Answer:
if the temperature of the sink is lowered by 6.7%, the efficiency of the Carnot engine will increase by 10%.
Step-by-step explanation:
The efficiency of a Carnot engine is given by the equation:
η = 1 - T_sink / T_source
where η is the efficiency, T_sink is the temperature of the sink, and T_source is the temperature of the source.
In this case, we are given that the efficiency of the engine is 40%. Therefore, we can write:
0.4 = 1 - T_sink / T_source
Rearranging this equation, we get:
T_sink / T_source = 1 - 0.4
T_sink / T_source = 0.6
Next, we are asked to find the temperature of the sink needed to increase the efficiency of the engine by 10%. Let's call this new efficiency η_new. Since the efficiency of a Carnot engine is given by the same equation as before, we can write:
η_new = 1 - T_sink_new / T_source
where η_new is the new efficiency, and T_sink_new is the new temperature of the sink.
We know that the new efficiency is 10% higher than the original efficiency, so we can write:
η_new = η + 0.1η = 0.4 + 0.1(0.4) = 0.44
Substituting this into the equation above, we get:
0.44 = 1 - T_sink_new / T_source
Rearranging this equation, we get:
T_sink_new / T_source = 1 - 0.44
T_sink_new / T_source = 0.56
Now we can set up an equation to find the new temperature of the sink:
0.56 = T_sink_new / T_source
T_sink_new = 0.56T_source
To find the temperature difference between the old and new sink temperatures, we can subtract the two equations we have derived:
T_sink_new - T_sink = 0.6T_source - 0.56T_source
T_sink_new - T_sink = 0.04T_source
Finally, we can substitute the given efficiency of 40% to find the source temperature:
0.4 = 1 - T_sink / T_source
T_source = T_sink / (1 - 0.4)
T_source = T_sink / 0.6
Substituting this expression for T_source into the equation for the temperature difference, we get:
T_sink_new - T_sink = 0.04(T_sink / 0.6)
Simplifying, we get:
T_sink_new - T_sink = 0.067T_sink
Multiplying both sides by 100, we get:
(T_sink_new - T_sink) * 100 = 6.7T_sink
Therefore, if the temperature of the sink is lowered by 6.7%, the efficiency of the Carnot engine will increase by 10%.