NEW EQUATIONS FOR DENSITY, POTENTIAL TEMPERATURE AND ENTROPY OF A SALINE THERMAL FLUID

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1 NEW EQUATIONS FOR DENSITY, POTENTIAL TEMPERATURE AND ENTROPY OF A SALINE THERMAL FLUID Hongbing Sun 1,Manfred Koch 2, Andrew Markoe 3, Rainer Feistel 4 ABSTRAT A set of fitted polynomial equations for calculating the physical variables density, entropy and potential temperature of a thermal saline fluid in response to a temperature( ), pressure ( MPa) and salinity (0-40) change is established in this study. The freshwater components of the equations were extracted from the recently released tabulated data of freshwater properties of Wagner and Prub (2002) which is recommended by the International Association for the Properties of Water and Steam (IAPWS). The salt water component of the equation, based on the near-linear relationship between density, salinity and temperature, is extracted from the classical lower temperature Equation of State of Seawater 1980 and Feistel (2003) data sets. The freshwater and salt water components are combined to establish a set of workable multipolynomial equation, whose coefficients are computed trough standard linear regression analysis. The results obtained in this way for the density, entropy and potential temperature are comparable with the existing models except that our new equations cover a wider range of temperature than the traditional (0 40 o ) temperature range. One can apply those equations to the calculation of in-situ or onboard density, specific volume (therefore, the porosity) and potential temperature change that were usually measured on shipboard by the Ocean Drilling Project (ODP). Keywords: Saline Fluid, State Equation, Fitting, Density, Salinity, Entropy, Potential Temperature 1 Department of Geological, Environmental and Marine Sciences, 3 Department of Mathematics, Lawrenceville, New Jersey, USA. 2 Department of Geohydraulics and Engineering Hydrology, University of Kassel, Kurt-Wolters Strasse 3, Kassel, Germany. 4 Institut für Ostseeforschung, Warnemünde, Germany

1. INTRODUTION Because of fundamental thermodynamic laws, a change in water pressure can bring a change in both temperature and other physical properties (such as density and specific volume) of the

For example, lifting up a high-pressured geothermal fluid or a regular seawater parcel from the bottom of the ocean to the surface---where atmospheric pressure prevails---, the temperature of the

2 1. INTRODUTION Because of fundamental thermodynamic laws, a change in water pressure can bring a change in both temperature and other physical properties (such as density and specific volume) of the water. For example, lifting up a high-pressured geothermal fluid or a regular seawater parcel from the bottom of the ocean to the surface---where atmospheric pressure prevails---, the temperature of the thermal fluid will change even under adiabatic conditions when there is no exchange of the heat of the fluid with its surroundings (Figure 1). Density, temperature and pressure variations can also drive the fluid movement. The change of such various water properties due to pressure, temperature and salinity is referred to as the state of the water. Figure 1. Environmental contrasts between a deep borehole and the seafloor results in changes in the physical properties of the fluid. Traditionally, state equations of seawater have been established by direct empirical fitting to experiment data (cf. Millero et al., 1980, 1981, Fofonoff,1985, Fofonoff and Millard, 1983 and Bryden, 1973). With the progress in thermal dynamics theory, more recent studies (Feistel and Hagen, 1995, Feistel, 2003, and McDougall et al., 2003) have been focusing on the theoretical derivation an subsequent computation of various physical properties of seawater, at least in the temperature range of 0 to 40 o. However, this 0 to 40 o temperature range rules out their application to thermal fluids where the temperature can be above 40 o. The well-established state of water equation at high pressure and temperature from Wagner and Pruß(2002) which was the base of the International Association for the Properties of Water and Stream (IAPWS) 95 release provides an adequate method for calculating the physical properties of freshwater. However, the lack of salinity consideration in the

3 freshwater equations limits their application in a geothermal fluid condition as well. This project is attempting to combine the freshwater data at high temperature and pressure from Wagner and Pruß (2002), and saline coefficients from EOS80 s experimental data by Millero et al. (1981), Bromley et al., (1967, 1970) and the entropy data of Feistel (2003) at temperatures 0-40 o and salinity of 0 to 40 to establish a preliminary equation that can be used to roughly estimate the density, potential temperature and entropy of saline thermal fluid. Density (g/cm 3 ) Legend numbers are temperature(0), from EOS 80, Table Salinity Figure 2. Relationships between density and salinity for several temperatures as based on the experimental data of EOS80 (Millero et al., 1981). 2. DENSITY OEFFIIENTS OF THE THERMAL FLUID The polynomial coefficients for the density of the freshwater at temperature of o and pressure of 0.1 to 100 MPa were extracted from the data of Wagner and Pruß (2002). The density of seawater was from EOS80 (equation of seawater 1980) by Unesco (Joint panel on oceanographic tables and standards, Millero et al., 1981) in a temperature range of 0 to 40 0 and a salinity range of 0 to 40. The density differences between fresh and saline water in this temperature range were used to extract the polynomial coefficients of the saline part of the 4 th order multi-polynomial equation in Table 1. The logical for extending the saline part of the coefficients based on the density difference at low temperature is the approximate linear relationship between density, salinity and temperature (Figure 2). The density polynomial equation results from the addition of polynomial coefficients listed in Table 1, where ρ is the density in g/cm 3, t is the potential temperature in o, p is the pressure in MPa, and S is the salinity (no units). The goodness of the polynomial fit is indicated the multiple regression coefficient R 2 listed in the table for the freshwater and salt water columns, left and right, respectively.

