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Ind. Eng. Chem. Res. 2008, 47, 1304-1309
Quantifying Leak Rates in Gas Handling Systems C. W. Extrand,* Don Ware,† John Kirchgessner, and Jon Schlueter Entegris, 3500 Lyman BouleVard, Chaska, Minnesota 55318
Gas-handling systems, subassemblies, and components constructed from metals and plastics alike need to be tested for pressure loss and leakage. In this study, the concept of conductance or characteristic leak rate has been further developed for pressurized scenarios and applied to a model system. Conductance was measured from both pressure-decay experiments and from constant pressure-flow experiments using three different gases at three different temperatures. Results from the two types of tests were compared using an analysis developed as part of this study. Conductance values from decay and steady-state measurements were in general agreement. Thus, in the evaluation of real components, subassemblies, or systems, conductance can be estimated from a simple pressure-decay measurement and then used to estimate steady-state performance. Introduction The number and quality of connections can be an important factor in gas loss. Imperfections in seals or connections allow gas-handling systems to leak. It is relatively easy to estimate gas-loss or leak rates via pressure decay, where a system is charged with a gas, closed, and the decline of pressure is monitored with time. If the inner volume of the system is known, one can calculate a characteristic leak rate or conductance value.1 However, in many cases, gas-handling systems are not operated in a closed fashion but remain pressurized at some constant or steady state value. If the leak rate is very low, it is difficult to directly quantify the loss of steady-state systems, for example, by means of gas-flow meter. Thus, it would be advantageous to be able to measure a conductance value by a pressure-decay measurement and then use that conductance value to estimate steady-state performance.2 Whereas much has been published on leak testing, most of that work has focused on vacuum systems.1,3-7 Leak testing of pressurized systems has received much less attention. There are semiempirical models that allow estimation of conductance values from pressurized systems, but comparison to constant pressure (i.e., steady state) measurements is difficult and limited to very small differential pressure between the gas-handling system and the environment.8,9 In this study, the concept of conductance, or characteristic leak rate, has been further developed for pressurized scenarios and applied to a model system. Analysis. Starting from fundamentals,10 an expression can be derived that describes the decay in pressure (P) for a closed gas-handling system as a function of time (t),
[ ( ) ( )]
Po -1 Pf C 1 P) exp - t Pf Po V
-1
(1)
where Po is the initial pressure, Pf is the final or ambient pressure, V is the volume of the system, and C is a characteristic gas-loss or leak rate that is sometimes referred to as conductance. An expression for the steady-state volumetric loss rate (Qc) can * To whom correspondence should be addressed. E-mail:
[email protected]. † Current address: ATMI, 10851 Louisiana Avenue South, Bloomington, MN 55438, USA.
be derived by differentiating eq 1 and then manipulating it with the ideal gas law.
Qc ) C
( ) Po -1 Pf
(2)
Conductance values are temperature dependent. If the gas in the system behaves ideally, then its conductance values can be reduced to a standard value (CS),
CS ) C
( ) P f TS PS T
(3)
where T is temperature and TS is a standard reference temperature (TS ) 25 °C ) 298 K).1 This model does not include the influence of adiabatic heating, viscous flow, pressure-induced volume change of the component, and so forth. A full derivation of eqs 1-3 is given in the Appendix. Experimental Details The experimental setup, shown in Figure 1, consisted of a model gas-handling component, a low-flow metering valve (Swagelok S Series, stainless steel, vernier dial, SS-SVCR4) to allow the system to leak at a controlled, reproducible rate, a series of pneumatically actuated bellows valves (Swagelok BK Series, stainless steel, normally closed) to charge or purge the model system, a pressure transducer (MKS 870B Baratron single-ended absolute pressure transducer, 100 psi capacity, Model 870B12PCD2GA1), and a gas-flow meter appropriate for the steady-state gas-loss rate (MKS Alta digital mass flow controller with maximum flow rate of 10 cm3/min, Model 1480A11CR1 µB or a Gilian Gilibrator 2 with a low-flow bubble generator for flow rates of 1-250 cm3/min or with a standardflow bubble generator for flow rates of 20-6000 cm3/min). The gases used in these measurements, helium, nitrogen, and oxygen (Industrial Grade, Toll Co., Minneapolis, MN), were introduced from gas cylinders (“T” size) via pressure regulators. The model gas-handling component was constructed from stainless steel (SmartWeld). It was cylindrical (approximately 9 cm long and 7.5 cm in diameter) with an ASA vacuum end flange on one end that was sealed by a copper gasket. Connections were made with VCR flanges to minimize uncontrolled leaks. The internal volume (V) of the assembled model system (model gas-handling component, valves, transducer, and connections) was 379.5 cm3.11
10.1021/ie071118s CCC: $40.75 © 2008 American Chemical Society Published on Web 01/23/2008
Ind. Eng. Chem. Res., Vol. 47, No. 4, 2008 1305
Figure 2. Pressure (P) versus time (t) for nitrogen gas at 30 °C. Decay curves for three different initial pressures. Po ) 31, 60, and 95 psi.
