Question

Two Carnot heat engines are operating in series such that the heat sink of the first engine serves as the heat source of the second on. If the source temperature of the first engine is 1300 K and the sink temperature of the second engine is 300 K and the thermal efficiencies of both engines are the same, the temperature of the intermediate reservoir is

Answer 1

The intermediate reservoir temperature for two Carnot heat engines operating in series with equal efficiencies is approximately 650 K.

Step-by-step explanation:

To find the intermediate reservoir temperature for two Carnot heat engines operating in series with equal efficiencies, we can use the equation for thermal efficiency of a Carnot engine, which is η = 1 - (Tc/Th). The source temperature of the first engine is 1300 K, and the sink temperature of the second engine is 300 K. Let's denote the intermediate temperature as Tm.

Since the efficiencies are the same, we set their efficiencies equal to each other which gives us:
1 - (Tm/1300) = 1 - (300/Tm)
Solving for Tm gives Tm = √(1300 * 300).

Calculating √(1300 * 300), we get an intermediate temperature Tm of approximately 650 K for the intermediate reservoir.

Answer 2

the temperature of the intermediate reservoir is 624.5 K

Step-by-step explanation:

Given the data in the question

The two Carnot heat engines are operating in series;

[ T
`_H` ]

↓

((1)) ⇒ W
`_{out`

↓

[ T
`_M` ]

↓

((2)) ⇒ W
`_{out`

[ T
`_L` ]

The maximum possible efficiency for any heat engine is the Carnot efficiency;

η
`_{rev` = 1 -
`(T_L)/(T_H)`

the thermal efficiencies if both engines are the same will be;

η
`_A` = η
`_B`

1 -
`(T_M)/(T_H)` = 1 -
`(T_L)/(T_M)`

1 - 1 -
`(T_M)/(T_H)` = -
`(T_L)/(T_M)`

-
`(T_M)/(T_H)` = -
`(T_L)/(T_M)`

`(T_M)/(T_H)` =
`(T_L)/(T_M)`

T
`_M`² = T
`_L` × T
`_H`

T
`_M` = √(T
`_L` × T
`_H`)

source temperature of the first engine T
`_H` = 1300 K

sink temperature of the second engine T
`_L` = 300 K

we substitute

T
`_M` = √(300 × 1300)

T
`_M` = √390000

T
`_M` = 624.4998 K ≈ 624.5 K

Therefore, the temperature of the intermediate reservoir is 624.5 K