Dynamic Thermodynamics with Internal Energy, Volume, and Amount

Ind. Eng. Chem. Res. 2009, 48, 3167–3176

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Dynamic Thermodynamics with Internal Energy, Volume, and Amount of Moles as States: Application to Liquefied Gas Tank A. R. J. Arendsen*,† and G. F. Versteeg‡ Procede Group BV, P. O. Box 328, 7500 AH Enschede, The Netherlands, and Faculty of Mathematics and Natural Sciences, UniVersity of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands

Dynamic models for process design, optimization, and control usually solve a set of heat and/or mass balances as a function of time and/or position in the process. To obtain more robust dynamic models and to minimize the amount of assumptions, internal energy, volume, and amount of moles are chosen as states for the conservation laws of the dynamic model. Temperature, pressure, and the amount and composition of the phases are calculated on the basis of these states at every time step. The Redlich-Kwong and Peng-Robinson (RK-PR) cubic equation of state is used as the thermodynamic model. This study describes the aspects of this approach and additionally gives a wide view over the whole internal energy and volume surface in specific phase diagrams. A complete separation between the dynamic balance model and the thermodynamic model is achieved. Several examples show the application of this approach for a liquefied gas tank and demonstrate that the method is applicable to one and two phases in a wide temperature and pressure range, from liquid and/or gas phase to supercritical conditions. 1. Introduction During the operation of industrial batch and continuous processes, temperature, pressure, and the amounts of phases can vary in time. Dynamic models for process design, optimization, and control usually solve simultaneously a set of heat and/or mass balances as a function of time and/or position in the process. Density and temperature are usually chosen as state variables for the differential equations, and pressure is assumed to be constant. To solve these dynamic equations, the thermodynamic model is often simplified to an ideal gas and/or incompressible liquid model. To obtain more robust models and to minimize the amount of assumptions, other sets of states for the dynamic models have been proposed. In Arendsen1-3 density and specific enthalpy are proposed as states for the modeling of dynamic processes in food industry and refrigeration cycles. Internal energy, volume, and the amount of moles are proposed by Saha and Carroll,4 Mu¨ller and Marquardt,5 Michelsen,6 and Gonçalves et al.7 Both approaches are comparable, although internal energy implicitly includes the pressure work done by the system. In the case of the enthalpy the dynamic internal energy balance needs an additional term for this work. In the present study, the internal energy is chosen as the state because of this advantage and the internal energy, volume, and the amount of moles per component directly follow from the first law, as a thermodynamically consistent combination of states. The present study describes the aspects of this approach and additionally gives a wide view over the internal energy and volume surface in specific phase diagrams. Furthermore, the three-parameter equation of state (EoS) proposed by Cismondi and Mollerup8 is used as the thermodynamic model because this equation is better suited for describing the volume than the two-parameter equation of state. The objective is not to improve on the accuracy of an EoS approach but to make better use of consistent thermodynamic models. This work will show that internal energy, volume, and the amount of moles suffice as states to describe dynamic models of multiple phase and component systems. A complete separation * To whom correspondence should be addressed. Tel.: +31 53 711 25 16. Fax: +31 53 711 25 99. E-mail: [email protected]. † Procede Group BV. ‡ University of Groningen.

between the dynamic balance model and the thermodynamic model is achieved. This makes it easy to use the same thermodynamic model for different applications or to switch between different thermodynamic models for a specific application. Several examples show the applicability of this approach for a liquefied gas tank. 2. Theory and Calculation 2.1. Basic Thermodynamics. The fundamental thermodynamic relation of an open multiphase system at equilibrium is given by9 dUT ) T dST - P dVT +

