Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
The equations describing the math model of the AGT Leakage System and subsequent numerical results
presented in this report are intended to serve as a basis for the selection of a final system. Note that
this analysis does not predict an accurate magnitude of fixture leakage for any of the summarized cases,
rather, it quantifies the improvements that may be expected upon implementation of the MSD system.
In addition, note that the focus of this math model has been on the leakage past the high helium
concentration Orings only. That is, the subsystem associated with the significantly larger low
concentration helium Orings was not included in the numerical approximation.
Reference Figure 1 for the cross section of the system considered.
Flow Sensor
0 psi
Case 1) 14.7 psi (ambient)
Case 2) 1 psi (Vacuum)
Case 3) Variable pressure/vacuum (nitrogen)
3000 psi
Figure 1: The cross section of the system. Cases 1-3 will be studied separately later in the report
The MSD team is responsible for controlling the input to the vent to minimize fixture leakage (as
specified in other documentation). This report will study the differences in leakage behavior for all
three of the vent conditions given in Figure 1.
In all three cases it was necessary to consider the mixture of nitrogen (ambient air) and the leakage
helium, and the consequent leakage into the can. For all cases, the vent was assumed to begin at an
atmospheric pressure of 100% Nitrogen. When the test is initialized helium enters the vent through
orifice 1 (Figure 2) and vent pressure builds slightly.
In case 1, the difference in pressure within the vent and the cleanroom is the only driving force for vent
flow. Case 2 is a similar scenario, except the vent condition is changed from atmospheric to a 1 psi
vacuum. Note that initially, while the difference in pressure is large, a large amount of vent gas is
removed from the vent. After the control volume (Figure 2) equalizes to slightly above 1 psi, the
pressure difference between the control volume and 1 psi vacuum source is all that drives flow. Case 3
considers a scenario where the vent is intentionally pulsed with 120 psi nitrogen, and subsequently
pulsed out with a 1 psi vacuum. High flow rates are expected into/out of the vent cavity due to the large
pressure differences in case 3.
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
It was immediately assumed that a perfect mixture of gases exists in the vent at all times. The vent was
treated as a bulk volume and the leakage past the Orings was modeled by using orifice equations.
Reference Figure 2 for the simplified schematic
0 psi (To Mass Spectrometer)
3000 psi
Orifice 1
Entire Vent
Volume
Orifice 2
Orifice 3
Figure 2: The simplified schematic
Assumptions:
•
•
•
Most Importantly: This is a pressure driven flow
– Permeability considerations were made (Parker equations from design review). The
leakage rates predicted through the Orings were too small.
Perfect gas mixture throughout the volume at all times
N2 and He are ideal gases
For all cases it was assumed that the vent starts out with 100% N2 concentration at atmospheric
pressure. A uniform mixture of He and N2 was added into the vent through the “high pressure orifice”
