Journal of Physics: Conference Series PAPER OPEN ACCESS A one-dimensional analytical calculation method for obtaining normal shock losses in supersonic real gas flows To cite this article: Maximilian Passmann et al 207 J. Phys.: Conf. Ser. 82 02004 Related content - Expansion and thermalization processes of plasma flow on the BSG-II deice T. Uchida, K. Miyamoto, J. Fujita et al. - Ionisation relaxation in shock-heated krypton at electron densities from - 50*0 9 m -3 U Frobe, B -H Muller and W Botticher - Self-alidated calculation of characteristics of a Francis turbine and the mechanism of the S-shape operational instability Z Zhang and M Titzschkau View the article online for updates and enhancements. This content was downloaded from IP address 37.44.96.25 on 20//207 at 8:3
International Conference on Recent Trends in Physics 206 (ICRTP206) Journal of Physics: Conference Series 755 (206) 000 IOP Publishing doi:0.088/742-6596/755//000 A one-dimensional analytical calculation method for obtaining normal shock losses in supersonic real gas flows Maximilian Passmann, Stefan aus der Wiesche, and Franz Joos 2 Department of Mechanical Engineering, Muenster Uniersity of Applied Sciences, 48565 Steinfurt, Germany 2 Laboratory of Turbomachinery, Department of Power Engineering, Helmut-Schmidt Uniersity, 22043 Hamburg, Germany E-mail: max.passmann@fh-muenster.de Abstract. The calculation of isentropic flow and normal shock waes of real gases are important, especially in the preliminary design of turbo-machinery and test rigs. In an ideal gas, the relations for onedimensional isentropic flow and normal shock waes are well known and can be found in standard textbooks. Howeer, for fluids exhibiting strong deiations from the ideal gas assumption uniersal relations do not exist due to complex equations of state. This paper presents a analytical method for the prediction of isentropic real gas flows and normal shock waes, based on the Redlich-Kwong (RK) equation of state. Explicit expressions based on a series expansion for describing isentropic flow of Noec TM 649 are compared to Refprop data and ideal gas equations. For moderate pressures the RK method is in ery good agreement with the Refprop data, while the ideal gas equations fail to predict the real gas behaiour. The same obserations are made for normal shock calculations, where both real gas methods yield ery close results. Especially the predicted stagnation pressure losses across a shock wae are in excellent agreement. Nomenclature Latin Symbols a Speed of sound [m/s] A Cross sectional area [m 2 ] A,B,C Constants in Eq(8) [ ] c Velocity [m/s] c Const. olume heat capacity [J/(kgK)] h Specific enthalpy [kj/kg] M Mach number [ ] M m Molar mass [kg/kmol] p Pressure [Pa] P Shock strength [ ] R Specific gas constant [J/(kgK)] R uni Uniersal gas constant [J/(kgK)] s Specific entropy [J/(kgK)] T Temperature [K] u Internal energy [m/s] Specific olume [m 3 /kg] Z Compressibility factor [ ] Greek Symbols α Constant in Eq() [ ] β Constant in Eq() [ ] ɛ Error [%] ζ Constant in Eq(5) [ ] Partial deriatie [ ] Γ Fundamental deriatie [ ] γ Adiabatic exponent [ ] Subscripts 0 Stagnation condition Value upstream from shock wae 2 Value upstream from shock wae c Value at critical point ig Ideal gas state rg Real gas state. Introduction Despite the rapid adancements in computational fluid dynamics (CFD), a design and optimization approach solely based on CFD simulations is still time consuming. Therefore onedimensional methods, especially in the preliminary design phase, are important. In an ideal gas, the relations for one-dimensional isentropic flow and normal shock waes are well known and can be found in standard textbooks. Howeer, for fluids exhibiting strong deiations from the ideal gas assumption, uniersal relations do not exist. It is the purpose of this paper to deelop a method for the description of one-dimensional isentropic flow and normal shock losses in supersonic real gas flows, using the analytical Redlich-Kwong (RK) equation of state (EoS). Content from this work may be used under the terms of the Creatie Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd
Fluids exhibiting real gas behaiour are of special interest for organic Rankine Cycle (ORC) applications. An ORC process is a thermodynamic cycle, based on the classical steam Rankine cycle that utilizes organic fluids instead of water as working fluid. The key parameter in terms of organic fluid properties is a low boiling point, making the usage of low-temperature heat sources possible. In recent years, research has focused on so called dry fluids, with an oerhanging saturation cure, which is faourable to aoid blade erosion []. The reliable design of ORC components requires both numerical and experimental work. For this reason, the Laboratory for Thermal and Power Engineering at Muenster Uniersity of Applied Sciences, Germany is building up a closed loop wind tunnel for organic apors, aiming at the inestigation of real-gas effects in nozzles and axial turbine blade cascades [2]. Howeer, to determine the working point of the test rig, the oerall losses at operating conditions hae to be estimated. The dominating contribution to the total losses occurs in the test section. Details on its design can be found in [3, 4]. To get a better quantitatie estimate about the pressure losses and hence the required pressure rise to drie the wind tunnel, a normal shock wae in the test section was considered. The computational method to calculate these losses for an organic apour are deeloped and tested in this contribution. Another important technical application for normal shock waes is the correction of total pressure measurements from supersonic Pitot-tubes. The paper is structured as follows. After a brief literature reiew on existing real gas flow models, the present approach is described in detail. First, the thermodynamic real gas model is presented, followed by methods to describe isentropic flow and normal shock calculations. The results are then presented and discussed. 2. Literature Reiew A number of models for describing the steady one-dimensional isentropic flow of a real gas can be found in the literature. An excellent historical reiew of isentropic flow models has been published by Sullian [5]. Reimer [6] and Johnson [7] presented isentropic relations based on modified ideal gas relations that accurately describe fluid behaiour. Howeer, in both cases only slight deiations from ideal gas behaiour occurred, which can be attributed to the fluids and the range of operating conditions selected for the study. Kouremenos and Kakatsios [8] chose a similar approach and defined explicit relations approximating real gas behaiour, with a similar mathematical form as the ideal gas equations. Later, they extended this work to the description of thermodynamic properties using the RK EoS [9]. One of the few experimental works has been carried out by Bier et al. [0]. They inestigated the flow of the refrigerant R22 in an axially symmetric de Laal nozzle for Mach numbers up to M = 2.2. The experimental results were compared to one-dimensional computations using an adanced EoS as well as an approximation procedure, that could reproduce all measured alues with deiations within a few percent. A different approach on isentropic flow makes use of the fundamental deriatie Γ as defined by Thompson []. The fundamental deriatie of gas dynamics is a quantitatie measure of the ariation of the speed of sound with respect to density in an isentropic change of state [2]. Thompson and Sullian [3] reported a method for predictions of the sonic conditions in a real gas utilizing the fundamental deriatie. The results are relatiely simple relations based on a Taylor series expansion. Howeer, the authors remark, that while the application of the fundamental deriatie leads to a simple and effectie description of isentropic real gas flows, the calculation of Γ itself is not necessarily simple. Colonna et al. [2] inestigated the possibility of computing the fundamental deriatie using different EoS. They found non-analytical or reference EoS to be superior to analytical formulations. Especially in the icinity of the critical point, analytical EoS fail to predict the correct physical behaiour. One of the few analytical approaches to calculate normal shock waes of real gases was published by Kouremenos [4]. He deried a non-linear equation using the RK EoS, but this first study lacks information on the results and accuracy of the method. Later on, Kouremenos et al. [5] presented a modified calculation procedure and inestigated the influence of different EoS and stagnation conditions on normal shock waes in steam. They conclude, that the RK EoS proides accurate solutions for moderate pressures, but becomes increasingly inaccurate for higher pressures and temperatures. 2
3. Thermodynamic Real Gas Model The description of the thermodynamic behaiour of a real gas requires a suitable EoS. The most accurate type are non-analytic or fundamental EoS [6], explicit in Helmholtz energy. Equations in this form are capable of describing the thermodynamic behaiour of pure substances with an accuracy that probably exceeds measurement uncertainties [6]. Unfortunately, this type of EoS has to be soled numerically and cannot be used to derie simple working equations. For the present work the widely used Redlich-Kwong EoS [7] has been selected. The equation in the form p(t, ) is represented by p = R T β α T 2 ( + β). () Herein, p,, T are pressure, specific olume, and temperature, and R is the specific gas constant R = R uni /M m. The constants α and β are functions of pressure and temperature at the critical point (p c, T c ) and are gien by α = 0.42748 R2 T 5 2 c, β = 0.08664 R T c. (2) p c p c If the compressibility factor Z is close to unity, the ideal gas EoS is reproduced by Eq [8]. A full description of the thermodynamic behaiour using a cubic EoS requires additional analytical expressions for enthalpy and entropy. Howeer, it can be shown, that a fluid model consisting of the three equations p(t, ), h(t, ), and s(t, ) is thermodynamically equialent to a fundamental equation of state [8]. For a thermal EoS with the independent ariables temperature and olume, the enthalpy of a real gas is defined as [8] where u(t, ) describes the specific internal energy, gien by T [ u(t, ) = u ig + T h(t, ) = u(t, ) + p(t, ), (3) T ig c ig (T ) dt + ( p T ) ] p d. (4) Herein, u ig is the specific internal energy of the corresponding ideal gas state (p ig 0). The first integral in Eq (4) describes the temperature dependency of u for an ideal gas state, whereas the second integral accounts for deiations from ideal behaiour. The specific entropy of a real gas can be dealt with in a similar manner, resulting in the expression s(t, ) = s ig + T T ig c ig (T ) dt T + R ln ig + [( p T ) R ] d, (5) where s ig is the specific entropy of the corresponding ideal gas state (p ig 0). The selection of the reference point is arbitrary. In the present work, internal energy u ig and entropy s ig were set to zero at T ig = 273.5 K and p ig = Pa. For the speed of sound a holds the exact relation [8] ( ) [ ( ) p a 2 = 2 = 2 T p 2 ( ) ] p. (6) s c T T Equation 6 shows that the speed of sound can be deried from the thermal EoS p(t, ) and the real gas isochoric specific heat capacity c (T, ): ( c (T, ) = c ig 2 ) p (T ) + T T 2 d. (7) Herein, the ideal gas specific heat capacity c ig (T ) is approximated by a second-order polynomial c ig (T ) = A + B T + C T 2. (8) 3
