SSRAO

Explanation:
A turbomachine is a device in which energy transfer between a fluid and a rotor occurs through dynamic (hydrodynamic) action — the fluid exerts forces on the rotating blades, and vice versa. This distinguishes turbomachines from positive displacement machines (pumps, compressors with pistons/gears). Turbomachines include turbines (fluid to rotor), pumps/compressors/fans (rotor to fluid), and torque converters. The energy transfer is continuous and involves changes in fluid momentum.

Explanation:
Euler's turbomachine equation: W = U1Vw1 - U2Vw2 (work output per unit mass for turbine), where U1, U2 are blade speeds at inlet and outlet, and Vw1, Vw2 are whirl (tangential) components of absolute velocity. For a pump/compressor: W_input = U2Vw2 - U1Vw1. This is derived from the angular momentum equation (moment of momentum). It applies to all turbomachines regardless of fluid type or machine geometry.

Explanation:
Specific speed of turbine: Ns = N√P / H^(5/4), where N is speed (rpm), P is power (kW or hp), H is head (m). It represents the speed of a geometrically similar turbine that produces unit power (1 kW or 1 hp) under unit head (1 m). Specific speed is used to select the type of turbine: Pelton (Ns = 10-35), Francis (Ns = 60-300), Kaplan (Ns = 300-900). Higher specific speed → higher flow rate, lower head.

Explanation:
Degree of reaction R = (static enthalpy drop in rotor) / (total enthalpy drop in stage) = (p1 - p2) / (p0_in - p0_out) for incompressible flow. For an axial turbine stage: R = 1 - (Vw1 + Vw2)/(2U). R = 0: impulse stage (all pressure drop in nozzle, none in rotor). R = 0.5: 50% reaction (equal pressure drop in nozzle and rotor). R = 1: all pressure drop in rotor. Most axial turbines use R = 0.5 for symmetric velocity triangles.

Explanation:
The velocity triangle relates three velocities vectorially: V = U + Vr, where V is absolute velocity of fluid, U is blade (peripheral) velocity, and Vr is relative velocity of fluid with respect to blade. This is a vector equation. The velocity triangle is fundamental to turbomachinery analysis. At inlet: V1, U1, Vr1 form the inlet triangle. At outlet: V2, U2, Vr2 form the outlet triangle. The Euler equation is derived from these triangles.

Explanation:
Flow coefficient φ = Vf/U = Cf/U, where Vf (or Cf) is the flow (meridional) component of absolute velocity and U is the blade peripheral speed. φ is a dimensionless parameter used in turbomachinery similarity and performance analysis. It is analogous to the flow coefficient in dimensional analysis. For geometrically similar machines at the same operating point, φ is constant. Typical values: φ = 0.1-0.4 for centrifugal pumps.

Explanation:
Head coefficient (pressure coefficient) ψ = gH/U², where g is gravitational acceleration, H is head (m), and U is blade tip speed (m/s). ψ is dimensionless and used in turbomachinery similarity. For geometrically similar machines: ψ = constant at the same operating point. From Euler equation: ψ = Vw2/U2 - Vw1/U1 (for pump with no pre-swirl: ψ = Vw2/U2). Typical values: ψ = 0.3-0.6 for centrifugal pumps.

Explanation:
Power coefficient π = P/(ρU³D²) = φ × ψ, where P is power, ρ is fluid density, U is blade speed, D is diameter. This follows from: P = ρQgH = ρ(φUD²)(ψU²) = ρU³D²(φψ). Therefore π = φψ. These three dimensionless coefficients (φ, ψ, π) completely characterise the performance of a turbomachine family. At the same operating point (same φ), geometrically similar machines have the same ψ and π.

Explanation:
Specific speed Ns = NQ^0.5/H^0.75 = 1500 × (0.1)^0.5 / (20)^0.75 = 1500 × 0.3162 / 9.457 = 474.3 / 9.457 ≈ 50.2. In SI units (N in rpm, Q in m³/s, H in m), Ns ≈ 50-53. This falls in the centrifugal pump range (Ns = 10-70 in SI). Note: specific speed values depend on the unit system used. In metric units (Q in m³/min), Ns would be different. The answer ≈ 53 is correct for the given SI formula.

