/** This class contains equations about the properties of water. It is primarily based on The International Association for the Properties of Water and Steam (IAPWS, http://iapws.org/) document IAPWS R6-95(2018), a PDF of which is usually available from: http://www.iapws.org/relguide/IAPWS-95.html Specifically, the file http://www.iapws.org/relguide/IAPWS95-2018.pdf All references to equation and table numbers are the equation numbers in this document. The development of this document is discussed in https://aip.scitation.org/doi/10.1063/1.1461829 There is an online calculator that can be used to validate these equations at: http://twt.mpei.ac.ru/mcs/worksheets/iapws/IAPWS95.xmcd IAPWS also produces a simpler formulation "for industry", called IAPWS-IF97 which is described at: http://www.iapws.org/relguide/IF97-Rev.html That page contains a PDF of IAPWS-IF97 and some "backward equations" that make it easier to calculate without iterating inverse solutions. These may be used to find good "first guesses" for inverse solutions. Simpler equations for vapor pressure and densities at the saturation point are found here: http://www.iapws.org/relguide/Supp-sat.html especially in the PDF file: http://www.iapws.org/relguide/Supp-sat.pdf These can be used to obtain good initial guesses for finding higher-accuracy results in the full IAPWS95 model. A scientific paper describing the derivation of the saturation properties can be found at: https://aip.scitation.org/doi/10.1063/1.555926 */ class Water { /** Constants as defined in IAPWS RS-95(2018). These may not exactly match the current best-known values of these constants in the SI, but they are used as defined there for purposes of matching its output exactly. */ /** The temperature of the critical point of water. (eq.1) */ class var Tc = 647.096 K /** The critical density of water. (eq.2) */ class var rhoc = 322 kg m^-3 /** The specific gas constant. Due to the use of the specific gas constant, Eq. (4) corresponds to a mass-based formulation. In order to convert values of specific properties to molar properties, an appropriate value for the molar mass should be used (eq. 3) */ class var R = 0.46151805 kJ kg^-1 K^-1 /** The critical pressure of water from IAPWS SR1-86(1992) */ class var pc = 22.064 MPa /** The triple point of water. */ class var TriplePoint = 273.16 K /** Numerical values of the coefficients and parameters of the ideal-gas part of the dimensionless Helmholtz free energy, Eq. 5. This is table 1 in IAPWS-95. The coluns are [ni0, gammai0] */ class var Table1 = [undef, // No element 0 -8.3204464837497, 6.6832105275932, 3.00632, [0.012436, 1.28728967], [0.97315, 3.53734222], [1.27950, 7.74073708], [0.96956, 9.24437796], [0.24873, 27.5075105]] /** Numerical values of the coefficients and parameters of the residual part of the dimensionless Helmholtz free energy, Eq. (6) */ class var Table2 = [undef, // No element 0 //[ci, di, ti, ni] [x, 1, -1/2, 0.12533547935523e-1], [x, 1, 7/8, 0.78957634722828e1], [x, 1, 1, -0.87803203303561e1], [x, 2, 1/2, 0.31802509345418], [x, 2, 3/4, -0.26145533859358], [x, 3, 3/8, -0.78199751687981e-2], [x, 4, 1, 0.88089493102134e-2], [1, 1, 4, -0.66856572307965], [1, 1, 6, 0.20433810950965], [1, 1, 12, -0.66212605039687e-4], [1, 2, 1, -0.19232721156002], [1, 2, 5, -0.25709043003438], [1, 3, 