7. Radial Flow Turbines

Question 1

what is the change in stagnation enthalpy through a radial compressors rotor dh_0

Answer 1

• dh_0 = delta * 1/2(U^2 - W^2 + V^2)
• U = blade speed
• V = absolute velocity
• W = relative velocity

Question 2

for a radial inflow turbine with N straight radial blades, what is the mass flow per passage

Answer 2

• mass flow per passage = 𝑚̇/N

Question 3

what is the blade pitch s

Answer 3

• s = 2pi*r/N

Question 4

what is the mean relative velocity W_bar

Answer 4

• W_bar = 𝑚̇ / 2pi*rρb

Question 5

if the difference in relative velocity between the suction and pressure surfaces of the N blades is dW_θ = W_pressure - W_suction, what is the actual formula for dW_θ

Answer 5

• dW_θ = -2pi / N * 2Ωr

Question 6

if the difference in pressure between the suction and pressure surfaces of the N blades is dp_θ = p_pressure - p_suction, what is the actual formula for dp_θ

Answer 6

• dp_θ = -ρW_bar*dW_θ

Question 7

what is the radial pressure gradient wrt radius, dp/dr

Answer 7

• dp/dr = ρΩ^2r

Question 8

what is the slip factor σ and what is its typical value

Answer 8

• σ = 1 - sqrt(cos(X2)) / N^0.7
• σ = 0.85

Question 9

what is the stage loading coefficient Ψ and flow coefficient φ for an impeller (radial compressor rotor)

Answer 9

• Ψ = σ(1 + φtan(X2))
• φ = V_r2/U_2

Question 10

for radial blades, X2 = 0 is common. what does this mean about the relationship between Ψ and σ

Answer 10

• Ψ = σ

Question 11

what is the inducer

Answer 11

• the part of the impeller near the leading edge (the entrance basically)

Question 12

what is the axial velocity at the inlet V_x1

Answer 12

• V_x1 = 𝑚̇ / ρ_1pi(r_t^2 - r_h^2)
• r_t = tip radius
• r_h = hub radius

Question 13

what is the relative tangential velocity at the inducer tip W_θ1

Answer 13

• W_θ1 = Ω*r_t

Question 14

what is the total relative velocity at the inducer tip W_1

Answer 14

• W_1^2 = W_θ1^2 + V_x1^2

Question 15

how do you find the r_t that you want if you have the other variables

Answer 15

• you differentiate and find the r_t that minimizes the relative velocity
• because you want r_t to be as large as possible

Question 16

if the most important parameter of the diffuser is its throat width, how must it be sized

Answer 16

• to pass the required mass flow rate without choking
• to limit the diffusion between the impeller exit and the throat

Question 17

what are the typical pressure recovery coefficients through the diffuser, dp/(p_02 - p_2)

Answer 17

• dp/(p_02 - p_2) is between 0.5 and 0.6

Question 18

what is the continuity and conservation of angular momentum equation for the vaneless space between the impeller and diffuser

Answer 18

• 𝑚̇ = ρV_r2pirb = constant
• b = diffuser width
• rV_θ = constant

Question 19

what does the scroll in a radial turbine do

Answer 19

• the scroll adds swirl to the flow before it enters the stator blades

Question 20

what is the continuity equation for the inlet of a radial turbine

Answer 20

• 𝑚̇ = ρA_inV_θin = ρ2pir_1b_1V_r1
• b_1 = passage height of stator
• in = entry of the scroll

Question 21

what is the conservation of angular momentum of a radial turbine

Answer 21

• r_inV_θin = r_1V_θ1

Question 22

what is the formula for tan(α1)

Answer 22

• tan(α1) = V_θ1 / V_r1

Question 23

what are the typical entry conditions to the radial turbines rotor for α2 and β2

Answer 23

• α2 is between 65 and 80 degrees
• β2 = α2,rel = 0

Question 24

what is the stage loading coefficient at the rotor exit σ (also being the slip factor in this case)

Answer 24

• σ = dh_0/U_2^2
• this is still about 0.85

Question 25

what is the wasted kinetic energy as a proportion of the enthalpy drop, exit KE/dh_0

Answer 25

• exit KE/dh_0 = (0.5U_3^2φ_3^2) / (σ*U_2^2)

Question 26

what kinds of losses dominate at low and high specific speeds

Answer 26

• at low specific speeds the rotor losses dominate due to the high wetted area to flow area ratio in very short blades
• at high specific speeds the exit kinetic energy, causing leaving loss, dominates