Explanation:
Thoma cavitation number σ = NPSH_A / H, where NPSH_A is the available Net Positive Suction Head and H is the pump head. Cavitation occurs when σ < σ_critical (σ_c). NPSH_A = (p_atm - p_v)/(ρg) - Z_s - h_f, where p_v is vapour pressure, Z_s is suction head, h_f is friction loss in suction pipe. NPSH_R (required) is determined by the pump manufacturer. For safe operation: NPSH_A > NPSH_R. Cavitation causes noise, vibration, erosion, and performance loss.

Explanation:
Manometric efficiency η_m = H_m / H_Euler = H_m / (U2Vw2/g), where H_m is the actual manometric head developed by the pump and H_Euler = U2Vw2/g is the theoretical (Euler) head. η_m accounts for hydraulic losses within the pump (friction, recirculation, shock). Typical η_m = 0.75-0.95. Overall efficiency η_o = η_m × η_v × η_mech, where η_v is volumetric efficiency and η_mech is mechanical efficiency.

Explanation:
Slip factor σ_s = Vw2_actual / Vw2_ideal = Vw2_actual / U2 (for radial blades with no pre-swirl). Due to slip (finite number of blades, relative eddy), the actual whirl velocity at impeller exit is less than the ideal (Vw2 < U2 for radial blades). Typical σ_s = 0.85-0.92. Stodola's slip factor: σ_s = 1 - π/z (where z is number of blades). Stanitz formula: σ_s = 1 - 0.63π/z. Slip reduces the work input and pressure rise.

Explanation:
Work done factor (λ) in axial compressors accounts for the non-uniform axial velocity profile across the annulus (boundary layer growth on hub and casing walls). The actual work input is less than the Euler work: W_actual = λ × U × ΔVw, where λ < 1 (typically 0.85-0.98). The non-uniform velocity profile means the mean axial velocity is less than the design value, reducing the whirl velocity change and hence work input. λ decreases with increasing number of stages.

Explanation:
A single stage of an axial flow compressor typically achieves a pressure ratio of 1.1 to 1.4 (sometimes up to 1.6 in advanced designs). This is much lower than centrifugal compressors (3-8 per stage) because axial stages have lower blade loading to avoid flow separation and stall. Multiple stages are used in series to achieve high overall pressure ratios (e.g., 30:1 in aircraft engines with 15-20 stages). The low per-stage pressure ratio allows high efficiency.

Explanation:
The tip Mach number M_tip = U_tip/a (where a is speed of sound) is critical in centrifugal compressors. When M_tip > 1, shock waves form in the inducer/impeller passages, causing: (1) Additional losses, (2) Reduced efficiency, (3) Choking of flow. Modern high-speed centrifugal compressors operate with M_tip = 1.2-1.8 using swept-back blades and careful aerodynamic design to manage shocks. The tip Mach number limits the maximum pressure ratio achievable in a single stage.

Explanation:
In an impulse turbine (degree of reaction R = 0): all the pressure drop (enthalpy drop) occurs in the nozzle/stator. The rotor operates at constant pressure — only the direction of the high-velocity jet changes. The Pelton wheel is a pure impulse turbine. In the de Laval steam turbine, steam expands fully in nozzles and the high-velocity jets impinge on the rotor buckets. The relative velocity in the rotor is constant (no pressure change), only direction changes.

Explanation:
Specific speed ranges for hydraulic turbines (metric units, N in rpm, P in kW, H in m): Pelton wheel: Ns = 10-35 (low specific speed, high head, low flow), Francis turbine: Ns = 60-300 (medium), Kaplan turbine: Ns = 300-900 (high specific speed, low head, high flow). The Pelton wheel is used for very high heads (100-2000 m) and low flow rates. Low specific speed corresponds to high head, low flow applications.

Explanation:
Euler head H_Euler = (U2Vw2 - U1Vw1)/g. With no pre-swirl at inlet (Vw1 = 0, radial entry): H_Euler = U2Vw2/g. From the velocity triangle at outlet: Vw2 = U2 - Vf2/tan(β2), where β2 is blade angle at outlet and Vf2 is flow velocity. For backward-curved blades (β2 < 90°): Vw2 < U2. For radial blades (β2 = 90°): Vw2 = U2. For forward-curved blades (β2 > 90°): Vw2 > U2.