Figure 1. Experimental setup used for both pressure decay and steadystate measurements.
To establish the flow rate of leaked gas, it was imperative that all of the gas exited the system through the metering valve. Therefore, the model system was tested for unwanted leaks (i.e., other than through the metering valve) by charging with He gas to 100 psi, closing all of the valves, and monitoring the pressure for 3 days. No significant pressure drop was observed (10 psi/ hour), approaching Pf in just a few hours. Similarly, during steady-state measurements, gas leaked readily through the open metering valve. Thus, in both the decay and steady-state experiments, we could confidently conclude that all of the gas exited the system through the open metering valve, allowing accurate measurement of the leak rate from the system. The model system, with the exception of the gas cylinder, regulator, and gas-flow meter, was placed inside a temperature chamber. The chamber is depicted in Figure 1 as a rectangular box. The metering valve was opened to a specific setting and then locked for all of the experimental data given in this article. Measurements were made with initial pressures (Po) ranging from 30 to 90 psi (absolute values) at three temperatures (T), T ) 30, 60, and 90 °C. The final pressure (Pf) for all of the measurements was atmospheric pressure, Pf ) 14.7 psi. The ambient temperature (Tf) in the room was 25 °C. Data acquisition and control were performed remotely with a personal computer. T and Po also were monitored over the duration of the experiment to ensure their constancy. Pf values were checked before and after each test run. The uncertainty in the pressures, temperatures, volume, and time were estimated to be: P ) Po ) Pf ) (0.5 psi, T ) Tf ) (1 °C, V ) (3 cm3, Qf ) Qc) (0.25 cm3/min, and t ) (0.02 min ((1 s). Error in calculated values of P, Qf, Qc, C, and CS were estimated using standard error propagation techniques.12 Decay Tests. To perform a decay test, the metering valve was open and the purge valve was closed. The experimental setup was heated to the appropriate temperature and charged with one of the gases. Once the desired initial pressure (Po) was reached, the charge valve was closed to start the test and acquisition of the time-dependent pressure (P) was initiated. Gas leaked from the system through the metering valve and P decreased with t. Decay measurements were made with and without the flow meter attached to the metering valve. However, this had no effect on the decay rates. Plots of P versus t were constructed and eq 1 was used to extract a decay-based C value.
Figure 3. Conductance coefficients (C) from decay measurements versus initial differential pressure (Po - Pf) for nitrogen gas at 30 °C.
