PROCEEDINGS, Thirtieth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 31-February 2, 2005 SGP-TR-176

VERMA PROCEDURE FOR THE DETERMINATION OF HEAT CAPACITY OF LIQUIDS

Mahendra P. Verma

Geotermia, Instituto de Investigaciones Eléctricas Av. Reforma 113, Col. Palmira, Apdo. 1-475

Cuernavaca, Morelos, 62490, México e-mail: mahendra@iie.org.mx

ABSTRACT

An ActiveX component, SteamTablesIIE, based on the IAPWS-95 formulation was written in Visual Basic 6.0 for calculating thermodynamic properties of pure water as a function of any two independent state variables from temperature (T), pressure (P), volume (V), internal energy (U), enthalpy (H), and Gibbs free energy (G). However, thermodynamic inconsistencies were found in the formulation. The thermodynamic properties like U, H, G, and S are calculated from the experimental values of heat capacity at constant pressure (CP) and/or at constant volume (CV). The measurements of CP of solids and CV of gases are feasible; however, it is difficult to measure CP or CV for all the conditions of T and P required for calculating the thermodynamic properties of liquids. The reported experimental values of CP and CV along the saturation curve increase drastically near to the critical point of water. Similarly, according the IAPWS-95 formulation, the value of CP or CV at the critical point of water is ~109 kJ/kg K, which means that the critical point acts as a heat sink. Additionally, the calculated values of other thermodynamic properties from CP or CV are not in agreement with their values obtained from the formulation. Heat capacity is not a state function. However, one can use the same trajectory for measuring the heat capacity and calculating the thermodynamic properties. Based on this criterion, a new Verma procedure is devised. Firstly, the heat capacity along the saturation curve, CSat is defined as the proportion of amount heat to the change in temperature. Similarly, the values of CV can be measured precisely in the liquid and vapor phases. Then using the PVT characteristics, CV and CSat, the thermodynamic properties of water are calculated. Accordingly, the internal energy for the compressed liquid and superheated steam are calculated as

1 1

1 . .,

2 2

2 . .,v .

, .

, . , . .

sat

sat ref T P liq

sat

ref refsat ref T P ap

T T para V V V

liq V liq sat liqT para V V T V

T T para V V V

vap V vap sat vap T PT para V V T V

U C dT C dT PdV

U C dT C dT PdV L H

=

=

=

=

= + −

= + − +

∫ ∫ ∫

∫ ∫ ∫

where the reference point is the triple point (T.P.) of water and L.H. is the latent heat at T.P.

INTRODUCTION

The geochemical modeling of aquatic systems like rain, rivers, lakes, groundwater aquifers, geothermal reservoirs, petroleum reservoirs, seas, oceans, etc. on our planet, Earth contemplates to understand the physical-chemical processes responsible for their origin and evaluation. The principle component of all these systems is water. Similarly, water is vital for the existence of life on the Earth. Additionally, water plays an important role in the geological processes like mass transportation, dissolution-precipitation, crust stratigraphy, etc. In the electric industry, water is used to generate electricity. Water is heated to produce vapor by different sources of heat like carbon, oil, natural gas, nuclear fuel, geothermal heat, etc. and the vapor is used to move turbines. Thus, to understand the above processes, the thermodynamic properties of water are of fundamental importance. The International Association for the Properties of Water and Steam (IAPWS) promotes scientific investigations for creating standard values for the thermodynamic properties of water. Wegner and Praβ (2000) developed an empirical formulation, IAPWS-95 on the basis of least square fittings of experimental data for the properties of water. Using the formulation, Verma (2003) wrote an ActiveX component, IAPWS95SteamTables in Visual Basic 6.0 for calculating 23 properties of pure water as a function of temperature (190 to 2000 K) and pressure (3.23x10-8 to 10,000 MPa). Since temperature (T), pressure (P), volume (V), internal energy (U), enthalpy (H), Gibbs free energy (G), etc. are state functions, Verma (2005) wrote a new ActiveX component, SteamTablesIIE in Visual Basic 6.0 to calculate the thermodynamic properties of pure water as a function of any two independent state variables from the state functions. However, some thermodynamic inconsistencies were found in the formulation, which were the limitations for the functionality of SteamTablesIIE for all the permissible ranges of the independent variables.

In this article we present the thermodynamic inconsistencies in the IAPWS-95 formulation. Similarly, a new Verma procedure for calculating thermodynamic properties of liquids is devised. An experimental setup for measuring the thermodynamic properties of water is presented.

STATE FUNCTION

Verma (2002) presented a method of interdependence of state variables in order to scrutinize the internal consistency among the thermodynamic datasets. Thermodynamic variables (e.g., T, P, V, G, U, H and S) including the equilibrium constant of a chemical reaction, solubility, viscosity, thermal conductivity, etc. are state functions. A state function does not depend on the past history of the substance or on the path it has followed in reaching a given state. It should be single valued and continuously differentiable unless there is a phase transition. On fixing the values of any two independent state functions (for example, T and P), the values of all the other state functions are uniquely defined. Any three state variables X, Y and Z obey the following mathematical relation for exact functions

1−=⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

XYZ Z

Y

X

Z

Y

X (1)

If Z is constant, we are looking for the dependence of X on Y and vice versa. Then

1 =⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

X

Y

Y

X (2)

If the same value of X exists for two values of Y (say Y1 and Y2) at constant Z, there should be at least one minimum or maximum in the behavior of X between Y1 and Y2. At the minimum or maximum, one can write

, 0, 0 and .X Y

Y ve XY X

∂ ∂⎛ ⎞ ⎛ ⎞∆ = ± ∆ = = = ∞⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

Thus,

0 indeterminate, but not 1.X Y

Y X

∂ ∂⎛ ⎞ ⎛ ⎞ = × ∞ ⇒ =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

In summary, there cannot be any maximum or minimum in the behavior of a state function with an independent thermodynamic variable when the other independent variable is constant. In other words, a state function (or an exact function) cannot be a multi-valued function. The mathematical concepts discussed above are explained through schematic diagrams in Figure 1. Let us consider a state function (say X) which is a function of independent thermodynamic variables P and T. Figure 1(a) shows two behaviors of X at a given value of P (i.e. at P1 and P2, respectively). In case I there are two values of X at a given T, whereas in case II there are two values of T for a value of X. Thus X(T) in case I and T(X) in case II are not single valued functions.

