<content> <p>Calculations</p> </content>

932969a8-f9c5-48ce-9cd4-282f870f29ca 48 Рыльский Александр Павлович Valery Ochkov 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use

Developed by Russian National Committee (RNC) of
International Association for the Properties of Water and Steam (IAPWS).

This calculation page is based on the
"Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use" [1]
provided by IAPWS.

Detailed information about used equations, constants, range of validity etc is presented in PDF version of IAPWS Release which can be downloaded from IAPWS web site www.iapws.org

Authors (and responsible persons):
Konstantin Orlov (orlov@twt.mpei.ac.ru)
Valery Ochkov (ochkov@twt.mpei.ac.ru)
Moscow Power Engineering Institute (MPEI).
Note that displayed last digits depends on numeric implementation of underlying formulations.

Input parameters

Density 838.025 kg m 3^/ * :

Temperature 500 K * :

<content> <p>Calculations</p> </content>

ρ Density :

T Temperature :

Formulation reference values

ρ.c 322 kg m 3^/ * :

R 461.51805 J kg K * / * :

T.c 647.096 K * :

Dimensionless input parameters

τ T.c T / : 1.294192

δ ρ ρ.c / : 2.60256211180124

Main equations

The specific Helmholtz free energy is represented by the equation

ρ T f R T * / δ τ ϕ δ τ ϕ.0 δ τ ϕ.r + ≡ (≡

The ideal-gas part of the dimensionless Helmholtz free energy is represented by the equation

δ τ ϕ.0 δ ln n0.1 + n0.2 + (τ * n0.3 τ ln * + n0 i el 1 e γ0 i el τ * -^- ln * (i 48 sum + ≡

The residual part of the dimensionless Helmholtz free energy is represented by the equation

δ τ ϕ.r n i el δ d i el^* τ t i el^* (n i el δ d i el^* τ t i el^* e δ c i el^-^* (n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* (- β i el τ γ i el - (2^* (-^* (n i el 1 τ - (A i el δ 1 - (2^(12 β i el * /^* + (2^(B i el δ 1 - (2^(a i el^* + (b i el^* (δ * e C i el δ 1 - (2^* - τ 1 - D i el 2^-^* (i 5556 sum + i 5254 sum + i 851 sum + i 17 sum ≡

Dimensionless Helmholtz Free Energy and it's derivatives

The ideal-gas part of the dimensionless Helmholtz free energy

Coefficients from Table 1 [1] (can be downloaded here in text format):

DATA i n0 y0 1 8.3204464837497 - - 2 6.6832105275932 - 3 3.00632 - 4 0.012436 1.28728967 5 0.97315 3.53734222 6 1.27950 7.74073708 7 0.96956 9.24437796 8 0.24873 27.5075105 93 mat :

Extracting data to vectors:

I DATA 2 DATA rows 1 - 1 + 11 submatrix :

n0 DATA 2 DATA rows 1 - 1 + 22 submatrix :

γ0 DATA 2 DATA rows 33 submatrix :

I 1234567881 mat

n0 8.3204464837497 - 6.6832105275932 3.00632 0.012436 0.97315 1.2795 0.96956 0.24873 81 mat

γ0 - - - 1.28728967 3.53734222 7.74073708 9.24437796 27.5075105 81 mat

The ideal-gas part of the dimensionless Helmholtz free energy

ϕ.0 δ ln n0 1 el + n0 2 el τ * + n0 3 el τ ln * + n0 i el 1 e γ0 i el τ * -^- ln * (i 48 sum + :

ϕ.0 2.04797733479595

The derivatives of ideal-gas part of the dimensionless Helmholtz free energy

ϕ.0_δ 1 δ / : 0.38423674711375

ϕ.0_δδ 1 - δ 2^/ : 0.147637877832556 -

ϕ.0_τ n0 2 el n0 3 el τ / + n0 i el γ0 i el * 11 e γ0 i el τ * -^- / 1 - (* (i 48 sum + : 9.0461110617524

ϕ.0_ττ n0 3 el τ 2^/ - n0 i el γ0 i el (2^* e γ0 i el τ * -^* 11 e γ0 i el τ * -^- / (2^* (i 48 sum - : 1.93249185013052 -

ϕ.0_δτ 0 :

The residual part of the dimensionless Helmholtz free energy

Coefficients from Table 2 [1] (can be downloaded here in text format):

