Question

a. Refrigerators make use of the heat cycles, such as the Carnot cycle, but in order to remove heat from the cold source (fridge) and put that energy into the hot source (the enrivonment), another energy source must be added to the system to make this occur. The minimum energy to be supplied follows the equation given below. Find an expression for the coefficient of performance in terms of T_h and T_c.

                        coefficient of performance (c) = heat transferred/work done = |qc| / |w|

     

      b. What is the coefficient of performance to run a household refrigerator (i.e. freeze water)?

Answer 1

(a). If a reversible heat engine extracted heat from hot reservoir and rejected to the cold reservoir , then it produces positive amount of work . If we suppose this is the forward cycle of the engine , then in refrigerator the cycle is reverse because , in it heat is withdrawn from cold reservoir and rejected to hot reservoir . In refrigerator work is destroyed , i.e. , W<0 .

Let , temperature of cold reservoir and hot reservoir are T_c and T_h respectively . And also let , heat extracted from low-temperature reservoir is Q_c and heat rejected to high temperature reservoir is Q_h .

Now the coefficient of performance , , of a refrigerator is the ratio of the heat extracted from low-temperature reservoir to the work destroyed , that is ,

   = Q_c / (- W)

= Q_c / - (Q_c+Q_h) {since W = Q_c+Q_h}

also we know that , (Q_c / Q_h) = - (T_c / T_h)

therefore , = (Q_c / Q_h) / - (Q_c/Q_h +1)

= - (T_c / T_h) / - ( -T_c / T_h +1) = T_c / (T_h - T_c)

This is the expression of the coefficient of performance in terms if T_h and T_c.