Transport Phenomena - short answer question from AMIE exams (Summer 2021)

Answer the following in brief (10 x 2)

Define free turbulence and wall turbulence.

Turbulence can be generated in two ways: (1) by friction forces at solid walls (or surfaces) 

and (2) by the flow of fluid layers with different velocities past the other. These two types 

of turbulence are, respectively, known as wall turbulence and free turbulence.

Give units of mass diffusivity and kinematic viscosity.

The unit of both is m²/s.

Differentiate between laminar flow and turbulent flow.

• Laminar flows are smooth and streamlined, whereas turbulent flows are irregular and chaotic. 
• A low Reynolds number (< 2000) indicates laminar flow while a high Reynolds number indicates turbulent flow. The flow behaviour drastically changes if it is laminar vs. turbulent.

Write expressions for Fourier’s law of heat conduction and the one-dimensional form of Fick’s law of diffusion.

Fourier’s law states that the negative gradient of temperature and the time rate of heat transfer is proportional to the area at right angles of that gradient through which the heat flows. Fourier’s law is the other name of the law of heat conduction.

q = -k∇T

Where,

q is the local heat flux density in W.m2

k is the conductivity of the material in W.m-1.K-1

▽T is the temperature gradient in K.m-1

Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration.

J = D(dφ/dx)

where

J is the diffusion flux, of which the dimension is the amount of substance per unit area per unit time. J measures the amount of substance that will flow through a unit area during a unit time interval.

D is the diffusion coefficient or diffusivity. Its dimension is area per unit of time.

φ (for ideal mixtures) is the concentration, of which the dimension is the amount of substance per unit volume.

x is position, the dimension of which is length.

Define molecular mass flux and molar flux.

Molecular mass

We defined the molecular mass flux of “a ” as the flow of mass of “a ” through a unit area per unit time

In Mathematical expression written as;

j_a = ρ_a(V_a - V)

Here we include only the velocity of species a: relative to the mass average velocity V.

Molar flux

The molar flux of species “a" as the no. of moles flowing through a unit area per unit time

In Mathematical expression written as;

j_a = C_a(V_a - V*)

Write relationships of the friction factor and drag coefficient with Reynolds number in the laminar region.

Drag Coefficient vs. Reynolds Number

C_f = (24/Re) + (4/√Re) + 0.4

Friction factor vs. Reynolds number

f = 64/Re

What is Reynolds's analogy?

Reynolds's analogy gives the interrelationship between fluid friction and newton’s law of viscosity.

The Reynolds analogy is given by,

N_u/R_eP_r = S_t = C_f/2

where

Nu = Local Nusselt number, Re = Local Reynolds number, St = Local Stanton number, Pr = Prandtl number, Cf = Skin friction coefficient

Define Schmidt number and Grashof number.

Schmidt number (Sc) is a dimensionless number defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes.

The Grashof number  (Gr) gives the ratio of buoyant forces to viscous forces and is used to determine the flow regime of fluid boundary layers in laminar systems.

Define Reynolds number.

The Reynolds number is the ratio of inertial forces to viscous forces within a fluid which is subjected to relative internal movement due to different fluid velocities. A region where these forces change behaviour is known as a boundary layer, such as the bounding surface in the interior of a pipe. A low Reynolds number indicates laminar flow while a high Reynolds number indicates turbulent flow.

Re = VDρ/μ

Write a continuity equation for an incompressible fluid.

A1V1 = A2V2

Write expressions for terminal settling velocity in Stokes's law regime and Newton’s law regime.

When an object falls through a fluid, it attains a constant velocity through its subsequent motion. This happens because the net force on the body due to gravity and fluid becomes zero.

This constant velocity is termed terminal velocity.

Stokes law states that the force of viscosity on a small sphere moving through a viscous fluid is given by:

F=6πμrv

Where,
F is the frictional force acting on the interface between the fluid and the particle. 
μ is the dynamic viscosity
R is the radius of the spherical object
V is the flow velocity relative to the object

Terminal velocity in Newton's law regime

V_t = 1.74√(gD_p(ρ₁ - ρ₂)/ρg

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