Level, pH, conductivity and viscosity

A liquid column of height $h$ and density $\rho$ produces a pressure $\rho gh$ at its base. A differential pressure (DP) transmitter converts this pressure to a level signal. SI units are used throughout: pressure in Pa, density in kg/m${}^3$, $g=9.81$ m/s${}^2$. The output is a 4-20 mA loop with 4 mA as the live zero.

In a bubbler (purge or dip-tube) system, air or nitrogen is fed slowly down a tube that reaches near the tank bottom. Once gas bubbles out of the tube end, the gas back-pressure equals the hydrostatic pressure $\rho gh$ of the liquid above the tube end (open tank), and a pressure gauge or transmitter reads it. Like any hydrostatic method, it depends on the liquid density.

In a closed (pressurised) tank, the vapour pressure acts on the liquid surface as well. The low side of the transmitter is connected to the top of the tank. With a dry low-side leg (vapour only, its own weight negligible and no condensation), the vapour pressure cancels and the DP still depends only on $\rho gh$. If vapour can condense in the leg, the leg is deliberately filled with liquid. A wet leg of height $H$ and density ${\rho }_w$ adds a constant pressure on the low side, so the DP becomes $\rho gh-{\rho }_{w}gH$, which is negative at low level. The transmitter is then given a zero elevation: its lower range value is set to $-{\rho }_{w}gH$ so that empty still gives 4 mA.

A displacer is a long cylinder hung in the liquid. By Archimedes' principle, the upward buoyant force equals the weight of liquid displaced. As the level rises, more of the displacer is submerged, the apparent weight falls, and a force or torque sensor reads the level. A float, by contrast, rides on the surface and gives the position directly.

A capacitive probe is a rod (radius $a$) inside a coaxial tube or the metal tank wall (inner radius $b$). The liquid replaces air as the dielectric over the wetted length $h$, so the capacitance increases linearly with level. For an insulating liquid of relative permittivity ${\epsilon }_r$ and probe length $L$:

For a conducting liquid, the probe rod is insulated and the liquid acts as the second plate, so capacitance again rises with the wetted length. Here the insulating sleeve's thickness and permittivity set the capacitance per unit length, so the reading depends very little on the liquid's own permittivity.

Ultrasonic and radar sensors are non-contact. A transmitter at the top of the tank, at height $H$ above the tank bottom, sends a pulse and times the echo from the liquid surface. The distance to the surface is half the round-trip path.

pH is defined as $\mathrm{pH}=-{\log }_{10}{a}_{{\mathrm{H}}^{+}}$, where $a_{{\mathrm{H}}^{+}}$ is the hydrogen-ion activity; in dilute solutions this is close to $-{\log }_{10}[{\mathrm{H}}^{+}]$ with $[{\mathrm{H}}^{+}]$ in mol/L. A glass electrode has a thin hydrogen-sensitive glass membrane. The potential across it, measured against a reference electrode, follows the Nernst equation. Convention used here: the measured potential falls as pH rises (the sign reverses if the cell's leads are swapped; the slope magnitude does not change).

The reference electrode (commonly silver/silver chloride) gives a fixed potential through a liquid junction; a combination electrode houses the glass and reference electrodes in one body. A meter is calibrated with two buffer solutions: one buffer (usually pH 7) sets the offset and a second sets the slope. The glass response departs from the ideal slope at very high pH (alkaline or sodium error, reading too low) and at very low pH (acid error).

Because the slope depends on temperature, a pH meter includes a temperature sensor, often a Pt100 or thermistor, and applies automatic temperature compensation. This corrects the slope. It does not correct the change of the solution's own pH with temperature, which is a real chemical change.

The glass membrane has a very high resistance, typically about 10 MΩ to 1000 MΩ. The measuring amplifier must therefore have an input impedance much larger still (of the order of $10^{12}\Omega$). Otherwise a current drawn from the electrode drops voltage across the membrane resistance and causes a loading error. Short, shielded cable is also needed to limit leakage and noise pickup.

The conductivity $\kappa$ of an electrolyte (SI unit S/m) is found by measuring the conductance $G=1/R$ between two electrodes in a cell. The cell constant $K$ depends only on geometry, assuming a uniform field between parallel electrodes of area $A$ separated by distance $l$.

Dynamic viscosity $\mu$ (Pa s) is the ratio of shear stress to shear rate in a Newtonian fluid; for a non-Newtonian fluid this ratio changes with shear rate and is called the apparent viscosity. Kinematic viscosity is $\nu =\mu /\rho$ (m${}^2$/s). Older units: 1 poise (P) $=0.1$ Pa s, so 1 cP $=1$ mPa s; 1 stokes (St) $=10^{-4}$ m${}^2$/s, so 1 cSt $=10^{-6}$ m${}^2$/s. Three standard principles are used, plus efflux cups.

In the capillary type, the flow rate or time for a fixed volume to pass is measured under a known pressure drop. In the falling-sphere type, the sphere is timed over a known distance to get $v_t$. In the rotational type, the torque at a fixed speed (or the speed at fixed torque) gives $\mu$ directly; it also suits non-Newtonian fluids because the shear rate can be varied. Efflux-cup (orifice) viscometers time the outflow of a fixed volume through a short orifice and convert the time $t$ to kinematic viscosity with an empirical relation of the form $\nu =At-B/t$, where $A$ and $B$ are instrument constants.