Entropy Variation in Isothermal Fluid Flow Considering Real Gas Effects

Home , Isothermal flow, Isothermal process, Real gas, Thermal expansion

doi: 10.5028/jatm.2011.03019210

Maurício Guimarães da Silva* Institute of Aeronautics and Space Entropy variation in isothermal fluid São José dos Campos – Brazil [email protected] flow considering real gas effects

Paulo Afonso Pinto de Oliveira Abstract: The present paper concerns on the estimative of the pressure (In memoriam) loss and entropy variation in an isothermal fluid flow, considering real Universidade Estadual Paulista gas effects. The 1D formulation is based on the isothermal compressibility Guaratinguetá – Brazil module and on the thermal expansion coefficient in order to be applicable for both gas and liquid as pure substances. It is emphasized on the simple * author for correspondence methodology description, which establishes a relationship between the formulation adopted for ideal gas and another considering real gas effects. A computational procedure has been developed, which can be used to determine the flow properties in duct with a variable area, where real gas behavior is significant. In order to obtain quantitative results, three virial coefficients for Helium equation of state are employed to determine the percentage difference in pressure and entropy obtained from different formulations. Results are presented graphically in the form of real gas correction factors, which can be applied to perfect gas calculations. Keywords: Isothermal flow, Real gas, Entropy, Petroleum engineering.

List of Symbols -1 αP Thermal expansion module K 2 βT Isothermal compressibility module m /N a Sound velocity m/s εP Pressure correction factor - 2 2 A Cross section m εS Entropy correction factor Pa A Wet area m2 γ Specific heat ratio - W 3 B Second virial coefficient cm3/mole ρ Density kg/m C Third virial coefficient cm3/mole2 µ Viscosity kg/ms Shear stress N/m2 Cp Specific heat at constant pressure J/kgK W Cv Specific heat at constant volume J/kgK ! f Friction coefficient - INTRODUCTION h Enthalpy J/kg M Mach number - The term “isothermal process” describes a thermodynamic !m Mass flow rate kg/s process that occurs at a constant temperature. There are a lot of examples of technical analysis using P Pressure Pa isothermal process. Isothermal compression is an pW Wet perimeter m 2 example to illustrate the position that isothermal q Heat exchange parameter W/m K c process takes along other thermodynamic processes. In R Gas constant J/kgK a real engine (compressor), the isothermicity condition 3 R Parameter of Equation (37) m /Pa/kg/K cannot be fulfilled and every real thermodynamic S Entropy J/kgK process demands more energy than the isothermal t Time s compression. In this context, the practical importance of isothermal compression lies in its use, as a reference T Temperature K process to evaluate the actual compression (Oldrich u Velocity m/s and Malijesvsky, 1992). The adiabatic frictional flow U Internal energy J/kg assumption is appropriate to high speed flow in short v Specific volume m3/kg ducts. For flow in long ducts, such as natural gas W Molecular weight of helium mole He pipelines, the gas state approximates more closely to the x Longitudinal coordinate m isothermal one. By doing this approach, it is possible Z Compressibility factor - to establish a relationship among all thermodynamic α Angle between the horizontal and rad quantities (White, 2005). the direction of flow In the flow field thermodynamic calculation of Received: 18/10/10 aeronautical devices, the real process is approximated Accepted: 02/11/10 to a suitable idealized one that can be mathematically

J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 29 Silva, M.G., Oliveira, P.A.P. described. Usually, the choice lies among isentropic, of many other models, such as Fanno flow (John polytropic, and isothermal processes. These and Keith, 2006). This analysis is applied to the thermodynamic processes are also used as reference 1D fluid flow (Fig. 1). It is assumed that the fluid ones in the internal flow field calculation of aeronautical undergoes an isothermal process during this process. devices. The isentropic and polytropic processes have In other words, its total energy remains unchanged been discussed by many authors, including their by the flow. The generalized form of this process applications to real gases (Oldrich and Malijesvsky, includes the possibility of the presence and effect

1992). In contrast, the isothermal process has not of viscosity (through µ and W ), gravity (through α– received such attention. Attempts at refining the results the angle between the horizontal! and the direction of by inserting values calculated from a virial equation flow), pressure (P), heat exchange ( qc ) and thermal of state for real gas have led to some improvement. conducting terms (k). is the shear stress due to wall W However, these efforts particularize the solution of the friction. Equations 1,! 2 and 3 show the generalized flow field issue. mathematical model.

The present paper is concerned over the estimative of the pressure loss and entropy variation in an isothermal fluid flow considering real gas effects. The formulation q is based on the isothermal compressibility module and ! W on the coefficient of thermal expansion, in order to be ! applicable for both gas and liquid as pure substances. Flow direction CV The description of the simple methodology, which establishes a relationship between the formulation adopted for ideal gas and that considers the real W dx gas effects, will be emphasized. A computational ! procedure has been developed, and it can be used to determine the flow properties in duct with a variable Figure 1: Differential control volume (CV). area where real gas behavior is significant. In order to obtain quantitative results, three virial coefficients for helium equation of state (Miller and Wilder, (i) Continuity Equation 1968; Schneider and Duffie, 1949) are employed to determine the percentage difference in pressure, and A uA " 0 ; (1) entropy between the different formulations. Results t x are graphically presented in the form of real gas correction factors, which can be applied to perfect gas (ii) Momentum Equation calculations. This paper is part of a continuing effort uA uAu P 4 u that is being carried out at the Institute of Aeronautics " A RA p and Space from the Brazilian Aerospace Technology t x x 3 x x W W ; (2) and Science Department (DCTA/IAE, acronyms in Portuguese) to develop flow analysis methods, (iii) Energy Equation: and design of aeronautical devices in a conceptual hA uAh P p T context. " A Au kA t x t x x x

2 4 u qpW MATHEMATICAL FORMULATION RA u W pW . 3 x cos (3) Isothermal flow is a model of compressible fluid flow ! whereby the flow remains at the same temperature In order to create practical relations for engineering while flowing in a conduit. In the kind of flow, heat use, steady state flow hypothesis is adopted, and the transferred through the walls of the conduit is offset thermal conduction and viscosity terms are neglected. by frictional heating back into the flow. Although This approach gives the following results: the flow temperature remains constant, a change in stagnation temperature occurs because of a change in (i) Continuity Equation the velocity. From this approximation, it is possible to demonstrate that, for ideal gas formulation, the flow d uA is choked at mach number (M) given by " 1/ , " 0 ; (4) and not at mach number equal to one as in! the case dx ! 30 J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 Entropy variation in isothermal fluid flow considering real gas fectsef

(ii) Momentum Equation v dh " Cp P ,T dT T v dP d uAu dP T . (10) " A p P dx dx W W ; (5) Now, take into consideration the momentum Eq. 5 in dP d uAu (iii) Energy Equation: Au " p the form dx dx w w , and substitute it in energy equation, considering the continuity equation, d uAh dP qp ! " Au u A W which will result in Eq. 11: dx dx W W cos (6) v The strategy chosen in this development is to convert d uAh d qp um 2 A " w (11) the dependent variable (ρ, u, h) in terms of pressure dx dx cos (P) and temperature (T). Thus, the equation system is reduced to two equations, whose dependent variables !Note that: are pressure and temperature, and the independent one is the longitudinal coordinate (x). It is important to d uAh dh d uA dh dh " uA h " uA " m highlight that during this development, any constitutive dx dx dx dx dx relation, such as virial equation of state, was adopted, in order to generalize the applicability of the method Thus, from Eq. 11, to all pure substances. dv dA dT v dP mv A v mCp P ,T m T v m 2 dx dx " Q , 2 dx T P dx A A Energy Equation