Explanation:
Cavitation occurs when the local static pressure in the flow falls below the vapour pressure of the liquid at that temperature. Vapour bubbles form (cavities) and when they move to higher pressure regions, they collapse violently, causing: (1) Noise and vibration, (2) Erosion of blade surfaces (pitting), (3) Reduction in efficiency and head, (4) Structural damage. In hydraulic turbines, cavitation occurs at the runner exit (draft tube inlet) where pressure is lowest. Thoma's cavitation coefficient σ is used to assess cavitation risk.

Explanation:
The draft tube serves two purposes: (1) Kinetic energy recovery — the diverging draft tube decelerates the flow, converting kinetic energy to pressure energy (diffuser action), increasing turbine efficiency by 10-15%, (2) Allows turbine installation above tailwater level — the draft tube creates a negative pressure (suction) at the turbine exit, effectively increasing the available head. Without a draft tube, the turbine must be installed at tailwater level. The draft tube efficiency η_dt = (V3² - V4²)/(2g × V3²/2g).

Explanation:
Pressure rise in a centrifugal pump impeller occurs due to two mechanisms: (1) Centrifugal action — the rotating fluid experiences centrifugal force, creating a pressure gradient from hub to tip: Δp_centrifugal = ρ(U2² - U1²)/2, (2) Diffusion — the relative velocity decreases from inlet to outlet (Vr1 > Vr2), converting kinetic energy (in relative frame) to pressure: Δp_diffusion = ρ(Vr1² - Vr2²)/2. The total pressure rise = centrifugal + diffusion components.

Explanation:
Stall in axial compressors occurs when the angle of attack (incidence angle) on the blades exceeds the stall angle, causing boundary layer separation on the suction surface of the blade. This happens at low flow rates (high incidence) or high flow rates (negative incidence). Stall causes: (1) Loss of pressure rise, (2) Increased losses, (3) Rotating stall (stall cells rotating around the annulus at 20-70% of rotor speed), (4) Surge (if stall propagates to full annulus). Stall is the precursor to surge.

Explanation:
Surge is a global instability in compressors characterised by: (1) Violent, periodic oscillations of mass flow (including flow reversal), (2) Large pressure fluctuations, (3) Intense noise and vibration, (4) Potential damage to blades, bearings, and seals. Surge occurs when the compressor cannot maintain the required pressure rise at low flow rates. The surge line on the compressor map defines the stability boundary. Anti-surge control systems (blow-off valves, recirculation) prevent surge. Surge is more dangerous than stall.

Explanation:
Hydraulic efficiency η_h = Runner power / Water power = (Power developed by runner) / (ρgQH). It accounts for hydraulic losses in the turbine (friction in passages, shock losses, leakage). Overall efficiency η_o = η_h × η_mech × η_v, where η_mech accounts for bearing friction and η_v for volumetric losses. For a Pelton wheel: η_h = 2(U/V1)(V1 - U)(1 + k·cos β), where k is blade friction coefficient and β is bucket angle.

Explanation:
Specific speed Ns = NQ^0.5/H^0.75 is used to: (1) Select the appropriate pump type — centrifugal (low Ns), mixed flow (medium Ns), axial flow (high Ns), (2) Predict the shape of the H-Q curve and efficiency curve, (3) Compare geometrically similar pumps, (4) Scale up/down pump designs. Low Ns (10-30): centrifugal, high head, low flow. Medium Ns (30-100): mixed flow. High Ns (100-300): axial flow, low head, high flow. Specific speed is constant for geometrically similar pumps at the same operating point.

Explanation:
The whirl velocity (Vw) is the tangential (circumferential) component of the absolute velocity V. It is the component in the direction of blade motion (U). The Euler equation uses whirl velocities: W = U1Vw1 - U2Vw2 (turbine) or W = U2Vw2 - U1Vw1 (pump). The flow velocity (Vf) is the meridional (axial or radial) component. The absolute velocity V = √(Vw² + Vf²). Whirl velocity determines the angular momentum change and hence the torque and work.

Explanation:
Blade loading coefficient (temperature drop coefficient) for axial turbine: ψ = W/U² = ΔVw/U = (Vw1 + Vw2)/U (for turbine with Vw1 and Vw2 in same direction). From Euler equation: W = U(Vw1 + Vw2) for turbine (assuming Vw2 is in same direction as Vw1). Therefore ψ = W/U² = (Vw1 + Vw2)/U. Typical values: ψ = 1.5-3.5 for turbines. Higher ψ means more work per stage but lower efficiency. ψ is related to degree of reaction: R = 1 - ψ/2 for symmetric velocity triangles.

