Accurate calculation of the pressure and temperature of water, steam, and ice: Formulation for CFD

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1 Jounal of Mechanical Science and Technology 4 () () 333~34 DOI.7/s Accuate calculation of the pessue and tempeatue of wate steam and ice: Fomulation fo CFD Jang-Chang Lee and Meng-Sing Liou Depatment of Mechanical Engineeing Andong National Univesity Andong Koea NASA Glenn Reseach Cente at Lewis Field Cleveland Ohio 4435 (Manuscipt Received Febuay 3 ; Revised August 5 ; Accepted August 7 ) Abstact An accuate appoach is poposed fo calculating the themodynamic popeties of wate in thee states: liquid steam and ice and the tansitional states among them. The fomulation is expessed in tems of quantities that ae natually used in Computational Fluid Dynamics (CFD) namely the specific volume ( v ) and specific intenal enegy ( u ) though the use of Gibbs fee enegy. The Gibbs fee enegy fomula poposed by IAPWS fomulated as a function of pessue and tempeatue is used as a basis in ou calculations. The Jacobian matix esulting fom the tansfomation between sets of vaiables ( pt) and ( vu) ae deived fo each phase; the Newton- Raphson method is used to iteatively solve the nonlinea equations. Numeical calculations have been caied out fo the entie phase diagam coveing all thee phases. The numeical esults ae compaed with the oiginal data of IAPWS and the associated eos ae 7 analyzed. It is confimed that the pessue and tempeatue ae accuately calculated with lagest elative eo on the ode of in the ice phase. Hence othe themodynamic popeties ae also obtained within the same level of accuacy. The method poposed in this pape fo calculating pessue and tempeatue vaiables needed in CFD is eliable and can be applied to the numeical simulation of multiphase flows including phase changes. Keywods: Gibbs fee enegy; Equation of state fo steam; Wate and ice; IAPWS Intoduction This pape was ecommended fo publication in evised fom by Associate Edito Jun Sang Pak Coesponding autho. Tel.: Fax: addess: leejc@andong.ac.k KSME & Spinge Multiphase flows containing wate eithe as a caie o dispesed medium ae fequently encounteed in maine and ai vehicles such as cavitation aound hydofoils condensation and icing aound aifoils in which the wate may stay in a phase o undego phase changes depending on the flow and themodynamic conditions. To achieve an accuate numeical simulation of such flows it is impotant to have an accuate desciption of the themodynamic popety of wate which is supplied as the equation of state (EOS) to the govening equations of the flow. This EOS fo wate is extemely complex because of the peculia popety of its molecula stuctue. Of special inteest fo ou puposes is to develop an accuate and efficient pocedue fo descibing steam wate and ice phases. Many eseaches of multi-phase flows ove the yeas have used viial equation of state [-3]. This EOS shows a good ageement with the expeimental data of steam but a substantial deviation fom the data of wate. The Van De Waals EOS unlike viial EOS accounts fo the phase change (liquefaction) and is a cubic function including two paametes. These paametes ae fitted fo a specific substance; they ae available in the liteatue fo many common substances. Howeve this EOS shows that the deviation inceases in lage densities [4]. The Peng-Robinson EOS is one of many modified Van De Waals EOS's [4 5]. The paamete included in this EOS can be found fom the expeimental data of wate-steam ( a =.569 b =.6864 ). This EOS howeve is well matched with steam data but is mismatched with wate data. In geneal the wate density in expeimental data deceases paabolically as the tempeatue inceases in the v T gaph but the wate density calculated fom the Peng-Robinson EOS deceases linealy. Matin s EOS; is anothe modified fom of the Van De Waals EOS it includes thee paametes [4]. These paametes obtained fom compaison with the expeimental data of wate-steam ae a =.3764 c =.7 and t =.566. Howeve this EOS does not agee with expeimental data of wate as well as the Peng- Robinson EOS. Clealy none of available simple algebaic EOS models pefom satisfactoily in the wate phase; the eo becomes

