Gravity's Impact on Perfect Gas Flow in Constant-Area Duct

[shair44]

Homework Statement

Consider a perfect gas flowing in a constant-area. Duct adiabatically and without friction. Changes in state come about as the result of changes in elevation in the Earth's gravity field. The z-direction is away from the center of the earth, and hence gravity acts in the negative z-direction.
(a) Starting from first principles, determine by analysis the direction of change (increase or decrease) of the Mach l\umber, gas speed, sound speed, density, pressure, stagnation temperature, and isentropic stagnation pressure, all for a positive increase in z,

(i) For subsonic speeds
(ii) For 3upersonic speeds

Homework Equations

continouty, momentum & energy equation
m=v/a

The Attempt at a Solution

 

[Valkarie]

(a) (i) For subsonic speeds, the Mach number will decrease, the gas speed will decrease, the sound speed will decrease, the density will increase, the pressure will increase, the stagnation temperature will increase, and the isentropic stagnation pressure will increase. (ii) For supersonic speeds, the Mach number will increase, the gas speed will increase, the sound speed will increase, the density will decrease, the pressure will decrease, the stagnation temperature will decrease, and the isentropic stagnation pressure will decrease.

 

[Mene]

I can provide a response to the given content by analyzing the impact of gravity on perfect gas flow in a constant-area duct.

Firstly, it is important to understand the concept of a constant-area duct. In a constant-area duct, the cross-sectional area remains constant throughout the flow, which means that the volume of the gas remains constant. This also implies that the mass flow rate remains constant.

Now, let us consider the effect of gravity on the flow of a perfect gas in this constant-area duct. Gravity acts in the negative z-direction, which means that as we move upwards (increasing z), we are moving against the direction of gravity. This results in a decrease in the weight of the gas, which in turn leads to a decrease in the density of the gas.

For subsonic speeds, the Mach number (M) is less than 1, which means that the flow is subsonic. As we move upwards, the decrease in density leads to a decrease in the velocity of the gas (v), as per the continuity equation (m = ρAv). This decrease in velocity also leads to a decrease in the sound speed (a), as per the equation a = √(γRT), where γ is the specific heat ratio, R is the gas constant, and T is the temperature.

As for the pressure and temperature, they are directly proportional to the density of the gas. Hence, as the density decreases, the pressure and temperature also decrease. However, in an adiabatic process, the stagnation temperature (T0) and isentropic stagnation pressure (P0s) remain constant. This implies that the decrease in temperature and pressure is compensated by an increase in velocity, as per the energy equation (T0 = T + (v^2/2Cp) and P0s = P(1 + (γ-1)/2M^2)^γ/(γ-1), where Cp is the specific heat at constant pressure).

For supersonic speeds, the Mach number (M) is greater than 1, which means that the flow is supersonic. As we move upwards, the decrease in density leads to an increase in the velocity of the gas, as per the continuity equation. This increase in velocity also leads to an increase in the sound speed, as per the equation a = √(γRT).

Similarly, the pressure and temperature also increase with an increase in velocity,