First Law of Thermodynamics-The Energy Equation (4) Work transfer can also occur at the control surface when a force associated with fluid normal stress acts over a distance.

Work transfer can also occur at the control surface because of tangential stress forces. Rotating shaft work is transferred by tangential stresses in the shaft material. Combining all the information Equation with total stored energy, For steady flow, and uniformly distributed properties,

For only one stream entering and leaving, When shaft work is involved, the flow is unsteady, at least locally, e.g., the velocity and pressure at a fixed location near the rotating blade of a fan is unsteady, while upstream and downstream of the machine, flow is steady.

One dimensional energy equation for steady-in-the mean- flow valid for both incompressible and compressible flows. Using enthalpy, following equation is obtained,

Example 1 A pump delivers water at a steady rate of 300 gal/min as shown in the figure. The change in water elevation across the pump is zero. The rise in internal energy of water associated with a temperature rise is 3000 ft. lb/slug. Determine the power (hp) required by the pump for an adiabatic process.

Example 2 Steam enters a turbine with a velocity of 30 m/s and enthalpy of 3348 kJ/kg. The steam leaves the turbine as a mixture of vapor and liquid having a velocity of 60 m/s and an enthalpy of 2550 kJ/kg. If the flow through the turbine is adiabatic and changes in elevation are negligible, determine the work output involved per unit mass of steam through-flow.

Comparison of the Energy Equation with the Bernoulli Equation When the one-dimensional energy equation for steady- in-the –mean flow is applied to a flow that is steady, the equation becomes, Dividing by the mass flow rate, and rearranging, whereHeat transfer rate per unit mass flow rate Comparing with Bernoulli’s equation, it can be concluded for incompressible frictionless flow,

Frictional loss Useful or available energy Loss of available energy

An important group of fluid mechanics problems involves one- dimensional, incompressible, steady-in-the-mean flow with friction and shaft work for pumps, blowers, fans and turbines. For this kind of flow, expressing shaft work per unit mass, the energy equation becomes, Mechanical energy equation or extended Bernoulli’s equation, involves energy per unit mass (ft. lb/slug=ft 2 /s 2 ; or N. m =m 2 /s 2 )

Example 3 Axial flow fan- delivers 0.4 kW power to the fan blades produce a 0.6 m axial stream of air at 12 m/s. Energy equation in energy per unit weight involves heads where

Energy Equation for Nonuniform Flows If the velocity profile at any section where flow crosses the control surface is not uniform, the energy equation is:

The difference in loss calculated assuming uniform velocity and actual velocity profiles in not large compared to w shaft net in