The ares rocket has a liquid oxygen tank that sits on top of the core stage. The turbopump inlet for the engines on the core stage must have a certain pressure to operate correctly. The engine designer has asked you to calculate the range of pressures he must account for.calculate the pressure at the inlet for both a full tan and a nearly empty tank. Assume the vehicle accelerates at a maximum of 4g's. The density of liquid oxygen is 71.14 lbm/ft3
Picture a rocket with liquid oxygen on top l= 50', and liquid hydrogen on bottom l=180'
To calculate the pressure at the turbopump inlet for both a full tank and a nearly empty tank, we'll first determine the weight of the liquid oxygen in the tank for both scenarios.
1. Full Tank:
Given:
Height of liquid oxygen (l) = 50 ft
Density of liquid oxygen (ρ) = 71.14 lbm/ft³
To find the weight of the liquid oxygen in the tank (W_f), we'll multiply the volume by the density and acceleration due to gravity (32.2 ft/s²):
W_f = ρ * V * g
The volume of the tank (V) can be calculated using the cylindrical volume formula:
V = π * r² * l
Assuming a cylindrical tank shape, we need to determine the radius of the tank. Unfortunately, the dimensions of the tank are not provided in the question. Please provide the radius of the tank so we can proceed with the calculations.
2. Nearly Empty Tank:
Given:
Height of liquid hydrogen (l) = 180 ft (assuming this is the length of the core stage)
To calculate the pressure at the turbopump inlet for a nearly empty tank, we need to consider the acceleration of the vehicle. The maximum acceleration given is 4g's, where 1g = 32.2 ft/s².
By multiplying the acceleration by the mass of the vehicle, we can determine the force acting on the turbopump inlet. Dividing this force by the cross-sectional area of the tank will give us the pressure.
Unfortunately, the mass or cross-sectional area of the vehicle is not provided. Please provide the necessary information so we can calculate the pressure at the turbopump inlet for a nearly empty tank.
To calculate the range of pressures at the turbopump inlet for a full and nearly empty tank, we can start by using the equation for pressure at a given depth in a liquid:
P = ρ * g * h
Where:
P is the pressure in pounds per square foot (psf),
ρ is the density of the liquid in pounds per cubic foot (lbm/ft^3),
g is the acceleration due to gravity in feet per second squared (ft/s^2), and
h is the height or depth in feet (ft).
In this case, we have liquid oxygen in the tank. Given that the density of liquid oxygen is 71.14 lbm/ft^3, we can substitute this value into the equation.
For a full tank, the height (h) is equal to the length of the tank, which is given as 50 feet. So the pressure at the turbopump inlet for a full tank is:
P_full = 71.14 * g * 50
Next, let's calculate the pressure at the turbopump inlet for a nearly empty tank. Since the tank is nearly empty, the height (h) of the liquid oxygen in the tank can be considered as negligible. Hence, the pressure at the inlet for a nearly empty tank is approximately zero.
Therefore, the pressure range the engine designer must account for is from nearly zero to P_full (calculated above). This is because the pressure at the turbopump inlet will decrease as the tank empties during the rocket launch.
Note: It's also worth mentioning that the acceleration due to gravity used in the calculation of pressure will be influenced by the acceleration of the rocket itself. Since the maximum acceleration mentioned is 4g's, you might need to consider this when calculating the actual pressure range during different stages of the rocket's acceleration.