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In order to calculate the mass flow rate of water heated by an electric resistance heater, the energy balance equation is invaluable. This equation expresses the conservation of energy principle. In simple terms, it states that the energy input into a system must equal the energy output. This is crucial for understanding how much energy is transferred into heating the water. The formula used is:\[P = \dot{m}c\Delta T\]- **P** represents the power provided by the heater in watts.- **\(\dot{m}\)** is the mass flow rate of the fluid, which is what you are solving for.- **c** is the specific heat capacity, a property of water that shows how much energy is needed to raise the temperature of a unit mass by one degree Kelvin.- **\(\Delta T\)** signifies the change in temperature from the inlet to the outlet.Using this simple yet powerful equation, we can determine how much water can be heated per second by the electrical power provided.

An electric resistance heater converts electrical energy into thermal energy. This appliance often uses metal wires that heat up when an electric current passes through them. This property makes electric resistance heaters efficient for heating fluids like water in this exercise. When current flows through the heater, it generates a specific power output measured in kilowatts (kW). In our case, the heater provides a power of 5 kW. For ease of calculation, it is important to remember that 1 kW equals 1,000 watts. The entire produced energy is utilized to heat the water as the process is insulated. Thus, the heat loss is negligible, which means the system efficiency is approximately 100%. This makes it ideal for calculating the energy input for the water heating, needed in conjunction with the energy balance equation to determine the mass flow rate.

The specific heat capacity (\(c\)) is an intrinsic property of any substance. For water, this is a functional parameter when calculating how much energy is needed to increase its temperature. In this exercise, the specific heat capacity of water is approximately 4.18 kJ/kg·K, which signifies that it requires 4.18 kilojoules to raise 1 kilogram of water by 1 Kelvin.This value is crucial in the energy balance equation, influencing how much water can be heated in a specific time given the energy available. This property allows us to convert energy input into a temperature change, helping to find the mass flow rate.

Temperature conversion is necessary when working with equations that require absolute temperature. In thermodynamics, Kelvin is the preferred scale because it starts at absolute zero, making calculations more straightforward. For instance, converting Celsius to Kelvin is easy and done by adding 273.15:- **Celsius to Kelvin conversion.**\[T(K) = T(^{\circ}\mathrm{C}) + 273.15\]In this exercise, you'll convert the water temperature from 15°C and 60°C to Kelvin before using them in the energy balance equation. This is critical for accuracy in thermodynamic equations, ensuring all energy unit calculations are consistent.