Under Pressure Part 1
Just The Facts

DOFPro Team

Under Pressure

Under Pressure Part 1

  • What pressure is
  • How you use it
  • How you measure it

Under Pressure Part 2

  • Atmospheric Pressure
  • Barometric Pressure
  • Gauge Pressure
  • Absolute Pressure
  • Vacuum Pressure

Pressure

\(\mathrm{Pressure} \equiv f_{\mathrm{normal}}/A\), with SI units of the Pascal (\(\mathrm{Pa}\))

\[1\ \mathrm{Pa} \equiv 1\ \mathrm{N}/\mathrm{m^2}\]

Other common units are the \(\mathrm{Torr}\) or \(\mathrm{mmHg}\), the atmosphere \((\mathrm{atm}),\) the kilopascal \((\mathrm{kPa})\), the \(\mathrm{bar}\), and the pound per square inch \((\mathrm{psi})\).

\[ \begin{aligned} 1\ \mathrm{atm} & = 1.01325\ \mathrm{bar} = 101.325\ \mathrm{kPa} = 101325\ \mathrm{Pa} \\ & = 760\ \mathrm{Torr} \\ & = 14.6959\ \mathrm{psi} \end{aligned}\]

Hydrostatic Pressure

\(h\)

\(D\)

\(A = \pi D^2 / 4\)

\(P_0\)

\(\rho\)

\(m = \rho A h,\ F = mg\)

\(\Delta P = \frac{mg}{A}=\frac{\rho A h g}{A} = \rho g h\)

\(P = \Delta P + P_0 = \rho g h + P_0\)

\(P\)

Some Hydrostatic Pressure Units
\(\mathrm{mmHg}\)
\(\mathrm{inHg}\)
\(\mathrm{inH_2O}\)
\(\mathrm{ftH_2O}\)
\(\mathrm{mH_2O}\)

Manometer

\(P_1 + \rho_\mathrm{A}g(h_2+h_1)\)

\(= \rho_\mathrm{B}g h_1 + \rho_\mathrm{C}g h_2 + P_2\)

\(\Delta P = P_1 - P_2\)

\(= [\rho_\mathrm{B} h_1 + \rho_\mathrm{C} h_2 - \rho_\mathrm{A}(h_2+h_1)] g\)

For \(\rho_\mathrm{A}=\rho_\mathrm{C}\)

\(\Delta P= (\rho_\mathrm{B}-\rho_\mathrm{A})g h_1\)

Example

\(P_\mathrm{Ar} = 1.493\ \mathrm{bar},\)

\(T = 68\ ^\circ\mathrm{F}\)

\(\rho_\mathrm{benzene} = 879\ \frac{\mathrm{kg}}{\mathrm{m^3}}\)

\(\rho_\mathrm{Hg} = 13,546\ \frac{\mathrm{kg}}{\mathrm{m^3}}\)

\(P_\mathrm{atm} = 1.000\ \mathrm{atm}\)

Solution

\(\Delta P = \rho g h\)

\(P_\mathrm{atm}+ \Delta P_\mathrm{piston} + \Delta P_\mathrm{C_6H_6} = \Delta P_\mathrm{Hg} + P_\mathrm{Ar}\)

\(101,325\ \mathrm{Pa}+\rho_\mathrm{piston}\ g\ (20\ \mathrm{cm})+879\ \frac{\mathrm{kg}}{\mathrm{m^3}}\ g\ (19.28\ \mathrm{ft})\)

\(=13,546\ \frac{\mathrm{kg}}{\mathrm{m^3}}\ g\ (7.480\ \mathrm{in})+149,300\ \mathrm{Pa}\)

\(\implies \rho_\mathrm{piston} = 11,500\ \frac{\mathrm{kg}}{\mathrm{m^3}}=11.50\frac{\mathrm{g}}{\mathrm{cm^3}}\)

Spreadsheet of Solution

The Takeaways

  1. Pressure is the second most commonly measured process variable.
  2. Hydrostatic pressure \(= \rho g h\).
  3. Manometers measure \(\Delta P\)’s with hydrostatic pressure.






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