Derrivations Flashcards
Starting with the heat balance equation
ma(1+f)(h03-h02) = mfHVnb
where f is FAR, the fuel air ratio,am
is the air mass flow rate, h0i is the stagnation
enthalpy at point i,fm
is the fuel mass flow rate, HV is the heating value of the fuel
and ηb is the combustion efficiency. Derive an expression for f ’ = f/(1+f) in terms of
Cpb (specific heat at constant pressure p, in the burner),HV, ηb and ΔT0 (=T03-T02)
where T0i is the stagnation temperature at point i.
2014 Q1c
ΔT0 will also be equal to T03-T01 in this ramjet. Derive an expression for ΔT0 in terms
of the inlet temperature T1 and the dimensionless coefficients Θ0 (=T03/T1) and
θ0 (=T01//T1).
2014 Q1d
Use this expression to show that f ’ = f1(Θ0 - θ0) where f1 = CpbT1/HVnb
2014 Q1e
The thrust, T, provided by a propeller is given by; T = m(V3-V1)
where m is the mass flow through the disc, V1 is the upstream (free stream) velocity,
V2 is the velocity at the propeller and V3 is the fully developed slipstream velocity. If
a = (V2-V1)/V1 b = (V3-V1)/V1
then derive an expression for the thrust in terms of the air density, ρ0, propeller area,
A, inlet velocity, V1, and b, and find the relationship between a and b.
2014 Q2a
The solution to the quadratic equation derived as part of a) is
b = -1+-sqrt(q+(8/pi)*Tc)
where Tc is a thrust coefficient. For a propeller diameter, D, of 2.5m operating at n =
3000rpm, at sea-level-standard conditions at, V1 = 200mph, use the equations for
propellers provided to find the fully developed slipstream velocity and hence the
thrust and thrust power. Assume the coefficient CT = 0.1 and that 1 mile = 1609m
2014
Beginning with the energy equation
hc = (Ve^2/2)+he
derive the following variation on the rocket equation, for the maximum velocity of a
rocket in terms of the ratio of specific heats, γ, the molecular mass,
M, of the
combustion products, the universal gas constant,
R, chamber pressure, pc, and exit
pressure, pe.
Ve = sqrt( 2(gamma/gamma - 1)(R/M)Tc*(1-(Pe/Pc)^(gamma-1/gamma))
2014
For 1-Dimensional steady flow, accounting for the contribution of both external andinternal forces on the control region, derive an expression for the thrust, T in terms ofthe mass flow rate, m , and the velocity increase (V4-V1).
2015
Derive the Tsiolkovsky relation for rockets by considering the force balance equation for purely vertical ascent
T-W-D = M(dV/dt)
where T is the thrust (= 𝑚̇ c), W the weight, D the air drag, M the rocket mass, and dV/dt the acceleration of the rocket
2016 Q2c
Using the Rankine-Froude actuator disk model derive an expression for the thrust, T,
produced by a propeller in terms of the air density,
, cross sectional area of the disk,
A, forward speed, V and slipstream factor, b.
2018