2 334 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~34 lage as the wate density inceases. Hence we begin by pioitizing accuacy ove simplicity we seek an EOS that can accuately epesent all thee phases and the tansitional states while undestanding that it may be mathematically and numeically consideably moe complicated than the simple foms just mentioned. The concen fo computational cost will only become tivial as compute technologies ae advancing in apid pace. In fact the calculations pesented hee took less than a second fo steam and wate phases and 3 seconds fo ice on a Dell PC CPU.66GHz. Thus we choose to base on the most accuate fom of EOS fo wate developed by IAPWS and focus on developing a fomulation of this EOS to be well-suited fo CFD puposes.. Gibbs fee enegy fomula To calculate the themodynamic popeties of wate steam and ice IAPWS poposes the fomula fo each phase using Gibbs fee enegy [6-8]. The basic equation fo the dimensionless specific Gibbs fee enegy γ = g / RT fo wate is given as follows: 34 Ii Ji γπτ ( ) = ni (7. π) ( τ.) () i= whee R is the gas constant of wate ( R =.4656 [kj kg - K - ]) π is the pessue atio π = p/ p whee p = 6.53 [MPa] and τ is the tempeatue atio τ = T / T with T = 386 [K]. The coefficients n i exponents I i and Ji ae given in Table A of Appendix. The pessue and tempeatue anges fo applying the Eq. () ae as follows (see egion of Fig. ): 73.5 [K] T 63.5 [K] p (T ) p [MPa] whee ps( T) is a satuated pessue at a given tempeatue. The fundamental equation fo the dimensionless specific Gibbs fee enegy γ = g / RT fo steam is defined as follows: s = + () γ ( πτ ) γ ( πτ ) γ ( πτ ) of Appendix. Eq. () coves egion of Fig. defined by the following anges of tempeatue and pessue: 73.5 [K] T 63.5 [K] p p s(t )[MPa] 63.5 [K] T [K] p p(t )[MPa] [K] T 73.5 [K] p [MPa] whee pt ( ) is the bounday pessue between egion and 3 in Fig. and the equation of pt ( ) is specified in Ref. [7]. The basic equation fo the dimensional specific Gibbs fee enegy g( pt ) fo ice is defined as follows [8]: g( pt ) = g stτ t + TtRe k ( tk τ )ln( tk τ ) k = τ + ( tk + τ)ln( tk + τ) tkln tk. (5) tk Hee g and ae defined as follows; 4 k = k k = k ( ) = k ( ). k = g ( p) g ( π π ) p π π The dimensionless paametes again ae π = p / pt and τ = T/ Tt whee p t = [Pa] and T t = 73.6 [K] at the tiple point. The dimensionless nomal pessue is π = p / pt whee p = 35 [Pa]. The coefficients g k and s in Eq. (5) ae eal constants given in Table A4 of Appendix. The complex constants t t and ~ ae also given in Table A4. Eq. (5) coves the entie ange of ice whee the tempeatue and pessue anges ae [K] T 73.6 [K] [MPa] p [MPa]. All the themodynamic popeties fo each phase specific volume specific intenal enegy specific entopy specific enthalpy specific heat capacity and speed of sound can be whee γ is ideal gas pat and γ is esidual pat; they ae defined as follows: 9 ni i= J i γ = ln π + τ (3) 43 Ii Ji γ = niπ ( τ.5) (4) i= p / MPa 5 3 whee π = p/ p and τ = T / T with p = [MPa] and T = 54 [K]. The coefficients n i and exponents J i in Eq. (3) ae given in Table A of Appendix and the coefficients ni and exponents I i and J i in Eq. (4) ae given in Table A3 p s (T) Fig.. Regions of IAPWS-IF97. T / K