4, 0.16074868486251], [1, 4, 2, -0.40092828925807e-1], [1, 4, 13, 0.39343422603254e-6], [1, 5, 9, -0.75941377088144e-5], [1, 7, 3, 0.56250979351888e-3], [1, 9, 4, -0.15608652257135e-4], [1, 10, 11, 0.11537996422951e-8], [1, 11, 4, 0.36582165144204e-6], [1, 13, 13, -0.13251180074668e-11], [1, 15, 1, -0.62639586912454e-9], [2, 1, 7, -0.10793600908932], [2, 2, 1, 0.17611491008752e-1], [2, 2, 9, 0.22132295167546], [2, 2, 10, -0.40247669763528], [2, 3, 10, 0.58083399985759], [2, 4, 3, 0.49969146990806e-2], [2, 4, 7, -0.31358700712549e-1], [2, 4, 10, -0.74315929710341], [2, 5, 10, 0.47807329915480], [2, 6, 6, 0.20527940895948e-1], [2, 6, 10, -0.13636435110343], [2, 7, 10, 0.14180634400617e-1], [2, 9, 1, 0.83326504880713e-2], [2, 9, 2, -0.29052336009585e-1], [2, 9, 3, 0.38615085574206e-1], [2, 9, 4, -0.20393486513704e-1], [2, 9, 8, -0.16554050063734e-2], [2, 10, 6, 0.19955571979541e-2], [2, 10, 9, 0.15870308324157e-3], [2, 12, 8, -0.16388568342530e-4], [3, 3, 16, 0.43613615723811e-1], [3, 4, 22, 0.34994005463765e-1], [3, 4, 23, -0.76788197844621e-1], [3, 5, 23, 0.22446277332006e-1], [4, 14, 10, -0.62689710414685e-4], [6, 3, 50, -0.55711118565645e-9], [6, 6, 44, -0.19905718354408], [6, 6, 46, 0.31777497330738], [6, 6, 50, -0.11841182425981], // [ci, di, ti, ni, alphai, betai, gammai, epsiloni] [x, 3, 0, -0.31306260323435e2, 20, 150, 1.21, 1], [x, 3, 1, 0.31546140237781e2, 20, 150, 1.21, 1], [x, 3, 4, -0.25213154341695e4, 20, 250, 1.25, 1], // [ai, bi, Bi, ni, Ci, Di, Ai, betai] [7/2, 85/100, 0.2, -0.14874640856724, 28, 700, 0.32, 3/10], [7/2, 95/100, 0.2, 0.31806110878444, 32, 800, 0.32, 3/10]] /** Numerical values for IAPWS97 table 34. These are used to solve saturation-pressure and saturation-temperature equations. */ class var Table34 = [undef, // No n0 0.11670521452767e4, // n1 -0.72421316703206e6, // n2 -0.17073846940092e2, // n3 0.12020824702470e5, // n4 -0.32325550322333e7, // n5 0.14915108613530e2, // n6 -0.48232657361591e4, // n7 0.40511340542057e6, // n8 -0.23855557567849, // n9 0.65017534844798e3] // n10 /** Calculates the pressure at the specified density and temperature. p = rho^2 (df/drho)_T See Table 3 for this equation. */ class pressure[rho is mass_density, T is temperature] := { delta = rho / rhoc return rho R T (1 + delta dphirdeltaDimensioned[rho, T]) } /** Calculates the compressibility factor Z = p / (rho R T) */ class compressibilityFactor[rho is mass_density, T is temperature] := { return pressure[rho, T] / (rho R T) } /** Calculates the specific internal energy of the system at the specified atmospheric pressure equals the saturated vapor pressure. You can use StandardAtmosphereTest.frink which calls StandardAtmosphere.frink to find the approximate atmospheric pressure for any altitude. */ class boilingPoint[ps is pressure] := { return saturatedTemperature[ps] } /** Returns the approximate saturated vapor pressure of water from IAPWS SR1-86(1992). This does not use the full IAPWS95 model, but it can be used to get a very good first guess. This is Section 3 */ class approxSaturatedVaporPressure[T is temperature] := { theta = T/Tc tau = 1 - theta p = pc exp[Tc/T (-7.85951783 tau + 1.84408259 tau^(3/2) + -11.7866497 tau^3 + 22.6807411 tau^(7/2) + -15.9618719 tau^4 + 1.80122502 tau^(15/2))] return p } /** Returns the approximate saturated vapor pressure