qpW dv v dT v dP Since h " h@P x,T xB , it can be written as: where : Q " , " , cos dx T P dx P T dx mv h h h and u " . dh " dT dP " Cp P ,T dT dP (7) A T P P T P T ! From algebraic manipulation of the previous equation, By definition: the energy equation is obtained with pressure and temperature as dependent variables, which is: h "U Pv , so dP dT dh " dU Pdv vdP . G P ,T H P ,T " I P ,T , (12) dx dx Considering an isothermal process: ! 2 m v v h U v where: G " v T v, " P v A P T (8) T P P T P T P T

! 2 m v The second law applied to the reversible process is given H " Cp P ,T v A T , and Q P by: " dS . Since Q " dU Pdv , the first law becomes: T TdS " dU Pdv . In this expression, the isothermal 2 Q m 2 d ln A S U v I " v condition is used: T " P . m A dx P T P T P T

Therefore, using Eq. 8, and, by the Maxwell relation Momentum Equation (Wyllen and Sonntag, 1987), the result is the following (Eq. 9): Since the cross section of the duct is a function of the longitudinal coordinate, A=A(x), it can be written: h v " T v P T (9) du d mv d v ; T P " " m dx dx A dx A Substituting! Eq. 9 in Eq. 7, it is obtained Eq. 10:

J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 31 Silva, M.G., Oliveira, P.A.P.

Therefore, by considering definitions of isothermal d uAu d v thus, " m 2 . compressibility module and thermal expansion dx dx A coefficients, it is possible to write the general equations as: By substituting the last equation into momentum Eq. 5 and dividing it by A, one obtains the representative dP dT G P ,T H P ,T " I P ,T equation of momentum conservation in terms of pressure dx dx and temperature, which can be seen in Eq. 13: dP dT J P ,T K P ,T " M P ,T , (16) dP dT dx dx J P,T K P,T " M P,T , (13) dx dx where,

2 m v where: J " 1, T G " v 1 K T P A P T P ,

2 m v K " , and H " Cp P ,T vK , A T P ! 2 ! 2 m m d ln A p J " v 1 , and M " v W W . A T A dx A ! u 2 ! K " v P . Generalized flow formulation ! Equations 12 and 13 represent the generalized Critical Conditions for Generalized Flow formulation written in terms of pressure and temperature, which include heat exchange, flow with friction (wall Applying Kramer rule in the system of equations friction), and flow in variable-area ducts influence. It is H I noteworthy that, up to this point, any approximation for dP K M HM IK (Eq. 16), it will provide: " " . equation of state was not used. In order to extend the dx H G HJ GK applicability of the mathematical model, it is interesting K J to write the parameters G, H, I, J, K and M in terms It is desirable, as in the isentropic case, to investigate of isothermal compressibility module (β ) and thermal T this relationship by the choice of a convenient reference expansion module (α ). In gas dynamics, compressibility P state. Since stagnation conditions are not constant, is a measure of the relative volume change of a fluid or the stagnation state is not suitable for this purpose. solid as a response to a pressure (or mean stress) change. However, the state corresponding to unity mach Since the compressibility depends strongly on whether number (“critical condition”) is suitable, because, as the process is adiabatic or isothermal, it is usually Eq. 16 shows, condition there is constant for a given defined as: flow. In this case, the critical condition is obtained from the expression: 1 v " T v P (14) T HJ GK " 0 . (17) The thermal expansion coefficient describes how the Since: size of an object changes with a change in temperature. Specifically, it measures the fractional change in 2 size per degree change in temperature at a constant m v HJ " Cp P ,T Cp P ,T pressure. In the general case of a gas, liquid, or solid, A P T the volumetric coefficient of thermal expansion is given by: 4 2 m v v m v v v A T P A T , 1 v P T P " P v T (15) P and

32 J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 Entropy variation in isothermal fluid flow considering real gas fectsef

