## 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