of water from IAPWS IF-97. This does not use the full IAPWS95 model, but it can be used to get a very good first guess. This is Section 8 and 8.1 of IAPWS IF-97 */ class saturatedVaporPressure[Ts is temperature] := { Tstar = 1 K Theta = Ts / Tstar + Table34@9 / ((Ts/Tstar) - Table34@10) // eq. 29b A = Theta^2 + Table34@1 Theta + Table34@2 B = Table34@3 Theta^2 + Table34@4 Theta + Table34@5 C = Table34@6 Theta^2 + Table34@7 Theta + Table34@8 p = 1 MPa ((2 C) / (-B + (B^2 - 4 A C)^(1/2)))^4 // eq. 30 return p } /** Returns the approximate saturated vapor temperature of water from IAPWS97. This does not use the full IAPWS95 model, but it can be used to get a very good first guess. This is Section 8.2 of IAPWS97. */ class saturatedTemperature[ps is pressure] := { // Equation 31 pstar = 1 MPa beta = (ps / pstar)^(1/4) // Eq. 29a E = beta^2 + Table34@3 beta + Table34@6 F = Table34@1 beta^2 + Table34@4 beta + Table34@7 G = Table34@2 beta^2 + Table34@5 beta + Table34@8 D = (2 G)/(-F - (F^2 - 4 E G)^(1/2)) Ts = 1 K ((Table34@10 + D - ((Table34@10 + D)^2 - 4(Table34@9 + Table34@10 D))^(1/2)) / 2) return Ts } /** Returns the approximate saturated liquid density of water from IAPWS SR1-86(1992). This does not use the full IAPWS95 model, but it can be used to get a very good first guess. This is Section 4.1 */ class saturatedLiquidDensity[T is temperature] := { theta = T/Tc tau = 1-theta rho = rhoc (1 + 1.99274064 tau^(1/3) + // b1 1.09965342 tau^(2/3) + // b2 -0.510839303 tau^(5/3) + // b3 -1.75493479 tau^(16/3) + // b4 -45.5170352 tau^(43/3) + // b5 -6.74694450e5 tau^(110/3)) // b6 return rho } /** Returns the approximate saturated vapor density of water from IAPWS SR1-86(1992). This does not use the full IAPWS95 model, but it can be used to get a very good first guess. This is Section 4.2 */ class saturatedVaporDensity[T is temperature] := { theta = T/Tc tau = 1-theta rho = rhoc exp[1 + -2.03150240 tau^(2/6) + // c1 -2.68302940 tau^(4/6) + // c2 -5.38626492 tau^(8/6) + // c3 -17.2991605 tau^(18/6) + // c4 -44.7586581 tau^(37/6) + // c5 -63.9201063 tau^(71/6)] // c6 return rho } /** The specific Helmholtz free energy f (also sometimes called A). The IAPWS formulation is implemented as a fundamental equation for the specific Helmholtz free energy f. This equation is expressed in dimensionless form, phi = f/(R T), or, conversely f = phi R T (see eq. 4 and following) */ class f[rho is mass_density, T is temperature, debug=false] := { return R T phiDimensioned[rho, T, debug] } /** The formulation of the phi equation is implemented as a dimensionless version phi[delta, tau] where delta = rho / rhoc tau = Tc / T This turns a dimensionally-correct call with density and temperature into the internal dimensionless form. (see eq.4 and following) */ class phiDimensioned[rho is mass_density, T is temperature, debug=false] := { return phi[rho / rhoc, Tc / T, debug] } /** The dimensionless phi function phi[delta, tau] is separated into two parts, an ideal-gas part phi0[delta, tau] and a residual part phir[delta, tau] (eq. 4) */ class phi[delta is dimensionless, tau is dimensionless, debug=false] := { phi0 = phi0[delta, tau] phir = phir[delta, tau] if debug println["phi0=$phi0\nphir=$phir\n"] return phi0 + phir } /** Calculates the first derivative of phi with respect to tau. This includes