The constants A, B, and C can be taken from the literature or from a fit procedure. For this paper, Refprop data was used. Equation 8 was fitted by means of a least-squares method and reproduces the exact alues within 0.5 %. After integration Eq (3) and Eq (5) become: s(t, ) = s ig +A ln T +B (T T ig )+ C T ig 2 (T 2 Tig)+R ln 2 + α ln ig 2bT 3 2 + β +R ln b, (9) h(t, ) = u ig +A (T T ig )+ B 2 (T 2 Tig)+ 2 C 3 (T 3 Tig)+ 3 3α ln 2βT 2 + β + RT β α T 2 ( + β). (0) Thus, Eq (,9,0) represent a closed thermodynamic real gas model, which has been used for the following inestigations. 4. Isentropic Flow Relations From a practical point of iew, simple working formula for the description of isentropic changes in a real gas are desirable. A method for obtaining relations in the form p/p 0, T/T 0, / 0, and A /A in terms of Mach number is outlined in the following. The goerning equations of steady one-dimensional isentropic flow are the continuity equation and the energy equation c 2 A = c 2 A = const., () h + c2 2 = h 2 + c2 2 2 = h 0 = const., (2) where c is the elocity, the specific olume, and h and s the specific enthalpy and entropy. In the following, an isentropic expansion process proceeding from a known set of stagnation conditions p 0, T 0, 0 along an isentrope s 0 = const. to an arbitrary point p, T, is considered. Furthermore, the stagnation or reseroir conditions are assumed to be known. Consequently, the isentrope s 0 passes through this point. To describe this isentrope we calculate the fluid properties along this path for an arbitrary number of points n i. The local Mach number M i at each point is defined by M i = c i. (3) a i Herein, the local speed of sound a i can be obtained from Eq 6 and the local elocity c i is related to the enthalpy by c i = 2 (h 0 h i ). (4) Start Input: T 0, 0, M max, n i Compute: p 0 (T 0, 0 ), s 0 (T 0, 0 ), h 0 (T 0, 0 ) Estimate p lb so that M lb M max Generate linearly spaced ector: T i =[T 0... T lb ] Compute properties along isentrope s 0 =const i =s(t i,s 0 ), p i =p(t i, i ), h i =h(t i, i ), α i =α(t i, i ) Compute Mach number at each point: M i =c i /a i Fit ratios to 6 th -order polynomial: p/p 0 = f(m), T/T 0 = f(m), 0 / = f(m) Determine alues at sonic conditions (M * =) p * =p/p 0 (M * ) p 0, T * =T/T 0 (M * ) T 0, α * =α(t *, * ), * = 0 / (M * ) 0 - Fit area ratio to 6 th -order polynomial: A * /A= * /i ci/c *, A * /A=f(M) Compute elocity at each point: c i =[2 (h 0 -h i )] /2 End Yes Check alues at p lb : M lb M max? No Restart with lower p lb Figure. Flow-chart showing the computational procedure for deriing one-dimensional isentropic relations based on a sixth-order polynomial. Bold ariables are ector quantities. 4
Pressure, temperature, specific olume and area ratios are then fitted to a polynomial in terms of Mach number by means of a least-squares fit. The present work has shown, that simple ideal-gas-like formulations, as used for instance by Kouremenos and Kakatsios [8], fail to predict real gas behaiour of fluids like Noec TM 649. Therefore the equations were extended by means of a series expansion. A sixth-order polynomial was found to be sufficient to describe real gas behaiour in the present case. The polynomials F (M) are of the generic form: F (M) = 6 ζ i M i. (5) i=0 Herein, ζ i describe the fitted constants. The computational procedure is summarised in Fig. The correct alue of the lower pressure limit for the fit p lb has to be determined iteratiely, based on a prescribed maximum Mach number M max. It is worth noting, that this algorithm is independent of the EoS and can be used with cubic as well as fundamental EoS. 5. Normal Shock Calculations A shock wae is a discontinuity in a ery small region that occurs between two boundaries in supersonic flow. Normal shock waes are perpendicular to the flow direction and transform a supersonic flow to a subsonic flow regime. A schematic sketch of a normal shock wae is shown in Fig 2 (left). Figure 2 (right) shows the idealised process in the h-s-diagram. Shock waes are dominated by iscous and heat transfer effects. Therefore the assumption of isentropic flow is no longer alid. As a result from the second law of thermodynamics the entropy across a shock wae has to increase, i.e. s 02 > s 0. The description of a normal shock wae inoles fie ariables, i.e. static