EGR 334 Thermodynamics Chapter 9: Sections 5-6 Lecture 35: Gas Turbine modeling with the Brayton Cycle Quiz Today?

Today’s main concepts: Be able to recognize Dual and Brayton Cycles Understand what system may be modeled using Brayton Cycle. Be able to perform a 1st Law analysis of the Brayton Cycle and determine its thermal efficiency. Be able to explain how regeneration may be applied to a Brayton Cycle model. Reading Assignment: Read Chapter 9, Sections 7-8 Homework Assignment: Problems from Chap 9: 42, 47, 55

OK….Quick Matching Quiz a) Carnot b) Rankine c) Otto d) Diesel p v . 4 1 3 1’ 2 2’ B C D A

Today you get to add two more cycles to your cycle repertoire Today you get to add two more cycles to your cycle repertoire. Dual Cycle Brayton cycle. Used as a hybrid cycle which includes elements of both the Otto and Diesel cycles. Used to model internal combustion engines Used as a model for gas turbines (such as jet engines).

Heat addition occurs in two steps Sec 9.4 : Air-Standard Duel Cycle Neither the Otto or Diesel cycle describe the actual P-v diagrams of an engine Heat addition occurs in two steps 2 – 3 : Constant volume heat addition 3 – 4 : Constant pressure heat addition (first part of power stroke) Process 1 – 2 : Isentropic compression Process 2 – 3 : Constant volume heat transfer Process 3 – 4 : Constant pressure heat transfer Process 4 – 5 : Isentropic expansion Process 5 – 1 : Constant volume heat rejection To set state 3: Use ideal gas law with V3 = V2. and

Dual Cycle analysis process 1-2: s1 = s2 process 2-3: v2 = v3 Sec 9.4 : Air-Standard Duel Cycle Dual Cycle analysis process 1-2: s1 = s2 process 2-3: v2 = v3 process 3-4: p3 = p4 process 4-5: s4 = s5 process 5-1: v5 = v1

The thermal efficiency. The cut off ratio Example (9.38): The pressure and temperature at the beginning of compression in an air-standard dual cycle are 14 psi, 520°R. The compression ratio is 15 and the heat addition per unit mass is 800 Btu/lbm. At the end of the constant volume heat addition process the pressure is 1200 psi. Determine, Wcycle, in BTU/lb. Qout, in BTU/lb. The thermal efficiency. The cut off ratio State 1 2 3 4 5\ T (R) p (psi) u (Btu/lb) h (Btu/lb) vr Pr State 1 2 3 4 5\ T (R) 520 p (psi) 14 1200 u (Btu/lb) h (Btu/lb) vr pr

Example (9.38): State 1 2 3 4 5\ T (R) 520 p (psi) 14 1200 u (Btu/lb) 88.62 h (Btu/lb) vr 158.58 Pr 1.2147 State 1 2 3 4 5\ T (R) 520 p (psi) 14 1200 u (Btu/lb) h (Btu/lb) vr Pr Given Information: compression ratio, r = 15 Qin= Q23 + Q34 = 800 Btu Qout = - Q51 Identify State Properties State 1: p1 = 14 psi, T1 = 520 R State 2: s2 = s1 v2 = v1/r State 3: v3 = v2 and p3 = 1200 psi State 4: p4 = p3 = 1200 psi State 5: s5 =s4 and v5 = v1 Use Table A22E to fill in many of the other properties.

and since 1-2 is isentropic find vr2 State 1 2 3 4 5\ T (R) 520 1468.8 p (psi) 14 594.26 1200 u (Btu/lb) 88.62 260.26 h (Btu/lb) 124.27 361.53 vr 158.58 10.572 pr 1.2147 51.561 State 1 2 3 4 5\ T (R) 520 p (psi) 14 1200 u (Btu/lb) 88.62 h (Btu/lb) 124.27 vr 158.58 pr 1.2147 Example (9.38): State 1: given T = 520 R look up u, h, vr, and pr State 2: use r to find v2 and since 1-2 is isentropic find vr2 then use Table A22E to look up T2, pr2, u2, and h2: Pressure p2, can then be calculated using

then use Table A22E to look up u3 and h3: State 1 2 3 4 5\ T (R) 520 1468.8 p (psi) 14 594.26 1200 u (Btu/lb) 88.62 260.26 h (Btu/lb) 124.27 361.53 vr 158.58 10.572 pr 1.2147 51.561 State 1 2 3 4 5\ T (R) 520 1468.8 2966 p (psi) 14 594.26 1200 u (Btu/lb) 88.62 260.26 577.4 h (Btu/lb) 124.27 361.53 780.7 vr 158.58 10.572 pr 1.2147 51.561 Example (9.38): State 3: given v3 = v2 and p3 = 1200 psi, use ideal gas law: then use Table A22E to look up u3 and h3:

State 4: Knowing p4=p3 and the heat in: 1 2 3 4 5\ T (R) 520 1468.8 2966 4577.6 p (psi) 14 594.26 1200 u (Btu/lb) 88.62 260.26 577.4 949.7 h (Btu/lb) 124.27 361.53 780.7 1263.6 vr 158.58 10.572 0.2848 pr 1.2147 51.561 5961.6 State 1 2 3 4 5\ T (R) 520 1468.8 2966 p (psi) 14 594.26 1200 u (Btu/lb) 88.62 260.26 577.4 h (Btu/lb) 124.27 361.53 780.7 vr 158.58 10.572 pr 1.2147 51.561 Example (9.38): State 4: Knowing p4=p3 and the heat in: Qin= 800 Btu/lb use the 1st Law: O Use Table A-22E to find T4 ,u4, pr4, and v4r

Replace V’s using ideal gas. State 1 2 3 4 5\ T (R) 520 1468.8 2966 4577.6 2299 p (psi) 14 594.26 1200 61.44 u (Btu/lb) 88.62 260.26 577.4 949.7 431.0 h (Btu/lb) 124.27 361.53 780.7 1263.6 601.48 vr 158.58 10.572 0.2848 2.768 pr 1.2147 51.561 5961.6 305.24 State 1 2 3 4 5\ T (R) 520 1468.8 2966 4577.6 p (psi) 14 594.26 1200 u (Btu/lb) 88.62 260.26 577.4 949.7 h (Btu/lb) 124.27 361.53 780.7 1263.6 vr 158.58 10.572 0.2848 pr 1.2147 51.561 5961.6 Example (9.38): State 5: process 4-5 is also isentropic Replace V’s using ideal gas. Use Table A-22E to look up T5, u5, h5, and pr5 and then find p5:

Example (9.38): Wcycle, in Btu/lb. Qout, in Btu/lb. The thermal eff. State 1 2 3 4 5\ T (R) 520 1468.8 2966 4577.6 2299 p (psi) 14 594.26 1200 61.44 u (Btu/lb) 88.62 260.26 577.4 949.7 431.0 h (Btu/lb) 124.27 361.53 780.7 1263.6 601.48 vr 158.58 10.572 0.2848 2.768 pr 1.2147 51.561 5961.6 305.24 Example (9.38): Wcycle, in Btu/lb. Qout, in Btu/lb. The thermal eff. The cut off ratio

Example (9.38): Wcycle, in Btu/lb. Qout, in Btu/lb. Thermal efficiency State 1 2 3 4 5\ T (R) 520 1468.8 2966 4577.6 2299 p (psi) 14 594.26 1200 61.44 u (Btu/lb) 88.62 260.26 577.4 949.7 431.0 h (Btu/lb) 124.27 361.53 780.7 1263.6 601.48 vr 158.58 10.572 0.2848 2.768 pr 1.2147 51.561 5961.6 305.24 Example (9.38): Wcycle, in Btu/lb. Qout, in Btu/lb. Thermal efficiency The cut off ratio Cut off ratio: from ideal gas equation at constant pressure:

Air-Standard analysis of Gas Turbine Power plants. Sec 9.5 : Modeling Gas Turbine Power Plants Air-Standard analysis of Gas Turbine Power plants. Gas power plants are lighter and more compact than vapor power plants. Used in aircraft propulsion & marine power plants. http://greenplanethero.com/Star_Engine/star_engine.html http://encyclozine.com/science/physics/thermodynamics?page=4

Air-Standard analysis: Working fluid is air Sec 9.5 : Modeling Gas Turbine Power Plants Air-Standard analysis: Working fluid is air Heat transfer from an external source (assumes there is no reaction) Jet engine: Suck (intake) Squeeze (compressor) Bang/Burn (combustion) Blow (turbine/exhaust) Heat Ex Process 1 – 2 : Isentropic compression of air (compressor). http://www.utpb.edu/research-grants/ht3r/description/components/brayton-cycle-lab/ Process 2 – 3 : Constant pressure heat transfer to the air from an external source (combustion) Process 3 – 4 : Isentropic expansion (through turbine) Process 4 – 1 : Completes cycle by a constant volume pressure in which heat is rejected from the air

Sec 9.5 : Modeling Gas Turbine Power Plants Gas Turbine Analysis process 1-2: s1 = s2 process 2-3: p2 = p3 process 3-4: s3 = s4 process 4-1: p4 = p1 For a gas turbine, the back work ratio is much larger than that in a steam cycle since vair>>vliquid bwr for a gas turbine power cycle is typically 40-80% vs. 1-2% for a steam power cycle.