Explanation:
Choking occurs when the Mach number reaches 1 (sonic condition) at some cross-section in the flow path (typically the throat of a nozzle or the minimum area section). Once choked, the mass flow rate cannot increase further regardless of the pressure ratio. On a compressor map, the choke line represents the maximum flow at each speed. Choking limits the operating range of the compressor. In turbines, choking of the nozzle limits the mass flow and power output.

Explanation:
Polytropic efficiency η_p is the small-stage (elemental) efficiency — the ratio of isentropic work to actual work for an infinitesimally small compression step, assumed constant throughout the process. For a compressor: η_p = (γ-1)/γ × ln(p2/p1) / ln(T2/T1). The overall isentropic efficiency η_is = [(p2/p1)^((γ-1)/γ) - 1] / [(p2/p1)^((γ-1)/(γη_p)) - 1]. For a compressor: η_is < η_p (preheat effect). For a turbine: η_is > η_p. Polytropic efficiency is more useful for comparing multi-stage machines.

Explanation:
Pre-whirl (inlet guide vanes giving swirl in the direction of rotation) reduces the relative velocity at the inducer tip: Vr1 = √(Vf1² + (U1 - Vw1)²). By adding positive pre-swirl (Vw1 in direction of U1), the relative velocity Vr1 decreases, reducing the relative Mach number M_r1. This allows higher rotational speed (higher U1) without exceeding the critical relative Mach number, enabling higher pressure ratios. However, pre-whirl reduces the work input (Euler equation: W = U2Vw2 - U1Vw1).

Explanation:
Turbomachines are classified by flow direction through the rotor: (1) Axial flow — fluid flows parallel to the rotation axis (axial compressors, axial turbines, propellers), (2) Radial flow — fluid flows perpendicular to the rotation axis (centrifugal pumps/compressors, radial turbines), (3) Mixed flow — fluid flows at an angle between axial and radial (mixed flow pumps, Francis turbines). This classification determines the velocity triangle geometry and performance characteristics.

Explanation:
The Pelton wheel (impulse turbine) is suitable for: high heads (typically 300-2000 m), low flow rates, and low specific speeds (Ns = 10-35). The high head creates high-velocity jets that impinge on the buckets. At low heads, the jet velocity would be too low for efficient operation. Pelton wheels are used in mountainous regions with high-head reservoirs. Francis turbines: 40-600 m head. Kaplan turbines: 2-40 m head. The choice depends on head, flow, and specific speed.

Explanation:
For a Pelton wheel, the hydraulic efficiency η_h = 2(U/V1)(1 - U/V1)(1 + k·cos β), where k is blade friction coefficient and β is bucket angle. Differentiating with respect to U/V1 and setting to zero: d(η_h)/d(U/V1) = 0 gives U/V1 = 0.5 for ideal case (k=1, β=180°). In practice, with friction (k < 1) and β ≈ 165°, the optimum U/V1 ≈ 0.46. At this speed ratio, the relative velocity at exit is nearly zero, meaning maximum energy is extracted from the jet.

Explanation:
The Francis turbine is a mixed flow, reaction turbine: (1) Mixed flow — water enters radially (inward flow) and exits axially, (2) Reaction — both pressure and velocity change in the runner (degree of reaction R = 0.5-0.7). Water enters the spiral casing, passes through guide vanes (wicket gates), enters the runner radially, and exits axially into the draft tube. Francis turbines are used for medium heads (40-600 m) and medium flow rates. They are the most widely used hydraulic turbines worldwide.

Explanation:
The Kaplan turbine is an axial flow, reaction turbine with adjustable (variable pitch) runner blades. The adjustable blades allow the turbine to maintain high efficiency over a wide range of flow rates and heads (unlike fixed-blade propeller turbines). Both the guide vanes and runner blades are adjustable. Kaplan turbines are used for low heads (2-40 m) and high flow rates (high specific speed Ns = 300-900). They are used in run-of-river plants and tidal power stations.