pressure, temperature, specific olume, enthalpy, and elocity. Therefore in addition to the thermal EoS p = p(t, ), enthalpy equation h = h(t, ), continuity of mass Eq (), and conseration of energy Eq (2), a fifth equation is necessary to compute the changes across a normal shock. This is achieed by introducing the conseration of momentum: p + c2 = p 2 + c2 2 2. (6) Herein, p describes the static pressure, c is the elocity, and the specific olume. For an ideal gas, this set of equations can be soled in closed form, resulting in the normal shock or Rankine-Hugoniot equations [9]. For non-ideal fluids this is usually impossible due to the complexity of the EoS. A way of finding an analytical solution is by means of cubic EoS. h 0 = h 02 0 p 0 T 0 02 p 02 T 02 Normal Shock c, p, T, h, s c 2, p 2, T 2, 2 h 2, s 2 Specific enthalpy h [J/kg] h 2 h p T Normal Shock 2 p 2 T 2 s 0 s 02 Specific entropy s [J/(kg K)] Figure 2. Schematic sketch of an idealised normal shock wae (left) and changes across a normal shock wae in the specific enthalpy ersus entropy plane (right). 5
Using Eq () and Eq (0) for thermal and enthalpy EoS, in principle allows for a solution of one analytical expression containing only one unknown, i.e. 2 = f(t 2 ). Howeer, the present study has shown that the resulting expression becomes quite cumbersome and not ery reliable to sole. Far better results were obtained from a system of two non-linear equations with two unknowns T 2 and 2 that can be soled numerically using a root finding algorithm, such as the Newton-Raphson method [20]. Supplying the corresponding ideal gas alues as initial data, results in a quick conergent behaiour. Using the conseration of momentum in combination with the RK EoS (Eq ()) to eliminate the unknown pressure p 2 results in: 0 = p (T, ) R T 2 2 β + α T 2 2 2 ( 2 + β) + c2 2 c 2. (7) Similarly the energy equation, Eq (2), in combination with the enthalpy equation, Eq (0), can be used to obtain the second expression: 0 = h (T, ) u ig + A (T 2 T ig ) + B 2 (T 2 2 T 2 ig) + C 3 (T 3 2 T 3 ig) + + RT 2 2 β α 2 3α 2βT 2 T 2 2 2 ( 2 + β) ln + c2 2 2 + β + 2 2 2 c2 2 2. (8) The numerical solution of Eq (7) and Eq (8) proides T 2 and 2. All other properties downstream from the shock wae can be obtained from the RK EoS (Eq ()) and enthalpy and entropy equations (Eq (0,9)). 6. Results and Discussion The calculation procedures presented in this paper are tested with the fluorinated ketone Noec TM 649, which is the working fluid for the closed loop wind tunnel [2]. Figure 3 shows an h-s-diagram for Noec TM 649 [2]. Of special interest for the present study are stagnation conditions in the superheated region with pressure and temperature in the range of p 0 = 4...6 0 5 Pa and T 0 = 370...420 K. This range was chosen because it represents the possible operating conditions of the planned closed loop wind tunnel. The fluid possesses the desired oerhanging saturation cure, as mentioned in Sec. Deiations from ideal gas behaiour can already be obsered for moderate pressures, i.e. at 5 0 5 Pa and 390 K, the compressibility factor is already as low as Z = 0.829. In the first step, accuracy and consistency of the RK real gas model were compared with the reference EoS (McLinden et al. [22]), which is implemented in Refprop [2]. For stagnation conditions of 5 0 5 Pa and 390 K an exemplary comparison between both EoS is presented in Fig 4. The RK predictions show deiations mostly lower than 0 %. The biggest errors occur in the prediction of the specific olume, with a maximum deiation of 20.9 %. Errors increase for higher pressures, near the saturation line, and in the icinity of the critical point. Howeer, for the range of interest in this work, the RK EoS is a reasonable approximation of the real fluid behaiour. It should also be noted, that some initial tests with other cubic EoS, such as the formulations by Aungier [23], Soae [24], and Peng and Robinson [25] showed no significant improement oer the