4 2 2 2 m v v m v m v I 1 1 GK " v T v " 1 T " 1 T A T P A T A T Mv 2 P P P T P P , m Ƣ M 2 1 v 1 Real (21) ! A T by substituting it in Eq. 17, it is obtained: Equation 21 presents an interesting mathematical format. Cp P ,T It is possible to distinguish the “driver” terms (left side of u " v 2 the equation), in a dimensionless form, which represents v v Cp P ,T T the physical conditions necessary to obtain an isothermal P T T P (18) flow. This term is built with the contributions of heat exchange, wall friction, area variation, and mass flow. It! should be noted that the relationship between the The right side of the Eq. 21 represents the thermodynamic specific heats at pressure and volume constant is given conditions obtained in an isothermal flow, with the “driver” by Van Wyllen and Sonntag (1987, p. 284): conditions defined by the left side. Another important thing to note about Eq. 21 is that the variation of the parameter 2 v v I / Mv , at constant temperature, can be found from the Cp P ,T Cv P ,T " T equation of state and mass flow rate. This information can T P P T be used in the control system design based on isothermal The! sound velocity (a), in an isothermal process, can fluid flow. The simplicity of this formulation is a great v attractive for conceptual design and engineering analysis. be written as (John and Keith, 2006, p. 45): a " . T

Substituting the relations above in Eq. 18, the following Critical conditions for isothermal flow equation is obtained by: Equation 21 shows that the critical condition can be Cp P ,T Cp P ,T v 2 obtained from the expression: u " v " " a v v Cv P ,T Cv P ,T 2 P P m T T 1 v " 0 (22) A T In other words, the critical condition in a generalized Considering the concepts of sound velocity and flow field, considering real gas effects, is given by the isothermal compressibility module, the second term 2 2 mach number equals to one. This value is in accordance m u T 2 of the left side becomes: v T " " . Then, with technical literature. 2 A v m 1 v " 1 2 " 0 . From that, it is obtained: A T Real ! Isothermal flow formulation 1 " (23) Real Isothermal flow can be characterized by the relation (Eq. 19): Analogously! to the ideal formulation (John and Keith, 2006, p. 366), Eq. 23 shows that the critical mach dT number for isothermal flow is not subcritical flow. It " 0 (19) T follows that because the adiabatic speed of sound is greater than the isothermal speed of sound, isothermal Considering! Eq. 19 and the general system (Eq. 16), it flow may be supercritical without being necessarily is obvious the relation: supersonic. It is important to highlight that constitutive relations are not used in this development. Thus, it is dP I M assumed that Eq. 23 is valid for all real gas or liquid " " dx G J (20) formulation, for the flow process does not involve change of physical state. From! this expression, it is possible to characterize isothermal flow in another, but similar, manner: IJ = MG. Pressure drop in isothermal flow

Using the previous definitions for parameters J and G, The most effective approach to flow problems of this it can be concluded that: type is to express ratios of the gas properties and flow

J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 33 Silva, M.G., Oliveira, P.A.P. parameters between any two points in the flow stream, as 2 Real function of the mach number and specific heat ratio of the dP fdx " 2 gas. Using these ratios, a reference state is defined, and 2 P Ideal 1 Real D the ratios of the variables at any mach number to those at (28) the reference state are tabulated. Following this reasoning, ! consider the case of duct with constant cross section, by In other words, it can be concluded that: applying Eq. 20: dP dP " T (29) W PW P Ideal dP M " " A (24) 2 The right side of the Eq. 29 can be related to the mach dx J 1 Real number. Using the definition of βT and an isothermal It! is a common practice to assume (John and Keith, process, we will have: 2006, p. 291): v v v dv " dT dP " dP 1 2 T P P , (30) " u f P T T W 8 (25) The expression (Eq. 26) can be written as: Thus,

2 P 4A 4 1 f Real W W " W " W " u 2 dv fdx A A D D D " 2 2 . 2 (31) v 1 Real D By substituting it in Eq. 24, it is obtained: However, from continuity equation, one has:

1 u 2 1 fdx dv du d dP " " " 2 . 2 v 1 Real D v u (32)

Since Thus:

2 2 2 1 u 1 Real v 1 Real dP d " " , dP " " v v T (33) 2 2 T 2 T P Ideal Ideal ! It is demonstrated that: It is easy to demonstrate that Eq. 33 can be used for flow in ducts with variable cross section.