∑ µ dn i

T i

(1)

i

A thermodynamic model that combines the relation between UT, ST, VT, and nT, describes the system completely. The partial derivatives of UT(ST,VT,nT), ST(UT,VT,nT), or VT(ST,UT,nT) give temperature, pressure, and the chemical potentials of the system and therefore complete the description of the system. Since an internal energy balance for the system gives UT, a momentum balance gives VT and a mole balance gives nT, a combination of these conservation laws with an equation of state that describes the ST(UT,VT,nT) relation suffice to describe the behavior of all thermodynamic properties in the system. This approach is applicable especially in systems where the temperature and/or pressure are not known a priori. There are three other surfaces that describe the complete system, namely, (HT, ST, P, nT), (GT, T, P, nT), and (AT, T, VT, nT), but they all imply that pressure and/or temperature are known, a priori. Another advantage of this approach is that UT, VT, and nT are extensive properties. They are first-order homogeneous functions, additive and proportional to the size of the system and thus applicable for the conservation laws of the system, contrary to T, P, and µ, which are intensive nonadditive properties. Finally, this approach makes a clear distinction between the intensive properties and its conjugated extensive properties. The intensive properties are the driving force of flow in irreversible processes where the transport of quantities occurs. The driving force of entropy is a temperature gradient, that of volume is a pressure gradient, and that of the

10.1021/ie801273a CCC: $40.75  2009 American Chemical Society Published on Web 01/29/2009

3168 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009

equilibrium are subject to the following constraints:

Table 1. Propane and Butane Properties property Tc Pc Zc ω AIG BIG CIG DIG EIG H° S°

dimension

propane

butane

K bar

369.83 42.48 0.276 0.1523 51.92 192450 1626.5 116800 723.6 -104680 270.20

425.12 37.96 0.274 0.2002 71.34 243000 1630 150330 730.42 -125790 309.91

(J/mol)/K K K K K J/mol (J/mol)/K

dUT ) dUL + dUG ) 0 dVT ) dVL + dVG ) 0 dnTi ) dnLi + dnGi ) 0 The entropy change of the total system is equal to dST )

propane (mol %)

Tcalc (K)

Texpt (K)

Pcalc (bar)

Pexpt (bar)

14.7 52.1 82.6

418.6 399.8 381.5

419.0 400.7 382.3

39.4 42.4 43.1

39.4 42.3 43.1

property dimension case 1 case 2 case 3 case 4 case 5 case 6

kW mol/s

20 5.07 1 0.5 0.01 54.6 0

20 5.02 1 0.5 0.04 28.2 0

20 4.15 1 0.5 0.5 2.76 0

20 6.25 1 0.7 0.1 30 0-7.6

20 3.67 1 0.3 0.1 30 0-7.6

20 6.25 1 0.7 0.1 0 0-2.1

amount of moles is a chemical potential gradient. The latter is often confused in diffusion processes and the exchange of the amount of moles between gases and liquids where concentration (moles) gradients are defined as the driving momentum, which is thermodynamically incorrect. The dynamic modeling of a vapor liquid system by means of this approach can be separated completely into a thermodynamic model and a dynamic balance model. The total internal energy, volume, and amount of moles of the total system are calculated on the basis of the appropriate dynamic conservation laws. Temperature, pressure, and the amount and composition of the phases are calculated on the basis of these states at every time step. The total system is assumed always to be at thermodynamic equilibrium. The different models used and the way they are combined are explained subsequently. This method works for multiple components and multiple phase systems respectively and is able to handle gas, liquid, and solid phases as long as thermodynamically consistent models are available. In this study a two-component vapor-liquid system is studied. A mixture of propane and butane is chosen as the example system. 2.2. Vapor-Liquid Equilibrium (VLE). An ideally isolated system exchanges no heat, work, and material with the surroundings. Spontaneous processes occurring in such systems result in an increasing entropy of the system. At equilibrium a system is at its maximum entropy. This means that (dST)UT,VT,niT ) 0

(2)

or dST )

P 1 dUT + dVT T T

µi

∑ T dn

T i

)0

(3)

i

If an isolated system consists of a liquid and a gas phase that can exchange internal energy, work, and moles with each other, then the extensive independent properties U, V, and n at

∑

If the constraints are substituted into this equation (combination of (4) and (5)), it follows that dST )

Table 3. Design and Initial Conditions Cases °C bar m3

1 PL L PG G 1 L G dU + dU + dV + dV TL TG TL TG µLi µGi L dn dnGi ) 0 (5) i L G i T i T

∑

Table 2. Calculated and Experimental Critical Temperature and Pressure

Tinitial Pinitial V zC3H8 βinitial Q φ

(4)

