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Chapter 14: Problem 114

Under what conditions will the normalized velocity, thermal, and concentration boundary layers coincide during flow over a flat plate?

Short Answer

Expert verified
Answer: The normalized velocity, thermal, and concentration boundary layers coincide during flow over a flat plate when the Prandtl number (Pr) is equal to the Schmidt number (Sc).

Step by step solution

01

Identify dimensionless parameters governing boundary layers

The three main dimensionless parameters governing boundary layers are Reynolds number (Re), Prandtl number (Pr), and Schmidt number (Sc). The Reynolds number represents the ratio of inertial forces to viscous forces, the Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity, and the Schmidt number represents the ratio of momentum diffusivity to mass diffusivity. Re = \(\frac{ρUL}{μ}\) Pr = \(\frac{μ}{α}\) Sc = \(\frac{μ}{D}\) where: ρ = fluid density U = free stream velocity L = length of the flat plate μ = dynamic viscosity α = thermal diffusivity D = mass diffusivity
02

Determine the conditions for coinciding boundary layers

For the boundary layers to coincide, their thicknesses should be related by the dimensionless parameters Re, Pr, and Sc. The thickness of the boundary layers can be related to the dimensionless parameters as follows: δ = \(\frac{L}{\sqrt{Re}}\) δ_t = \(\frac{δ}{\sqrt{Pr}}\) δ_c = \(\frac{δ}{\sqrt{Sc}}\) where: δ = thickness of the velocity boundary layer δ_t = thickness of the thermal boundary layer δ_c = thickness of the concentration boundary layer For the boundary layers to coincide, we need to have: δ = δ_t = δ_c Which means: \(\frac{L}{\sqrt{Re}} = \frac{δ}{\sqrt{Pr}} = \frac{δ}{\sqrt{Sc}}\)
03

Find the relation between Pr and Sc for coinciding boundary layers

Now, we can derive the relation between Pr and Sc by setting the expressions for δ_t and δ_c equal: \(\frac{δ}{\sqrt{Pr}} = \frac{δ}{\sqrt{Sc}}\) Then, solving for Pr: Pr = \(\frac{δ^2}{δ^2}\) Sc Thus, Pr = Sc
04

State the conditions for coinciding boundary layers

The conditions under which the normalized velocity, thermal, and concentration boundary layers coincide during flow over a flat plate are when the Prandtl number (Pr) is equal to the Schmidt number (Sc).

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Most popular questions from this chapter

What is a concentration boundary layer? How is it defined for flow over a plate?

The basic equation describing the diffusion of one medium through another stationary medium is (a) \(j_{A}=-C D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) (b) \(j_{A}=-D_{A B} \frac{d\left(C_{A} / C\right)}{d x}\) (c) \(j_{A}=-k \frac{d\left(C_{A} / C\right)}{d x}\) (d) \(j_{A}=-k \frac{d T}{d x}\) (e) none of them

What are the adverse effects of excess moisture on the wood and metal components of a house and the paint on the walls?

Air at \(40^{\circ} \mathrm{C}\) and 1 atm flows over a \(5-\mathrm{m}\)-long wet plate with an average velocity of \(2.5 \mathrm{~m} / \mathrm{s}\) in order to dry the surface. Using the analogy between heat and mass transfer, determine the mass transfer coefficient on the plate.

What is the relation \((f / 2) \mathrm{Re}=\mathrm{Nu}=\mathrm{Sh}\) known as? Under what conditions is it valid? What is the practical importance of it? \(\mathrm{St}_{\text {mass }} \mathrm{Sc}^{2 / 3}\) and what are the names of the variables in it? Under what conditions is it valid? What is the importance of it in engineering?
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