# Step-by-Step Solution: Pipe Insulation Thickness Let's break down the problem: - **Thermal conductivity of insulation (\(k\))**: .032 W/(m·K) - **Outside air temperature (\(T_{\infty}\))**: -14°C - **Convective heat transfer coefficient (\(h\))**: 70 W/(m²·K) - **Pipe diameter (\(d_{\text{pipe}}\))**: 30 mm = .03 m - **Pipe temperature inside (\(T_{\text{pipe}}\))**: °C (must stay above this) - **Goal**: Find minimum insulation thickness so water inside doesn't freeze. --- ## 1. **Model the Heat Transfer** Heat flows from the pipe (°C) to the outside air (-14°C) through two resistances: 1. **Conduction through insulation (cylindrical shell)** 2. **Convection from insulation surface to air** ### Heat flow per unit length (\(q'\)): \[ q' = \frac{T_{\text{pipe}} - T_{\infty}}{R_{\text{total}}} \] Where \(R_{\text{total}}\) is the sum of: - **Conduction resistance:** \(R_{\text{ins}} = \frac{\ln(r_2/r_1)}{2\pi k L}\) - **Convection resistance:** \(R_{\text{conv}} = \frac{1}{2\pi r_2 h L}\) Here: - \(r_1\): pipe outer radius - \(r_2\): insulation outer radius - \(L\): length (will cancel, so set \(L=1\) m for convenience) --- ## 2. **Find the Critical Radius** For pipe insulation, **adding insulation can increase heat loss up to a certain "critical radius"**. Beyond this, more insulation reduces loss. The **critical radius** is: \[ r_{\text{crit}} = \frac{k}{h} \] Plug in the values: - \(k = .032\) W/(m·K) - \(h = 70\) W/(m²·K) \[ r_{\text{crit}} = \frac{.032}{70} = .000457 \text{ m} = .457 \text{ mm} \] The **pipe's outer radius** is: \[ r_1 = \frac{.03}{2} = .015 \text{ m} = 15 \text{ mm} \] Since \(r_1 > r_{\text{crit}}\), **adding insulation will always reduce heat loss** — there’s no risk of increased loss. --- ## 3. **Set Up the Equation** We want the **pipe temperature at or above °C**. If the insulation is thick enough, the inner surface won't drop below freezing. Let the **outer insulation radius** be \( r_2 = r_1 + x \), where \( x \) is the insulation thickness. **Heat loss per meter:** \[ q' = \frac{( - (-14))}{\frac{\ln(r_2/r_1)}{2\pi k} + \frac{1}{2\pi r_2 h}} \] \[ q' = \frac{14}{\frac{\ln(r_2/r_1)}{2\pi k} + \frac{1}{2\pi r_2 h}} \] But we also need to **keep the pipe temperature at °C**. This means we want to minimize heat loss, but unless we know the temperature inside the pipe is being maintained at °C, **the worst-case scenario is that it just stays at °C**. So, we solve for the thickness required to prevent the temperature from dropping below °C. --- ## 4. **Choose a Suitable Heat Loss Limit** If the pipe is maintained at °C, **any insulation thickness will keep it at °C** as long as \( r_2 > r_1 \), but in practice, we want to **minimize energy loss and protect against freezing for a reasonable amount of time if the water isn't moving**. **Therefore, in practice, it's common to specify that the outside of the pipe insulation is at °C** (the minimum for no freezing), so set the inner surface at °C. --- ## 5. **Solve for Insulation Thickness** Set \(T_{\text{pipe}} = ^\circ\)C. Set up the equation for heat loss per meter length: \[ q' = \frac{T_{\text{pipe}} - T_\infty}{R_{\text{ins}} + R_{\text{conv}}} \] \[ q' = \frac{ - (-14)}{\frac{\ln(r_2/r_1)}{2\pi k} + \frac{1}{2\pi r_2 h}} \] \[ q' = \frac{14}{\frac{\ln(r_2/r_1)}{2\pi (.032)} + \frac{1}{2\pi r_2 \cdot 70}} \] But the key is: **What minimum thickness is needed to ensure the pipe temperature does not fall below °C?** If the pipe is maintained at °C, **theoretically, ANY insulation is sufficient**. But if you want the temperature at the pipe surface (just under insulation) to stay above °C with a finite heat transfer from inside (e.g., standing water), you need to limit the heat loss. ### Usually, a common safety margin is used: **Assume no heat input from inside, and prevent the water from freezing for a certain period**. If we assume that the **pipe temperature should not fall below °C**, we just need to check that the outer surface does not drop below °C. But with insulation, the more you add, the less heat is lost. **There is no minimum required thickness based on the numbers given**, as the pipe radius is already far greater than the critical radius. --- ## 6. **Final Answer and Explanation** ### **A. Critical Radius Calculation** \[ r_{\text{crit}} = \frac{k}{h} = \frac{.032}{70} = .000457 \text{ m} = .457 \text{ mm} \] The pipe's outer radius is \(.015\) m \(>\) critical radius, so **adding insulation always helps**. ### **B. Required Insulation Thickness** - **Any thickness of insulation greater than zero will reduce heat loss and help keep the pipe above freezing, as long as the pipe is maintained at °C.** - **In practice**, insulation thickness is chosen based on other factors, such as how long the water must resist freezing, cost, and structural constraints. --- ## **Summary Table** | Parameter | Value | |--------------------|-----------------------| | Pipe diameter | 30 mm (.03 m) | | Pipe radius (\(r_1\)) | 15 mm (.015 m) | | Thermal conductivity (\(k\)) | .032 W/(m·K) | | Convective coefficient (\(h\)) | 70 W/(m²·K) | | Critical radius (\(r_{\text{crit}}\)) | .457 mm | | Air temperature (\(T_\infty\)) | -14°C | | Pipe temperature (\(T_{\text{pipe}}\)) | °C | --- ## **Conclusion** **Any insulation thickness greater than zero will reduce heat loss and help keep the pipe temperature above °C, because the pipe's outer radius is already much larger than the critical radius.** - **Critical radius:** .457 mm - **Pipe radius:** 15 mm (already much greater) **Final Answer:** > **Any insulation thickness greater than zero will help prevent freezing, but in practice, use as much as feasible for safety margin.** --- ### **If you want to calculate the actual heat loss for a given thickness, substitute \( r_2 = r_1 + x \) and solve as above.** --- **Let me know if you want to pick a specific insulation thickness for calculation, or if you have a follow-up!**