Science in China Series E: Technological Sciences © 2008
SCIENCE IN CHINA PRESS
Springer
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The effects of gas models on the predicted performance and flow of a centrifugal refrigeration compressor stage WANG ZhiHeng & XI Guang† School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
In this paper three perfect gas models with constant specific heat or with variable specific heat and one real gas model based on the gas property tables are respectively considered to implement into the three-dimensional CFD (computational fluid dynamics) analysis of a centrifugal refrigeration compressor stage. The results show that the gas models applied to the CFD code have significant influences on the performance of stage and the flow structures in the stage. Although the thermodynamics operating condition of evolving fluid in the centrifugal refrigeration compressor has a significant deviation from the perfect gas, the perfect gas model with the modified value of gas constant and the variable specific heat offers a good prediction of stage performance. To predict some basic fluid flow parameters and flow structure accurately, the real gas effects should be considered and the reasonably accurate thermodynamic properties based on the analytical gas equation of state or numerical interpolation of gas tables should be applied to the CFD code. gas model, numerical analysis, centrifugal refrigeration compressor
1 Introduction With the advancement of numerical methods and computing capability, the numerical simulation of turbomachinery flows using computational fluid dynamics (CFD) techniques has now been widely accepted as a key tool for aerodynamic design. To continually improve the capabilities of CFD, the significant development in three aspects must be required: the robust and reliable algorithms, the turbulence model and the fluid model. In these three topics, lots of efforts have been made and much work should be further done. In this paper we focus our attention on the third topic. In the industrial compressors or turbines, the working fluids are different and working over a wide range of thermodynamics operating condition. Nowadays it is common to carry out the CFD analysis with the perfect gas (also called the ideal gas) model in these turbomachines. In some Received May 6, 2008; accepted May 16, 2008 doi: 10.1007/s11431-008-0165-y † Corresponding author (email:
[email protected]) Supported by the National Natural Science Foundation of China (Grant No. 50725621)
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cases this method is unlikely to introduce the significant error. However, for some applications the assumption of the perfect gas will lead to the inaccurate predictions of the flow structure. Many researchers have investigated the effects of the gas model on the CFD calculations and presented the numerical models to account for real gas effects. Cravero and Satta[1] described a real fluid model based on Redlich-Kwong-Aungier equation of state and its implementation into a Navier-Stokes solver. The CFD analysis with the real gas model was performed in an industrial turbine nozzle to assess the impact of the real fluid evaluation on the fluid. Boncinelli et al.[2] included gas properties into an RANS code using interpolation from gas tables to analyze a transonic centrifugal impeller and a three-stage low-pressure steam turbine. Liu and Xu[3] used the perfect gas and the real gas respectively to simulate the flow in a centrifugal compressor stage and showed that the results of the computation with the real gas model were closer to the experimental data. Yao and Amos[4] evaluated the aerodynamic performance characteristics and the flow of industrial gas turbines by using a CFD code with a variable gas property (cp) model and recommended that the variable gas property effect in 3D calculations of industrial gas turbine flow should be included to achieve an accurate analysis of the flow and performance. However, Northall[5] added variable gas properties to a 3D multistage code to analyze a multistage turbine flow and concluded that the effect on turbine performance prediction of including variable gas properties in 3D CFD is small. In this paper the flow in a centrifugal refrigeration compressor stage is evaluated by the 3D CFD code with four different definitions to the property of the evolving fluid. The attention is mainly focused on the influence of gas properties’ variation on the stage performance and the flow structure in the stage.
