# The Precharge Calculator

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2 Notes on Gas Behavior The chart at the right shows just how much difference there is between the ideal gas law and real nitrogen. The chart was obtained by taking a certain mass of nitrogen and forcing it to occupy a certain volume at a certain temperature. Given the mass, volume and temperature, the pressure was calculated from the ideal gas law and compared to the actual pressure listed in the NIST tables 4. The difference between the two was plotted as a percent error according to the formula: Error% = 100*(P calc -P NIST ) / P NIST. Error % N 2 - Pressure using the Ideal Gas Law (10.00) (20.00) (30.00) (40.00) (50.00) (60.00) (70.00) Temperature (K) 1 Atm 2 Atm 50 Atm 75 Atm 100 Atm 150 Atm 200 Atm 300 Atm 700 Atm 1000 Atm 1200 Atm You can see that the error is less than ±10% for pressures below 200 atm (3,000 psia) (except at very cold temperatures). However, there is a dramatic increase in error at 300 atm (5,000 psia) and by 700 atm (10,000 psia) the calculated pressure is 30% to 40% too low. At higher pressures the error is even worse. Clearly, trying to use the ideal gas law at pressures over 5,000 psia (345 bara) will result in unacceptable errors. All systems for water deeper than 5,000 ft (1,500 meters) are in this range. Helium is much closer to ideal than nitrogen and therefore performs better than nitrogen at high pressures. This can allow the use of fewer accumulator bottles in a given high pressure application. However, even though it is closer to ideal, helium still has large deviations from ideal behavior at high pressures. As seen in the plot to the right, at 900 atm (13,000 psia) the calculated pressure using the ideal gas law is 25% to 35% low when compared to the NIST tables. Clearly, the ideal gas law has problems with helium as well as nitrogen. Error % He - Pressure using Ideal Gas Law (5.00) (10.00) (15.00) (20.00) (25.00) (30.00) (35.00) (40.00) Temperature (K) 1 Atm 50 Atm 75 Atm 100 Atm 150 Atm 200 Atm 300 Atm 700 Atm 900 Atm A Nearly Perfect Gas Law The equations of state used in The Precharge Calculator are among the best that exist for nitrogen and helium. They deviate from published NIST tables by only 0.005% over the entire temperature and pressure range. And according to NIST, their tables are normally well within 0.6% of the real gas. By starting with such an accurate equation of state, adiabatic discharges are normally well within 0.1% of carefully solved problems using the NIST tables when solving for pressure and useful fluid. Similarly, the calculations are normally far more accurate than the uncertainty in your input data 5. 3 Windows is a registered trademark of Microsoft Corporation. 4 See footnote 1 and the NIST Webbook. 5 Of course, regardless of the accuracy of the mathematical model, any real design must be supported by careful field tests to insure that the real accumulator performs like the model predicts before being put into service. Revised 9 February 2005 Page 2 of 5

3 Comparison with real tests Interlink Systems, Inc. was invited to participate in tests of accumulators under conditions designed to simulate operation at about 10,000 ft water depth. The results were written up in detail and presented at the 2001 Offshore Technology Conference 6. The tests covered pressures up to 10,000 psia using both nitrogen and helium. Some of the test results are repeated here. Figure 1 A test case using nitrogen. The volume of fluid in the accumulator was recorded directly as a function of time, as were the pressure and temperature of the gas. The fluid volume was used as input into the Precharge Calculator, which computed pressure and temperature, both as functions of time. The Precharge Calculator then created the graphs shown here. As you can see, the agreement for nitrogen is very good. A similar set of graphs follows for helium. They show excellent agreement between the calculations and the experiment. Note that these simulations include heat transfer over time. Figure 2 A Test case using helium. 6 Mathematical Prediction and Experimental Verification of Deep Water Accumulator Capacity, Craig A. Good and James P. McAdams, Offshore Technology Conference, Houston Texas, OTC 13234, May Revised 9 February 2005 Page 3 of 5

4 It is impossible to get all of the peaks and dips correct when using standard, simple models based on the ideal gas law. Ignoring heat flow also can introduce significant errors, depending on the situation. For instance, the following graphs show the effect of using the simple, adiabatic, ideal gas formula PV 1.4 = k in the nitrogen example above (Figure 1). At first glance, the computed pressure curve does not look too bad. However, note that even the slowly increasing pressures are over 750 psi low and are only OK after settling for 20 minutes or more. Also, upon discharge the actual pressure falls over 1000 psi below the pressure predicted by the simple model. Finally, the extreme inaccuracy in the temperature curve indicates that there is something badly wrong with the simple model. In fact, there is no way to always guarantee that the simple model will get even this close to the right answer. Figure 3 Errors introduced by a simple, adiabatic, ideal gas model. An Example Problem Using The Precharge Calculator Suppose we want to put an accumulator bank in 9,500 ft of water. It will be part of a hydraulic system in which the pumps on the surface normally supply 5,000 psi. The hydraulic fluid used has a specific gravity of when measured at the temperature of the seawater. The HPU is located about 50 ft above sea level. The accumulators will be precharged on the surface at 90 F, but will operate on the ocean floor at 4 C. Each bottle has a gas volume of 9.4 liters (when there is no fluid present). The functions being operated require a total of 25 gallons of fluid. That 25 gallons will be discharged over a period of 35 seconds and the pressure must remain at least 2600 psi above the sea hydrostatic pressure. The customer specifications demand that, when isolated from the main supply line, the accumulator bank must be designed to deliver 1.1 times the required amount of fluid. 1) How much fluid can each bottle supply, and how many bottles are necessary in the bank? 2) What is a good precharge pressure to use, as measured at the time the bottles are charged? The following is the actual Precharge Calculator output for the problem above. The left column is input data. Data was exported in rich text format and then imported directly into Word. Revised 9 February 2005 Page 4 of 5

5 Comments This is an illustration of the output from the example problem in the Precharge Calculator Features sheet. Precharge Calculator Example Capacity Calculations Initial Pressure + Head --- Calculated Results --- Fluid Head 9, ft Computed Head Pressure Specific Gravity Initial Static Pres 4, psi Initial Gauge Pres 5, psi Min Static Pres 4, psi Min Pressure + Head Fluid Head 9, ft... at Operating Conditions Specific Gravity Precharge 7, psi Min Gauge Pres 2, psi Initial Gas Vol % Gas Max Density 0.50 g/cc Precharge Conditions Empty Volume 9.40 L Discharged - Dynamic Pressure 9, psi Useful Fluid 0.22 GAL Temperature F Remaining Fluid 0.00 GAL Precharge Gas Nitrogen Table Final Pressure 6, psi Temperature C Operating Conditions Initial Pressure 9, psi Discharged - Isothermal Min. Pressure 6, psi Useful Fluid 0.22 GAL Minimum Fluid 0.00 L Remaining Fluid 0.00 GAL Ambient Temp C Final Pressure 7, psi Starting Temp C Computed Bank Size Dynamic Conditions Required Volume GAL Time constant sec Bottles Needed 126 Discharge Time sec Resulting Volume GAL Gas Needed 15,247 cu-ft Required Volume of Fluid Desired Volume GAL Safety Factor Notes: Unless otherwise noted, all pressures are absolute. Gas Needed is measured at STP (0 degrees C and 1 atm) Interlink Systems, Inc. Precharge Calculator 1.308/1 11/19/2004 9:30:03 PM C:\Precharge\Manual\DataSheetExample.pcg Revised 9 February 2005 Page 5 of 5

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