Water permittivity at temperature

What is the dielectric constant of the water in the cell once it has warmed up?

ε = e \left( 1 - 0.004\,(T - 20) \right)

Method

1. Take the water’s relative permittivity at 20 °C — 80.1 for distilled, 78.5 for tap, 76 for mineral, 74 for sea water.
2. Find how far the temperature is from 20 °C. Above it the permittivity falls; below it, rises.
3. Water loses about 0.4 % of its permittivity per degree of warming. Multiply the difference by 0.004 and subtract that fraction.
4. The cell’s capacitance is directly proportional to this, so the same fraction moves the capacitance. Resonant frequency goes as one over the square root of capacitance, so it moves about half as far, and the other way.

Assumptions

• The 0.4 % per degree coefficient is linear. It is a good approximation between roughly 0 °C and 60 °C and increasingly poor outside that; near boiling the real curve is steeper than this straight line.
• The water is at one temperature throughout. A cell under load is warmer at the plates than at the wall, and what matters for capacitance is the average across the gap.
• This is the static (low-frequency) permittivity. Above about a gigahertz water’s permittivity falls away regardless of temperature — see the Cole-Cole calculation for that.
• Dissolved solids are not modelled here. They matter much more for conductivity and loss than for permittivity, which is why the four profiles differ by a few per cent in ε and by six orders of magnitude in σ.