VacTran 3 Manual, страница 38

The general formula for linear interpolation isFor our example, Y is the interpolated speed, and Given X is the average pressure for the increment. Note thatsince the pressure tends to vary by orders of magnitude, linear interpolation will give poor results. In order to weightthe interpolation on a log scale, all values are converted to log, interpolated, and converted back again. Therefore,our interpolation formula becomes:For example, for increment 1,= 55.6 liters/secondInterpolated pumps speeds for each increment are summarized below:incrementpressureincrementsaveragepressure1E+03pumpspeedinterpolatedspeed5011E+025.50E+027555.621E+015.50E+0110080.831E+005.50E+00150111.141E-015.50E-01150150.051E-025.50E-02100135.061E-035.50E-035083.571E-045.50E-041032.9Using HandCalc.VTSIS, we plot Pump speed vs pressure.© 2011 Professional Engineering Computations378VacTran 3The graph if this data is shown below:© 2011 Professional Engineering ComputationsHand calculating a pump down curve37919.1.4 Calculate delivered speedStep 4) Calculate delivered speedThe pumping speed of the pump at the vacuum vessel, considering the loss of the conductance, is called thedelivered speed.The formula for delivered speed iswhereS= Delivered pump speed at chamber in liters/secondSp = Pump speed at pump inlet in liters/secondC= total conductance connecting pump to chamber in liters/secondUsing the data for our pump and the calculated conductance data for our orifice, we plug into the above formula toobtain the following:incrementavgpressureconductanceinterpolatedpump speeddel.

speed15.50E+02168486.055.655.625.50E+0116856.080.880.435.50E+001693.0111.1104.345.50E-01176.7150.081.355.50E-0225.1135.021.865.50E-039.983.59.675.50E-048.432.97.3© 2011 Professional Engineering Computations380VacTran 3Try this calculation using thesystemmodelexample,HANDCALC.SIS,andgraphDelivered speed vs. pressure.

Thefollowing data is shown in the MainText Window:© 2011 Professional Engineering ComputationsHand calculating a pump down curve381Effect of changing number of incrementsThis graph shows both the speed at the pump for each pressure increment, along with the delivered speed at thevessel.

Delivered speed will always be less than pump speed because of conductance losses.© 2011 Professional Engineering Computations382VacTran 3You may notice a difference in the shape of the pump speed curve from the original curve that was entered. Thisdifference is due to the interpolated values that were calculated for each pressure increment. If we choose moreincrements, the curve gets smoother. The curve below is the same as before, but with 50 increments instead of 7.© 2011 Professional Engineering ComputationsHand calculating a pump down curve38319.1.5 Calculate pump down timeStep 5) Calculate pump down timeFinally, we are able to calculate pump down time for this simple vacuum system. For a system with no gas load,the time to pump down a given increment of pressure ist =PVx ln iSPwhereV= volume in litersS= delivered pump speed in liters/secondPi= initial pressure in TorrP= final pressure for increment in TorrAt each increment, the value of ln (Pi/P) = ln(10) = 2.3.For the first increment,t = 100 liters/55.5 liters/second x 2.3= 4.1 secondsPump down calculations are summarized below:incrementavgpressuredel.

speedincrementtimeelapsedtime15.50E+0255.54.14.125.50E+0180.42.97.035.50E+00104.32.29.245.50E-0181.12.812.055.50E-0221.110.522.665.50E-039.623.946.575.50E-047.331.678.1© 2011 Professional Engineering Computations384VacTran 3This system will pump down in 78.1 seconds with no gas load, as shown in the graph plotted using the Pumpdown time command on the Graphs menu.© 2011 Professional Engineering ComputationsHand calculating a pump down curve19.2385The role of the gas loadThe calculations in the previous section are straightforward without considering gas loads. For vacuum systemsthat operate at relatively high pressures, where gas loads usually have less affect, an unloaded system calculationmay be all that is necessary.

However, if one does not perform the calculation, an assumption that gas loads arenegligible can be a poor one.The above flow diagram demonstrates the addition of a gas load source. This source can come from within thevessel, such as surface outgassing from the vessel walls, or from outside the vessel through permeation or leaks.Schematically, the above diagram combines all gas load sources into a single input stream that must eventually bepumped by the vacuum system in order to achieve the desired pressure. Assume that the valve shown in thediagram offers no impedance to the flow of gas into the vacuum vessel, but serves only as a visual aid to showwhether the gas load is active.With the addition of the gas load, the vacuum pump now has two main sources of gas to evacuate:1) The original volume of gas in the vessel when pumping began2) The additional gas load that continuously tends to raise the pressure.

From the flow diagram, one can see thatflow out through the vacuum pump must be greater than flow in through the gas load in order to achieve a lowerpressure. It is also apparent that with the additional gas load, pump down time will always be greater than thatwith the "clean" vacuum system.© 2011 Professional Engineering Computations38619.3VacTran 3Hand calculation steps - with gas loadExample: outgassing of the vessel surfacematerial: fresh aluminumIn the previous example, a vessel volume of 100 liters was given.