(

)

(

)

1 PL 1 PG L dU + dVL L G L G T T T T µGi µLi - G dnLi ) 0 (6) L T T i

∑

(

)

This equation must be satisfied for any changes in the extensive independent properties. This can only be satisfied if the terms in each of the brackets are equal to zero; i.e., TL ) TG PL ) PG µLi ) µGi

(7)

The temperature, pressure, and chemical potential of the individual species must be equal for all the phases of the system at equilibrium. At equilibrium, the gradients in the intensive properties are zero and there is no exchange of the conjugated extensive properties (internal energy, volume, and the amount of moles) between the phases. The conditions for thermodynamic equilibrium can be applied for any system with any amount of phases and components, even if reactions take place between the different components. The chemical potential is usually described as a function of temperature, pressure, and the amount of components per phase. The thermodynamic model is divided in an ideal gas reference state chemical potential as a function of temperature and at a reference pressure. The nonideal part of the model is described by a fugacity model.

(

µi(T, P, n) ) µIG i (T, P0) + RT ln

φi(T, P, n)niP

∑nP

i 0

i

)

(8)

In the case of vapor-liquid equilibrium without reactions, temperature, pressure, and reference states for all components in each phase are equal, so equal chemical potential equals φLi xi ) φGi yi

(9)

2.3. Equation of State. The fugacity coefficient of both phases is calculated by means of an equation of state. There are no equations of state available that describe the ST(UT,VT,nT) relation directly. Cismondi and Mollerup8 suggested a new three-parameter cubic EoS that combines Redlich-Kwong and Peng-Robinson EoS to the RK-PR EoS. This EoS improves the volumetric predictions, which are important when volume is used as a state. The residual Helmholtz internal energy formulation, Ar(T,V,n), can be used to derive all thermodynamic properties:

Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3169

F)

r

B A (T, V, n) ) -n ln 1 RT V

(

)

( )

D ln RTB(∆1 - ∆2)

B V (10) B 1 + ∆2 V 1 + ∆1

The following mixing rules are applied: n)

∑n

∑n ∑na

D)

i

i

i

B)

j ij

i

∑nb

j

ni

∑ nδ

∆1 )

i i

(11)

i

i

i

with aij ) aji ) √aiiajj(1 - kij) a ) ac

(

3 2 + Tr

)

k

∆2 )

1 - ∆1 1 + ∆1 (12)

Cismondi and Mollerup8 give relations to calculate ac, k, b, and δ as a function of Tc, Pc, Zc, and ω. All other thermodynamic properties are calculated as partial derivatives of the Helmholtz function. The pressure is the partial derivative to volume: ∂F ( ∂V )

P ) -RT

nRT V

+

T,n

(13)

The compressibility factor is equal to Z)

PV nRT

(14)

The fugacity coefficient of each component in the mixture is the partial derivative to the amount of moles: ln φi )

( ) ∂F ∂ni

- ln Z

( ( )) ( ( ))

2 2 CIG EIG i i T T IG IG CIG + DIG (24) P,i(T) ) Ai + Bi i IG Ci EIG i sinh cosh T T The system can contain one or two phases. In the case of one phase all properties can be calculated when the temperature, pressure, and total composition are known. In the case of two phases the vapor must be in equilibrium with the liquid. 2.4. Specific Internal Energy and Volume Surface of the Total System. To calculate all thermodynamic properties at every time step on the basis of UT, VT, and nT an UVn-flash calculation is required. Two different approaches are presented in literature for the UVn-flash. Both methods try to find the global maximum of the total entropy of the system. Michelsen6 and Gonçalves et al.7 perform a constrained global minimization of an objective nonlinear function. With this first approach a priori knowledge of the amount of phases and components is not necessary. Nevertheless, this approach is numerically difficult and can lead to convergence problems. Arendsen,1-3 Saha and Carroll,4 and Mu¨ller and Marquardt5 solve a set of nonlinear equations. This second approach requires knowledge of the number of phases and components but has less numerical problems and needs less computational time. The last method is used in this work and explained in the following. Since UT and VT are linear proportional to nT, the states are made specific to reduce the amount of independent states.

