Turbomachinery and Rocket Propulsion

Axial turbomachine velocity triangles

The velocity triangle is the vector diagram that relates the absolute velocity \mathbf{C}, the relative velocity \mathbf{W} seen by a blade, and the blade speed U = \omega r at a given radius through \mathbf{C} = \mathbf{W} + \mathbf{U}. In textbook problems the mean radius is used, and the axial velocity C_x is usually taken as constant through a stage. With all angles measured from the axial direction, the tangential components are C_\theta = C_x\tan\alpha and W_\theta = C_x\tan\beta, and the magnitudes are C = C_x/\cos\alpha and W = C_x/\cos\beta. The rotor relation W_\theta = C_\theta - U holds in the direction of rotation. When the two tangential components lie on opposite sides of the axis, the blade speed is their sum, and when they lie on the same side it is their difference. Given two speeds and a blade angle, the unknown axial velocity follows from C^2 = C_x^2 + C_\theta^2 at each station.

• blade angle — the angle between the relative flow and the axial direction at a blade edge. The blade metal angle is designed to match it, so a relative flow at a given angle needs a blade set at that angle.

The degree of reaction R of a turbine stage is the ratio of the static enthalpy change in the rotor to the static enthalpy change in the whole stage. With constant axial velocity it can be written as R = \frac{C_x}{2U}(\tan\beta_3 - \tan\beta_2) for the stated angle convention, so the axial velocity follows directly once the blade angles, the speed and R are known. The mass flow is then \dot m = \rho A C_x.

The absolute velocity leaving a nozzle row is resolved into axial and tangential parts. If it makes angle \alpha_1 with the axis, then C_x = C_1\cos\alpha_1 and C_\theta = C_1\sin\alpha_1. The rotor inlet relative tangential component is W_\theta = C_\theta - U, and the blade angle at the leading edge satisfies \tan\beta = W_\theta/C_x. A larger tangential relative component therefore requires a steeper blade.

For an ideal axial compressor stage with 50 % reaction and symmetrical blading, the static enthalpy rise is shared equally by the rotor and the stator. The rotor supplies the entire stagnation enthalpy rise because the stator does no work on the flow, and the static pressure rises in rotor and stator are approximately equal. The axial velocity stays constant through an axial stage, so it does not halve across the rotor. In such a stage the rotor inlet relative angle equals the stator exit absolute angle.

For a fixed rotational speed the blade speed U is fixed, so the flow coefficient \phi = C_x/U changes only through the axial velocity. With fixed density and annulus area, \dot m = \rho C_x A rises with the flow coefficient. A stationary nozzle row does no work, so the absolute stagnation enthalpy is unchanged across it. In an axial machine the blade speed is the same at rotor inlet and exit, so the rothalpy I = h + W^2/2 - U^2/2 is conserved, and the relative stagnation enthalpy is constant through the rotor.

Incidence is the difference between the actual relative flow angle entering a blade and its design blade angle. At fixed mass flow, raising the blade speed increases the tangential relative velocity and therefore the relative inlet angle, which can give positive incidence and flow separation on the suction surface. Raising the mass flow at fixed speed increases the axial velocity and produces negative incidence instead. Surge is linked to low flow at high pressure ratio, and raising the speed at constant mass flow pushes the operating point toward the surge line.

For an overall pressure ratio \pi and isentropic efficiency \eta, the actual stagnation temperature rise is \Delta T_0 = \frac{T_{01}}{\eta}\left(\pi^{(\gamma-1)/\gamma} - 1\right), and the specific work is w = c_p\,\Delta T_0. The Euler relation w = U(C_{\theta 2} - C_{\theta 1}) then gives the change in tangential velocity once U is known. Combined with a degree-of-reaction condition, it fixes both tangential components, and the outlet blade angle follows from \tan\beta_2 = W_{\theta 2}/C_x.

Across a compressor stage with losses, the rotor raises both total pressure and total temperature. The stator does no work, so its total temperature is nearly unchanged, but viscous losses lower its total pressure. The rotor accelerates the flow in the absolute frame and the stator diffuses it, so the absolute speed at the rotor exit is greater than at the rotor inlet, and the speed after the stator is lower than after the rotor.

The degree of reaction of a turbine stage is the ratio of the static temperature drop in the rotor to the static temperature drop across the whole stage. If station 1 is the stage inlet, station 2 the nozzle exit and station 3 the stage exit, then R = (T_2 - T_3)/(T_1 - T_3). For a gas with known c_p and \gamma, the gas constant is R = c_p(\gamma-1)/\gamma, and the static inlet temperature follows from p = \rho R T.

Rocket thrust, specific impulse and propellant mass

The thrust of a nozzle is F = \dot m V_e + (p_e - p_a)A_e, the sum of the momentum thrust and the pressure thrust. For an ideally expanded nozzle p_e = p_a, so F = \dot m V_e. Specific impulse is I_{sp} = F/\dot m in N·s/kg (equivalent to m/s), or I_{sp} = F/(\dot m g_0) in seconds. Hence the propellant mass flow rate is \dot m = F/I_{sp} when I_{sp} is in N·s/kg, and the propellant burned in time t is m_p = \dot m\,t. A common mistake is to stop at the flow rate and forget to multiply by the burn time.

With I_{sp} in seconds, the thrust relation becomes F = \dot m\, I_{sp}\, g_0. Solving for the mass flow gives \dot m = F/(I_{sp} g_0).

For an isentropic expansion from stagnation temperature T_0 to exit pressure p_e, the exit velocity is V_e = \sqrt{\frac{2\gamma}{\gamma-1}RT_0\left[1-(p_e/p_0)^{(\gamma-1)/\gamma}\right]}, where R = \mathcal{R}_u/M uses the molar mass M. When the nozzle is ideally expanded, I_{sp} = V_e/g_0 in seconds.

The thrust can also be written F = C_F\,p_c A_t, where the thrust coefficient C_F depends only on the pressure ratio and the area ratio. The characteristic velocity c^* = p_c A_t/\dot m gives I_{sp} = C_F c^*/g_0, and rearranging gives the chamber pressure p_c = \dot m c^*/A_t.

An underexpanded nozzle has exit pressure above ambient, so an extra pressure thrust term (p_e - p_a)A_e/(p_c A_t) is added to C_F. At a fixed area ratio, thrust therefore does not scale exactly with chamber pressure. An optimally expanded nozzle has p_e = p_a and no pressure term.

For a rocket with a constant effective exhaust velocity and no external forces, the ideal rocket equation is \Delta V = I_{sp} g_0 \ln(m_0/m_f). When two vehicles share the same I_{sp}, their velocity ratio depends only on their mass ratios. A constant burn rate reduces the mass as m_f = m_0 - \dot m t, which changes the burnout mass and therefore the ratio of logarithms.