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THERMODYNAMICS - CASE STUDY SOLUTION


 

To complete the test on the gear box, the work output from the gear box, the exergy transfer, and the exergy destruction need to be determined.

Assumptions:

  • The temperature on the outer surface is uniform.
  • The test process is steady.
     


System

 

(1) Determine the work delivered by the gearbox

Take the gear box as a closed system. The gearbox satisfies the energy balance, which is

      Ein - Eout = ΔU

where
      Ein = total energy transferred into the system
      Eout = total energy transferred out of the system
      ΔU = Internal change of the gearbox

Since the test process is steady, properties inside the gearbox do not change. Work Win is received through the high-speed shaft, work Wout is delivered through the low-speed shaft, and heat is transferred from the outer surface of the gearbox to the ambient air. Therefore, the energy balance becomes,

      Win - Wout - Q = 0

where Win, Wout, and Q are positive numbers. In order to determine Wout, heat transferred from the gearbox (Q) needs to be determined first.

For convection heat transfer, Q equals,

      Q = Ah(Tb - T0) = 0.8(1,800)(40 - 25) = 21.6 kW

Win is given as 60 kW. Substitute Q and Win into the energy balance gives,

      60 - Wout - 21.6 = 0
      Wout = 38.4 kW

(2) Explore the exergy transfer of the gearbox and the exergy destruction

Exergy can be transferred by heat, work, and mass. Since this system is a closed system, exergy of this gearbox is only transferred by heat and work.

Exergy transferred by heat is given as

      XQ = (1 - T0/T)Q

Heat is transferred from the system to the surroundings, hence, Q is negative in the above equation.

      

Heat transferred from the system results in exergy transferred from the system.

Exergy transfer by the shaft work is equal to the work itself. Hence,

      Xw,in = 60.0 kW

      Xw,out = 38.4 kW

Exergy destruction is given as

      Xdestroyed = T0Sgen

where
      Sgen = entropy generation during the test process
      Tb = ambient temperature, is given as 25oC

The entropy generation during the test process can be determined by the entropy balance of the gearbox. That is,

      Sin - Sout + Sgen = ΔSsystem

Since the test process is steady, no entropy change occurs inside the system.

      ΔSsystem = 0

Then the entropy balance for the gearbox is simplified to:

      Sgen = Sout - Sin

Note that entropy can not be transferred by work. It can only be transferred by heat and mass. Since the gearbox is a closed system, entropy is only transferred by heat.

      Sgen = Sout - 0 = Q/Tb

where
      Q = heat transfer from the gearbox, Q = 21.6 kW
      Tb = temperature at the outer surface, is given
             as 40oC

     
   

Substitute Q and Tb to the entropy balance yields,

      Sgen = 21.6/(40+273)

Since Xdestroyed = T0Sgen, Exergy destruction can be determined as

      Xdestroyed = T0Sgen = (25 + 273)(21.6/(40 + 273))
                    = 20.6 kW

Another way to determine the exergy destruction is: exergy destruction equals the difference between the exergy in and exergy out, which is

      Xdestroyed = Xin - Xout = 60.0 - 38.4 - 1.0 = 20.6 kW

The exergy analysis is summarized in the following table.

Rate of Exergy in:

      high-speed shaft

60 kW (100%)
Rate of Exergy out:

      low-speed shaft

38.4 kW (64%)

      heat transfer

1.0 kW (1.7%)
Rate of Exergy Destruction:
  20.6 kW (34.3%)