T

A

∞⇒∂∂

T

A

∞⇒∂∂AT

T1 T2T3

A1

A2

A3

A

T1

A1

∞⇒∂∂=

∂∂

TP

XP

P1

P2

T

A

T1

A1

Point ofInflexion

T

a

b

c

I

II

V

P

T1

T2

T3

T

V1

P

V

T

d

e

f

Increase of T

V2

V3

Increase of V

P1

P2

P3

Increase of P

P1<P2<P3

V1<V2<V3

T1<T2<T3

T

A

∞⇒∂∂

T

A ∞⇒∂∂

T

A

∞⇒∂∂AT ∞⇒

∂∂AT

T1 T2T3

A1

A2

A3

A

T1

A1

∞⇒∂∂=

∂∂

TP

XP ∞⇒

∂∂=

∂∂

TP

XP

P1

P2

T

A

T1

A1

Point ofInflexion

T

a

b

c

I

II

V

P

T1

T2

T3

T

V1

P

V

T

d

e

f

Increase of T

V2

V3

Increase of V

P1

P2

P3

Increase of P

P1<P2<P3

V1<V2<V3

T1<T2<T3

Figure. 1. A schematic diagram to explain the behavior of a state function with independent thermodynamic variable (T and P). According to the definition, the behaviors in the Figs. 1(a, b, c) are impermissible for a state function. Similarly, the Figs. 1(d, e, f) represents the permissible behaviors of a state function, which were derived from the state equation of ideal gas (PV=nRT).

Similarly, ∞=∂∂

XT in case I and ∞=∂

∂T

X in case

II. It means that T or X are not continuously differentiable. In other words, neither T nor X is a thermodynamic state function in the respective cases. Figure 1(b) presents the behaviors of X with T at two pressures P1 and P2. The functions are crossing at temperature T1. Then at T=T1, P

X∂ = ∞∂

and

∞=∂∂

TP . It means that P is not a state function.

Figure 1(c) shows a point of inflection in the variation of X with T at constant P. A point of inflection is not always singular. However, a rotation axes such that one of the axis is parallel to the tangent at the point of inflection, makes the point of inflection as singular in the new coordinate system (i.e. the gradient x

T′∂

′∂ is positive or negative on

both sides and 0 or infinite at the point of inflection). A linear transformation of state functions produces new state functions. So, there cannot be any point of inflection in the behavior of a state function. In summary, the behaviors of a state function with an independent state variable when the other independent variable is constant, as shown in Figures 1 (a, b and c), are impermissible. The ideal gas equation is PV=nRT. For an ideal gas system of constant mass, Figures 1(d, e and f) present

the behaviors of P with V at constant T, of T with P at constant V and of V with T at constant P, respectively. In the Figures 1(e and f) the behaviors converge to the origin (i.e., P or T is tending to zero); however, the classical thermodynamics is not valid when P or T is tending to zero. In these situations both dependent and independent variables are zero, which means the convergence point is indeterminate. Thus the behaviors presented in the Figures (d, e and f) are valid for a state function. Additionally, thermodynamics does not impose any restriction on the behavior of a state function (say X) with respect to independent state variables (say T and P). However, if we know the behavior of T with P and behavior of X with T (or P) for a system, we can predict the behavior of X with P (or T). For an ideal gas system of constant mass and V, T increases with increasing P and vice versa. It means if V increases with T, it should decrease with P. On considering T and P as independent variables and V as constant, all the other state functions should be constant (i.e., uniquely defined) under the situation. Thus, the behavior of P and T should be similar for all the other state functions as that for V. For example, if H increases with T, it should decrease with P. Let us rewrite the equation (1) in X, T and P as

XPT P

T

T

X

P

X⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂−=⎟

⎠⎞

⎜⎝⎛

∂∂

(3)

If T increases with increasing P in a system, then ( ) veP

TX

+=∂∂ . If X increases with T, it should

decrease with P in order to fulfill the equation (3). Similarly, if we consider V and P as independent thermodynamics variables, V decreases generally with increasing P in a phase and vice versa. ( ) veP

VX

−=∂∂ . If X increases (or decreases) with P, it

should also increase (or decrease) with V. In summary, the behavior of a state function with an independent state variable (say T) at the constant of another independent stable variable (say P) should be single valued and continuous differentiable in a phase. It is very similar to the behavior of electromagnetic fields, which never cross and are parallel for short displacements. The scientific literature contains many experimental and theoretical determinations of a state function which are violating its definition. Here the basic aspects of a state function and its dependence with the independent thermodynamic variables will be used to scrutinize as an example the internal consistencies in the water steam tables.

