System Design Equations

 System Design Equations

 Key System Design Equations

 1 Total Dynamic Head (TDH) Equation

Definition: Total energy added to fluid by pump, including static, pressure, velocity, and friction components.

$H_{total} = H_{static} + H_{pressure} + H_{velocity} + H_{friction}$

$H_{total} = (z_2 - z_1) + \frac{P_2 - P_1}{\rho g} + \frac{V_2^2 - V_1^2}{2g} + h_f$

Variables:

·       $H_{total}$ = Total dynamic head (m)

·       $H_{static}$ = Static head (elevation difference) (m)

·       $H_{pressure}$ = Pressure head difference (m)

·       $H_{velocity}$ = Velocity head difference (m)

·       $H_{friction}$ = Friction head loss (m)

·       $z_1, z_2$ = Elevation at suction/discharge (m)

·       $P_1, P_2$ = Pressure at suction/discharge (Pa)

·       $V_1, V_2$ = Velocity at suction/discharge (m/s)

·       $\rho$ = Fluid density (kg/m³)

·       $g$ = Gravitational acceleration (9.81 m/s²)

·       $h_f$ = Total friction losses (m)

Recommended CFD Variables:

·       Pressure distribution along suction/discharge piping
·       Velocity profiles at pump inlet/outlet
·       Friction factor variations with Reynolds number

Range for Our Example:

·       $H_{total}$: 45-55 m (design: 50 m)
·       $H_{static}$: 10-15 m
·       $H_{pressure}$: 25-30 m
·       $H_{friction}$: 5-10 m
·       Flow rate: 80-120 m³/h

Engineering Insights:

·       Friction losses typically 10-20% of total head
·       Velocity head usually negligible (<1m) for large pipes
·       CFD in ANSYS Fluent can predict exact $h_f$ using k-ε turbulence model

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 2 Darcy-Weisbach Friction Loss Equation

Definition: Calculates head loss due to friction in pipes.

$h_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}$

Variables:

·       $h_f$ = Friction head loss (m)
·       $f$ = Darcy friction factor (dimensionless)
·       $L$ = Pipe length (m)
·       $D$ = Pipe diameter (m)
·       $V$ = Average flow velocity (m/s)

Recommended CFD Variables:

·       Friction factor $f$ vs. Reynolds number
·       Wall shear stress distribution
·       Pressure drop per unit length

Range for Our Example:

·       $f$: 0.015-0.025 (turbulent flow, SS316 pipe)
·       $L$: 10-50 m (suction + discharge)
·       $D$: 0.08-0.10 m
·       $V$: 2-4 m/s
·       $h_f$: 3-8 m total

Engineering Insights:

·       Use Moody chart or Colebrook equation for $f$
·       CFD in COMSOL can model exact pipe roughness effects
·       Minor losses (fittings, valves) add 20-50% to straight pipe losses

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 3 System Curve Equation

Definition: Relationship between head and flow rate for the entire piping system.

$H_{system} = H_{static} + K \cdot Q^2$

Where $K$ is the system resistance coefficient.

Variables:

·       $H_{system}$ = System head requirement (m)

·       $H_{static}$ = Static head (m)

·       $K$ = System resistance coefficient (m/(m³/h)²)

·       $Q$ = Flow rate (m³/h)

Recommended CFD Variables:

·       System curve intersection with pump curve
·       Operating point stability analysis
·       Head-flow relationship at various valve positions

Range for Our Example:

·       $H_{static}$: 10-15 m
·       $K$: 0.002-0.004 m/(m³/h)²
·       $Q$: 0-150 m³/h
·       $H_{system}$: 10-70 m

Engineering Insights:

·       Operating point = intersection of pump curve and system curve
·       MATLAB/Simscape can model dynamic system curve changes
·       Valve throttling increases $K$, shifting operating point

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 4 Specific Speed Equation

Definition: Dimensionless parameter characterizing pump geometry and performance.

Metric Form: $N_s = \frac{N \cdot \sqrt{Q}}{H^{3/4}}$

US Form: $N_s(US) = \frac{N \cdot \sqrt{Q_{gpm}}}{H_{ft}^{3/4}}$

Variables:

·       $N_s$ = Specific speed (dimensionless)

·       $N$ = Rotational speed (RPM)

·       $Q$ = Flow rate at BEP (m³/h or GPM)

·       $H$ = Head at BEP (m or ft)

Recommended CFD Variables:

·       Impeller geometry optimization vs. $N_s$
·       Efficiency variation with specific speed
·       Cavitation characteristics

Range for Our Example:

·       $N$: 2900 RPM
·       $Q$: 100 m³/h (440 GPM)
·       $H$: 50 m (164 ft)
·       $N_s$ (metric): 2800
·       $N_s$ (US): 1450
·       Interpretation: Radial-flow impeller (optimal range: 500-4000 metric)

Engineering Insights:

• ·       $N_s$ < 2000: Radial flow (high head, low flow)
• ·       $N_s$ = 2000-8000: Mixed flow
• ·       $N_s$ > 8000: Axial flow (low head, high flow)
• ·       Python/SciPy can optimize $N_s$ for maximum efficiency

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 5 Suction Specific Speed

Definition: Parameter indicating pump susceptibility to cavitation.

$S = \frac{N \cdot \sqrt{Q}}{NPSH_R^{3/4}}$

Variables:

• ·       $S$ = Suction specific speed
• ·       $NPSH_R$ = Net Positive Suction Head Required (m)

Range for Our Example:

• ·       $S$: 8000-11000 (typical range)
• ·       $NPSH_R$: 3-5 m
• ·       Warning: $S$ > 12000 indicates high cavitation risk

Engineering Insights:

• ·       Lower $S$ = better suction performance
• ·       CFD in ANSYS can predict cavitation inception
• ·       Design target: $S$ < 10000 for reliable operation