Assume that this represents acylindrical vacuum vessel with the following rough dimensions:height:radius:50 cm25 cmtotal surface area = 2 P r h = 7850 cm2rate of outgassing is given byOutgassing rate (Q)qnt nfor time < 1 hour, Q is in torr - liters/secondfor time >1 hour and <10 hoursfor time > 10 hoursQ10 h o u r s3600010x Q 10 x tnwhere t is in second unitsand 3600 seconds = 1 hour, 36000 seconds = 10 hoursQ is in torr - liters/secondfor fresh aluminum,Q1 = 6.3e-9 torr-liters/second/cm21= 1Q10 = 6e-10 torr-liters/second/cm210 = 1The nature of the out gassing equation is such that the gas load will continually decrease over time, butwill never become zero.

Therefore, a decision must be made as to the time extent to calculate the gasload. Since our example system pumps down to the desired pressure in about 80 seconds, 3600seconds would be a reasonable amount of time to calculate gas load.Consistent with previous calculations, assume that seven increments will be used for the system pumpdown calculation with the gas load. Based on the above formula, the following gas load values arecalculated at seven time increments up to 3600 seconds).Time increments are divided on a log scale.

For time increment 1,Tim e 10log(3600) / 7 = 3.22 secondsGas loads for our aluminum vessel are summarized as follows:© 2011 Professional Engineering ComputationsHand calculating a pump down curveincrementtimegas load135.53E-022101.72E-023335.33E-0341081.65E-0353475.13E-04611181.59E-04736004.95E-05Using the HANDCALC.VTSYS example,and selecting Gas load vs. time from theGraphs menu, gives the following resultsin the Main Text Window. The numberscorrespond to the hand calculationsshown previously.© 2011 Professional Engineering Computations387388VacTran 3The corresponding graph for HANDCALC.VTSYS:© 2011 Professional Engineering ComputationsHand calculating a pump down curve38919.3.1 Calculate pump down time with gas loadThe gas-loaded calculation is identical to the ideal pump down calculation, except that the gas load adds additionalvolume to the initial volume that has to be pumped out.

In other words, add the gas load volume generated duringeach increment. Base the gas load value for each increment on the elapsed time of the previous increment. Dividethe gas load by the average pressure for the increment to obtain liters/second, and then multiply by the time of theprevious increment to get liters.The results are as follows:avg pressuredel.speedincrem entclean tim eelapsed cleanGas loadtim etorr-liters/secGasquantityAddedvolum eAdded Tim efrom gas loadtotal tim etorr-liters55055.64.14.14.15580.42.97.04.30E-021.78E-013.24E-030.007.05.5104.32.29.22.54E-027.28E-021.32E-020.009.20.5581.32.812.01.93E-024.27E-027.76E-020.0012.00.05521.810.522.61.48E-024.19E-027.61E-010.0822.70.00559.623.946.57.85E-038.34E-021.52E013.6350.20.000557.331.678.13.55E-039.76E-021.78E0256.0137.8Where the values come from:Gas Load (torr-liters/second): Log interpolation of gas load data, based on the previous increment "total time"value.Gas quantity (torr-liters): Product of gas load and previous increment's increase in "Total time" valueAdded volume: Gas load divided by average pressure for incrementt =Added time from gas quantity:PVx ln iSPusing Added volume from gas loadTotal time: Previous total time + increment clean time + added time from gas load© 2011 Professional Engineering Computations390VacTran 3This system pumped was calculated to pump down in 78.1 seconds without a gas load.

With the aluminum surfaceoutgassing added to the model, pump down will now take about 138 seconds.The graph below is our example, after selecting the Pump down time command from the Graphs menu. Pumpdown curves with and without gas load are graphed together.© 2011 Professional Engineering ComputationsHand calculating a pump down curve391The following data is the associated text window output.

Note that the values from VacTran match thehand calculation.Sy s t em Gas Load v s Ti me:Sec onds , Tor r - Li t er s / Sec ond1) 3. 2249893E+00, 5. 5267458E- 022) 1. 0380949E+01, 1. 7156405E- 023) 3. 3433082E+01, 5. 3257781E- 034) 1. 0769298E+02, 1. 6532550E- 035) 3. 4691300E+02, 5. 1321180E- 046) 1. 1175338E+03, 1. 5931381E- 047) 3. 6000000E+03, 4. 9455000E- 05Ti me, Ves s el pr es s ur e:Sec onds , Tor r1) 4. 1463789E+00, 1. 0000000E+022) 7. 0091908E+00, 1.

0000000E+013) 9. 2176245E+00, 1. 0000000E+004) 1. 2048597E+01, 1. 0000000E- 015) 2. 2593106E+01, 1. 0000000E- 026) 4. 6499220E+01, 1. 0000000E- 037) 7. 8064908E+01, 1. 0000000E- 04Pr es s ur e v s Ti meSec onds , Tor r1) 4. 1463789E+00,2) 7. 0092835E+00,3) 9. 2180094E+00,4) 1. 2051179E+01,5) 2. 2675958E+01,6) 5. 0208630E+01,7) 1. 3781079E+02,( us i ng gas l oad) :1.1.1.1.1.1.1.0000000E+020000000E+010000000E+000000000E- 010000000E- 020000000E- 030000000E- 04© 2011 Professional Engineering Computations392VacTran 319.3.2 Effect of number of incrementsBy using small number of increments, the accuracy of the preceding calculations was compromised.

Setting thenumber of increments to a low number such as 7 is only useful for checking hand calculations. In real situations,choose at least 50 to 500 increments. The 50-increment calculation is shown below. Notice the dramaticdifference between this graph and the previous one.The following graph shows how the number of increments can change the results for this problem, and when toexpect convergence on results.