The exergy destruction associated with each process of the Brayton cycle and the exergy of the exhaust gases at the exit of the regenerator.

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Answer 1

Question

Chapter 9.12, Problem 144P

To determine

The exergy destruction associated with each process of the Brayton cycle and the exergy of the exhaust gases at the exit of the regenerator.

Expert Solution & Answer

Answer to Problem 144P

The exergy destruction associated with process 1-2 of the given Brayton cycle is 38.91 kJ/kg.

The exergy destruction associated with process 3-4 of the given Brayton cycle is 35.83 kJ/kg.

The exergy destruction associated with regeneration process of the given Brayton cycle is 4.37 kJ/kg.

The exergy destruction associated with process 5-3 of the given Brayton cycle is 58.09 kJ/kg.

The exergy destruction associated with process 6-1 of the given Brayton cycle is 142.6 kJ/kg.

The exergy of the exhaust gases at the exit of the regenerator is 143.2 kJ/kg.

Explanation of Solution

Show the regenerative Brayton cycle with air as the working fluid, on T−s diagram as in Figure (1).

Thermodynamics: An Engineering Approach, Chapter 9.12, Problem 144P

For the given Brayton cycle with regeneration and air as the working fluid Ti, Pi, hi and Pri are the temperature at ith state, pressure at ith state, specific enthalpy at ith state, and relative pressure at ith state respectively.

Write the pressure and relative pressure relation for the process 1-2.

Pr2=P2P1Pr1 (I)

Write the pressure and relative pressure relation for the process 3-4.

Pr4=P4P3Pr3 (II)

Write the expression of efficiency of the compressor (ηC).

ηC=h2s−h1h2−h1 (III)

Write the expression of efficiency of the turbine (ηT).

ηT=h3−h4h3−h4s (IV)

Write the expression of net work output by the gas turbine (wnet).

wnet=wT,out−wC,in

wnet=(h3−h4)−(h2−h1) (V)

Here, work done by the turbine is wT,out, and the work input to the compressor is wC,in.

Write the expression of effectiveness of the regenerator (ε).

ε=h5−h2h4−h2 (VI)

Write the expression of heat input to the regenerative Brayton cycle (qin).

qin=h3−h5 (VII)

Write the expression of heat rejected by the regenerative Brayton cycle (qout).

qout=qin−wnet (VIII)

Write the expression of thermal efficiency of the given turbine (ηth).

ηth=wnetqin (IX)

Write the energy balance equation on the heat exchanger.

h5−h2=h4−h6 (X)

Write the expression of exergy destruction associated with the process 1-2 of the given Brayton cycle (xdestroyed,12).

xdestroyed,12=T0(s2−s1)

xdestroyed,12=T0(s2∘−s1∘−RlnP2P1) (XI)

Here, the temperature of the surroundings is T0, the gas constant of air is R, entropy of air at state 2 as a function of temperature alone is s2∘, and entropy of air at state 1 as a function of temperature alone is s1∘.

Write the expression of exergy destruction associated with the process 3-4 of the given Brayton cycle (xdestroyed,34).

xdestroyed,34=T0(s4−s3)

xdestroyed,34=T0(s4∘−s3∘−RlnP4P3) (XII)

Here, entropy of air at state 3 as a function of temperature is s3∘, and entropy of air at state 4 as a function of temperature is s4∘.

Write the expression of exergy destruction associated with the regeneration process of the given Brayton cycle (xdestroyed,regen).

xdestroyed,regen=T0[(s5−s2)+(s6−s4)]

xdestroyed,regen=T0[(s5∘−s2∘)+(s6∘−s4∘)] (XIII)

Here, entropy of air at state 5 as a function of temperature alone is s5∘, and entropy of air at state 6 as a function of temperature alone is s6∘.

Write the expression of exergy destruction associated with the process 5-3 of the given Brayton cycle (xdestroyed,53).

xdestroyed,53=T0(s3−s5+−qinTH)

xdestroyed,53=T0(s3∘−s5∘−RlnP3P5−qinTH) (XIV)

Here, the temperature of the heat source is TH.

