoadParameter("AcentricFactor", m_omega);
 
    m_a  = 0.45724 * m_gasConstant * m_gasConstant * m_Tc * m_Tc / m_Pc;
    m_b  = 0.0778 * m_gasConstant * m_Tc / m_Pc;
    m_fw = 0.37464 + 1.54226 * m_omega - 0.2699 * m_omega * m_omega;
}

References m_a, m_b, m_fw, Nektar::EquationOfState::m_gasConstant, m_omega, m_Pc, and m_Tc.

◆ ~PengRobinsonEoS()

Nektar::PengRobinsonEoS::~PengRobinsonEoS ( void )

inlineoverrideprivate

Definition at line 97 of file PengRobinsonEoS.h.

97{};

Member Function Documentation

◆ Alpha()

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value || tinysimd::is_vector_floating_point<T>::value>::type>

T Nektar::PengRobinsonEoS::Alpha ( const T & temp )

inlineprivate

Definition at line 103 of file PengRobinsonEoS.h.

    {
        T sqrtAlpha = 1.0 + m_fw * (1.0 - sqrt(temp / m_Tc));
        return sqrtAlpha * sqrtAlpha;
    }

References m_fw, m_Tc, and tinysimd::sqrt().

Referenced by GetPressureKernel(), v_GetDPDe_rho(), v_GetDPDrho_e(), v_GetEFromRhoP(), v_GetEntropy(), and v_GetRhoFromPT().

◆ create()

static EquationOfStateSharedPtr Nektar::PengRobinsonEoS::create ( const LibUtilities::SessionReaderSharedPtr & pSession )

inlinestatic

Creates an instance of this class.

Definition at line 57 of file PengRobinsonEoS.h.

    {
        EquationOfStateSharedPtr p =
            MemoryManager<PengRobinsonEoS>::AllocateSharedPtr(pSession);
        return p;
    }

References Nektar::MemoryManager< DataType >::AllocateSharedPtr(), and CellMLToNektar.cellml_metadata::p.

◆ GetPressureKernel()

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value || tinysimd::is_vector_floating_point<T>::value>::type>

T Nektar::PengRobinsonEoS::GetPressureKernel	(	const T &	rho,
		const T &	e
	)

inlineprivate

Definition at line 147 of file PengRobinsonEoS.h.

    {
        T temp    = GetTemperatureKernel(rho, e);
        T oneOrho = 1.0 / rho;
        T p       = m_gasConstant * temp / (oneOrho - m_b) -
              m_a * Alpha(temp) /
                  (oneOrho * oneOrho + 2.0 * m_b * oneOrho - m_b * m_b);
 
        return p;
    }

References Alpha(), GetTemperatureKernel(), m_a, m_b, Nektar::EquationOfState::m_gasConstant, and CellMLToNektar.cellml_metadata::p.

Referenced by v_GetPressure().

◆ GetTemperatureKernel()

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value || tinysimd::is_vector_floating_point<T>::value>::type>

T Nektar::PengRobinsonEoS::GetTemperatureKernel	(	const T &	rho,
		const T &	e
	)

inlineprivate

Definition at line 122 of file PengRobinsonEoS.h.

    {
        // First we need to evaluate the log term
        //    ln[(1/rho + b - b*sqrt(2)) / (1/rho + b + b*sqrt(2))]
        NekDouble sqrt2 = sqrt(2.0);
        T logTerm       = LogTerm(rho);
 
        // The temperature can be expressed as an equation in the form
        //      A * (T^1/2)^2 + B * T^1/2 + C = 0
 
        NekDouble A  = m_gasConstant / (m_gamma - 1.0);
        NekDouble f1 = m_a / (m_b * 2.0 * sqrt2);
        NekDouble f2 = (1.0 + m_fw);
        T B          = -f1 * logTerm / sqrt(m_Tc) * m_fw * f2;
        T C          = f1 * logTerm * f2 * f2 - e;
 
        // Solve for T^1/2 (positive root)
        T sqrtT = (sqrt(B * B - 4 * A * C) - B) / (2 * A);
        // Calculate the temperature
        return sqrtT * sqrtT;
    }

References LogTerm(), m_a, m_b, m_fw, Nektar::EquationOfState::m_gamma, Nektar::EquationOfState::m_gasConstant, m_Tc, and tinysimd::sqrt().

Referenced by GetPressureKernel(), and v_GetTemperature().

