USING AN ALL-SPEED ALGORITHM TO SIMULATE UNSTEADY FLOWS THROUGH MIXED HYDRO-PNEUMATIC SYSTEMS

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1 HEFAT4 th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics 4 6 July 4 Orlando, Florida USING AN ALL-SEED ALGORITHM TO SIMULATE UNSTEADY FLOWS THROUGH MIXED HYDRO-NEUMATIC SYSTEMS Masoud Darbandi *, Amir A. Beige, MohammadAli Akbari and Gerry E. Schneider * Author for Correspondence Center of Excellence in Aerospace Systems, Department of Aerospace Engineering, Sharif University of Technology,.O.Box , Tehran, Iran Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, NL 3G, Canada darbandi@sharif.edu ABSTRACT In this paper, e take to important steps to analyse hydropneumatic systems and their components. In this regard, e first extend a ne all-speed approach to solve the full onedimensional Navier-Stokes equations. The extended approach can be equally used to simulate both gas and liquid flos, considering an equivalent gas constant for the latter case. We further apply the extended approach to analyse both liquid ater and air flos through to separate hydro-pneumatic circuits. To verify the accuracy of our solution, e compare our solutions ith those of Lax-Wendroff method. The comparison shos that the current proposed all-speed algorithm provides solutions ith excellent accuracy. One important advantage of this ne extended algorithm is in treating the to-phase fluid flo problems, here there are mixed liquid ater and vapour behaviours, e.g., diaphragm pumps. Therefore, as a second contribution, e simulate a multi-component diaphragm pump, hose parts consist of check valves, signal subsystems, conveys, and air distribution system. As a final step, e obtain the limit of cavitation temperature at the suction side of the check valve for a sample diaphragm pump. INTRODUCTION High pressure hydro-pneumatic systems need to be constructed ith high quality materials. These systems carry different flo rates and tolerate high pressures during their operations. To carry out careful simulations for such important systems, it is required to consider a multi-element frameork ith sufficient accuracies and capable of modeling the transient flo through very complex components at various gas/liquid flo conditions. Evidently, an efficient all-speed algorithm can predict the transient behavior of a hydro-pneumatic system ithout the need to apply for to individual or separate compressible and incompressible solvers and ithout the need to sitch beteen them. It should be mentioned that the meaning of lo-mach number flo solution is totally different from the current all-speed algorithm, hich solves pure incompressible flo in addition to different compressible flo cases using one unique algorithm. Indeed, simulation of high pressure hydro-pneumatic systems is typically a difficult job because of the structural and functional complexities associated ith such systems []. As an example, a diaphragm pump is a multi-component hydropneumatic system consisting of several simpler hydraulic and pneumatic parts, exhibiting unsteady behaviour due to its moving diaphragms. This system can be also subject to the risk for cavitation and a serious reduction in its life performance and endurance. As is knon, the cavitation phenomenon can normally occur hen the vapour pressure of the process fluid reaches the orking chamber pressure. If this happens, the subsequent cavitation (e.g., ear, pressure shocks) ould heavily reduce the pump s performance and lifetime. Since the numerical analysis of cavitation is often faced ith crucial instabilities; due to using different numerical methods to achieve the interaction beteen to compressible and incompressible phases, the use of an all-speed method for both liquid and gas parts can greatly improve the multi-phase flo solution stability in treating complex hydro-pneumatic systems. Simulating a diaphragm pump as a black box and ithout imposing the details of unsteady behaviour of its moving components ould result in some disadvantages, e.g., ignoring some important mechanical behaviours like pressure pulsations, hich can dramatically affect the correct modeling of a real diaphragm pump. Since e simulate a complete diaphragm pump, e can find the cavitation temperature limit for it and can consequently extend suitable instructions to avoid it. Additionally, by modeling a complete diaphragm pump, e ould be readily able to take into consideration the unsteady behaviour of any hydro-pneumatic sub-components, hich should be taken in modeling. 9