2 Real 1 fdx dP " 2 2 D Entropy variation in isothermal flow T 1 Real (26)

Since S " S P ,T , it can be written: Equation 26 establishes the relationship between the pressure drop and the parameters ‘mach number’, ‘wall friction’, and ‘thermodynamic properties of the S S dS " dT dP (34) gas’, when the longitudinal direction considered. Note T P P T that for ideal gas: Using the concepts of specific heat at constant pressure and Maxwell relation: 1 v 1 v 1 " " " T v P v P P (27) T dT v dS " Cp P ,T dP T T (35) Substituting Eq. 27 into 26, the pressure drop P formulation developed for ideal gas (John and Keith, Considering an isothermal process and the definition 2006, p. 366) is recovered, which is: of αP, it is obtained:

34 J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 Entropy variation in isothermal fluid flow considering real gas fectsef

" terms of R . According to Wyllen and Sonntag (1987, dS PvdP . (36) p. 285):

Substituting Eq. 26 in 36, it is obtained: Cp P ,T Cv P ,T " TR P ; (42)

2 ! Real thus, the expression for specific heat ratio is given by: dS fdx d " 2 " 2 , (37) R 1 Real D Ideal TR " 1 P Real Cv P ,T (43) Pv where: R " T This value is used in Eq. 44, which is: ! By definition: 2 Real dP fdx d " 2 " v P 2 D (44) v Ideal 1 Real Ideal T R " P v Equation 44 can be integrated in terms of mach P (38) number, considering the friction factor f constant T in the segment dx . From this result, it is possible to Since evaluate the pressure drop for ideal gas formulation. Thus, applying the multiplicative factor T on the last result, the drop pressure considering real gas effects v P T " 1, is obtained. Regarding to the entropy variation, it is P T v T v P possible to write from Eq. 37 and 44: or dS dP " , (45) R P Ideal v P T " P , (39) CORRECTION FACTORS T v v

P T In this section, the pressure correction factor ( P )

and entropy correction factor ( S ) are defined.! The parameter is calculated from the use of Eq. 33 for P ! the value of R can be written as: ideal and real! gases. By substituting Eq. 14 in Eq. 33, the following result is given:

P R " v dP P dP dP " " P , (46) T v (40) P Real P T P Ideal P Ideal where the correction factor is given by: Note that for ideal gas: P ! 1 " R P , (47) R " v " R , (41) P v T In! analogous fashion, it is defined the entropy which is in accordance with results from technical correction factor. In accordance with Eq. 37 and 44, literature (John and Keith, 2006, p. 367). the entropy correction factor is given by:

" R / R , (48) EXAMPLE OF APPLICATION S From Eq. 29 and 37, it is possible to define In order to obtain quantitative results, an equation of multiplicative correction factors that can be used in state for helium, based on a three virial coefficient an ideal gas formulation in order to solve problems (Miller and Wilder, 1968; Schneider and Duffie, 1949), associated with the real gas formulation. However, it is is employed to determine the correction factor for important to define the concept of specific heat ratio in pressure and entropy variation. The equation used to

