Question

An absorption refrigeration system is to remove heat from the refrigerated space at \(2^{\circ} \mathrm{C}\) at a rate of \(28 \mathrm{kW}\) while operating in an environment at \(25^{\circ} \mathrm{C}\). Heat is to be supplied from a solar pond at \(95^{\circ} \mathrm{C}\). What is the minimum rate of heat supply required?

Short answer

Answer: The minimum rate of heat supply required from the solar pond is 30.34 kW.

Step-by-step solution

Calculate the Coefficient of Performance (COP) of the refrigeration system

The Coefficient of Performance (COP) can be calculated using the temperature of the refrigerated space \(T_{low}\) and the temperature of the environment \(T_{high}\): $$ COP = \frac{T_{low}}{T_{high} - T_{low}} $$ Note that the temperatures should be in Kelvin. First, convert the temperatures to Kelvin: $$ T_{low} = 2^{\circ}\mathrm{C} + 273.15 = 275.15\,\mathrm{K} $$ $$ T_{high} = 25^{\circ}\mathrm{C} + 273.15 = 298.15\,\mathrm{K} $$ Next, calculate the COP: $$ COP = \frac{275.15\,\mathrm{K}}{298.15\,\mathrm{K} - 275.15\,\mathrm{K}} = \frac{275.15}{23} = 11.96 $$

Calculate the minimum rate of heat supply required

Now that we have the COP, we can use the heat extraction rate given, \(Q_{low}\), and the COP to find the minimum rate of heat supply \(Q_{high}\): $$ COP = \frac{Q_{low}}{Q_{high} - Q_{low}} $$ So we can rearrange the formula to solve for \(Q_{high}\): $$ Q_{high} = \frac{Q_{low}}{COP} + Q_{low} $$ Where: \(Q_{low} = 28\,\mathrm{kW}\), rate at which heat is removed from the refrigerated space \(COP = 11.96\) Now, plug in the values and calculate \(Q_{high}\): $$ Q_{high} = \frac{28\,\mathrm{kW}}{11.96} + 28\,\mathrm{kW} = 2.34\,\mathrm{kW} + 28\,\mathrm{kW} = 30.34\,\mathrm{kW} $$ So the minimum rate of heat supply required from the solar pond is \(30.34\,\mathrm{kW}\).

Key concepts

Coefficient of Performance (COP)

Understanding the Coefficient of Performance (COP) is crucial when studying an absorption refrigeration system. It's a metric that indicates the efficiency of a refrigeration system by comparing the amount of heat removed from the refrigerated space to the energy put in to remove that heat. Put simply, COP is the ratio of the cooling (or heating) provided to the work required.

In the case of the exercise, the COP was calculated using the temperatures of the refrigerated space and the environment, both converted to Kelvin. This straightforward formula: \[COP = \frac{T_{low}}{T_{high} - T_{low}}\] reveals the inherent efficiency of the system. The higher the COP, the less energy is required to remove a certain amount of heat, making it a key factor in designing eco-friendly and cost-effective refrigeration systems. In practical terms, if you have a refrigeration system with a COP of 12, it means that for every unit of energy used, 12 units of heat are removed from the space – representing an efficient system.

It's essential to note that COP is dependent on the temperature conditions of the refrigerated space and the environment and can vary significantly under different operating conditions.

Heat Supply Rate

The heat supply rate is a term that refers to the amount of heat energy supplied to a system per unit of time. In thermodynamics, it's often denoted as a power (because power is the rate at which energy is transferred), and in this problem, it is represented in kilowatts (kW). The heat supply rate is a pivotal element in the analysis of the absorption refrigeration system's operation since it affects the performance and efficiency.

To maintain continuous operation of an absorption refrigeration system, heat must be supplied constantly. In this exercise, the minimum heat supply rate needed for the refrigeration system to function effectively was found by rearranging the COP formula to solve for the heat supply rate, resulting in: \[Q_{high} = \frac{Q_{low}}{COP} + Q_{low}\]

This equation shows us that the heat supply must be at least sufficient to cover the heat removed from the refrigerated space, plus the heat needed to drive the refrigeration cycle. Knowing the minimum rate of heat supply is crucial for ensuring that the energy source, such as a solar pond, is adequate to maintain the desired temperature inside the refrigerated space.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature, and their relation to energy and work. The principles of thermodynamics are key to understanding how refrigeration systems, such as the absorption refrigeration system in our exercise, work. There are four laws of thermodynamics that govern the physical properties of heat exchange processes and set constraints on the efficiency of refrigeration systems.

The first and second laws of thermodynamics play a major role in analyzing such systems. The first law, also known as the law of energy conservation, implies that energy cannot be created or destroyed, only transferred or changed from one form to another. This law is the foundation for the energy balance calculations in the exercise, ensuring that the total energy supplied to the system equates the sum of the energy extracted from the refrigerated space and the work input.

The second law introduces the concept of entropy, a measure of disorder, stating that systems tend toward increased entropy. This law explains why heat naturally flows from hot to cool spaces and sets inherent limitations on the efficiency of heat engines and refrigerators, hence affecting the COP. Absorption refrigeration systems take advantage of these thermodynamic principles to transfer heat against its natural flow without moving parts, typically using a heat-absorbing fluid to facilitate the process.