One-inlet-one-exit Nozzle

The energy balance for a steady-flow device (nozzle, compressor, turbine and pump) with one inlet and one exit is:

Its differential form is:

      δq - δw = dh + dke + dpe

If the device undergoes an internally reversible process, the heat transfer term δq can be replaced by dh - vdP since

      δq_rev = Tds
      Tds = dh - vdP (the second Tds relation)

Then the energy balance becomes

      dh - vdP - δw_rev = dh + dke + dpe

By rearranging the above equation, the reversible steady-flow work can be expressed as

      - δw_rev = vdP + dke + dpe

Water Flowing through a
Hydraulic Turbine

Integrating it form location 1 to location 2 yields,

The above equation is the relation for the reversible work output associated with an internally reversible process in a steady-flow device. When the changes in kinetic and potential energies are negligible, the relation reduces to

The above equation states that the larger the specific volume, the larger the reversible work produced or consumed by a steady-flow device. To minimize the work input during a compression process, one should keep the specific volume of the working fluid as small as possible. In the same manner, to maximize the work output during an expansion process, one should keep the specific volume of the working fluid as large as possible.

One needs to know the relationship between the specific volume v and the pressure P for the given process to perform the integration in the above relation. If an incompressible fluid is used as the working fluid, the specific volume v is a constant. The relation for the reversible work output associated with an internally reversible process in a steady-flow device is simplified to give

      w_rev = -v(P₂ - P₁) - Δke - Δpe

Hydraulic turbines used in hydroelectric power plants run in a steady-flow process with incompressible fluid, i.e., water, as the working fluid.

If no work interactions are involved, like nozzle or pipe section, the above equation is reduced to

      
where V is the velocity of the fluid. This equation is known as the Bernoulli equation in fluid mechanics.

The steady-flow devices deliver the most and consume the least work when it undergoes a reversible process. Consider two steady-flow devices, one is reversible and the other is irreversible (actual process), operating between the same inlet and exit states. The differential forms for the energy balance of these two devices are

      δq_act - δw_act = dh + dke + dpe

      δq_rev - δw_rev = dh + dke + dpe

The right hand sides of these two equations are the same. It gives,

      q_act - δw_act = q_rev - δw_rev

Rearranging this equation gives,

      δw_rev - δw_act = q_rev - q_act

Since q_rev = Tds, the above equation becomes,

      δw_rev - δw_act = Tds - q_act

the increase of entropy principle gives

Thus,

      δw_rev - δw_act 0 or δw_rev δw_act

That is, for the same inlet and exit conditions, when the device undergoes a reversible process, a work-produce device like turbine produces the most work (w is positive), or a work-consuming device like compressor consumes the least work (w is negative).