J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 35 Silva, M.G., Oliveira, P.A.P. represent the pressure-density-temperature relationship and of real helium gas is the virial equation of state: P " RT 1 2 B T 3 2C T 2 (53) P " Z RT " RT 1 B T C T (49) T ! Considering the virial equation of state, Eq.49 and 47, A relatively large amount of experimental data on the pressure correction factor ( ) is given by: the second virial coefficient, B(T), exist for low P ! and moderate temperatures (up to about 1,000 K). 1 " 1 2 B T 3 2C T Experimental results for the third virial coefficient, P Z (54) C(T), are few and show a great deal of scatter. At the conditions dealt within this paper, the contribution of C(T) to the state equation is relatively small. In Entropy correction factor (εS) this context, the expression derived for the virial coefficients is given by: Using Eq. 40 and 48, it is demonstrated that:

1 3 5 7 dB T dC T 2 B T " b bT 4 b T 4 b T 4 b T 4 , (50) " 1 B T T C T T 0 1 2 3 4 S dT dT . (55) and

1 3 5 " 4 4 4 C T c0 c1T c2T c3T (51) RESULTS The coefficients of virial equation are given in Tables The values of B(T) and C(T) computed from Eq. 1 and 2. The units of T, B(T) and C(T) are K, cm3/mole 2 and 3, for real helium gas, are plotted in Figs. 2 and (cm3/mole)2, respectively. and 3. Note that the virial coefficients are plotted in 3 3 terms of [cm /mole], where [cm /mole] = WHe/1000 (m3/kg). The accuracy of these coefficients was Table 1: Second virial coefficient –B(T) checked by comparing the values of the resulting Coefficient T < 1300 K T > 1300 K compressibility coefficient, Z, with those given in b0 -13.4067 1.178236 the tabulation of helium properties prepared by the

b1 165.4459 -7.57134 International Union of Pure and Applied Chemistry (IUPAC). Figure 4 shows the compressibility b2 -1357.92 5225.701 b 5959.061 -188923 coefficient Z, which was in very close agreement 3 with IUPAC for all densities. b4 -12340.8 2460461 Figure 5 shows the multiplicative correction factor for pressure drop. It is a common practice the use of Table 2: Third virial coefficient –C(T) relation (Eq. 26), in a context of ideal gas, in order to obtain the solution for isothermal flow of real gas Coefficient T by only using a real specific heat ratio. However, it c -13.7898 0 can be noted that the specific heat ratio is not the c 139.7339 1 only parameter of influence. In fact, the correction c 8114.259 2 factor must also be considered in this procedure. c -17456.9 P 3 In this! context, although the first methodology is easy to conceive, the latter, more complete, requires the knowledge of more details about the physical properties of the pure substance. Another interesting Pressure correction factor (εP) aspect of this result is related to the value of P . When a gas undergoes a reversible process, in which! there From the Eq. 49, it is obtained: is heat transfer, the process frequently takes place in such a manner that a plot of log P versus v is a straight dB T dC T P 2 " R 1 B T T C T T , line. This is called polytropic process (Wyllen and T dT dT (52) Sonntag, 1987, p. 167), in other words: !

36 J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 Entropy variation in isothermal fluid flow considering real gas fectsef

11.5

10.5 ] o l m / 3 10 m c [

B 9.5

8.5

8 200 400 600 800 1000 1200 1400 1600 T [K] T= 1300 K Figure 2: Second virial coefficient –B(T).

120

110

100

90 ] 2 o l m /

3 80 m c [

C 70

40 200 400 600 800 1000 1200 1400 1600 T [K] Figure 3: Third virial coefficient –C(T).

1.11 = 10 kg/m3 1.1 = 20 kg/m3 = 30 kg/m3 1.09

1.08

1.07 Z 1.06

1.05

1.04

1.03

1.02 200 400 600 800 1000 1200 1400 1600 T [K] Figure 4: Compressibility coefficient – Z.

J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 37 Silva, M.G., Oliveira, P.A.P.

