Problem 9 - University Physics with Modern Physics

Question

20.9. A refrigerator has a coefficient of performance of \(2.10 .\) In cach cycle it absorbs \(3.40 \times 10^{4} \mathrm{J}\) of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

Step-by-Step Solution

Answer

(a) 16190.48 J; (b) 50190.48 J.

Step 1: Understand Coefficient of Performance

The coefficient of performance (COP) for a refrigerator is defined as the ratio of the heat absorbed from the cold reservoir, \( Q_C \), to the work input, \( W \). This can be expressed as:\[ COP = \frac{Q_C}{W} \]Given: \( COP = 2.10 \), and \( Q_C = 3.40 \times 10^4 \text{ J} \).

Step 2: Solve for Work Input

Rearrange the COP formula to solve for the work input, \( W \):\[ W = \frac{Q_C}{COP} \]Plug in the given values:\[ W = \frac{3.40 \times 10^4 \text{ J}}{2.10} \]Calculate \( W \) to find the mechanical energy required each cycle.

Step 3: Calculate Mechanical Energy Required

Calculate:\[ W = \frac{3.40 \times 10^4 \text{ J}}{2.10} \approx 16190.48 \text{ J} \]Thus, approximately \( 16190.48 \text{ J} \) of mechanical energy is required each cycle to operate the refrigerator.

Step 4: Determine Heat Discarded

According to the first law of thermodynamics, the total energy must be conserved. Thus, the heat discarded to the high-temperature reservoir, \( Q_H \), is the sum of work done and the heat absorbed:\[ Q_H = W + Q_C \]Using the calculated \( W \):\[ Q_H = 16190.48 \text{ J} + 3.40 \times 10^4 \text{ J} \]

Step 5: Calculate Heat Discarded

Calculate:\[ Q_H = 16190.48 \text{ J} + 3.40 \times 10^4 \text{ J} = 50190.48 \text{ J} \]Thus, about \( 50190.48 \text{ J} \) of heat is discarded during each cycle to the high-temperature reservoir.

Key Concepts

Coefficient of Performance

The Coefficient of Performance (COP) is a crucial concept in thermodynamics, particularly when dealing with refrigeration cycles. It essentially measures efficiency. The COP is defined as the ratio of useful heating or cooling provided to the work necessary to produce that heating or cooling. For refrigerators, it is specifically the ratio of the heat absorbed from the cold reservoir to the work input required:\[ COP = \frac{Q_C}{W} \]

• High COP: Indicates a more efficient refrigerator. More heat is absorbed with less work input.
• Low COP: Suggests lower efficiency. More work is needed to absorb the same amount of heat.

In the exercise, the given COP of 2.10 means that for every joule of work, the refrigerator absorbs 2.10 joules of heat.

Energy Conservation

Energy Conservation is a foundational principle in thermodynamics. In this context, it relates to the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed. In the refrigerator cycle, the energy input as work and the energy added as heat must equal the energy output as heat:\[ Q_H = Q_C + W \]Here,

• \(Q_H\): Total heat discarded to the high-temperature reservoir.
• \(Q_C\): Heat absorbed from the cold reservoir.
• \(W\): Work done (mechanical energy) required each cycle.

This relationship helps us understand how energy flows through the refrigerator system, ensuring energy conservation.

Heat Transfer

Heat Transfer is the movement of thermal energy from one object or substance to another. In the world of refrigerators, efficient heat transfer is paramount to effective cooling. A refrigerator operates by absorbing heat from the interior (the cold reservoir) and transferring it to the outside air (the high-temperature reservoir). This is achieved through the flow of refrigerant inside coils:

• Heat Absorption: Occurs inside the fridge, where the refrigerant absorbs heat from the stored items, making the air inside progressively cooler.
• Heat Rejection: Happens outside the fridge, where the absorbed heat is released to the surroundings.

This cycle of heat transfer is repeated continuously, maintaining a consistent cool temperature within the refrigerator.

Refrigerators

Refrigerators are devices designed to move heat from a cold location to a warmer one, effectively keeping perishable goods cool and fresh. They rely on the principles of thermodynamics and heat transfer to function efficiently.

• Components: Comprised of compressors, condensers, evaporators, and refrigerant fluid to cycle heat effectively.
• Purpose: By extracting heat from the inside and expelling it outside, refrigerators lower the internal temperature, creating a cold environment suitable for food preservation.

By understanding how a refrigerator operates, you can appreciate the engineering that goes into maintaining a cold space in warm surroundings.

Mechanical Energy

Mechanical Energy in the context of a refrigerator refers to the work done to run the compressor and other components. This energy is derived from electrical energy that powers the refrigerator's mechanical processes.

• Work Requirement: Essential to move refrigerant through the system, enabling it to absorb and dissipate heat efficiently.
• Energy Transformation: Electrical energy is converted to mechanical energy, driving the compressor to circulate the refrigerant.

The mechanical energy required for each cycle is a critical factor in determining the refrigerator’s efficiency, represented by the work input \( (W) \) and directly affecting the COP of the system.