STEAM TABLES OF PURE WATER

The IAPWS-95 formulation for the thermodynamic properties of pure water apparently provides more consistent values of the properties of water than the earlier formulations (Wegner and Praβ, 2000). As mentioned earlier that any two state functions are

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

Superheated Steam

Triple Point

Critical Point

Satu

ratio

n Cur

ve

Mel

ting

Cur

ve

Ice V

Ice

III

Ice

I

Ice VI Ice VII

Subl

imat

ion

Cur

ve

Sol

id

I

II

Supercritical Fluid Region

Compressed Liquid

Critical IsochorIII

IV

V

Solid

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

Liq

uid

I Liquid

Vapor

Ice

I

MinimumSpecific Volume

Curve

I

IV

II

VI

V

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

Superheated Steam

Triple Point

Critical Point

Satu

ratio

n Cur

ve

Mel

ting

Cur

ve

Ice V

Ice

III

Ice

I

Ice VI Ice VII

Subl

imat

ion

Cur

ve

Sol

id

I

II

Supercritical Fluid Region

Compressed Liquid

Critical IsochorIII

IV

V

Solid

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

Liq

uid

I Liquid

Vapor

Ice

I

MinimumSpecific Volume

Curve

I

IV

II

VI

V

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

Superheated Steam

Triple Point

Critical Point

Satu

ratio

n Cur

ve

Mel

ting

Cur

ve

Ice V

Ice

III

Ice

I

Ice VI Ice VII

Subl

imat

ion

Cur

ve

Sol

id

I

II

Supercritical Fluid Region

Compressed Liquid

Critical IsochorIII

IV

V

Solid

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

Liq

uid

I Liquid

Vapor

Ice

I

MinimumSpecific Volume

Curve

I

IV

II

VI

V

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

1.E-8

1.E-6

1.E-4

1.E-2

1.E+0

1.E+2

1.E+4

Pre

ssur

e/M

Pa

0 200 400 600 800 1000 1200

Temperature /K

Superheated Steam

Triple Point

Critical Point

Satu

ratio

n Cur

ve

Mel

ting

Cur

ve

Ice V

Ice

III

Ice

I

Ice VI Ice VII

Subl

imat

ion

Cur

ve

Sol

id

I

II

Supercritical Fluid Region

Compressed Liquid

Critical IsochorIII

IV

V

Solid

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

1.0E-3

1.0E-1

1.0E+1

1.0E+3

250 260 270 280 290

Liq

uid

I Liquid

Vapor

Ice

I

MinimumSpecific Volume

Curve

I

IV

II

VI

V

Figure. 2. PT-space of pure water according to the

IAPWS-95 formulation, which is divided by six separation boundaries: (i) sublimation curve, (ii) saturation curve, (iii) critical isochor, (iv) ice I melting curve, (v) melting (Ice III to Ice VII) curve and (vi) minimum specific volume curve. The inserted figure depicts the location of liquid I and the minimum specific volume boundary.

sufficient to define completely the values of all the other state functions in a phase for a pure system of constant mass. Using the interdependence among the state variables T, P, V, U, H, G and S on considering any two as independent variables, we will illustrate the thermodynamic inconsistencies in the IAPWS-95 formulation.

PVT Characteristics of Water Verma (2003) demonstrated the PT relationship for pure water according to the IAPWS-95 formulation (Figure 2). He presented that the critical isochor (i.e., total specific volume V= 0.003106 m3/kg or density = 322.0 kg/m3) in the supercritical fluid region acts as a phase boundary between the liquid-like and vapor-like fluid. The saturation curve is the locus of T and P values where liquid and vapor coexist in equilibrium: it starts from the ice-liquid-vapor triple point and terminates at the critical point. Water has many solid forms, depending on the conditions of P and T. All the Ice curves (I, III, V, VI and VII) define the melting curve. The supercritical fluid region, existing at T and P higher than those of critical point, is shown by diagonal solid lines, which do not represent phase changes but depend on arbitrary definition of what constitutes the liquid and vapor phases. According to the IAPWS-95 formulation the supercritical fluid region also contains solid (ice) water. The region of vapor and liquid, bounded by the PT-space, will be drawn later in terms of different independent thermodynamic variables.

It is well known that the variation of specific volume of liquid water with T at P=0.1 MPa has a minimum volume at 277.127 K. We know that P, T and V are state functions. If we consider P and V as independent state variables, there are two values of T at a given value of P and V. But according to the definition of state function, it should be uniquely defined, if T is a state function. This cannot be explained thermodynamically if we consider the whole liquid water region as a single phase. It was called as an anomalous behavior since the development of thermodynamics. It is anomalous to the thermodynamic laws, if the whole liquid water is considered as single phase. Recently, Sciortino et al. (2003) presented the physics of liquid-liquid transition. The editorial of the journal, Physical Review Letters (2003) summarized the work of Sciortino et al. as “A Tale of Two Liquids”. It says, “Any liquid which eventually expands as it cools must have a liquid-liquid critical point.” Thus there are two types of liquid water, which are represented by liquid and liquid I in Figure 2. Additionally, other works suggest that there is an effect of hydrogen bonding on the molecular structure of water at low temperatures (IAPWS, 2004). There are two types of structure: one is associated with hydrogen bonding (below 4ºC) and other is without hydrogen bonding. According to the definition of “phase”, a change in the structure of water molecule will produce a different phase. Therefore, there is a phase transition along the minimum volume curve (Figure 2), but there are still many questions to be answered about the type of phase transition, etc. However, T is uniquely defined on fixing P and V in each liquid phase. The inserted figure in Figure 2 shows that the minimum specific volume curve acts as a separation boundary between the liquid and liquid I phases of water. It is derived from the IAPWS-95 formulation, so its exactitude depends only on the formulation. Figure 3a shows the VP-space for pure water. There is a triple line where liquid-vapor-solid (ice) coexists. The two phase region (liquid and/or liquid I and vapor) is bounded among the triple line, liquid saturation curve and vapor saturation curve. The vapor and liquid saturation curves meet at the critical point. The critical isochor, represented by a vertical line is the separation boundary between liquid and vapor. The liquid phase is up to the minimum specific volume curve. The liquid I phase will be bounded by the minimum specific volume curve, melting curve, and triple line. The solid (ice) will be on the higher volume side at the melting curve. A question comes in mind what type of water will be on the lower volume side at the minimum specific volume curve. The region is shown in the inserted figure with a question mark on phase. Probably, there may exist some solid phase. Here we are not interested in the thermodynamic properties of solid (ice) water; therefore, we will not discuss it further.

