Question

An engine block with a surface area measured to be \(0.95 \mathrm{~m}^{2}\) generates a power output of \(50 \mathrm{~kW}\) with a net engine efficiency of 35 percent. The engine block operates inside a compartment at $157^{\circ} \mathrm{C}\(, and the average convection heat transfer coefficient is \)50 \mathrm{~W} / \mathrm{m}^{2}\(. \)\mathrm{K}$. If convection is the only heat transfer mechanism occurring, determine the engine block surface temperature. Answer: \(841^{\circ} \mathrm{C}\)

Short answer

Based on the given parameters - power output, surface area, efficiency, and the convection heat transfer coefficient - the surface temperature of the engine block, assuming only convection heat transfer is occurring, is calculated to be \(841^{\circ} \mathrm{C}\).

Step-by-step solution

Analyze the given data

To find the surface temperature of the engine block, we are given the following quantities: - Surface area (A) = \(0.95 \mathrm{~m}^{2}\) - Power output (P) = \(50 \mathrm{~kW}\) = \(50 \times 10^{3} \mathrm{~W}\) - Net engine efficiency (η) = 35% = 0.35 - Compartment temperature (T1) = \(157^{\circ}\mathrm{C}\) - Convection heat transfer coefficient (h) = \(50 \mathrm{~W} / \mathrm{m}^{2}\cdot \mathrm{K}\)

Calculate the total heat generated

To find the total heat generated (Q), we must consider the net engine efficiency. The formula for heat generated is: $$ Q = \frac{P}{\eta} $$ Plugging in the given values, we get: $$ Q = \frac{50 \times 10^{3} \mathrm{~W}}{0.35} $$

Determine the convection heat transfer

Since convection is the only heat transfer mechanism, we can use the formula for convection heat transfer (Q) to find the surface temperature (T2) of the engine block: $$ Q = h \cdot A \cdot (T2 - T1) $$ Where T1 is the compartment temperature. Rearranging the formula for T2: $$ T2 = T1 + \frac{Q}{h \cdot A} $$

Calculate the surface temperature

Now let's plug in the values and find T2: $$T2 = 157 + \frac{50 \times 10^{3} \mathrm{~W} / 0.35}{50 \mathrm{~W} / \mathrm{m}^{2}\cdot \mathrm{K} \cdot 0.95 \mathrm{~m}^{2}}$$ Calculating the values, we get: $$ T2 = 841^{\circ} \mathrm{C} $$ So, the surface temperature of the engine block is \(841^{\circ} \mathrm{C}\).