Kinetic energy in laminar flow

At a point where the velocity is vx, the kinetic energy per unit volume is equal to pvx2/2. The volumetric flow rate through an element of area of infinitesimal width Sr is equal to 2irrbr.vx. Thus the flow rate of kinetic energy S KE) through the element of area is given by 

The total flow rate of kinetic energy through the whole cross section is obtained by integrating equation 2.36  

Equation 2.37 is valid for any symmetric velocity profile. In the case of laminar flow of a Newtonian fluid, the velocity profile is given by equation 1.67, so the kinetic energy flow rate is found as 

The mass flow rate M is equal to pTrr2u, so the average kinetic energy per unit mass is given by 

Thus the kinetic energy per unit mass of a Newtonian fluid in steady laminar flow through a pipe of circular cross section is u2. In terms of head this is u2/g. Therefore for laminar flow, a = i in equation 1.14. 

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Kinetic energy in laminar flow. In application of the total mechanical-energy balance in Eq. (2.7-28), the average kinetic energy per unit mass of fluid is needed. For fluids, this is (S2)... 

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