Explanation:
NPSH_Required (NPSH_R) is the minimum head (above vapour pressure head) that must be available at the pump inlet to prevent cavitation. NPSH_R is determined by the pump manufacturer from tests. NPSH_Available = (p_atm - p_v)/(ρg) - Z_s - h_f_suction, where Z_s is suction lift and h_f is friction loss. For safe operation: NPSH_A > NPSH_R (typically NPSH_A > NPSH_R + 0.5-1.0 m safety margin). NPSH_R increases with pump speed and flow rate.

Explanation:
Affinity laws for centrifugal pumps (same pump, variable speed): Q ∝ N (flow proportional to speed), H ∝ N² (head proportional to speed squared), P ∝ N³ (power proportional to speed cubed). These follow from dimensional analysis: Q = φND³, H = ψN²D², P = πρN³D⁵. For variable diameter (same speed): Q ∝ D³, H ∝ D², P ∝ D⁵. These laws are used for pump speed control (VFD) — reducing speed by 20% reduces power by 51% (energy savings).

Explanation:
50% reaction (R = 0.5) means equal static enthalpy (pressure) drop in the stator (nozzle) and rotor. This results in symmetric velocity triangles (inlet and outlet triangles are mirror images). Advantages of 50% reaction: (1) Symmetric blade profiles (same blade shape for stator and rotor), (2) No tendency for boundary layer separation, (3) High efficiency. The axial velocity is constant (Vf1 = Vf2), and the absolute velocity at rotor exit equals the absolute velocity at stator inlet (V3 = V1), enabling repeating stages.

Explanation:
Isentropic efficiency of compressor η_is = W_isentropic / W_actual = (h2s - h1) / (h2 - h1), where h2s is the enthalpy at the end of isentropic compression and h2 is the actual exit enthalpy. Since actual compression involves irreversibilities (friction, heat transfer), h2 > h2s, so η_is < 1. For an ideal gas: η_is = (T2s - T1)/(T2 - T1) = [(p2/p1)^((γ-1)/γ) - 1] / (T2/T1 - 1). Typical η_is = 0.75-0.90 for centrifugal compressors.

Explanation:
For a Pelton wheel: W = U(Vw1 - Vw2). At inlet: Vw1 = V1 (jet velocity, all tangential). Relative velocity at inlet: Vr1 = V1 - U. At outlet: Vr2 = k·Vr1 = k(V1-U) (k = blade friction coefficient). Vw2 = U - Vr2·cos β = U - k(V1-U)cos β. Therefore: W = U(V1 - U + k(V1-U)cos β) = U(V1-U)(1 + k·cos β). Hydraulic efficiency: η_h = W/(V1²/2) = 2(U/V1)(1-U/V1)(1+k·cos β). Maximum at U/V1 = 0.5.

Explanation:
Multiple jets in a Pelton wheel increase the power output (P ∝ number of jets) without increasing the runner diameter or speed. Each jet contributes independently to the power. Multiple jets also increase the specific speed (Ns ∝ √(number of jets)). Pelton wheels can have 1-6 jets. Multiple jets are used when: (1) Single jet would be too large, (2) Higher specific speed is needed, (3) Higher power is required at the same head and speed. The jets are equally spaced around the runner.

Explanation:
Dimensional analysis of a pump (variables: Q, H, N, D, ρ, μ, g) yields dimensionless groups: (1) Flow coefficient φ = Q/(ND³), (2) Head coefficient ψ = gH/(N²D²), (3) Power coefficient π = P/(ρN³D⁵), (4) Reynolds number Re = ρND²/μ. For geometrically similar pumps at high Re (turbulent, Re effect negligible): φ, ψ, π are sufficient. Specific speed Ns = φ^0.5/ψ^0.75 is a combination of φ and ψ. These groups form the basis of pump similarity laws and performance prediction.

Explanation:
Pressure recovery coefficient Cp = (p2 - p1)/(ρV1²/2) = (p2 - p1)/(q1), where q1 = ρV1²/2 is the dynamic pressure at diffuser inlet. For an ideal (isentropic) diffuser: Cp_ideal = 1 - (A1/A2)² = 1 - (V2/V1)² (from continuity). Actual Cp < Cp_ideal due to friction and flow separation. Diffuser efficiency η_d = Cp/Cp_ideal. Diffusers are used in centrifugal compressors (vaneless and vaned diffusers) to convert kinetic energy to pressure.