DATA1 i ci di ti ni 1 - 1 0.5 - 0.012533547935523 2 - 1 0.875 7.8957634722828 3 - 11 8.7803203303561 - 4 - 2 0.5 0.31802509345418 5 - 2 0.75 0.26145533859358 - 6 - 3 0.375 7.8199751687981 103 -^* - 7 - 41 8.8089493102134 103 -^* 8114 0.66856572307965 - 9116 0.20433810950965 101112 6.6212605039687 105 -^* - 11121 0.19232721156002 - 12125 0.25709043003438 - 13134 0.16074868486251 14142 0.040092828925807 - 151413 3.9343422603254 107 -^* 16159 7.5941377088144 106 -^* - 17173 5.6250979351888 104 -^* 18194 1.5608652257135 105 -^* - 1911011 1.1537996422951 109 -^* 201114 3.6582165144204 107 -^* 2111313 1.3251180074668 1012 -^* - 221151 6.2639586912454 1010 -^* - 23217 0.10793600908932 - 24221 0.017611491008752 25229 0.22132295167546 262210 0.40247669763528 - 272310 0.58083399985759 28243 4.9969146990806 103 -^* 29247 0.031358700712549 - 302410 0.74315929710341 - 312510 0.47807329915480 32266 0.020527940895948 332610 0.13636435110343 - 342710 0.014180634400617 35291 8.3326504880713 103 -^* 36292 0.029052336009585 - 37293 0.038615085574206 38294 0.020393486513704 - 39298 1.6554050063734 103 -^* - 402106 1.9955571979541 103 -^* 412109 1.5870308324157 104 -^* 422128 1.6388568342530 105 -^* - 433316 0.043613615723811 443422 0.034994005463765 453423 0.076788197844621 - 463523 0.022446277332006 4741410 6.2689710414685 105 -^* - 486350 5.5711118565645 1010 -^* - 496644 0.19905718354408 - 506646 0.31777497330738 516650 0.11841182425981 - 525 mat :

DATA2 i c d t n alpha beta gamma epsilon 52 - 30 31.306260323435 - 20150 1.21 153 - 31 31.546140237781 20150 1.21 154 - 34 2521.3154341695 - 20250 1.25 149 mat :

DATA3 i a b B n C D A beta 55 3.5 0.85 0.2 0.14874640856724 - 28700 0.32 0.3 56 3.5 0.95 0.2 0.31806110878444 32800 0.32 0.3 39 mat :

Extracting data to vectors:

i1 DATA1 2 DATA1 rows 1 - 1 + 11 submatrix :

c1 DATA1 2 DATA1 rows 22 submatrix :

d1 DATA1 2 DATA1 rows 1 - 1 + 33 submatrix :

t1 DATA1 2 DATA1 rows 1 - 1 + 44 submatrix :

n1 DATA1 2 DATA1 rows 1 - 1 + 55 submatrix :

i2 DATA2 2 DATA2 rows 1 - 1 + 11 submatrix :

d2 DATA2 2 DATA2 rows 1 - 1 + 33 submatrix :

t2 DATA2 2 DATA2 rows 1 - 1 + 44 submatrix :

n2 DATA2 2 DATA2 rows 1 - 1 + 55 submatrix :

α2 DATA2 2 DATA2 rows 1 - 1 + 66 submatrix :

β2 DATA2 2 DATA2 rows 1 - 1 + 77 submatrix :

γ2 DATA2 2 DATA2 rows 1 - 1 + 88 submatrix :

ε2 DATA2 2 DATA2 rows 1 - 1 + 99 submatrix :

i3 DATA3 2 DATA3 rows 1 - 1 + 11 submatrix :

a3 DATA3 2 DATA3 rows 1 - 1 + 22 submatrix :

b3 DATA3 2 DATA3 rows 1 - 1 + 33 submatrix :

B3 DATA3 2 DATA3 rows 1 - 1 + 44 submatrix :

n3 DATA3 2 DATA3 rows 1 - 1 + 55 submatrix :

C3 DATA3 2 DATA3 rows 1 - 1 + 66 submatrix :

D3 DATA3 2 DATA3 rows 1 - 1 + 77 submatrix :

A3 DATA3 2 DATA3 rows 1 - 1 + 88 submatrix :

β3 DATA3 2 DATA3 rows 1 - 1 + 99 submatrix :

