Transport Phenomena - short answer questions from AMIE exams (Summer 2020)

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Answer the following in brief (10 x 2)

Reynolds analogy

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

The Reynolds analogy is given by,


Nu/RePr = St = Cf/2


where

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

Stefan-Boltzmann law

Stefan-Boltzmann law states that the total radiant heat power emitted from a surface is proportional to the fourth power of its absolute temperature.

u/A = \sigma {T^4}

where  σ is Stefan’s constant = 5.67 × 10-8 W/m2 k4

Langevin equation

Consider a large particle (the Brownian particle) immersed in a fluid of much smaller particles (atoms).

v(t) = {e^{ - t/{\tau _B}}}v(0) + \frac{1}{m}\int_0^t {{e^{ - (t - s)/{\tau _B}}}} dW(s)

Comparison between turbulent thermal conductivity and turbulent viscosity

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 turbulent thermal conductivity of the material in W.m-1.K-1

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

As per Newton's law of viscosity

\tau = - \mu \frac{{dv}}{{dy}}

Here μ is turbulent viscosity.

Equation of motion

Similarly in other directions.

The factors that affect the diffusion of gases

The rate of diffusion is affected by the concentration gradient, membrane permeability, temperature, and pressure.
  • The difference in concentration affects the rate of diffusion. The greater the concentration gradient, the quicker diffusion takes place. 
  • The temperature affects the rate of diffusion. As the temperature increases, particles gain more kinetic energy and so can diffuse across a membrane more quickly. Therefore, as the temperature increases, the rate of diffusion increases.
  • The surface area of the membrane affects the rate of diffusion. As the surface area of the membrane increases, the rate of diffusion also increases, as there is more space for molecules to diffuse across the membrane.
  • Distance. The shorter the distance the substances have to move, the faster the rate of diffusion.

Fick’s law of binary diffusion

Fick’s law of diffusion explains the diffusion process (movement of molecules from higher concentration to lower concentration region).

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.


Fick's 2nd law of diffusion describes the rate of accumulation (or depletion) of concentration within the volume as proportional to the local curvature of the concentration gradient. 

\frac{{\partial C}}{{\partial t}} = D\frac{{{\partial ^2}C}}{{\partial {x^2}}}

The Boltzmann equation

The Boltzmann equation can be in use to determine how physical quantities change i.e., heat energy and the momentum when a fluid is in transport. The Boltzmann equation is a nonlinear integrodifferential equation, and the unknown function in the equation is a probability density function in the six-dimensional space of a particle position and momentum.

Boltzmann equation is:

\frac{{{P_{sb}}}}{{{P_{sa}}}} = \frac{{{N_b}}}{{{N_a}}} = \frac{{gb{e^{( - Eb/kT)}}}}{{ga{e^{( - Ea/kT)}}}} = \frac{{gb}}{{ga}}{e^{ - (Eb - Ea)/kT}}

Significance of Dimensional Analysis

The advantages of dimensional analysis are:
  • Preliminary test for the correctness of the given equation
  • Convert from one unit to another
  • Derive the relationship between the physical quantities.
  • Determine the dimensions of constant or unknown physical quantities
  • The number of experiments conducted is reduced

Three levels at which transport phenomena can be studied

These three areas of study are: 
  • Fluid Mechanics
  • Heat Transfer
  • Mass Transfer
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