RK EoS. In the next step, the isentropic relations were fitted to the fluid properties according to the procedure described in Sec 4 and Fig. Isentropic expansions using the RK and Refprop EoS were compared to each other oer a wide range of stagnation conditions, up to the critical point region. Table shows the maximum error for the RK EoS with respect to the corresponding Refprop data. The results show, that the prediction of the specific olume is afflicted with the highest deiations up to 40 % near the critical point, followed by the static pressure with deiations up to 8 %. The lowest errors were obsered for the temperature with maximum deiations around 2 %. Furthermore, the results indicate, that the RK EoS becomes less reliable 6
for higher pressures and temperatures, especially for expansions from stagnation conditions aboe p 0 = 2 0 5 Pa, errors higher than 0 % are being obsered. Deiations were also found to increase in the icinity of the saturation cure. As shown in Tab, for a stagnation pressure of 5 0 5 Pa three different stagnation temperatures were considered. For T 0 = 378 K, the point closest to the saturation cure, the deiations are larger than for the two points with higher temperatures, that are farther away from the saturation cure. A similar behaiour, albeit with higher errors, has been obsered for higher pressures. For the range of interest in this work (p 0 = 4...6 0 5 Pa, T 0 = 370...420 K), the RK EoS has been found to delier re- 420, 460 K 40 K Enthalpy (kj/kg) Specific enthalpy h [kj/kg] 400, 380, 450 K 440 K 430 K 3 bar 4 bar 6 bar 8 bar 400 K 390 K 380 K 370 K 360 K bar.5 bar 2 bar 0.5 bar 0.25 bar 0. bar 360,,50.50,53.53,55.55,58.58,60.60,63.63,65.65 Specific entropy s [kj/(kg K)] Entropy (kj/kg-k) 0.05 bar 0.025 bar 0.0 bar 350 K Figure 3. Specific enthalpy h s. specific entropy s plot with lines of constant temperature (blue) and pressure (red) for Noec TM 649 calculated from [2]. Pressure p [Pa] s 0 =const p 0 =5 0⁵ Pa T 0 =390 K REFPROP ϵ max = 3.2 % ϵ min =.2 % Redlich-Kwong Specific olume [m³/kg] ϵ max = 20.9 % ϵ min = 2.9 % REFPROP s 0 =const p 0 =5 0⁵ Pa T 0 =390 K Redlich-Kwong Temperature T [K] Temperature T [K] Speed of sound a [m/s] s 0 =const p 0 =5 0⁵ Pa T 0 =390 K REFPROP Temperature T [K] Redlich-Kwong ϵ max =.9 % ϵ min = 0.7 % Enthalpy difference Δh=h-h0 [kj/kg] ϵ max = 2.4 % ϵ min =.8 % REFPROP Redlich-Kwong Temperature T [K] s 0 =const p 0 =5 0⁵ Pa T 0 =390 K Figure 4. Comparison of isentropic lines (s 0 =const) for Noec TM 649 calculated from Redlich- Kwong and Refprop [2] for stagnation conditions of p 0 = 5 0 5 Pa, T 0 = 390 K and maximum and minimum error for Redlich-Kwong predictions. 7
Table. Maximum error of isentropic fits for the RK EoS with respect to Refprop data for expansions from different stagnation conditions up to a Mach number of M max = 2.2. Critical pressure and temperature for Noec TM 649 are p c = 8.69 0 5 Pa and T c = 44.8 K. p 0 /p c T 0 /T c p 0 T 0 Error p/p 0 Error T/T 0 Error 0 / MPa K % % % 0.9952.0027.86 443 7.59.80 39.48 0.9096 0.989.7 437.73 2.07 2.5 0.856 0.9846.6 435 2.4.75 7.53 0.642 0.9506.2 49.90.33 9.49 0.5350 0.9280 40.7.8 8.9 0.320 0.8737 0.6 386.3 0.83 5.2 0.8556 378.73 0.74 4.89 0.2675 0.874 0.5 385.04 0.63 3.85 0.8827 390 0.98 0.57 2.88 0.240 0.8375 0.4 370.02 0.63 4.62 0.605 0.848 0.3 360 0.92 0.52 3.48.0 0.9 p/p 0, T/T 0, 0 /, A*/A [-] 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. p/p 0 Noec TM 649 (Redlich-Kwong) p/p 0 Noec TM 649 (REFPROP) p/p 0 Air (REFPROP) T/T 0 Noec TM 649 (Redlich-Kwong) T/T 0 Noec TM 649 (REFPROP) T/T 0 Air (REFPROP) 0 / Noec TM 649 (Redlich-Kwong) 0 / Noec TM 649 (REFPROP) 0/ Air (REFPROP) A*/A Noec TM 649 (Redlich-Kwong) A*/A Noec TM 649 (REFPROP) A*/A Air (REFPROP) 0.0 0.0 0.2 0.4 0.6 0.8.0.2.4.6.8 2.0 Mach number M [-] Figure 5. Comparison of isentropic relations (s 0 =const) for Noec TM 649 and air for stagnation conditions of p 0 = 5 0 5 Pa and T 0 = 390 K. liable predictions with