using the same amount of energy.
Variables
What is the strategy?
- Calculate the maximum coefficient of performance of the heat pump
- Use the definition of coefficient of performance with the actual COP to calculate the maximum amount of heat transferred to the interior.
Physics principles and equations
This inequality specifies the maximum possible COP of a heat pump with given temperature reservoirs:
(Be sure to use Kelvin temperatures.)
The definition of COP
COP = Qh/W
Step-by-step derivation
We first calculate the maximum possible coefficient of performance for the heat pump.
The maximum possible COP is over ten times higher than the actual (and more realistic) COP of 5.48 as stated in the problem. We now
calculate the heat transferred into the house, using this actual value for the coefficient of performance.
The heat pump transfers significantly more heat than the furnace would. As you might expect, this does not violate the principle of conservation
of energy. The device “pumps” thermal energy from outdoors to indoors, further cooling the outdoors in order to warm the indoors.
temperature of cold reservoir Tc = 15.0°C
temperature of hot reservoir Th = 19.0°C
work done on heat pump W = 1250 J
maximum COP of pump COPmax
actual COP of pump COP = 5.48
heat flowing to interior QhStep Reason
1. maximumCOP
2. substitute values
3. COPmax = 73.0 evaluate
Step Reason
4. COP = Qh/W definition of COP
5. Qh = (COP)W solve for Qh
6. Qh = (5.48)(1250 J) substitute values
7. Qh = 6850 J evaluate
21.12 - Interactive summary problem: efficiency of an automobile engine
In the simulation to the right, you get to put an engine to work in a common situation. You are the driver of a car. You specify how much
gasoline you want the engine to consume. When you press GO, the car will accelerate at full engine power until the gasoline is gone, at which
point the simulation will stop and your calculations will be evaluated.
Your goal is to specify the amount of gasoline that will allow you to reach a speed of 90.0 km/h. Here, we will consider the engine as supplying
an external force doing work on the car. The engine must do 9.35×10^5 J of work to accelerate the car to 90.0 km/h. This figure is a good
approximation of the amount of work a real engine must do to both increase the car’s kinetic energy to the desired amount and overcome
forces such as air resistance, friction, and so forth.
To decide how much gasoline you need, you must first calculate the efficiency of the engine. You can do this by adding any amount of gasoline
and pressing GO to see the resulting heat added and heat expelled while the engine runs. Use these values to calculate the engine’s