Question

A \(30-\) cm-long, \(1500-W\) electric resistance heating element whose diameter is \(1.2 \mathrm{cm}\) is immersed in \(70 \mathrm{kg}\) of water initially at \(20^{\circ} \mathrm{C}\). Assuming the water container is wellinsulated, determine how long it will take for this heater to raise the water temperature to \(80^{\circ} \mathrm{C}\). Also, determine the minimum work input required and exergy destruction for this process, in kJ. Take \(T_{0}=20^{\circ} \mathrm{C}\).

Short answer

Solution: Step 1: Calculate the energy required to heat the water: \(Q = 70 kg \cdot 4.18 \frac{kj}{kg\cdot K} \cdot (80^{\circ} C - 20^{\circ} C) = 70 \cdot 4.18 \cdot 60 = 17556 kj\) Step 2: Calculate the time required to heat the water: Since we have Q in kJ and P in Watts, we need to convert P or Q to the same unit: \(1500W = 1.5 \frac{kj}{s}\) Now we can find the time required: \(t = \frac{Q}{P} = \frac{17556kj}{1.5\frac{kj}{s}} = 11704 s \) The time required to heat the water from 20°C to 80°C is 11,704 seconds. Step 3: Calculate the minimum work input: The minimum work input required is equal to the energy required to heat the water (Q): Minimum work input = 17,556 kJ In summary, it takes 11,704 seconds to raise the temperature of the water from 20°C to 80°C, and the minimum work input required for this process is 17,556 kJ.

Step-by-step solution

Calculate the energy required to heat the water

We know that the energy required to change the temperature of a substance can be calculated as follows: \(Q = mc(T_{final} - T_{initial})\) Where \(Q\) is the energy required, \(m\) is the mass, \(c\) is the specific heat capacity, and \(T_{final}\) and \(T_{initial}\) are the final and initial temperatures, respectively. For water, the specific heat capacity (\(c\)) is \(4.18 \frac{kj}{kg\cdot K}\). Therefore, to find the energy required to heat the water, we have: \(Q = 70 kg \cdot 4.18 \frac{kj}{kg\cdot K} \cdot (80^{\circ} C - 20^{\circ} C) \)

Calculate the time required to heat the water

We are given the power of the heating element (\(P\)) as \(1500 W\). Since power is defined as the rate of energy transfer, we have: \(P = \frac{Q}{t}\) Where \(t\) is the time it takes to heat the water. Solving for \(t\), we get: \(t = \frac{Q}{P}\) Using the values calculated in Step 1, we can find the time required to heat the water.

Calculate the minimum work input and exergy destruction

We know that for an adiabatic process with no external work, the change in the system's internal energy is equal to the heat exchanged, which is the work done by the heating element. Therefore, the minimum work input required is equal to the energy required to heat the water, which we calculated in Step 1. The exergy destruction can be calculated using the following equation: \(Ex_{destruction} = T_{0} \cdot (S_{final} - S_{initial})\) Where \(T_{0}\) is the reference temperature (20°C) and \(S_{final}\) and \(S_{initial}\) are the final and initial entropies of the system, respectively. Since we don't have enough information to calculate the entropy change, we can't find the exergy destruction for this process. However, we know that the exergy destruction is always greater than zero, as it represents the irreversible energy losses during the process.

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity is a fundamental concept in thermodynamics that describes how much heat energy is required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin). It is an intrinsic property of the material and is often denoted with the symbol 'c'. The formula used to calculate the energy, Q, required to change a substance's temperature is:

\[ Q = mc(T_{final} - T_{initial}) \]

In the provided exercise, the specific heat capacity of water is given as \(4.18 \frac{kJ}{kg \cdot K}\), which is relatively high, indicating that water can absorb a lot of heat before its temperature increases significantly. This property makes water an excellent coolant and is why it is commonly used in heating systems. Imagine you have a pot of water on a stove; the high specific heat capacity means that it will take some time before the water starts boiling, even though the stove is transferring heat at a constant rate. Similarly, this concept helps explain how long it will take the heat from an electric resistance heater to raise the temperature of the water from 20°C to 80°C.

The concept of specific heat capacity is critical for engineers and scientists as it is essential for calculating thermal energy changes in processes and designing systems that involve heat transfer.

Energy Transfer Rate and Heating Time

The energy transfer rate is the amount of energy moved from one system to another per unit time, and it is synonymous with the term 'power' in the context of heat transfer. The power of a heating element is the rate at which it can transfer heat to its surroundings, measured in watts (W). One watt is equivalent to one joule per second.

When calculating the time it takes for a heater to warm water, as in our exercise, the formula you would use is:

\[ P = \frac{Q}{t} \]

Here, 'P' stands for power, 'Q' for the energy needed to raise the water's temperature, and 't' for time. By rearranging the formula to solve for time \(t = \frac{Q}{P}\), we can determine how long the electric resistance heating element will take to achieve the desired temperature increase. If you were heating a pool, a bathtub, or even an industrial vat in a factory, using these calculations would be crucial for timing and energy efficiency. Your electric bill might provide a real-world context, as it reflects the energy transfer rate from the electrical company to your home appliances, including heating elements like the one in this exercise.

Exergy Destruction in Thermodynamic Processes

Exergy destruction is a thermodynamic concept that measures the irreversibility of a process and represents the portion of energy that cannot be used to do work due to entropy increase or energy dispersal. It is an important measure in assessing the efficiency of thermal systems.

In any real-world process, there is always some degree of exergy destruction due to friction, electrical resistance, or other factors that disperse energy. This is closely tied to the concept of entropy, a measure of disorder or randomness in a system.

The initial and final states of entropy in a process would allow us to compute exergy destruction, but these values are often complex to calculate and require additional details such as the properties of materials and the process dynamics. The equation referencing exergy destruction is:

\[ Ex_{destruction} = T_{0} \cdot (S_{final} - S_{initial}) \]

In practical terms, minimizing exergy destruction is key to making processes more sustainable by reducing energy waste. The relevance of exergy destruction can also be understood through everyday examples like an old inefficient light bulb, which loses much of the electricity as heat instead of light, thereby destroying exergy and increasing energy costs.