3. ENTROPY (δ) The polynomial coefficients of freshwater entropy for temperatures between 0 and 374 o and pressures between 0.1 and 100 MPa were fitted to the data from Wagner and Prob (2002).

The entropy differences between the freshwater and sea water at temperature 0 to 40 0 and salinity 0 to 40 were used to extract the polynomial coefficients for the saline part of the coefficients in

4 3. ENTROPY (δ) The polynomial coefficients of freshwater entropy for temperatures between 0 and 374 o and pressures between 0.1 and 100 MPa were fitted to the data from Wagner and Prob (2002). The entropy of seawater was extracted from the calculation of Feistel (2003) s Gibb s free energy at temperature 0 to 40 0 and salinity 0 to 40. The entropy differences between the freshwater and sea water at temperature 0 to 40 0 and salinity 0 to 40 were used to extract the polynomial coefficients for the saline part of the coefficients in Table 2. Denotation of the symbols is the same as in Table POTENTIAL TEMPERATURE The potential temperature θ is defined as the temperature that an element of seawater would have if it is moved adiabatically and with no change of salinity from an initial pressure p to a reference pressure p r.in relation to the entropy, the potential temperature θ(s,t,p,p r ) at pressure p in reference to pressure p r is given implicitly by (Feistel, 2003), δ ( S, θ, pr ) = δ ( S, t, p) (1) where δ is the entropy, θ is the potential temperature, t is the in-situ temperature in o, p is the in-situ pressure, p r is the reference pressure in MPa, and S is the salinity. The calculation of the potential temperature follows McDougall et al.(2003) s practice by using

5 the equation of entropy derived in Table 2. The potential temperature is given by: θ ( S, t, p, pr ) = to + Γ( S, θ ( So, to, po, pr ), p)dp (2) where Γ is the adiabatic lapse rate calculated by Γ = G = δ δ (3) Tp G TT p T where δ p and δ T are the derivatives of entropy to pressure P and temperature T (or t) respectively. A FORTRAN program that solves Eq. (1) through (3) to calculate the potential temperature from the entropy coefficients of Table 2 is attached at the end of the paper. The density calculation as based on the coefficients of Table (1) is programmed in the same module as well. One can simply compile the FORTRAN code and produce a DOS executable file to run the calculation. A DOS executable file is available upon request from the first author. The parameters for the program input require in-situ pressure p, reference pressure p r, temperature t, and the salinity S, and the program output provides the density ρ, entropy δ, and the potential temperature θ. For users who have difficulty running a DOS program, an alternative is provided. The potential temperature can be calculated directly from the polynomial coefficients of Table 3. Because the coefficients of Table 3 have been derived from the fitting of the data calculated with the attached program, and only a limited number of coefficients listed, its accuracy will not be as good as the direct program calculation. 5. VALIDITY RANGE OF THE EQUATIONS The equations in this paper will be valid only for a liquid above the saturation line and beyond the melting curve and underneath the critical point, i.e. within the area outlined in Figure 3. This is because of the limitation of the data used in the derivation of the equations. As the coefficients for the saline part above 40 o in the equations in this paper were extended from the regression of saline data between 0 to 40 o based on the approximate linear relationship between salinity, density and Figure 3. Temperature and Pressure Validity Range of the Proposed Equations (onfined by the solid lines) temperature, and since there is a lack of experimental data for the thermal saline fluid, a precise estimation of the error range for the saline coefficients is difficult. A comparison of the polynomial equations obtained in Tables 1 and 2 of this paper with various existing models is provided in Table 4. From this one can get an approximate idea on how well the equations derived in this paper perform. All the densities in the table were calculated with the in-situ temperature. Users can also use the numbers in Table 4 as

6 check values. The potential temperature and density for This project in Table 4 originates directly from the attached program, and not from the potential temperature coefficients of Table 3. The users are encouraged to use the program directly if possible. The equations from this project are only a rough estimation of saline thermal fluid and would need more saline thermal fluid experimental data to improve their accuracy.

