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

      E_in - E_out = ΔU

where
      E_in = total energy transferred into the system
      E_out = 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 W_in is received through the high-speed shaft, work W_out 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,

      W_in - W_out - Q = 0

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

For convection heat transfer, Q equals,

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

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

      60 - W_out - 21.6 = 0
      W_out = 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

      X_Q = (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,

      X_w,in = 60.0 kW

      X_w,out = 38.4 kW

Exergy destruction is given as

      X_destroyed = T₀S_gen

where
      S_gen = entropy generation during the test process
      T_b = ambient temperature, is given as 25^oC

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

      S_in - S_out + S_gen = ΔS_system

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

      ΔS_system = 0

Then the entropy balance for the gearbox is simplified to:

      S_gen = S_out - S_in

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.

      S_gen = S_out - 0 = Q/T_b

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

Substitute Q and T_b to the entropy balance yields,

      S_gen = 21.6/(40+273)

Since X_destroyed = T₀S_gen, Exergy destruction can be determined as

      X_destroyed = T₀S_gen = (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

      X_destroyed = X_in - X_out = 60.0 - 38.4 - 1.0 = 20.6 kW

The exergy analysis is summarized in the following table.