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2 Compressible Fluid Flow: Pressurizing and Discharging of Tanks Under Adiabatic and Isothermal Conditions 06-363 Transport Process Laboratory Carnegie Mellon University May 22, 2006 Group 6: Alan Abel Phil Lowe Rachel Clair Daqian Wu 2 Abstract Models describing the isothermal or adiabatic filling and discharge of a tank were analyzed. Two tanks were used in this experiment; the larger was 7 gallons, and the smaller was 1.1 gallons. The smaller one was used to test the adiabatic model and the larger was used to test the isothermal model. Due to possible malfunctions in the pressure transducers, poor response times in the pressure transducers, and/or inadequate data-sampling rate the experimental data could not be used to prove or disprove the models in question. However first hand observations did qualitatively confirm the accuracy of the models tested. 1 Table of Contents Introduction 1 Theory 1 Experimental 3 Results 4 Discussion 6 Conclusion 7 References 7 Appendix A 8 Appendix B 9 Appendix C 11 1 Introduction This experiment was set up to examine the relationship between pressure and discharge rate through a needle valve. It allows us to examine the phenomena of choked flow which is when the gas velocity becomes independent of downstream pressure. However by changing the upstream pressure one can increase the mass flow rate out of the nozzle. This is important in industry because allows manipulation of mass flow with the use of valves and orifice plates under choked flow. Our setup allowed us to examine both isothermal and adiabatic discharge conditions and data was collected using Labview software. Theory Discharge To simplify our analysis of the flow we made several assumptions about the behavior of the gas in the tank: 1. The gas molecules operate under isentropic thermodynamic conditions. 2. The discharge of the gas can be either characterized as adiabatic or isothermal. 3. There is ideal isentropic flow from the tank to the nozzle throat. Assumption 2 relies on the time allowed for the discharge to occur. If the gas exits very rapidly there is not enough time heat transfer to occur between the tank walls and the gas. Therefore we modeled the adiabatic exit case with a small tank of about 5 liters. A large tank, 21 liters in our case, provides sufficient space and discharge time for temperature to be stabilized and allows examination of isothermal discharge. Under our assumptions of adiabatic and stagnant gas, we can apply isentropic thermodynamic conditions to our analysis of the gas flow in the small tank. We can derive pressure difference as a function of time by starting from mass flow rate and simplify it to Fleinger’s formula. dm vA (1) dt With the stagnant gas conditions we assumed, the mass balance becomes: 1 dm 2 1 0.5 APo ( ( ) ) (2) dt RTo 1 In the case of air this can be simplified to Fliegner’s formula: dm P 0.04042 A o (3) dt To This can then be substituted into a continuity equation to solve for a relationship between pressure and time. udV (4) compressible flow v dV (5) incompressible flow t V In equations (4) and (5) u is the velocity of the gas and V is the volume of the gas. Equation (3) gives us the initial instantaneous flow rate and by combining it with either (4) or (5), depending on the operating conditions, we can derive a relationship between pressure and time. 2 ( 1) 2 1 1 2 ( 1) 1 P [1 ( )( ) t ] (6) choked adiabatic 2 2 ( 1) 1 P exp[ ( ) 2( 1) t ] (7) choked isothermal 2 P P (8) Pi t t (9) t char V t char (10) At ai Once enough gas has exited the nozzle, the flow becomes unchoked and we can transfer our analysis into the unchoked region by applying a similar derivation as we did for the choked region. We would use the unchoking points ( Punchoke, t unchoke) as our initial conditions and occurs when the static nozzle pressure equals atmospheric pressure. Equation (11) and (12) show the relationship between dimensionless time and pressure over the unchoked portion for adiabatic and isothermal conditions, respectively. ( 1) 2 0.5 At x 3 5x 3 t t unchoke ( ) ( PB ) 2 {( )( x 2 1) 0.5 ln[ x ( x 2 1) 0.5 ]}xunchoke (11) 1 Ae x 4 8 8 2 2 0.5 At ( xunchoke x 5 ) 2( xunchoke x 3 5 3 t t unchoke ( ) [ ) ( xunchoke x)] (12) 1 Ae 5 3 1 P x [( ) 1]0.5 (13) PB PB (backpressure) PB (14) Pi Charging The analysis in this case the almost the same as that performed for the discharging