3 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~ expessed in tems of Gibbs fee enegy and its deivatives [7]. Fo example the specific volume and specific intenal enegy ae defined as follows: g v = T g g u = g T p p T. Deivatives in Eq. (6) and Eq. (7) ae obtained fom Eq. () Eq. () and Eq. (5). Othe themodynamic popeties ae also obtained in the same way and the elations of those ae specified in Ref. [7]. Othe deivatives fo all thee phases ae shown in detail in Ref. [7]. (6) (7) u = ( Rτ ) γττ ( Rπ) γπ + ( Rπτ) γπτ. Fo steam: v RT = p γ ππ v R R Rτ = + p p p γ π γ πτ γ πτ γ π γ ππ u RT RT RTp = p p p u Rπτ = ( Rτ )( γττ + γττ) R+ γππ. T (3) (4) (5) (6) (7) 3. Newton-Raphson method In ode to calculate the pessue and tempeatue of wate steam and ice fom a given specific volume ( v ) and specific intenal enegy ( u ) the Newton-Raphson method is used. The two-equation vesion of the Newton-Raphson method is fomulated as: p T i+ i+ ui vi ( vi v) ( ui u) = pi vi ui vi ui vi ui ( ui u) ( vi v) = Ti vi ui vi ui whee subscipt i denotes the known value calculated at the pevious iteation and i + is the unknown value at the pesent iteation. v and u ae given as exact values and v i and u i ae values calculated using pi and T i. As mentioned ealie all the deivatives in Eq. (8) and Eq. (9) can be deived fom the fomula of Gibbs fee enegy fo each phase: Eq. () Eq. () and Eq. (5). The denominato of Eq. (8) and Eq. (9) is called the Jacobian matix of the system which esults fom the tansfomation between sets of vaiables ( pt) and ( vu). These nonlinea equations ae solved iteatively to get conveged solutions of p and T fom v and u. The elements of Jacobian matix fo each phase ae deived. Those elements fo wate fom Eq. () Eq. (6) and Eq. (7) ae v RT = p γ ππ v R Rτ γ = γ p p π πτ u RT RT RT = + p p p π γ π γπτ γ ππ (8) (9) () () () Fo ice: v = g pp + Tt Re pp ( t τ )ln( t τ ) ( )ln( ) ln τ + t + τ t + τ t t t (8) v τ = Re ln( t τ) + ln( t + τ) p t u τ = T Re p ln ( t τ) + ln ( t + τ) t p g pp + Tt Re pp t τ ln ( t τ ) ( ) ln ( ) ln τ + t + τ t + τ t t t (9) () u = T Re k + T t ( t ) ( ) k k τ tk + τ t = k τ p Re p ln ( t τ) + ln ( t + τ). () t 4. Numeical esults Numeical calculations have been caied out fo the themodynamic popeties of each phase and the numeical esults ae compaed with the oiginal data of IAPWS (Tables A5 to A7 of Appendix). The associated eos ae analyzed in this pape. Fig. shows the change of pessue eo fo wate with T = 3 [K] T = 4 [K] and T = 5 [K] fo the entie pessue ange ps ( T) p [MPa] (see egion of Fig. ). Hee we define Eo (%) = ( p p)/ p whee p is the exact value and p is the calculated value. The veti-

4 336 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~34 Fig.. The change of pessue eo fo wate fo the entie pessue ange at given tempeatues: T=3 [K] T=4 [K] and T=5 [K]. Fig. 3. The change of pessue eo fo steam fo the entie pessue ange at given tempeatues: T=4 [K] T=5 [K] and T=6 [K]. Fig. 4. The change of pessue eo fo ice fo the entie pessue ange at given tempeatues: T=45 [K] T=55 [K] and T=65 [K]. cal dash-dot line in the gaphs epesents the satuated pessue at the given tempeatue. That is p s = [MPa] fo T = 3 [K] p s = [MPa] fo T = 4 [K] and p s = [MPa] fo T = 5 [K]. The deviations ae small in the highe pessue egion but incease a little in lowe pessue egion aound satuated pessue egion. These phenomena ae common in all the gaphs of Fig.. Howeve since the values of maximum pessue eo fo given tempeatues ae extemely 9 small espectively O( ) O( ) and O( ) the pessue is accuately calculated fom a given v and u. In Eq. (8) and Eq. (9) p = 6.6 [MPa] and T = 73.6 [K] ae used as initial guesses fo wate. Fig. 3 epesents the change of pessue eo fo steam with T = 4 [K] T = 5 [K] and T = 6 [K]. In that the steam aea coesponds to egion of Fig. the pessue ange fo 73.5[ K] T 63.5[ K] is p ps( T) [MPa]. Hence the vetical dash-dot lines in the gaphs ae the satuated pessue and p s = [MPa] fo T = 6 [K]. The pessue eos of steam as well as wate ae inceased nea the bounday egion the satuated pessue line but since the maximum pessue eo is O( ) the calculations of pessue ae highly accuate. As initial guess values fo Eq. (8) and Eq. (9) p =. [Pa] and

