Hydrodynamics and Fluid Machinery - Cheat Sheet
Final 30-minute reference for jet forces, pipe bends, pump and turbine power calculations, and affinity laws. Focus on momentum principle, energy conversion, and exam-critical formulas.
Jet Force & Momentum Principle
Formulas
F = ρQv or F = ρAv²
Meaning
ρ = density (1000 kg/m³), Q = discharge (m³/s), v = velocity (m/s), A = jet area (m²)
Watch out
Do NOT use γ (specific weight) here — momentum uses ρ (density). Common error: confusing 1000 kg/m³ with 9.81 kN/m³
When to use
Jet strikes a stationary flat plate normally (perpendicular impact)
F_n = ρQv sin(α)
Meaning
α = angle between jet and plate surface; F_n = normal force component
Watch out
α is measured from the plate surface, NOT from the jet direction. If plate is at 30° to jet, α = 30°, not 60°
When to use
Jet strikes an inclined flat plate; α = 90° recovers the normal-impact formula
F = ρQv(1 − cos θ)
Meaning
θ = angle through which vane turns the jet (deflection angle)
Watch out
This formula is for a STATIONARY vane. Moving vane uses relative velocity (v − u) where u = vane speed
When to use
Jet strikes a curved vane (stationary); θ = 180° gives max force (F = 2ρQv)
F_relative = ρQ_rel(v_rel)(1 − cos θ)
Meaning
v_rel = (v − u), Q_rel = A(v − u); jet moves with vane at speed u
Watch out
If v < u, the jet does not strike the vane — guard against this in problem setup
When to use
Vane is moving (e.g., turbine bucket moving at tangential speed u)
Common values
Value
1000 kg/m³
Symbol
ρ
Quantity
Water density
Value
9.81 kN/m³ or 9810 N/m³
Symbol
γ
Quantity
Water specific weight
Value
θ ≈ 165° to 170° (not 180° due to bucket geometry)
Symbol
θ_max
Quantity
Maximum deflection angle (Pelton bucket)
Important facts
- Momentum is conserved: inlet momentum − outlet momentum = force × time
- For a jet splitting on a smooth vane, assume discharge splits proportionally to deflection angles (conservation of Q)
- Pressure force pA acts at pipe inlet/outlet; must be included in momentum balance for pipe bends
- Kinetic energy loss = 0.5ρQ(v² inlet − v² outlet) when flow undergoes sudden expansion or collision
Key definitions
Momentum principle
Example
Jet hitting a plate: the plate exerts force on jet to stop it; by Newton's 3rd law, jet exerts equal/opposite force on plate
Force on a fluid = mass rate × change in velocity: ΣF = ρQ(v_out − v_in)
Relative velocity (moving vane)
Example
Jet at 20 m/s, vane moving at 5 m/s → relative velocity = 15 m/s
Velocity of jet relative to the vane frame = (v_jet − v_vane)
Deflection angle (θ)
Example
Pelton bucket turns jet 180° → θ = 180°, cos(180°) = −1, so F = 2ρQv (maximum)
Angle through which the vane changes the direction of the jet
Diagrams to know
- Jet normal to flat plate (force perpendicular to plate)
- Jet on inclined plate at angle α (resolve force into normal and tangential components)
- Curved vane deflecting jet by angle θ (force along bisector of deflection for symmetric vane)
- Moving vane with relative velocity triangle (jet velocity, vane velocity, relative velocity)
Reactions or equations
ΣF_x = ρQ(v_x,out − v_x,in)
Note
Positive direction is chosen; negative result means force opposes chosen direction
Conditions
Applies to control volume; subscripts x, y for coordinate directions
Force on Pipe Bends
Formulas
F_x = ρQ(v_{2x} − v_{1x}) + (p_1 A_1)_x − (p_2 A_2)_x
Meaning
Apply momentum in x-direction: ρ, Q as before; p_i = pressure (Pa), A_i = area (m²); subscript indicates component
Watch out
Pressure forces pA have direction components — a 90° bend has (p₁A₁) horizontal and (p₂A₂) vertical. Sign matters!
