3. Preliminary Design

Question 1

what is the specific speed

Answer 1

• specific speed is a dimensionless group
• whose numerical value is a means of deciding what type of machine is most suitable for an application

Question 2

what is the formula for the volume flow rate, Q

Answer 2

• Q = 𝑚̇/ρ

Question 3

what is the incompressible expression for the flow coefficient, φ

Answer 3

• φ = Q / ΩD^3

Question 4

what is the incompressible expression for the stage loading coefficient, Ψ

Answer 4

• Ψ = dp(0) / ρ Ω^2 D^2

Question 5

what is the formula for the specific speed, N(s), aka the shape factor

Answer 5

• N(s) = (𝑚̇/ρ_exit)^1/2 * Ω(dh_0)^-3/4

Question 6

what is the specific speed and specific diameter mainly used for

Answer 6

• choosing the type of machine at the preliminary design stage

Question 6

what is the formula for the specific diameter, D(s)

Answer 6

• D(s) = (dh_0)^1/4 * ρ_exit * D / 𝑚̇

Question 7

for a repeating stage axial machine, what are the 4 conditions that are satisfied for it to work, using 1 rotor and stator stage for reference

Answer 7

• V(x) = constant
• a(1) = a(3)
• V(1) = V(3)
• the change in V(θ) through the rotor is equal and opposite to that through the stator

Question 8

what is the formula for the stage loading coefficient in terms of the flow coefficient for a repeating axial machine, Ψ

Answer 8

• Ψ = φ(tan(a2) - tan(a1))

Question 9

what is the formula relating the flow coefficient and reaction to the stage loading coefficient, for a repeating axial machine Ψ

Answer 9

• Ψ = 2(1 - Λ - φtan(a1))

Question 10

what is the purpose of preliminary stage design

Answer 10

• to fix the velocity triangles by choosing the values of three dimensionless groups
• then by matching to the overall requirements, the layout of the machine can be determined

Question 11

what are the variables that a designer must fundamentally fix in order to fix the velocity triangles for a repeating stage machine

Answer 11

• three from a1, a2, b1 and b2
• fixing 3 automatically fixes the other

Question 12

rather than fixing a1, a2, b1 or b2 directly, which variables are better to fix

Answer 12

• φ, Ψ and Λ
• the missing flow angles can be found from other formulas relating them

Question 13

what is the formula for the reaction, Λ, in terms of φ

Answer 13

• Λ = 1 - φ/2(tan(a2) - tan(a1))

Question 14

what is the formula for tan(β)

Answer 14

• tan(β) = tan(a) - 1/φ

Question 15

how are the fixed values for φ, Ψ and Λ chosen

Answer 15

• from best practice and experience testing previous designs
• typical values for each are explored later

Question 16

what is the formula for the number of stages, n(stage) and how are the variables determined

Answer 16

• n(stage) = dh(0,overall) / (U^2 * Ψ)
• dh(0,overall) is fixed by the operating requirements
• U is determined by stress limits of operating requirements

Question 17

if the blade cross-section is constant from hub to tip, what is the formula for peak centrifugal stress in a rotor, σ

Answer 17

• σ = ρ(b) Ω^2 A(x) / 2pi

Question 18

what is the formula for the annulus area, A(x), if the mass flow rate and flow coefficient is known

Answer 18

• A(x) = 𝑚̇ / ρφU

Question 19

what is the formula for dh_0,stage if you can work out dh_0,machine and are told the number of stages N

Answer 19

• dh_0,stage = dh_0,machine / N

Question 20

what are the three repeating stage conditions

Answer 20

• Vx = constant
• a1 = a3
• V1 = V3

Question 21

for the reaction Λ = dh_rotor/dh_stage = 1 - dh_stator / dh_stage, what is dh_stage and dh_stator

Answer 21

• dh_stage = u^2*Ψ
• dh_stator = 1/2(V_2^2 - V_1^2)

Question 22

if the reaction is 0.5 in a repeating axial turbine stage, what is the relationship between the blade angles from a1 to a3 and b1 to b3

Answer 22

• b3 = -a2
• b2 = -a1

Question 23

what is zweifels expression Z in words

Answer 23

• Z = actual blade force / ideal blade force

Question 24

when investigating blade loading and zweifels rule, what is the first plot you draw

Answer 24

• V/V2 against axial distance
• youre drawing the pressure and suction surface distributions for high pitch (s) and spacing (s/c), low pitch and optimum

Question 25

how does the blade loading vary with high and low blade pitch s

Answer 25

• high pitch and spacing gives a high peak V
• boundary layer loss is proportional to V^3 so this is high
• low pitch and spacing gives a low peak V
• but losses due to high wetted area
• generally, as pitch increases blade loading increases

Question 26

what is the plot for showing the significance of the Z coefficient

Answer 26

• Cp against x/Cx
• Cp = p - p01 / p01 - p2
• you draw the same shape of the PS and SS distributions and shade it in
• but also with a rectangle from y = - to the end of the shaded plot
• this rectangle represents ideal loading and the shaded is ideal

Question 27

what does Z actually indicate

Answer 27

• the ideal loading is impossible because its a rectangle
• so Z indicates the area of overshoot necessary to achieve loading
• such that theres an optimum balance of loss due to wetted area and boundary layer loss