both the ideal gas and residual parts. */ class dphidtau[delta is dimensionless, tau is dimensionless] := { return dphi0dtau[delta, tau] + dphirtau[delta, tau] } /** Calculates the second derivative of phi with respect to tau. This includes both the ideal gas and residual parts. */ class d2phitau[delta is dimensionless, tau is dimensionless] := { return d2phi0dtau[delta, tau] + d2phirtau[delta, tau] } /** Calculates the second derivative of phi with respect to delta. This includes both the ideal gas and residual parts. */ class d2phidelta[delta is dimensionless, tau is dimensionless] := { return d2phi0ddelta[delta, tau] + d2phirdelta[delta, tau] } /** The ideal gas part phi0 of the dimensionless Helmholtz free energy. This is obtained from an equation for the specific isobaric heat capacity in the ideal-gas state developed by Cooper [6] This implements eq. 5 */ class phi0[delta is dimensionless, tau is dimensionless] := { sum = ln[delta] + Table1@1 + /* n1 0 */ Table1@2 tau + /* n2 0 */ Table1@3 ln[tau] /* n3 0 */ for i = 4 to 8 { [ni, gammai] = Table1@i sum = sum + ni ln[1 - exp[-gammai tau]] } return sum } /** The first partial derivative of the ideal gas part of the dimensionless Helmholtz free energy with respect to delta. */ class dphi0ddelta[delta is dimensionless, tau is dimensionless] := { return 1/delta } /** The second partial derivative of the ideal gas part of the dimensionless Helmholtz free energy with respect to delta. */ class d2phi0ddelta[delta is dimensionless, tau is dimensionless] := { return -1 / delta^2 } /** The first partial derivative of the ideal gas part of the dimensionless Helmholtz free energy with respect to tau. */ class dphi0dtau[delta is dimensionless, tau is dimensionless] := { sum = Table1@2 + Table1@3 / tau for i=4 to 8 { [ni, gammai] = Table1@i sum = sum + ni gammai ((1 - exp[-gammai tau])^-1 - 1) } return sum } /** The second partial derivative of the ideal gas part of the dimensionless Helmholtz free energy with respect to tau. */ class d2phi0dtau[delta is dimensionless, tau is dimensionless] := { sum = -Table1@3 / tau^2 for i=4 to 8 { [ni, gammai] = Table1@i sum = sum - ni gammai^2 exp[-gammai tau] (1 - exp[-gammai tau])^-2 } return sum } /** The second partial derivative of the ideal gas part of the dimensionless Helmholtz free energy with respect to delta then with respect to tau. */ class d2phi0ddeltadtau[delta is dimensionless, tau is dimensionless] := { return 0 } /** The residual part phir[delta, tau] of the dimensionless Helmholtz free energy. This implements eq. 6 */ class phir[delta is dimensionless, tau is dimensionless] := { sum = 0 for i=1 to 7 { [ci, di, ti, ni] = Table2@i sum = sum + ni delta^di tau^ti } for i=8 to 51 { [ci, di, ti, ni] = Table2@i sum = sum + ni delta^di tau^ti e^(-delta^ci) } for i=52 to 54 { [ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i sum = sum + ni delta^di tau^ti * exp[-alphai (delta - epsiloni)^2 - betai(tau - gammai)^2] } for i=55 to 56 { [ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2] theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai)) Delta = theta^2 + Bi ((delta-1)^2)^ai sum = sum + ni Delta^bi delta Psi } return sum } /** The partial derivative of phir[delta, tau] with respect to delta, with correct dimensions. */ class dphirdeltaDimensioned[rho is mass_density, T