Steady-State Tests. To perform a steady-state test, the metering valve was left open and the purge valve was closed. The experimental setup was heated and charged with a gas. Once the desired initial pressure (Po) was reached and the system had equilibrated, the charge valve was left open to ensure that gas was supplied at a constant pressure from the gas cylinder and regulator. Gas leaked from the system through the metering valve to the gas-flow meter. The volumetric flow rate (Qf) was read from the flow meter at Pf. Values of Qf, Pf, Po, and T were recorded. The volumetric leak rate depends upon the pressure at which the measurement was made. Because the flow meter was outside the oven at ambient pressure and temperature (Pf ) 14.7 psi and Tf ) 25 °C), Qc values were adjusted to match the pressure (Po) and temperature (T) inside the model component using the following expression:
Qc ) Qf
( ) Pf T P o Tf
(4)
Equation 2 was used to calculate steady-state C values from measured values of Po, Pf, and Qc. Results and Discussion Decay Tests. Figure 2 shows pressure (P) versus time (t) for nitrogen gas at 30 °C. Three different initial pressures were used, Po ) 31, 60, and 95 psi. Once the desired Po value was attained, the charge valve was closed. Gas leaked from the system through the metering valve. With the passage of time, pressure decayed toward Pf ) 14.7 psi. The greater the value of Po, the faster the initial decay rate. The circles are experimental data; the solid curves were calculated with eq 1 using C values of 2.8 cm3/ min for 31 psi, 2.0 cm3/min for 60 psi, and 1.6 cm3/min for 95 psi. The measured and estimated decay curves generally agreed well. Figure 3 shows conductance coefficients (C) from decay measurements versus initial differential pressure (Po - Pf) for nitrogen gas at 30 °C. The symbols shown in Figure 3 represent the curve fits from the three decay curves shown in Figure 2.
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Ind. Eng. Chem. Res., Vol. 47, No. 4, 2008
Figure 4. Steady-state volumetric loss rate (Qf) versus differential pressure (Po - Pf) from steady-state measurements made with nitrogen at 30 °C.
Figure 5. Conductance (C) vs differential pressure (Po - Pf) for decay and steady-state measurements made with nitrogen at 30 °C. The filled symbols are from decay measurements; the open ones are from steadystate measurements.
Conductance coefficients decreased with pressure, falling from 2.8 to 1.6 cm3/min as Po - Pf increased from 16 to 80 psi. Steady-State Tests. Figure 4 shows steady-state volumetric loss rate (Qf) versus differential pressure (Po - Pf) from steadystate measurements made with nitrogen at 30 °C. During these experiments, the charge valve is open, Figure 1. Therefore, at a given Po - Pf value, gas leaked from the system through the metering valve at a constant rate of Qf. Qf increased with Po Pf. At each differential pressure, Qf was used to calculate a steady-state C value. These C values from the steady-state measurements are compared to those from the decay measurements in the next section. Comparison of Conductance Values. Figure 5 shows conductance (C) versus differential pressure (Po - Pf) for decay and steady-state measurements made with nitrogen at 30 °C. The filled symbols are from decay measurements; the open ones are from steady-state measurements. Within experimental error, estimates of conductance values from the steady-state measurements were the same as those from decay. This suggests that conductance values estimated from pressure-decay measurements can indeed be used to estimate steady-state performance. Conductance values for different temperatures and gases are discussed in the two following sections. Temperature Dependence of Conductance Values. Figure 6 shows conductance (C) versus differential pressure (Po - Pf) for nitrogen measured at two temperatures (T). The filled symbols are from decay measurements; the open ones are from steady-state measurements. For a given pressure differential, conductance coefficients increased slightly with temperature. The increase in conductance values appeared to be proportional to the absolute temperature, following Charles’ Law. This is not surprising because nitrogen is expected to behave ideally under our test conditions. Equation 3 was used to construct a
Figure 6. Conductance (C) vs differential pressure (Po - Pf) for nitrogen measured at two temperatures (T). The filled symbols are from decay measurements; the open ones are from steady-state measurements.
Figure 7. Standard conductance (CS) vs differential pressure (Po - Pf) for nitrogen measured at three temperatures (T). The filled symbols are from decay measurements; the open ones are from steady-state measurements.
Figure 8. Standard conductance (CS) vs differential pressure (Po - Pf) for all three gases measured at three temperatures (T). The filled symbols are from decay measurements; the open ones are from steady-state measurements.
standard conductance (CS) curve for the various temperatures, employing a standard reference temperature, Ts ) 273 K, as a basis. Figure 7 shows standard conductance (CS) versus differential pressure (Po - Pf) for nitrogen measured at three temperatures (T). The filled symbols are from decay measurements; the open ones are from steady-state measurements. The slight pressure dependence of the CS values can be most simply described by a linear relation.
