Question

In a food processing plant, hot liquid water is being transported in a pipe \(\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i}=2.5 \mathrm{~cm}\right.\), \(D_{o}=3 \mathrm{~cm}\), and \(\left.L=10 \mathrm{~m}\right)\). The hot water flowing with a mass flow rate of \(0.15 \mathrm{~kg} / \mathrm{s}\) enters the pipe at \(100^{\circ} \mathrm{C}\) and exits at \(60^{\circ} \mathrm{C}\). The plant supervisor thinks that since the hot water exits the pipe at \(60^{\circ} \mathrm{C}\), the pipe's outer surface temperature should be safe from thermal burn hazards. In order to prevent thermal burn upon accidental contact with skin tissue for individuals working in the vicinity of the pipe, the pipe's outer surface temperature should be kept below \(45^{\circ} \mathrm{C}\). Determine whether or not there is a risk of thermal burn on the pipe's outer surface. Assume the pipe outer surface temperature remains constant.

Short answer

Answer (short): To determine if there's a risk of thermal burn, calculate the heat loss rate, the pipe's thermal resistance, the temperature difference between inside and outside of the pipe, and finally the pipe's outer surface temperature. If the outer surface temperature is below \(45^{\circ}C\), there is no risk of thermal burn.

Step-by-step solution

Calculate the heat loss rate through the pipe.

The heat loss rate (\(Q\)) through the pipe can be calculated using the following formula: $$ Q = m \cdot c_p \cdot (T_{in} - T_{out})$$ where: \(m\) is the mass flow rate of water in kg/s, \(c_p\) is the specific heat of water (approximately \(4.18\ \mathrm{kJ/kg\cdot K}\)), \(T_{in}\) is the water temperature at the pipe entrance in \(^{\circ}C\), \(T_{out}\) is the water temperature at the pipe exit in \(^{\circ}C\). Using the provided data, the heat loss rate through the pipe can be calculated as: $$ Q = 0.15\ \mathrm{kg/s} \times 4.18\ \mathrm{kJ/kg\cdot K} \times (100^{\circ}C - 60^{\circ}C)$$

Calculate the thermal resistance of the pipe.

The thermal resistance (\(R_{thermal}\)) of the pipe can be calculated using the following formula: $$ R_{thermal} = \dfrac{\ln \dfrac{D_o}{D_i}}{2 \pi k L}$$ where: \(D_o\) is the outer diameter of the pipe (in meters), \(D_i\) is the inner diameter of the pipe (in meters), \(k\) is the thermal conductivity of the pipe (in \(\mathrm{W/m\cdot K}\)), \(L\) is the length of the pipe (in meters). Converting the dimensions to meters and substituting the given values, we find the thermal resistance: $$ R_{thermal} = \dfrac{\ln \dfrac{0.03\ \mathrm{m}}{0.025\ \mathrm{m}}}{2 \pi \times 15\ \mathrm{W/m\cdot K} \times 10\ \mathrm{m}}$$

Calculate the temperature difference between inside and outside of the pipe.

The temperature difference between the water inside the pipe and the pipe's outer surface can be found using the heat loss rate and the thermal resistance: $$ \Delta T = Q \times R_{thermal}$$

Calculate the pipe's outer surface temperature.

Knowing the temperature difference, we can determine the pipe's outer surface temperature (\(T_{surface}\)) by subtracting the calculated difference from the water's exit temperature: $$ T_{surface} = T_{out} - \Delta T$$ If the calculated \(T_{surface}\) is below \(45^{\circ}C\), there would be no risk of thermal burn on the pipe's outer surface. By following these steps, you can determine if there's a risk of thermal burn on the pipe's outer surface in this food processing plant.

Key concepts

Heat Loss Rate Calculation

Understanding the heat loss rate is crucial when assessing the efficiency and safety of thermal systems, such as the transport of hot liquid in pipes. In a practical context, like the one provided in our exercise, the heat loss rate quantifies the amount of heat being transferred from the hot water inside the pipe to its surroundings over a given period.

The mathematical representation for calculating the heat loss rate, Q, is given by the formula \( Q = m \cdot c_p \cdot (T_{in} - T_{out}) \), wherein m is the mass flow rate of the fluid, c_p is the specific heat capacity of the fluid, and \(T_{in} - T_{out}\) represents the temperature drop of the fluid as it travels through the pipe. When the specifics of the fluid flow and its properties are known, calculating the heat loss rate can pinpoint how much thermal energy is being lost, which is a precursor to understanding whether exterior surfaces might pose a burn hazard.

In the case of our example, by substituting the given values into the formula, you can determine whether the pipe's surface temperature could potentially cause burns.

Thermal Resistance

The concept of thermal resistance, often symbolized as \( R_{thermal} \), is analogous to electrical resistance but applied to the context of heat transfer. It is a measure of a material's ability to resist the flow of heat. The higher the thermal resistance, the less heat is allowed to pass through a material over time.

The resistance of a cylindrical object like a pipe can be calculated by the formula \(R_{thermal} = \frac{\ln \frac{D_o}{D_i}}{2 \pi k L}\), incorporating the pipe's inner \(D_i\) and outer \(D_o\) diameters, its thermal conductivity \(k\), and length \(L\). The exercise involves applying this formula to compute \(R_{thermal}\), which we use to assess whether the heat loss through the pipe's material is sufficiently low to prevent the surface from reaching temperatures that might cause thermal burns upon contact.

Heat Transfer in Pipes

Analyzing heat transfer in pipes is essential for engineers and safety experts in order to maintain both process efficiency and safety. The heat transfer process in pipes is governed by the principles of thermodynamics and heat transfer mechanisms which include conduction, convection, and sometimes radiation.

For our scenario, the primary mode of heat transfer from the hot water to the pipe wall is convection, and from the pipe wall to the surroundings is conduction. The insulating effect of the pipe material is represented by its thermal resistance, which hinders the flow of heat. The lower the thermal resistance, the greater the heat transfer to the environment, potentially leading to a hotter pipe surface.

The calculation of heat transfer in pipes involves knowing the properties of both the fluid and the pipe material. In the industrial setting given, this knowledge assists in guaranteeing that the pipe surface temperature remains within safe limits to prevent accidental burns.

Temperature Difference Calculation

The temperature difference calculation is a straightforward but vital part of our analysis. It's the temperature gradient that drives heat transfer – heat flows naturally from a warmer area to a cooler one. In the given problem, calculating the temperature difference between the inside and the outside of the pipe aids in determining the outer surface temperature of the pipe.

This temperature difference, denoted as \( \Delta T \), is found by multiplying the heat loss rate by the pipe's thermal resistance. Expressed mathematically, \(\Delta T = Q \times R_{thermal}\). Once \( \Delta T \), is established, it can be subtracted from the water's exit temperature to yield the surface temperature of the pipe.

It is this final calculation that wraps up our assessment, allowing us to conclude whether the pipe's surface temperature falls within a range considered safe for human contact, thus preventing thermal burns.