Question

Air at \(15^{\circ} \mathrm{C}\) flows over a flat plate subjected to a uniform heat flux of \(240 \mathrm{~W} / \mathrm{m}^{2}\) with a velocity of $3.5 \mathrm{~m} / \mathrm{s}\(. The surface temperature of the plate \)6 \mathrm{~m}$ from the leading edge is (a) \(40.5^{\circ} \mathrm{C}\) (b) \(41.5^{\circ} \mathrm{C}\) (c) \(58.2^{\circ} \mathrm{C}\) (d) \(95.4^{\circ} \mathrm{C}\) (e) \(134^{\circ} \mathrm{C}\) (For air, use $k=0.02551 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7296, \nu=1.562 \times\( \)10^{-5} \mathrm{~m}^{2} / \mathrm{s}$ )

Short answer

Question: The surface temperature of the flat plate at \(6 \mathrm{~m}\) from the leading edge is closest to: a) 21°C b) 24°C c) 27°C d) 30°C e) 33°C

Step-by-step solution

Calculate Reynolds number at x = 6m

To find the Reynolds number at the specific point on the plate where we need to find the surface temperature, we use the formula, $$Re_x = \frac{u \cdot x}{ν}$$ where \(u = 3.5 \mathrm{~m} / \mathrm{s}\) (velocity of air), \(x = 6 \mathrm{~m}\) (distance along the plate), \(ν = 1.562 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) (kinematic viscosity of air).

Check if the flow is laminar or turbulent

A critical Reynolds number of approximately \(5 \times 10^5\) is used to differentiate between laminar and turbulent boundary layers. If \(Re_x < 5 \times 10^5\), the flow is laminar; otherwise, it's turbulent.

Calculate Nusselt number

If the flow is laminar, then the Nusselt number for a flat plate subjected to constant heat flux can be calculated using the formula: $$Nu_x=\sqrt{0.037 Re_x Pr^{1/3}-0.037·1000}$$ If turbulent, the Nusselt number can be calculated using an appropriate correlation.

Find the convection heat transfer coefficient (h)

To find the heat transfer coefficient (h), we use the formula, $$h = \frac{Nu_x * k}{x}$$ where \(Nu_x\) is the Nusselt number, \(k = 0.02551 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) is the thermal conductivity of air.

Calculate the temperature difference ΔT

The temperature difference between the plate and the air along the length of the plate can be found using the formula $$ΔT = \frac{q_s}{h}$$ where \(q_s = 240 \mathrm{~W} / \mathrm{m}^{2}\) is the heat flux over the flat plate.

Find the surface temperature of the plate (T_plate)

Finally, the surface temperature of the plate at \(6 \mathrm{~m}\) from the leading edge can be calculated using the formula $$T_{plate} = T_{air} + ΔT$$ where \(T_{air} = 15^{\circ} \mathrm{C}\) is the temperature of the air. Compare the calculated surface temperature with the options given (a, b, c, d, e) to find the correct answer.