I have a question about conservation of energy in my system A heat pump allows you to move a quantity of heat with a smaller energy input. It is most efficient for small temperature differences. A heat engine allows you to convert a temperature difference into mechanical work. It is most efficient for large temperature differences. Since their efficiencies are inversely proportional, if you use a heat pump to create a temperature gradient you won't be able to get as much energy back out with a heat engine. Now imagine we have 500 kJ of high temperature thermal energy from an industrial process. The heat was 'created' through regular heating, let's assume at 100% efficiency for simplicity. The industrial heat drives a heat engine and the temperature difference is such that we get a 50% efficiency, i.e. 250 kJ of mechanical power out. The heat from the industrial process is added via a heat exchanger. Let's say we preheat the heat exchanger via heat pump, but by only a small amount and therefore at high efficiency e.g. COP=4. e.g. we have added 100 kJ heat at a cost to us of 25 kJ .I'm guesstimating the numbers for the purpose of the explanation, but we now add 400 kJ from the industrial process so that there is 500 kJ in the heat exchanger and get the same power output of 250 kJ, of which 20% is attributed to the preheating. In the latest scenario instead of getting 250 kW mechanical power out at a cost of 500 kW, we achieve the same 250 kW at a cost of 425 kW. We have therefore increased efficiency. What I am worried about is that when I look at the preheat on its own, it feels like we have violated the conservation of energy. Can someone please provide explain either where the concept breaks the laws of thermodynamics or if not, where does the energy come from? Don't get too stuck up on why we are doing this, i've left out the application because of IP owned by someone else.