CS ) CS,0 - k(Po - Pf)
(5)
The solid line represents the linear fit of the nitrogen data from all three temperatures as described by eq 4, where the slope or conductance-pressure factor is k ) 1.6 × 10-2 standard cm3/min‚psi and the intercept or zero-pressure conductance is CS,0 ) 2.9 standard cm3/min. Comparison of the Various Gases. The other two gases, helium and oxygen, behaved similarly. Figure 8 shows standard conductance (CS) versus differential pressure (Po - Pf) for all
Ind. Eng. Chem. Res., Vol. 47, No. 4, 2008 1307 Table 1. Intercepts (CS,0) and Slopes (k) for the Various Gases. CS,0 Gas
(std
He N2 O2 overall
cm3/min)
k (std
2.65 ( 0.10 2.92 ( 0.10 2.18 ( 0.08 2.60 ( 0.08
cm3/s)
0.044 ( 0.002 0.049 ( 0.002 0.036 ( 0.001 0.043 ( 0.001
three gases. The filled symbols are from decay measurements; the open ones are from steady-state measurements. By viewing the entire data set on a single plot, it appears qualitatively that three gases fall on the same curve. Data sets for each gas (that included both decay and steady-state measurements) were examined quantitatively using linear regression. Table 1 lists intercepts (CS,0) and slopes (k) from eq 4. Even though all three gases are expected to behave ideally under the test conditions employed here, CS,0 and k values differed slightly for each gas. Overall averages of CS,0 and k were intermediate to the values of the individual gases. The solid line in Figure 8 represents the linear fit of all three gases measured at all three temperatures. It is important to note that the conductance values given here are specific to the particular loss or leak rate of the experimental setup. If the metering valve were adjusted to a different setting, then the conductance values would have changed. Opened further, the conductance values would have been larger. Conversely, had it been more restricted, conductance values would have been smaller. This experimental approach and model could be applied to a series of connected components, a subassembly, or an entire system with multiple loss or leak points. If the conductance values were known for all of the components in a system, then a system conductance could be estimated by simply summing the components’ values.1 Use of the Model to Estimate Leak Rates. The main goal of this study was to develop a model and an experimental protocol that would allow for the estimation of conductance values from a pressure-decay test and then use those values to estimate leak rates under steady-state conditions. Let us work through an example. By combining eqs 2-5, we arrive at an expression that will allow us to determine the ambient leak rate from our model system.
Qf ) [CS,0 - k(Po - Pf)]
( )( )( )( ) Tf P s P o P o -1 Ts Pf Pf Pf
(6)
If Tf ) Ts and Pf ) Ps, as was the case in our study, then eq 6 reduces to
Qf ) [CS,0 - k(Po - Pf)]
( )( ) Po Po -1 Pf Pf
(7)
Consider the leak rate of nitrogen from our model component under steady state conditions. From the decay measurements with nitrogen gas at T ) 30, 60, and 90 °C, average values of Cs,0 and k were respectively 2.9 std cm3/min and 1.6 × 10-2 std cm3/min‚psi. Assume the initial and ambient pressures are Po ) 75 psi and Pf ) 14.7 psi. If nitrogen gas leaks from the system at T ) 30 °C, using eq 7 and the data from nitrogen gas decay measurements listed above, we estimated the steady-state leak rate to be Qf ) 40 std cm3/min. This value agreed well with the actual leak rate measured with a flow meter, Qf ) 42 std cm3/min. Similarly, if nitrogen in the system were heated to T ) 90 °C and an inlet pressure of Po ) 65 psi were applied, then eq 7 predicted a steady-state leak rate of Qf ) 32 std cm3/ min; the actual steady-state value was 34 std cm3/min.