Processing data in vectors for correct equations representation as in IAPWS Release [1]:

i i1 i2 i3 stack :

c c1 :

d d1 d2 stack :

t t1 t2 stack :

n n1 n2 n3 stack :

α c1 α2 stack :

β c1 β2 β3 stack :

γ c1 γ2 stack :

ε c1 ε2 stack :

a d a3 stack :

b d b3 stack :

B d B3 stack :

C d C3 stack :

D d D3 stack :

A d A3 stack :

Additional values from Table [5] in [1] for further calculations of residual part of the dimensionless Helmholtz free energy and derivatives

i 5556 range :

θ i el 1 τ - (A i el δ 1 - (2^(12 β i el * /^* + (:

θ 55 el 1.24703360102646

θ 56 el 1.24703360102646

ψ i el e C i el δ 1 - (2^* - D i el τ 1 - (2^* -^:

ψ 55 el 0

ψ 56 el 0

Δ i el θ i el (2^B i el δ 1 - (2^(a i el^* + (:

Δ 55 el 6.98427093741785

Δ 56 el 6.98427093741785

dψdδ i el 2 C i el * δ 1 - (* ψ i el * - :

dψdδ 55 el 0

dψdδ 56 el 0

dΔdδ i el δ 1 - (A i el θ i el * 2 β i el / * δ 1 - (2^(12 β i el * / 1 -^* 2 B i el * a i el * δ 1 - (2^(a i el 1 -^* + (* :

dΔdδ 55 el 31.7100435492422

dΔdδ 56 el 31.7100435492422

dΔbdδ i el b i el Δ i el (b i el 1 -^* dΔdδ i el * :

dΔbdδ 55 el 20.1371313288037

dΔbdδ 56 el 27.3347049089122

d2ψdδ2 i el 2 C i el * δ 1 - (2^* 1 - (2 * C i el * ψ i el * :

d2ψdδ2 55 el 0

d2ψdδ2 56 el 0

d2Δdδ2 i el 1 δ 1 - / dΔdδ i el * δ 1 - (2^4 B i el * a i el * a i el 1 - (* δ 1 - (2^(a i el 2 -^* 2 A i el (2^* 1 β i el / (2^* δ 1 - (2^(12 β i el * / 1 -^(2^* + A i el θ i el * 4 β i el / * 12 β i el * / 1 - (* δ 1 - (2^(12 β i el * / 2 -^* + (* + ((:

d2Δdδ2 55 el 120.982842559292

d2Δdδ2 56 el 120.982842559292

d2Δbdδ2 i el b i el Δ i el (b i el 1 -^d2Δdδ2 i el * b i el 1 - (Δ i el (b i el 2 -^* dΔdδ i el (2^* + (* :

d2Δbdδ2 55 el 63.1148731080083

d2Δbdδ2 56 el 98.0844177399398

dΔbdτ i el 2 θ i el * b i el * Δ i el (b i el 1 -^* - :

dΔbdτ 55 el 1.58383127770261 -

dΔbdτ 56 el 2.14993684525363 -

dψdτ i el 2 D i el * τ 1 - (* ψ i el * - :

dψdτ 55 el 0

dψdτ 56 el 0

d2ψdτ2 i el 2 D i el * τ 1 - (2^* 1 - (2 * D i el * ψ i el * :

d2ψdτ2 55 el 0

d2ψdτ2 56 el 0

d2Δbdτ2 i el 2 b i el * Δ i el (b i el 1 -^* 4 θ i el (2^* b i el * b i el 1 - (* Δ i el (b i el 2 -^* + (:

d2Δbdτ2 55 el 1.18524169131415

d2Δbdτ2 56 el 1.68565395367722

d2ψdδdτ i el 4 C i el * D i el * δ 1 - (* τ 1 - (* ψ i el * :

d2ψdδdτ 55 el 0

d2ψdδdτ 56 el 0

d2Δbdδdτ i el A i el b i el * 2 β i el / * Δ i el (b i el 1 -^* δ 1 - (* δ 1 - (2^(12 β i el * / 1 -^* - 2 θ i el * b i el * b i el 1 - (* Δ i el (b i el 2 -^* dΔdδ i el * - (:

d2Δbdδdτ 55 el 2.99292154616275 -

d2Δbdδdτ 56 el 5.03879143044079 -

The residual part of the dimensionless Helmholtz free energy

ϕ.r n i el δ d i el^* τ t i el^* (i 17 sum n i el δ d i el^* τ t i el^* e δ c i el^-^* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* (i 5254 sum + n i el Δ i el (b i el^* δ * ψ i el * (i 5556 sum + : 3.42693205681559 -