is temperature] := { return dphirdelta[rho / rhoc, Tc / T] } /** The partial derivative of phir[delta, tau] with respect to delta. This implements eq. 6 */ class dphirdelta[delta is dimensionless, tau is dimensionless] := { sum = 0 for i=1 to 7 { [ci, di, ti, ni] = Table2@i sum = sum + ni di delta^(di-1) tau^ti } for i=8 to 51 { [ci, di, ti, ni] = Table2@i sum = sum + ni exp[-delta^ci] (delta^(di-1) tau^ti(di - ci delta^ci)) } for i=52 to 54 { [ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] (di/delta - 2 alphai (delta - epsiloni)) } for i=55 to 56 { [ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2] theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai)) Delta = theta^2 + Bi ((delta-1)^2)^ai dPsiddelta = -2 Ci (delta-1) Psi dDeltaddelta = (delta-1) (Ai theta 2 / betai ((delta-1)^2)^(1/(2 betai)-1) + 2 Bi ai((delta-1)^2)^(ai-1)) dDeltabiddelta = bi Delta^(bi-1) dDeltaddelta sum = sum + ni (Delta^bi (Psi + delta dPsiddelta) + dDeltabiddelta delta Psi) } return sum } /** The second partial derivative of phir[delta, tau] with respect to delta, with correct dimensions. */ class d2phirdeltaDimensioned[rho is mass_density, T is temperature] := { return d2phirdelta[rho / rhoc, Tc / T] } /** The second partial derivative of phir[delta, tau] with respect to delta. */ class d2phirdelta[delta is dimensionless, tau is dimensionless] := { sum = 0 for i=1 to 7 { [ci, di, ti, ni] = Table2@i sum = sum + ni di (di-1) delta^(di-2) tau^ti } for i=8 to 51 { [ci, di, ti, ni] = Table2@i sum = sum + ni exp[-delta^ci] (delta^(di-2) tau^ti ((di - ci delta^ci)(di-1-ci delta^ci) - ci^2 delta^ci)) } for i=52 to 54 { [ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i sum = sum + ni tau^ti exp[-alphai (delta-epsiloni)^2 - betai(tau-gammai)^2] * (-2 alphai delta^di + 4 alphai^2 delta^di (delta-epsiloni)^2 - 4 di alphai delta^(di-1) (delta-epsiloni) + di (di-1) delta^(di-2)) } for i=55 to 56 { [ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2] theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai)) Delta = theta^2 + Bi ((delta-1)^2)^ai dPsiddelta = -2 Ci (delta-1) Psi dDeltaddelta = (delta-1) (Ai theta 2 / betai ((delta-1)^2)^(1/(2 betai)-1) + 2 Bi ai((delta-1)^2)^(ai-1)) d2Psiddelta = (2 Ci (delta-1)^2 -1) 2 Ci Psi dDeltabiddelta = bi Delta^(bi-1) dDeltaddelta d2Deltaddelta = 1 / (delta-1) dDeltaddelta + (delta-1)^2 (4 Bi ai (ai-1)((delta-1)^2)^(ai-2) + 2 Ai^2 (1/betai)^2 (((delta-1)^2)^(1/(2 betai) - 1))^2 + Ai theta 4/betai (1 / (2 betai) - 1)((delta-1)^2)^(1/(2 betai) - 2)) d2Deltabiddelta = bi (Delta^(bi-1) d2Deltaddelta + (bi - 1) Delta^(bi-2) dDeltaddelta^2) sum = sum + ni (delta^bi (2 dPsiddelta + delta d2Psiddelta) + 2 dDeltabiddelta (Psi + delta dPsiddelta) + d2Deltabiddelta delta Psi) } return sum } /** The partial derivative of phir[delta, tau] with respect to tau, with correct dimensions. */ class dphirtauDimensioned[rho is mass_density, T is temperature] := { return dphirtau[rho / rhoc, Tc / T] } /** The partial derivative of phir[delta, tau] with respect to tau. */ class dphirtau[delta is dimensionless, tau is dimensionless] := { sum = 0 for i=1 to 7 { [ci, di, ti, ni] = Table2@i sum = sum + ni ti delta^di tau^(ti-1) } for i=8 to 51 { [ci, di, ti, ni] = Table2@i sum = sum + ni ti delta^di tau^(ti-1) exp[-delta^ci] } for i=52 to 54 { [ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] (ti/tau - 2 betai(tau - gammai)) } for i=55 to 56 { [ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2] theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai)) Delta = theta^2 + Bi ((delta-1)^2)^ai dDeltabidtau = -2 theta bi Delta^(bi-1) dPsidtau = -2 Di (tau-1) Psi sum = sum + ni Delta (dDeltabidtau Psi + Delta^bi dPsidtau) } return sum } /** The partial second derivative of phir[delta, tau] with respect to tau, with correct dimensions. */ class d2phirtauDimensioned[rho is mass_density, T is temperature] := { return d2phirtau[rho / rhoc, Tc / T] } /** The partial second derivative of phir[delta, tau] with respect to tau. */ class d2phirtau[delta is dimensionless, tau is dimensionless] := { sum = 0 for i=1 to 7 { [ci, di, ti, ni] = Table2@i sum = sum + ni ti (ti-1) delta^di tau^(ti-2) } for i=8 to 51 { [ci, di, ti, ni] = Table2@i sum = sum + ni ti (ti-1) delta^di tau^(ti-2) exp[-delta^ci] } for i=52 to 54 { [ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] ((ti/tau - 2 betai(tau - gammai))^2 - ti/tau^2 - 2 betai) } for i=55 to 56 { [ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2] theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai)) Delta = theta^2 + Bi ((delta-1)^2)^ai dDeltabidtau = -2 theta bi Delta^(bi-1) dPsidtau = -2 Di (tau-1) Psi d2Psidtau = (2 Di (tau-1)^2 - 1) 2 Di Psi d2Deltabidtau = 2 bi Delta^(bi-1) + 4 theta^2 bi (bi-1) Delta^(bi-2) sum = sum + ni delta (d2Deltabidtau Psi + 2 dDeltabidtau dPsidtau + Delta^bi d2Psidtau) } return sum } /** The partial second derivative of phir[delta, tau] with respect to tau, with correct dimensions. */ class d2phirdeltatauDimensioned[rho is mass_density, T is temperature] := { return d2phirdeltatau[rho / rhoc, Tc / T] } /** The partial second derivative of phir[delta, tau] with respect to delta then to tau. */ class d2phirdeltatau[delta is dimensionless, tau is dimensionless] := { sum = 0 for i=1 to 7 { [ci, di, ti, ni] = Table2@i sum = sum + ni di ti delta^(di-1) tau^(ti-1) } for i=8 to 51 { [ci, di, ti, ni] = Table2@i sum = sum + ni ti delta^(di-1) tau^(ti-1) (di - ci delta^ci) exp[-delta^ci] } for i=52 to 54 { [ci, di, ti, ni, alphai, betai, gammai, epsiloni] = Table2@i sum = sum + ni delta^di tau^ti exp[-alphai (delta-epsiloni)^2 - betai (tau-gammai)^2] (di/delta - 2 alphai (delta - epsiloni))(ti/tau - 2 betai(tau - gammai)) } for i=55 to 56 { [ai, bi, Bi, ni, Ci, Di, Ai, betai] = Table2@i Psi = exp[-Ci (delta-1)^2 - Di (tau - 1)^2] theta = (1 - tau) + Ai ((delta-1)^2)^(1/(2 betai)) Delta = theta^2 + Bi ((delta-1)^2)^ai dDeltabidtau = -2 theta bi Delta^(bi-1) dPsidtau = -2 Di (tau-1) Psi d2Psidtau = (2 Di (tau-1)^2 - 1) 2 Di Psi dPsiddelta = -2 Ci (delta-1) Psi d2Deltabidtau = 2 bi Delta^(bi-1) + 4 theta^2 bi (bi-1) Delta^(bi-2) dDeltaddelta = (delta-1) (Ai theta 2 / betai ((delta-1)^2)^(1/(2 betai)-1) + 2 Bi ai((delta-1)^2)^(ai-1)) dDeltabiddelta = bi Delta^(bi-1) dDeltaddelta d2Psiddeltadtau = 4 Ci Di (delta-1)(tau-1) Psi d2Deltabiddeltadtau = -Ai bi 2 / betai Delta^(bi-1) (delta-1)((delta-1)^2)^(1/(2 betai) - 1) - 2 theta bi (bi-1) Delta^(bi-2) dDeltaddelta sum = sum + ni (Delta^(bi) (dPsidtau + delta d2Psiddeltadtau) + delta dDeltabiddelta dPsidtau + dDeltabidtau (Psi + delta dPsiddelta) + d2Deltabiddeltadtau delta Psi) } return sum } }