When to use
Pipe bend with inlet 1 and outlet 2; must account for both momentum change AND pressure forces
F_resultant = √(F_x² + F_y²)
Meaning
Magnitude of anchoring force (what the support/anchor must provide to hold the pipe)
Watch out
Direction of resultant: θ = arctan(F_y/F_x) — this is often asked in exams
When to use
After computing F_x and F_y separately; gives total magnitude
Common values
Value
101.325 kPa (absolute)
Symbol
p_atm
Quantity
Atmospheric pressure (gauge = 0)
Value
98.1 kPa (approximately 100 kPa)
Symbol
p = ρgh
Quantity
Pressure from 10 m water column
Important facts
- For a horizontal pipe at same elevation: p₁ = p₂ if no friction (ideal case), so pressure forces may cancel on certain bend orientations
- A 90° bend results in maximum force when inlet/outlet diameters are equal and pressures are equal
- In practice, pressure at outlet may differ due to depth change (use hydrostatic pressure if pipe rises or falls)
- Anchor reaction is perpendicular to the resultant force direction
Key definitions
Anchoring force
Example
90° horizontal bend: anchoring force = √(F_x² + F_y²) directed opposite to the resultant momentum + pressure force
Reaction force exerted by pipe supports/anchors to hold the bend against the dynamic force of flowing water
Control volume
Example
Includes the bend itself and the fluid within it; exterior forces (pressure at ends, anchor reaction) balance momentum change
Imaginary fixed region around the pipe bend; momentum balance applies across inlet and outlet faces
Diagrams to know
- 90° horizontal bend: identify pressure forces at inlet (pointing right) and outlet (pointing up), plus momentum change
- Reducing bend (larger to smaller diameter): velocity increases at outlet, pressure decreases (Bernoulli); both affect force
- Vertical bend (pipe rising): account for weight of water in bend, change in elevation head, and hydrostatic pressure difference
Reactions or equations
F_y = ρQ(v_{2y} − v_{1y}) + (p_1 A_1)_y − (p_2 A_2)_y
Note
Common case: inlet vertical (v_1y ≠ 0), outlet horizontal (v_2y = 0) → requires both terms
Conditions
Same momentum principle applied to y-direction; complete vector analysis
Pump Power & Performance
Formulas
P_input = (γQH)/η
Meaning
γ = 9.81 kN/m³, Q = discharge (m³/s), H = head added by pump (m), η = pump efficiency (decimal 0–1)
Watch out
η goes in the denominator (divide by efficiency). Small η means large input power needed. Common error: multiply by η instead of dividing
When to use
Calculate motor/engine power required to drive a pump; H is the total head rise across pump
P_hydraulic = γQH
Meaning
Useful water power (output) delivered to the fluid; same units (kW if γ in kN/m³, Q in m³/s, H in m)
Watch out
This is NOT the input power — it is the energy imparted to water. Always multiply by efficiency to get actual motor power
When to use
When only the hydraulic power is asked (ideal case, η = 1); or to verify efficiency: η = P_hydraulic / P_input
Q_1 / Q_2 = N_1 / N_2
Meaning
Affinity law for discharge: proportional to rotational speed; N in rpm
Watch out
This is LINEAR: if speed doubles, discharge doubles. Do NOT square or cube
When to use
Pump speed changes (e.g., variable-frequency drive); find new discharge or speed from this ratio
H_1 / H_2 = (N_1 / N_2)²
Meaning
Affinity law for head: proportional to speed squared
Watch out
Square the speed ratio — very different from discharge law. If N doubles, H increases 4×
When to use
Pump speed changes; find new head. This is QUADRATIC
P_1 / P_2 = (N_1 / N_2)³
Meaning
Affinity law for power: proportional to speed cubed
Watch out
Cube the speed ratio. If N doubles, P increases 8×. This is why variable-speed drives save energy
When to use
Pump speed changes; find new power requirement. This is CUBIC (most aggressive)