part, expect that we can no longer assume isentropic behavior in the gas. There is some kinetic energy in the inlet gas and it has to be accounted in our analysis. This is done by including it in the definition of the dimensionless definitions of time and pressure. P P (15) Ps 3 V t char (16) As a s For equations (15) and (16), s denotes the conditions in the source. Choked filling can occur if the difference in the pressure between the tank and the source is great enough. ( 1) 1 P Pi ( ) 2 ( 1) t (17) choked adiabatic 2 ( 1) 1 2 ( 1) P Pi Ti ( ) t (18) choked isothermal 2 To derive the unchoked we perform a similar procedure as mentioned for the discharging phase. The only difference is that we now set the nozzle pressure equal to the tank pressure at unchoking, instead of atmospheric. 1 1 Ae P {1 [(1 ( P unchoke ) 0.5 ) ( ) 0.5 (t t unchoke)] 2 } 1 (19) 2 At 1 1 A P {1 [(1 ( P unchoke ) 0.5 ) ( ) Ti e (t t unchoke)]2 } 1 0.5 (20) 2 At Equations (19) and (20) are solutions for unchoked adiabatic and isothermal, respectively. Experimental A diagram of the apparatus used for the experiment is shown below. 4 The equipment used includes two vessels, the first with a volume of 7 gallons and the second with a volume of 1.1 gallons. Each vessel is equipped with a release toggle valve on top, connected to a thermocouple. The thermocouple allows the computer to collect and record the temperature changes during discharging with Labview software. Also on the top of each tank, is a safety release valve; this valve will open if the pressure inside the tank exceeds 150 psi. On the front of each tank is a three-way valve, used for the charging experiment. The three positions of the valve are used as follows: the first, to allow air to flow into vessel from the air compressor, the second, to calibrate the pressure inside the vessel, and the third, to vent the gas inside the vessel to the atmosphere. A pressure transducer is located next to the three-way valve to record the pressure inside the vessel throughout the experiment. Data was collected from several trials using both the 7 gallon vessel and the 1.1 gallon vessel. Data was collected for both charging and discharging the vessels. Several trials on each vessel allowed for the compilation of plenty of data to create accurate conclusions. Labview software recorded the experimental information and plotted a graph for each successful run. Data was also recoded by hand. To ensure the most accurate accumulation of experimental information, environmental conditions were measured and recorded to keep the data consistent. Results Figure 1 is a plot of tank pressure vs. time for filling of the large tank (isothermal). Similar plots for a data set of each operation (filling and discharging for both size tanks) can be found in Appendix B. Pressure vs. Time, Big Tank Charging 120 100 80 Pressure (psi) 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Time (s) Figure 1: pressure vs time, isothermal charging large tank 5 From a mathematical analysis of the charging and discharging tanks, the tank pressure was calculated and plotted versus time. An example of this is shown in Figure 2. 8 10 5 6 10 5 P 1( t ) 5 P 2( t )4 10 2 10 5 0 0 1 2 3 4 5 t Figure 2: plot of calculated pressure vs. time for isothermal filling Of note in the figure is the separation of flow regimes. On the above plot, the red region of pressure values indicate the region of choked flow, whereas the green section represents the region of unchoked flow. Calculated pressure values were then compared to measured pressure values at the corresponding times during the different flow regimes. It is important to note, however, that during the region of unchoked flow for tank discharge, only the time can be calculated as an explicit function of pressure, not vice versa. This prevents any possible pressure comparison for that section of data. The calculated time required to charge the small tank was so short, however, that our equipment could not effectively collect enough data points during that time. The complete results of the pressure error comparison can be found in Appendix B. The error for the isothermal tank charging is shown in Figure 3: 6 t pmeas pcalc % error 0.521 15.375 32.08 52.07294 1.052 15.592 49.793 68.68636 1.222 15.652 55.463 71.77938 1.572 16.055 67.1 76.07303 2.093 16.142 83.465 80.66016 2.624 16.587 97.32 82.95623 2.964 16.965 103.943 83.67855 3.325 17.118 108.621 84.24062 measured tfinal: 77.201 calculated tfinal: 3.767 % error: 1949.403 Figure 3: Error amounts of measured pressure versus calculated pressure in isothermal tank charging The most striking aspect of the measured pressure data is the significantly slower response than expected. For the pressurizing of the large tank, the measured final time is approximately 20 times longer than the calculated time. Discussion The results clearly show that the models employed do not agree with our data for any of the filling or draining experiments. There are three possible causes of this. 1) The pressure transducers were broken. 