Calculator

Gas Flow Status

• Viscous flow: Known as continuous flow, frequent collisions between gas molecules occur much more than collisions with the walls of pipe. Mean free path of a gas molecule is significantly shorter than the inner diameter of pipe.
• Transition flow: Known as knudsen flow, mean free path of a gas molecule is roughly equal to the inner diameter of pipe. This flow happens in between viscous flow and molecular flow.
• Molecular flow: Mean free path of a gas molecule is significantly greater than the inner diameter of pipe.

Conductance: C

There is the physical effect of pipe, fittings, valves in the vacuum system. All these components connected between chamber and pump are resistance to gas flow and will reduce the pumping speed of pump ($S$).

• Conductance of circular straight pipe for air at 20 ℃:

Viscous flow: $C_{v} = 135 (D^{4} / L) * P_{ave}$ (L/sec)

Transition flow: $C_{t} = 135 (D^{4} / L) * P_{ave} + 12.1 D^{3}/L$ (L/sec)

Molecular flow: $C_{m} = 12.1 D^{3} / L$ (L/sec)

$C_{v}$: Conductance of pipe in viscous flow
$C_{t}$: Conductance of pipe in transition flow in between viscous and molecular (L/sec)
$C_{m}$: Conductance of pipe in molecular flow
$D$: Pipe inner diameter (cm)
$P_{ave}$: Average pressure in pipe
$L$: Length of pipe (cm)

Pumping Speed of Pump: S

A pump name usually uses the peak pumping speed, so it can be misunderstood as having this value in all pressure ranges. But the pumping speed of pump is not a constant value.

For example, as shown below, if there is a pump named FV300, this pump has a peak pumping speed of 300 m³/hr only at ATM and does not have a value of 300 m³/hr in all pressure ranges.

Please keep in mind that the pumping speed is not a constant value, but a variable value depending on the pressure.

Effective Pumping Speed at Chamber: S_eff

By the flow resistance in a pipe, the effective pumping speed working at chamber ($S_{eff}$) is always less than the pumping speed of pump ($S$).

$1/S_{eff}=1/S+1/C$

$S_{eff}=(S·C)/(S+C)$

$S_{eff}$: Effective pumping speed at chamber
$S$: Pumping speed of pump
$C$: Total conductance of pipe

$C$ is the total conductance of piping in the vacuum system consisted of various components such as pipes, fittings, valves.

• In serial connection, the total conductance $C$ is:

$1/C=1/C_{1}+1/C_{2}+1/C_{3}+\ldots+1/C_{N}$

• In parallel connection, the total conductance $C$ is:

$C=C_{1}+C_{2}+C_{3}+C_{4}+\ldots+C_{N}$

Pump Down Time Calculation

The estimation of pump down time from the start pressure to the target pressure is one of the important tasks in configuring and designing a vacuum system. Through the pump down time simulation of the configured system, it can be estimated whether the pump down time is within the desired time. Gas is based on N2 at room temp.

Formula

It is used the equation below to calculate pump down time:

$T=(V/S_{eff})*\displaystyle\ln(p1/p2)$

$T$: Pump down time
$V$: Chamber volume
$S_{eff}$: Effective pumping speed at chamber
$p1$: Start pressure
$p2$: Target pressure

Note: The effective pumping speed at chamber ($S_{eff}$) is actual pumping speed working at chamber combined with pumping speed of pump ($S$) and total conductance of pipe ($C$).

Pump Down Time Manual Calculation
without Pipe Conductance

• Chamber: 70 L
• Pump: FV300
• Requirement:
  • - Start pressure: 1013 mbar (atmospheric pressure, ATM)
  • - Target pressure: 5.0E-2 mbar
  • - Takt time: Within 25 sec. from ATM to 5.0E-2 mbar

Simple Calculation with Peak Pumping Speed, 300 m³/hr

$T=(V/S_{eff})*\displaystyle\ln(p1/p2)$

$=(70/300000)*\displaystyle\ln(1013/0.05)$

$=0.002314\ \mathrm{hr}=8.329\ \mathrm{sec}$

• The peak pumping speed, 300 m³/hr is considered as a constant value in all pressure range.
• Direct connection between chamber and pump without pipe connection, $S_{eff}=S$

