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THERMODYNAMICS - THEORY

    Mass and Energy Balance of Unsteady-flow Processes



Unsteady-flow Processes
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In the previous sections, it was noted that nozzles, diffusers, turbines, compressors and other devices undergo a steady-flow process because of their long-time running consideration. But their startup and shutdown periods undergo transient operations since their states change with time. The flow processes involved are called unsteady-flow processes, or transient-flow processes. Unlike steady-flow processes, unsteady-flow processes start and end over some finite time period (Δt). An additional example of an unsteady-flow process is filling or discharging a tank.

   

 

 

 

 

 

 



Mass Balance for Unsteady-flow Processes

 

During an unsteady-flow process, the mass in the control volume changes with time. The mass balance for a system undergoing any process, can be used for control volume as

      

where
      i = inlet
      e = exit
      ΔmCV = mCV@final - mCV@initial
      mi = the mass flow into the control volume
            through one inlet
      me = the mass flow out of the control volume
              through one exit

Or in rate form
      
where
      i = the rate of mass flow into the control
             volume through an inlet
      e = the rate of mass flow out of the control
               volume through an exit
       = the rate of change of mass within the
                      control volume

Also, the energy content of a control volume changes with time during an unsteady-flow process. The general energy balance can be used for the control volume as

      Ei - Ee = ΔECV

where
      Ei = the total energy transferred into the control
             volume by heat, work, and mass
      Ee = the total energy transferred out of the control
             volume by heat, work, and mass
      ΔECV = energy change in the control volume in
                 forms of internal, kinetic, potential, etc.,
                 energies

or in rate form

      

where
      = rate of energy transferred into the control
               volume by by heat, work, and mass
       = rate of energy transferred out of the
                  control volume by heat, work, and mass
       = rate of energy change in the control
                      volume in forms of internal, kinetic,
                      potential, etc., energies

Noting that the energy can be transferred by heat, work, and mass only, the energy balance can be rewritten as

      

where
      θ = h + v2/2 + gz , the total energy of a flowing fluid per unit mass
      ΔECV = (ΔU + ΔKE + ΔPE)CV

     
    Unform-flow Processes


Uniform-flow Processes
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Uniform-flow processes are special cases of unsteady-flow processes. During a unform-flow process, the state of the control volume changes with time, but it does so uniformly. That is,

  • At any instant during the process, the state of the control volume is the same throughout. Hence, at an instant, the state of the mass leaving from the exit is the same as the state of the mass in the control volume.
  • The fluid flows at an opening is uniform and steady. That is, the properties do not change with time or position over the cross section of an inlet or exit. But they are different at different openings.

With these identifications, the mass and energy balances for uniform-flow processes become

     

     

where
      2 = final state of the control volume
      1 = initial state of the control volume  
      i = inlet
      e = exit   

     

System Reduces to a Closed System when all the Inlets and Exits are Closed
 

When no mass is entering or leaving the control volume, and the kinetic and potential energy changes associated with the control volume are negligible, the energy equation can be reduced to the first law relation for closed system.

    m2 = m1= m