Question

(a) Water flows through a shower head steadily at a rate of \(10 \mathrm{L} / \mathrm{min}\). An electric resistance heater placed in the water pipe heats the water from 16 to \(43^{\circ} \mathrm{C}\). Taking the density of water to be \(1 \mathrm{kg} / \mathrm{L},\) determine the electric power input to the heater, in \(\mathrm{kW}\), and the rate of entropy generation during this process, in \(\mathrm{kW} / \mathrm{K}\). (b) In an effort to conserve energy, it is proposed to pass the drained warm water at a temperature of \(39^{\circ} \mathrm{C}\) through a heat exchanger to preheat the incoming cold water. If the heat exchanger has an effectiveness of 0.50 (that is, it recovers only half of the energy that can possibly be transferred from the drained water to incoming cold water), determine the electric power input required in this case and the reduction in the rate of entropy generation in the resistance heating section.

Short answer

How do these values change when a heat exchanger is introduced? Answer: The electric power input to the heater is 18.81 kW, and the rate of entropy generation is 0.0195 kW/K. When a heat exchanger is introduced, the electric power input reduces to 9.4 kW, and the rate of entropy generation decreases by 0.00975 kW/K.

Step-by-step solution

Calculate the mass flow rate of the water

Since water flows at a rate of 10 L/min and the density of water is given as 1 kg/L, we can determine the mass flow rate as follows: Mass flow rate = Flow rate × Density= 10 L/min × 1 kg/L = 10 kg/min

Calculate the energy required to heat the water

We will use the specific heat capacity formula to find the energy required to heat the water: Q = m × c × ΔT where Q is the energy required, m is the mass of water, c is the specific heat capacity of water (4.18 kJ/kg·K), and ΔT is the temperature difference. We can substitute the values: ΔT = 43°C - 16°C = 27°C Q = 10 kg/min × 4.18 kJ/kg·K × 27°C = 1128.6 kJ/min

Calculate the electric power input

To calculate the electric power input, we will use the formula Power (P) = Energy (Q) / Time (t) P = 1128.6 kJ/min ÷ 60 s/min = 18.81 kW

Calculate the rate of entropy generation

The rate of entropy generation can be calculated as follows: Rate of entropy generation = m × c × ln(T2 / T1)/T Entropy rate = 10 kg/min × 4.18 kJ/kg·K × ln((43+273)/(16+273)) / (43-16) Entropy rate = (1/60)×10 kg/min × 4.18 kJ/kg·K × ln((316)/(289)) / 27 = 0.0195 kW/K

Using the heat exchanger

When introducing a heat exchanger with an effectiveness of 0.50, the reduction in energy provided to the heater will be: Q_reduced = 0.5 × Q Q_reduced = 0.5 × 1128.6 kJ/min = 564.3 kJ/min

Calculate the new electric power input

The new electric power input will be: P_new = 564.3 kJ/min ÷ 60 s/min = 9.4 kW

Calculate the reduction in the rate of entropy generation

The reduction in entropy rate will be half the original rate: Entropy rate reduction = 0.5 × Entropy rate Entropy rate reduction = 0.5 × 0.0195 kW/K = 0.00975 kW/K The electric power input to the heater is 18.81 kW, and the rate of entropy generation is 0.0195 kW/K. With the heat exchanger installed, the electric power input reduces to 9.4 kW, and the rate of entropy generation decreases by 0.00975 kW/K.

Key concepts

Mass Flow Rate

Understanding the mass flow rate is crucial when examining processes involving fluid flow, such as heating water in a shower system. It quantifies the amount of mass passing through a given surface per unit time. Commonly expressed in kilograms per second (kg/s), mass flow rate is vital for calculating the energy required to heat a fluid and, subsequently, the heater's power requirement.

According to the exercise, to determine the mass flow rate of water, we multiply the flow rate (in liters per minute) by the density of water (in kilograms per liter). Since the density is 1 kg/L, the mass flow rate directly corresponds to the flow rate in liters, resulting in a mass flow rate of 10 kg/min. This value serves as the foundation for subsequent calculations involving the heating process of water.

Specific Heat Capacity

The concept of specific heat capacity is fundamental in thermodynamics. It represents the amount of energy required to raise the temperature of one kilogram of a substance by one degree Celsius (or Kelvin). The specific heat capacity of water, typically 4.18 kJ/kg·K, allows us to calculate the amount of energy required to heat a specific quantity of water.

To calculate the energy (Q) needed to heat the water from one temperature to another, multiply the mass (m), specific heat capacity (c), and temperature change (ΔT). In the exercise, the specific heat capacity plays a pivotal role in determining both the electric power input to the heater and the rate of entropy generation, as it relates energy to the temperature change in water.

Entropy Generation

The term entropy generation is intertwined with the second law of thermodynamics and signifies the degree of energy dispersion or irreversibility in a process. It's a measure of how much useful energy is lost in the form of unrecoverable heat. In heating applications, higher entropy generation typically indicates less efficiency since more energy is unusable for work.

In the exercise, the rate of entropy generation during the water heating process demonstrates the system's inefficiency. It's calculated by multiplying the mass flow rate, specific heat capacity, and the natural logarithm of the temperature ratio, all divided by the average temperature over which the heating occurs. This metric is significant in identifying opportunities for energy conservation, such as using a heat exchanger to preheat incoming water.

Heat Exchanger Effectiveness

The heat exchanger effectiveness is a measure of how well the heat exchanger performs relative to its theoretical maximum performance. An effectiveness of 1 implies that the heat exchanger transfers all the available thermal energy from the hot side to the cold one, which is often not achievable due to practical limitations like imperfect thermal contact or finite element sizes.

In our exercise, a heat exchanger with an effectiveness of 0.50 means it can only recover half the energy that could be transferred. When integrating such a heat exchanger into the system, we see a reduction in the required electric power input for heating the water. Effectiveness is a key design parameter that affects not only energy efficiency but also the size and cost of a heat exchanger. The usage of this concept helps students understand the impact of design choices and system efficiency on energy usage and conservation.