◆ LogTerm()

template<class T , typename = typename std::enable_if< std::is_floating_point<T>::value || tinysimd::is_vector_floating_point<T>::value>::type>

T Nektar::PengRobinsonEoS::LogTerm ( const T & rho )

inlineprivate

Definition at line 113 of file PengRobinsonEoS.h.

    {
        return log((1.0 / rho + m_b - m_b * sqrt(2)) /
                   (1.0 / rho + m_b + m_b * sqrt(2)));
    }

References tinysimd::log(), m_b, and tinysimd::sqrt().

Referenced by GetTemperatureKernel(), v_GetDPDe_rho(), v_GetEFromRhoP(), and v_GetEntropy().

◆ v_GetDPDe_rho()

NekDouble Nektar::PengRobinsonEoS::v_GetDPDe_rho	(	const NekDouble &	rho,
		const NekDouble &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 126 of file PengRobinsonEoS.cpp.

{
    NekDouble T       = GetTemperature(rho, e);
    NekDouble logTerm = LogTerm(rho);
 
    // First calculate the denominator 1/rho^2 + 2*b/rho - b^2
    //    and sqrt(Alpha) = 1+f_w*(1-sqrt(Tr))
    NekDouble denom = 1.0 / (rho * rho) + 2.0 * m_b / rho - m_b * m_b;
    NekDouble sqrtA = sqrt(Alpha(T));
 
    // Compute cv = dedT_rho
    NekDouble cv = m_gasConstant / (m_gamma - 1) -
                   m_a / (2 * m_b * sqrt(8)) * logTerm * (m_fw * (1 + m_fw)) /
                       sqrt(T * m_Tc);
 
    // Now we obtain dPdT_rho
    NekDouble dPdT = m_gasConstant / (1.0 / rho - m_b) +
                     m_a / sqrt(T * m_Tc) * m_fw * sqrtA / denom;
 
    // The result is dPde_rho = dPdT_rho / cv
    return dPdT / cv;
}

References Alpha(), Nektar::EquationOfState::GetTemperature(), LogTerm(), m_a, m_b, m_fw, Nektar::EquationOfState::m_gamma, Nektar::EquationOfState::m_gasConstant, m_Tc, and tinysimd::sqrt().

◆ v_GetDPDrho_e()

NekDouble Nektar::PengRobinsonEoS::v_GetDPDrho_e	(	const NekDouble &	rho,
		const NekDouble &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 101 of file PengRobinsonEoS.cpp.

{
    NekDouble T    = GetTemperature(rho, e);
    NekDouble dPde = GetDPDe_rho(rho, e);
 
    // First calculate the denominator 1/rho^2 + 2*b/rho - b^2
    //    and alpha = [1+f_w*(1-sqrt(Tr))]^2
    NekDouble denom = 1.0 / (rho * rho) + 2.0 * m_b / rho - m_b * m_b;
    NekDouble alpha = Alpha(T);
 
    // Calculate dPdrho_T
    NekDouble dPdrho_T =
        m_gasConstant * T / ((1.0 - m_b * rho) * (1.0 - m_b * rho)) -
        2 * m_a * alpha * rho * (1 + m_b * rho) /
            ((denom * rho * rho) * (denom * rho * rho));
 
    // Calculate dedrho_T
    NekDouble dedrho_T =
        -m_a * sqrt(alpha) * (1.0 + m_fw) / (denom * rho * rho);
 
    // The result is dPdrho_e = dPdrho_T - dPde_rho * dedrho_T
    return dPdrho_T - dPde * dedrho_T;
}

References Alpha(), Nektar::EquationOfState::GetDPDe_rho(), Nektar::EquationOfState::GetTemperature(), m_a, m_b, m_fw, Nektar::EquationOfState::m_gasConstant, and tinysimd::sqrt().

◆ v_GetEFromRhoP()

NekDouble Nektar::PengRobinsonEoS::v_GetEFromRhoP	(	const NekDouble &	rho,
		const NekDouble &	p
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 150 of file PengRobinsonEoS.cpp.

