Bernoulli's equation is a steady-state energy balance. It states that the sum of a fluid's kinetic energy, potential energy and pressure are constant along a streamline. For incompressible flow, Bernoulli's equation is: $$ \frac{ \rho u^{2} }{ 2 } + \rho gz + P = constant $$ where \( u \) is fluid velocity, \( g \) is the gravitational constant (9.81 m/s²), \( P \) is pressure, and \( \rho \) is fluid density. Thus, the properties at the outlet can be determined given the properties at the inlet: $$ \frac{ \rho u^{2}_{in} }{ 2 } + \rho gz_{in} + P_{in} = \frac{ \rho u^{2}_{out} }{ 2 } + \rho gz_{out} + P_{out} $$ A mass balance is required to solve for \( u_{out} \). The volumetric flowrate is constant because the fluid is assumed to be incompressible, meaning velocity times cross-sectional area remains constant: $$ \frac{ \pi D_{ in }^{ 2 } }{4} u_{in} = \frac{ \pi D_{ out }^{ 2 } }{4} u_{out} $$ where \( D_{i} \) is pipe diameter. Solving for the fluid velocity at the outlet, $$ u_{out} = u_{in} \frac{ D^{2}_{in} }{ D^{2}_{out} } $$