uR )

(15) zi )

T,V,nj

The partial derivative to temperature gives the entropy of the mixture:

( ∂F ∂T )

Sr(T, V, n) ) -RT

The ideal gas enthalpy and entropy are calculated by means of the Design Institute for Physical Properties (DIPPR)12 correlation for the ideal gas heat capacity as a function of temperature:

V,n

- RF

(16)

UR , nR nTi nT

xi )

nLi nL

VR nR

R ) T, G, or L

yi )

nGi nG

β)

nG nT

uT ) u(T, VT, z)

Ur(T, V, n) ) Ar(T, V, n) + TSr(T, V, n)

(17)

Ar(T, P, n) ) Ar(T, V, n) - nRT ln Z

(18)

Sr(T, P, n) ) Sr(T, V, n) + nR ln Z

(19)

Ur(T, P, n) ) Ur(T, V, n)

(20)

Hr(T, P, n) ) Ur(T, P, n) + PV - nRT

(21)

Gr(T, P, n) ) Ar(T, V, n) + PV - nRT(1 + ln Z)

(22)

The reference state of the residual properties is a hypothetical perfect gas mixture at the specific temperature and pressure. All bulk properties are given by the sum of the ideal mixture part and the residual part of the property. For example the enthalpy is given by

∑H

IG

i

(T, P, ni)

(23)

(25)

(26)

Two different sets of equations need to be solved in the cases of one and two phases, respectively. In the case of one phase the set reduces to one equation with temperature as the unknown:

Other residual bulk properties are

H(T, P, n) ) Hr(T, P, n) +

VR )

(27)

Subsequently all other thermodynamic properties can be calculated on the basis of the Helmholtz formulation of the EoS. It is not necessary to distinguish between the liquid and the vapor phases because the total specific volume directly corresponds to the specific volume of the actual phase. In the case of two phases the following sets of equations need to be solved:

{

uT ) βuG(T, P, y) + (1 - β)uL(T, P, x) VT ) βVG(T, P, y) + (1 - β)VL(T, P, x)

∑ (y

i

- xi) ) 0

i

βyi + (1 - β)xi - zi ) 0 φGi (T, P, y)yi - φLi (T, P, x)xi ) 0

(28) The unknowns are T, P, β, xi, and yi. All equations are implemented in MATLAB, and the functions fzero and fsolVe are used to solve the equations. The initial guesses for the solvers are based on the values of the previous time step. The initial internal energy and volume are calculated on the basis of the set initial temperature and vapor fraction of the system. It is not necessary to calculate the critical point of


Figure 1. Boundaries between zones in uT, VT plane.

Figure 3. Calculated phase envelopes at different compositions.

Figure 2. Schematic representation of a heated tank.

the mixture before all other calculations can be done. If the critical point is reached during the dynamic process, the same procedure can be used to calculate all thermodynamic properties. Different approaches can be applied to determine the amount of phases. One approach is to solve always the set of equations for the two phases. If the calculated β is between 0 and 1, then the two phases occur; otherwise, only one phase exists. Another approach is to determine the amount of phases on the basis of uT and VT. The boundaries between the zones with different amounts of phases are visualized in the uT, VT plane in Figure 1. The reciprocal specific volume is given in logarithmic scale on the y-axis, and the internal energy is given on the x-axis. The plane is divided into two zones, with one and two phases. The one-phase zone gradually changes from liquid to gas phase through supercritical conditions, in the upper right corner of Figure 1. Beneath the critical volume the boundary is given by the following set of equations:

{

Figure 4. Entropy contour plot as function of internal energy and volume.

uB ) uG(T, P, z) VB ) VG(T, P, z)

∑ (y - x ) ) 0 i

i

i

yi - zi ) 0 φGi (T, P, y)yi - φLi (T, P, x)xi ) 0

(29)

Figure 5. Temperature contour plot as function of internal energy and volume.


Figure 6. Pressure contour plot as function of internal energy and volume.

Figure 9. Temperature contour plot as a function of internal energy and volume for cases 1-3.