2 Description of the compressor stage The stage considered in this paper is the first stage of a two-stage centrifugal refrigeration compressor consisting of a shrouded impeller, an airfoil diffuser, a bend and a return channel. The evolving fluid is HFC-134a, whose behavior in the operating conditions is far from ideal (the gas compressibility factor is about 0.85). The impeller is a typical modern three-dimensional one with splitter blades. The design rotation speed is 8200 r/min and the mass flow rate of 14 kg/s. The center lines of the diffuser vane and the return channel vane are both a circular arc and the blade profiles both coming from NACA65. Some technical data of the stage are summarized in Table 1. Table 1 Main technical data of the stage Stage
Impeller
Vaned diffuser
Return channel
Evolving fluid design rotation speed (r/min) inlet total pressure (kPa) inlet total temperature (K) exit radius (mm) blade swept angle (deg) blade exit width (mm) blade number inlet radius (mm) inlet blade angle (deg) outlet radius (mm) outlet blade angle (deg) blade number inlet radius (mm) inlet blade angle (deg) outlet radius (mm) outlet blade angle (deg) blade number
HFC-134a 8200 350 280 175 65 17 11+11 262.5 22 297.5 35 37 297.5 37 120 90 17
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3 Gas models It is a common practice in 3D CFD codes to model the evolving fluid as a perfect gas with a constant specific heat. In the case of compressors acting on refrigerant gases or multistage steam turbines it is recommended that the variable specific heat with temperature or the real gas effects should be included into the CFD codes. In this paper we describe the properties of the evolving fluid using four different models to investigate the gas properties’ influences on the flow in the compressor stage addressed above. 3.1 Perfect gas model In this model the perfect gas law is used: p = ρ Rg T with Rg the gas constant for a particular gas under consideration: Rg = c p − cv
(1) (2)
and the internal energy and the enthalpy per unit mass are both the function only of temperature de = cv dT and dh = c p dT . (3) 3.1.1 Model 1: calorically perfect gas. It is a common practice in most turbomachinery 3D CFD codes to model the perfect gas with constant cp, cv and Rg. In this case, the internal energy and the enthalpy per unit mass are given by e = cvT and h = c pT . (4) In the perfect gas equation the gas constant of HFC-134a is modified based on the equation of state including an average compressibility factor, defined as follows: Rg = 72.01 J/(kg·K). Additionally, the heat conductivity k and the viscosity μ are given by a constant. In the CFD analysis, an appropriate constant specific heat at constant pressure cp is chosen and the specific ratio is calculated: cp c p = 958.04 J/(kg·K) and γ = = 1.08. c p − Rg Factually, the specific heat at constant pressure cp of a perfect gas varies with temperature. And in this case, the perfect gas is called thermally perfect gas. In this study two methods below are used to take the variable cp into consideration. 3.1.2 Model 2: perfect gas with c p (T ) and Rg defined. According to the operating condition of the evolving fluid, the specific heat at constant pressure cp over the temperature range [250 K, 450 K] is represented as a polynomial function of static temperature of the form: c p (T ) = 3.8452T − 162.32 [ J (kg ⋅ K) ] . (5) The heat conductivity k and the dynamic viscosity μ of the gas are given by
k (T ) = (0.0899T − 13.0724) × 10−3 [W (m ⋅ K)] ,
(6)
μ (T ) = (0.0402T − 0.2607) × 10−6 [Pa ⋅ s] .
(7)
In this model the gas constant is taken as the modified value equivalent to Rg in Model 1.
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3.1.3 Model 3: perfect gas with c p (T ) and γ (T ) defined. In this model the gas constant is not specified, and the specific heat ratio is given in terms of a polynomial function of local temperature instead: γ (T ) = 0.001258T + 0.82915 . (8) It must be pointed out that, the polynomial function above is obtained by fitting the specific heat ratio table of R134a. 3.2 Mode 4: real gas model using gas property tables Different gas models implementing the real gas effects into the CFD codes have been reported and they can be divided into two categories: the analytical equations of state, numerical curve fittings or interpolation of gas properties. As reviewed by Cravero and Satta[1], many equations of state have been reported and applied for different applications. The advantage of an analytical equation of state is that it describes the gas properties with a good accuracy in a wide range of pressure and temperature for many substances. However, their applications within a CFD code need time-consuming iterative calculations and may generate overmuch unacceptable computation because most of the equations of state are complicated and usually implicit equations. On the other hand, the latter model accounting for the real gas effects can give a higher accuracy level with time-efficient implementation, despite that they must be tuned for a specific fluid. In this paper the real thermodynamic properties of the fluid are reproduced by means of interpolation of the variables from the gas property tables. Every table is based on that a certain thermodynamic property can be defined as a function of two independent variables. In a table the values of function are given on the Cartesian mesh in the plane of the variables. Based on the tables and the bilinear interpolation, the gas properties can be obtained efficiently. The accuracy can be enhanced by increasing the number of nodes in the tables. According to the characteristics of the CFD solver, ten tables are provided: p (e, ρ ), T (e, ρ ), μ (e, ρ ), k (e, ρ ), e( p, T ),
ρ ( p, T ), p(h, s ), ρ (h, s ), h( s, p), s (h, p ).
