Gas Pressure Units

  • 1 bar = 100 000 Pa
    • Non-SI
    • Used in EU, deprecated in the US
    • Standard atmospheric pressure 1 atm = 101325 Pa
  • 1 Torr = 1mmHg raises a column of mercury by 1 mm at 0°C
    • Non-SI
    • 760 Torr = 1 atm
    • Order of magnitude 1 Torr $\approx$ 1 mbar
  • 14.7 psi (poundforce/$\mathrm{in^2}$) = 1 atm

Pressure regimes

Rough vacuum$1-1000$ mbar
Medium vacuum$1-10^{-3}$ mbar
High vacuum (HV)$10^{-3}-10^{-7}$ mbar
Ultra high vacuum (UHV)$10^{-7}-10^{-14}$ mbar

Ideal Gas Law

$$p = n k_B T$$ where $n$ is particle density

So what is the pressure in a 10L chamber containing 1 gram of $\mathrm{^4He}$ at 20$^\circ$C?

$$p = \left( \frac{ N_A \frac{\mathrm{particles}}{\mathrm{mol}} }{4 \frac{\mathrm{gr}}{\mathrm{mol}}} \times \frac{1 \mathrm{gr}}{10 \mathrm{L}} \right) k_B (293 \mathrm{K}) = \\ 60.9 \mathrm{kPa} =457 \mathrm{Torr} = 8.8 \mathrm{psi} $$

Pumping speed vs. throughput

Speed = $\frac{\mathrm{Volume}}{\mathrm{Time}}$ $$S = \frac{V}{t}$$

Throughput = $\frac{\textrm{Quantity of gas}}{\mathrm{Time}}$ $$ Q = \frac{p.V}{t}$$

Conductance

of a pipe, hose, valve, etc.

$$Q = C(p_2 - p_1)$$
  • C has units of L/s
  • C is determined by the geometry of the pipe
  • C is independent of pressure only at ultra-low pressures
  • Equivalent to Ohm's law
    • Connection in series: $R_\Sigma = R_1 + R_2 + ...$
    • in parallel: $C_\Sigma = C_1 + C_2 + ...$

Types of flow

Continuum flow: mean free path of molecules $\lambda \ll d$ diameter of the pipe. Only in rough vacuum regime

Navier-Stoke equation is commonly used to understand this type of flow $$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla \bar{p} + \mu \, \nabla^2 \mathbf u + \tfrac13 \mu \, \nabla (\nabla\cdot\mathbf{u}) + \rho\mathbf{g} $$

The ultimate reference on solving this equation is An Introduction to Fluid Dynamics by G. K. Batchelor

Reynolds number

$$\textrm{Re} = \frac{\rho u L}{\mu}$$
Re < 500 (Laminar)Re > 2200 (Turbulent)

Types of flow

Molecular flow: $\lambda \le d$ or Kn $= \frac{\lambda}{d} > 1$

Relevant in medium to ultra-high vacuum regimes.

Velasco, A. et. al. (2012) Phys. Rev. E. 86. 10.1103

Knudsen equation

For large Kn $=\frac{\lambda}{d}$ in a cylindrical pipe of length $L$ and diameter $d$

$$ Q = p\frac{dV}{dt} = \frac{\sqrt{2 \pi}}6 \Delta P \frac{d^3}{ L \sqrt{\rho}}$$

Wikipedia/Knudsen_equation

Pumps

Page 20

1. Rotary pumps periodically increase/decrease pumping chamber volume

  • Good for $10^{-2} - 10^{-4}$ mbar
  • Leaky without oil, oily without leak
  • Usually used as back-up pump
  • Water can condense inside the chamber

2. Roots pumps were invented by Francis and Philander Roots.

  • Spacing between roots < 0.1 mm
  • Can operate very fast without mechanical wear
  • Good for displacing large volumes of gas ( over 10,000 $\mathrm{m^3/hr}$)
  • No oil, leaky. Small compression ratio (10-100)

3. Diaphragm pumps use a flexible membrane to compress gas.

  • 1 - 100 mbar after 2-3 stages
  • Oil-free, and coated with TEFLON
  • Suitable for pumping otherwise condensable gases
  • Good for chemists
  • Limited deformability of membrane means small pumping speed

4. Turbomolecular pumps kick gas molecules in the desired direction.

Gas$\bar{c}$ (m/s)
$H_2$1761
$He$1245
$H_2O$587
$N_2$471
$O_2$440
$CO_2$375
  • Effective when speed $\approx$ average thermal velocity $$\bar{c} = \sqrt{\frac{8RT}{\pi M}} $$
  • 30k - 80k rpm
  • Need back-up pump
  • Can reach $10^{-10}$ mbar
  • High speed implies unreliable:
    • Oil lubrication/Steal ball bearings
    • Grease lubrication/hybrid bearings
    • No lubrication/magnetic suspension