and removed by the “low pressure orifice”. The addition/removal of fluid to/from the vent is case
sensitive (see Figure 1)
Nomenclature:
𝑚 = mass
𝑚̇ = mass flow rate
𝑛 = moles
𝑛̇ = molar flow rate
𝜌 = fluid density
𝑀𝑊 = fluid molecular weight
𝐴𝑜 = orifice cross sectional area
𝐶𝑑 = orifice coefficient
𝑃𝑣𝑒𝑛𝑡 = vent pressure
𝑃1 = pressure input into the vent
𝑅 = Universal gas constant
𝑅𝑠 = Species specific gas constant
𝑉 = Vent volume
𝑑𝑃 = Delta pressure (either side of an
orifice)
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
General Equations:
𝑑
𝑚 = 𝑚̇𝑡𝑜𝑡 = ∑ 𝑚̇𝑖𝑛 − ∑ 𝑚̇𝑜𝑢𝑡
𝑑𝑡
The mass balance must now be converted to a molar flow balance to account for the mixing of gasses in
the vent:
𝑚̇
𝑀𝑊
𝑛̇ 𝑡𝑜𝑡 = ∑ 𝑛̇ 𝑖𝑛 − ∑ 𝑛̇ 𝑜𝑢𝑡
= 𝑛̇
The molar flow rate of a gas through an orifice may be written as:
𝑛̇ =
𝜌𝐶𝑑 𝐴𝑜 2𝑑𝑃
√
𝑀𝑊
𝜌
But for multiple species (i.e. helium and nitrogen gas) the molar flow rate of each gas within the control
volume must be specified:
𝑛̇ 𝐻𝑒,𝑡𝑜𝑡 = ∑ 𝑛̇ 𝐻𝑒,𝑖𝑛 − ∑ 𝑛̇ 𝐻𝑒,𝑜𝑢𝑡
𝑛̇ 𝑁2,𝑡𝑜𝑡 = ∑ 𝑛̇ 𝑁2,𝑖𝑛 − ∑ 𝑛̇ 𝑁2,𝑜𝑢𝑡
If leakage past both the high and low pressure Orings may be modeled orifice flow, then the molar flow
rate of helium and nitrogen into and out of the control volume can be split up as follows:
𝑛̇ 𝑂𝑟𝑖𝑓𝑖𝑐𝑒,𝐻𝑒 = %𝐻𝑒 ∗
𝜌𝑀𝑖𝑥 𝐶𝑑 𝐴𝑜 2𝑑𝑃
√
𝑀𝑊𝐻𝑒
𝜌𝑀𝑖𝑥
𝑛̇ 𝑂𝑟𝑖𝑓𝑖𝑐𝑒,𝑁2 = %𝑁2 ∗
𝜌𝑀𝑖𝑥 𝐶𝑑 𝐴𝑜 2𝑑𝑃
√
𝑀𝑊𝑁2
𝜌𝑀𝑖𝑥
Where the delta pressure term is the difference between the control volume pressure and the pressure
on the other side of the respective orifice. The signum function was applied to the equations to
determine whether flow was entering or leaving the control volume.
The orifice area, Ao is adjusted for each of the Orings to approximate realistic flow/concentration values
based on empirical data. For the vent orifice Ao is simply the area of the vent outlet. Square edge
orifices (Cd = 0.6) were assumed throughout the analysis. Note that mixed density of the gas within the
control volume calculated using the partial pressures with the ideal gas law.
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
𝑃𝑝𝑎𝑟𝑡𝑖𝑎𝑙 = 𝜌𝑅𝑠 𝑇
𝜌𝑚𝑖𝑥 =
𝜌=
𝑃𝑝𝑎𝑟𝑡𝑖𝑎𝑙
𝑅𝑠 𝑇
𝑃𝑣𝑒𝑛𝑡 %𝑁2 %𝐻𝑒
(
+
)
𝑇
𝑅𝑁2
𝑅𝐻𝑒
Reference Table 1 on the following page for a summary of relevant system parameters and the
associated units.
Value
Units
Notes
Patm
101300
Pa
N/A
T amb
300
K
N/A
MW He
4
amu
N/A
MW N2
14
amu
N/A
Cd 1,2,3
0.6
N/A
All orifices
Ao (1)
7.51E-15
m^2
Orifice 1 (Figure 2)
Ao (2)
7.51E-15
m^2
Orifice 2 (Figure 2)
Ao (3)
2.45E-06
m^2
Orifice 3 (Figure 2)
V
85
mm^3
N/A
Table 1: All relevant system constants, dimensions, and other parameters
Simulation/Results:
A screenshot of the Simulink block diagram which was constructed per the given equations is given in
Figure 3. Note that this block diagram is capable of calculating the helium flow rate and total volume of
helium into the can for all three cases.
Molar flow rate of
He Integrator
Molar flow rate of gas into can calculator
N2 Integrator
Calculates Mixed Density
helium into vent
(3000 psi)
Molar flow rate of gas into/out of vent
Calculates % Moles
Figure 3: The Simulink block diagram
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
The plots of Figures 4-6 were generated considering cases 1 and 2. The plots of Figure 4 were generated
for the percentage of helium within over time.
