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Cheatsheet: Flow Systems
1. Fundamental Equations
1.1 Continuity Equation
| Form | Equation |
|---|---|
| Incompressible Flow | Q = A₁v₁ = A₂v₂ (constant volumetric flow rate) |
| Compressible Flow | ṁ = ρ₁A₁v₁ = ρ₂A₂v₂ (constant mass flow rate) |
| Mass Flow Rate | ṁ = ρQ = ρAv |
1.2 Bernoulli Equation
| Form | Equation |
|---|---|
| Standard Form | P₁/ρg + v₁²/2g + z₁ = P₂/ρg + v₂²/2g + z₂ + hL |
| Energy Form | P₁/ρ + v₁²/2 + gz₁ = P₂/ρ + v₂²/2 + gz₂ + losses |
| Total Head Form | H₁ = H₂ + hL, where H = P/ρg + v²/2g + z |
- P/ρg = pressure head (m)
- v²/2g = velocity head (m)
- z = elevation head (m)
- hL = head loss due to friction and fittings (m)
1.3 Energy Equation with Pumps/Turbines
| Component | Equation |
|---|---|
| With Pump | P₁/ρg + v₁²/2g + z₁ + hp = P₂/ρg + v₂²/2g + z₂ + hL |
| With Turbine | P₁/ρg + v₁²/2g + z₁ = P₂/ρg + v₂²/2g + z₂ + ht + hL |
| Pump Power | P = ρgQhp or P = ṁghp |
| Turbine Power | P = ρgQht or P = ṁght |
2. Head Loss Calculations
2.1 Major Losses (Pipe Friction)
| Parameter | Equation |
|---|---|
| Darcy-Weisbach | hL = f(L/D)(v²/2g) or ΔP = f(L/D)(ρv²/2) |
| Hazen-Williams | v = 0.849CHWRh0.63S0.54 (SI units) |
| Manning Equation | v = (1/n)Rh2/3S1/2 (SI units) |
- f = Darcy friction factor (dimensionless)
- L = pipe length (m)
- D = pipe diameter (m)
- v = average velocity (m/s)
- CHW = Hazen-Williams coefficient (120-140 for most pipes)
- Rh = hydraulic radius = A/P (m)
- S = slope of energy grade line = hL/L
- n = Manning roughness coefficient
2.2 Friction Factor Determination
| Flow Regime | Friction Factor |
|---|---|
| Laminar (Re <> | f = 64/Re |
| Turbulent Smooth Pipes | 1/√f = 2.0 log(Re√f) - 0.8 (Prandtl) |
| Turbulent Rough Pipes | 1/√f = -2.0 log(ε/3.7D) (fully rough) |
| Colebrook Equation | 1/√f = -2.0 log(ε/3.7D + 2.51/Re√f) |
| Swamee-Jain | f = 0.25/[log(ε/3.7D + 5.74/Re0.9)]² |
- Re = Reynolds number = ρvD/μ = vD/ν
- ε = absolute roughness (m)
- ε/D = relative roughness
2.3 Minor Losses (Fittings and Components)
| Method | Equation |
|---|---|
| Loss Coefficient | hL,minor = K(v²/2g) or ΔP = K(ρv²/2) |
| Equivalent Length | hL,minor = f(Leq/D)(v²/2g) |
| Total Minor Losses | hL,minor = Σ Ki(vi²/2g) |
2.3.1 Common K Values
| Component | K Value (Approximate) |
|---|---|
| Entrance (sharp-edged) | 0.5 |
| Entrance (rounded) | 0.1 |
| Exit | 1.0 |
| 90° Elbow (standard) | 0.9 |
| 90° Elbow (long radius) | 0.6 |
| 45° Elbow | 0.4 |
| Gate Valve (fully open) | 0.2 |
| Globe Valve (fully open) | 10.0 |
| Check Valve (swing type) | 2.0 |
| Tee (line flow) | 0.6 |
| Tee (branch flow) | 1.8 |
3. Flow Regimes and Reynolds Number
3.1 Reynolds Number
| Configuration | Equation |
|---|---|
| Circular Pipe | Re = ρvD/μ = vD/ν |
| Non-circular Conduit | Re = ρvDh/μ where Dh = 4A/P |
| Open Channel | Re = vRh/ν where Rh = A/P |
- ρ = density (kg/m³)
- v = velocity (m/s)
- D = diameter (m)
- μ = dynamic viscosity (Pa·s)
- ν = kinematic viscosity (m²/s)
- Dh = hydraulic diameter (m)
- A = cross-sectional area (m²)
- P = wetted perimeter (m)
3.2 Flow Classification
