Assume isentropic ratios for temperature and pressure at constant specific heat

Question

Assume isentropic ratios for temperature and pressure at constant specific heat ratios. Produce two plots where temperature is a function of the pressure ratio and pressure is a function of the temperature ratio. On both plots, include multiple curves where the specific heat ratio ranges from 1.1 < gamma < 1.8 (you may select resolution of step size). What do you observe about the functional forms as gamma changes?

Now consider the total temperature and total pressure properties as a STP air (gamma =1.4) accelerates to 333 m/s. What do you observe about the functional form of the total properties To and Po?

Explanation / Answer

In general specific heat(C) gives us an idea of the amount of energy(heat) we need to provide to a system in order to bring about a unit rise in the temperature of the system. It's value may vary depending on the process you are providing this energy. Hence we have two values of C namely Cv and Cp .

Cv for a gas is the change in internal energy (U) of a system with respect to change in temperature at a fixed volume of the system i.e. Cv =( U/ T)v  whereas Cp for a gas is the change in the enthalpy (H) of the system with respect to change in temperature at a fixed pressure of the system i.eCp = ( H/ T)p.
We know that, H = U + PV (+ VP, P=0 for constant pressure) . So the enthalpy term is greater than the internal energy term because of the PV term i.e in case of a constant pressure process more energy is needed, to be provided to the system as compared to that of a constant volume process to achieve the same temperature rise, as some energy is utilized in the expansion work of the system. And the relation that correlates these two is Cp = Cv + R

But since liquids and solids can practically assumed to be incompressible, Cp and Cv for them have almost same values and hence only a single value of specific heat is used for them.

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