9. Pompa Vana Kompressör Kongresi 5-7 Mayıs 2016, İstanbul MATHEMATICAL MODELING OF PERFORMANCE OF A LIQUD PISTON COMPRESSOR Süleyman Doğan Öner Email: oner@ug.bilkent.edu.tr İbrahim Nasuh Yıldıran Email: ibrahim.yildiran@ug.bilkent.edu.tr Okan Deniz Yılmaz Email: okany@ug.bilkent.edu.tr Barbaros Çetin Associate Professor Email: barbaros.cetin@bilkent.edu.tr ABSTRACT The compressed air produced by compressors has been using in many different industries, and the efficiency of a compressor is important regarding the both the economical and environmental aspects. In a conventional compressor, a solid piston compresses the air. During this compression process, an air leak occurs which diminishes the performance of a compressor. The leakage becomes even greater as the operation time increases. Moreover, the compression process of the solid piston performed at a relatively high speed (several hundred rpms) to minimize the air leakage. Since the gas does not have enough time to have enough heat transfer, compression at high speed increases in the gas temperature, which leads to an inefficient compression process. One alternative to overcome this issue is to use liquid piston in which the solid piston is replaced by the liquid rising in a column. This way, both the air leakage can be avoided and the compression process can be performed slower. Moreover, due to the circulating nature of the water, and the superior heat transfer characteristic of the flowing water, a better heat transfer can take place which leads to a lower temperature rise of the gas, and hence more efficient compression process. With a more efficient compression process, the environmental foot print of the process can be minimized. In this study, a mathematical model to predict the performance of a liquid piston is developed. The liquid piston considered in this study includes also small metal tubings inside the main cylinder to enhance the heat transfer between the liquid and the compressed gas. The performance of the liquid piston compression is presented as a function of number of tubes packed in the main cylinder. NOMENCLATURE m mass (kg) A heat transfer area (m 2 ) A c cross-sectional area (m 2 ) C Sutherland constant (120 K) h convective heat transfer coefficient k thermal conductivity of the gas (W/m K) C v specific heat capacity (kj/kgk) L axial extent of heated region (m) Nu Nusselt number, hl char /k Re Reynolds number, V L char /µ p Pressure (Pa) P Wetted perimeter (m) Pr Pranl number, µ/α volume of the gas (m 3 )
t tube wall thickness (m) D i inner tube diameter (m) D o outer tube diameter (m) D pc outer diameter of the cylinder (m) T temperature (K) T w wall temperature (K) T o reference temperature (K) U internal energy (kj) V Average velocity (m/s) f friction factor W f frictional work (J) W compression work (J) E stored energy (J) Greek ν kinematic viscosity ρ density µ o reference dynamic viscosity η efficiency η comp compression efficiency ε surface roughness Subscript c.v. control volume g Gas INTRODUCTION Pressurized air is commonly used in industry. Compressors with reciprocating pistons are preferred to increase pressure of air. These compressors are using mechanical power to move the metal piston in a cylinder in which the pressure of air is increased. At the end of compression process, the air is delivered to a storage tank through valves. Then, another valve is opened to suck atmospheric air in to the cylinder. This cycle runs over and over again till the filling of storage tank with air at desired pressure. During the compression process, an air leak occurs. To minimize the amount of air leak, the compression process should be performed at a relatively high speed (typically several hundred rpms). Although increased piston velocity prevents gas leakage up to some extent, it also restricts the amount of heat transfer from the pressurized gas to the environment. Hence, temperature of air is increased. This situation hinders to obtain pressurized air with high efficiency, because some amount of work delivered by the motor is used to increase temperature of the gas [1]. In addition to this, when air is cooled to room temperature in storage tank, its pressure decreases significantly. As an alternative technology, the replacement of the solid piston with a liquid piston has been suggested [2]. In a liquid piston compressor, water, oil or some another appropriate liquid can be pumped into a cylinder to pressurize the air. When desired pressure is maintained, the check valve let the air be transferred to storage tank. There are major improvements coming with liquid piston compressor. The most significant advantage of this new concept is to prevent gas leakage inherently by the nature of the liquid [3]. Since the air is not allowed to leak, the cycle can be operated slower then conventional compressors which allows the heat transfer to take place which will decrease the temperature rise during the compression process. Hence, the efficiency of compressor is increased by using the power delivered by the pump almost only to increase the pressure of air. Moreover, inclusion of some metal inserts can further enhance the heat transfer between the pressurized air and the water, hence the efficiency of the compression process [4]. Therefore, liquid piston compressors serves as a more environmentally friendly alternative compared to conventional compressors by supplying the same amount of air with desired pressure with a less power. In this study, a mathematical model to predict the performance of a liquid piston is developed. The liquid piston considered in this study includes also small metal tubings inside the main cylinder to enhance the heat transfer between the liquid and the compressed gas. The performance of the liquid piston