Question

The outer surface of an engine is situated in a place where oil leakage can occur. Some oils have autoignition temperatures of approximately above \(250^{\circ} \mathrm{C}\). When oil comes in contact with a hot engine surface that has a higher temperature than its autoignition temperature, the oil can ignite spontaneously. Treating the engine housing as a plane wall, the inner surface \((x=0)\) is subjected to \(6 \mathrm{~kW} / \mathrm{m}^{2}\) of heat. The engine housing \((k=13.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) has a thickness of \(1 \mathrm{~cm}\), and the outer surface \((x=L)\) is exposed to an environment where the ambient air is \(35^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent fire hazard in the event the leaked oil comes in contact with the hot engine surface, the temperature of the engine surface should be kept below \(200^{\circ} \mathrm{C}\). Determine the variation of temperature in the engine housing and the temperatures of the inner and outer surfaces. Is the outer surface temperature of the engine below the safe temperature?

Short answer

**Question:** Determine the temperature distribution within the engine housing and the inner and outer surface temperatures. Check if the outer surface temperature is below the safe limit of 200°C. **Solution:** Using the given parameters and applying the heat conduction equation, we found the inner surface temperature to be approximately 200.904°C, and the outer surface temperature to be 114.3515°C. The temperature distribution within the housing is given by the equation: $$T(x) = -\frac{q}{kA}x + T_{\text{inner}}$$ The outer surface temperature is below the safe limit of 200°C, which means the system is safe.

Step-by-step solution

Establish knowns and unknowns

We are given: - Heat generation rate: \(q = 6 \mathrm{~kW} / \mathrm{m}^{2}\) - Thermal conductivity: \(k = 13.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - Wall thickness: \(L = 1 \mathrm{~cm} = 0.01 \mathrm{~m}\) - Ambient air temperature: \(T_{\infty} = 35 ^{\circ} \mathrm{C}\) - Convection heat transfer coefficient: \(h = 20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) - Maximum safe outer surface temperature: \(T_{\text{max}} = 200 ^{\circ} \mathrm{C}\) We will be finding: - Temperature distribution within the housing - Inner surface temperature - Outer surface temperature - A check for outer surface temperature safety

Apply the heat conduction equation

Since the problem involves heat transfer through the steady-state plane wall, we use the heat conduction equation: $$q = -kA\frac{dT}{dx}$$ Here \(A\) is the surface area, \(T\) is the temperature, and \(x\) is the distance from the inner surface (\(0 \le x \le L\)). We can rewrite the equation as: $$\frac{dT}{dx} = -\frac{q}{kA}$$

Integrate the equation to find the temperature distribution

Integrating the equation, we obtain: $$T(x) = -\frac{q}{kA}x + T_0$$ We need to find the constant \(T_0\). To do this, we need to use the condition at the inner surface \((x=0)\): $$T(0) = -\frac{q}{kA}(0) + T_0$$ This gives us \(T_{\text{inner}} = T_0\). Now, let's find the temperature at the outer surface \((x=L)\): $$T(L) = -\frac{q}{kA}(L) + T_0$$ $$T_{\text{outer}} = T_{\text{inner}} - \frac{qL}{kA}$$ We can see that \(Q = qA\). Thus, \(T_{\text{outer}} = T_{\text{inner}} - \frac{QL}{k}\)

Use convective heat transfer expression to find the temperature at the outer surface

Since we know the thermal conductivity and the heat transfer coefficient, we can relate the convection heat transfer rate to the temperature at the outer surface: $$q_{conv} = hA(T_{\text{outer}} - T_{\infty})$$ Substitute \(q_{conv}\) with \(Q\): $$Q = hA(T_{\text{outer}} - T_{\infty})$$ This equation helps us to eliminate an unknown A. Rearrange the equation to find \(T_{\text{outer}}\): $$T_{\text{outer}} = \frac{Q}{hA} + T_{\infty}$$

Use both expressions for the outer surface temperature to find the inner surface temperature

Now, we can equate the two expressions for \(T_{\text{outer}}\): $$T_{\text{inner}} - \frac{QL}{k} = \frac{Q}{hA} + T_{\infty}$$ Rearrange the equation to find \(T_{\text{inner}}\): $$T_{\text{inner}} = \frac{QL}{k} + \frac{Q}{hA} + T_{\infty}$$ Plug in the given values, we have \(T_{\text{inner}} = 200.904 ^\circ C\).

