Question

A heat pump is under consideration for heating a research station located on an Antarctica ice shelf. The interior of the station is to be kept at 158C. Determine the maximum theoretical rate of heating provided by a heat pump, in kW per kW of power input, in each of two cases: The role of the cold reservoir is played by (a) the atmosphere at 2208C, (b) ocean water at 58C

Answer 1

Answer:

a

The heat provided is  $E_t = 0.1480 kW$

b

The heat provided is  $E_T = 0.5656 kW$

Explanation:

From the  question we are told that

        The temperature of interior is $T_h = 158 ^oC = 158 + 273 = 431\ K$

The coefficient of performance of the heater is  

               $COP= \frac{T_h}{T_h +T_c}$

Where $T_c$ is the temperature of the cold reservoir with a value  $T_c = 2208 ^oC = 2208+273 =2481 K$

          So    

                 $COP= \frac{431}{431 + 2481 }$

                 $COP= 0.1480$

The coefficient of performance is also represented mathematically as

           $COP = \frac{Heat \ provide \ to\ the \ research }{Workdone}$

Let assume the workdone is  = 1 N m

            $Heat \ provide \ to\ the \ research (E_t) = 1 * 0.1480$

                                                           $E_t = 0.1480 kW$

 The coefficient of performance of the heater is  

               $COP= \frac{T_h}{T_h +T_c}$

Where $T_c$ is the temperature of the cold reservoir with a value  $T_c = 58 ^oC = 58+273 =331 K$

          So    

                 $COP= \frac{431}{431 + 331 }$

                 $COP= 0.5656$

The coefficient of performance is also represented mathematically as

           $COP = \frac{Heat \ provide \ to\ the \ research }{Workdone}$

Let assume the workdone is  = 1 N m

            $Heat \ provide \ to\ the \ research (E_T) = 1 * 0.5656$

                                                           $E_T = 0.5656 kW$

Answer 2

Answer:

a. 8.228 kW

b. 28.8 kW

Explanation:

a) Carnot COP = T_high/(T_high - T_low) = (15+273)/((15+273)-(-20+273)) = 8.228

Heating provided / Work input = COP

Heating provided = 8.228*1 = 8.228 kW

b)

Carnot COP = T_high/(T_high - T_low) = (15+273)/((15+273)-(5+273)) = 28.8

Heating provided / Work input = COP

Heating provided = 28.8*1 = 28.8 kW