{
    NekDouble denom   = 1.0 / (rho * rho) + 2.0 * m_b / rho - m_b * m_b;
    NekDouble logTerm = LogTerm(rho);
    // First we solve for the temperature, which can be expressed as
    //      A * (T^1/2)^2 + B * T^1/2 + C = 0
    NekDouble A, B, C;
 
    A = m_gasConstant / (1.0 / rho - m_b) -
        (m_a * m_fw * m_fw) / (denom * m_Tc);
    B = 2 * m_a / denom * m_fw * (1.0 + m_fw) / sqrt(m_Tc);
    C = -m_a * (1.0 + m_fw) * (1 + m_fw) / denom - p;
 
    // Solve for T^1/2 (positive root)
    NekDouble T = (-B + sqrt(B * B - 4 * A * C)) / (2 * A);
    T           = T * T;
 
    // Calculate alpha(T))
    NekDouble alpha = Alpha(T);
    // sqrt(Tr)
    NekDouble sqrtTr = sqrt(T / m_Tc);
    // Calculate internal energy
    return m_gasConstant * T / (m_gamma - 1) +
           m_a / (m_b * sqrt(8)) * logTerm *
               (alpha + sqrt(alpha) * m_fw * sqrtTr);
}

References Alpha(), LogTerm(), m_a, m_b, m_fw, Nektar::EquationOfState::m_gamma, Nektar::EquationOfState::m_gasConstant, m_Tc, CellMLToNektar.cellml_metadata::p, and tinysimd::sqrt().

◆ v_GetEntropy()

NekDouble Nektar::PengRobinsonEoS::v_GetEntropy	(	const NekDouble &	rho,
		const NekDouble &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 82 of file PengRobinsonEoS.cpp.

{
    NekDouble T       = GetTemperature(rho, e);
    NekDouble logTerm = LogTerm(rho);
    // Entropy for an ideal gas
    NekDouble sIg =
        m_gasConstant / (m_gamma - 1) * log(T) - m_gasConstant * log(rho);
 
    // First sqrt(Alpha) = 1+_w*(1-sqrt(Tr)) and sqrt(Tr)
    NekDouble sqrtA  = sqrt(Alpha(T));
    NekDouble sqrtTr = sqrt(T / m_Tc);
 
    NekDouble deltaS = m_gasConstant * log(1 - m_b * rho);
    deltaS += m_a * sqrtA * m_fw * logTerm * (sqrtTr / T) / (m_b * sqrt(8));
 
    return sIg + deltaS;
}

References Alpha(), Nektar::EquationOfState::GetTemperature(), tinysimd::log(), LogTerm(), m_a, m_b, m_fw, Nektar::EquationOfState::m_gamma, Nektar::EquationOfState::m_gasConstant, m_Tc, and tinysimd::sqrt().

◆ v_GetPressure() [1/2]

NekDouble Nektar::PengRobinsonEoS::v_GetPressure	(	const NekDouble &	rho,
		const NekDouble &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 71 of file PengRobinsonEoS.cpp.

{
    return GetPressureKernel(rho, e);
}

References GetPressureKernel().

◆ v_GetPressure() [2/2]

vec_t Nektar::PengRobinsonEoS::v_GetPressure	(	const vec_t &	rho,
		const vec_t &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 77 of file PengRobinsonEoS.cpp.

{
    return GetPressureKernel(rho, e);
}

References GetPressureKernel().

◆ v_GetRhoFromPT()

NekDouble Nektar::PengRobinsonEoS::v_GetRhoFromPT	(	const NekDouble &	rho,
		const NekDouble &	p
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 178 of file PengRobinsonEoS.cpp.

{
    // First solve for the compressibility factor Z using the cubic equation
    //    Z^3 + k1 * Z^2 + k2 * Z + k3 = 0
    //    for PengRobinson:
    //        k1 = B-1, k2 = A - 2*B - 3*B^2,  k3 = - AB + B^2 + B^3
    //    where A = a*alpha(T)*P/(RT)^2, B = bP/(RT)
    NekDouble A = m_a * Alpha(T) * p / (m_gasConstant * m_gasConstant * T * T);
    NekDouble B = m_b * p / (m_gasConstant * T);
 
    NekDouble k1 = B - 1.0;
    NekDouble k2 = A - 2 * B - 3 * B * B;
    NekDouble k3 = -A * B + B * B + B * B * B;
 
    // Use ideal gas (Z=1) as starting guess for iteration
    NekDouble Z = 1.0;
    // Newton-Raphson iteration to find Z
    NekDouble tol      = 1e-6;
    NekDouble maxIter  = 100;
    NekDouble residual = 1;
    NekDouble f, df;
    unsigned int cnt = 0;
    while ((fabs(residual) > tol) && cnt < maxIter)
    {
        f        = Z * Z * Z + k1 * Z * Z + k2 * Z + k3;
        df       = 3 * Z * Z + 2 * k1 * Z + k2;
        residual = f / df;
        Z -= residual;
        ++cnt;
    }
    if (cnt == maxIter)
    {
        cout << "Newton-Raphson in PengRobinsonEoS::v_GetRhoFromPT did not "
                "converge in "
             << maxIter << " iterations (residual = " << residual << ")"
             << endl;
    }
 
    // Now calculate rho = p/(ZRT)
    return p / (Z * m_gasConstant * T);
}

References Alpha(), m_a, m_b, Nektar::EquationOfState::m_gasConstant, and CellMLToNektar.cellml_metadata::p.

