A Flow Element

Work is needed to push the fluid into or out of the boundaries of a control volume if mass flow is involved. This work is called the flow work (flow energy). Flow work is necessary for maintaining a continuous flow through a control volume.

Consider a fluid element of volume V, pressure P, and cross-sectional area A as shown left. The flow immediately upstream will force this fluid element to enter the control volume, and it can be regarded as an imaginary piston. The force applied on the fluid element by the imaginary piston is:

      F = PA

The work done due to pushing the entire fluid element across the boundary into the control volume is

      W_flow = FL = PAL = PV

For unit mass,

      w_flow = Pv

The work done due to pushing the fluid element out of the control volume is the same as the work needed to push the fluid element into the control volume.

Total Energy of a Flowing Fluid

The total energy of a simple compressible system consists of three parts: internal, kinetic, and potential energy.

      E = U + KE + PE

For unit mass,

      e = u + ke + pe = u + v²/2 + gz

where
      e = total energy
      u = internal energy
      v = velocity of the system
      z = the elevation of the fluid

The fluid entering or leaving a control volume possess an additional energy, the flow work (Pv). Hence, the total energy of a flowing fluid becomes

      θ = Pv + u + v²/2 + gz

where
      θ = methalpy, the total energy of a flowing fluid

The definition of enthalpy gives

      h = Pv + u

Replacing Pv + u by h yields

      θ = h + v²/2 + gz

By using the enthalpy instead of internal energy, flow work is not a concern.

Steady-flow Process

Steady flow process is a process where: the fluid properties can change from point to point in the control volume but remains the same at any fixed point during the whole process. A steady-flow process is characterized by the following:

• No properties within the control volume change with time. That is
  m_cv = constant       E_cv = constant
• No properties change at the boundaries with time. Thus, the fluid properties at an inlet or exit will remain the same during the whole process. They can be different at different opens.
• The heat and work interactions between a steady-flow system and its surroundings do not change with time.

The conservation of mass principle, which has been previously introduced, in rate format, is:

During a steady-flow process, the total amount of mass contained within a control volume does not change with time. That is,

      dm_system/dt = 0

Hence the conservation of mass principle gives the total amount of mass entering a control volume equal to the total amount of mass leaving it. In an equation format, it is

(Total mass entering the control volume per unit time)
=
(Total mass leaving the control volume per unit time)

or,

where
      i = inlet
      e = exit

Also, the energy balance for a process, which has been previously introduced, in rate format, is:

For a steady-flow process, the total energy content of a control volume remains constant. That is,

      dE_system/dt = 0

Thus, the amount of energy entering a control volume in all forms (heat, work, mass transfer) must be equal to the amount of energy leaving it for a steady-flow process. In an equation format, it is

(Rate of net energy transfer in by heat, work and mass)
=
(Rate of net energy transfer out by heat, work and mass)

or,

For a general steady-flow process, the energy balance can be written as

If the sign introduced previously for heat and work is used, the energy balance for a general steady-flow process can be rewritten as: