Sure, let's solve this step by step:
1. First, let's understand the given measures:
- The heat pump delivers 3100 Joules of heat into the house.
- The Coefficient of Performance (COP) for this heat pump is 3.0. The COP is the ratio of heat delivered to the work input required by the pump.
- The indoor temperature in the house is 22 ∘ C.
- We are given two scenarios for the outdoor temperatures: 0 ∘ C and -15 ∘ C.
2. In order to use these temperatures in our calculations, we will have to convert them to the Kelvin scale from the Celsius scale by adding 273.15. The reason we use the Kelvin scale is because all temperature differences used in thermodynamics calculations must be expressed in Kelvin.
- The indoor temperature is therefore 22 + 273.15 = 295.15K.
- The outdoor temperature for the first scenario is 0 + 273.15 = 273.15K.
- The outdoor temperature for the second scenario is -15 + 273.15 = 258.15K.
3. Now, let's calculate the work input needed by the heat pump at two different outdoor temperatures using the given COP:
(a) For an outdoor temperature of 0 ∘ C (or 273.15K), using the formula Work Input = Heat Delivered / CoP, the work input required would be 3100 Joules / 3.0 = 1033.33 Joules.
(b) Similarly for an outdoor temperature of -15 ∘ C (or 258.15K), the work input required would be again 3100 Joules / 3.0 = 1033.33 Joules. Notice that with a fixed CoP, the work input remains the same regardless of the outdoor temperature.
4. Now, let's calculate the work input assuming an ideal Carnot COP, which describes the maximum possible efficiency of a thermodynamic engine. The Carnot CoP is calculated using the formula CoP_Carnot = T_H / (T_H - T_L), where T_H is the high temperature reservoir (here the house) and T_L is the low temperature reservoir (outside).
(a) For an outdoor temperature of 0 ∘ C (or 273.15K), the Carnot CoP would be 295.15K / (295.15K - 273.15K) = 22.07. The work input would therefore be 3100 Joules / 22.07 = 231.07 Joules.
(b) For an outdoor temperature of -15 ∘ C (or 258.15K), the Carnot CoP would be 295.15K / (295.15K - 258.15K) = 6.98. The work input would therefore be 3100 Joules / 6.98 = 388.62 Joules.
Observe that the Carnot CoP decreases as the outdoor temperature drops, and consequently the work input required increases.
That’s all. I hope this explanation was clear enough. If you need anything else, feel free to ask.