Pressure and manometry

Every morning Ray Mendoza walks the Northside station and reads its gauges: clearwell level, suction and discharge pressures on each pump, the differential across the venturi meter and the air pressure in the hydropneumatic surge tank. Each reading depends on the same idea: in a fluid at rest, pressure depends only on depth. That one fact lets a simple U-shaped tube of liquid measure pressures and pressure differences with remarkable accuracy, and it sets the loads on every tank wall and gate in the plant. This lesson develops the hydrostatic equation and the art of reading manometers.

Pressure at a point

Pressure is the normal compressive force per unit area that a fluid exerts on a surface. In a fluid at rest there is no shear stress, and a force balance on a tiny wedge of fluid shows that the pressure at a point is the same in every direction. A pressure sensor on the side of a pipe and one facing upstream read the same value if the water is still. That is Pascal's law, and it is why we can speak of the pressure at a point without naming a direction. In moving fluids, viscous stresses add small directional differences, but for most engineering purposes pressure remains a single number at each point.

Pressure is measured relative to some reference. Absolute pressure is measured from a perfect vacuum. Gauge pressure is measured from the local atmospheric pressure, which is what a Bourdon gauge on a pipe shows. A vacuum reading is how far below atmospheric pressure the fluid is. Hydrostatic calculations in open tanks are usually done in gauge pressure, because atmospheric pressure acts on both sides of every wall and cancels. Cavitation checks and gas calculations need absolute pressure.

The hydrostatic equation

Consider a small vertical cylinder of fluid of cross-sectional area \(dA\) and height \(dz\), with \(z\) measured upward. The pressure on its bottom face pushes up, the pressure on its top face pushes down, and gravity pulls the fluid's weight \(\rho g\,dA\,dz\) down. For the fluid to stay at rest these forces must balance, which gives the basic equation of fluid statics: pressure decreases upward at a rate equal to the specific weight.

For a liquid of constant density, integrating between two elevations gives the familiar result. Measured downward from a free surface at atmospheric pressure, as depth \(h\), the gauge pressure is simply \(\gamma h\). Pressure rises by about 9.79 kPa for every meter of depth in water at 20 °C. At the bottom of the 6 m deep clearwell, the gauge pressure is about 59 kPa, whatever the shape or size of the tank. A narrow standpipe and a wide reservoir connected at the bottom show the same water level, a result once called the hydrostatic paradox.

Because pressure depends only on elevation in a continuous body of the same fluid at rest, all points at the same level in that fluid have the same pressure. This is the key rule for manometers. It holds only within one continuous fluid: if you cross an interface into a different liquid, or pass through a closed valve, the rule no longer connects the two sides.

Liquid columns are often used as pressure units. A pressure head is the height of a column of a given liquid that the pressure would support, \(h = p/\gamma\). Operators speak of feet or meters of water: the discharge gauge on a Northside pump reads 540 kPa (78 psi), which is a head of 55 m (181 ft) of water. One psi is equivalent to 2.31 ft of water, a conversion every US water operator learns early. Barometers once used inches of mercury, and 29.92 in Hg (760 mm Hg) is standard atmospheric pressure.

Atmospheric pressure and the barometer

The atmosphere is itself a fluid at rest, held down by gravity. At sea level its pressure averages 101.325 kPa, and it falls with elevation, quickly at first because air near the ground is densest. A mercury barometer measures it directly: a glass tube closed at the top is filled with mercury and inverted into a dish. The mercury falls until the column's weight balances the atmosphere's push on the dish, leaving a nearly perfect vacuum above it. Standard atmospheric pressure supports 760 mm of mercury, or about 10.3 m of water, which is why no suction pump can lift water more than about 10 m and why a pump should sit low enough that the water arrives at its inlet under positive pressure.

In Omaha, at an elevation near 330 m (about 1080 ft), the average barometric pressure is about 97.4 kPa (14.1 psia, or 28.8 in Hg). Weather systems move it by a few kilopascals either way. Ray's station log records the barometer reading each day, because the absolute pressure at the pump inlet, which governs cavitation, changes with it.

In a gas, density changes with pressure, so the hydrostatic equation must be integrated with the ideal-gas law. For an isothermal atmosphere this gives an exponential decay of pressure with height. Over the height of a building or a tank, however, the change in air pressure is tiny, about 12 Pa per meter at ground level, so we treat atmospheric pressure as uniform around any one piece of equipment.

Manometers

A manometer measures pressure by balancing it against a column of liquid. The simplest is a piezometer, a vertical tube tapped into a pipe or tank and open at the top; the liquid rises until its height above the tap equals the pressure head. Piezometers work only for modest pressures in liquids, because a pressure of 100 kPa would need a tube 10 m tall. A U-tube manometer is more versatile. It holds a heavier gauge fluid, often mercury or a colored oil, in the bend, and the difference in its levels on the two sides indicates the pressure difference.

Reading any manometer, however complicated, follows one method. Start at a point where the pressure is known or wanted. Move through the fluids one leg at a time, adding \(\gamma h\) each time you go down and subtracting \(\gamma h\) each time you go up. Jump horizontally across a continuous fluid freely, because points at the same level in the same fluid share the same pressure. Finish at the other point and set the sum equal to its pressure. Written out, the method turns a confusing sketch into a single line of algebra.