The total internal energy and volume are equal to the internal energy and volume of the pure vapor at equilibrium with the liquid. At a given VT, the corresponding uB can be calculated and if uT < uB, the system has two phases. The same procedure can be followed above the critical point.

{

Figure 7. Vapor fraction contour plot as function of internal energy and volume.

uT ) uL(T, P, z) VT ) VL(T, P, z)

∑ (y

i

- xi) ) 0

i

xi - zi ) 0 φGi (T, P, y)yi - φLi (T, P, x)xi ) 0

(30) The total internal energy and volume are equal to the internal energy and volume of the pure liquid at equilibrium with the vapor. 2.5. Dynamic Conservation Laws (Liquefied Gas Tank). The total internal energy, volume, and amount of moles of the system are calculated on the basis of the appropriate dynamic conservation laws. Temperature, pressure, and the amount and composition of the phases are calculated on the basis of the states at every time step. The whole model is solved as a differential algebraic equation (DAE) problem. A heated tank filled with a mixture of propane and butane is taken as an example. A pressure release valve is optional. A schematic representation of the situation is given in Figure 2. The dynamic internal energy balance includes the accumulation term on the left-hand side. The first term on the right-hand side represents the heat supplied to the system. The second term represents the gas flow released from the pressure valve, if it is opened; otherwise, it is equal to zero. This term includes the enthalpy of the gas phase, since it includes the pressure work done by the expansion. dUT ) Q - hGφ dt

(31)

The tank is assumed to be rigid, so the volume does not change in time. Figure 8. Gas and liquid composition contour plot as function of internal energy and volume.

dVT )0 dt

(32)


Figure 10. Temperature, pressure, liquid fraction, and propane fraction as a function of time for case 1.


If the conservation of momentum is included for any system, it is possible to account for expansion and compression of the system. The mole balance is solved for every component separately. It is assumed that the gas that flows out of the tank has exactly the composition of the gas phase in the tank. dnTi ) -yiφ dt

(33)

3. Results 3.1. Equation of State. Martinez and Hall11 tested the RKPR EoS successfully on light synthetic natural gas mixtures. The used properties of propane and butane are given in Table 1. The binary interaction parameter kij is set to zero. Since the objective of this study is not to improve on the accuracy



Figure 13. Temperature, pressure, flow, and liquid and propane fractions as a function of time for case 4.

of an EoS, the accuracy of the EoS predictions to experimental values is not tested. However, to test the implementation of the model, some results are presented and compared with experimental values. Figure 3 shows the calculated phase envelopes at different compositions of the propane and butane mixtures.

The calculated critical temperature and pressure of these mixtures follow from these results, are listed in Table 2, and are compared with experimental data presented by Kreglewski and Kay.10 From Table 2 it can be concluded that the model is properly implemented.




3.2. Specific Internal Energy and Volume Surface. In Figure 4 the sT(uT,VT) surface for a 50 mol % mixture of propane and butane based on the RK-PR EoS is presented. Since the sT(uT,VT) diagram suffices to describe the whole thermodynamic system, all other thermodynamic properties can basically be derived from this diagram. Since the x- and y-axes are based on extensive properties, all points in the plane have a unique solution. There are no overlapping regions, as for instance in a P-T phase diagram of a pure component where the boiling line only represents the

relation between pressure and temperature, but does not tell anything about the amount of vapor and liquid. The sT(uT,VT) diagram even suffices if a solid phase is added. Figures 5-8 give temperature, pressure, vapor fraction, and vapor and liquid composition as a function of the specific internal energy and specific volume for a 50 mol % mixture of propane and butane. 3.3. Dynamic Simulations. For the liquefied gas tank different cases are calculated. In the first three cases a constant