Figure 3b shows the VT-space. It is similar to VP-space and it also shows the location of Liquid I. Thus the behaviors of P, T and V in the IAPWS-95 formulation for the steam tables of pure water are thermodynamically consistent.

Behaviors of H, G and S of Water Figure 4 shows (a) isobaric behavior of H with T and (b) isothermal behavior of H with P. In the pure water system of constant mass and V, P increases with increasing T and vice versa. It means if H increases with T, it should decrease with P. It can be observed in the Figure 4(a) that the behavior of H in the vapor phase is consistent with the definition of a state function. However, the behavior of H in the liquid phase is inconsistent. In the low temperature range H increases with both T and P and is overlapping with the two phase region, whereas it increases with T and decreases with P in the high temperature range. The tendencies are crossing in the middle temperature range. Thus the values of H in the low and middle temperature ranges are violating the definition of a state function. Figure 4(b) shows the isothermal variation of H with P. There are two pressures for a given value of H in the compressed liquid region. For example, H=2000 kJ/kg for the 700 K isotherm is at P1=63.458 MPa and P2=456.356 MPa. If we consider T and H as independent variables, it is impossible to predict the right value of P. The meaning of two values of P is that P is not an exact solution. This way- thermodynamics is not an Exact Science. Everyone knows that thermodynamic is an exact science, therefore the two values of P in the above situation are incorrect. Let us further exemplify the above situation. We have a container filled with water at T=700 K and P=63.485 MPa. Since there is same H (2000 kJ/kg) for P= 63.458 MPa and 456.356 MPa at T= 700 K (see Figure 4b), it means we can pressurize (from 63.458 to 465.356 MPa) the container without any work. In other words, we can increase the pressure in a system of constant V and T without doing any work. If it is possible, we can develop a Carnot cycle to solve the world energy problem with no expense. If we consider a change in volume in the above example (from V1 to V2), one has to do some work

2

1

V

VW PdV= ∫ (4)

W cannot be zero; however, 0H∆ = means that W should be zero. This is contradictory to the basic thermodynamics. Similar inconsistency was observed in the behavior of U. In summary there is a violation of fundamental laws of thermodynamics for H and U in the IAPWS-95 formulation. Figure 5 shows (a) isobaric behavior of G with T and (b) isothermal behavior of G with P. The behavior of G is consistent with the definition of state function. It increases with P, but it decreases with T. It has a

P

ress

ure

/MP

a

Sublimation Curve

Vapor Saturation

Cri

tical

Iso

chor

Mel

ting

Cur

ve

Triple Line Vapor

Solid

Liq

uid

1.0E-6

1.0E-4

1.0E-2

1.0E+0

1.0E+2

1.0E+4

Two Phase Region

Liq

uid

Satu

ratio

n

Critical Point

0.0001

0.001

0.01

0.1

1

10

9.980E-04 9.990E-04 1.000E-03 1.001E-03

Mel

ting

Cur

ve

Phase? Liquid I

Liquid

Solid

Minimum Specific Volume Curve

Liquid saturation curve

1.E-4 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6 1.E+8

Volume /(m3/kg)

100

300

500

700

900

1100

1300

1500

1700

1900

Tem

per

atur

e /K

268

270

272

274

276

278

280

9.80E-04 9.85E-04 9.90E-04 9.95E-04 1.00E-03 1.01E-03

Triple LineSublimation Curve

Vapor

Solid

Liquid I

Liquid

Solid

Liquid

272

274

276

278

280

1.00000E-03 1.00010E-03 1.00020E-03

Vapor

Liquid I

Solid

Saturation Curve

Two Phase Region

Vap

or

Liq

uid

b

a

Pre

ssu

re/M

Pa

Sublimation Curve

Vapor Saturation

Cri

tical

Iso

chor

Mel

ting

Cur

ve

Triple Line Vapor

Solid

Liq

uid

1.0E-6

1.0E-4

1.0E-2

1.0E+0

1.0E+2

1.0E+4

Two Phase Region

Liq

uid

Satu

ratio

n

Critical Point

0.0001

0.001

0.01

0.1

1

10

9.980E-04 9.990E-04 1.000E-03 1.001E-03

Mel

ting

Cur

ve

Phase? Liquid I

Liquid

Solid

Minimum Specific Volume Curve

Liquid saturation curve

1.E-4 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6 1.E+8

Volume /(m3/kg)