The derivatives of residual part of the dimensionless Helmholtz free energy

ϕ.r_δ n i el d i el * δ d i el 1 -^* τ t i el^* (i 17 sum n i el e δ c i el^-^* δ d i el 1 -^τ t i el^* d i el c i el δ c i el^* - (* (* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* d i el δ / 2 α i el * δ ε i el - (* - (* (i 5254 sum + n i el Δ i el (b i el^ψ i el δ dψdδ i el * + (* dΔbdδ i el δ * ψ i el * + (* (((i 5556 sum + (:

ϕ.r_δ 0.364366650363885 -

ϕ.r_δδ n i el d i el * d i el 1 - (* δ d i el 2 -^* τ t i el^* (i 17 sum n i el e δ c i el^-^* δ d i el 2 -^τ t i el^* d i el c i el δ c i el^* - (d i el 1 - c i el δ c i el^* - (* c i el (2^δ c i el^* - (* (* (i 851 sum + n i el τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* 2 α i el * δ d i el^* - 4 α i el (2^* δ d i el^* δ ε i el - (2^* + 4 d i el * α i el * δ d i el 1 -^* δ ε i el - (* - d i el d i el 1 - (* δ d i el 2 -^* + ((* (i 5254 sum + n i el Δ i el (b i el^2 dψdδ i el * δ d2ψdδ2 i el * + (* 2 dΔbdδ i el * ψ i el δ dψdδ i el * + (* + d2Δbdδ2 i el δ * ψ i el * + (* (i 5556 sum + :

ϕ.r_δδ 0.856063700974615

ϕ.r_τ n i el t i el * δ d i el^* τ t i el 1 -^* (i 17 sum n i el t i el * δ d i el^* τ t i el 1 -^* e δ c i el^-^* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* t i el τ / 2 β i el * τ γ i el - (* - ((* (i 5254 sum + n i el δ * dΔbdτ i el ψ i el * Δ i el (b i el^dψdτ i el * + (* (i 5556 sum + :

ϕ.r_τ 5.81403435238416 -

ϕ.r_ττ n i el t i el * t i el 1 - (* δ d i el^* τ t i el 2 -^* (i 17 sum n i el t i el * t i el 1 - (* δ d i el^* τ t i el 2 -^* e δ c i el^-^* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* t i el τ / 2 β i el * τ γ i el - (* - (2^t i el τ 2^/ - 2 β i el * - (* (i 5254 sum + n i el δ * d2Δbdτ2 i el ψ i el * 2 dΔbdτ i el * dψdτ i el * + Δ i el (b i el^d2ψdτ2 i el * + (* (i 5556 sum + :

ϕ.r_ττ 2.23440736884337 -

ϕ.r_δτ n i el d i el * t i el * δ d i el 1 -^* τ t i el 1 -^* (i 17 sum n i el t i el * δ d i el 1 -^* τ t i el 1 -^* d i el c i el δ c i el^* - (* e δ c i el^-^* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* d i el δ / 2 α i el * δ ε i el - (* - (* t i el τ / 2 β i el * τ γ i el - (* - (* (i 5254 sum + n i el Δ i el (b i el^dψdτ i el δ d2ψdδdτ i el * + (* δ dΔbdδ i el * dψdτ i el * + dΔbdτ i el ψ i el δ dψdδ i el * + (* + d2Δbdδdτ i el δ * ψ i el * + (* (i 5556 sum + :

