The air conditioner refrigerated air unit for a house in Alb

The air conditioner (refrigerated air unit) for a house in Albuquerque can be run backwards and used as a heat pump during cool spring weather. Suppose that the outside temperature is 50 degreeF, and the inside temperature is to be maintained at 70 degreeF. Assuming the unit operates reversibly, how much electrical work must be performed for every kJ liberated in the house?

Solution

To calculate the electrical work done we need to calculate the efficiency of the given engine. This can be calculted by the basic knowledge of carnot\'s machines which says that ideal carnot enigne is the one which works on carnot cycle

So from that we know the basic expression to calculate the efficiency for the given machine is:

(eff) = W/Q(h) (1)

where W= Q(h)-Q(l)

and after solving further by considering temperature we get the expression as: (expression is available in all basic thernmodynamics books)

(eff) = T(h)-T(l)/T(h) (2)

(eff) = 1 - T(l)/T(h) (3)

and for our case we are given that machine/engine is working in reverse order so the above expression comes to the form:

(eff) = 1/1 - T(l)/T(h) (4)

We are given temperature values, after converting to Celcius we can proced further

Using the above obtained expression( eq-4) for our case:-

(eff) = 1 / (1-323.5/343.5)

we get, (eff) = 18.18%

Now we can calculate the work done in kJ

(eff) = W/T(h)

W = (eff)*T(h)

W = 18.18*70

W=1272.72k J