Write the expression of exergy destruction associated with the process 6-1 of the given Brayton cycle (xdestroyed,61).

xdestroyed,61=T0(s1−s6+qoutTL)

xdestroyed,61=T0(s1∘−s6∘−RlnP1P6+qoutTL) (XV)

Here, the temperature of the sink is TL.

Write the expression of stream exergy at the exit of the regenerator (state 6) (ϕ6).

ϕ6=(h6−h0)−T0(s6−s0) (XVI)

Here, the specific enthalpy of the surroundings is h0.

Write the expression of change entropy for the exit of the regenerator (s6−s0).

s6−s0=s6∘−s0∘−RlnP6P1 (XVII)

Here, entropy of air at the surroundings as a function of temperature alone is s0∘, and the pressure of air at the surroundings is P0.

Conclusion:

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a temperature of 310 K (T1).

h1=310.24 kJ/kgPr1=1.5546s10=1.73498 kJ/kg⋅K

Substitute 7 for P2P1, and 1.5546 for Pr1 in Equation (I).

Pr2=(7)(1.5546)=10.88

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a relative pressure of 10.88 (Pr2).

h2s=541.26 kJ/kg

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a temperature of 1150 K (T3).

h3=1219.25 kJ/kgPr3=200.15s30=3.129 kJ/kg⋅K

Substitute 17 for P4P3, and 200.15 for Pr3 in Equation (II).

Pr4=(17)(200.15)=28.59

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a relative pressure of 28.59 (Pr4).

h4s=711.8 kJ/kg

Rearrange Equation (III), and substitute 310.24 kJ/kg for h1, 541.26 kJ/kg for h2s, and 0.75 for ηC.

h2=h1+h2s−h1ηC=310.24 kJ/kg+(541.26−310.24) kJ/kg0.75=618.26 kJ/kg

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a enthalpy of 618.26 kJ/kg (h2).

s20=2.42763 kJ/kg⋅K

Rearrange Equation (IV), and substitute 1219.25 kJ/kg for h3, 711.8 kJ/kg for h4s, and 0.82 for ηT.

h4=h3−ηT(h3−h4s)=1219.25 kJ/kg−(0.82)(1219.25−711.8)kJ/kg=803.14 kJ/kg

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a enthalpy of 803.14 kJ/kg (h4).

s40=2.69407 kJ/kg⋅K

Substitute 1219.25 kJ/kg for h3, 618.26 kJ/kg for h2, 803.14 kJ/kg for h4, and 310.24 kJ/kg for h1 in Equation (V).

wnet=(1219.25−803.14)kJ/kg−(618.26−310.24)kJ/kg=108.09 kJ/kg

Substitute 0.65 for ε, 803.14 kJ/kg for h4, and 618.26 kJ/kg for h2 in Equation (VI).

0.65=(h5−618.26 kJ/kg)(803.14−618.26) kJ/kg h5=738.43 kJ/kg

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a enthalpy of 738.43 kJ/kg (h5).

s50=2.60815 kJ/kg⋅K

Substitute 1219.25 kJ/kg for h3, and 738.43 kJ/kg for h5 in Equation (VII).

qin=(1219.25−738.43) kJ/kg=480.82 kJ/kg

Substitute 108.09 kJ/kg for wnet, and 480.82 kJ/kg for qin in Equation (VIII).

qout=(480.82−108.09) kJ/kg=372.73 kJ/kg

Substitute 108.09 kJ/kg for wnet, and 480.82 kJ/kg for qin in Equation (IX).

ηth=108.09 kJ/kg480.82 kJ/kg=0.225=22.5 %

Substitute 738.43 kJ/kg for h5, 803.14 kJ/kg for h4, and 618.26 kJ/kg for h2 in Equation (X).