maximum deiations for pressure and temperature mostly around % and slightly higher alues up to 5 % for the specific olume. As an example, Fig 5 presents the isentropic flow relations for Noec TM 649 for stagnation conditions of 5 0 5 Pa and 390 K. The results obtained from RK and Refprop are in good agreement, with maximum deiations gien in Tab. Finally, the method for calculating the normal shock was tested, using the following procedure: For a gien set of stagnation conditions p 0, T 0 the fluid properties for a specified Mach number M are computed, using the isentropic relations described in Sec 4. The next step is the normal shock calculation. In case of the RK EoS the procedure described in Sec 5 is followed, whereas for the Refprop case three goerning equations (Eq (,2,6)) in combination with the EoS by McLinden et al. [22] were soled numerically, using a Newton-Raphson method. Subsequently, the stagnation conditions behind the shock are determined from the EoS and the results are checked for consistency, i.e. the stagnation enthalpy has to remain constant h 02 = h 0, whereas the entropy has to increase s 02 > s 0. Figure 6 shows the ariation of M 2, p /p 2, p 02 /p 0, T /T 2 and 2 / in terms of upstream 8
Mach number M. The results for the RK EoS were obtained from the procedure described aboe, while the solutions with Refprop were computed by directly soling the three goerning equations (Eq (,2,6)) in combination with the EoS by McLinden et al. [22]. A stable conergent behaiour was achieed by setting the corresponding ideal gas solution as initial alues for the iteratie solution. Again, excellent agreement was found between RK and Refprop predictions, while the ideal gas formulations fail to predict the non-ideal fluid behaiour. An additional parameter in the description of normal shock waes is the strength of a shock wae P, which is defined as the ratio of the increase in static pressure to the initial static pressure upstream from the shock wae [9]: P = p 2 p p = p 2 p. (9) The results shown in Fig 7 (right) indicate, that normal shock waes calculated from real gas formulations are weaker than the solutions obtained from the ideal gas equations. Howeer, the absolute loss in stagnation pressure for the real gas model is up to 40 % higher than the corresponding losses for an ideal gas. Of major interest is also the consistency of the calculations. Mach number M 2, p /p 2, p 02 /p 0 [-] 0.8 0.6 0.4 0.2 M 2 Redlich-Kwong M 2 REFPROP M 2 Ideal Gas p /p 2 Redlich-Kwong p /p 2 REFPROP p /p 2 Ideal Gas p 02/p 0 Redlich-Kwong p 02/p 0 REFPROP p 02/p 0 Ideal Gas 0.5 2 2.5 3 Mach number M [-] T /T 2 2 / [-] 0.8 0.6 0.4 0.2 T /T 2 Redlich-Kwong T /T 2 REFPROP T /T 2 Ideal Gas 2/ Redlich-Kwong 2/ REFPROP 2/ Ideal Gas 0.5 2 2.5 3 Mach number M [-] Figure 6. Variation of M 2, p /p 2 (left) and T /T 2, 2 / (right) in terms of M for normal shock waes in Noec TM 649. Stagnation conditions were set at p 0 = 5 bar and T 0 = 390 K. Stagnation pressure loss Δp 0 =p 0 -p 02 [Pa] 4.5 x 05 4 3.5 3 2.5 2.5 0.5 Δp 0 Redlich-Kwong Δp 0 REFPROP Δp 0 Ideal Gas 0.5 2 2.5 3 Mach number M [-] Shock strength P [-] 8 7 6 5 4 3 2 P Redlich-Kwong P REFPROP P Ideal Gas 0.5 2 2.5 3 Mach number M [-] Figure 7. Variation of shock losses (left) and shock strength (right) in terms of M for normal shock waes in Noec TM 649. Stagnation conditions were set at p 0 = 5 bar and T 0 = 390 K. 9