1.11 = 10 kg/m3 1.1 = 20 kg/m3 = 30 kg/m3 1.09

1.08

1.07 P 1.06

1.05

1.04

1.03

1.02 200 400 600 800 1000 1200 1400 1600 T [K] Figure 5: Correction factor for pressure drop.

not clear in B(T) variation (Fig. 2), this discontinuity dP is probably associated with the model adopted for gas P " n, equation. The fit curve coefficients for high temperature dv (56) (Table 1) are not consistent when it is considered high v Ideal variation in density. where, n = 1 (one) is an ideal isothermal process. COMENTS AND CONCLUSION From Eq. 32, 33 and 52, it is possible to note that: The primary purpose of this investigation was to develop a method required to study the behavior of real pure dP substance in an isothermal fluid flow. Particular emphasis 1 P " " was given to develop useful procedures and techniques dv P T P (57) in order to study the general types of gas, which are v Real encountered in aeronautical applications, such as, wind tunnel, combustors, and so on. More complicated systems

Thus, the pressure corrector factor εP establishes can be studied, in a 1D context, with little additional a direct comparison of ideal and real isothermal difficulty. From this research, it is possible to draw several processes. Results presented in Fig. 5 can be used as a conclusions: good indicative of the tolerance that must be adopted in a system specification, based on isothermal fluid • The mathematical formulations developed for pressure flow results. and entropy variation, Eq. 26 and 36, respectively, can be used for different pure substances. the

pressure correction factors (εP) that must be adopted Figure 6 shows the entropy correction factor, εS. It in an isothermal gas flow are a function of isothermal is clear that ε is more temperature sensitive than ε . S P compressibility module, βT , and static pressure. These Internal energy and entropy are not directly physically factors establish a direct comparison of ideal and real measurable, whereas certain of the intensive variables isothermal processes (Eq. 47). (e.g. T,P) are. Thus, the entropy formulation presented in this paper is an important mathematical model • The entropy correction factor (εS) is a function to know the accuracy of entropy correction factor, of thermal expansion module (αP), isothermal obtained indirectly from the 1D analysis using ideal gas compressibility module (βT), and specific volume formulation. Another aspect that must be considered (v). It is given by the Eq. 48. Similar to the during flow analysis of helium gas is related to the pressure correction factor, it depends on the state discontinuity observed at T = 1300 K. Although it is equation.

38 J. Aerosp.Technol. Manag., São José dos Campos, Vol.3, No.1, pp. 29-40, Jan. - Apr., 2011 Entropy variation in isothermal fluid flow considering real gas fectsef

9 = 10 kg/m3 3 8 = 20 kg/m = 30 kg/m3

% S 5

1 200 400 600 800 1000 1200 1400 1600 T [K] Figure 6: Correction factor for entropy variation.

ACKNOWLEDGMENTS Oldrich, J., Malijesvsky, A., 1992, “The Isothermal Change of a Real Gas”, Technology Today, No. 2, p. 56-60. The first author gratefully acknowledges the partial support for this research, which was provided by BASF Schneider, W.G., Duffie, J.A.H., 1949, “Compressibility – The Chemical Company. This paper is dedicated to my of Gases at High Temperature. II. The Second Virial friend Paulo Afonso Pinto de Oliveira (In memoriam). Coefficient of Helium in The Temperature range of o0 C to 600oC”, Journal of Chemical Physics, Vol. 17, No 9, p. 751. doi:10.1063/1.1747394b REFERENCES John, J.G.A., Keith, T.G., 2006, “Gas Dynamics”, 3th Van Wyllen, G., Sonntag, R., 1987, “Fundamentos da edition, pp. 520. Termodinâmica Clássica”, 2th edition, p. 320.

Miller, C.G., Wilder, S.E., 1968, “Real Helium Hypersonic Whyte, F.M., 2005, “Fluid Mechanics”, McGraw Hill, Flow Parameters for Pressures and Temperatures to 3600 New York, 5th edition, p. 866. atm and 15,000 K”, NASA TN-D 4869.

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