A steam engine delivers 5.4×10⁸J of work per minute and services 3.6 × 10⁹J of heat per minute from its boiler. What is the efficiency of the engine? How much heat is wasted per minute?

The steam engine absorbs energy in form of heat from boiler and converts some of it into mechanical work and return rest to the surrounding or we can say some part of heat absorbed is converted into work and rest is wasted, suppose Q₁ heat is absorbed from Boiler (Source), W amount of work is done by the engine and Q₂ heat is returned to the surroundings (Sink)

As has been shown diagrammatically

So we have

Q₁ = W + Q₂

Now since engine is absorbing 3.6 × 10⁹J of heat per minute from its boiler so each minute amount of heat absorbed by engine from Boiler is

Q₁ = 3.6 × 10⁹J

steam engine delivers 5.4×10⁸J of work per minute, so each minute work done by engine is

W = 5.4×10⁸J

Now efficiency is defined as the amount of work done per unit supplies Energy , or it is the ratio of work output and the heat input of the system , mathematically

Where W is the work done by engine and Q₁ is the amount heat absorbed

So efficiency of the engine is

In percentage form

So the efficiency of the engine is 15%

Now we need to find the amount of heat wasted, which is the amount of heat or energy transferred back to Surrounding (Sink) without doing any work, denoted by Q₂

We already know

Q₁ = W + Q₂

Or we can say

Q₂ = Q₁ – W

Now since engine is absorbing 3.6 × 10⁹J of heat per minute from its boiler so each minute amount of heat absorbed by engine from Boiler is

Q₁ = 3.6 × 10⁹J

steam engine delivers 5.4×10⁸J of work per minute, so each minute work done by engine is

W = 5.4×10⁸J

So we get

Q₂ = 3.6 × 10⁹J – 5.4 × 10⁸J

= 36 × 10⁸ J - 5.4 × 10⁸J = 30.6 × 10⁸J

=3.06 ×10⁹ J

i.e. 3.06 ×10⁹ J of heat is wasted each minute or we can say

3.06 ×10⁹ J heat per minute is being wasted