CELE Hydraulics & Fluid Mechanics — Hydrodynamics and Fluid MachineryCheat Sheet
Hydrodynamics and Fluid Machinery cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Hydrodynamics and Fluid Machinery for CELE Hydraulics & Fluid Mechanics. Download, print, revise.
Exam context
On the CELE 2026, the Hydraulics & Fluid Mechanics subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Hydrodynamics and Fluid Machinery lands at position 9th out of 10 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Hydraulics & Fluid Mechanics on a typical CELE paper.
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.
Sections
Formulas
Formula
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)
Formula
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
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)
Formula
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)
Section Title
Jet Force & Momentum Principle
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
Term
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
Definition
Force on a fluid = mass rate × change in velocity: ΣF = ρQ(v_out − v_in)
Term
Relative velocity (moving vane)
Example
Jet at 20 m/s, vane moving at 5 m/s → relative velocity = 15 m/s
Definition
Velocity of jet relative to the vane frame = (v_jet − v_vane)
Term
Deflection angle (θ)
Example
Pelton bucket turns jet 180° → θ = 180°, cos(180°) = −1, so F = 2ρQv (maximum)
Definition
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
Note
Positive direction is chosen; negative result means force opposes chosen direction
Equation
ΣF_x = ρQ(v_x,out − v_x,in)
Conditions
Applies to control volume; subscripts x, y for coordinate directions
Formulas
Formula
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
Formula
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
Section Title
Force on Pipe Bends
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
Term
Anchoring force
Example
90° horizontal bend: anchoring force = √(F_x² + F_y²) directed opposite to the resultant momentum + pressure force
Definition
Reaction force exerted by pipe supports/anchors to hold the bend against the dynamic force of flowing water
Term
Control volume
Example
Includes the bend itself and the fluid within it; exterior forces (pressure at ends, anchor reaction) balance momentum change
Definition
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
Note
Common case: inlet vertical (v_1y ≠ 0), outlet horizontal (v_2y = 0) → requires both terms
Equation
F_y = ρQ(v_{2y} − v_{1y}) + (p_1 A_1)_y − (p_2 A_2)_y
Conditions
Same momentum principle applied to y-direction; complete vector analysis
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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)
Formula
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)
Section Title
Pump Power & Performance
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
Term
Head (H)
Example
Pump lifts water 30 m vertically at low speed: H ≈ 30 m + suction losses + discharge losses
Definition
Height equivalent of pressure or energy per unit weight of water; combines pressure head, velocity head, and elevation head
Term
Pump efficiency (η)
Example
Pump with η = 80%: input 100 kW → delivers 80 kW to water (20 kW lost as heat, friction, turbulence)
Definition
Ratio of hydraulic power output to input (shaft) power: η = (γQH) / P_input; typically 70–90% for centrifugal pumps
Term
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
Definition
Pressure head available above vapor pressure at pump inlet; ensures fluid stays liquid inside pump
Term
Cavitation
Example
High-speed pump at high altitude or with long suction line: low NPSH → cavitation → pitting of impeller vanes
Definition
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
Note
Rearrange to find input power given η, Q, H or to find hydraulic power given input power and η
Equation
η = (γQH) / P_input = P_hydraulic / P_input
Conditions
Energy balance across pump; efficiency always between 0 and 1
Formulas
Formula
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
Formula
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
Formula
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
Section Title
Turbine Power & Classification
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
Term
Impulse turbine
Example
Small flow, high head (>500 m); single or multiple jets on a bucket-wheel; efficient at part-load
Definition
Jet(s) strike moving buckets; momentum transfer drives rotor (NOT pressure drop across blades). Example: Pelton wheel
Term
Reaction turbine
Example
Large flow, medium head (5–300 m); submerged runner in spiral casing; requires draft tube for efficiency
Definition
