EBK THERMODYNAMICS: AN ENGINEERING APPR
EBK THERMODYNAMICS: AN ENGINEERING APPR
8th Edition
ISBN: 8220100257056
Author: CENGEL
Publisher: YUZU
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Textbook Question
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Chapter 9.12, Problem 64P

An air-standard cycle, called the dual cycle, with constant specific heats is executed in a closed piston– cylinder system and is composed of the following five processes:

1-2 Isentropic compression with a compression ratio, r = V1/V2

2-3 Constant-volume heat addition with a pressure ratio, rp = P3/P2

3-4 Constant-pressure heat addition with a volume ratio, rc V4/V3

4-5 Isentropic expansion while work is done until V5 = V1

5-1 Constant-volume heat rejection to the initial state

  1. (a)   Sketch the P-ν and T-s diagrams for this cycle.
  2. (b)   Obtain an expression for the cycle thermal efficiency as a function of k, r, rc, and rp.
  3. (c)   Evaluate the limit of the efficiency as rp approaches unity, and compare your answer with the expression for the Diesel cycle efficiency.
  4. (d)   Evaluate the limit of the efficiency as rc approaches unity, and compare your answer with the expression for the Otto cycle efficiency.

(a)

Expert Solution
Check Mark
To determine

Draw the Pv and Ts diagrams for the given cycle.

Answer to Problem 64P

The Pv and Ts diagrams for the given cycle are shown as in Figure (1).

Explanation of Solution

Draw the Pv and Ts diagram for the given cycle.

EBK THERMODYNAMICS: AN ENGINEERING APPR, Chapter 9.12, Problem 64P

Thus, the Pv and Ts diagrams for the given cycle are shown as in Figure (1)

(b)

Expert Solution
Check Mark
To determine

The expression for the back work ratio as a function of k and r.

Answer to Problem 64P

The expression for the back work ratio as a function of k,r,rc,andrp is 1rckrp1rk1(rp1)+krprk1(rc1)_.

Explanation of Solution

Apply first law to the closed system for processes 2-3, 3-4, and 5-1 to get the expression of qinandqout.

qin=cv(T3T2)+cp(T4T3)

qout=cv(T5T1)

Here, heat added to the system and heat rejected from the system is qinandqout, constant volume specific heat is cv, temperature at state 1, 2, 3, 4, and 5 are T1,T2,T3,T4,andT5, and constant pressure specific heat is cp.

Express the cycle thermal efficiency.

ηth=1qoutqin (I)

Conclusion:

Process 1-2: Isentropic

Calculate the ratio of T2/T1.

T2T1=(v1v2)k1=rk1

Here, volume at states 1 and 2 is v1,v2, compression ratio is r, and specific heat ratio is k.

Process 2-3: Constant volume

Calculate the expression for T3/T2.

T3T2=P3v3P2v2T3T2=P3P2=rp

Here, pressure at state 1 and 2 is P1andP2, and pressure ratio is rp

Process 3-4: Constant pressure

Calculate the expression for T4/T3.

P4v4T4=P3v3T3T4T3=v4v3T4T3=rc

Here, compression ratio is rc.

Process 4-5: Isentropic

Calculate the expression for T5/T4.

T5T4=(v4v5)k1T5T4=(v4v1)k1T5T4=(rcv3v1)k1T5T4=(rcv2v1)k1

T5T4=(rcr)k1

Process 5-1: Constant volume

Calculate the expression for T5/T1.

T5T1=T5T4T4T3T3T2T2T1

Substitute (rcr)k1 for T5T4, rc for T4T3, rp for T3T2, and rk1 for T2T1.

T5T1=(rcr)k1(rc)rp(rk1)=rckrp

Calculate the ratio of T3/T1.

T3T1=T3T2T2T1

Substitute rp for T3T2 and rk1 for T2T1.

T3T1=rprk1

Substitute cv(T3T2)+cp(T4T3) for qin and cv(T5T1) for qout in equation (I).

ηth=1cv(T5T1)cv(T3T2)+cp(T4T3)=1T1(T5/T11)T2(T3/T21)+kT3T1(T4/T31)=1(T5/T11)T2T1(T3/T21)+kT3T1(T4/T31) (II)

Substitute rprk1 for T3/T1, rp for T3T2, rc for T4T3, rk1 for T2T1, and rckrp for T5T1 in Equation (II).

ηth=1(rckrp1)rk1(rp1)+krprk1(rc1)

Thus, the expression for the back work ratio as a function of k,r,rc,andrp is 1rckrp1rk1(rp1)+krprk1(rc1)_.

(c)

Expert Solution
Check Mark
To determine

The limit of the efficiency as rp approaches unity.

Answer to Problem 64P

The limit of the efficiency as rp approaches unity is 11rk1[rck1k(rc1)].

Explanation of Solution

Recall the expression for the back work ratio as a function of k,r,rc,andrp and apply the limit as rp approaches unity.

limrp1ηth=1{limrp1rckrp1rk1(rp1)+krprk1(rc1)}=11rk1[rck1k(rc1)]

Thus, the limit of the efficiency as rp approaches unity is 11rk1[rck1k(rc1)].

The limit of the efficiency as rp approaches unity is same as the thermal efficiency of diesel cycle.

(d)

Expert Solution
Check Mark
To determine

The limit of the efficiency as rc approaches unity.

Answer to Problem 64P

The limit of the efficiency as rc approaches unity is 11rk1.

Explanation of Solution

Recall the expression for the back work ratio as a function of k,r,rc,andrp and apply the limit as rc approaches unity.

limrp1ηth=1{limrp1rckrp1rk1(rp1)+krprk1(rc1)}=1[rp1rk1(rp1)]=11rk1

Thus, the limit of the efficiency as rc approaches unity is 11rk1.

The limit of the efficiency as rc approaches unity is same as the thermal efficiency of Otto cycle.

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Chapter 9 Solutions

EBK THERMODYNAMICS: AN ENGINEERING APPR

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