compression is presented as a function of number of tubes packed in the main cylinder. The polytropic constant, compression efficiency, total efficiency and the total work per stroke of the compression process is determined as a function of number of tubes inserted in the main cylinder. THEORY Taking the control volume as the gas inside the cylinder (see Figure ), by neglecting kinetic and potential energies, first law of thermodynamics reads as: du c.v. The term on left hand side can be written as: du c.v. = C v T g dm c.v. = Q Ẇ (1) + m c.v. C v dt g where T is the temperature of the gas inside the cylinder at time t. The stroke of a piston composed of two phases. In Phase-I, only the compression process takes with a closed valve. When the desired critical pressure (P c ) is reached the value opens and the compressed gas is transferred out of the cylinder at constant pressure, P c (Phase-II). The rate of change of the mass inside the control volume can be written as: dm c.v. 0 t < t c (Phase-I) = ρa c ẋ P c dt g R g Tg 2, t > t c (Phase-II) (2) (3)
x(t) Valve T g (t) P g (t) 8(t) 8 Control volume T g (t + t) P g (t + t) 8(t + t) H o The second term on the right hand side of Eq. 1 describes the rate of work done on the control volume and can be calculated as: Ẇ = P g d Moreover, the temperature, pressure and the density of the gas is related to each other through the Ideal Gas Law at any instant: (7) Water Water P g = ρr g T g (8) FIGURE 1. Time = t Time = t + Schematics of the cylinder with metal tubes where t c is the time when the critical value is reached. The first term on the right hand side of Eq. 1 indicates that convective heat transfer between the gas and solid walls which are assumed to be kept at constant temperature of T (neglecting heat transfer occurring the at the interface of the water and the gas): Q conv = ha(t T s ) (4) To estimate the heat transfer coefficient, the gas flow within the cylinder can be modeled as a fully-developed pipe flow, and the Nusselt number can determined as [1]: Nu = hl char k = are m Pr n (5) where a, m and n are constant which depend on the flow regime. Within the operational temperature range of the process, the thermo-pyhsical properties of the gas is temperature dependent. However, considering only the kinematic viscosity as a function of temperature following Sutherland s law ν = µ 0 ρ T 0 +C T +C ( ) 3/2 T (6) T o the heat transfer coefficient can be determined by the following constants with Pr = 0.7 [1]: a = 0.664, m = 1/2, n = 1/3 laminar flow (Re < 2300) a = 0.023, m = 0.8, n = 0.3 turbulent flow (Re > 2300) t Eq. (1) can be integrated numeically with the following initial conditions: T g = 300 K at t = 0 (9) P g = 1 bar at t = 0 (10) o = A c H o at t = 0 (11) To determine the compression efficiency, the energy stored in the compressed gas needs to be calculated. The stored energy equation can be stated as [1]: E = P c f inal ln p c p atm (12) The work delivered to the gas during compression can be determined once the area under the P- curve of compression process is calculated numerically. Then the compression efficiency, η comp, can be determined as: η comp = o 0 E P g d (13) For the calculation of the total efficiency, the frictional work supplied to generate the liquid flow up needs to be calculated together with the work required for the compression. The frictional work can be obtained by multiplying the pressure drop with the rate of change of the volume of the water: o W f = Pd (14) 0
Similarly, the calculation of the frictional work can also be modified to take into account the flow inside the tubes and the outside the tubes. FIGURE 2. Schematics of the cylinder with metal tubes where pressure drop can be determined as: P = f H o x ρẋ 2 L char 2 where f represents the friction factor and can be obtained as: (15) 64/Re (for laminar flow) 1.325 f = [ ( ε ln + 5.74 )] 2 (for turbulent flow) (16) 3.7L char Re 0.9 Once the pressure drop is calculated, the total efficiency which takes frictional losses into account can be stated as follows; η = E W +W f (17) So far, the mathematical model has been developed for a liquid piston compressing an air in the cylinder. The compression efficiency of the process can be further increased by introducing metal tubes inside the cylinder. Once the model tubes are introduced, the heat transfer takes place both in the inner surface of the tubes and the outer surface of the tubes. The heat transfer occurring at the inner surface can be taken into account by calculating the appropriate heat transfer area (the heat transfer coefficient is same for each tube). For the outer surface, the heat transfer coefficienct can be determined by defining the hydraulic diameter as follows (refering to Fig. ): D h = A c P w = A co N t π(d 2 o D 2 i )/4 N t πd o (18) TABLE 1. Number and Dimensions of Tubes D i t D 0 1.8 3.8 2000 4.8 6.8 600 RESULTS AND DISCUSSION The compression process can be characterized as, PV n = C where n being the polytropic constant. The process has two limits. One limit is the reversible and adiabatic (i.e. isentropic) case, where n becomes the k (=1.4 for air) being the specific heat ratio. The other limit is the isothermal limeit where the n becomes unity. Isothermal case allows the highest heat transfer and requires the minimum work input for the compression process. The compression process can be modeled mathematically as discussed previously. In this section, the polytropic constant, work input per stroke, total process time to fill a 80 L air tank, compression and total efficiency of the process are presented as a function of number of tubes inserted into the cylinder and the different flow rates. The orientation of the tubes inserted into the cylinder is depicted in Fig. 2. The geometric parameters