The choice of gauge fluid sets the sensitivity. Mercury's high density keeps the manometer short for large differences, but it is toxic and is being phased out in water utilities in favor of electronic transmitters. For small differences, such as the draft in a duct, a light oil with a specific gravity near 0.8 gives a large deflection. An inclined manometer, with one leg tilted at a small angle \(\theta\) to the horizontal, stretches the reading along the tube by a factor \(1/\sin\theta\): at 10°, a 10 mm vertical change spreads over 58 mm of scale.

# Pressure at depth and a multi-fluid manometer
g = 9.81
rho_w, rho_hg, rho_oil = 998.0, 13550.0, 850.0
patm = 97.4   # kPa, typical barometer reading at the station (elevation about 330 m)

# Hydrostatic pressure in the clearwell (open to atmosphere)
for depth in (0.0, 2.0, 4.0, 6.0):
    p_gauge = rho_w * g * depth / 1000
    print(f"Clearwell depth {depth:3.1f} m: {p_gauge:6.2f} kPa gauge, {p_gauge + patm:6.2f} kPa abs")

# U-tube mercury manometer across the venturi on the intake main.
# Start at A (upstream tap), go down through water to the mercury surface,
# across the mercury, and up through water to B (throat tap).
h_mercury = 0.185   # m, difference in mercury levels
dp = (rho_hg - rho_w) * g * h_mercury / 1000   # kPa, taps at the same elevation
print()
print(f"Mercury manometer, deflection {h_mercury*1000:.0f} mm: pA - pB = {dp:.2f} kPa")

# Walk a chain of fluids: list of (density, change in elevation going DOWN is +)
def walk(p_start, legs):
    p = p_start
    for rho, dz_down in legs:
        p += rho * g * dz_down / 1000
    return p

# Tank of oil over water, gauge at the bottom side, manometer to air
p_bottom = walk(0.0, [(rho_oil, 1.20), (rho_w, 0.80)])
print(f"Oil 1.20 m over water 0.80 m: bottom pressure = {p_bottom:.2f} kPa gauge")

# Air pressure in a closed hydropneumatic tank read by an open water manometer
# Water stands 2.35 m above the tank's water surface in the open leg.
p_air = rho_w * g * 2.35 / 1000
print(f"Hydropneumatic tank air pressure = {p_air:.2f} kPa gauge = {p_air + patm:.2f} kPa abs")

Clearwell depth 0.0 m:   0.00 kPa gauge,  97.40 kPa abs
Clearwell depth 2.0 m:  19.58 kPa gauge, 116.98 kPa abs
Clearwell depth 4.0 m:  39.16 kPa gauge, 136.56 kPa abs
Clearwell depth 6.0 m:  58.74 kPa gauge, 156.14 kPa abs

Mercury manometer, deflection 185 mm: pA - pB = 22.78 kPa
Oil 1.20 m over water 0.80 m: bottom pressure = 17.84 kPa gauge
Hydropneumatic tank air pressure = 23.01 kPa gauge = 120.41 kPa abs

Mechanical and electronic gauges

Manometers are accurate and need no calibration beyond the density of the gauge fluid, but they are bulky. In daily operation, Ray relies on Bourdon gauges: a curved tube of oval cross-section that straightens slightly when pressurized, turning a pointer through a gear. Bourdon gauges read gauge pressure and are typically accurate to 1 to 2 percent of full scale. They should be chosen so normal readings fall in the middle third of the dial, and they need snubbers or liquid filling where pressure pulses, such as on a pump discharge.

Electronic transmitters use a thin diaphragm whose deflection is sensed by strain gauges, a change in capacitance or a vibrating element. They can measure absolute, gauge or differential pressure, send a signal to the control system, and resolve changes of a few pascals. The venturi meter at Northside now uses a differential pressure transmitter instead of a mercury U-tube, but the transmitter is checked against a water manometer during calibration, and the principle of the measurement has not changed.

Pressure in a stratified tank

Many tanks hold more than one liquid, or a liquid under a gas. The hydrostatic equation applies layer by layer, with each layer's specific weight. A sludge holding tank might have a layer of scum on top, clear water in the middle and dense sludge at the bottom; the pressure at the floor is the sum of \(\gamma h\) for every layer. A closed hydropneumatic tank has compressed air above the water. The air pressure, nearly uniform because air is so light, adds to the hydrostatic pressure at every depth in the water.

Fluids in rigid-body motion

The hydrostatic equation also applies to a liquid that moves as a rigid body, with no relative motion between layers and therefore no shear. If a tank truck carrying water accelerates horizontally at \(a\), the free surface tilts at an angle whose tangent is \(a/g\), and lines of constant pressure tilt with it. A cylindrical tank spinning about its vertical axis at angular speed \(\omega\) develops a paraboloid free surface, rising by \(\omega^2 r^2/(2g)\) from the center to radius \(r\). These cases appear in tanker design and centrifugal separators. They also explain why a centrifugal pump's casing pressure rises toward the outer edge: the water spinning with the impeller behaves much like a liquid in rigid rotation.