NPSH_required ≤ NPSH_available
Meaning
NPSH = net positive suction head; R = required (depends on pump design), A = available (depends on site/conditions)
Watch out
If NPSH_available < NPSH_required, cavitation occurs (damage, noise, loss of efficiency). Site elevation and suction line losses reduce available NPSH
When to use
Check for cavitation risk on pump suction side; typically given in pump datasheets
Common values
Value
70–90%
Symbol
η
Quantity
Typical centrifugal pump efficiency
Value
9.81 kN/m³
Symbol
γ
Quantity
Specific weight of water
Value
2.34 kPa (absolute)
Symbol
p_v
Quantity
Vapor pressure of water at 20°C
Value
6–8 m (atmospheric pressure limits absolute)
Symbol
h_s,max
Quantity
Typical suction lift limit (centrifugal pump)
Important facts
- Affinity laws assume pump operates at same point on efficiency curve (geometrically similar flow patterns)
- When speed changes: Q ∝ N, H ∝ N², P ∝ N³ — power is most sensitive to speed changes
- Pump operates at intersection of pump curve (P, H, η vs Q) and system curve (head losses vs Q)
- Impeller diameter change also obeys affinity laws: Q ∝ D, H ∝ D², P ∝ D³
- NPSH_available = p_atm/γ − h_s − h_f,suction − p_v/γ (where h_s = suction lift, h_f = friction loss, p_v = vapor pressure)
- At high altitude, p_atm decreases → NPSH_available decreases → cavitation risk increases
Key definitions
Head (H)
Example
Pump lifts water 30 m vertically at low speed: H ≈ 30 m + suction losses + discharge losses
Height equivalent of pressure or energy per unit weight of water; combines pressure head, velocity head, and elevation head
Pump efficiency (η)
Example
Pump with η = 80%: input 100 kW → delivers 80 kW to water (20 kW lost as heat, friction, turbulence)
Ratio of hydraulic power output to input (shaft) power: η = (γQH) / P_input; typically 70–90% for centrifugal pumps
NPSH (net positive suction head)
Example
Atmospheric pressure 101.325 kPa, vapor pressure at 20°C ≈ 2.3 kPa, suction line elevation +2 m → NPSH_available calculated from Bernoulli
Pressure head available above vapor pressure at pump inlet; ensures fluid stays liquid inside pump
Cavitation
Example
High-speed pump at high altitude or with long suction line: low NPSH → cavitation → pitting of impeller vanes
Vapor bubble formation when local pressure drops below vapor pressure; bubbles collapse, causing erosion and noise
Diagrams to know
- Pump performance curve: H and η vs discharge Q (typical bell-shaped η curve, H curve may slope down or flat)
- System curve: H_total = H_static + H_friction(Q); intersection with pump curve = operating point
- NPSH curve for a pump: shows NPSH_required increasing with Q; plot against NPSH_available from site
- Affinity law comparison: three speed-ratio examples showing linear (Q), quadratic (H), and cubic (P) relationships
Reactions or equations
η = (γQH) / P_input = P_hydraulic / P_input
Note
Rearrange to find input power given η, Q, H or to find hydraulic power given input power and η
Conditions
Energy balance across pump; efficiency always between 0 and 1
Turbine Power & Classification
Formulas
P_output = η × γ × Q × H
Meaning
η = turbine efficiency (0–1), γ = 9.81 kN/m³, Q = discharge (m³/s), H = net head available (m)
Watch out
Multiply by η (opposite of pump formula where you divide). High efficiency and high head/flow = large power. Common exam trap: reversing pump/turbine formulas
When to use
Calculate electrical/mechanical power generated by a turbine from flow and head
H_net = H_gross − H_losses
Meaning
H_gross = total available head (elevation difference), H_losses = friction in intake, penstock, tailrace (m)
Watch out
H_losses can be 5–15% of H_gross in real projects; ignoring losses overestimates power
When to use
