Pressure & flow formulas

Gauge pressure is negative below atmospheric; absolute pressure is never negative.

SI units. p is the pressure difference between two points a vertical height h apart in a fluid of density $\rho$ at rest.

Both pressure taps at the same elevation. ${\rho }_m$ is the manometer liquid, ${\rho }_f$ the fluid above it in the limbs, h the difference of levels. For a gas, ${\rho }_f\ll {\rho }_m$ and $\Delta p\approx {\rho }_{m}gh$.

L is the reading along the inclined tube of area a, $\theta$ its angle to the horizontal, A the area of the well; a gas sits above the liquid (otherwise use ${\rho }_m-{\rho }_f$). For $A\gg a$ the vertical height is $h=L\sin \theta$, so the reading is $1/\sin \theta$ times what a vertical tube would show. The magnification rises as $\theta$ falls, so it suits small pressures.

h is the rise read on the vertical tube of area a, A the area of the well. The well level falls by $x=ah/A$; only the tube reading is taken. For $A\gg a$ it reduces to $\rho gh$.

Circular flat diaphragm clamped at its edge, radius R, thickness t, Young's modulus E, Poisson's ratio $\nu$, uniform pressure difference p. Valid while the deflection is small compared with the thickness; the deflection is then linear in p.

The gas of total trapped volume V at unknown pressure p is compressed into a closed capillary of area a. The mercury in the open reference capillary is raised level with the closed end, so the trapped gas column length equals the level difference h. Boyle's law gives $pV=(p+h)ah$ with pressures in mmHg; use consistent units (a in mm², h in mm, V in mm³ gives p in mmHg).

m is the total mass of the piston and the weights on it, $A_p$ the effective piston area; the oil pressure that floats the piston is applied to the gauge under test. SI units.

Q volumetric flow rate, v mean velocity, $\dot{m}$ mass flow rate.

Steady, incompressible, frictionless flow along a streamline at the same elevation.

$A_2$ is the throat or orifice area, $\beta =d/D$, $\rho$ the fluid density; incompressible fluid. $1/\sqrt{1-{\beta }^4}$ is the velocity-of-approach factor, and $K=C_d/\sqrt{1-{\beta }^4}$ is called the flow coefficient. Q is proportional to $\sqrt{\Delta p}$. For a gas with an appreciable pressure drop, multiply by an expansibility factor $\epsilon <1$.

h is the manometer level difference, ${\rho }_m$ the manometer liquid, $\rho$ the flowing liquid filling the leads; from $\Delta p=({\rho }_m-\rho )gh$. The manometer reads the piezometric difference, so the same relation holds for an inclined or vertical meter.

$p_0$ stagnation pressure, $p_s$ static pressure, $C_p\approx 1$; incompressible flow (low Mach number for a gas). Gives the local velocity at the tip, not the mean velocity.

$A_a$ annular area between float and tube, $V_b$ float volume, $A_b$ float cross-section, ${\rho }_b$ float density, $\rho$ fluid density. With a small linear taper, and the bore at the zero mark equal to the float diameter, $A_a$ and so Q are nearly proportional to float height, giving a nearly linear scale. The calibration holds only for the fluid density (and viscosity) it was made for.

Uniform field B; v is the mean velocity for axisymmetric flow; SI units (B in tesla). The emf is linear in flow and, above the minimum conductivity, independent of density, viscosity, temperature and pressure.

From $t_{down}=L/(c+v\cos \theta )$ and $t_{up}=L/(c-v\cos \theta )$. Since $v\ll c$ the approximation is used, so $v=c^2\Delta t/(2L\cos \theta )$. It needs a clean fluid, and this form depends on c, which varies with temperature.

Exact, and independent of the speed of sound c, so a meter computing v this way is not affected by temperature changes in c.

$f_0$ is the transmitted frequency and $\theta$ the angle between the beam and the flow, with transmitter and receiver both at that angle. The beam is reflected by particles or bubbles moving with the fluid, so the fluid must contain scatterers.

f is the pulse frequency, K the K-factor in pulses per unit volume; the total count gives total volume. Linear over its working range; needs clean, low-viscosity fluid (K changes with viscosity), and wears in the bearings.