Tests with different EoS, i.e. using Refprop for the description of isentropic flow and the RK EoS for normal shock calculations, resulted in inconsistencies and non-physical solutions. 7. Conclusion and Outlook A method for calculating one-dimensional isentropic flow and the normal shock of a real gas is presented. A thermodynamic real gas model based on the cubic RK EoS accurately predicts the fluid properties of Noec TM 649 for moderate pressures, but becomes less accurate for higher pressures and near the critical point. Simple working formulae based on a sixth-order polynomials are sufficient to describe the isentropic flow of real gases. The agreement with Refprop data is excellent, while the ideal gas formulation fails to predict real gas behaiour. A system of two non-linear equations, using the RK EoS, is soled to calculate normal shock waes. The results are compared to Refprop and ideal gas data. While both real gas models are in ery good agreement, the ideal gas shows strong deiations. Compared to the ideal gas model, both real gas models predict higher absolute pressure losses, but lower alues for the shock strength. This paper shows, that a analytical real gas model based on the RK EoS accurately predicts isentropic flow and normal shocks for moderate pressures. Therefore it can be used as an alternatie to a purely empirical approach using Refprop. Although, the prediction of the specific olume by the RK model is afflicted with relatiely high errors, the normal shock losses are in ery good agreement with Refprop data. The model has been found to be reliable for moderate pressures and temperatures, higher errors occur near the critical point region. Also the behaiour at supercritical conditions has to be inestigated. The results of the present paper suggest the following areas for future research: So far only Noec TM 649 has been considered. An extension to other fluids is necessary to further inestigate the reliability of the method. While initial tests with other cubic EoS hae not shown significant improement oer the RK EoS, the inestigation of other cubic EoS (e.g. Soae [24]) could further improe the method. Subsequently, an extension to oblique shock waes will be of interest. 8. References [] Chen H, Goswami D Y and Stefanakos E K 200 Renewable and Sustainable Energy Reiews 4 3059 3067 [2] Reinker F, Hasselmann K, aus der Wiesche S and Kenig E Y 206 ASME Journal of Engineering for Gas Turbines and Power 38 05260 [3] Hasselmann K, Reinker F, aus der Wiesche S and Kenig E Y 205 ASME/JSME/KSME 205 Joint Fluids Engineering Conference (American Society of Mechanical Engineers) pp V00T5A002 V00T5A002 [4] Passmann M, Reinker F, Hasselmann K, aus der Wiesche S and Joos F 206 ASME Turbo Expo 206: Turbine Technical Conference and Exposition (American Society of Mechanical Engineers) pp 8 [5] Sullian D 98 ASME Journal of Fluids Engineering 03 258 267 [6] Reimer R M 964 ASME Journal of Basic Engineering 86 69 73 [7] Johnson R C 964 ASME Journal of Basic Engineering 86 59 525 [8] Kouremenos D A and Kakatsios X K 985 Forschung im Ingenieurwesen 5 69 74 [9] Kouremenos D and Kakatsios I X 99 Forschung im Ingenieurwesen 57 25 27 [0] Bier K, Ehrler F and Kissau I G 977 Forschung im Ingenieurwesen 43 75 84 [] Thompson P A 97 Physics of Fluids 4 843 849 [2] Colonna P, Nannan N, Guardone A and Van der Stelt T 2009 Fluid Phase Equilibria 286 43 54 [3] Thompson P and Sullian D 977 ASME Journal of Fluids Engineering 99 27 225 [4] Kouremenos I D A 986 Forschung im Ingenieurwesen A 52 23 3 [5] Kouremenos I D, Kakatsios I X and Krikkis R 990 Forschung im Ingenieurwesen 56 54 57 [6] Poling B E, Prausnitz J M, John Paul O and Reid R C 200 The properties of gases and liquids ol 5 (McGraw-Hill New York) [7] Redlich O and Kwong J N 949 Chemical Reiews 44 233 244 [8] Baehr H D and Kabelac S 202 Thermodynamik: Grundlagen und technische Anwendungen (Springer-Verlag) [9] Shapiro A H 953 New York: Ronald Press, 953-54 [20] Press W H 2007 Numerical recipes 3rd edition: The art of scientific computing (Cambridge Uniersity Press) [2] Lemmon E W, Bell I H, Huber M L and McLinden M O 205 NIST Reference Fluid Thermodynamic and Transport PropertiesREFPROP, Beta Version 9..2.5 [22] McLinden M O, Perkins R A, Lemmon E W and Fortin T J 205 Journal of Chemical & Engineering Data 60 3646 3659 [23] Aungier R 995 ASME Journal of Fluids Engineering 7 277 28 [24] Soae G 972 Chemical Engineering Science 27 97 203 [25] Peng D Y and Robinson D B 976 Industrial & Engineering Chemistry Fundamentals 5 59 64 0