Pressure drop across moving blades accelerates flow through runner; both pressure and momentum contribute to torque. Examples: Francis, Kaplan
Term
Net head (H_net)
Example
Reservoir at +100 m, tailwater at +10 m, losses = 8 m → H_net = 100 − 10 − 8 = 82 m
Definition
Effective pressure head delivered to turbine after subtracting intake, penstock, and tailrace friction losses
Term
Specific speed (N_s)
Example
Francis turbine typically N_s = 60–300; Pelton typically N_s = 1–4
Definition
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
Note
Typical η = 85–92% for modern turbines; Pelton best (90–95%), Kaplan varies (85–92% depending on design head)
Equation
η = P_output / P_available = P_output / (γQH_net)
Conditions
Defines turbine efficiency; product of hydraulic, mechanical, and volumetric efficiencies
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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)
Section Title
Practical Hydraulic Calculations & Unit Conversions
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
Term
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
Definition
Sum of elevation, pressure, and velocity heads: H_total = z + p/γ + v²/(2g) (Bernoulli constant along streamline)
Term
Static head
Example
Reservoir 40 m above discharge, zero gauge pressure: H_static = 40 m (ignoring velocity and losses)
Definition
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
Note
If no friction (ideal): total head is constant. Real flow: head decreases downstream due to h_f
Equation
Bernoulli: (p₁/γ) + z₁ + (v₁²/2g) = (p₂/γ) + z₂ + (v₂²/2g) + 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
Rows
Values
- >500 m (very high)
- 5–300 m (medium)
Property
Head range
Values
- Small to medium
- Large
Property
Discharge range
Values
- 1–4
- 60–300 (Francis); 200–900 (Kaplan)
Property
Specific speed (N_s, SI)
Values
- Potential → kinetic (nozzle) → momentum transfer
- Pressure drop across blades; pressure + momentum
Property
Energy conversion
Values
- Buckets on wheel rim
- Submerged curved blades in spiral casing
Property
Runner type
Values
- No (atmospheric discharge)
- Yes (extracts tail water energy)
Property
Draft tube required?
Values
- 90–95%
- 88–92% (varies)
Property
Efficiency at rated load
Values
- Good (maintains high efficiency down to ~30% load)
- Francis: moderate; Kaplan: excellent (adjustable blades)
Property
Part-load efficiency
Values
- High head: mountain reservoirs, run-of-river
- Francis: medium head hydropower; Kaplan: tidal, low-head run-of-river
Property
Common applications
Columns
- Feature
- Impulse (Pelton)
- Reaction (Francis/Kaplan)
Table Title
Impulse vs Reaction Turbines
Rows
Values
- Adds head (lifts fluid)
- Extracts head (generates power)
Property
Function
Values
- Mechanical → hydraulic
- Hydraulic → mechanical
Property
Energy conversion
Values
- P_input = (γQH) / η
- P_output = η × γ × Q × H
Property
Power formula
Values
- Divide (denominator)
- Multiply (coefficient)
Property
Efficiency location
Values
- 70–90% (centrifugal)
- 85–95% (modern turbines)
Property
Typical efficiency
Values
- Q ∝ N (linear)
- Q ∝ N (linear)
Property
Affinity law: Q
Values
- H ∝ N² (quadratic)
- H ∝ N² (quadratic)
Property
Affinity law: H
Values
- P ∝ N³ (cubic)
- P ∝ N³ (cubic)
Property
Affinity law: P
Values
- Yes — suction side (NPSH_required check)
- No — pressure side; operates under submergence
Property
Cavitation risk?
Columns
- Aspect
- Pump
- Turbine
Table Title
Pump vs Turbine — Power & Efficiency
Rows
Values
- F = ρQv = ρAv²
- v = jet velocity, A = jet area; result is force perpendicular to plate
Property
Jet on stationary flat plate (normal impact)
Values
- F_n = ρQv sin(α)
- α is angle between jet and plate surface; α = 90° gives normal impact formula
Property
Jet on inclined plate at angle α
Values
- F = ρQv(1 − cos θ)
- θ = 180° → F = 2ρQv (max force); θ = 90° → F ≈ 1.41ρQv
Property
Jet on curved vane (stationary), deflection θ
Values
- F = ρ(v−u)Q_rel(1 − cos θ)
- Use relative velocity (v − u); if v < u, jet misses vane. Typical in turbines
Property
Jet on moving vane at speed u, deflection θ
Columns
- Scenario
- Formula
- Notes
Table Title
Jet Force Formulas — Quick Reference
Rows
Values
- Q₂ = Q₁ × (N₂/N₁)
- Q doubles (×2)
Property
Discharge (Q)
Values
- H₂ = H₁ × (N₂/N₁)²
- H increases ×4
Property
Head (H)
Values
- P₂ = P₁ × (N₂/N₁)³
- P increases ×8 (cubic!)
Property
Power (P)
Values
- NPSH_required ∝ N²
- NPSH requirement increases ×4; risk increases at high speed
Property
Cavitation/NPSH
Columns
- Parameter
- Relationship
- Example: if N doubles (×2)
Table Title
Affinity Laws — Scaling Pump/Turbine Performance
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