Question

Consider a hot automotive engine, which can be approximated as a \(0.5-\mathrm{m}\)-high, \(0.40\)-m-wide, and \(0.8\)-m-long rectangular block. The bottom surface of the block is at a temperature of \(100^{\circ} \mathrm{C}\) and has an emissivity of \(0.95\). The ambient air is at $20^{\circ} \mathrm{C}\(, and the road surface is at \)25^{\circ} \mathrm{C}$. Determine the rate of heat transfer from the bottom surface of the engine block by convection and radiation as the car travels at a velocity of $80 \mathrm{~km} / \mathrm{h}$. Assume the flow to be turbulent over the entire surface because of the constant agitation of the engine block.

Short answer

Based on the given information (dimensions, temperature, and emissivity of a hot automotive engine block, car's velocity), and following the step by step solution provided, calculate the rate of heat transfer from the bottom surface of the engine block by convection and radiation.

Step-by-step solution

Calculate the area of the bottom surface of the engine block

To determine the rate of heat transfer, we first need to find the area of the bottom surface of the engine block. The given dimensions are 0.5 m (height), 0.40 m (width), and 0.8 m (length). The area of the bottom surface can be calculated as follows: Area \(A = width \times length \)

Convert the car's velocity from km/h to m/s

We are given the car's velocity in km/h, but we need it in m/s. To convert the velocity, use the following conversion factor: \(1 \mathrm{~km/h} = (1000 \mathrm{~m/km}) / (3600 \mathrm{~s/h})\)

Calculate the Reynolds number

The Reynolds number is an important parameter in fluid flow problems, particularly in the calculation of the convection heat transfer coefficient. It can be calculated using the following formula: \(Re = \frac{VD}{\nu}\) Where \(V\) is the velocity (m/s), \(D\) is the characteristic length (m), and \(\nu\) is the kinematic viscosity (m²/s). For this problem, we can assume the characteristic length as the length of the engine block. We also need the kinematic viscosity of air at \(20^{\circ}\mathrm{C}\), which is approximately \(1.5 \times 10^{-5} \mathrm{m²/s}\).

Calculate the convection heat transfer coefficient using the Dittus-Boelter equation

Since the flow is turbulent, we use the Dittus-Boelter equation to calculate the convection heat transfer coefficient: \(h = 0.0296 \times {Re}^{0.8} \times {Pr}^{0.4} \times k / D \) Where \(h\) is the convection heat transfer coefficient (W/m² K), Re is the Reynolds number, Pr is the Prandtl number (unitless), and \(k\) is the thermal conductivity (W/m K). The Prandtl number for air at \(20^{\circ}\mathrm{C}\) is approximately 0.7, and the thermal conductivity of air at \(20^{\circ}\mathrm{C}\) is approximately \(0.026 \mathrm{W/m \cdot K}\).

Calculate the heat transfer by convection

Using the convection heat transfer coefficient calcuǀlated in Step 4 and the temperature difference between the engine block and the ambient air, we can calculate the heat transfer by convection: \(q_{conv} = h A \Delta T\) Where \(q_{conv}\) is the heat transfer by convection (W), \(h\) is the convection heat transfer coefficient (W/m² K), \(A\) is the area of the bottom surface of the engine block (m²), and \(\Delta T\) is the temperature difference between the engine block and the ambient air (K).

Calculate the heat transfer by radiation using the Stefan-Boltzmann law

To calculate the heat transfer by radiation, we use the Stefan-Boltzmann law: \(q_{rad} = \epsilon A \sigma (T_{1}^4 - T_{2}^4)\) Where \(q_{rad}\) is the heat transfer by radiation (W), \(\epsilon\) is the emissivity of the bottom surface (0.95), \(A\) is the area of the bottom surface of the engine block (m²), \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W/m² \cdot K^4}\)), and \(T_{1}\) and \(T_{2}\) are the temperatures of the engine block (K) and the road surface (K). Note that temperatures should be converted to Kelvin.

Determine the total heat transfer

Finally, add the heat transfer by convection and radiation to determine the total heat transfer from the bottom surface of the engine block: \(q_{total} = q_{conv} + q_{rad}\)