-3
7
Percentage of Helium In Vent
x 10
-6
4.5
Case 1: Ambient Applied Pressure
Case 2: Constant 1 psi Vacuum
6
4
Case 1: Ambient Applied Pressure
Case 2: Constant 1 psi Vacuum
3.5
% Helium In Vent
5
% Helium In Vent
Percentage of Helium In Vent
x 10
4
3
2
3
2.5
2
1.5
1
1
0
0.5
0
50
100
150
200
Time (seconds)
250
300
350
0
400
0
0.2
0.4
0.6
0.8
1
1.2
Time (seconds)
1.4
1.6
1.8
2
Figure 4: A comparison of the helium percentages in the vent for the (left) full test duration (right)
“beneficial zone” of case 2
Upon inspection of the plot to the left of Figure 4 it is seen that when a vacuum is pulled over the entire
test duration, the % of helium within the vent is able to grow at a faster rate. The beneficial region of
pulling a vacuum is only seen when the pressure difference between the vent and the vacuum source is
large. With the given parameters, this has been calculated to be about 2 seconds (plot to the right of
Figure 4). Similar results may be expected for the volume of leakage helium vs time plots of Figure 5.
-6
2.5
-11
Comparison of Cases 1 and 2
x 10
1.2
Case 1: Ambient Applied Pressure
Case 2: Constant 1 psi Vacuum
Volume of Leakage He (cc)
Volume of Leakage He (cc)
1.5
1
0.5
0
Case 1: Ambient Applied Pressure
Case 2: Constant 1 psi Vacuum
1
2
Beneficial Range of Case 3
x 10
0.8
0.6
0.4
0.2
0
50
100
150
200
Time (seconds)
250
300
350
400
0
0
0.5
1
1.5
2
2.5
Time (seconds)
Figure 5: A comparison of the helium leakage volume into the AGT can for the (left) full test duration ,
and (right) “beneficial zone” of case 2.
It has been concluded that pulling a constant vacuum within the vent will not aid in reducing AGT fixture
leakage. For further discussion of this somewhat counter-intuitive phenomena reference the final MSD I
detail design review powerpoint posted on the EDGE website.
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
Case 3 is now taken into consideration. For the methodology of determining a pulse/purge duty cycles
(presented in Table 2) and a detailed explanation of the case 3 concept refer to the powerpoint posted
on the EDGE website. Only results for case 3 will be presented in this report.
Period (s)
Duty Cycle (%)
10
20
20
10
30
6.7
40
5
Table 2: The pulse/purge periods and duty cycles considered within the simulateion. Duty cycles
were calculated considering the 2 second “beneficial vacuum” time calculated in case 2.
The plots of Figures 6 and 7 depict the volumetric flow rate of helium into the can and total volume of
helium leaked into the can respectively. Results are provided for several simulations in which differing
pulse/purge cycle periods and duty cycles were input into the model.
-13
7
Period =
Period =
Period =
Period =
6
He Can Leakage (cc/s)
Volumetric Flow Rate of Helium into Can Over Time
x 10
5
4
10 s
20 s
30 s
40 s
3
2
1
0
0
50
100
150
200
250
Time (seconds)
300
350
Figure 6: Case 3 volumetric flow rate of helium into the can over time
400
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
-10
He Can Leakage Volume (cc)
He Can Leakage Volume Over Time
x 10
1.4
1.2
Period =
Period =
Period =
Period =
1
0.8
10 s
20 s
30 s
40 s
0.6
0.4
0.2
0
0
50
100
150
200
250
Time (seconds)
300
350
400
Figure 7: Case 3 total leaked volume vs time for several pulse/purge periods and duty cycles
The plot of the % helium in the vent for the 10 second pulse/purge period is provided in Figure 8:
-7
1.6
% Helium For a 10 Second Pulse Purge Period
x 10
1.4
1.2
% Helium
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
Time (seconds)
250
300
350
400
Figure 8: the % Helium in the vent for the 10 second pulse/purge period
Finally a comparison between the baseline case 1 and case 3 is provided in Figure 9. It is seen that a
significant improvement may be expected upon implementation of the case 3 system.
Austin Frazer
AGT Fixture Leakage Prevention Analysis
Report Generated: 11/3/2012
-11
He Can Leakage (cc/s)
1
Volumetric Flow Rate of Helium into the Can Over Time
x 10
Case 3
Case 1 (Baseline)
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
Time (seconds)
300
350
400
Figure 9: The comparison of the helium leakage into can for the baseline (case 1) and case 3.