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

Answer the following briefly (20 marks)

Dependence of boundary thickness with the distance x from x = 0 for a flat plate.

The thickness of the boundary layer increases as the square root of the distance from the leading edge of the plate.

$\delta (x) = 1.72\sqrt {\left( {\frac{{vx}}{{{U_0}}}} \right)}$

Express Nu as the ratio of two temperature gradients.

$Nu = \frac{{hl}}{k} = \frac{{ - (\partial t/\partial y)}}{{({t_s} - {t_\infty })/l}}$

The physical significance of St; Ra.

Stanton number indicates the degree of amount of heat delivered by the fluid when there is heat transfer between solid surface and fluid. The greater the Stanton number is, the more effectively heat is transferred.

Rayleigh number is defined as the product of Grashof number and Prandtl number. The rayleigh number is denoted by the symbol Ra. Rayleigh number also to find the type of fluid flow,

If Ra < 10⁹, then the flow is laminar and If Ra > 10⁹ then the flow is turbulent.

Momentum equation (one-directional)

$\sum {{F_x} = \frac{{dM}}{{dt}}}$

Similarities of kinematic viscosity and mass diffusivity.

The kinematic viscosity is also referred to as the momentum diffusivity of the fluid, i.e. the ability of the fluid to transport momentum.

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.

$Sc = \upsilon /D$

Fick’s second law of diffusion.

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.

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

Creeping flow.

Stokes flow also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. Re << 1. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large.

Prandtl’s mixing length.

According to Prandtl, the mixing length l, is that distance between two layers in the transverse direction such that lumps of fluid particles from one layer could reach the other layer and the particles are mixed in the other layer in such a way that the momentum of the particles in the direction of x is same.

The relation between shear stress and mixing length is given by

$\tau = \rho {l^2}{\left( {\frac{{du}}{{dy}}} \right)^2}$

Express Grashof no as a function of three forces.

Grashof number, Gr, as the ratio between the buoyancy force and the viscous force:

$Gr = \frac{{{l^3}{\rho ^2}\beta \Delta T}}{{{\mu ^2}}}$

It can be written as

$Gr = (\rho {l^3}\beta \Delta T)\left( {\frac{{\rho {v^2}{l^2}}}{{{{(\mu vl)}^2}}}} \right)$

= bupyant force x (inertia force/visous force²)

Express Nu/Bi as a ratio of two conductivities.

$Bi = hl/{k_s}$

$Nu = \frac{{h{k_f}}}{l}$

From these

$Nu/Bi = {k_f}/{k_s}$

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