Calculation with Pumping Speed Curve

$T=(V/S_{ave})*\displaystyle\ln(p1/p2)$

$t1=(70/265000)·ln(1013/60)=0.0007465\ \mathrm{hr}=2.688\ \mathrm{sec}$

$t2=(70/190000)·ln(60/2)=0.001253\ \mathrm{hr}=4.511\ \mathrm{sec}$

$t3=(70/115000)·ln(2/0.2)=0.001401\ \mathrm{hr}=5.046\ \mathrm{sec}$

$t4=(70/60000)·ln(0.2/0.08)=0.0010690\ \mathrm{hr}=3.848\ \mathrm{sec}$

$t5=(70/25000)·ln(0.08/0.052)=0.0012061\ \mathrm{hr}=4.342\ \mathrm{sec}$

$t6=(70/5500)·ln(0.052/0.05)=0.0004991\ \mathrm{hr}=1.797\ \mathrm{sec}$

$T=t1+t2+t3+t4+t5+t6$

$T=2.688+4.511+5.046+3.848+4.342+1.797$

$T=22.232\ \mathrm{sec}$

• Requirement: Chamber pressure from ATM to 5.0E-2 mbar within 25 sec.
• With FV300, the pump down time to 5.0E-2 mbar is 22.232 sec. which is within 25 sec. So it meets the requirement.
• Direct connection between chamber and pump without pipe connection, $S_{eff}=S$

Pump Down Time Manual Calculation
with Pipe Conductance

• Chamber: 70 L
• Pump: FV300
• Pipe: 500 m³/hr (constant conductance*)
• Requirement:
  • - Start pressure: 1013 mbar (atmospheric pressure, ATM)
  • - Target pressure: 5.0E-2 mbar
  • - Takt time: Within 25 sec. from ATM to 5.0E-2 mbar

Simple Calculation with Peak Pumping Speed, 300 m³/hr with Pipe, 500 m³/hr Conductance

$S_{eff}=(S·C)/(S+C)$

$=(300·500)/(300+500)$

$=187.5\ \mathrm{m^{3}/hr}$

$T=(V/S_{eff})·\displaystyle\ln(p1/p2)$

$=(70/187500)·\displaystyle\ln(1013/0.05)$

$=0.003702\ \mathrm{hr}=13.327\ \mathrm{sec}$

• The peak pumping speed, 300 m³/hr and 500 m³/hr pipe conductance are considered as a constant value in all pressure range. So the pump down time calculation in this way is still far from the reality.
• Please note that the pumping speed of pump and the conductance of pipe are not constant but variable values depend on the pressure.

Calculation with Effective Pumping Speed

$S_{eff}=(S·C)/(S+C)$

$S_{eff}@1.0E+03\ \mathrm{mbar}=188\ \mathrm{m^{3}/hr}$

$S_{eff}@6.0E+01\ \mathrm{mbar}=158\ \mathrm{m^{3}/hr}$

$S_{eff}@2.0E+00\ \mathrm{mbar}=115\ \mathrm{m^{3}/hr}$

$S_{eff}@2.0E-01\ \mathrm{mbar}=69\ \mathrm{m^{3}/hr}$

$S_{eff}@8.0E-02\ \mathrm{mbar}=37\ \mathrm{m^{3}/hr}$

$S_{eff}@5.2E-02\ \mathrm{mbar}=9.8\ \mathrm{m^{3}/hr}$

$S_{eff}@5.2E-02\ \mathrm{mbar}=0.99\ \mathrm{m^{3}/hr}$

$T=(V/S_{ave})·\displaystyle\ln(p1/p2)$

$t1=(70/172517)·ln(1013/60)=0.001147\ \mathrm{hr}=4.1284\ \mathrm{sec}$

$t2=(70/136459)·ln(60/2)=0.001744\ \mathrm{hr}=6.2810\ \mathrm{sec}$

$t3=(70/92175)·ln(2/0.2)=0.001748\ \mathrm{hr}=6.2951\ \mathrm{sec}$

$t4=(70/53001)·ln(0.2/0.08)=0.001210\ \mathrm{hr}=4.3566\ \mathrm{sec}$

$t5=(70/23420)·ln(0.08/0.052)=0.001287\ \mathrm{hr}=4.6351\ \mathrm{sec}$

$t6=(70/5400)·ln(0.052/0.05)=0.000508\ \mathrm{hr}=1.8299\ \mathrm{sec}$

$T=t1+t2+t3+t4+t5+t6$

$T=4.128+6.281+6.2951+4.3566+4.6351+1.8299$

$T=27.5261\ \mathrm{sec}$

• Requirement: Within 25 sec. from ATM to 5.0E-2 mbar
• With FV300 & 500 m³/hr pipe, the pump down time to 5.0E-2 mbar is 27.5261 sec. which is out of specification. (within 25 sec.) So it doesn’t meets the requirement.
• The pipe conductance, 500 m³/hr is considered as a constant value in all pressure range. So the pump down time calculation in this way is much closer than the previous way but still gap from the reality.
• Conductance of pipes are variable values depend on pressure with normally 3 different gas flow status. $C_{v}$, $C_{t}$, $C_{m}$ should be considered for more accurate calculation of pipe conductance