Explanation:
Specific speed ranges (metric units, N in rpm, P in kW, H in m): Pelton: Ns = 10-35, Francis: Ns = 60-300, Kaplan: Ns = 300-900. Ns = 300 is at the boundary between Francis and Kaplan, but is typically classified as Kaplan (axial flow, adjustable blades). At Ns = 300, the turbine would be a slow Kaplan or fast Francis. The Kaplan turbine is preferred at this specific speed for its adjustable blades and high efficiency over a wide flow range.

Explanation:
Work input to axial compressor stage: W = U(Vw2 - Vw1) = U·ΔVw, where U is blade speed (constant for axial machine), Vw1 is whirl velocity at rotor inlet, Vw2 is whirl velocity at rotor exit. From velocity triangles: Vw1 = U - Vf·cot β1, Vw2 = U - Vf·cot β2. Therefore: ΔVw = Vf(cot β1 - cot β2) = Vf(cot α2 - cot α1). W = U·Vf(cot β1 - cot β2). With work done factor λ: W_actual = λ·U·ΔVw.

Explanation:
The Euler turbomachine equation W = U1Vw1 - U2Vw2 is derived from the angular momentum (moment of momentum) equation: Torque T = ṁ(r1Vw1 - r2Vw2). Power P = Tω = ṁω(r1Vw1 - r2Vw2) = ṁ(U1Vw1 - U2Vw2), where U = ωr. Work per unit mass W = U1Vw1 - U2Vw2. This is Newton's second law applied to angular momentum. The equation applies to all turbomachines regardless of fluid type, flow direction, or machine geometry.

Explanation:
In the rotating frame of reference, the rothalpy I = h + Vr²/2 - U²/2 is constant along a streamline (for adiabatic, inviscid flow). This is the rotating frame equivalent of stagnation enthalpy. I = h0_rel - U²/2 = h + Vr²/2 - U²/2. The rothalpy is constant across the rotor (I1 = I2), which gives: h1 + Vr1²/2 - U1²/2 = h2 + Vr2²/2 - U2²/2. This is used to analyse flow in the rotating passages of turbomachines.

Explanation:
Specific speed ranges for pumps (SI units, N in rpm, Q in m³/s, H in m): Centrifugal (radial): Ns = 10-70, Mixed flow: Ns = 70-170, Axial flow: Ns = 170-400. Ns = 50 falls in the centrifugal pump range. Centrifugal pumps are used for high head, low flow applications. As specific speed increases, the pump transitions from centrifugal to mixed flow to axial flow, with decreasing head and increasing flow rate. The impeller shape changes from narrow radial to wide axial as Ns increases.

Explanation:
The volute casing (spiral casing) in a centrifugal pump: (1) Collects the fluid discharged from the impeller at high velocity, (2) Converts kinetic energy to pressure energy through its gradually increasing cross-sectional area (diffuser action), (3) Guides the fluid to the discharge pipe. The volute is designed so that the fluid velocity decreases gradually (constant angular momentum: Vw × r = constant), minimising losses. Alternatively, a diffuser ring (vaned or vaneless) can be used between impeller and volute.

Explanation:
Maximum efficiency of a Pelton wheel occurs when U/V1 ≈ 0.5 (bucket speed = half jet velocity). At this condition, the relative velocity at exit is approximately zero, meaning maximum energy is extracted from the jet. The ideal hydraulic efficiency at U/V1 = 0.5 with β = 180° and k = 1: η_h = 2(0.5)(0.5)(1+1) = 1.0 (100%). In practice, with β ≈ 165° and k ≈ 0.9: η_h ≈ 0.85-0.90. The optimum U/V1 ≈ 0.46 in practice.

Explanation:
The compressor map (performance map) shows: (1) Pressure ratio (p2/p1) on Y-axis vs. corrected mass flow rate (ṁ√T01/p01) on X-axis, (2) Lines of constant corrected speed (N/√T01), (3) Surge line (left boundary — instability limit), (4) Choke line (right boundary — maximum flow), (5) Efficiency contours (islands). The map is used to: select operating point, assess surge margin, predict off-design performance, and design control systems. Corrected parameters account for varying inlet conditions.

