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Because the conventional formula for turbulent orifice flow rate has an infinite derivative when the pressure difference is zero, ODE solvers may fail during numerical simulation of fluid power circuits. To remedy this, a two-regime orifice flow formula is proposed in which an empirical polynomial laminar flow function is used for small pressure differences. The proposed formula has a smooth transition between laminar and turbulent regimes, and its derivative does not have any singularities.
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Ellman et al. (1999) studied this question.
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