2) The pressure transducers could not respond fast enough to the pressure change. 3) The rate at which the computer acquired the data was too slow. The 15 psi pressure transducer was used in the initial leak tests and was certainly broken when it was exposed to 93 psi. This renders all of the filling and draining experiments conducted with this sensor suspect. The 150 psi transducer is, based on the model, not providing accurate answers either. It is not very likely that this sensor was broken during testing or any subsequent experiment since the highest pressures used were on the order of 100 psi. It is, however, likely that the response time of this sensor or the data-sampling rate of the computers was not fast enough to accurately record what was taking place in the tanks during filling or draining. The predictions of the time required for draining or filling of the tanks by the various models previously presented was confirmed by observations made on the tanks during the experiments. Specifically the time it took for the small tank to drain from 93 psi to atmospheric pressure was determined by listening to the air escaping from the tank. The time required for most of the air to escape was recorded on a stopwatch. This time was found to be on the order of 1 second, which agrees closely with the time predicted by the adiabatic tank-draining model. This type of observation compared well with what was predicted by the model in each case of tank filling and draining. It is then reasonable to conclude that the model for each type of flow is accurate. 7 Conclusions The tank draining and filling models presented in this report are accurate The pressure transducers and data acquisition equipment need to be checked to make sure they are capable of operating at a high enough speed to give accurate results in the data References: 1) J. Craig Dutton and Robert E. Coverdill. “Experiments to Study Gaseous Discharge and the Filling of Vessels.” Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign. Full-text available at http://www.ijee.dit.ie/articles/999982/experime.htm 8 Appendix A: Nomenlature Symbol/abbrev. Meaning Units m mass kg P pressure PSI Ps source pressure PSI Pb back pressure PSI Pi initial pressure PSI ρ density kg/m3 v gas velocity m/s V tank volume gal T temperature K Ti initial temperature K Ts source temperature K γ heat coefficient ratio dimensionless Ae nozzle exit area m2 At nozzle throat area m2 ai initial sonic velocity of gas in tank m/s 9 Appendix B: Pressure-time Data Pressure vs. Time, Small Tank Discharge 120 100 80 Pressure (psi) 60 40 20 0 0 5 10 15 20 25 30 35 40 Time (s) Pressure vs. Time, Small tank charging 120 100 80 Pressure (psi) 60 40 20 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) 10 Pressure vs. Time, Big Tank Discharge 120 100 80 Pressure (psi) 60 40 20 0 0 10 20 30 40 50 60 Time (s) Pressure vs. Time, Big Tank Charging 120 100 80 Pressure (psi) 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Time (s) 11 Appendix C: Sample Calculation for tend J v 26.498 L M 29.1 r 287.05 Ti 295.81K 1.4 kg K m ai 340 s Ts 295.81K 2 d .283in A t d t char v A t ai Ti 2 Tiplus Ts Ae At tchar 1.92 s 5 2 At 4.058 10 m Ps 110.7psi t Pi t plus ( t ) Pi 14.7psi t char Piplus Ps Punch Ps .528 Piplus 0.133 choked isothermal charge: Punch 58.45psi Punch ( 1) Pplusunch 1 2( 1) Ps Pplus ( t ) Piplus Tiplus tplus ( t) 2 Pplusunch 0.528 Pplusunch Piplus tplusunch ( 1) 1 2( 1) Tiplus 2 tplusunch 0.683 tunch tplusunch tchar P( t) Pplus ( t) Ps tunch 1.312 s 1 2 1 2 1 1 2 1 T At Pplusunchoked ( t) 1 1 Pplusunch 2 iplus A tplus ( t) tplusunch 2 e 12 1 Pplusunchoked( t )0.5 0 0 5 10 t Guess g 3s Given Pplusunchoked ( g) 1 x Find ( g) x 3.767 s tend x

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