The flow at the entrance and exit of an axial-flow compressor rotor is in radial equilibrium. The

distributions of the tangential components of absolute velocity with radius are

c?1 ar b=r, before the rotor,

c?2 ar b=r, after the rotor,

where a and b are constants. What is the variation of work done with radius? Deduce expressions

for the axial velocity distributions before and after the rotor, assuming incompressible flow theory and that the radial gradient of stagnation pressure is zero. At the mean radius, r 0.3 m, the

stage loading coefficient, ? ?W/U2

t is 0.3, the reaction ratio is 0.5, and the mean axial velocity

is 150 m/s. The rotor speed is 7640 rev/min. Determine the rotor flow inlet and outlet angles at a

radius of 0.24 m given that the hub-tip ratio is 0.5. Assume that at the mean radius the axial

velocity remained unchanged (cx1 cx2 at r 0.3 m). (Note: ?W is the specific work and Ut

the blade tip speed.)

6. An axial-flow turbine stage is to be designed for free-vortex conditions at exit from the nozzle

row and for zero swirl at exit from the rotor. The gas entering the stage has a stagnation temperature of 1000 K, the mass flow rate is 32 kg/s, the root and tip diameters are 0.56 m and 0.76 m,

respectively, and the rotor speed is 8000 rev/min. At the rotor tip the stage reaction is 50% and

the axial velocity is constant at 183 m/s. The velocity of the gas entering the stage is equal to that

leaving. Determine

(i) the maximum velocity leaving the nozzles;

(ii) the maximum absolute Mach number in the stage;

(iii) the root section reaction;

(vi) the power output of the stage;

(v) the stagnation and static temperatures at stage exit.

Take R 0.287 kJ/(kg K) and Cp 1.147 kJ/(kg K).

7. The rotor blades of an axial-flow turbine stage are 100 mm long and are designed to receive gas

at an incidence of 3 deg from a nozzle row. A free-vortex whirl distribution is to be maintained

between nozzle exit and rotor entry. At rotor exit the absolute velocity is 150 m/s in the axial

direction at all radii. The deviation is 5 deg for the rotor blades and zero for the nozzle blades

at all radii. At the hub, radius 200 mm, the conditions are as follows:

Nozzle outlet angle 70

Rotor blade speed 180 m/s

Gas speed at nozzle exit 450 m/

Using the definitions of parameters given below, choose the appropriate procedure for each situation described. PM=proportion of all male passengers who survived " PF= proportion of all female passengers who survived . HM = mean age of all male passengers . HF = mean age of all female passengers Determine whether survival rates differed between male and female ^ A confidence interval for PM passengers B. A confidence interval for PF Estimate the difference in average ages of male and female passengers. C. A confidence interval for the difference UM - HF Determine whether more than half of all female passengers survived. D. A confidence interval for the difference PM - PF Estimate the average age of all male passengers E. A hypothesis test with Ho: HM = 30, Ha: HM # 30 F. A hypothesis test with Ho: p = 0.5, Ha: p > 0.5 G. A hypothesis test with Ho: PM = HF. Ha: HM # HF H. A hypothesis test with Ho: PM = PF. Ha: PM # PF1. (12pts) The following 16 random samples; 5.33, 4.25, 3.15, 3.70, 1.61, 6.40, 3.12, 6.59, 3.53, 4.74, 0.11, 1.60, 5.49, 1.72, 4.15, 2.30; came from normal distribution with mean / and variance o', i.e., X1, X2, ..., X16 ~ N(1, o'), with the density function f(I) : 262 V2TO (a) (4pts) Find the maximum likelihood estimates of u and o', denoted with a and a2. (b) (4pts) Based on above a and o', construct 95% confidence intervals for p and of separately. (c) (4pts) Based on part (b) and the duality of confidence intervals and the hypothesis tests, carry out a hypothesis test of Ho : p = 4 versus Hi : u # 4 at a = 0.05.Describe the connection between 100(1 - o)% confidence intervals and two-sided significance tests with significance level o for population means. Choose the correct answer below. (i) A 100(1 - @)% confidence interval is consistent with a two-sided test with significance level a because if a two-sided test with this significance level does not reject the null hypothesis, then the confidence interval contains the value in the null hypothesis. (ii) A 100(1 - @)% confidence interval is consistent with a two-sided test with significance level a because if a two-sided test with this significance level does not reject the null hypothesis, then the confidence interval does not contain the value in the null hypothesis. (iii) A 100(1 - @)% confidence interval is consistent with a two-sided test with significance level 20 because if a two-sided test with this significance level does not reject the null hypothesis, then the confidence interval contains the value in the null hypothesis. (iv) A 100(1 - o)% confidence interval is consistent with a two-sided test with significance level 20 because if a two-sided test with this significance level does not reject the null hypothesis, then the confidence interval does not contain the value in the null hypothesis. (v) A 100(1 - @)% confidence interval is consistent with a two-sided test with significance level a/2 because if a two-sided test with this significance level does not reject the null hypothesis, then the confidence interval does not contain the value in the null hypothesis