100

300

500

700

900

1100

1300

1500

1700

1900

Tem

per

atur

e /K

268

270

272

274

276

278

280

9.80E-04 9.85E-04 9.90E-04 9.95E-04 1.00E-03 1.01E-03

Triple LineSublimation Curve

Vapor

Solid

Liquid I

Liquid

Solid

Liquid

272

274

276

278

280

1.00000E-03 1.00010E-03 1.00020E-03

Vapor

Liquid I

Solid

Saturation Curve

Two Phase Region

Vap

or

Liq

uid

Pre

ssu

re/M

Pa

Sublimation Curve

Vapor Saturation

Cri

tical

Iso

chor

Mel

ting

Cur

ve

Triple Line Vapor

Solid

Liq

uid

1.0E-6

1.0E-4

1.0E-2

1.0E+0

1.0E+2

1.0E+4

Two Phase Region

Liq

uid

Satu

ratio

n

Critical Point

0.0001

0.001

0.01

0.1

1

10

9.980E-04 9.990E-04 1.000E-03 1.001E-03

Mel

ting

Cur

ve

Phase? Liquid I

Liquid

Solid

Minimum Specific Volume Curve

Liquid saturation curve

Pre

ssu

re/M

Pa

Sublimation Curve

Vapor Saturation

Cri

tical

Iso

chor

Mel

ting

Cur

ve

Triple Line Vapor

Solid

Liq

uid

1.0E-6

1.0E-4

1.0E-2

1.0E+0

1.0E+2

1.0E+4

Two Phase Region

Liq

uid

Satu

ratio

n

Critical Point

0.0001

0.001

0.01

0.1

1

10

9.980E-04 9.990E-04 1.000E-03 1.001E-03

Mel

ting

Cur

ve

Phase? Liquid I

Liquid

Solid

Minimum Specific Volume Curve

Liquid saturation curve

1.E-4 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6 1.E+8

Volume /(m3/kg)

100

300

500

700

900

1100

1300

1500

1700

1900

Tem

per

atur

e /K

268

270

272

274

276

278

280

9.80E-04 9.85E-04 9.90E-04 9.95E-04 1.00E-03 1.01E-03

Triple LineSublimation Curve

Vapor

Solid

Liquid I

Liquid

Solid

Liquid

272

274

276

278

280

1.00000E-03 1.00010E-03 1.00020E-03

Vapor

Liquid I

Solid

Saturation Curve

Two Phase Region

Vap

or

Liq

uid

1.E-4 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6 1.E+8

Volume /(m3/kg)

100

300

500

700

900

1100

1300

1500

1700

1900

Tem

per

atur

e /K

1.E-4 1.E-2 1.E+0 1.E+2 1.E+4 1.E+6 1.E+8

Volume /(m3/kg)

100

300

500

700

900

1100

1300

1500

1700

1900

Tem

per

atur

e /K

268

270

272

274

276

278

280

9.80E-04 9.85E-04 9.90E-04 9.95E-04 1.00E-03 1.01E-03

Triple LineSublimation Curve

Vapor

Solid

Liquid I

Liquid

Solid

Liquid

272

274

276

278

280

1.00000E-03 1.00010E-03 1.00020E-03

Vapor

Liquid I

Solid

Saturation Curve

Two Phase Region

Vap

or

Liq

uid

b

a

Figure. 3. P-V and T-V spaces for pure water

according to the IAPWS-95 formulation. The inserted figures illustrate the location of Liquid I phase.

500

1500

2500

3500

1 10 100 1000

P /(MPa)

H /(

kJ/k

g)

0

1000

2000

3000

4000

250 500 750 1000

T /(K)

H /(

kJ/k

g)

0

100

200

300

400

275 300 325 350

P=0.1 MPa

P=10

P=50

P=100

P=PC

Vapor Saturation

Liquid Saturation

C.P.

Critical Isochor

T=500 K

T=800

T=600

T=TC

Vapor Saturation

Liquid Saturation

C.P.

Crit

ical I

soch

or

Two Phase Region

Liquid Region

Vapor Region

P1P2

H=2000

T=700

T=900 T=1000

T=TC

Liquid Saturation

BA

Liquid Region

Vapor Region

a

b

500

1500

2500

3500

1 10 100 1000

P /(MPa)

H /(

kJ/k

g)

0

1000

2000

3000

4000

250 500 750 1000

T /(K)

H /(

kJ/k

g)

0

100

200

300

400

275 300 325 350

P=0.1 MPa

P=10

P=50

P=100

P=PC

Vapor Saturation

Liquid Saturation

C.P.

Critical Isochor

T=500 K

T=800

T=600

T=TC

Vapor Saturation

Liquid Saturation

C.P.

Crit

ical I

soch

or

Two Phase Region

Liquid Region

Vapor Region

P1P2

H=2000

T=700

T=900 T=1000

T=TC

Liquid Saturation

BA

Liquid Region

Vapor Region

a

b

500

1500

2500

3500

1 10 100 1000

P /(MPa)

H /(

kJ/k

g)

0

1000

2000

3000

4000

250 500 750 1000

T /(K)

H /(

kJ/k

g)

0

100

200

300

400

275 300 325 350

P=0.1 MPa

P=10

P=50

P=100

P=PC

Vapor Saturation

Liquid Saturation

C.P.

Critical Isochor

T=500 K

T=800

T=600

T=TC

Vapor Saturation

Liquid Saturation

C.P.

Crit

ical I

soch

or

Two Phase Region

Liquid Region

Vapor Region

P1P2

H=2000

T=700

T=900 T=1000

T=TC

Liquid Saturation

BA

Liquid Region

Vapor Region

a

b

500

1500

2500

3500

1 10 100 1000

P /(MPa)

H /(

kJ/k

g)

0

1000

2000

3000

4000

250 500 750 1000

T /(K)

H /(

kJ/k

g)

0

100

200

300

400

275 300 325 350

P=0.1 MPa

P=10

P=50

P=100

P=PC

Vapor Saturation

Liquid Saturation

C.P.

Critical Isochor

T=500 K

T=800

T=600

T=TC

Vapor Saturation

Liquid Saturation

C.P.

Crit

ical I

soch

or

Two Phase Region

Liquid Region

Vapor Region

P1P2

H=2000

T=700

T=900 T=1000

T=TC

Liquid Saturation

BA

Liquid Region

Vapor Region

a

b

Figure. 4. (a) Isobaric and (b) Isothermal variations

of Enthalpy (H).