used in the analysis is given in Table 1. Two different diameters for the tubes are used. Depending on the diameters of the tubes, the maximum number of tubes that can stacked int he cylinder differs. The maximum number of tubes for a closely packed condition are also given in the table. Fig. 3 shows the polytopic constant as a function of number tubes and the flow rate. Two limiting cases are also indicated on the figure. It is clear that when number of tubes are increased, the compression process is getting closer to the isothermal case since the heat transfer area increases. As the flow rate decreases, the process is also getting closer to the isothermal case which is quite expected since the process getting slower and slower gas Polytropic constant (n) 1.4 1.3 1.2 1.1 FIGURE 3. Isentropic limit Isothermal limit Polytropic constant of the compression process
Total time [s] 100 10 Compression Efficiency ( η comp ) 0.9 0.8 0.7 0.1 2000 FIGURE 4. Total time to fill 80 L air tank 0.6 # of Tube (a) Total Work [J/stroke] 1750 1500 1250 1000 Total Efficiency (η) 0.9 0.8 0.7 750 500 0.6 (b) FIGURE 5. Total work per stroke of the compression process FIGURE 6. Efficiency of the compression process can have a better heat transfer which leads to lower gas temperature during the process. This fact is also true for conventional compressors. If the process gets slower and slower, the process would be closer to the isothermal case. However, in conventional compressors, the slow process is not option due to the gas leakage occurring during the compression process. On the other hand, the gas leakage is not an issue for a liquid piston compressor. However, slower process also means that the time duration for the filling of the tank would be higher. Fig. 4 shows the total time required for the filing of the 80-L tank. For the tubes with larger diameter, the filling process take longer, since the amount of gas compressed is smaller. Moreover, as expected, the total time increases as the flow rate decreases. To assess the performance of the compression process, the total work required per stroke is illustrated in Fig. 5. As expected for the cases in which the polytropic constant is getting closer to unity, the work required also decreases. The lower the flow rate the less is the required work. Of course, there should be an optimum point depending on the application. If the slower process is acceptable, the compression can be performed with a minimum total work. However, if the process time is critical, depending on the desired process time, a different set of input parameters can be configured. The results presented in Figs. 4 and 5 enables an optimum selection for the input parameters depending on the application. Fig. 6 (a) shows the compression efficiency of the process, and Fig. 6 (b) shows the total efficiency of the process. As seen
previously, slower the compression process higher the compression efficiency. Similar trend is also true for the total efficiency except for the solid red curve in Fig. 6 (b). The trend is similar since work accounted for the frictional losses is not the dominant one. However as the flow rate increases, and more and more tubes are presented, now the area of the gas decreases which results in higher velocities, and the frictional losses dominates. This is the reason for the solid red curve has a diminishing total efficiency with increasing number of tubes. near isothermal liquid piston air compressor in a compressed air energy storage system. American Control Conference, pp. 2934 2939. [4] Zhang, C., Simon, T. W., and Li, P. Y., 2013. Optimization of the axial porosity distribution of porous inserts in a liquid-piston gas compressor using a one-dimensional formulation. Heat Transfer and Thermal Engineering. CONCLUSION Liquid piston compressor is an alternative technology which aims to increase efficiency by preventing gas leakages and high heat transfer rates. Heat transfer is favorable since it allows to keep temperature of air constant at a certain level, hence power delivered by the pump is consumed almost only to pressurize the air. This is accomplished by decelerating the compression. Conventional compressor should operate at high frequencies in order to minimize the gas leakage. Since leakage is not a concern in liquid piston compressor, the cycle may operate at low frequencies. This enables the system to work with increased heat transfer from air to the surroundings which leads to a process being close to an isothermal one. A mathematical model has been developed and presented in this paper which assess the effect of different design parameters on the process. It has been shown that introducing metal tubing inserts into the cylinder, the process becomes more and more closer to being an isothermal one. One limiting case for the cylinder with the metal inserts is the total process time. Depending on the nature of the process, the design parameters can be optimized to obtain the desired performance within the certain limit of process time. In this study, only the compression process and the discharge of the compressed gas is modeled. The modeling of the expansion step and the inclusion of the heat transfer occurring at the metal tube walls will be our future research direction. ACKNOWLEDGMENT Financial support from the Turkish Scientific and Technical Research Council (Project No: 1139B411501759) is greatly appreciated. REFERENCES [1] de Ven, J. D. V., and Li, P. Y., 2009. Liquid piston gas compression. Applied Energy, 86(10), pp. 2183 2191. [2] C. Qin, E. L., 2014. Liquid piston compression efficiency with droplet heat transfer. Applied Energy, 114, pp. 539 550. [3] Shirazi, F. A., Saadat, M., Yan, B., Li, P. Y., and Simon, T. W., 2013. Iterative optimal and adaptive control of a