Question

A pipe is used for transporting boiling water in which the inner surface is at \(100^{\circ} \mathrm{C}\). The pipe is situated in surroundings where the ambient temperature is \(10^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The wall thickness of the pipe is \(3 \mathrm{~mm}\) and its inner diameter is \(30 \mathrm{~mm}\). The pipe wall has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.23 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.002 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). For safety reasons and to prevent thermal burn to workers, the outer surface temperature of the pipe should be kept below \(50^{\circ} \mathrm{C}\). Determine whether the outer surface temperature of the pipe is at a safe temperature so as to avoid thermal burn.

Short answer

Answer: Yes, the outer surface temperature is approximately 47.6°C, which is below the safety limit of 50°C, making it safe for workers.

Step-by-step solution

Calculate the average pipe wall conductivity (k)

We are given the variable thermal conductivity formula: \(k(T)=k_{0}(1+\beta T)\) with \(k_{0}=1.23\,\mathrm{W}/(\mathrm{m}\cdot\mathrm{K})\), \(\beta=0.002\,\mathrm{K}^{-1}\), and \(T\) in \(\mathrm{K}\). We first have to estimate the average temperature between the inner surface \((100^{\circ}\mathrm{C})\) and the outer surface \((50^{\circ}\mathrm{C})\), so \(T_\text{avg}=\frac{100 + 50}{2}=75^{\circ}\mathrm{C}\). In Kelvin, we have \(T_\text{avg} = 75 + 273.15 = 348.15\,\mathrm{K}\). Now, we can calculate the average pipe wall conductivity: \(k = k_{0}(1 + \beta T_\text{avg}) = 1.23(1 + 0.002\cdot 348.15) = 1.23(1+0.6963) \approx 2.08\,\mathrm{W}/(\mathrm{m}\cdot\mathrm{K})\)

Compute the heat transfer per unit length (Q')

Now we can find the conduction heat transfer per unit length \((Q')\) using the conduction formula \(Q' = \frac{-k\cdot A(T_\text{in}-T_\text{out})}{L}\) and knowing \(A = \pi d_i\) (circumference of the inner pipe). We have: \(d_i = 0.03\,\mathrm{m}\) (inner diameter) \(L = 0.003\,\mathrm{m}\) (pipe wall thickness) Given the ambient temperature \((T_\text{amb} = 10^{\circ}\mathrm{C})\) and the inner surface temperature \((T_\text{in} = 100^{\circ}\mathrm{C})\): \(Q' = \frac{-2.08\pi d_i(100 - 10)}{0.003} \approx -1792.3\,\mathrm{W}/\mathrm{m}\)

Calculate the outer surface temperature (T_out)

Now we can find the outer surface temperature \((T_\text{out})\) using the convection formula \(Q'=hA(T_\text{out}-T_\text{amb})\) and \(A = \pi d_o\) (circumference of the outer pipe): \(h = 70\,\mathrm{W}/(\mathrm{m}^2\cdot\mathrm{K})\) (convection heat transfer coefficient) \(d_o = d_i + 2L = 0.03 + 2\cdot 0.003 = 0.036\,\mathrm{m}\) (outer diameter) Solving for \(T_\text{out}\), we have: \(-1792.3\,\mathrm{W}/\mathrm{m} = 70(\pi d_o)(T_\text{out} - 10)\) \(T_\text{out} = \frac{-1792.3\,\mathrm{W}/\mathrm{m}}{70(\pi d_o)} + 10 \approx 47.6^{\circ}\mathrm{C}\) Since the outer surface temperature is \(47.6^{\circ}\mathrm{C}\), which is below the safety limit of \(50^{\circ}\mathrm{C}\), the pipe is safe and prevents thermal burn to workers.

Key concepts

Understanding Convection Heat Transfer

Convection heat transfer is a mechanism of heat transfer where thermal energy is carried away by the movement of a fluid, which could be a liquid or a gas. This is particularly important in the context of the exercise we're discussing, where hot water is transported through a pipe and the ambient air is at a much lower temperature.

In our scenario, the heat from the boiling water within the pipe transfers to the pipe's inner surface and from there to the outer surface, eventually dissipating into the surrounding air. This transition from hotter to cooler regions occurs due to the difference in temperature—driving a heat flow from the pipe to the ambient environment.

To quantify this process, we use the 'convection heat transfer coefficient,' denoted as 'h' in the exercise. This coefficient reflects how effective a fluid is at transporting heat. The higher the value of 'h', the more efficient the convection process, meaning more heat will be transferred for a given surface area and temperature difference.

Heat Transfer in Pipes

Heat transfer in pipes is a complex process affected by the material's thermal conductivity, the fluid inside, and the environmental conditions.

Thermal conductivity, 'k', is a property that indicates how well a material can conduct heat. In the given exercise, the thermal conductivity varies with temperature, adding a layer of complexity when calculating the amount of heat loss through the pipe's walls. An accurate prediction of heat transfer is crucial for ensuring that the pipe's temperature remains within safe operating limits.

The students must understand how to calculate the correct average thermal conductivity for the wall's material knowing that it changes with temperature. By doing so, we can determine whether the pipe's outer surface temperature is within a safe range. This involves mathematical computation to estimate the average temperature across the pipe's thickness and then using the specific conductivity formula provided.

Safety in Thermal Systems

Safety is paramount when designing and operating thermal systems. The exercise highlights a typical safety concern—preventing burns from surfaces that are too hot to touch.

Here, the safety threshold for the pipe's outer surface temperature has been set at no more than 50 degrees Celsius. There are many ways to manage and mitigate these safety concerns, including insulation, material selection with specific thermal properties, and sometimes regulating the operating temperatures of the fluid inside the systems.

When solving thermal problems, students must be aware of excluding unrealistic results or conditions that might compromise safety. Exercises like these help to reinforce the practical application of heat transfer principles in designing safe thermal system components, such as pipes in this case, that are both efficient and avoid hazards like thermal burns in the workplace.