| Flow Type | Reynolds Number Range |
|---|---|
| Laminar | Re <> |
| Transitional | 2000 < re=""><> |
| Turbulent | Re > 4000 |
4. Pipe Networks
4.1 Series Pipes
- Same flow rate through all pipes: Q₁ = Q₂ = Q₃ = Q
- Total head loss: hL,total = hL1 + hL2 + hL3
- Same mass flow rate: ṁ₁ = ṁ₂ = ṁ₃
4.2 Parallel Pipes
- Total flow rate: Qtotal = Q₁ + Q₂ + Q₃
- Same head loss across all branches: hL1 = hL2 = hL3
- Same pressure drop: ΔP₁ = ΔP₂ = ΔP₃
4.3 Hardy Cross Method (Pipe Networks)
| Step | Procedure |
|---|---|
| 1. Initial Assumption | Assume flow distribution satisfying continuity at nodes |
| 2. Calculate Head Loss | For each pipe: hL = rQn where n ≈ 2 for turbulent flow |
| 3. Loop Correction | ΔQ = -ΣhL / (n·Σ|hL|/Q) |
| 4. Update Flows | Qnew = Qold + ΔQ (add for clockwise, subtract for counterclockwise) |
| 5. Iterate | Repeat until ΔQ becomes negligible |
4.4 Node Analysis Method
- Continuity at each node: Σ Qin = Σ Qout
- Express flows in terms of pressure heads
- Solve system of equations for unknown nodal pressures
5. Pump Systems
5.1 Pump Performance Parameters
| Parameter | Equation |
|---|---|
| Total Head | hp = (P₂ - P₁)/ρg + (v₂² - v₁²)/2g + (z₂ - z₁) |
| Water Power | Pw = ρgQhp = ṁghp |
| Brake Power | Pbrake = 2πNT/60 (N in rpm, T in N·m) |
| Pump Efficiency | ηp = Pw/Pbrake = ρgQhp/Pbrake |
| Specific Speed | Ns = NQ0.5/hp0.75 (dimensionless form) |
5.2 System Curve and Operating Point
- System curve: hsystem = hstatic + KQ² (where K depends on pipe friction)
- Operating point: intersection of pump curve and system curve
- hstatic = elevation difference + pressure head difference
5.3 Affinity Laws (Scaling)
| Parameter | Same Impeller, Different Speed |
|---|---|
| Flow Rate | Q₂/Q₁ = N₂/N₁ |
| Head | h₂/h₁ = (N₂/N₁)² |
| Power | P₂/P₁ = (N₂/N₁)³ |
| Parameter | Different Impeller, Same Speed |
|---|---|
| Flow Rate | Q₂/Q₁ = D₂/D₁ |
| Head | h₂/h₁ = (D₂/D₁)² |
| Power | P₂/P₁ = (D₂/D₁)³ |
5.4 NPSH (Net Positive Suction Head)
| Type | Definition |
|---|---|
| NPSH Available | NPSHA = (Patm - Pvapor)/ρg + zs - hL,suction |
| NPSH Required | NPSHR = value provided by pump manufacturer |
| Cavitation Prevention | NPSHA > NPSHR (safety factor of 1.1-1.3) |
- zs = suction elevation (positive if above pump, negative if below)
- hL,suction = head loss in suction line
5.5 Pumps in Series and Parallel
| Configuration | Characteristics |
|---|---|
| Series | Qsystem = Q; hsystem = h₁ + h₂ |
| Parallel | Qsystem = Q₁ + Q₂; hsystem = h₁ = h₂ |
6. Open Channel Flow
6.1 Hydraulic Parameters
| Parameter | Definition/Equation |
|---|---|
| Hydraulic Radius | Rh = A/P (A = area, P = wetted perimeter) |
| Hydraulic Depth | Dh = A/T (T = top width) |
| Froude Number | Fr = v/√(gDh) or Fr = v/√(gy) for rectangular channels |
| Specific Energy | E = y + v²/2g = y + Q²/2gA² |
6.2 Flow Classification
| Flow Type | Froude Number |
|---|---|
| Subcritical | Fr < 1="" (slow,="" deep=""> |
| Critical | Fr = 1 |
| Supercritical | Fr > 1 (fast, shallow flow) |
6.3 Manning Equation
| Form | Equation |
|---|---|
| Velocity | v = (1/n)Rh2/3S1/2 |