2 In this paper, e extend the simulation capabilities of an existing multi-element numerical frameork and apply it to analyse a high pressure hydro-pneumatic circuit. In this regard, e first extend an all-speed flo solver capable of solving both liquid ater and air flos through different hydro-pneumatic assemblies. Next, e compare the achieved solutions ith those of Lax-Wendroff method. It should be noted that since the hydro-pneumatic systems deal ith both gas and liquid phases simultaneously, the classical numerical methods can only be effective in solving certain flo regimes. Hoever, an all-speed method, hich can simultaneously solve both gas (compressible or incompressible) and liquid (incompressible) phases ithout any extra considerations, ould help to achieve a more stable convergence procedure and reduce the solution time for the entire system. It is because e avoid sitching beteen to compressible and incompressible solvers during the solution procedure. Eventually, e use our extended allspeed algorithm and simulate the flo behaviour through a real super-component diaphragm pump. Then, e sho ho e can avoid cavitation by specifying a orking temperature limit for the suction side of the pump s check valve. NOMENCLATURE A [ ] ipe cross-sectional area A e [ ] Effective area C [-] Courant number D [m] ipe diameter dx [m] Displacement of diaphragm middle point dv [ ] Volume change e [ / ] Specific energy F [kg/( )] Wall friction k [-] Linearization constant k [-] Linearization constant p [kg/( )] ressure [m] ipe perimeter r [-] randtl number Q [kg/( )] Heat transfer from all Re [-] Reynolds number t [s] Time T [K] Wall temperature u [m/s] Velocity x [m] osition in pipe t [s] Time step x [m] Cell size ε [m] ipe roughness λ [ / ] Heat conductivity λ T [-] Turbulent friction factor [kg/ ] Density SUBSCRITS-SUERSCRITS-ACCENTS e [-] east o [-] old [-] Wall or est overbar [-] Lagged from previous iteration COMUTATIONAL MODELING The one-dimensional flo governing equations are given by ( u) + = t x ( ) ( ) u u p + = F t x x ( e) ( ue + pu) + = Q t x We choose the finite volume method to treat these equations. Darbandi and Schneider [,3] developed an all-speed algorithm in hich the discretized governing equations ere simultaneously solved for pressure, temperature, and momentum component ( u) variables. Comparing ith the orks of other investigators, ho normally used the velocity component as the primary dependent variables in their algorithms, they shoed that the use of momentum component ( u) ould result in several important advantages including a strong flo analogy creation beteen compressible and incompressible formulations. This enabled them to use any incompressible flo procedure to solve compressible flos [4]. Additionally, the momentum component ould provide more simplification in linearization procedure [5], and a more effective suppression of oscillations for flos passing through discontinuities []. The finite volume treatment of flo equations ould lead to x( p p ) + ( u) e ( u) = t x ( u) ( u) + ue ( ue) u( u) + pe p = () t x ( E) ( E) + ( ue) e ( ue) + ( pu) e ( pu) = t in hich the flo cross section area is taken unity. Since the density variable is considered as a secondary unknon in our pressure-based algorithm, the transient term in the discretized continuity equation needs suitable linearization ith respect to our chosen unknon variables. We employ a Taylor series expansion to consider active roles for both pressure and temperature terms as follos [3]: + ( ) + ( T T ) T here () (3) = = RT T RT (4) here the variables ith an over bar indicate that they are approximated from the previous iteration. 9

