Let's break down the problem step by step, focusing on the energy balance for the water heating loop and the implications of changes in flow rate and heater power on the time taken to reach a steady state.

Energy Balance Equation for the Tank

To establish the energy balance equation for the water tank, we need to consider the energy entering and leaving the tank. The key components of the energy balance include:

• The energy input from the electric heater.
• The energy output due to the flow of water out of the tank.
• The heat loss from the tank to the environment.

Let’s denote:

• T_o: Temperature of the water in the tank (°C)
• T_in: Inlet water temperature (°C)
• T_out: Outlet water temperature (°C)
• Q_in: Heat input from the heater (W)
• Q_loss: Heat loss from the tank (W)

The energy balance equation can be expressed as:

m * C_p * (dT_o/dt) = Q_in - Q_loss - q * C_p * (T_o - T_in)

Where:

• Q_in = P = 400 W (power of the heater)
• Q_loss = U * A * (T_o - T) = 5 W/m²K * 0.3 m² * (T_o - 25°C)
• q = 0.1 kg/s (mass flow rate)

Impact of Increased Flow Rate

When considering the effect of increasing the flow rate (q), we can logically deduce the following:

With a higher flow rate, more water is being cycled through the heater and back to the tank in a given time period. This means that the tank will receive water at a lower temperature more quickly, which can lead to a more rapid adjustment of the tank temperature towards the steady state. Therefore, the time taken to reach the steady state will be:

Effect of Increased Heater Power

Now, let’s analyze what happens if we increase the power of the electric heater (P). A higher heater power means that more energy is being supplied to the water in the tank per unit time. This increased energy input will heat the water more quickly, allowing the tank temperature to rise faster towards the steady state.

In summary, both increasing the flow rate and increasing the heater power will reduce the time taken for the water temperature in the tank to reach a steady state. This is because both changes enhance the rate at which energy is transferred to the water, either by increasing the mass flow of water or by increasing the energy supplied to the water. Understanding these dynamics is crucial for optimizing heating systems in practical applications.