◆ v_GetTemperature() [1/2]

NekDouble Nektar::PengRobinsonEoS::v_GetTemperature	(	const NekDouble &	rho,
		const NekDouble &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 60 of file PengRobinsonEoS.cpp.

{
    return GetTemperatureKernel(rho, e);
}

References GetTemperatureKernel().

◆ v_GetTemperature() [2/2]

vec_t Nektar::PengRobinsonEoS::v_GetTemperature	(	const vec_t &	rho,
		const vec_t &	e
	)

finalprotectedvirtual

Implements Nektar::EquationOfState.

Definition at line 66 of file PengRobinsonEoS.cpp.

{
    return GetTemperatureKernel(rho, e);
}

References GetTemperatureKernel().

Friends And Related Function Documentation

◆ MemoryManager< PengRobinsonEoS >

friend class MemoryManager< PengRobinsonEoS >

friend

Definition at line 1 of file PengRobinsonEoS.h.

Member Data Documentation

◆ className

std::string Nektar::PengRobinsonEoS::className

static

Initial value:

=
    GetEquationOfStateFactory().RegisterCreatorFunction(
        "PengRobinson", PengRobinsonEoS::create,
        "Peng-Robinson equation of state.")

Name of the class.

Definition at line 66 of file PengRobinsonEoS.h.

◆ m_a

NekDouble Nektar::PengRobinsonEoS::m_a

protected

Definition at line 69 of file PengRobinsonEoS.h.

Referenced by GetPressureKernel(), GetTemperatureKernel(), PengRobinsonEoS(), v_GetDPDe_rho(), v_GetDPDrho_e(), v_GetEFromRhoP(), v_GetEntropy(), and v_GetRhoFromPT().

◆ m_b

NekDouble Nektar::PengRobinsonEoS::m_b

protected

Definition at line 70 of file PengRobinsonEoS.h.

Referenced by GetPressureKernel(), GetTemperatureKernel(), LogTerm(), PengRobinsonEoS(), v_GetDPDe_rho(), v_GetDPDrho_e(), v_GetEFromRhoP(), v_GetEntropy(), and v_GetRhoFromPT().

◆ m_fw

NekDouble Nektar::PengRobinsonEoS::m_fw

protected

Definition at line 74 of file PengRobinsonEoS.h.

Referenced by Alpha(), GetTemperatureKernel(), PengRobinsonEoS(), v_GetDPDe_rho(), v_GetDPDrho_e(), v_GetEFromRhoP(), and v_GetEntropy().

◆ m_omega

NekDouble Nektar::PengRobinsonEoS::m_omega

protected

Definition at line 73 of file PengRobinsonEoS.h.

Referenced by PengRobinsonEoS().

◆ m_Pc

NekDouble Nektar::PengRobinsonEoS::m_Pc

protected

Definition at line 72 of file PengRobinsonEoS.h.

Referenced by PengRobinsonEoS().

◆ m_Tc

NekDouble Nektar::PengRobinsonEoS::m_Tc

protected

Definition at line 71 of file PengRobinsonEoS.h.

Referenced by Alpha(), GetTemperatureKernel(), PengRobinsonEoS(), v_GetDPDe_rho(), v_GetEFromRhoP(), and v_GetEntropy().

Static Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Private Member Functions

Friends

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ PengRobinsonEoS()

◆ ~PengRobinsonEoS()

Member Function Documentation

◆ Alpha()

◆ create()

◆ GetPressureKernel()

◆ GetTemperatureKernel()

◆ LogTerm()

◆ v_GetDPDe_rho()

◆ v_GetDPDrho_e()

◆ v_GetEFromRhoP()

◆ v_GetEntropy()

◆ v_GetPressure() [1/2]

◆ v_GetPressure() [2/2]

◆ v_GetRhoFromPT()

◆ v_GetTemperature() [1/2]

◆ v_GetTemperature() [2/2]

Friends And Related Function Documentation

◆ MemoryManager< PengRobinsonEoS >

Member Data Documentation

◆ className

◆ m_a

◆ m_b

◆ m_fw

◆ m_omega

◆ m_Pc

◆ m_Tc