3 In current method, the nonlinear convection terms are linearized ith respect to the momentum component ( u), hich is a primary dependent variable in our algorithm. One sophisticated linearization scheme suggests [6]: uu k u ( u) k u (5) here k and k are to arbitrary constants. If k = k =, it results in the Neton-Raphson linearization scheme as follos: density is inserted in the ideal gas equation (see Figure ) to find equivalent gas constant for liquid ater condition. Although this constant is not realistic, it is considered in our formulations to simulate liquid ater flo in the same ay as for gases. Indeed, the computation time for solving the liquid ater EOS to obtain an equivalent gas constant is small comparing ith the total time required to solve the linear algebraic system of equations. u( u) u( u) + ( u) u uu (6) If k = and k =, it ould yield a simple linearization as follos: uu u ( u) (7) The flo variables are interpolated at the cell faces using the physical influence scheme (IS), hich utilizes the flo governing equations to derive the integration point expression. Using the IS, the momentum component and temperature are derived at cell faces as follos: C ( A / A + / ) C ( u) ( u) ( u) ( p p ) e e e e e e = + E Ce ue ( + 4 Ce ) + 4Ce (8) C C p ( u) ( u) ( T) Te ( ) ( + ) + e e e E e T ( ) Ce u ecv Ce E Ce (9) here the Courant number is defined as C = u t / x in this formulations. Also, the pressure term is linearly interpolated at the control volume cell faces. Regarding the spatial accuracy, the current formulations perform second-order accurate in lo Reynolds number flos. Hoever, at very high Reynolds numbers, it reduces to a firstorder scheme. Also, the current method is first-order accurate in time. It is orth to emphasize that the accuracy of the present numerical solution is excellent in spite of using a first-order scheme to treat high Reynolds number flos even on coarse grid distributions. It is because the use of inclusive cell-face expressions provides strong couplings beteen the pressure and velocity fields and minimizes the false diffusion in the domain [7]. Using the above described method of discretization, liquid incompressible flos can be simulated ithout adding any artificial compressibility. The detail algorithm for incompressible flo treatment has been shon in Figure. As is seen, the Jeffery and Austin [8] equation of state for liquid ater is solved numerically in this process using the Neton- Raphson interpolation scheme. This provides the density as a function of pressure and temperature. This equation of state ould be much more accurate than an equivalent simple cubic equations of state over ide ranges of pressure (.-3 bar) and temperature (-34- C) fields, hich are typically encountered in hydro-pneumatic systems. The derived value of Figure The procedure to solve liquid ater flo using the current all-speed algorithm The friction losses are considered in the current formulations using the Zigrang-Sylvester friction factor estimation. It is given by F = λ V /( d) () here T ε / D.5 ε.5 = log + [.4 log ( )].9 λ 3.7 Re D Re T () We also use some specific relations to compute the convection heat transfer coefficient and the corresponding heat transfer from the alls using [9]: h.8.33 =.3 Re r λ/d () Q = h( T T) / A (3) 9

4 A double diaphragm pump as a super-component system is modeled in this paper. It is consisted of four parts of diaphragm, signal system, conveys, and check valves. Figure presents a schematic representation of the current chosen double diaphragm pump. To model the mechanical aspects of this pump, it is assumed that no net mechanical force is exerted on the diaphragms and that the diaphragm motion is described through an effective area folloing specific relations beteen the volume change and the diaphragm middle point displacement. This relation is given by flo through the diaphragm pump considering the diaphragm structure interaction. The effective diameter corresponding to the effective area is assumed to be constant ( cm) in this ork. It is also obvious from the pump performance maps (e.g., see Figure 3) that the pump outlet pressure ould be a function of air distribution system flo variables. In order to simulate this behaviour, variable loss orifices are added to the double diaphragm pump model to simulate pressure drop in a ay that it is consistent ith its real behaviour. dv = A e(x) dx (4) Figure 3 erformance map for the simulated double diaphragm pump [] To model the unsteady characteristics of the simulated pump, the frequency of diaphragm motion is computed assuming that the sept volume per stroke does not change during the pump operation. So, the pump frequency is a linear function of its flo rate in our modeling. Figure Schematic of the investigated double diaphragm pump and its equivalent hydro-pneumatic circuit RESULTS AND DISCUSSION To test cases are considered to evaluate the current method in treating both gas and liquid flos. A schematic of the investigated air (pneumatic) system is shon in Figure 4. 