heat input is supplied to the system and the pressure relief valve is not opened. The different cases start with different liquid fractions (1-β). The liquid fraction is presented in mole fractions and not in volumetric fractions. So, the liquid fraction is not equal to the liquid level but is strongly related. The design and initial conditions are given in Table 3. Since in cases 1-3 the valve does not open, the specific volume is constant and represents a horizontal line in the phase diagrams at different heights. These lines are shown in Figure 9. The different development of the temperature in time for the three cases follows from the contour lines and can be compared with the plots in the next paragraphs. The same procedure can be followed for the other properties. All cases start at the 20 °C contour line. The initial pressure for each case is different because the initial vapor fraction is different and the overall composition is constant. Subsequently, the initial vapor and liquid compositions are different. This is clearly observable in Figure 8 because the starting points of the three cases are situated on different contour lines of the vapor and liquid compositions. Depending on the heat flow, the internal energy of the tank will increase in time. It is not possible to present cases 4-6 in the diagram, because they start at different overall compositions and the overall composition changes in time. Therefore, the diagram changes for every time step. 3.3.1. Case 1. Figure 10 shows the temperature, pressure, liquid fraction, and composition of the gas and liquid phases as a function of time for the case where the tank is filled with a high liquid fraction. While temperature and pressure increase, the liquid fraction drops first, due to evaporation. Subsequently the expansion of the liquid becomes dominant and the liquid fraction increases, until a liquid full situation. The increase in temperature and pressure accelerates; especially, the pressure increases and the tank will probably explode in a boiling liquid expanding vapor explosion (BLEVE). The fast increase in pressure is visible in Figure 6, also. The pressure contour lines are close to each other in the upper part of the plot. 3.3.2. Case 2. In this case the tank is filled with a liquid fraction, so that the total specific volume of the system corresponds to the critical specific volume of the mixture. Temperature and pressure increase, while the liquid fraction drops slowly (Figure 11). The composition of the gas gradually resembles more the composition of the liquid. At a certain point, the difference between the gas and liquid phases disappears and the liquid fraction drops rapidly. The whole mixture becomes supercritical. The increase in temperature and pressure accelerates, but in the same manners. The fast drop of the liquid fraction is visible in Figure 7, where all the vapor fraction contour lines come together in the supercritical point of the mixture. 3.3.3. Case 3. In this case the tank is half-filled. The liquid fraction decreases rapidly, due to the evaporation (Figure 12). The gas composition changes to the total composition. When all the liquid is evaporated, one phase is left. In this case the temperature increases rapidly. This rapid change is visible in Figure 9, where the temperature contour lines bend to almost vertical lines at the boundary of the gas-phase zone. 3.3.4. Case 4. In the next two cases a pressure relief valve is installed, which opens at 20 bar and closes at 18 bar. The mole flow is assumed to be proportional to the pressure difference between the pressure in the tank and the ambient pressure. This case represents a 0.7/0.3 (mol/mol) propane/ butane mixture. The temperature and pressure increase until the pressure relief valve opens for the first time. The temperature stays almost constant (Figure 13), while the pressure drops slowly to 18 bar. This cycle repeats several times. However,

the temperature increases slowly, due to the fact that more propane flows out of the system, since the vapor pressure of propane is higher than butane. The boiling out of the propane is visible in the liquid composition, also. 3.3.5. Case 5. This case represents a 0.3/0.7 (mol/mol) propane/butane mixture. Figure 14 shows that the temperature and pressure increase until the pressure relief valve opens for the first time. It takes less time to evaporate all liquid than in case 4, and the final temperature is higher. 3.3.6. Case 6. This case shows what happens if the pressure relief valve is opened after 2 min, without any heat input. This case represents a 0.7/0.3 (mol/mol) propane/butane mixture. The mole flow is assumed to be proportional to the pressure difference between the pressure in the tank and the ambient pressure. Figure 15 shows that the temperature and pressure decrease due to the evaporating gas. The temperature drops to -30 °C, which is equal to the boiling temperature of the mixture at 1 bar. Although the total amount of moles in the tank decreases, the liquid mole fraction increases due to the decreasing pressure, which lowers the amount of moles in the gas phase drastically. 4. Conclusions This work shows that internal energy, volume, and the amount of moles suffice as states for describing dynamic models of multiple phase and component systems. A complete separation between the dynamic balance model and the thermodynamic model is achieved. This makes it easy to use the same thermodynamic model for different applications, or to switch between different thermodynamic models for a specific application. Different examples show that the method is applicable to one and two phases in a wide temperature and pressure range, from liquid and/or gas phase to supercritical conditions. Further research is done on the addition of a consistent solid-phase model to study condensation and/or sublimation processes by expansion or cooling. The method will also be used to describe the flow in a multidimensional domain of a multiple phase and component mixture, which will be able to describe coupled heat and mass flow enhanced by temperature, pressure, and/or chemical potential gradients. Nomenclature a ) attractive term in equation of state (J m3 mol-1) A ) Helmholtz internal energy (J) b ) repulsive term in equation of state (m3 mol-1) B ) repulsive term of mixture in equation of state (m3) D ) attractive term of mixture in equation of state (J m3 mol) F ) residual Helmholtz internal energy (mol) G ) Gibbs internal energy (J) h ) specific enthalpy (J mol-1) H ) enthalpy (J) k ) parameter for temperature dependence of attractive term in equation of state n ) amount of moles (mol) P ) pressure (Pa) R ) universal gas constant (J mol-1 K-1) S ) entropy (J K-1) t ) time (s) T ) temperature (K) u ) specific internal energy (J mol-1) U ) internal energy (J) V ) specific volume (m3 mol-1) V ) volume (m3) x ) gas-phase mole fraction