-4000

-3000

-2000

-1000

0

250 500 750 1000

T /(K)

G /(

kJ/k

g)

-2000

-1500

-1000

-500

0

1 10 100 1000

P /(MPa)

G /(

kJ/k

g)

-100

-50

0

50

100

250 300 350 400 450 P=0.1MPa

P=PC

P=5

P=100

P=50

P=15

P=100P=50

P=PC

Saturation curve

C.P.

Critical Isochor

Liquid Region

Vapor Region

T=500 K

T=800

T=600

T=TC

T=700

T=900

T=1000

Liquid Region

Vapor Region

Saturation curve

a

b

C.P.Critical isochor

-4000

-3000

-2000

-1000

0

250 500 750 1000

T /(K)

G /(

kJ/k

g)

-2000

-1500

-1000

-500

0

1 10 100 1000

P /(MPa)

G /(

kJ/k

g)

-100

-50

0

50

100

250 300 350 400 450 P=0.1MPa

P=PC

P=5

P=100

P=50

P=15

P=100P=50

P=PC

Saturation curve

C.P.

Critical Isochor

Liquid Region

Vapor Region

T=500 K

T=800

T=600

T=TC

T=700

T=900

T=1000

Liquid Region

Vapor Region

Saturation curve

a

b

C.P.Critical isochor

-4000

-3000

-2000

-1000

0

250 500 750 1000

T /(K)

G /(

kJ/k

g)

-2000

-1500

-1000

-500

0

1 10 100 1000

P /(MPa)

G /(

kJ/k

g)

-100

-50

0

50

100

250 300 350 400 450 P=0.1MPa

P=PC

P=5

P=100

P=50

P=15

P=100P=50

P=PC

Saturation curve

C.P.

Critical Isochor

Liquid Region

Vapor Region

T=500 K

T=800

T=600

T=TC

T=700

T=900

T=1000

Liquid Region

Vapor Region

Saturation curve

a

b

C.P.Critical isochor

-4000

-3000

-2000

-1000

0

250 500 750 1000

T /(K)

G /(

kJ/k

g)

-2000

-1500

-1000

-500

0

1 10 100 1000

P /(MPa)

G /(

kJ/k

g)

-100

-50

0

50

100

250 300 350 400 450 P=0.1MPa

P=PC

P=5

P=100

P=50

P=15

P=100P=50

P=PC

Saturation curve

C.P.

Critical Isochor

Liquid Region

Vapor Region

T=500 K

T=800

T=600

T=TC

T=700

T=900

T=1000

Liquid Region

Vapor Region

Saturation curve

a

b

C.P.Critical isochor

Figure. 5. (a) Isobaric and (b) Isothermal variations

of Gibbs free energy (G).

0.0001

0.001

0.01

0.1

1

10

100

-1500 -1000 -500 0 500

LiquidRegion

Two-phase

Region

Vapor Region

LiquidRegion

Two-phase

Region

Vapor

Region

LiquidRegion

Two-phaseRegion

Vapor Region

-2

0

2

4

6

8

10

12

14

-1000 -600 -200 200

a

b

c

Sublimationcurve

Sublimationcurve

Sublimationcurve

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000

-5 0 5 10 15 20

S/(kJ/kg)

V/(

m3/

kg)

Liquid SaturationLiquid Critical IsochorLiquid Max TLiquid Max PVapor SaturationVapor Critical IsochorVapor Max TVapor Min PTwo-phase Boundary

-2

0

2

4

6

8

10

12

14

16

18

20

-30000 -20000 -10000 0 10000

G/ (kJ/kg)

S/ (

kJ/k

g)

-2

0

2

4

6

8

10

12

14

16

18

20

-30000 -20000 -10000 0 10000

G/ (kJ/kg)

S/ (

kJ/k

g)

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000

-30000 -20000 -10000 0 10000

G/ (kJ/kg)

V /

(m3 /k

g)

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

1000000

10000000

100000000

-30000 -20000 -10000 0 10000

G/ (kJ/kg)

V /

(m3 /k

g)

Figure. 6. V-G, V-S and G-S spaces for pure water

according to the IAPWS-95 formulation.

discontinuity along the saturation curve and a point of inflection along the critical isochor. It means that there is a first order liquid-vapor phase transition along the saturation curve, while a second order liquid-vapor phase transition exists along the critical isochor. Figure 6 shows the V-G, V-S and S-G spaces for pure water according to the IAPWS-95 formulation. It covers the region corresponding to T (190 to 2000 K) and P (3.23x10-8 to 10000 MPa) for the liquid and vapor phases as shown in Figure 2. It can be observed that the V-G and V-G spaces do not overlap and are single valued, whereas the compressed liquid region is overlapping to the two phase and vapor regions in the case of S-G space. It means that G and S for the compressed liquid according to the IAPWS-95 formulation are multi-valued function and it is a violation of the definition of state function. Thus the values of U, H, G and S in the IAPWS-95 formulation are inconsistent thermodynamically. There is need to revise the procedure and the experimental values, used in the development of the IAPWS-95 formulation.