Refine power calculation to account for pipe/intake friction losses in hydropower plants
Specific speed: N_s = (N × √Q) / H^(3/4)
Meaning
N = rpm, Q = discharge (m³/s), H = head (m); N_s is dimensionless turbine characteristic; N_s in SI ≈ 0.87 × N_s (US)
Watch out
Specific speed determines turbine type — don't confuse with ordinary rpm; also be careful with unit consistency (SI vs US)
When to use
Classify turbine type: Pelton (low N_s ≈ 1–4), Turgo (N_s ≈ 4–10), Crossflow (N_s ≈ 10–100), Francis (N_s ≈ 60–300), Kaplan (N_s ≈ 200–900)
Common values
Value
1–4
Symbol
N_s (SI)
Quantity
Pelton wheel specific speed range
Value
60–300
Symbol
N_s (SI)
Quantity
Francis turbine specific speed range
Value
200–900
Symbol
N_s (SI)
Quantity
Kaplan turbine specific speed range
Value
90–95%
Symbol
η
Quantity
Typical Pelton efficiency
Value
88–92%
Symbol
η
Quantity
Typical Francis efficiency
Value
85–92% (varies with design head)
Symbol
η
Quantity
Typical Kaplan efficiency
Important facts
- Impulse turbines: all available head converts to kinetic energy in jet; nozzle designs crucial; efficient single or multi-jet operation
- Reaction turbines: submerged in pressurized casing; require draft tube to extract tail water energy; efficient over wider operating range
- Francis turbine: curved runner blades, medium head/flow; most common for hydropower; runaway speed ≈ 1.5–1.8 × rated speed
- Kaplan (propeller) turbine: adjustable blade pitch allows efficiency over wide flow range; large discharge, low head (tidal, run-of-river)
- Pelton bucket design: enters at center, exits to sides (180° deflection ≈ 165–170° practical) to maximize momentum transfer with minimal splash
- Power increases with H^1 (linear) and Q^1 (linear), so doubling head OR doubling flow doubles power; N_s = (N√Q)/H^0.75 guides selection
Key definitions
Impulse turbine
Example
Small flow, high head (>500 m); single or multiple jets on a bucket-wheel; efficient at part-load
Jet(s) strike moving buckets; momentum transfer drives rotor (NOT pressure drop across blades). Example: Pelton wheel
Reaction turbine
Example
Large flow, medium head (5–300 m); submerged runner in spiral casing; requires draft tube for efficiency
Pressure drop across moving blades accelerates flow through runner; both pressure and momentum contribute to torque. Examples: Francis, Kaplan
Net head (H_net)
Example
Reservoir at +100 m, tailwater at +10 m, losses = 8 m → H_net = 100 − 10 − 8 = 82 m
Effective pressure head delivered to turbine after subtracting intake, penstock, and tailrace friction losses
Specific speed (N_s)
Example
Francis turbine typically N_s = 60–300; Pelton typically N_s = 1–4
Dimensionless parameter combining speed, flow, and head; characterizes turbine geometry and performance
Diagrams to know
- Pelton wheel: bucket geometry, jet entry/exit, tangential velocity vector diagram
- Francis runner: spiral casing, curved blades, discharge direction; draft tube below
- Kaplan runner: adjustable blades, hub, ring structure; axial flow; draft tube configuration
- Turbine classification chart: N_s on x-axis (log scale), turbine type and head range labeled
- Energy conversion: reservoir elevation → penstock kinetic energy → runner rotation → generator power
Reactions or equations
η = P_output / P_available = P_output / (γQH_net)
Note
Typical η = 85–92% for modern turbines; Pelton best (90–95%), Kaplan varies (85–92% depending on design head)
Conditions
Defines turbine efficiency; product of hydraulic, mechanical, and volumetric efficiencies
Practical Hydraulic Calculations & Unit Conversions
Formulas
γ = ρ × g; γ = 9.81 kN/m³ or 9810 N/m³
Meaning
ρ = density (1000 kg/m³), g = gravitational acceleration (9.81 m/s²); γ relates mass and weight
Watch out
Do NOT use 9.81 kN/m³ in momentum formulas — use ρ = 1000 kg/m³. Two different contexts!
