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In most practical cases the pump model
| LaTeX Math Block Reference |
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depends on the difference between intake and discharge pressure | LaTeX Math Inline |
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| body | --uriencoded--p_%7B\rm out%7D - p_%7B\rm in%7D |
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and called Pump Characteristic Curve (see Fig. 1):| LaTeX Math Block |
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q = q(p_{\rm out} - p_{\rm in}) |
A popular pump proxy model is given by the quadratic equation:
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| anchor | q_pump |
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| alignment | left |
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| q = \frac{q_{\rm max}}{2 \cdot k_f} \cdot \left[ -1 + k_f + \sqrt{ (1 + k_f)^2 - 4 \cdot k_f \cdot (p_{\rm out}- p_{\rm in})/\delta p_{\rm max}) \ } \, \right] |
| | LaTeX Math Block |
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| anchor | q_pump |
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| alignment | left |
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| p_{\rm out} = p_{\rm in} + \delta p_{\rm max} \cdot \left[ 1+
(k_f -1 ) \cdot \frac{q}{q_{\rm max}} - k_f \cdot \left( \frac{q}{q_{\rm max}} \right)^2
\right ] |
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| LaTeX Math Block |
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\eta(q) = 4 \, \eta_{\rm max} \cdot q/q_{\rm max} \cdot ( 1 - q/q_{\rm max}) |
where
| LaTeX Math Inline |
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| body | --uriencoded--\delta p_%7B\rm max%7D |
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| maximum pressure gain that pump can exert over the input pressure |
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| maximum flowrate that pump can produce |
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| total hydraulic pump friction (dimensionless) |
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| pump efficiency |
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| maximum pump efficiency |
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