A converging-diverging nozzle with a throat area of 2 in² is connected to a large tank in which air is kept at a pressure of 80 psia and a temperature and a temperature of 60°F. If the nozzle is to operate at design conditions (flow is isentropic) and the ambient pressure outside the nozzle is 12.9 psia, calculate the ext area of the nozzle and the mass flow rate.
Use isentropic flow tables for supersonic flow (with k = 1.4) to achieve a pressure reduction factor of P/Po = 12.9/80 = 0.16125
You should get a Mach number of 1.85 and a flow area ratio A/A* = 1.495
The exit area should be 1.495 times the throat area, or 3.0 square inches.
The inside temperature is not needed.
To calculate the exit area of the nozzle and the mass flow rate, we need to use the isentropic flow equations for ideal gases.
1. First, let's find the specific gas constant for air:
The specific gas constant for air, R, is given by R = R_specific/M, where R_specific is the specific gas constant for air and M is the molar mass of air. The specific gas constant for air is approximately 1716.5 ft∙lbf/lbm∙°R, and the molar mass of air is approximately 28.97 lbm/lbmole. Therefore, R ≈ 1716.5/28.97 ≈ 59.4 ft∙lbf/lbm∙°R.
2. Next, let's calculate the stagnation temperature, T0:
The stagnation temperature is the temperature of the air if it were brought to rest isentropically. Since we know the tank temperature is 60°F, we need to convert it to the Rankine scale (°R) by adding 460. Thus, T0 = 60+460 = 520°R.
3. Now, let's calculate the stagnation pressure, P0:
The stagnation pressure is the pressure of the air if it were brought to rest isentropically. In this case, it is equal to the pressure in the tank, which is given as 80 psia.
4. To find the sonic velocity, a0, we can use the equation:
a0 = sqrt(g * R * T0), where g is the specific heat ratio (gamma) and is equal to 1.4 for air.
Plug in the values: a0 = sqrt(1.4 * 59.4 * 520) ≈ 1119.65 ft/s.
5. Next, we need to calculate the sonic area, A*:
A* = m_dot / (rho * a0), where m_dot is the mass flow rate and rho is the density of the air.
Since the flow is isentropic, density ratio (rho/rho0) can be represented as:
(rho/rho0) = (P/P0)^ (1/gamma), where P is the local pressure and rho0 is the density at the nozzle throat.
Plug in the values: A* = 2 in² = (m_dot / (rho0 * a0)) * (80 psia / 12.9 psia)^ (-1/1.4).
6. Rearrange the equation to solve for the mass flow rate, m_dot:
m_dot = A* * rho0 * a0 * (P/P0)^(1/1.4).
By solving the equation in step 6, you will obtain the mass flow rate. Then, you can substitute this value back into step 5 to find the exit area of the nozzle.