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Chapter 14: Problem 112

What is the physical significance of the Schmidt number? How is it defined? To what dimensionless number does it correspond in heat transfer? What does a Schmidt number of 1 indicate?

Short Answer

Expert verified
Answer: A Schmidt number of 1 indicates that the momentum diffusion (kinematic viscosity) is equal to the mass diffusivity of the fluid. This means that the transport of momentum and mass occurs at the same rate, implying that both diffusion processes are equally effective. In such cases, a balance between momentum and mass transport is achieved.

Step by step solution

01

Physical significance of Schmidt number

The Schmidt number (Sc) is a dimensionless number used to characterize the relative importance of momentum diffusion and mass diffusion in fluid dynamics and mass transfer processes. It represents the ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity. The physical significance of the Schmidt number is to provide a measure of how efficiently a solute (such as pollutant or a dissolved substance) is transported by diffusion compared to how efficiently it is transported by the fluid's turbulence or viscous effects. High Schmidt numbers indicate that mass diffusion is less effective compared to momentum diffusion, while low Schmidt numbers imply that mass diffusion dominates.
02

Definition of Schmidt number

The Schmidt number is defined as the ratio of the kinematic viscosity (ν) of a fluid to its mass diffusivity (D). Mathematically, it is represented as: \[Sc = \frac{\nu}{D}\] where: - ν (nu) is the kinematic viscosity of the fluid (m^2/s), - D is the mass diffusivity (m^2/s).
03

Corresponding dimensionless number in heat transfer

In heat transfer, the analogous dimensionless number is the Prandtl number (Pr). The Prandtl number is defined as the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity (α). Mathematically, it is represented as: \[Pr = \frac{\nu}{\alpha}\] where: - ν (nu) is the kinematic viscosity of the fluid (m^2/s), - α (alpha) is the thermal diffusivity of the fluid (m^2/s).
04

Schmidt number of 1

A Schmidt number of 1 indicates that the momentum diffusion (kinematic viscosity) is equal to the mass diffusivity of the fluid. This means that the transport of momentum and mass occurs at the same rate, implying that both diffusion processes are equally effective. In such cases, a balance between momentum and mass transport is achieved. For example, air usually has a Schmidt number close to 1, indicating a balance between the effects of viscosity and diffusion on the transport of solute particles.

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Most popular questions from this chapter

Consider a nickel wall separating hydrogen gas that is maintained on one side at \(5 \mathrm{~atm}\) and on the opposite at \(3 \mathrm{~atm}\). If the temperature is constant at \(85^{\circ} \mathrm{C}\), determine \((a)\) the mass densities of hydrogen gas in the nickel wall on both sides and \((b)\) the mass densities of hydrogen outside the nickel wall on both sides.

Using the analogy between heat and mass transfer, explain how the mass transfer coefficient can be determined from the relations for the heat transfer coefficient.

A recent attempt to circumnavigate the world in a balloon used a helium-filled balloon whose volume was \(7240 \mathrm{~m}^{3}\) and surface area was \(1800 \mathrm{~m}^{2}\). The skin of this balloon is \(2 \mathrm{~mm}\) thick and is made of a material whose helium diffusion coefficient is \(1 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). The molar concentration of the helium at the inner surface of the balloon skin is \(0.2 \mathrm{kmol} / \mathrm{m}^{3}\) and the molar concentration at the outer surface is extremely small. The rate at which helium is lost from this balloon is (a) \(0.26 \mathrm{~kg} / \mathrm{h}\) (b) \(1.5 \mathrm{~kg} / \mathrm{h}\) (c) \(2.6 \mathrm{~kg} / \mathrm{h}\) (d) \(3.8 \mathrm{~kg} / \mathrm{h}\) (e) \(5.2 \mathrm{~kg} / \mathrm{h}\)

A rubber object is in contact with nitrogen \(\left(\mathrm{N}_{2}\right)\) at \(298 \mathrm{~K}\) and \(250 \mathrm{kPa}\). The solubility of nitrogen gas in rubber is \(0.00156 \mathrm{kmol} / \mathrm{m}^{3}\).bar. The mass density of nitrogen at the interface is (a) \(0.049 \mathrm{~kg} / \mathrm{m}^{3}\) (b) \(0.064 \mathrm{~kg} / \mathrm{m}^{3}\) (c) \(0.077 \mathrm{~kg} / \mathrm{m}^{3}\) (d) \(0.092 \mathrm{~kg} / \mathrm{m}^{3}\) (e) \(0.109 \mathrm{~kg} / \mathrm{m}^{3}\)

What is diffusion velocity? How does it affect the mass-average velocity? Can the velocity of a species in a moving medium relative to a fixed reference point be zero in a moving medium? Explain.
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