Question

An ideal, or Carnot, heat pump is used to heat a house to a temperature of 294 K (21 oC). How much work must the pump do to deliver 3000 J of heat into the house (a) on a day when the outdoor temperature is 273 K (0 oC) and (b) on another day when the outdoor temperature is 252 K (-21 oC)

Answer 1

a)
`W_(in) = 214.286\,J`, b)
`W_(in) = 428.571\,J`

Step-by-step explanation:

a) The performance of a Carnot heat pump is determined by the Coefficient of Performance, which is equal to the following ratio:

`COP_(HP) = (T_(H))/(T_(H)-T_(L))`

Where:

`T_(L)` - Temperature of surroundings, measured in Kelvin.

`T_(H)` - Temperature of the house, measured in Kelvin.

Given that
`T_(H) = 294\,K` and
`T_(L) = 273\,K`. The Coefficient of Performance is:

`COP_(HP) = (294\,K)/(294\,K-273\,K)`

`COP_(HP) = 14`

Besides, the performance of real heat pumps are determined by the following form of the Coefficient of Performance, that is, the ratio of heat received by the house to input work.

`COP_(HP) = (Q_(H))/(W_(in))`

The input work to deliver a determined amount of heat to the house:

`W_(in) = (Q_(H))/(COP_(HP))`

If
`Q_(H) = 3000\,J` and
`COP_(HP) = 14`, the input work that is needed is:

`W_(in) = (3000\,J)/(14)`

`W_(in) = 214.286\,J`

b) The performance of a Carnot heat pump is determined by the Coefficient of Performance, which is equal to the following ratio:

`COP_(HP) = (T_(H))/(T_(H)-T_(L))`

Where:

`T_(L)` - Temperature of surroundings, measured in Kelvin.

`T_(H)` - Temperature of the house, measured in Kelvin.

Given that
`T_(H) = 294\,K` and
`T_(L) = 252\,K`. The Coefficient of Performance is:

`COP_(HP) = (294\,K)/(294\,K-252\,K)`

`COP_(HP) = 7`

Besides, the performance of real heat pumps are determined by the following form of the Coefficient of Performance, that is, the ratio of heat received by the house to input work.

`COP_(HP) = (Q_(H))/(W_(in))`

The input work to deliver a determined amount of heat to the house:

`W_(in) = (Q_(H))/(COP_(HP))`

If
`Q_(H) = 3000\,J` and
`COP_(HP) = 7`, the input work that is needed is:

`W_(in) = (3000\,J)/(7)`

`W_(in) = 428.571\,J`