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Chapter 6: Problem 43

Consider a flat plate positioned inside a wind tunnel, and air at \(1 \mathrm{~atm}\) and \(20^{\circ} \mathrm{C}\) is flowing with a free stream velocity of \(60 \mathrm{~m} / \mathrm{s}\). What is the minimum length of the plate necessary for the Reynolds number to reach \(2 \times 10^{7}\) ? If the critical Reynolds number is \(5 \times 10^{5}\), what type of flow regime would the airflow experience at \(0.2 \mathrm{~m}\) from the leading edge?

Short Answer

Expert verified
Short answer: The minimum length of the plate necessary for the Reynolds number to reach \(2 \times 10^{7}\) is approximately \(50.43\mathrm{~m}\). At \(0.2 \mathrm{~m}\) from the leading edge, the flow regime would be laminar.

Step by step solution

01

Given values and formula for Reynolds number

The given values include atmospheric pressure (\(1 \mathrm{~atm}\)), air temperature (\(20^{\circ} \mathrm{C}\)), free stream velocity (\(60 \mathrm{~m} / \mathrm{s}\)), and critical Reynolds number (\(5 \times 10^{5}\)). The formula for Reynolds number (Re) based on length is: Re \(= \frac{\rho uL}{\mu}\) where \(\rho\) is the air density, \(u\) is the free stream velocity, \(L\) is the length of the plate, and \(\mu\) is the dynamic viscosity of the air.
02

Calculate air density and dynamic viscosity

We need the air properties for density (\(\rho\)) and dynamic viscosity (\(\mu\)) at the given temperature (\(20^{\circ} \mathrm{C}\)). The values can be found in air properties tables or online resources. In this case, we will use the values \(\rho = 1.204\mathrm{~kg/m^3}\) for air density and \(\mu = 1.825 \times 10^{-5} \mathrm{~kg/(m\cdot s)}\) for dynamic viscosity.
03

Calculate the minimum length for Re to reach \(2 \times 10^{7}\)

Using the given target Reynolds number Re \(= 2\times10^7\), we can rearrange the Reynolds number formula to solve for the minimum length \(L\): L \(= \frac{\text{Re} \cdot \mu}{\rho u}\) Plugging in the known values: L \(= \frac{(2\times10^7) (1.825 \times 10^{-5}\mathrm{~kg/(m\cdot s)})}{(1.204\mathrm{~kg/m^3})(60 \mathrm{~m/s})} \) Calculating the result: L \(\approx 50.43 \mathrm{~m}\) The minimum length of the plate necessary for the Reynolds number to reach \(2 \times 10^{7}\) is approximately \(50.43\mathrm{~m}\).
04

Determine the flow regime at \(0.2 \mathrm{~m}\) from the leading edge

Now we need to determine the type of flow regime at \(0.2 \mathrm{~m}\) from the leading edge. To do this, we will calculate the Reynolds number using the same formula but changing the value of the length L to 0.2m: Re \(= \frac{\rho uL}{\mu}\) Re \(= \frac{(1.204\mathrm{~kg/m^3})(60 \mathrm{~m/s})(0.2\mathrm{~m})}{1.825 \times 10^{-5} \mathrm{~kg/(m\cdot s)}}\) Calculating the result: Re \(\approx 7.92\times10^4\) Now we compare this value to the critical Reynolds number, \(5 \times 10^{5}\). Because the calculated Reynolds number (\(7.92\times10^4\)) is less than the critical Reynolds number (\(5 \times 10^{5}\)), the flow regime at \(0.2 \mathrm{~m}\) from the leading edge would be laminar.

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Most popular questions from this chapter

Determine the heat flux at the wall of a microchannel of width \(1 \mu \mathrm{m}\) if the wall temperature is \(50^{\circ} \mathrm{C}\) and the average gas temperature near the wall is \(100^{\circ} \mathrm{C}\) for the cases of (a) \(\sigma_{T}=1.0, \gamma=1.667, k=0.15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \lambda / \operatorname{Pr}=0.5\) (b) \(\sigma_{T}=0.8, \gamma=2, k=0.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \lambda / \operatorname{Pr}=5\)

What is the physical significance of the Reynolds number? How is it defined for external flow over a plate of length \(L\) ?

When is heat transfer through a fluid conduction and when is it convection? For what case is the rate of heat transfer higher? How does the convection heat transfer coefficient differ from the thermal conductivity of a fluid?

Most correlations for the convection heat transfer coefficient use the dimensionless Nusselt number, which is defined as (a) \(h / k\) (b) \(k / h\) (c) \(h L_{c} / k\) (d) \(k L_{c} / h\) (e) \(k / \rho c_{p}\)

A metallic airfoil of elliptical cross section has a mass of \(50 \mathrm{~kg}\), surface area of \(12 \mathrm{~m}^{2}\), and a specific heat of \(0.50 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). The airfoil is subjected to air flow at \(1 \mathrm{~atm}, 25^{\circ} \mathrm{C}\), and \(5 \mathrm{~m} / \mathrm{s}\) along its 3 -m-long side. The average temperature of the airfoil is observed to drop from \(160^{\circ} \mathrm{C}\) to \(150^{\circ} \mathrm{C}\) within \(2 \mathrm{~min}\) of cooling. Assuming the surface temperature of the airfoil to be equal to its average temperature and using momentum-heat transfer analogy, determine the average friction coefficient of the airfoil surface.
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