4 Numerical method In the paper the Navier-Stokes code EURANUS/TURBO from NUMECA International is used to solve three-dimensional Reynolds averaged Navier-Stokes equations to simulate the flow field at the centrifugal refrigeration compressor stage. The gas models addressed above are manually defined and implemented into the CFD codes. The Spalart-Allmaras model is chosen as the turbulence model. The calculations are performed with a second-order centered scheme, with second and fourth order artificial dissipation terms and a W-cycle multigrid technique. The numerical procedure applies a four-stage Runge-Kutta scheme, together with local time stepping and implicit residual smoothing for convergence acceleration[6]. One impeller passage with a main blade and a splitter, one passage for both diffuser and return channel are considered as computational domains. For the interface between the components the full non-matching mixing plane approach is used. Figure 1 shows the grids of the impeller, the diffuser and the return channel, respectively, which are created with the auto-grid generation software IGG/AutoGrid of NUMECA. The mesh consists of a total of 1401075 nodes. Particular care has been taken in order to ensure sufficient resolution in the end walls and blade boundary WANG ZhiHeng et al. Sci China Ser E-Tech Sci | Aug. 2008 | vol. 51 | no. 8 | 1160-1168
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Figure 1 Computational mesh.
layers by controlling the position of the mesh points close to the solid surfaces. The grid location near the blade surfaces and the end walls produces Y-plus for the first cell center being less than 5. At the inlet plane, the absolute flow angles, the uniform total pressure and total temperature are fixed. The axial velocities are obtained by extrapolation from the inner cells. At the outlet pane, the design mass flow is restricted. The hub and shroud walls of the impeller are moving with the rotor blade. All the solid wall boundaries are assumed to be adiabatic. Periodic conditions are enforced along the boundaries upstream and downstream of the passage. The computations are performed at a CFL number of 3 and are converged to near machine accuracy, that is 5 orders of residual reduction and constancy of mass flow. The error in mass flow between inlet and outlet of the computational domain is less than 0.5%. It is noted that the thermodynamic condition of the evolving fluid at the inlet of stage is very close to the saturation line. To improve the computation convergence, the computation with Model 1 is performed firstly. This computation result is taken as the initial solution for the computations with Model 2, Model 3 and Model 4.
5 Numerical results and discussion Four computations are performed to evaluate the influence of gas models on the centrifugal refrigeration stage’s performance and flow at the design mass flow rate respectively using the different gas models addressed above. The gas models are compared in terms of compressor’s stage performance at the design mass flow rate in Table 2. From this table, it can be seen that: (1) Compared to the results of Model 4, slightly higher values of the stage efficiency are predicted by the CFD analysis with Model 1 and Model 2. This indicates that using the perfect gas model with a modified gas constant is unlikely to introduce a significant error in the prediction of stage’s efficiency and the variation in cp with the temperature does not affect the stage’s efficiency too much. (2) In Model 2 the variable cp is considered. This improves the accuracy of stage’s efficiency a little, but improves the accuracy of the pressure ratio a lot. (3) The use of Model 3 introduces a significant error in the prediction of stage performance. This is mainly because the perfect gas model is adopted and cp (T) and γ (T ) are however specified according to the real gas’s thermodynamic properties. In Model 1 and Model 2, by introducing an averaged compressibility factor to the perfect gas model the errors between the real thermodynamic properties and the perfect gas model are effectively reduced. This indicates that, to have an accurate prediction of the performance of the stage in which the evolving fluid’s behavior in the operating 1164
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Table 2 Comparison of stage performance predicted using different gas models Gas model
ηis − ηisModel 4
ε ε Model 4
Additional computational time versus Model 1
Model 1
1.71%
0.911
0
Model 2
1.30%
1.021
34.4%
Model 3
−9.65%
0.698
34.5%
Model 4
0
1
20.4%
condition is far from ideal, the real gas model or the modified perfect gas model is necessary to describe the evolving fluid’s thermodynamic properties. (4) Compared with Model 1, the overall computational time is increased by about 35% when either Model 2 or Model 3 is used. This is because the use of Model 2 or Model 3 in the CFD code needs many extra iterative calculations to determine the gas properties. However, since the implementation of Model 4 needs no extra iterative calculation through using the interpolations of thermodynamic properties, the increase of computational time using Model 4 is smaller. To further investigate the influence of gas model on the flow structure, the flows in the stage are analyzed and compared. Figure 2 gives the comparison of the nondimensional pressure distribution
Figure 2 Comparisons of the nondimensional static pressure distribution and the stream lines in the averaged meridian plane.