Behaviors of CP and CV of Water The behaviors of P, V and T are consistent, while the behaviors of U, H, G and S are inconsistent thermodynamically in the IAPWS-95 formulation. To understand the thermodynamic inconsistencies we will analyze the experimental data for the heat capacity (Cp) at constant pressure. Figure 7 shows the experimental values of Cp, used in the derivation of the IAPWS-95 formulation. The values for the liquid and vapor phases along the saturation curve are shown by solid and dashed curves, respectively. They are increasing with T for both the phases. The values of Cp are very high near to critical point (i.e., ~109 kJ/kg K). According to the behavior of Cp, the critical point acts as a heat sink, this cannot be explained with thermodynamics. The values of Cp in the compressed liquid and superheated steam regions for the selected pressure ranges are also shown in Figure 7. The vertical lines are the positions of liquid-vapor separation boundary, obtained by the PVT characteristics of pure water. The blue vertical line is corresponding to the critical conditions. The red lines in the left of blue line are for the separation boundaries along the saturation curve, while the red lines in the right are the separation boundaries along the critical isochor. The maximum in the behavior of CP at a particular pressure is on the vertical lines or in the compressed liquid region. The heat capacity is not state function; however, CP is defined along a specific path. So, it should have definite tendency, but it does not have. The multiple values in the compressed liquid region indicate that CP increases first with T and then decreases. It cannot be explained with the basic laws of thermodynamics. Thus, the development of the

Vapour region

Liquid region

1

10

100

400 600 800T (K)

Cp

(kJ

/kg

/K)

Vapor along saturationLiquid along saturationP= 4.0 - 6.0 MPaP= 9.0 - 11.0P=14.0 - 16.0P=19.0 - 21.0P=24.0 - 26.0P=29.0 - 31.0P=34.0 - 36.0P=39.0 - 41.0P=44.0 - 46.0P=49.0 - 51.0 Vapour region

Liquid region

1

10

100

400 600 800T (K)

Cp

(kJ

/kg

/K)

Vapor along saturationLiquid along saturationP= 4.0 - 6.0 MPaP= 9.0 - 11.0P=14.0 - 16.0P=19.0 - 21.0P=24.0 - 26.0P=29.0 - 31.0P=34.0 - 36.0P=39.0 - 41.0P=44.0 - 46.0P=49.0 - 51.0

5 10 15 20 25 30

PC

R

50454035

Vapour region

Liquid region

1

10

100

400 600 800T (K)

Cp

(kJ

/kg

/K)

Vapor along saturationLiquid along saturationP= 4.0 - 6.0 MPaP= 9.0 - 11.0P=14.0 - 16.0P=19.0 - 21.0P=24.0 - 26.0P=29.0 - 31.0P=34.0 - 36.0P=39.0 - 41.0P=44.0 - 46.0P=49.0 - 51.0 Vapour region

Liquid region

1

10

100

400 600 800T (K)

Cp

(kJ

/kg

/K)

Vapor along saturationLiquid along saturationP= 4.0 - 6.0 MPaP= 9.0 - 11.0P=14.0 - 16.0P=19.0 - 21.0P=24.0 - 26.0P=29.0 - 31.0P=34.0 - 36.0P=39.0 - 41.0P=44.0 - 46.0P=49.0 - 51.0

5 10 15 20 25 30

PC

R

50454035

Figure. 7. Experimental data of Cp for pure water

used in the development of the IAPWS-95 formulation.

IAPWS-95 formulation is based on the thermodynamically inconsistent experimental data. The CP and CV of water have been measured along the saturation curve; however, their physical significance has never been explained. For example, there is need to know a relation between the increase in T and amount of heat given to a system at constant P in order to measure CP. There will be a change in P on changing T along the saturation curve. It is not possible to keep constant P. Thus, the measurement of CP or CV along the saturation curve is not feasible.

VERMA PROCEDURE FOR MEASURING THERMODYNAMIC PROPERTIES OF LIQUID

Till now, the thermodynamic properties have been calculated from CP and/or CV. As it has been discussed above that it is not possible to measure CP and CV of liquids at all conditions of independent variables (say T and P), required for calculating the thermodynamic properties. Therefore, a new Verma procedure is devised. The heat capacity along the saturation curve, CSat is defined as the proportion of amount heat to the change in temperature. Thus, CSat is a function of only one independent variable (say T). Heat capacity is not a state function. Its value depends on the trajectory between two points. Therefore, one has to use the same trajectory (along the saturation curve and then the constant V path) for the calculation of the thermodynamic properties like U, H, G, S, etc. The values of CV in the compressed liquid region and superheated steam region can be measured quite precisely. Actually, the liquid I region is based only on the experimental data of T and V at P=0.1 MPa. There are no other experimental data for T and V at constant P in this region. Therefore, we will not consider here the liquid I region and the triple point for liquid water is considered as the reference point.

3

5

0.01 0.1

-5

5

15

25

35

45

0.0001 0.01 1 100 10000

V (m3/kg)

P (M

Pa)

-5

5

15

25

35

45

250 350 450 550 650

T (K)

P (

MP

a)

T.P.

P. C.

V1

A

BV2

Saturation curve

P. C.

V1

Ref. P.

Compressed liquid region

Superheated steam region

BV2

Saturation

curve

T.P. VapT.P. Liq

L. T.

A

Liqu

idsa

tura

tion

curv

e

Ref. P.

3

5

0.01 0.1

-5

5

15

25

35

45

0.0001 0.01 1 100 10000

V (m3/kg)

P (M

Pa)

-5

5

15

25

35

45

250 350 450 550 650

T (K)

P (

MP

a)

T.P.

P. C.

V1

A

BV2

Saturation curve

P. C.

V1

Ref. P.

Compressed liquid region

Superheated steam region

BV2

Saturation

curve

T.P. VapT.P. Liq

L. T.

A

Liqu

idsa

tura

tion

curv

e

Ref. P.

Figure. 8. A schematic diagram to define trajectories for measuring the thermodynamic properties of pure water in the compressed liquid and superheated steam regions and along the saturation curve.

The thermodynamic properties of water are calculated using the PVT characteristics, CV and CSat,. Accordingly, the internal energy for the compressed liquid and superheated steam are expressed as (Figure 8)

1

1

1

2

2

2

, .

, .

, .

