# Why is pressure not considered in calculation of 'Maximum achievable velocity of a gas'?

While reviewing the basics of Gas dynamics, I came across the eq for the maximum achievable velocity by gas.

Ref from Wikipedia

This is the velocity that is achievable by the gas by converting the heat content part of its internal energy into kinetic energy. Here, we are considering the process to be at constant pressure. If we don't consider a constant pressure, wont the max achievable velocity be higher?

Expanded Question: 2a) Can internal energy be directly converted into kinetic energy at constant pressure (adiabatic process without any work interaction)? 2b) Is a constant pressure adiabatic process possible (without any work interaction)?

From the Steady Flow Energy equation (SFEE),

$$h_{1}+\frac{{c_{1}}^2}{2}+ gz_{1}+q = h_{2}+\frac{{c_{2}}^2}{2}+ gz_{2}+w$$ $$...........eq1$$

Where, h is enthalpy per unit mass, u+PV

c is velocity

z is the height

q is the heat in to the system

w is the work done by the system P is the pressure V is the volume

Now, del(q) and del(w) are 0 (adiabatic and no work interaction), and if there is no change in the potential energy, del(gz) = 0.

Then eq1 bacomes

$$h_{1}+\frac{{c_{1}}^2}{2} = h_{2}+\frac{{c_{2}}^2}{2}$$

$$u_{1}+P_{1}V_{1}+\frac{{c_{1}}^2}{2} = u_{2}+P_{2}V_{2}+\frac{{c_{2}}^2}{2}$$

This still holds true if we consider the constant pressure process.

$$(u_{1}-u_{2})+P(V_{1}-V_{2})= \frac{{c_{2}}^2-{c_{1}}^2}{2}$$

From this the answer to both 2a and 2b is YES internal energy (considering constant pressure) can be converted to Kinetic energy. and YES constant pressure adiabatic process is possible.

But from

LINK

I know constant pressure adiabatic process is not possible and I cant find an example of internal energy (considering constant pressure) converted to Kinetic energy (adiabatic nozzle is not at constant pressure).

What am I missing here? How is max velocity achieved considering constant pressure?

• Instead of Wiki, why not use some of the material about gases and fluids by Feinmann? feynmanlectures.caltech.edu/I_39.html and feynmanlectures.caltech.edu/I_45.html Many of the lectures are also available as audio Commented Jun 27, 2023 at 9:04
• @SolarMike Thank you. Those lectures where excellent!. But I still don't understand how the Maximum achievable velocity is defined only by internal energy/temperature? Is this Wiki article not giving the complete picture? Commented Jun 27, 2023 at 12:45

## 1 Answer

Total temperature or stagnation temperature $$T_t$$ is related to total (stagnation) pressure $$P_t$$ by the ideal gas law. So you can convert one to the other. But the easiest way to see it is that your formula shows the internal energy $$c_pT_t$$ fully converted to kinetic energy $$\frac 1 2 v^2$$ (on a unit mass basis).

• Considering the ideal gas here. The total enthalpy (ht)remains constant as the static enthalpy (h)decreases and kinetic energy(KE) increases. In this hypothetical situation, the volume would be zero, and so too would enthalpy(as enthalpy is a function of temperature for perfect gas). The pressure has still remained constant Why cant this pressure be expanded to reach a higher velocity? Commented Jun 30, 2023 at 4:27
• Idk what you mean by volume being zero. That never happens. This situation of achieving maximum velocity means that you open a tank of gas directly into a vacuum, e.g. in the nozzle of a spacecraft. The pressure also goes to zero. Commented Jun 30, 2023 at 4:41
• The Wiki article shows the max velocity is achieved at constant pressure. By zero volume, I mean, by using the Perfect gas eq. Since the Pressure remains constant the volume has to go to zero as the temp goes to zero..? Commented Jun 30, 2023 at 16:00