Explanation:
Reaction turbines: (1) Both pressure and velocity change in the rotor (degree of reaction R > 0), (2) The rotor is fully immersed in the pressurised fluid (not open to atmosphere like Pelton), (3) The pressure drop in the rotor creates a reaction force on the blades (like a rocket nozzle), (4) Examples: Francis turbine, Kaplan turbine, steam reaction turbines. Impulse turbines: all pressure drop in nozzle, rotor at constant pressure, open to atmosphere (Pelton wheel, de Laval turbine).

Explanation:
Corrected mass flow rate ṁ_corr = ṁ√T01/p01 (or ṁ√(T01/T_ref)/(p01/p_ref) for normalised form). This parameter accounts for varying inlet conditions (temperature and pressure) and allows comparison of compressor performance at different operating conditions. Similarly, corrected speed N_corr = N/√T01. These corrected parameters collapse the performance data onto a single map. They arise from dimensional analysis of compressible flow in turbomachines.

Explanation:
Stagger angle (setting angle) ξ is the angle between the chord line of the blade and the plane of rotation (tangential direction) or equivalently, the angle between the chord and the circumferential direction. It determines the orientation of the blade in the cascade. Camber angle θ = α1 - α2 (difference between inlet and outlet blade angles). Stagger angle ξ = (α1 + α2)/2 for symmetric camber. The stagger angle affects the blade loading and the cascade performance.

Explanation:
Both Francis and Kaplan turbines use adjustable guide vanes (wicket gates/stay vanes) to control the flow rate and hence the power output. The wicket gates rotate about their own axes to vary the flow area and the angle of flow entering the runner. In the Kaplan turbine, additionally, the runner blades are also adjustable (double regulation). In the Pelton wheel, flow is controlled by a needle valve (spear valve) in the nozzle. Guide vane adjustment allows efficient operation over a wide range of loads.

Explanation:
Isentropic efficiency of turbine η_is = W_actual / W_isentropic = (h1 - h2) / (h1 - h2s), where h2s is the enthalpy at the end of isentropic expansion. Since actual expansion involves irreversibilities, h2 > h2s, so W_actual < W_isentropic, and η_is < 1. For an ideal gas: η_is = (T1 - T2)/(T1 - T2s) = (T1 - T2) / (T1[1 - (p2/p1)^((γ-1)/γ)]). Typical η_is = 0.85-0.92 for well-designed turbines.

Explanation:
Specific speed (dimensionless form) Ω_s = NQ^0.5/(gH)^0.75 (for pumps) or Ω_s = NP^0.5/(ρ^0.5(gH)^1.25) (for turbines) is the same for all geometrically similar machines operating at the same efficiency point (homologous point). It is independent of machine size. This is the fundamental property that makes specific speed useful for: (1) Selecting machine type, (2) Scaling designs, (3) Comparing performance. The dimensional specific speed (Ns) depends on the unit system used.

Explanation:
In a vaneless diffuser, there are no blades to guide the flow. The flow follows free vortex law: rVw = constant (conservation of angular momentum in the absence of external torques). As radius increases, Vw decreases. The radial velocity Vr also decreases (continuity: Vr × 2πr × b = constant, where b is diffuser width). The total velocity V = √(Vw² + Vr²) decreases, converting kinetic energy to pressure. The flow path is a logarithmic spiral. Vaneless diffusers have wider operating range but lower efficiency than vaned diffusers.

Explanation:
Rotating stall consists of one or more stall cells (regions of separated, low-momentum flow) that rotate around the compressor annulus at typically 20-70% of the rotor speed. The stall cells rotate in the direction of rotor rotation. Rotating stall: (1) Reduces compressor performance (pressure rise and efficiency), (2) Causes alternating blade loading (fatigue), (3) Is a precursor to surge, (4) Can be detected by pressure fluctuations. Unlike surge (global instability), rotating stall is a local instability that may be stable. It is more common in axial compressors than centrifugal.

Explanation:
The Euler work can be expressed in three equivalent forms: W = U2Vw2 - U1Vw1 (whirl velocity form). Using velocity triangle: V² = U² + Vr² - 2UVr·cos(angle), the Euler work can be shown to equal: W = (V2² - V1²)/2 + (U2² - U1²)/2 + (Vr1² - Vr2²)/2. This is the 'three-term' form of the Euler equation: (1) Change in absolute KE, (2) Centrifugal effect (change in blade speed), (3) Change in relative KE (diffusion in rotor). This form is useful for understanding the pressure rise mechanisms.