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Thermodynamics is the science that deals with heat and energy transfer. In this problem, we examine the heating process of water in an electric water heater by considering the principles of thermodynamics. When you heat water, you transfer energy from the electrical heating element to the water molecules. This process increases the water's internal energy, raising its temperature.
The First Law of Thermodynamics is essential here. This law states that energy cannot be created or destroyed, only transferred or converted from one form to another. In our scenario, electrical energy is converted into thermal energy, heating the water.
We also discuss the concept of exergy, which refers to the useful portion of energy. When analyzing the exergy supplied to the water heater, we assess how much of the input energy becomes usable heat energy. Since there are negligible losses, almost all supplied electrical energy is converted into heat energy used to warm the water.

Calculating heat energy (Q) is crucial in solving thermodynamic problems involving temperature changes. The formula used is:
\[ Q = m \times c \times \triangle T \]
Here,

• Q is the heat energy required (in kJ),
• m is the mass of the substance (in kg),
• c is the specific heat capacity (in kJ/kg K), and
• \triangle T is the temperature change (in K).

In this exercise, we first calculate the mass of the water, which turns out to be 200 kg. The temperature change (\triangle T) is 32°C (or 32 K, since ΔT in Celsius is the same as in Kelvin). The specific heat capacity (c) of water is given as 4.18 kJ/kg K.
Using the formula, we compute the heat energy as:
\[ Q = 200 \text{ kg} \times 4.18 \text{ kJ/kgâ‹…K} \times 32 \text{ K} = 26752 \text{ kJ} \]
This shows the total energy needed to raise the water temperature from 23°C to 55°C.

Specific heat capacity (c) is a property of a substance that indicates how much heat energy is required to change the temperature of a unit mass by one degree (either Celsius or Kelvin). In our exercise, the specific heat capacity of water is provided as 4.18 kJ/kg K.
This means to raise 1 kg of water by 1°C, 4.18 kJ of energy is needed. Understanding this concept helps explain how much energy is required for heating processes. The higher the specific heat, the more energy is needed to raise the temperature.
In the water heater problem, knowing the specific heat capacity is essential for calculating the total heat energy needed. As we have shown, using the specific heat capacity, the mass, and the temperature change, we find out the exact amount of energy required to heat the water to the desired temperature.

In thermodynamics, incompressible fluids are those whose density does not change with pressure variations. Water is often modeled as an incompressible fluid, especially when dealing with moderate temperature and pressure ranges. This assumption simplifies calculations as we don't need to account for volume changes due to pressure.
In the exercise, modeling water as an incompressible fluid allows us to use a constant density (1 kg/L) to find the mass of water from its volume. This simplification is crucial for accurately performing calculations without introducing complex variables.
By assuming water is incompressible, the solution focuses on the direct relationship between heat energy, mass, and temperature change, making the calculations straightforward and accurate.