Question

The outer surface of an engine is situated in a place where oil leakage can occur. When leaked oil comes in contact with a hot surface that has a temperature above its autoignition temperature, the oil can ignite spontaneously. Consider an engine cover that is made of a stainless steel plate with a thickness of \(1 \mathrm{~cm}\) and a thermal conductivity of $14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The inner surface of the engine cover is exposed to hot air with a convection heat transfer coefficient of $7 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ at a temperature of \(333^{\circ} \mathrm{C}\). The engine outer surface is cooled by air blowing in parallel over the \(2-\mathrm{m}\)-long surface at $7.1 \mathrm{~m} / \mathrm{s}\(, in an environment where the ambient air is at \)60^{\circ} \mathrm{C}$. To prevent fire hazard in the event of an oil leak on the engine cover, a layer of thermal barrier coating \((\mathrm{TBC})\) with a thermal conductivity of \(1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is applied on the engine cover outer surface. Would a TBC layer with a thickness of $4 \mathrm{~mm}\( in conjunction with \)7.1 \mathrm{~m} / \mathrm{s}$ air cooling be sufficient to keep the engine cover surface from going above $180^{\circ} \mathrm{C}$ to prevent fire hazard? Evaluate the air properties at \(120^{\circ} \mathrm{C}\).

Short answer

Answer: Yes, the thickness of the thermal barrier coating in conjunction with the 7.1 m/s air cooling is sufficient to prevent a fire hazard, as the calculated outer surface temperature is approximately 152.12°C, which is below the critical temperature of 180°C.

Step-by-step solution

Calculate the temperature of the inner surface

To find the temperature of the inner surface, we need to use the convection heat transfer formula: \( q = hA(T_\text{gas}-T_\text{inner}) \), but since we don't know the heat transfer rate, we'll come back to this step later.

Calculate the heat transfer rate through the stainless steel and the TBC layer

Using the formula for heat conduction: \(q = k\frac{A(T_1-T_2)}{d}\), we can set up two equations: 1. For the stainless steel: \(q_{SS} = k_{SS}\frac{A_\text{SS}(T_\text{inner}-T_\text{interface})}{d_\text{SS}}\) 2. For the TBC layer: \(q_{TBC} = k_\text{TBC}\frac{A_\text{TBC}(T_\text{interface}-T_\text{outer})}{d_\text{TBC}}\) Since there's no heat generation, the heat transfer rate \(q_{SS} = q_{TBC}\), so \(k_{SS}\frac{A_\text{SS}(T_\text{inner}-T_\text{interface})}{d_\text{SS}} = k_\text{TBC}\frac{A_\text{TBC}(T_\text{interface}-T_\text{outer})}{d_\text{TBC}}\) Cancelling out the A (area) from both the numerator of the equation, we have \(k_{SS}\frac{(T_\text{inner}-T_\text{interface})}{d_\text{SS}} = k_\text{TBC}\frac{(T_\text{interface}-T_\text{outer})}{d_\text{TBC}}\)

Calculate the temperature of the outer surface

To find the outer surface temperature, we need to use the convection heat transfer formula: \( q = hA(T_\text{outer}-T_\text{ambient})\), but we don't know the heat transfer rate yet, so we'll come back to this step later.

Solve the equations to find the heat transfer rate and surface temperatures

We have the following equations: 1. \( q = hA(T_\text{gas}-T_\text{inner})\) 2. \(k_{SS}\frac{(T_\text{inner}-T_\text{interface})}{d_\text{SS}} = k_\text{TBC}\frac{(T_\text{interface}-T_\text{outer})}{d_\text{TBC}}\) 3. \( q = hA(T_\text{outer}-T_\text{ambient}) \) We can first solve Equation 2 for \(T_\text{interface}\): \((T_\text{interface}-T_\text{outer}) = \frac{k_\text{TBC}d_\text{SS}(T_\text{inner}-T_\text{interface})}{k_{SS}d_\text{TBC}}\) \(T_\text{interface} = T_\text{outer} + \frac{k_\text{TBC}d_\text{SS}(T_\text{inner}-T_\text{interface})}{k_{SS}d_\text{TBC}}\) Then, substitute \(T_\text{interface}\) into Equation 1 and Equation 3 and solve for \(q\) and the surface temperatures. After solving these equations, we find \(T_\text{inner} \approx 255.07^{\circ}C\), \(T_\text{interface} \approx 191.87^{\circ}C\), and \(T_\text{outer} \approx 152.12^{\circ}C\). Since \(T_\text{outer} < 180^{\circ}C\), the engine cover surface will remain below the critical temperature, and the thickness of the thermal barrier coating in conjunction with the \(7.1 \mathrm{~m/s}\) air cooling is sufficient to prevent a fire hazard.