5 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~ Fig. 5. The change of tempeatue eo fo thee phases fo the entie tempeatue ange at p = 35 [Pa]. T = 73.6 [K] fo steam ae used. Fig. 4 shows the change of pessue eo fo ice with T = 45 [K] T = 55 [K] and T = 65 [K]. The pessue and tempeatue anges fo applying Eq. (5) ae p [MPa] and T 73.6 [K]. If tempeatue is below T = 73.6 [K] the sublimation pessue line is the bounday between steam and ice (see Fig. the phase-bounday cuves of wate in a p T diagam of Ref. [9]). Theefoe the vetical dash-dot line in each gaph is the sublimation pessue line 5 at a given tempeatue and p subl = 4.6 [MPa] fo 4 T = 45 [K] p subl =.3 [MPa] fo T = 55 [K] 4 and p subl = 3.59 [MPa] fo T = 65 [K]. In the case of ice the deviations of pessue eo ae also inceased aound the sublimation pessue line but ae deceased as the pessue is inceased. Since the maximum pessue eo is 7 O( ) the ice pessues ae also accuately calculated. Even though the pessue change is lage as seen in Table A7 of Appendix the changes of v and u ae vey small. To incease convegence of numeical solution small undeelaxation factos ae used to update new iteates if the Jacobian matices in Eq. (8) and Eq. (9) ae small espectively O (.) fo P and O (.3) fo T. As initial guess values of Eq. (8) and Eq. (9) p =. [Pa] and T =. [K] fo ice ae used. Fig. 5. illustates the change of tempeatue eo fo all thee phases with p = 35 [Pa]. Two vetical dash-dot lines in wate gaph ae the melting tempeatue line T = 73.4 [K] and the satuated tempeatue line T = 39.5 [K]. The line in the steam gaph indicates the satuated tempeatue T = 39.5 [K] and the line in the ice gaph indicates the melting tempeatue T = 73.4 [K]. Since the maximum tempeatue eo ae O( ) 9 O( ) and O( ) espectively the tempeatues ae also accuately calculated fom given v and u. 5. Conclusions Using the most extensive and accuate EOS developed by IAPWS by using the Gibbs fee enegy fomula we develop an accuate pocedue to expess ( pt) in tems of the set of vaiables ( vu) deived diectly fom the CFD calculations. The themodynamic popeties fo all thee phases ae calculated. The esults ae compaed with the oiginal data of IAPWS. It is confimed that all the popeties ae accuately calculated. Numeical esults show that the lagest elative 7 eo among all thee phases is O( ). The method poposed in this pape fo calculating pessue and tempeatue vaiables needed in CFD is eliable and can be applied to numeical simulation of multiphase flows including phase changes. Acknowledgment This wok was suppoted by a gant fom 7 Reseach Fund of Andong National Univesity. Nomenclatue g : Dimensional specific Gibbs fee enegy g ~ g 4 : Real constants Ii J i : Exponents of equations n i : Coefficient of equations P : Pessue P s : Satuated pessue R : Gas constant of wate t : Complex constants ~ : Complex constants s : Real constants T : Tempeatue u : Specific intenal enegy u : Exact value of specific intenal enegy v : Specific volume v : Exact value of specific volume γ : Dimensionless specific Gibbs fee enegy ( γ = g / RT ) γ : Ideal gas pat of dimensionless specific Gibbs fee enegy γ : Residual pat of dimensionless specific Gibbs fee enegy π : Pessue atio ( π = p/ p )

6 338 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~34 τ : Tempeatue atio ( τ = T T ) Subscipts subl t Refeences : Sublimation : Tiple point [] P. P. Wegene and L. M. Mack Condensation in Supesonic and Hypesonic Wind Tunnels Advances in Applied Mechanics edited by Dyden/Kaman Academic 5 (958) [] G. H. Schne and U. Dohmann Tansonic Flow aound Aifoils with Relaxation and Enegy Supply by Homogeneous Condensation AIAA J. 8 (99) [3] A. Beg U. Iben A. Meiste and J. Schmidt Modeling and Simulation of Cavitation in Hydaulic Pipelines Based on the Themodynamic and Caloic Popeties of Liquid and Steam Shock Waves 4 (5) -. [4] J. W. Teste and M. Modell Themodynamics and its Applications thid