Heat engine efficiency

1.       Why it is not possible to have an engine with 100% efficiency?

The efficiency of a heat engine is given by:

Ƞ = 1 – T_sink/T_{source                  Where T is temperature (in K) of source and sink}

    = (T_source– T_sink)/T_source

Thus, to attain 100% efficiency, T_sink MUST be equal to zero.

Let us assume that we make a heat engine with source at 300K and sink at 0K. As soon as the first cycle is complete, the engine will release some heat into sink, thus elevating its temp. Let us assume that this rise in temperature is as low as 1 K.

Then,

Ƞ= 1 – 1/300

   = 300 – 1/300

   = 299/300

   = 99.6%; which is less than 100%!

Therefore, even if we make the sink temp 0K, it will remain so till 1^st cycle only. From the very next cycle, the efficiency of the engine will become less than 100%.

 

2.       Effect of increasing T_source or decreasing T_sink on efficiency.

Let us assume that a heat engine operates between 400 K and 200 K.

The current efficiency is 50%

Let us assume that the source temp is increased by 100K (to 500K).

Then,

Efficiency = (500 – 200)/500

                  = 60%

Let us assume that the sink temp is decreased by 100K (to 100K).

Then,

Efficiency = (400 – 100)/400

                  = 75%

Thus, we see that decreasing the sink temperature has a greater effect on efficiency as compared to increasing the source temperature!