ϕ.r_δτ 1.12176914670306 -

The dimensionless Helmholtz free energy and it's derivatives

ϕ ϕ.0 ϕ.r + : 1.37895472201964 -

ϕ.δ ϕ.0_δ ϕ.r_δ + : 0.019870096749865

ϕ.δδ ϕ.0_δδ ϕ.r_δδ + : 0.70842582314206

ϕ.τ ϕ.0_τ ϕ.r_τ + : 3.23207670936825

ϕ.ττ ϕ.0_ττ ϕ.r_ττ + : 4.16689921897389 -

ϕ.δτ ϕ.0_δτ ϕ.r_δτ + : 1.12176914670306 -

Thermodynamic properties

Pressure:

p 1 ϕ.r_δ δ * + (ρ * R * T * : MPa 10.0003858009206

P T Temperature D Density H2O CoolProp_Props MPa 10.0003858009219

Compressibility factor

Z p ρ R * T * / : 0.051713160959023

The internal energy

u τ ϕ.0_τ ϕ.r_τ + (* R * T * : kJ kg / 965.248345538733

The specific entropy

s τ ϕ.0_τ ϕ.r_τ + (* ϕ.0 - ϕ.r - (R * : kJ kg K * / 2.56690918542226

S T Temperature D Density H2O CoolProp_Props kJ kg K * / 2.56690918542213

The specific enthalpy

h 1 τ ϕ.0_τ ϕ.r_τ + (* + ϕ.r_δ δ * + (R * T * : kJ kg / 977.181624141303

H T Temperature D Density H2O CoolProp_Props kJ kg / 977.181624141256

The specific isochoric heat capacity

c.v τ 2^ϕ.0_ττ ϕ.r_ττ + (* R * - : kJ kg K * / 3.22106218674058

CVMASS T Temperature D Density H2O CoolProp_Props kJ kg K * / 3.22106218674041

The specific isobaric heat capacity

c.p τ 2^ϕ.0_ττ ϕ.r_ττ + (* - 1 δ ϕ.r_δ * + δ τ * ϕ.r_δτ * - (2^12 δ * ϕ.r_δ * + δ 2^ϕ.r_δδ * + / + (R * : kJ kg K * / 4.60222448139032

C T Temperature D Density H2O CoolProp_Props kJ kg K * / 4.6022244813901

Speed of sound

w 12 δ * ϕ.r_δ * + δ 2^ϕ.r_δδ * + 1 δ ϕ.r_δ * + δ τ * ϕ.r_δτ * - (2^τ 2^ϕ.r_ττ ϕ.0_ττ + (* / - (R * T * sqrt : 1271.28440914763 m s / *

Joule-Thomson coefficient

μ δ ϕ.r_δ * δ 2^ϕ.r_δδ * + δ τ * ϕ.r_δτ * + (- 1 δ ϕ.r_δ * + δ τ * ϕ.r_δτ * - (2^τ 2^ϕ.r_ττ ϕ.0_ττ + (* 12 δ * ϕ.r_δ * + δ 2^ϕ.r_δδ * + (* - / (1 R ρ * / * : K MPa / 0.056690812295974 -

Isothermal throttling coefficient

δ.T 11 δ ϕ.r_δ * + δ τ * ϕ.r_δτ * - 12 δ * ϕ.r_δ * + δ 2^ϕ.r_δδ * + / - (1 ρ / * : 0.000260903844218 m 3^kg / *

Isentropic temperature-pressure coefficient

β.s 1 δ ϕ.r_δ * + δ τ * ϕ.r_δτ * - 1 δ ϕ.r_δ * + δ τ * ϕ.r_δτ * - (2^τ 2^ϕ.r_ττ ϕ.0_ττ + (* 12 δ * ϕ.r_δ * + δ 2^ϕ.r_δδ * + (* - / (1 R ρ * / * : K MPa / 0.202592894563573

1. Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. Doorwerth, The Netherlands, September 2009, available at http://www.iapws.org

n i el δ d i el^* τ t i el^* (i 17 sum n i el δ d i el^* τ t i el^* e δ c i el^-^* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* (i 5254 sum + n i el Δ i el (b i el^* δ * ψ i el * (i 5556 sum + 3.427 -

δ τ ϕ.r n i el δ d i el^* τ t i el^* (i 17 sum n i el δ d i el^* τ t i el^* e δ c i el^-^* (i 851 sum + n i el δ d i el^* τ t i el^* e α i el δ ε i el - (2^* - β i el τ γ i el - (2^* -^* (i 5254 sum + n i el Δ i el (b i el^* δ * ψ i el * (i 5556 sum + 11 line :

δ τ ϕ.r 3.427 -

δ τ ϕ1.r n i el δ d i el^* τ t i el^* (i 17 sum 11 line :

n i el δ d i el^* τ t i el^* (i 17 sum 3.121 -

δ τ ϕ1.r 3.121 -