When to use
When power formula requires γ instead of ρ; converting between weight and mass flows
Q [m³/s] = A [m²] × v [m/s]
Meaning
Discharge = area × velocity; fundamental continuity equation for incompressible flow
Watch out
Cross-sectional area must be perpendicular to flow direction; for pipe, A = π(D/2)²
When to use
Convert between velocity and discharge; find jet area from discharge and velocity
1 kW = 1 kN/m³ × 1 m³/s × 1 m = γ [kN/m³] × Q [m³/s] × H [m]
Meaning
Power in kW when using γ in kN/m³, Q in m³/s, H in m
Watch out
If using N/m³ for γ, power comes out in watts (W); if using kN/m³, power is in kW. Always verify units
When to use
Pump/turbine power calculations; simple dimensional check
Pressure head: h = p / γ [m]
Meaning
Convert pressure (Pa or kPa) to equivalent water column height (m); h = p / ρg
Watch out
γ = 9.81 kN/m³; if p = 98.1 kPa, h = 98.1 kPa / 9.81 kN/m³ = 10 m (exactly)
When to use
Total head analysis (elevation + pressure + velocity head); NPSH calculations
Velocity head: h_v = v² / (2g) [m]
Meaning
Kinetic energy expressed as equivalent head; g = 9.81 m/s²
Watch out
Even at high velocity (e.g., 10 m/s), velocity head = 100/(2×9.81) ≈ 5.1 m — often small compared to pressure/elevation head
When to use
Bernoulli equation; total head in pipes with significant velocity
Common values
Value
1000 kg/m³
Symbol
ρ
Quantity
Water density
Value
9.81 kN/m³
Symbol
γ
Quantity
Water specific weight (SI)
Value
9.81 m/s²
Symbol
g
Quantity
Gravitational acceleration
Value
9.81 kPa
Symbol
p = γh
Quantity
Pressure equivalent of 1 m water
Value
101.325 kPa = 10.33 m water
Symbol
p_atm
Quantity
Atmospheric pressure (standard)
Important facts
- Bernoulli is valid along a streamline for frictionless, steady, incompressible flow
- In real pipes, add friction head loss: H_friction ≈ f(L/D)(v²/2g) (Darcy-Weisbach); reduces net available head
- Pressure units: 1 kPa = 1000 Pa = 0.00987 m of water (approximately); 100 kPa ≈ 10.2 m water column
- Discharge in pipes remains constant (continuity): A₁v₁ = A₂v₂; if pipe reduces, velocity increases and pressure decreases (Bernoulli)
Key definitions
Total head (H_total)
Example
Pump inlet at −2 m elevation, p = 40 kPa, v = 1 m/s: total head = −2 + 40/9.81 + 1²/19.62 ≈ −2 + 4.08 + 0.05 ≈ 2.1 m
Sum of elevation, pressure, and velocity heads: H_total = z + p/γ + v²/(2g) (Bernoulli constant along streamline)
Static head
Example
Reservoir 40 m above discharge, zero gauge pressure: H_static = 40 m (ignoring velocity and losses)
Elevation difference + pressure head (no velocity component); H_static = Δz + Δp/γ
Diagrams to know
- Hydraulic grade line (HGL): plot of (p/γ + z) vs distance; shows available pressure head at each point
- Energy grade line (EGL): plot of (p/γ + z + v²/2g) vs distance; lies above HGL by velocity head; drops due to friction
- Pump and system curve: H vs Q; operating point at intersection; affinity law scaling of pump curve for different speeds
Reactions or equations
Bernoulli: (p₁/γ) + z₁ + (v₁²/2g) = (p₂/γ) + z₂ + (v₂²/2g) + h_f
Note
If no friction (ideal): total head is constant. Real flow: head decreases downstream due to h_f
Conditions
Steady, incompressible flow along a streamline; h_f = friction loss between points 1 and 2
Must remember
- Jet force on stationary flat plate: F = ρQv = ρAv² (MOMENTUM, NOT pressure). Use ρ = 1000 kg/m³, not γ = 9.81 kN/m³.