3 Source, S, charges a 4m chamber, i.e., C, to a pressure of Bars. The velocity and pressure magnitudes are computed along pipe using the described all-speed finite-volume algorithm. The velocity and pressures have been plotted along pipe length in Figures 5 and 6, respectively. We have also used the second-order Lax-Wendroff method to verify the current solutions [,]. The comparisons sho that the current method has been accurate enough to present reliable solution in this test case. This is here e have not applied any artificial numerical damping into our algorithm. This is here the second-order Lax-Wendroff method needed such side considerations. Our experience shoed that using different friction calculations and different discretization types ould lead to some discrepancies in our results. The effective area can be estimated either using experimental observation or the finite element modeling of the 93

5 75 t=.4 s 7 65 Figure 4 Schematic of the investigated pneumatic system 3 Mass Flo Rate (Kg/s) All-Speed Algorithm, t=3. s Amesim Lax-Wendroff, t=3. s All-Speed Algorithm, t=.4 s Amesim Lax-Wendroff, t=.4 s 5 45 Current All-speed algorithm, grid points LMS AMEsim, grid points Current All-speed algorithm, 4 grid points LMS AMEsim, 4 grid points Velocity (m/s) X/L Figure 5 Velocity distribution in a gas line and comparison ith that of the Lax-Wendroff method at to different time levels X/L Figure 7 Evaluating the mass conservation through pneumatic line using the current method and that of a commercial softare The second test case is to simulate the liquid ater flo through a pipe ith an inflo pressure of 87. Bars, a temperature of 8 K, and imposing different outlet pressure magnitudes. Figure 8 shos the velocity distribution along this pipe considering different pressure outlet conditions. The current results are compared ith the solutions derived from the LMS Amesim softare. The agreement beteen them is excellent..8 x 7 5 ressure(pa).6.4. Velocity(m/s) LMS Amesim (Lax Wendroff), ε/d=., =84.6 bar Current Algorithm, ε/d=., =84.6 bar LMS Amesim (Lax Wendroff), ε/d=., Current Algorithm, ε/d=., LMS Amesim (Lax Wendroff), ε/d=., Current Algorithm, ε/d=., All-Speed Algorithm, t=3. Amesim Lax-Wendroff, t=3. All-Speed Algorithm, t=.4 Amesim Lax-Wendroff, t= Figure 6 ressure distributions in a gas line and comparison ith that of the Lax-Wendroff method at to different time levels Figure 7 verifies that the current algorithm fully conserves the mass flo through circuit. We can achieve a more accurate mass conservation results by increasing the number of grid nodes from to 4; hoever, the finite difference Lax- Wendroff method does not sho any change in its mass flo distribution ith more grid refinements x(m) Figure 8 Velocity distributions in a liquid line and comparison ith that of the Lax-Wendroff method Figures 9 and present the results for a simulated double diaphragm pump including its outlet pressure and the volume flo rate ith time, respectively. The results presented in these figures ere obtained for air pressure of 6.9 bars and an air flo rate of 4 l/sec. As is seen in this figure, there are some oscillations near the mean value found in the pump manufacturer performance map (see Figure 3) for the specified air flo rate and pressure. 94

6 ressure after diaphragm check valve ressure after diaphragm check valve erformance Map Average ressure ressure after suction check valve Vapor ressure at T=4 C Vapor ressure at T=6 C Vapor ressure at T=8 C 4 ressure (Bar) ressure (bar) Cavitation Time (s) Figure 9 Variation of the simulated diaphragm pump outlet pressure ith respect to time and comparison ith the pump performance map data Volume Flo Rate (L/min) Volume Flo Rate of Diaphragm Volume Flo Rate of Diaphragm erformance Map Average Volume Flo Rate Time (s) Figure Variation of the simulated diaphragm pump volume flo rate ith respect to time and comparison ith the pump performance map data As is knon, the check valves of diaphragm pumps are subject to different pressure magnitudes during their