7 REFERENES: Bromley, L. A., Desaussure, V. A., lipp, J.., & Wright, J. S. (1967). Heat capacities of sea water solutions at salinities of 1 to 12% and temperatures of 2 to 80. Journal of hemical and Engineering Data, 12, Bromley, L. A., Diamond, A. E., Salami, E., & Wilkins, D. G. (1970). Heat capacities and enthalpies of sea salt solutions to 200. Journal of hemical and Engineering Data, 15, Bryden, H. L. (1973). New polynomials for the thermal expansion, adiabatic temperature gradient and potential temperature of sea water. Deep-Sea Research, 20, Feistel, R and E Hagen, On the GIBBS thermodynamic potential of seawater. Prog. in Oceanogr., 36, Feistel, R. (2003).A new extended Gibbs thermodynamic potential of seawater, Progress in Oceanography 58, Fofonoff, N. P. (1985). Physical Properties of Seawater: A New Salinity Scale and Equation of State of Seawater. Journal of Geophysical Research, 90, Fofonoff, N. P., & Millard, R.. (1983). Algorithms for the computation of fundamental properties of seawater. Unesco technical papers in marine science, 44. McDougall, T. J., Jackett, D. R., Wright, D. G., & Feistel, R. (2003). Accurate and omputationally Efficient Algorithms for Potential Temperature and Density of Seawater. Journal of Atmospheric and Oceanic Technology, 20, Millero, F. J., hen,. -T., Bradshaw, A., & Schleicher, K. (1980). A new high pressure equation of state for seawater. Deep-Sea Research, 27A, Millero, F. J., hen,. -T., Bradshaw, A., & Schleicher, K. (1981). Summary of data treatment for the international high pressure equation of state for seawater. Unesco technical papers in marine science, 38, Kell, G.S., 1975: Density, thermal expansivity, and compressibility of liquid water from 0 to 150 o : correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. Journal of hemical and Engineering Data. Vol., 20, No.1. p Slesarenko, V., and Shtim, A. (1989). Determination of seawater enthalpy and entropy during the calculation of thermal desalination plants. Desalination, 71, Sun. H., Koch, M., Markoe, A. (this Proceedings Volume). Tidally induced changes of fluid pressure, geothermal temperature and rock seismicity. Wagner, W., & Pruß, A The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. Journal of Physical and hemical Reference Data, 31,

8 APPENDIX Fortran Program for calculating the density, entropy and potential temperature of thermal fluid. The executable DOS program is available upon request. PROGRAM Potential temperature entropy and density * Program to calculate the potential temperature, entropy and density ##################################################################### # # # June , Modified 9/12/2006 # # # # FORTRAN program to compute potential temperature # # density and entropy of a geothermal fluid # # # # Dr. Hongbing Sun # # Rider University # # New Jersey # # # # Tel: # # hsun@rider.edu # # # ##################################################################### c S: unitless or, psu, t:deg, p: Mpa, pr:mpa, pr is the reference pressure ======================================================= IMPLIIT REAL*8 (a-g,p-t) doubleprecision s, t, p,pr,pottemp,e_freshwater character answer*1 open (22, file='potout.dat',status='unknown') * '#####################################################' '# This program will compute the density, entropy, #' '# and potential temperature of thermal saline #' '# fluid for LIQUID from MPA, T:0-374o and #' '# Salinity #' '# #' '# Dr. Hongbing Sun, Rider University, NJ #' '#####################################################' 2 Write (*,*)' Enter the salinity value' read (*,*) S write(*,*)' Enter the temperature value in degree ' read (*,*) t 'Enter the in-situ pressure value in Mpa' read (*,*) p 'Enter the reference pressure value in Mpa' read (*,*) pr e1=-entropy(s,t,p) theta=t * Newton Iteration 5 dt=-(e1+entropy(s,theta,pr))/gtt(s,theta, pr) theta=theta+dt *Loop for potential temperature theta kcount=kcount+1 *End loop when it runs 200 times

9 * if (kcount.gt.200) then pottemp=theta kcount=1 goto 100 endif if (abs(dt).lt ) then pottemp=theta goto 100 endif goto continue write (22,*) 'Your Input Data:' write (22,*)'s-psu ','p-mpa ','pr-mpa ','t- ' write (22,*) 'S ','P ',' Pr ', 'T-c ' write (22,30)S,p,pr,T 30 format(f10.5, 3f12.4) write (22,*) write (22,*) 'Output Data:' write (22,*) 'Potential Temperature',pottemp,' degree ' write (22,*) 'entropy:',-e1, ' Kj/kg/K' e2=entropy(s,pottemp,pr) den=density(s,t,p) den2=density(s,pottemp,pr) write(22,*)'density (kg/m^3)' write(22,*) den, & ' density with potential temperature of',pottemp,' =', den2 * Print out the data on screen '====================================================' ' Your input data:' ' s-psu ',' p-mpa ',' pr-mpa ',' t- ' write (*,50)S,p,pr,T ' Output Data:' 'Potential Temperature', pottemp,' degree ' 'Entropy=',-e1, ' Kj/kg/K' 50 format(f10.5,3f12.4) write(*,*) write(*,*)'density (kg/m^3)' write(*,*) den &, ' density with potential temperature of',pottemp,' =', den2 '=====================================================' 'Run another set of data? Select Y/N' read (*,'(a1)')answer if (answer.eq.'y'.or.answer.eq.'y') goto 2 Write (*,*) 'The output data is also written in file potout &.dat' stop END DOUBLEPREISION FUNTION entropy(s,t,p)