- Moving vane force uses RELATIVE VELOCITY (v − u), not absolute jet velocity v. If vane speed u > jet speed v, vane misses the jet.
- Curved vane deflects jet by angle θ: force F = ρQv(1 − cos θ). Maximum at θ = 180° (F = 2ρQv). Always check deflection angle in exam.
- Pump INPUT power formula: P_input = (γQH) / η — divide by efficiency. Turbine OUTPUT power: P_output = η × γ × Q × H — multiply by efficiency. OPPOSITE conventions!
- Affinity laws (with speed change): Q ∝ N (linear), H ∝ N² (quadratic), P ∝ N³ (cubic). Doubling speed quadruples head and octuples power — massive effect on motor sizing.
- Impulse turbines (Pelton): high head (>500 m), small discharge, N_s = 1–4. Reaction turbines (Francis/Kaplan): medium head (5–300 m), large discharge, N_s = 60–900. Turbine TYPE determined by specific speed.
- Pipe bend: include BOTH momentum change (ρQΔv) AND pressure forces (pA) in each direction. Forgetting pressure forces is a common exam killer.
- NPSH_required > NPSH_available → cavitation (pump damage, noise, loss of prime). High altitude, high speed, long suction line ALL reduce NPSH_available — guard against cavitation.
- Pressure head conversion: 1 kPa ≈ 0.102 m water; 100 kPa ≈ 10.2 m. Velocity head at 10 m/s = 100/(2×9.81) ≈ 5.1 m. Always include velocity head in Bernoulli if flow speed is high (>3 m/s).
- Specific speed N_s = (N√Q) / H^(0.75) (SI units) uniquely identifies turbine type. Mixing SI and US units for N_s is a common error; always confirm unit system in exam question.
Last minute tips
- ρ vs γ CRITICAL: Momentum formulas use ρ (kg/m³); power formulas use γ (kN/m³ or N/m³). Wrong choice = wrong answer by factor of 10. Double-check every formula substitution.
- Pump vs Turbine Power: Pump divides by η (input/efficiency ratio), Turbine multiplies by η (output = efficiency × available power). Reverse it = instant wrong answer. Write both formulas side-by-side before solving.
- Affinity Law Exponents: Remember Q is LINEAR (×N), H is QUADRATIC (×N²), P is CUBIC (×N³). If you guess wrong on exponent, entire calculation fails. Visual mnemonic: Q-line, H-square, P-cube.
- Deflection Angle θ in Jet-Vane: θ is total deflection (how much the vane turns the jet), NOT the angle from horizontal. Pelton bucket deflects ~165° (not 180° due to geometry). Read problem carefully for angle definition.
- Exam Time Saver: Always verify units before plugging numbers in. If γ in kN/m³, Q in m³/s, H in m → power in kW directly. If units mismatch, recalculate. One minute spent on unit check saves 5 minutes of debugging wrong answers.
