When compressing liquid, how much energy is converted to increased temperature vs increased pressure?

Question

If you compress liquid in an infinitely stiff and infinity insulated cylinder such that the cylinder does not expand and no heat can transpire, how much of the energy will converted to increase in pressure and how much will be converted into increase in temperature?

This is something that J.P Joule himself documented here https://www.jstor.org/stable/pdf/108692.pdf in in his paper "On the Thermal Effects of Compressing Fluids" in the year 1858, which is a little hard to follow. Hoping someone can help and/or point me the right direction.

In most application the increase in temperature is ignored since it is pretty insignificant to the increase in pressure. But in the case of high pressure applications, it is significant. The application of interest is high pressure tests, where the process of pressurizing the liquid does heat the system such that upon reaching the test pressure the liquid is now hotter than it's environment and thus begins to cool. Yes some of the heat does come from the pump itself via conduction and viscous effects, but this only affects the fluid added, not the fluid that was already present which I can assure you does increase in temperature as well.

Whilst you can use the thermal expansion coefficient $\alpha\left[1 /^{\circ} \mathrm{C}\right]$ and Bulk Modulus of Elasticity $\beta [P a]$ to define the relationship between change in Pressure, Volume (or density) & Temp (PVT):

$$\frac{d V}{V}=-\frac{d \rho}{\rho} \cong \alpha \cdot d T-\frac{1}{\beta} \cdot d P$$

This is only of use if you know two of the three parameters. So I cold only use this to calculate the change in temperature if I know with a high level of accuracy in both the change in pressure and change in volume, which in most applications you only know the change in pressure with any level of accuracy since you don't know how much the pressure body expanded, if there is a leak, and if a flow meters is being used they are inherently inaccurate.

I'm sure there must be a way of calculating the how much energy used to compress a liquid goes to pressure and how much to temperature, such that if you know how much the pressure has increased, then you can calculate how much the temperature will increase. I suspect it will have something to do with Heat Capacities at constant volume $c_{v}$ and constant pressure $c_{p}$, but not yet sure how to implement them for this scenario.

Important to point out that for this scenario, neither the volume, pressure or temperature are constant, and typically speaking $\alpha$, $\beta$ and heat capacity values are listed/recorded to relation to one of these dimensions being constant i.e. constant volume, constant pressure or constant temperature.

Answer 1

I think, the solution you are looking for is right at the beginning of the paper, that you refer to. You just have to convert to modern units:

$$\Delta T = \frac{Te}{\rho K} \Delta p$$

with $T$ as the system's temperature in Kelvin, $e$ the liquid's thermal expansion coefficent, $\rho$ its density and $K$ its gravimetric heat capacity. For water at $300 \mathrm{K}$, I calculate a temperature increase of $1.5 \mathrm{K}$ at a pressure of 1000 Bar.

Answer 2

how much of the energy will converted to increase in pressure and how much will be converted into increase in temperature?

Since liquids are relatively incompressible, meaning their volume is relatively constant with varying pressure, pressure and temperature would not be independent. That said, it takes significant pressure to increase the temperature of, say, liquid water.

For example, adiabatically compressing saturated liquid water (water at the boiling point) from 10 KPa to liquid at 10 MPa (a 1000 fold increase) raises the temperature by only 5 C (based on the properties of compressed liquid water).

Hope this helps.