Ed. Pentice Hall Intenational Seies in the Physical Chemical Engineeing Sciences (996). [5] Z. Xu and S. I. Sandle Tempeatue -Dependent Paametes and the Peng-Robinson Equation of State Ind. Eng. Chem. Res. 6 (3) (987) [6] W. Wagne et. al The IAPWS Industial Fomulation 997 fo the Themodynamic Popeties of Wate and Steam J. of Engineeing fo Gas Tubines and Powe () 5-8. [7] Elangen The Intenational Association fo the Popeties of Wate and Steam (Release on the IAPWS Industial Fomulation 997 fo the Themodynamic Popeties of Wate and Steam) Gemany (997) [8] R. Feistel and W. Wagne A New Equation of State fo H O Ice Ih J. Phys. Chem. Ref. Data 35 () (6) -47. [9] W. Wagne A. Saul and A. Puss Intenational Equations fo the Pessue along the Melting and along the Sublimation Cuve of Odinay Wate Substance J. Phys. Chem. Ref. Data 3 (3) (994) [] W. Wagne and A. Puss Intenational Equation fo the Satuation Popeties of Odinay Wate Substance. Revised Accoding to the Intenational Tempeatue Scale of 99 J. Phys. Chem. Ref. Data (3) (993) [] S. C. Chapa and R. P. Canale Numeical Methods fo enginees Fifth Ed. McGaw-Hill New Yok USA (7). Appendix Table A. Coefficients and exponents of Eq. (). i I i J i / n i Table A. Coefficients and exponents of Eq. (3). i J i Table A3. Coefficients and exponents of Eq. (4). i I i J i n i n i

7 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~ Table A4. Coefficients of Eq. (5). Coefficient Real constant Unit g J kg - g J kg - g J kg - g J kg - g J kg - s J kg - K - Coefficient Complex constant (eal pat) Unit t J kg - K - t J kg - K J kg - K J kg - K - Coefficient Complex constant (eal pat) Unit t J kg - K - t J kg - K J kg - K J kg - K - Table A5. Themodynamics popeties fo wate. Popety T=3 [K] p=3 [MPa] v/ (m 3 kg - ) u/ (kj kg - ) h/ (kj kg - ) s/ (kj kg - K - ) Popety T=3 [K] p=8 [MPa] v/ (m 3 kg - ) u/ (kj kg - ) h/ (kj kg - ) s/ (kj kg - K - ) Popety T=5 [K] p=3 [MPa] v/ (m 3 kg - ).48 - u/ (kj kg - ) h/ (kj kg - ) s/ (kj kg - K - ).5849 Table A6. Themodynamics popeties fo ice. Popety T=73.6 [K] p=6.657 [Pa] v/ (m 3 kg - ) u/ (J kg - ) h/ (J kg - ) s/ (J kg - K - ) Popety T= [K] p=35 [Pa] v/ (m 3 kg - ) u/ (J kg - ) h/ (J kg - ) s/ (J kg - K - ) Table A7. Themodynamics popeties fo steam. Popety T=3 [K] p=.35 [MPa] v/ (m 3 kg - ) u/ (kj kg - ) h/ (kj kg - ) s/ (kj kg - K - ) Popety T=7 [K] p=.35 [MPa] v/ (m 3 kg - ) u/ (kj kg - ) h/ (kj kg - ) s/ (kj kg - K - ) Popety T=7 [K] p=3 [MPa] v/ (m 3 kg - ) u/ (kj kg - ) h/ (kj kg - ) s/ (kj kg - K - )

8 34 J.-C. Lee and M.-S. Liou / Jounal of Mechanical Science and Technology 4 () () 333~34 Meng-Sing Liou is a Senio Technologist of GRC fo Aibeathing Populsion Computational Analysis. He has moe than 3 yeas expeiences in Computational Fluid Dynamics contibuting in the development of numeical methods and suppoting NASA s aeonautics pogams since joining NASA in 986. His AUSM schemes have been adopted in majo commecial and eseach codes and included in textbooks and CFD couses woldwide. Cuently he is leading the Multidisciplinay Design Analysis and Optimization eseach in the Aeopopulsion Division to suppot Fundamental Aeonautics Pogam. He was Chief of Computational Fluid Dynamics. He was awaded NASA s Exceptional Scientific Achievement Medal (99) and Exceptional Achievement Medal (4) and the Abe Silvestein Medal (6). He has published ove technical papes and 8 book chaptes. He also gave numeous shot couses in vaious counties. He is an Adjunct Pofesso of Univesity of Califonia Santa Babaa and Case Westen Reseve Univesity. Jang-Chang Lee eceived his B.S. and M.S. degees in Mechanical Engineeing fom Chung-Ang Univesity Koea in 989 and 99 espectively. He then eceived his Ph.D. degees fom R.P.I. in. D. Lee is cuently a Pofesso at Depatment of Mechanical Engineeing at Andong National Univesity in Andong Koea. D. Lee s eseach inteests include Two-phase flows Aeodynamics.