Gwhiz wrote:
"I came to this same question when thinking about the ideal gas equation, PV=NkT. If temperature is a kind of average kinetic energy per gas molecule, then the right side of the ideal gas equation is essentially number of molecules times average kinetic energy per molecule, which is the total kinetic energy in your system of gas. If one side of the equation represents total kinetic energy, it would make sense for the other side also to represent total kinetic energy, and the units of bear that out. If you divide both sides by volume, your left with P=NkT/V."
Indeed, there is a very close association with pressure and temperature, temperature and energy density, and thus pressure and energy density.
In statistical mechanics the following molecular equation is derived from first principles: P = n k_B T for a given volume.
Therefore T = (P / (n k_B)) for a given volume.
Where:
k_B = Boltzmann Constant (1.380649e−23 J·K−1);
T = absolute temperature (K);
P = absolute pressure (Pa);
n = number of particles
If n = 1, then T = P / k_B in units of K / m³ for a given volume.
Now, some may protest "Temperature does not have units of K / m³ !!!"... note the 'for a given volume' blurb. We will cancel volume in a bit.
We can relate velocity to kinetic energy via the equation:
v = √(v_x² + v_y² + v_z²) = √((DOF k_B T) / m) = √(2 KE / m)
As particle velocity increases, kinetic energy increases. As kinetic energy increases, kinetic energy over volume increases. As kinetic energy over volume increases, energy density increases.
Kinetic theory gives the static pressure P for an ideal gas as: P = ((1 / 3) (n / V)) m v² = (n k_B T) / V
Combining the above with the ideal gas law gives: (1 / 3)(m v²) = k_B T
∴ T = mv² / 3 k_B for 3 DOF
∴ T = 2 KE / k_B for 1 DOF
∴ T = 2 KE / DOF k_B
See what I did there? I equated kinetic energy to pressure over that volume, thus canceling that volume, then solved for T.
A fluid moving with fewer than 3 DOF has a higher temperature (for the same kinetic energy) than a fluid with 3 DOF. This is especially relevant in, for instance, high-pressure system pressure relief piping, where the stagnation temperature can be much higher than the static temperature.
You will note that the above aligns with the Bernoulli Principle.
This is how Sandia National Laboratories calculates temperature.