10 ======================================================= omputes specific entropy S unitless or in psu, T in, and P in MPA, Etropy in kj/kgk ======================================================= IMPLIIT REAL*8 (A-H,O-Z) doubleprecision E_freshwater,E_freshsaltDiff Array of polynomial coefficients error flag DATA ErrorReturn/ D+99/ preset: flag for error return Entropy = ErrorReturn exclude crudely erratic calls: IF (S.LT. 0.OR. T.LT. 0.OR. P.LT. 0) RETURN E_freshwater= E-03+t* E-02 & +t**2* e-05+t**3* e-08 & +t**4* e-11+t**5* e-15 & +p*( e-04+t* e-06+t**2* e-09 & +t**3* e-11+t**4* e-15) & +p**2*( e-06+t* e-08+t**2* e-11 & +t**3* e-13) & +p**3*( e-09+t* e-11+t**2* e-13 & +t**3* e-17) & +p**4*( e-15+t* e-17) E_freshsaltDiff=s* E-04+s**2* E-05 & +s**3* e-07 +s**4* e-09 & +s*t* e-05+s*t**2* e-07 & +s*p* e-06+s*p**2* e-08 & +p*s*t* e-07+p*s**2*t* e-10 & +p*s*t**2* e-09+p**2*s*t* e-10 Total Entropy =Freshwater - Freshsaltwater difference entropy=e_freshwater-e_freshsaltdiff RETURN END DOUBLEPREISION FUNTION GTT(S,T,P) ======================================================= omputes partial differential of Gibb's eneryg. It equals the differential of entropy, dsigma/dt. S in unitless or psu, T in, and P in MPA, Entropy in kj/kgk ======================================================= IMPLIIT REAL*8 (a-g,p-t) Array of polynomial coefficients error flag DATA ErrorReturn/ D+99/ preset: flag for error return GTT = ErrorReturn exclude crudely erratic calls: IF (S.LT.0.OR. T.LT. 0.OR. P.LT. 0) RETURN

11 de_freshwater= e-02 & +t*2* e-05+t**2*3* e-08 & +t**3*4* e-11+t**4*5* e-15 & +p*( e-06+t*2* e-09 & +t**2*3* e-11+t**3*4* e-15) & +p**2*( e-08+t*2* e-11 & +t**2*3* e-13) & +p**3*( e-11+t*2* e-13 & +t**2*3* e-17) & +p**4*( e-17) de_freshsaltdiff= & +s* e-05+s*t*2* e-07 & +p*s* e-07+p*s**2* e-10 & +p*s*t*2* e-09+p**2*s* e-10 * Total Entropy =Freshwater - Freshsaltwater difference gtt=de_freshwater-de_freshsaltdiff RETURN END DOUBLEPREISION FUNTION density(s,t,p) c===================================================== omputes density of the thermal fluid. kg/m^3 S in psu, T in, and P in MPA, Entropy in kj/kgk ======================================================= IMPLIIT REAL*8 (a-g,p-t) Array of polynomial coefficients error flag DATA ErrorReturn/ D+99/ preset: flag for error return density = ErrorReturn exclude crudely erratic calls: IF (S.LT.0.OR. T.LT. 0.OR. P.LT. 0) RETURN dens_freshwater= e+02 & +t* e-02+t**2* e-03 & +t**3* e-05+t**4* e-08 & +p*( e-01+t**2* e-05 & +t**3* e-07+t**4* e-10) & +p**2*( e-03+t* e-04+t**2* e-06 & +t**3* e-09) & +p**3*( e-05+t* e-07+t**2* e-08 & +t**3* e-11+t**4* e-14) dens_freshsaltdiff=s* e-01+s**2* e-03 & +s**3* e-05+s**4* e-07 & +s*t* e-03+s*t**2* e-05 & +p*s* e-04+p**2*s* e-06 & +p*s*t* e-06 density=dens_freshwater-dens_freshsaltdiff RETURN END

Notes on the function gsw_internal_energy(sa,ct,p)

Notes on gsw_internal_energy Notes on the function gsw_internal_energy(sa,ct,p) The function gsw_internal_energy(sa,ct,p) evaluates the specific internal energy of seawater as a function of Absolute Salinity,