3176 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 y ) liquid-phase mole fraction z ) total system mole fraction Z ) compressibility factor Greek Letters β ) total gas-phase mole fraction δ ) third parameter in equation of state ∆ ) third parameter of mixture in equation of state φ ) fugacity coefficient φ ) molar flow rate (mol s-1) µ ) chemical potential (J mol-1) ω ) acentric factor Subscripts/Superscripts B ) gas- and liquid-phase boundary C ) supercritical G ) gas phase IG ) ideal gas L ) liquid phase r ) reduced value R ) residual part T ) total system 0 ) reference state

Literature Cited (1) Arendsen, A. R. J.; van Berkel A. I. Simulation of thermal food processing with phase transitions by means of the TNO density-enthalpy phase diagram. FOODSIM, Cork, Ireland, 2002.

(2) Arendsen, A. R. J.; van Berkel, A. I.; Heesink, A. B. M.; Versteeg, G. F. Dynamic modelling of thermal processes with phase transitions by means of a density-enthalpy phase diagram. 7th World Congress of Chemical Engineering, Glasgow, Scotland, 2005. (3) Arendsen, A. R. J.; Versteeg, G. F. Dynamic modelling of refrigeration cycles using density and enthalpy as state variables. 17th International Congress of Chemical and Process Engineering, Prague, The Czech Republic, 2006. (4) Saha, S.; Carroll, J. J. The isoenergetic-isochoric flash. Fluid Phase Equilib. 1997, 138, 23. (5) Mu¨ller, D.; Marquardt, W. Dynamic multiple-phase flash simulation: Global stability analysis versus quick phase determination. Comput. Chem. Eng. 1997, 21, S817. (6) Michelsen, M. L. State function based flash specifications. Fluid Phase Equilib. 1999, 617, 158. (7) Gonçalves, F. M.; Castier, M.; Arau´jo, O. Q. F. Dynamic simulation of flash drums using rigorous physical property calculations. Braz. J. Chem. Eng. 2007, 24, 277. (8) Cismondi, M.; Mollerup, J. Development and application of a threeparameter RK-PR equation of state. Fluid Phase Equilib. 2005, 232, 74. (9) Michelsen, M. L.; Mollerup, J. Thermodynamic Models: Fundamentals and Computational Aspects, 1st ed.; Tie-Line Publications: Holte, Denmark, 2004. (10) Kreglewski, A.; Kay, W. B. Critical constants of conformed mixtures. J. Phys. Chem. 1969, 73, 3359. (11) Martinez, S. A.; Hall, K. R. Thermodynamic properties of light synthetic natural gas mixtures using the RK-PR cubic equation of state. Ind. Eng. Chem. Res. 2006, 45, 3684. (12) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Hemisphere: New York, 2003.

ReceiVed for reView August 21, 2008 ReVised manuscript receiVed December 17, 2008 Accepted December 19, 2008 IE801273A

Dynamic Thermodynamics with Internal Energy, Volume, and Amount

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