3. Preliminary Design Flashcards
what is the specific speed
- specific speed is a dimensionless group
- whose numerical value is a means of deciding what type of machine is most suitable for an application
what is the formula for the volume flow rate, Q
- Q = 𝑚̇/ρ
what is the incompressible expression for the flow coefficient, φ
- φ = Q / ΩD^3
what is the incompressible expression for the stage loading coefficient, Ψ
- Ψ = dp(0) / ρ Ω^2 D^2
what is the formula for the specific speed, N(s), aka the shape factor
- N(s) = (𝑚̇/ρ_exit)^1/2 * Ω(dh_0)^-3/4
what is the specific speed and specific diameter mainly used for
- choosing the type of machine at the preliminary design stage
what is the formula for the specific diameter, D(s)
- D(s) = (dh_0)^1/4 * ρ_exit * D / 𝑚̇
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
- 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
what is the formula for the stage loading coefficient in terms of the flow coefficient for a repeating axial machine, Ψ
- Ψ = φ(tan(a2) - tan(a1))
what is the formula relating the flow coefficient and reaction to the stage loading coefficient, for a repeating axial machine Ψ
- Ψ = 2(1 - Λ - φtan(a1))
what is the purpose of preliminary stage design
- 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
what are the variables that a designer must fundamentally fix in order to fix the velocity triangles for a repeating stage machine
- three from a1, a2, b1 and b2
- fixing 3 automatically fixes the other
rather than fixing a1, a2, b1 or b2 directly, which variables are better to fix
- φ, Ψ and Λ
- the missing flow angles can be found from other formulas relating them
what is the formula for the reaction, Λ, in terms of φ
- Λ = 1 - φ/2(tan(a2) - tan(a1))
what is the formula for tan(β)
- tan(β) = tan(a) - 1/φ
how are the fixed values for φ, Ψ and Λ chosen
- from best practice and experience testing previous designs
- typical values for each are explored later
what is the formula for the number of stages, n(stage) and how are the variables determined
- 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
if the blade cross-section is constant from hub to tip, what is the formula for peak centrifugal stress in a rotor, σ
- σ = ρ(b) Ω^2 A(x) / 2pi
what is the formula for the annulus area, A(x), if the mass flow rate and flow coefficient is known
- A(x) = 𝑚̇ / ρφU
what is the formula for dh_0,stage if you can work out dh_0,machine and are told the number of stages N
- dh_0,stage = dh_0,machine / N
what are the three repeating stage conditions
- Vx = constant
- a1 = a3
- V1 = V3
for the reaction Λ = dh_rotor/dh_stage = 1 - dh_stator / dh_stage, what is dh_stage and dh_stator
- dh_stage = u^2*Ψ
- dh_stator = 1/2(V_2^2 - V_1^2)
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
- b3 = -a2
- b2 = -a1
what is zweifels expression Z in words
- Z = actual blade force / ideal blade force
when investigating blade loading and zweifels rule, what is the first plot you draw
- 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
how does the blade loading vary with high and low blade pitch s
- 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
what is the plot for showing the significance of the Z coefficient
- 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
what does Z actually indicate
- 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