738.43 kJ/kg−618.26 kJ/kg=803.14 kJ/kg−h6h6=682.97 kJ/kg

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a enthalpy of 682.97 kJ/kg (h6).

s60=2.52861 kJ/kg⋅K

Substitute 290 K for T0, 1.73498 kJ/kg⋅K for s1∘, 2.42763 kJ/kg⋅K for s2∘, 0.287 kJ/kg⋅K for R, and 7 for P2P1 in Equation (XI).

xdestroyed,12=(290 K)[(2.42763−1.73498)kJ/kg⋅K−(0.287kJ/kg⋅K)ln(7)]=38.91 kJ/kg

Thus, the exergy destruction associated with process 1-2 of the given Brayton cycle is 38.91 kJ/kg.

Substitute 290 K for T0, 3.129 kJ/kg⋅K for s3∘, 2.69407 kJ/kg⋅K for s4∘, 0.287 kJ/kg⋅K for R, and 17 for P4P3 in Equation (XII).

xdestroyed,34=(290 K)[(2.69407−3.129)kJ/kg⋅K−(0.287kJ/kg⋅K)ln(17)]=35.83 kJ/kg

Thus, the exergy destruction associated with process 3-4 of the given Brayton cycle is 35.83 kJ/kg.

Substitute 290 K for T0, 2.60815 kJ/kg⋅K for s5∘, 2.69407 kJ/kg⋅K for s4∘, 2.42763 kJ/kg⋅K for s2∘, and 2.52861 kJ/kg⋅K for s6∘ in Equation (XIII).

xdestroyed,regen=(290 K)[(2.60815−2.42763)kJ/kg⋅K+(2.52861−2.69407) kJ/kg⋅K]=4.37 kJ/kg

Thus, the exergy destruction associated with regeneration process of the given Brayton cycle is 4.37 kJ/kg.

Substitute 290 K for T0, 3.129 kJ/kg⋅K for s3∘, 2.60815 kJ/kg⋅K for s5∘, 0.287 kJ/kg⋅K for R, 1500 K for TH, and 480.82 kJ/kg for qin in Equation (XIV). For the process 5-3, pressure remains constant, hence substitute 0 for lnP3P5.

xdestroyed,53=(290 K)[(3.129−2.60815)kJ/kg⋅K−(0.287kJ/kg⋅K)(0)−480.82 kJ/kg1500 K]=58.09 kJ/kg

Thus, the exergy destruction associated with process 5-3 of the given Brayton cycle is 58.09 kJ/kg.

Substitute 290 K for T0, 1.73498kJ/kg⋅K for s1∘, 2.52861 kJ/kg⋅K for s6∘, 0.287 kJ/kg⋅K for R, 290 K for TL, and 372.73 kJ/kg for qout in Equation (XV). For the process 6-1, pressure remains constant, hence substitute 0 for lnP1P6.

xdestroyed,61=(290 K)[(1.73498−2.52861)kJ/kg⋅K−(0.287kJ/kg⋅K)(0)+372.73 kJ/kg290 K]= 142.6 kJ/kg

Thus, the exergy destruction associated with process 6-1 of the given Brayton cycle is 142.6 kJ/kg.

Refer Table A-17, “Ideal gas properties of air”, obtain the properties of air at a temperature of 290 K (T0).

h0=290.16 kJ/kgs00=1.66802 kJ/kg⋅K

Substitute 2.52861 kJ/kg⋅K for s6∘, and 1.66802 kJ/kg⋅K for s0∘ in Equation (XVII). Also, at the exit of the regenerator, pressure remains constant, hence substitute 0 for lnP6P1.

s6−s0=(2.52861−1.66802) kJ/kg⋅K=0.86059 kJ/kg⋅K

Substitute 682.97 kJ/kg for h6, 290.16 kJ/kg for h0, 290 K for T0, and 0.86059 kJ/kg⋅K for (s6−s0) in Equation (XVI).

ϕ6=(682.97−290.16) kJ/kg−(290 K)(0.86059 kJ/kg⋅K )=143.2 kJ/kg

Thus, the exergy of the exhaust gases at the exit of the regenerator is 143.2 kJ/kg.

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