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. $H_{total} = H_{static} + H_{pressure} + H_{velocity} + H_{friction}$ $H_{total} = (z_2 - z_1) + \frac{P_2 - P_1}{\rho g} + \frac{V_2^2 - V_1^2}{2g} + h_f$
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:
· Velocity profiles at pump inlet/outlet
· Friction factor variations with Reynolds number
Range
for Our Example:
· $H_{static}$: 10-15 m
· $H_{pressure}$: 25-30 m
· $H_{friction}$: 5-10 m
· Flow rate: 80-120 m³/h
Engineering
Insights:
· Velocity head usually negligible (<1m) for large pipes
· CFD in ANSYS Fluent can predict exact $h_f$ using k-ε turbulence model
* * *
2 Darcy-Weisbach Friction Loss Equation
$h_f
= f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}$
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:
· Wall shear stress distribution
· Pressure drop per unit length
Range
for Our Example:
· $L$: 10-50 m (suction + discharge)
· $D$: 0.08-0.10 m
· $V$: 2-4 m/s
· $h_f$: 3-8 m total
Engineering
Insights:
· CFD in COMSOL can model exact pipe roughness effects
· Minor losses (fittings, valves) add 20-50% to straight pipe losses
* * *
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:
· Operating point stability analysis
· Head-flow relationship at various valve positions
Range
for Our Example:
· $K$: 0.002-0.004 m/(m³/h)²
· $Q$: 0-150 m³/h
· $H_{system}$: 10-70 m
Engineering
Insights:
· MATLAB/Simscape can model dynamic system curve changes
· Valve throttling increases $K$, shifting operating point
* * *
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:
· 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
* * *
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