(std
cm3/min‚psi)
(std cm3/s‚atm)
10-2
(1.25 ( 0.19) × (1.56 ( 0.19) × 10-2 (0.93 ( 0.15) × 10-2 (1.27 ( 0.14) × 10-2
(3.1 ( 0.5) × 10-3 (3.8 ( 0.5) × 10-3 (2.3 ( 0.4) × 10-3 (3.1 ( 0.4) × 10-3
Alternatively, the leak rate of nitrogen could have been estimated using the full set of decay data from the master curve (i.e., all three gases measured at all three temperatures), where Cs,0 and k were 2.6 std cm3/min and 1.3 × 10-2 std cm3/min‚ psi. In this case, the steady-state leak rate of nitrogen gas at T ) 30 °C with Po ) 75 psi and Pf ) 14.7 psi was estimated to be Qf ) 38 std cm3/min. This differed from the experimentally measured value (Qf ) 42 std cm3/min) by only 10%. Using decay data from helium or oxygen alone under the same conditions yielded similar estimates for nitrogen, 40 and 35 std cm3/min, which were within 20% of the measured steady-state value. As hoped, we were indeed able to estimate steady-state leak rates using data generated from decay testing. It seems that, if a higher degree of accuracy is required, then decay measurements should be made with the same gas that will ultimately be used in steady-state operation. On the other hand, if lessaccurate estimates will suffice, then one could extract conductance values over the appropriate pressure range with a single gas and use those conductance values to estimate leak rates for a wide range of gases, pressures, and temperatures, so long as the gas or gases in question behave ideally. Conclusions Gas loss from a model component/system was measured under both decay and steady-state conditions and then characterized with a modified conductance model. Estimates of conductance from decay measurements were the same as those from the steady state. Conductance values decreased slightly with pressure and increased slightly with temperature (following Charles’ Law). The various gases employed here showed similar conductance values. Thus, it was possible to construct a master conductance curve (using a standard reference temperature as a basis) for the model system. This master curve represented a complete map of gas-loss or leak rate characteristics of our model system over a broad pressure-temperature range. Because our analysis gave the same conductance values from the two different test methods (pressure decay and steady-state), it should allow for the estimation of leak rates of real components, subassemblies, or systems operated under steady-state conditions using conductance values estimated from pressure-decay measurements. Appendix Derivation of Decay and Steady-State Equations. The change in mass (m) with time (t) of a gas-handling system (subassembly or component) is equal to the mass of gas exiting the system (m˘ out).10
∂m - ) m˘ out ∂t
(8)
Using the ideal gas law, the left side of eq 8 can be rewritten as
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Ind. Eng. Chem. Res., Vol. 47, No. 4, 2008
VM ∂P ∂m )∂t RT ∂t
-
(9)
where V is the volume of the system, P is the time-dependent pressure inside the system, T is temperature, M is the atomic or molecular mass of the gas, and R is the ideal gas constant. Similarly, the right side of eq 8 can be written as
m˘ out )
V˙ outM (P - Pf) RT
(10)
where V˙ out is the volumetric flow rate of gas exiting the system and Pf is the final or ambient pressure.13 Because this is a gas and not a liquid, the volume of gas flow out (V˙ out) is not constant. It changes with pressure. Again, using the ideal gas law, we can write a relation that describes a small element of gas containing a number atoms or molecules (n) at a given temperature (T). Inside the gas-handling system, the small element has a characteristic volume (Vc) that varies with pressure (P).
PVc ) nRT
PfVout ) nRT
(12)
These two expressions can be equated and rearranged to yield a relation that gives the volume of the element that has exited the system (Vout) in terms of the characteristic volume (Vc) of the element inside the system,
P Vout ) Vc Pf
(13)
P V˙ out ) C Pf
(14)
where C is a characteristic gas loss or leak rate () Vc/t) that is sometimes referred to as conductance. Combining eqs 8-10 and 14 gives a differential equation that describes the timedependent pressure decay of the system due to gas loss or leaks.