pumping cycles. This can lead to a cavitation appearance inside the pump. The phase change from liquid to gas can harm the pump [3]. In this ork, e predict the fluid thermodynamic properties, hich can be used to specify the cavitation point at donstream of the suction valve. Figure illustrates the time history of pressure variation behind the pump suction check valve. Inspecting this figure, it is obvious that the pump ould experience the cavitation condition, if the inflo temperature exceeds 8 o C. That is because the valve local pressure ould reduce under the saturation vapour pressure at the achieved temperature Time (s) Figure Calculating the cavitation temperature limit at the suction check valve of the simulated diaphragm pump CONCLUSION We developed an all-speed algorithm to treat to-phase flos through mixed hydraulic-pneumatic systems. The formulation as extended for solving the one-dimensional full Navier-Stokes equations. As as expected, the extended method and formulation ould be able to treat both liquid and gas flos as ell as mixed to-phase flos. To prove this, e used this method and solved a number of test cases including a diaphragm pump. This method helped us to take into consideration all moving parts in the diaphragm pump assembly in our simulation. Therefore, e could model the diaphragm pump and its subcomponents at different diaphragm pump operating conditions. Our numerical results shoed that it as possible to model both air and ater phases in the same ay. Indeed, the current numerical method can help to analyse the hydro-pneumatic systems ithout the need to apply for to different compressible and incompressible solvers. The use of to different solvers can lead to inappropriate convergences during the fluid flo solutions. The current results shoed that the extended finite-volume method ould perform better mass conservation in comparison ith the Lax-Wendroff finitedifference method. We used our extended solver and simulated a diaphragm pump. We ere able to predict the cavitation condition limit for this pump using thermodynamic variables. In future, e ill be able to predict the cavitation behaviour by covering gas-liquid interactions, using the current extended finite-volume method. ACKNOWLEDGMENTS The authors ould like to acknoledge the grant received from the Deputy of Research and Technology in Sharif University of Technology. This financial help is highly acknoledged and appreciated. REFERENCES [] Mobley R. K., Fluid oer Dynamics, Nenes,. 95

7 [] Darbandi M. and Bostandoost S.M., A ne formulation toard unifying the velocity role in collocated variable arrangement, Numerical Heat Transfer B 47, 5, pp [3] Darbandi M. and Schneider G.E., Analogy-based method for solving compressible and incompressible flos, Journal of Thermophysics and heat Transfer, Vol., 998, pp [4] Darbandi M. and Hosseinizadeh S.F., General pressure-correction strategy to include density variation in incompressible algorithms, Journal of Thermophysics and Heat Transfer, Vol.7, 3, pp [5] Darbandi M. and Schneider G.E., Comparison of pressure-based velocity and momentum procedures for shock tube problem, Numerical Heat Transfer, art B 33, 998, pp [6] Darbandi M., Roohi E., and Mokarizadeh V., Conceptual Linearization of Euler Governing Equations to Solve High Speed Compressible Flo Using a ressure-based Method, Numerical Methods for artial Differential Equations, Vol. 4, No., 8, pp [7] Darbandi M. and Vakilipour S., Developing implicit pressureeighted upinding scheme to calculate steady and unsteady flos on unstructured grids, International Journal for Numerical Methods in Fluids, Vol.56, 8, pp [8] Jeffery C.A., Austin.H., A ne analytic equation of state for liquid ater, J. Chem. hys., 999, () pp [9] INEL,, RELA5/MOD3 Code Manual, Models and Correlations NUREG/CR-5535, Rev-Vol, vol. IV. [] ARO pump performance curves, Ingersoll Rand Company Ltd,. [] Scavarda S., and Bideaux E., A neumatic Library for AMESim, roceedings of ASME International Mechanical Engineering Conference and Exhibition, IMECE'98 Anaheim, California, Nov. 998 pp [] LMS Amesim Hydraulic Library Manual, LMS Imagine.lab,. [3] Vetter G., Depmeier L., and Schubert W., Design and Installation Conditions of Diaphragm umps for High-ressure and Supercritical Fluids, The Journal of Supercritical Fluids, Vol. 5,99,pp