| Flow Rate | Q = (A/n)Rh2/3S1/2 |
6.3.1 Manning's n Values
| Surface | n Value |
|---|---|
| Glass, plastic | 0.010 |
| Concrete (finished) | 0.012 |
| Concrete (unfinished) | 0.014 |
| Clay tile | 0.014 |
| Corrugated metal | 0.024 |
| Earth canal (clean) | 0.022 |
| Earth canal (weedy) | 0.030 |
| Natural stream (clean) | 0.030 |
| Natural stream (major vegetation) | 0.075 |
6.4 Critical Depth
| Channel Type | Critical Depth Equation |
|---|---|
| Rectangular | yc = (Q²/gb²)1/3 where b = width |
| General | Q²/g = A³/T where A = area, T = top width |
| Specific Energy at Critical | Emin = 1.5yc (rectangular channel) |
6.5 Gradually Varied Flow
- Surface profile classification: M (mild), S (steep), C (critical), H (horizontal), A (adverse)
- Direct step method: Δx = (E₂ - E₁)/(S₀ - Sf)
- Standard step method: iterative solution for water surface profile
6.6 Hydraulic Jump
| Parameter | Equation (Rectangular Channel) |
|---|---|
| Sequent Depth Ratio | y₂/y₁ = 0.5(-1 + √(1 + 8Fr₁²)) |
| Energy Loss | ΔE = (y₂ - y₁)³/4y₁y₂ |
| Jump Length | Lj ≈ 6(y₂ - y₁) (approximate) |
7. Compressible Flow
7.1 Mach Number and Speed of Sound
| Parameter | Equation |
|---|---|
| Mach Number | Ma = v/c |
| Speed of Sound | c = √(kRT) for ideal gas |
| Speed of Sound (air) | c = √(γRT) where γ = 1.4 for air |
7.2 Flow Regimes
| Regime | Mach Number |
|---|---|
| Incompressible | Ma <> |
| Subsonic | 0.3 < ma=""><> |
| Sonic | Ma = 1 |
| Supersonic | 1 < ma=""><> |
| Hypersonic | Ma > 5 |
7.3 Isentropic Flow Relations
| Ratio | Equation |
|---|---|
| Temperature | T₀/T = 1 + [(γ-1)/2]Ma² |
| Pressure | P₀/P = [1 + [(γ-1)/2]Ma²]γ/(γ-1) |
| Density | ρ₀/ρ = [1 + [(γ-1)/2]Ma²]1/(γ-1) |
| Area Ratio | A/A* = (1/Ma){[2/(γ+1)][1 + [(γ-1)/2]Ma²]}(γ+1)/[2(γ-1)] |
- Subscript 0 = stagnation (total) conditions
- Subscript * = sonic conditions (Ma = 1)
- γ = specific heat ratio (1.4 for air)
7.4 Normal Shock Relations
| Parameter | Equation |
|---|---|
| Mach Number | Ma₂² = [Ma₁² + 2/(γ-1)] / [2γMa₁²/(γ-1) - 1] |
| Pressure Ratio | P₂/P₁ = 1 + [2γ/(γ+1)](Ma₁² - 1) |
| Density Ratio | ρ₂/ρ₁ = [(γ+1)Ma₁²] / [(γ-1)Ma₁² + 2] |
| Temperature Ratio | T₂/T₁ = [1 + [(γ-1)/2]Ma₁²][2γMa₁²/(γ-1) - 1] / [Ma₁²(γ+1)²/(2(γ-1))] |
| Stagnation Pressure | P₀₂/P₀₁ < 1="" (entropy="" increases="" across=""> |
- Subscript 1 = upstream (supersonic)
- Subscript 2 = downstream (subsonic)
8. Fluid Measurement
8.1 Venturi Meter
| Parameter | Equation |
|---|---|
| Theoretical Flow | Qth = A₂√[2(P₁-P₂)/ρ(1-β⁴)] where β = D₂/D₁ |
| Actual Flow | Q = CdQth where Cd ≈ 0.98 |
8.2 Orifice Meter
| Parameter | Equation |
|---|---|
| Theoretical Flow | Qth = A₂√[2(P₁-P₂)/ρ(1-β⁴)] |
| Actual Flow | Q = CdQth where Cd ≈ 0.61 |
8.3 Flow Nozzle
- Similar to venturi but shorter
- Cd ≈ 0.95-0.99
- Higher head loss than venturi, lower than orifice
8.4 Pitot-Static Tube
| Application | Equation |
|---|---|
| Incompressible | v = √[2(P₀-P)/ρ] = √(2ΔP/ρ) |
| Compressible (subsonic) | Ma = √{(2/(γ-1))[(P₀/P)(γ-1)/γ - 1]} |
8.5 Weirs
| Weir Type | Equation |
|---|---|
| Sharp-Crested Rectangular | Q = (2/3)Cdb√(2g)H3/2 where Cd ≈ 0.62 |
