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Chapter 7: Problem 23

In an experiment, the local heat transfer over a flat plate were correlated in the form of local Nusselt number as expressed by the following correlation $$ \mathrm{Nu}_{x}=0.035 \mathrm{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3} $$ Determine the ratio of the average convection heat transfer coefficient \((h)\) over the entire plate length to the local convection heat transfer coefficient \(\left(h_{x}\right)\) at \(x=L\).

Short Answer

Expert verified
Answer: The ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient at x=L is given by: $$ \frac{h_{avg}}{h_x} = \frac{Nu_{avg}}{Nu_L} $$ Find the values of \(Nu_{avg}\) and \(Nu_L\) by integrating the given Nusselt number correlation and using the resulting expressions for local and average convection heat transfer coefficients.

Step by step solution

01

Understanding Nusselt number

The Nusselt number (Nu) is a dimensionless number that represents the ratio of convective heat transfer to conductive heat transfer. The higher the Nusselt number, the better the heat transfer performance. In this exercise, we have the local Nusselt number \(Nu_x\) given as a function of the local Reynolds number, \(Re_x\), and the Prandtl number, Pr: $$ Nu_x = 0.035Re_x^{0.8}Pr^{1/3} $$
02

Express local convection heat transfer coefficient in terms of Nusselt number

To find the local convection heat transfer coefficient (\(h_x\)) in terms of the Nusselt number, we can use the following formula: $$ h_x = \frac{Nu_x \cdot k}{x} $$ Where \(k\) is the thermal conductivity and \(x\) is the distance from the leading edge of the flat plate.
03

Calculate the average Nusselt number using the given correlation

To find the average Nusselt number (\(Nu_{avg}\)) for the entire flat plate length, we need to integrate the given Nusselt number correlation over the plate length and divide by the plate length: $$ Nu_{avg} = \frac{1}{L} \int_0^L Nu_x \, dx $$ Substitute the given correlation for \(Nu_x\) into the formula above: $$ Nu_{avg} = \frac{1}{L} \int_0^L \big( 0.035Re_x^{0.8}Pr^{1/3} \big) \, dx $$
04

Express average convection heat transfer coefficient in terms of Nusselt number

Similar to the local convection heat transfer coefficient, we can express the average convection heat transfer coefficient (\(h_{avg}\)) in terms of the average Nusselt number using the following formula: $$ h_{avg} = \frac{Nu_{avg} \cdot k}{L} $$
05

Find the ratio of \(h_{avg}\) to \(h_x\)

We now have the expressions for both the local and average convection heat transfer coefficients, so we can calculate the desired ratio: $$ \frac{h_{avg}}{h_x} = \frac{\frac{Nu_{avg} \cdot k}{L}}{\frac{Nu_x \cdot k}{x}} $$ Simplify the equation by canceling out \(k\) and then substitute \(x=L\): $$ \frac{h_{avg}}{h_x} = \frac{Nu_{avg}}{Nu_L} $$ Determine the ratio after integrating in step 3 and substituting the results into this equation.

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Most popular questions from this chapter

Air (1 atm, \(\left.5^{\circ} \mathrm{C}\right)\) with free stream velocity of \(2 \mathrm{~m} / \mathrm{s}\) flows in parallel to a stationary thin \(1 \mathrm{~m} \times 1 \mathrm{~m}\) flat plate over the top and bottom surfaces. The flat plate has a uniform surface temperature of \(35^{\circ} \mathrm{C}\). Determine \((a)\) the average friction coefficient, \((b)\) the average convection heat transfer coefficient, and \((c)\) the average convection heat transfer coefficient using the modified Reynolds analogy and compare with the result obtained in \((b)\).

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 \(10^{\circ} \mathrm{C}\).

What is the effect of surface roughness on the friction drag coefficient in laminar and turbulent flows?

Wind at \(30^{\circ} \mathrm{C}\) flows over a \(0.5\)-m-diameter spherical tank containing iced water at \(0^{\circ} \mathrm{C}\) with a velocity of \(25 \mathrm{~km} / \mathrm{h}\). If the tank is thin-shelled with a high thermal conductivity material, the rate at which ice melts is (a) \(4.78 \mathrm{~kg} / \mathrm{h} \quad\) (b) \(6.15 \mathrm{~kg} / \mathrm{h}\) (c) \(7.45 \mathrm{~kg} / \mathrm{h}\) (d) \(11.8 \mathrm{~kg} / \mathrm{h}\) (e) \(16.0 \mathrm{~kg} / \mathrm{h}\) (Take \(h_{i f}=333.7 \mathrm{~kJ} / \mathrm{kg}\), and use the following for air: \(k=\) \(0.02588 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7282, v=1.608 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \mu_{\infty}=\) \(\left.1.872 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}, \mu_{\mathrm{s}}=1.729 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\right)\)

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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