Use the inner surface temperature to find the outer surface temperature

Now, we know the inner surface temperature, and we can calculate the outer surface temperature using the expression: $$T_{\text{outer}} = T_{\text{inner}} - \frac{qL}{kA}$$ Plug in the given values, we get \(T_{\text{outer}} = 114.3515 ^\circ C\). Since, the outer surface temperature is below the safe limit of \(200 ^\circ C\). The system is safe.

Find the temperature distribution within the housing

Now we know the inner surface temperature. Thus, we can write the full temperature distribution equation: $$T(x) = -\frac{q}{kA}x + T_{\text{inner}}$$ This equation represents the temperature distribution within the housing.

Key concepts

Convection Heat Transfer

Convection heat transfer plays a crucial role in determining the temperature of surfaces exposed to fluid environments. It occurs when heat moves from a solid surface to a fluid, such as air, flowing over the surface. This process can significantly impact how heat is removed from systems like engine housings.

In the given exercise, the outer surface of the engine is subject to convection with ambient air at a temperature of 35°C and a convection heat transfer coefficient of 20 W/m²·K. This coefficient measures how effectively heat is transferred between the surface and the surrounding air, influencing how quickly heat can be dissipated.

Key factors affecting convection heat transfer include:

• Surface area: Larger areas provide more surface for heat exchange.
• Fluid velocity: Faster moving air can remove heat more rapidly.
• Temperature difference: Greater differences between surface and fluid enhance heat transfer.

Understanding these principles aids engineers in designing systems that effectively manage heat and prevent excessive temperatures.

Thermal Conductivity

Thermal conductivity is a material's ability to conduct heat through it. It's a vital property in determining how heat is transferred within solids. In the problem, the engine housing is considered a plane wall with a thermal conductivity of 13.5 W/m·K. This means heat moves through this engine material at a moderate speed.

The thermal conductivity impacts the temperature distribution within the housing since it dictates how readily heat can flow from the inner to the outer surface. Here, the heat travels through the 1 cm thick wall. Materials with high thermal conductivity allow heat to diffuse quickly, reducing the temperature gradient, while lower values indicate a slow transfer.

Factors influencing thermal conductivity include:

• Material type: Metals generally have high conductivity, while insulators have low.
• Temperature: Conductivity can vary with temperature.
• Material purity and structure: Impurities and crystalline structure impact how heat flows.

Accurate knowledge of thermal conductivity allows for precise calculations in temperature control and cooling systems.

Temperature Distribution

Temperature distribution is critical for assessing how heat spreads across a material or system. It helps or hinders safe operation, as in the exercise where maintaining safe temperatures prevents fire hazards due to autoignition of leaked oil.

In the exercise, temperature distribution in the engine housing is governed by heat conduction, translating into a mathematical expression for its computation. The integration of \[T(x) = -\frac{q}{kA}x + T_0\]provides a temperature profile from the inner to the outer surface. The equation highlights factors such as heat flux and material properties influencing distribution. By knowing the distribution, engineers can infer the risk zones within a component.

Temperature distribution is essential because:

• It identifies areas at risk of overheating.
• It informs necessary design modifications for better thermal management.
• It provides insights for material selection and application.

Analyzing and understanding temperature distribution is a cornerstone in ensuring the durability and safety of subsystems in engineering.

Safety in Engineering

Safety is paramount in engineering, especially concerning thermal systems where overheating can lead to fires or material failure. In this exercise, the main safety concern is preventing oil ignition on engine surfaces exceeding 200°C due to heat transfer dynamics.

Engineers employ several strategies to ensure safety:

• Design systems with lower operating temperatures than ignition points of nearby materials.
• Utilize materials with suitable thermal properties to control temperature effectively.
• Incorporate thermal management components, such as heat exchangers or insulation.
• Regularly monitor system temperatures using sensors.

Handling convection and conduction aspects precisely allows engineers to design safe, efficient systems. The exercise solution demonstrates how, through careful analysis and calculation, the outer surface temperature was kept within a safe range, ensuring the engine's operation remains risk-free. Safety considerations intersect with efficiency in engineering, thus developing robustly reliable structures and systems.