Do you want to send a rocket to space? Use this tool to estimate the duration the rocket engine should burn.
Assuming exhaust gasses speed ve, mass flow rate of the rocket engine equal to dm/dt, empty rocket mass M0, fuel mass Mf, v the rocket speed at time t, then for a rocket with non-constant mass (fuel is burnt), the total momentum after burn time t, is:
Σ ( Mf – dm/dt t + M0) dv
The total thrust (engine force minus gravity minus aerodynamic resistance) after burn time t is:
Σ (ve dm/dt – ( Mf – dm/dt t + M0) g – c v2) t
For the rocket to reach altitude hmax it needs to obtain a speed equal to √(2 g (hmax – h0 ) ) at altitude h0. Therefore at this altitude it should be:
Σ dv/dt = √(2 g h)
For simplification, it is assumed that the total rocket mass is constant equal to:
Mav = ⅔ dm/dt t+ M0
The required momentum is assumed to be ½ Mav √(2 g h).
The total thrust on the rocket after a time t is assumed to be (ve dm/dt – Mav g ) t.
Therefore, for your rocket to reach altitude h, the following should be satisfied:
½ √(2 g h) Mav = (ve dm/dt – Mav g ) t
This tool can be used to graphically solve the previous equation. The plot gives the required momentum and total thrust as function of the burn time. Play with the parameters to make the two curves touch each other. The x-value of the touch point is the time (the duration) your engine should burn.
Notes:
a typical ve value is 3 km/s,
for low orbit, the rocket should be able to reach a virtual altitude of 3000 km (corresponds to 7.8 km/s).
For space shuttle give ve=4.3e3 m/s, dm/dt=6000 kg/s, h=3e6 m, M0=150e3 kg
For technical specs of space shuttle see [1]
Suppose pressure of superheated steam equal to 100 bars or 10 MPa. According to [2], the density at 350 °C is close to 50 kg/m3. In the tank: velocity is 0 at 10 MPa, at the nozzle: velocity is ve and the pressure is 0.1 MPa. From Bernoulli Equation [3], ve= √(2×10 MPa / (50 kg/m3) )= 632 m/s.
The pressure of 10 MPa in a boiler tank with diameter of 2 m requires thickness (see [4, 5]) of 10 MPa 2 m / (2 × 300 MPa) = 3.33 cm. The cross-section area equals π (22 - 1.972 )/4 m2 = 935.41 cm2. The specific weight of this tank is 748 kg/m (see [6]). The specific storage of this tank is 3.05 m3/m, or, for superheated steam, 152.4 kg/m.
How to calculate the weight of the tank. For example, if dm/dt.=110 kg/s and the required burn time is 17 sec then the tank weight should be 17×110×152.4= 284992 kg. You can use duralumin, it has similar tensile strength with steel [7], and is much lighter [8]. In this case the tank weight will be 99747 kg.
Play with the previous tool to calculate the required mass rate. After obtaining the mass rate you can estimate the thrust force, which is equal to ve dm/dt. For example, if dm/dt.=110 kg/s the thrust force is 110×623=73.7 kN. The tank opening to the nozzle should be 73.7 kN / 10 MPa= 73.7 cm2.