| V-Notch (90°) | Q = (8/15)Cd√(2g) tan(θ/2)H5/2 where Cd ≈ 0.58 |
| Broad-Crested | Q = CdbH3/2 where Cd ≈ 1.7 (SI units) |
- b = weir width (m)
- H = head above weir crest (m)
- θ = notch angle (degrees or radians)
9. Dimensional Analysis and Similitude
9.1 Common Dimensionless Numbers
| Number | Definition |
|---|---|
| Reynolds Number | Re = ρvL/μ = vL/ν (inertia/viscous) |
| Froude Number | Fr = v/√(gL) (inertia/gravity) |
| Euler Number | Eu = ΔP/ρv² (pressure/inertia) |
| Weber Number | We = ρv²L/σ (inertia/surface tension) |
| Mach Number | Ma = v/c (flow velocity/sound velocity) |
| Cavitation Number | Ca = (P - Pv)/(ρv²/2) |
9.2 Similarity Requirements
| Similarity Type | Requirement |
|---|---|
| Geometric | Lr = constant for all dimensions |
| Kinematic | vr = constant, flow patterns similar |
| Dynamic | Force ratios equal between model and prototype |
9.3 Scale Ratios
- Length ratio: Lr = Lp/Lm
- Velocity ratio: vr = √Lr (Froude scaling)
- Flow ratio: Qr = Lr5/2 (Froude scaling)
- Time ratio: tr = √Lr (Froude scaling)
- Force ratio: Fr = Lr³ (Froude scaling)
10. Drag and Lift
10.1 Drag Force
| Parameter | Equation |
|---|---|
| Drag Force | FD = CDA(ρv²/2) |
| Drag Coefficient | CD = FD/(A·ρv²/2) |
- A = frontal area for bluff bodies, wetted area for streamlined bodies
10.2 Typical Drag Coefficients
| Object | CD |
|---|---|
| Flat plate (perpendicular) | 1.28 |
| Flat plate (parallel) | 0.001-0.01 |
| Sphere (Re > 10⁴) | 0.4 |
| Cylinder (Re > 10³) | 1.2 |
| Streamlined body | 0.04-0.1 |
| Automobile | 0.3-0.5 |
10.3 Lift Force
| Parameter | Equation |
|---|---|
| Lift Force | FL = CLA(ρv²/2) |
| Lift Coefficient | CL = FL/(A·ρv²/2) |
10.4 Boundary Layer Parameters
| Parameter | Laminar (Rex <> |
|---|---|
| Thickness | δ/x ≈ 5/√Rex |
| Skin Friction Coefficient | Cf = 0.664/√Rex |
| Parameter | Turbulent (Rex > 5×10⁵) |
|---|---|
| Thickness | δ/x ≈ 0.37/Rex1/5 |
| Skin Friction Coefficient | Cf = 0.059/Rex1/5 |
11. Unsteady Flow
11.1 Water Hammer
| Parameter | Equation |
|---|---|
| Pressure Wave Speed | c = √(K/ρ) for rigid pipe; c = √(K/ρ)/√[1 + (K/E)(D/t)] for elastic pipe |
| Pressure Rise (Joukowsky) | ΔP = ρcΔv |
| Maximum Pressure | Pmax = P₀ + ρcΔv (instantaneous valve closure) |
- K = bulk modulus of fluid (2.2 GPa for water)
- E = elastic modulus of pipe material
- D = pipe diameter, t = wall thickness
- Δv = velocity change
11.2 Surge Tank Analysis
- Purpose: reduce water hammer effects in long pipelines
- Oscillation period: T = 2π√(LAs/gAp)
- L = pipeline length, As = surge tank area, Ap = pipe area
12. Key Formulas Summary
12.1 Pressure and Head
- Hydrostatic pressure: P = ρgh
- Pressure to head: h = P/ρg
- Gauge pressure: Pgauge = Pabsolute - Patmospheric
12.2 Velocity and Flow Rate
- Average velocity: v = Q/A
- Volumetric flow rate: Q = Av
- Mass flow rate: ṁ = ρQ = ρAv
- Weight flow rate: W = γQ where γ = ρg
12.3 Fluid Properties
- Specific weight: γ = ρg (water: 9810 N/m³)
- Specific gravity: SG = ρ/ρwater
- Kinematic viscosity: ν = μ/ρ (water at 20°C: 1.0×10⁻⁶ m²/s)
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