Complexities of PdV and VdP in Thermodynamics

PdV and VdP are both components of thermodynamic work PV. PdV is a change in volume at constant pressure while VdP is a change in pressure at constant volume.

We are familiar with the 1st law of thermodynamics equation, dH = dU + PdV. But remember, this equation applies to only constant pressure. This equation is not a universal truth for all processes. It has become a universal 1st law of thermodynamics because most of our processes occur at constant pressure.

Example: In Joule Thomson expansion when a real gas barring a few gases expands both pressure and volume changes. If the expansion is below the inversion temperature the gas cools. dH = dU + PdV equation, therefore, does not apply to the JT process. Here you must consider both PdV and VdP. The equation expands to dH = dU + PdV + VdP

The next obvious question is how these two terms contribute to total energy.

What is PdV? It is quite straight forwards

General description

PdV is volume change at constant pressure. If the volume of a system expands at constant pressure, the energy for this expansion comes from the internal energy of the system. The internal energy of a system consists of both its kinetic energy (KE) and potential energy (PE). In the case of constant pressure, there is usually no change in potential energy because most systems are not affected by changes in height or gravity. Therefore, the energy for expansion primarily comes from the kinetic energy of the particles within the system.

When the volume expands, the particles move apart and the average distance between them increases. This results in a decrease in the kinetic energy of individual particles, as they have to spread out over a larger space. As a result, the internal energy of the system decreases, and this energy is used to perform the expansion work against the external pressure. The net result is temperature decreases.

Example

Saturated steam making in the boiler

In a closed system like a boiler when saturated water is heated at constant pressure, at a constant temperature, of 100 degc, it breaks its H bonds and becomes saturated steam. The enthalpy of saturated water increases from 419 kj/kg to about 2260 kj/kg. PdV at a constant temperature. Thus PdV gives primarily enthalpy.

What is VdP?

Example

The main difference is PdV is the specific work for a closed system, as in a piston and cylinder, while ∫ VdP is the specific work for an open system, as in a steam or gas turbine. This is because steam (and gas) turbines are volumetric flow devices, described by ∫ VdP, not pistons described by ∫ PdV.

How does it work in a turbine?

The turbine is a typical example of an open system. In a turbine, the system boundary is the turbine housing. The turbine sucks in air through the boundary, which results in a?mass flow. The air is then compressed and mixed with fuel and heated (combustion of fuel), which then leads to a strong expansion of the gas. The expansion of the gas sets/keeps the turbine shaft in motion. The burned fuel-air mixture then leaves the turbine through the boundary. Part of the work done by the expansion is used to drive the compressor.

VdP?is isentropic shaft work from a flowing device.

Now if the device is isentropic, i.e., adiabatic-reversible. The Gibbs equation provides:

dH = dS + VdP

dH = VdP [ since dS =0 in isentropic process]

Therefore, delta W = dH = VdP

Therefore, VdP?is isentropic shaft work from a flowing device.