Heat and Thermodynamics 5 Question 11

11. Three Carnot engines operate in series between a heat source at a temperature T1 and a heat sink at temperature T4 (see figure). There are two other reservoirs at temperatures T2 and T3, as shown with T1>T2>T3>T4. The three engines are equally efficient if

(Main 2019, 10 Jan I)

(a) T2=(T13T4)1/4;T3=(T1T43)1/4

(b) T2=(T12T4)1/3;T3=(T1T42)1/3

(c) T2=(T1T4)1/2;T3=(T12T4)1/3

(d) T2=(T1T42)1/3;T3=(T12T4)1/3

Show Answer

Solution:

  1. Given, Carnot engines operates as,

As, efficiency of a Carnot’s engine is given by

We have,

η=1Tsink Tsource 

η1= efficiency of engine ε1=1T2T1

η2= efficiency of engine ε2=1T3T2 η3= efficiency of engine ε3=1T4T3

For equal efficiencies,

η1=η2=η31T2T1=1T3T2=1T4T3T2T1=T3T2=T4T3T22=T1T3 and T32=T2T4T24=T12T32T24=T12T2T4 or T23=T4T12 Also, T2=(T12T4)13T34=T22T42=T1T3T42T33=T1T42 or T3=(T1T42)13

Key idea In a Carnot engine the heat flow from higher temperature source ( at T1) to lower temperature sink(atT2) and give the work done equal to the W=Q1Q2.

For the given condition, Carnot engine A and B are operated in series as shown below

where,

Q1= heat rejected by engine A at T1K,

Q2= heat received by engine B at T2K and

Q3= heat rejected by engine B to source C at T3K.

According to Carnot engine principle,

W1=Q1Q2 (work-output from source A and B )

W2=Q2Q3 (work output from source B and C )

As per the given condition, if the work outputs of the two engines are equal, then

Q1Q2=Q2Q3Q1+Q3=2Q2

Q1Q2+Q3Q2=2 Therefore T1T2+T3T2=2 So, T1+T3T2=2T2=T1+T32=600+4002T2=500K



जेईई के लिए मॉक टेस्ट

एनसीईआरटी अध्याय वीडियो समाधान

दोहरा फलक