, . .

sat

sat ref

ref

sat

sat ref

ref refPT Vap

T T para V V

liq V liq sat liqT para V V T

V

V

T T para V V

vap V vap sat vapT para V V T

V

T PV

U C dT C dT

PdV

U C dT C dT

PdV L H

=

=

=

=

= +

−

= +

− +

∫ ∫

∫

∫ ∫

∫

(5)

where reference point (ref.) is the triple point (T.P.) for liquid water and L.H. is the latent heat at T.P. We are working to create new experimental data of the steam tables for pure water using this approach.

Experimental Setup Figure 9 shows a schematic diagram for an experimental setup to measure the heat capacity of

Temperature Controller

Power Supply

Interface and data acquisition

T-Tref

Cyl

indr

ical

hea

terRef. Cell Sam. Cell

Temperature Controller

Power Supply

Interface and data acquisition

T-Tref

Cyl

indr

ical

hea

terRef. Cell Sam. Cell

Figure. 9. Experimental setup for measuring heat capacity in the compressed liquid (CV,Liq) and superheated steam (CV,Vap) regions and along the saturation curve (CSat,Liq and CSat,Vap ).

pure water along the saturation curve for liquid (CSat,Liq) and vapor (CSat,vap) and in the compressed liquid (CV,Liq) and superheated steam (CV,Vap) regions. Basically, its principle of measurement is based on the Differential Scanning Calorimetry (DSC). There are two identical cells which are placed on a pair of identically positioned platforms in a cylindrical furnace. There is sample (water) in one cell and other cell is empty. The platforms have heaters and thermocouples to measure temperature. The cylindrical furnace is switched on to heat the cells at a specific rate. There will be same temperature in the cells, if both cells are empty. Since the cell 1 has sample, it will be heated with a lower heating rate. To get the same temperature at every instant, the cell 1 is heated by its platform heater. Let the heat flow rate of the heater be Q. The values of Q and T at every instant are stored in a computer. Figure 10 shows a relation between Q and T for an experiment with total water mass m (=mLiq+mVap). For a temperature change (∆T), some liquid will convert to vapor (-∆mLiq =∆mVap). We can write a mass and energy balance equation as

).(.

).(.

.. TTLmCmCm

TTLmT

Qm

T

Qm

T

Q

VapLiqSatLiqVapSatVap

VapLiq

LiqVap

Vap

⋅∆++=

⋅∆+∆

∆+

∆∆

=∆∆

(6) In general there are three unknowns CSat.Vap, CSat.Liq and L.T.(T). So we run the same experiment three times with different total mass in order to solve the equation (6). The fundamental problem in this approach is that there is relatively very small amount of vapor in the cell at low T (and P). Therefore, it may have high analytical uncertainty in the values of CSat.Vap. The alternative way to solve this problem is to measure the heat capacity of vapor according to the procedure used for pure gases. The vapor acts as a perfect gas at very low T and P. So, there is need to study the deviation in the behavior of CSat.Vap from that of perfect gas near to the saturation curve and critical isochor.

CONCLUSIONS

An interrelation among thermodynamic properties (state functions) of a system is first step for creating the internal consistent thermodynamic dataset, which is fundamental to understand the physical-chemical processes in the laboratory and in nature. This work shows that the IAPWS-95 formulation for thermodynamic properties of pure water is incorrect. It is found that the experimental values of heat capacity at constant pressure (CP) and at constant volume (CV) along the saturation curve are measured by a physically incorrect concept. It is unfeasible to measure the values of CP or CV for all the conditions of temperature and pressure, required for calculating the thermodynamic properties of liquids. Therefore, the IAPWS-95 formulation is obtained by fitting the incorrect experimental dataset for the thermodynamic properties of pure water. A new Verma procedure is devised. The heat capacity along the saturation curve, CSat is defined as the proportion of amount heat to the change in temperature. Additionally, it is shown that the values of CV of liquid can be measured precisely in the liquid and vapor phases. Using the PVT characteristics, CV and CSat, all thermodynamic properties of water can be calculated. An experimental setup is presented to measure the heat capacity of liquid (CSat.Liq) and vapor (CSat.Vap) along the saturation curve. The same system can be used for measuring the heat capacity of compressed liquid (CV,Liq) and superheated steam (CV,Vap) at constant volume. We are working on the development of a laboratory to measure the heat capacity of liquids. Hope to present the new internal consistent thermodynamic data of pure water in the forthcoming Sanford Geothermal Symposium.

T

Q

∆T

∆Q

Figure. 10. A schematic relation between heat flow rate (Q) and temperature (T) for a run with the total mass of water (m) in the cell.

ACKNOWLEDGEMENT

The author is grateful for the financial and technical support provided by his institute authority to conduct this project.

REFERENCES

IAPWS (2004), “Release notes on the updated thermodynamic properties of liquid and vapor of water. International Association for the Properties of Water and Steam”, Web page http://www.iapws.org/. Physical Review Letters (2003), “A tale of Two Liquids”, Web page http:\focus.aps.org\story\v12\st11#author. Sciortino F., La Nave E., Tartaglia P., “Physics of liquid-liquid critical point”, Physical Review Letters, 91, 155701-155704. Verma, M.P. (2002), “Geochemical techniques in geothermal development”, In D. Chandrasekharam, and J. Bundschuh (eds.), Geothermal Energy Resources for Developing Countries. The Sweets & Zeitlinger Publishers, Netherlands, 225-251. Verma, M.P. (2003), “Steam Tables for pure water as an ActiveX component in Visual Basic 6.0”, Computers and Geosciences, 29, 1155-1163. Verma, M.P. (2005), “SteamTablesIIE: an approach of multiple variable sets for steam tables of pure water”, Computers and Geosciences, In revision. Wegner, W. and Pruβ, A. (2002), “The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use”, Journal of Physical and Chemical Reference Data, 31, 387-535.