(
)
(18)
Then, with eqs 9 and 18, we can write an expression for steadystate mass loss rate (m˘ loss)
m˘ loss ) -
(
∂m CM 2 1 1 Po t)0 ) ∂t RT Pf Po
)
(19)
Similarly, steady-state volumetric loss rate (Qloss) as a function of the conductance is,
Qloss ) -
( )
∂Vout Po -1 t)0 ) C ∂t Pf
(2)
Whereas we have focused on steady-state conditions where the pressure remains constant at the initial value (Po), expressions can be derived for the instantaneous loss rate that occurs during pressure decay. For example, the instantaneous mass loss rate (m˘ loss,t) at any time t during decay is
m˘ loss,t ) -
∂m ) ∂t
( ) ( )[ ( ) ( )]
Po Po -1 -1 Pf C 1 C CM Pf exp - t exp - t RT Po V Pf Po V
-2
(20)
and the instantaneous volumetric loss rate (Qloss,t) is
where (P) is the time-dependent pressure and (Pf) is the final pressure. Written in terms of time, eq 13 becomes
C P ∂P )- P -1 ∂t V Pf
(
∂P C 1 1 ) - Po2 ∂t t ) 0 V Pf Po
(11)
As this element exits the system and encounters ambient conditions, both its pressure and volume will change.
-
Expressions describing gas loss or leaks under steady-state conditions where the gas supply to the system is left on and the pressure remains constant at the initial value (Po) can be derived as follows. First, we find the derivative of the timedependent pressure and then set t ) 0.
)
(15)
∂Vout ) ∂t
( ) ( )[ ( ) ( )]
Po Po -1 -1 Pf Pf C 1 C exp - t exp - t C Po V Pf Po V
-1
(21)
Acknowledgment Thanks to Huaping Wang for supporting our efforts and allowing John and Jon to spend some time on this project during their 2004 summer stay at Entegris. Literature Cited
Integrating eq 15 yields,
[ ]
P -1 Pf C )- t+Φ ln P V
(16)
where Φ is a constant of integration. Implementing the initial condition
P ) Po at t ) 0
(17)
produces the working equation that describes pressure decay as a function of time,
[ ( ) ( )]
Po -1 Pf 1 C P) exp - t Pf Po V
Qloss,t ) -
-1
(1)
(1) Moore, P. O. Leak Testing, 3rd ed.; American Society for Nondestructive Testing: Columbus, OH, 1998; Vol. 1. (2) For Entegris products, this approach will be useful for semiconductor gas handling components, semiconductor device handling closures (e.g., microenvironments, particularly those that employ gas purge) as well as fuel cell components, subassemblies, or systems. (3) O’Hanlon, J. F. A User’s Guide to Vacuum Technology, 2nd ed.; Wiley & Sons: New York, 1989. (4) Roth, A. Vacuum Technology, 3rd ed.; Elsevier North-Holland: New York, 1990. (5) Little, J. C.; Gordon, L. B. Automated, Differentially Pumped, Massspectrometer Sampling System. ReV. Sci. Instrum. 1991, 62 (2), 334-341. (6) Mohan, P. Vacuum Gauge Calibration at the NPL (India) Using Orifice Flow Method. Vacuum 1998, 51 (1), 69-74. (7) dos Santos, J. M. F. Simple Vacuum Experiments for Undergraduate Student Laboratories. Vacuum 2005, 80 (1-3), 258-263. (8) Wendt, H. R.; Brown, C. A. SMIF Pod Seal Leakage; IBM Almaden Research Center: San Jose, CA, May 23, 1997. (9) Mikkelsen, K.; Niebeling, T. Characterizing FOUPs and Evaluating Their Ability to Prevent Wafer Contamination. Micro 2001, 19 (3).
Ind. Eng. Chem. Res., Vol. 47, No. 4, 2008 1309 (10) Bennett, C. O.; Myers, J. E. Momentum, Heat, and Mass Transfer, 3rd ed.; McGraw-Hill: New York, 1982; pp 169-171. (11) The internal volume of the assembled model system was determined by massing it, filling it with ethanol, massing again, subtracting the unfilled mass from the filled mass, and then using the density of ethanol to convert the ethanol mass to a volume. (12) Taylor, J. R. An Introduction to Error Analysis, 2nd ed.; University Science Books: Sausalito, CA, 1997. (13) Equation 10 is generally an approximation. If Pf ) 0, then it is exact. If we were to use only P in eq 10, then we would expect the gas to
continue to flow until P decays all the way to zero. This is not correct for the case studied here, as flow ceases when P ) Pf.
ReceiVed for reView August 15, 2007 ReVised manuscript receiVed November 13, 2007 Accepted November 15, 2007 IE071118S