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and the stream lines in the averaged meridian plane. The nondimensional pressure Pre is defined as the ratio of the local static pressure to the total pressure of the stage inlet. It can be seen that in the case of Model 3 the static pressure in the diffuser and the return channel is much lower than the pressure predicted in the cases of other models. Results show the generally good agreements between the predictions using Model 2 and Model 4. A strong recirculation close to the shroud in the vaneless diffuser is detected in the above two models. But Model 1 cannot predict this flow structure accurately. Additionally, different from the other three models, Model 2 predicts that there is a slight recirculation close to the shroud at the diffuser outlet. The comparison of the magnitude of relative velocity in the stage mid-span section is shown in Figure 3. It also can be seen that the remarkable error arises when Model 3 is used. Comparisons between the other three models show that the results agree well in the impeller while the differences of the velocity magnitude can be observed in the passages of diffuser and return channel. The nondimensional static pressure distributions on blade surface at 50% of blade span are compared in Figure 4. It shows that the predicted static pressure using Model 3 is obviously lower than that using the other three models. In the case of using Model 3, on the rear part of the diffuser
Figure 3 Comparisons of the magnitude of relative velocity in the mid-span section.
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Figure 4 Comparisons of nondimensional static pressure distribution on blade surface at 50% of blade span. (a) Impeller full blade; (b) diffuser blade; (c) return channel Blade.
blade the pressure close to the concave is lower than that close to the convex due to the flow acceleration close to the concave (Figure 3(a)). Compared between Model 1, Model 2 and Model 4, the significant differences in the quantities of static pressure are observed although the trends of static pressure distribution are very similar. Compared with Model 1, Model 2 has a great improvement of the accuracy to predict the static pressure on the blade surface. But the differences between the predicted results from Model 2 and Model 4 are remarkable in particular on the blade surfaces of diffuser and return channel. Through the flow analysis and comparison, it shows that the gas models have important effects not only on the stage performances but also on the flow structure in the centrifugal refrigeration compressor stage. Model 2 can be used to predict the stage’s performance with acceptable accuracy and the difference between the results from Model 2 and Model 4 is about 2%. But Model 2 can be misleading in predicting some basic flow parameters, such as the pressure and the temperature. This leads to poor evaluations of flow structures in the stage. Therefore, it is necessary to account for the real gas effects more exactly in the CFD analysis using the real gas state equation or thermodynamic property tables to improve the prediction accuracy of flow structures in the centrifugal refrigeration compressor stage.
6 Conclusions In this paper the perfect gas models with different definitions and the real gas model based on the WANG ZhiHeng et al. Sci China Ser E-Tech Sci | Aug. 2008 | vol. 51 | no. 8 | 1160-1168
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gas property tables are respectively used in the 3D CFD analysis of a centrifugal compressor stage with the evolving fluid HFC-134a. The effects of the different gas models on the stage’s performance and flow structure are compared and analyzed. It can be concluded that the gas model of the refrigerant adopted in the CFD codes has significant effects on the stage’s aerodynamic performances and the flow structures in the stage. Although the evolving fluid in the centrifugal refrigeration compressor is far from perfect gas, the perfect gas model with the gas constant modified and the specific heat varying with the temperature can be used to predict the stage’s performance with acceptable accuracy. But this may lead to poor prediction of the basic flow parameters, and then errors in the flow structure in the stage. For these reasons, the real gas effects should be effectively considered in the CFD analysis to accurately predict the flow structure in the centrifugal refrigeration compressor and improve its design reliability. Nomenclature Rg : gas constant
e: specific internal energy
γ : specific heat ratio
T : temperature
h: specific enthalpy
ε : total pressure ratio
Z : compressibility factor
k: thermal conductivity
η : efficiency
cp: specific heat at constant pressure
p: static pressure
μ : dynamic viscosity
cv: specific heat at constant volume
s : specific entropy
ρ : density
Subscript 0: stagnation or start condition
1,2: inlet, outlet condition
is: isentropic
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