Given T1 & T3  use table to find h1 & h3 . Sec 9.3 : Air-Standard Diesel Cycle Gas Turbine Analysis Given T1 & T3  use table to find h1 & h3 . Find state 2. Find state 4. Compressor pressure ratio: For Cold-Air Standard analysis: For state 2. For state 4.

Effect of Compressor pressure on efficiency. Sec 9.3 : Air-Standard Diesel Cycle Gas Turbine Analysis Effect of Compressor pressure on efficiency. with Max T3 is approximately 1700 K

The thermal efficiency The back work ratio. The net power developed. Example : Air enters the compressor of an ideal cold air-standard Brayton cycle at 500°R with an energy input of 3.4x106 Btu/hr. The compression ratio is 14 and the max T is 3000°R. For k=1.4 calculate The thermal efficiency The back work ratio. The net power developed. State 1 2 3 4 T (R) 500 3000 h (BTU/lb) Similar to 9.41

The thermal efficiency The back work ratio. The net power developed. Example : Air enters the compressor of an ideal cold air-standard Brayton cycle at 500°R with an energy input of 3.4x106 BTU/hr. The compression ratio is 14 and the max T is 3000°R. For k=1.4 calculate The thermal efficiency The back work ratio. The net power developed. State 1 2 3 4 T (R) 500 3000 Since we are given k=1.4, use a cold-air standard analysis. Temperatures for states 1 and 3 are given. For state 2. Similar to 9.41 For state 4.

The thermal efficiency The back work ratio. The net power developed. Example : Air enters the compressor of an ideal cold air-standard Brayton cycle at 500°R with an energy input of 3.4x106 BTU/hr. The compression ratio is 14 and the max T is 3000°R. For k=1.4 calculate The thermal efficiency The back work ratio. The net power developed. State 1 2 3 4 T (R) 500 1063 3000 1411 Similar to 9.41

The thermal efficiency The back work ratio. The net power developed. Example : Air enters the compressor of an ideal cold air-standard Brayton cycle at 500°R with an energy input of 3.4x106 BTU/hr. The compression ratio is 14 and the max T is 3000°R. For k=1.4 calculate The thermal efficiency The back work ratio. The net power developed. State 1 2 3 4 T (R) 500 1063 3000 1411 But need the mass flow rate. Similar to 9.41

The thermal efficiency The net power developed. Example (9.43): The rate of heat addition to an air-standard Brayton cycle is 3.4x109 BTU/hr. The pressure ratio for the cycle is 14 and the minimum and maximum temperatures are 520°R and 3000°R, respectively. Determine The thermal efficiency The net power developed. State 1 2 3 4 T (R) 520 3000 Pr h (Btu/lb) Similar to 9.41

The thermal efficiency The net power developed. Example (9.43): The rate of heat addition to an air-standard Brayton cycle is 3.4x109 BTU/hr. The pressure ratio for the cycle is 14 and the minimum and maximum temperatures are 520°R and 3000°R, respectively. Determine The thermal efficiency The net power developed. State 1 2 3 4 T (R) 520 1092 3000 1573 pr 1.2147 17.01 941.4 67.24 h (Btu/lb) 124.27 264.12 790.68 388.63 State 1 2 3 4 T (R) 520 3000 pr 1.2147 941.4 h (Btu/lb) 124.27 790.68 Temperatures for states 1 and 3 are given. Relative pressure and enthalpy values from Table A-22E Find state 2. Similar to 9.41 Find state 4.

The thermal efficiency The net power developed. Example (9.43): The rate of heat addition to an air-standard Brayton cycle is 3.4x109 BTU/hr. The pressure ratio for the cycle is 14 and the minimum and maximum temperatures are 520°R and 3000°R, respectively. Determine The thermal efficiency The net power developed. State 1 2 3 4 T (R) 520 1092 3000 1573 pr 1.2147 17.01 941.4 67.24 h (Btu/lb) 124.27 264.12 790.68 388.63 Similar to 9.41

The thermal efficiency The net power developed. Example (9.43): The rate of heat addition to an air-standard Brayton cycle is 3.4x109 BTU/hr. The pressure ratio for the cycle is 14 and the minimum and maximum temperatures are 520°R and 3000°R, respectively. Determine The thermal efficiency The net power developed. State 1 2 3 4 T (R) 520 1092 3000 1573 pr 1.2147 17.01 941.4 67.24 h (Btu/lb) 124.27 264.12 790.68 388.63 But need the mass flow rate. Similar to 9.41

End of Slides for Lecture 35