Question

Water is to be heated from \(10^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\) as it flows through a 2 -cm-internal-diameter, 13 -m-long tube. The tube is equipped with an electric resistance heater, which provides uniform heating throughout the surface of the tube. The outer surface of the heater is well insulated, so that in steady operation all the heat generated in the heater is transferred to the water in the tube. If the system is to provide hot water at a rate of \(5 \mathrm{~L} / \mathrm{min}\), determine the power rating of the resistance heater. Also, estimate the inner surface temperature of the pipe at the exit.

Short answer

Answer: To find the power rating of the electric resistance heater, follow these steps: 1. Calculate the mass flow rate of water. 2. Calculate the energy transfer to the water. 3. Determine the power rating of the resistance heater. To estimate the inner surface temperature of the pipe at the exit, divide the energy transfer rate by the heat transfer coefficient and the pipe's exit area. Add the resulting difference to the exit water temperature.

Step-by-step solution

Calculate mass flow rate of water

First, we need to calculate the mass flow rate of water. Knowing the flow volume rate (\(5 \mathrm{~L} / \mathrm{min}\)) and the density of water (ρ = approximately \(1000 \mathrm{~kg} / \mathrm{m^3}\)), we can convert the volume flow rate to mass flow rate (\(\dot{m}\)) using the following equation: \(\dot{m} = \rho \times \mathrm{Volume \, Flow \, Rate}\). Convert the volume flow rate from L/min to m³/s.

Calculate energy transfer to water

We need to determine the energy transfer between the heater and the water (\(\dot{Q}\)). To do this, we use the energy equation: \(\dot{Q} = \dot{m} \times c \times (T_{out} - T_{in})\), where \(c\) is the specific heat capacity of water (\(c = 4.18 \mathrm{~kJ} / (\mathrm{kg} \cdot \mathrm{K})\), \(T_{out} = 80^\circ \mathrm{C}\), and \(T_{in} = 10^\circ \mathrm{C}\).

Calculate the power rating of the resistance heater

Now, we need to find the power rating of the resistance heater (\(P\)). The electric power and energy transfer have the same value because of the perfect insulation and steady operation. So, we have \(P = \dot{Q}\).

Estimate the inner surface temperature of the pipe at the exit

To estimate the inner surface temperature of the pipe at the exit, we note that the water heat transfer rate is constant along the pipe. Divide the energy transfer rate by the heat transfer coefficient (h) and the pipe's exit area (A). This will give us the difference between the inner surface temperature (\(T_s\)) and the water temperature (\(T_{out}\)). We can then add the difference to the exit water temperature (\(80^\circ \mathrm{C}\)) to get the inner surface temperature. Note that you may need to assume a value for the heat transfer coefficient based on given conditions or typical values for similar systems.

Key concepts

Energy Transfer

In thermal physics, energy transfer refers to the process of energy moving from one body to another. In the context of heating water, this energy transfer involves heat going from the heater to the water.
The heater generates thermal energy, which is absorbed by the water as it travels through the tube. The amount of energy transferred depends on several factors, including the temperature difference between the heater and the water, and the specific heat capacity of the water.
The specific heat capacity is a measure of how much energy it takes to raise the temperature of a unit mass of a substance by one degree Celsius. Utilizing the equation for energy transfer \[ \dot{Q} = \dot{m} \times c \times (T_{out} - T_{in}), \] means we can calculate the total energy required to heat the water from its initial to its final temperature as it passes through the tube. Here, \(\dot{Q}\) is the rate of heat transfer, \(\dot{m}\) is the mass flow rate, \(c\) is the specific heat capacity, and \(T_{out} - T_{in}\) is the change in temperature.
By knowing this parameter, we can understand how much energy the heater needs to supply to achieve the desired water temperature.

Mass Flow Rate

The mass flow rate is a crucial part of understanding how much water is being heated over time. It's defined as the mass of a substance passing through a surface per unit of time.
To compute the mass flow rate of water, you use the equation \[ \dot{m} = \rho \times \text{Volume Flow Rate}. \] Here, \(\rho\) is the density of water, typically around \(1000 \mathrm{~kg/m^3}\), and the volume flow rate is given.
The volume flow rate in this problem is initially provided in liters per minute (L/min). To align with our calculations, we convert this to cubic meters per second \(\text{m}^3/ ext{s}\), which provides a more standard unit for scientific calculations. Each liter equals \(0.001 \text{m}^3\) and each minute equals 60 seconds.
Calculating this gives us the mass flow rate, which tells us how much mass of water is being heated every second. This step is fundamental because it affects the total energy required for heating the water.

Specific Heat Capacity

Specific heat capacity is a vital concept in thermodynamics and heat transfer. It is defined as the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius.
Different substances have different specific heat capacities. Water, due to its hydrogen bonding, has a relatively high specific heat capacity of approximately \(4.18 \mathrm{~kJ/(kg \, K)}\). This property makes water an excellent medium for thermal energy storage and heat transfer.
In the exercise, the specific heat capacity is used in the energy transfer formula to find how much energy the heater needs to supply to raise the water's temperature from \(10^{\circ}\mathrm{C}\) to \(80^{\circ}\mathrm{C}\).
This means the formula: \[ \dot{Q} = \dot{m} \times c \times (T_{out} - T_{in}) \] illustrates how much heat is required depending on the amount of water, the change in temperature, and the specific heat capacity of water.

Resistance Heating

Resistance heating is a method of heating by converting electrical energy into thermal energy. In this exercise, the tube is equipped with an electric resistance heater, which provides a consistent amount of heat throughout the tube length.
Resistance heaters work based on the principle of Joule's Law, where the power (energy per unit time) is proportional to the square of the current times the electrical resistance: \[ P = I^2 R. \] Here, \(P\) stands for power, \(I\) is the electric current, and \(R\) is the resistance.
This constant heating ensures the water is uniformly heated throughout its journey in the pipe. The power rating of a resistance heater, in this context, must match the total energy transfer needed to reach the desired water temperature. This balance ensures that the system operates effectively without unnecessary energy wastage.
This efficient energy transfer, thanks to proper insulation and controlled resistance, is key to maintaining a steady operation with effective heating reachability.