Comparison tables
Impulse vs Reaction Turbines
| Feature | Impulse (Pelton) | Reaction (Francis/Kaplan) |
|---|---|---|
| Head range | >500 m (very high) | 5–300 m (medium) |
| Discharge range | Small to medium | Large |
| Specific speed (N_s, SI) | 1–4 | 60–300 (Francis); 200–900 (Kaplan) |
| Energy conversion | Potential → kinetic (nozzle) → momentum transfer | Pressure drop across blades; pressure + momentum |
| Runner type | Buckets on wheel rim | Submerged curved blades in spiral casing |
| Draft tube required? | No (atmospheric discharge) | Yes (extracts tail water energy) |
| Efficiency at rated load | 90–95% | 88–92% (varies) |
| Part-load efficiency | Good (maintains high efficiency down to ~30% load) | Francis: moderate; Kaplan: excellent (adjustable blades) |
| Common applications | High head: mountain reservoirs, run-of-river | Francis: medium head hydropower; Kaplan: tidal, low-head run-of-river |
Pump vs Turbine — Power & Efficiency
| Aspect | Pump | Turbine |
|---|---|---|
| Function | Adds head (lifts fluid) | Extracts head (generates power) |
| Energy conversion | Mechanical → hydraulic | Hydraulic → mechanical |
| Power formula | P_input = (γQH) / η | P_output = η × γ × Q × H |
| Efficiency location | Divide (denominator) | Multiply (coefficient) |
| Typical efficiency | 70–90% (centrifugal) | 85–95% (modern turbines) |
| Affinity law: Q | Q ∝ N (linear) | Q ∝ N (linear) |
| Affinity law: H | H ∝ N² (quadratic) | H ∝ N² (quadratic) |
| Affinity law: P | P ∝ N³ (cubic) | P ∝ N³ (cubic) |
| Cavitation risk? | Yes — suction side (NPSH_required check) | No — pressure side; operates under submergence |
Jet Force Formulas — Quick Reference
| Scenario | Formula | Notes |
|---|---|---|
| Jet on stationary flat plate (normal impact) | F = ρQv = ρAv² | v = jet velocity, A = jet area; result is force perpendicular to plate |
| Jet on inclined plate at angle α | F_n = ρQv sin(α) | α is angle between jet and plate surface; α = 90° gives normal impact formula |
| Jet on curved vane (stationary), deflection θ | F = ρQv(1 − cos θ) | θ = 180° → F = 2ρQv (max force); θ = 90° → F ≈ 1.41ρQv |
| Jet on moving vane at speed u, deflection θ | F = ρ(v−u)Q_rel(1 − cos θ) | Use relative velocity (v − u); if v < u, jet misses vane. Typical in turbines |
Affinity Laws — Scaling Pump/Turbine Performance
| Parameter | Relationship | Example: if N doubles (×2) |
|---|---|---|
| Discharge (Q) | Q₂ = Q₁ × (N₂/N₁) | Q doubles (×2) |
| Head (H) | H₂ = H₁ × (N₂/N₁)² | H increases ×4 |
| Power (P) | P₂ = P₁ × (N₂/N₁)³ | P increases ×8 (cubic!) |
| Cavitation/NPSH | NPSH_required ∝ N² | NPSH requirement increases ×4; risk increases at high speed |
Diagrams
Hydrodynamics & Fluid Machinery — Complete Concept Map
Complete concept hierarchy showing jet forces, pipe bends, pump and turbine fundamentals, affinity laws, and turbine classification by specific speed.
Pump Power Calculation Decision Tree
Decision flowchart for selecting correct pump power formula (input vs hydraulic) and verifying units before calculation.
Turbine Type Selection by Specific Speed
Decision tree for turbine selection based on calculated specific speed, including typical head ranges, discharge categories, and efficiency values.
Cavitation Risk Assessment (Pump Suction)
State diagram showing cavitation risk assessment process, threshold condition, and mitigation strategies for pump suction-side problems.
Power Loss Breakdown in Pump System
Pie chart showing typical allocation of input motor power in a pump system, highlighting that only 75% reaches the water (at η ≈ 75-80%).
Historical Turbine Development & Efficiency Progression
Timeline showing development of turbine types from early Pelton invention through modern high-efficiency designs, illustrating 140+ years of engineering advancement.