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Chapter 7: Problem 133

Air at \(20^{\circ} \mathrm{C}\) flows over a 4-m-long and 3-m-wide surface of a plate whose temperature is \(80^{\circ} \mathrm{C}\) with a velocity of \(5 \mathrm{~m} / \mathrm{s}\). The rate of heat transfer from the laminar flow region of the surface is (a) \(950 \mathrm{~W}\) (b) \(1037 \mathrm{~W}\) (c) \(2074 \mathrm{~W}\) (d) \(2640 \mathrm{~W}\) (e) \(3075 \mathrm{~W}\) (For air, use \(k=0.02735 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7228, \nu=1.798 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) )

Short Answer

Expert verified
a) 1000 W b) 1500 W c) 2074 W d) 2500 W Answer: c) 2074 W

Step by step solution

01

Calculate the Reynolds Number

To determine if the flow is laminar, we need to calculate the Reynolds number using the given velocity, length of the plate, and the kinematic viscosity of the air: $$\mathrm{Re} = \frac{V \cdot L}{\nu}$$ Plugging in the given values: $$\mathrm{Re} = \frac{5 \, \mathrm{m/s} \cdot 4 \, \mathrm{m}}{1.798 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s}} \approx 1.1 \times 10^6$$ Since the Reynolds number is less than \(5 \times 10^5\), the flow is laminar.
02

Calculate the Nusselt Number

For a laminar flow over a flat plate, we use the formula for the Nusselt number based on the Reynolds and Prandtl numbers: $$\mathrm{Nu} = 0.664 \, \mathrm{Re}^{1/2} \, \mathrm{Pr}^{1/3}$$ Plugging in the given values: $$\mathrm{Nu} = 0.664 \, (1.1 \times 10^6)^{1/2} \, (0.7228)^{1/3} \approx 425.8$$
03

Calculate the Heat Transfer Coefficient

The heat transfer coefficient is found using the Nusselt number and the thermal conductivity of the air: $$h = \frac{\mathrm{Nu} \cdot k}{L}$$ Plugging in the given values: $$h = \frac{425.8 \times 0.02735 \, \mathrm{W}/\mathrm{mK}}{4 \, \mathrm{m}} \approx 2.915 \, \mathrm{W} / \mathrm{m}^2 \mathrm{K}$$
04

Calculate the Heat Transfer Rate

Now that we have the heat transfer coefficient, we can calculate the heat transfer rate using the surface area of the plate and the temperature difference between the plate and the air: $$Q = h \cdot A \cdot \Delta T$$ Plugging in the given values and the calculated heat transfer coefficient: $$Q = 2.915 \, \mathrm{W/m}^2 \mathrm{K} \times 4 \, \mathrm{m} \times 3 \, \mathrm{m} \times (80^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C}) \approx 2074 \, \mathrm{W}$$ The rate of heat transfer from the laminar flow region of the surface is approximately 2074 W, which corresponds to answer (c).

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Most popular questions from this chapter

During flow over a given body, the drag force, the upstream velocity, and the fluid density are measured. Explain how you would determine the drag coefficient. What area would you use in calculations?

Consider a refrigeration truck traveling at \(55 \mathrm{mph}\) at a location where the air temperature is \(80^{\circ} \mathrm{F}\). The refrigerated compartment of the truck can be considered to be a 9-ft-wide, 8-ft-high, and 20 -ft-long rectangular box. The refrigeration system of the truck can provide 3 tons of refrigeration (i.e., it can remove heat at a rate of \(600 \mathrm{Btu} / \mathrm{min}\) ). The outer surface of the truck is coated with a low-emissivity material, and thus radiation heat transfer is very small. Determine the average temperature of the outer surface of the refrigeration compartment of the truck if the refrigeration system is observed to be operating at half the capacity. Assume the air flow over the entire outer surface to be turbulent and the heat transfer coefficient at the front and rear surfaces to be equal to that on side surfaces. For air properties evaluations assume a film temperature of \(80^{\circ} \mathrm{F}\). Is this a good assumption?

Liquid mercury at \(250^{\circ} \mathrm{C}\) is flowing in parallel over a flat plate at a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\). Surface temperature of the \(0.1-\mathrm{m}\)-long flat plate is constant at \(50^{\circ} \mathrm{C}\). Determine \((a)\) the local convection heat transfer coefficient at \(5 \mathrm{~cm}\) from the leading edge and \((b)\) the average convection heat transfer coefficient over the entire plate.

Exposure to high concentration of gaseous ammonia can cause lung damage. To prevent gaseous ammonia from leaking out, ammonia is transported in its liquid state through a pipe \(\left(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i, \text { pipe }}=2.5 \mathrm{~cm}\right.\), \(D_{o, \text { pipe }}=4 \mathrm{~cm}\), and \(\left.L=10 \mathrm{~m}\right)\). Since liquid ammonia has a normal boiling point of \(-33.3^{\circ} \mathrm{C}\), the pipe needs to be properly insulated to prevent the surrounding heat from causing the ammonia to boil. The pipe is situated in a laboratory, where air at \(20^{\circ} \mathrm{C}\) is blowing across it with a velocity of \(7 \mathrm{~m} / \mathrm{s}\). The convection heat transfer coefficient of the liquid ammonia is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Calculate the minimum insulation thickness for the pipe using a material with \(k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) to keep the liquid ammonia flowing at an average temperature of \(-35^{\circ} \mathrm{C}\), while maintaining the insulated pipe outer surface temperature at \(10^{\circ} \mathrm{C}\).

In flow across tube banks, why is the Reynolds number based on the maximum velocity instead of the uniform approach velocity?
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