# Transport Phenomena 2006-07

1.  Attempt any four parts of the following :

(a) Explain the continuum hypothesis.

(b) Water at 298 K discharges from a nozzle and travels horizontally hitting a flat vertical wall. The nozzle has a diameter of 12 mm and the water leaves the nozzle with a flat velocity profile at a velocity of 6 m/s. Neglecting frictional resistance of the air on the jet, calculate the force in N on the wall.

(c) What are thixotropic and rheopectic fluids ? Give two examples of each of these fluids.

(d) Give the following models of non-Newtonian fluids :

(i) Bingham model

(ii) Ostwald – de Waele model

(iii) Eyring model

(iv) Ellis model

(v) Reiner-Philippoff model.

(e) Fluid A has a viscosity that is twice that of fluid B. Which fluid would flow more rapidly through a horizontal tube of length L under the same pressure drop.

(f)  Explain theory of viscosity of gases at low density.

2. Attempt any two parts of the following :

(a) For laminar flow of a Newtonian fluid in a horizontal pipe, derive the expressions for the distribution of momentum flux and velocity.

List all assumptions and define the terms.

(b) For turbulent flow in a smooth circular tube with a radius R, the velocity profile varies according to the following expression at a Reynolds number of about 105 ad „\1v = %ax R-r R where r is the radial distance from the centre and fmax is the maximum velocity at the centre. Derive the equation relating the average velocity to ^max for an incompressible fluid.

(c)  The velocity components for a flow field are as follows : vx =a(x2 – y2) vy =-2axy

Prove that it satisfies the conservation of mass and determine stream functionv|/.

3. Attempt any two parts of the following :

(a) A fluid in a U-tube manometer, initially at rest, is set in motion by suddenly imposing a pressure difference Pa~ P^. Determine the differential equation for the motion of the manometer fluid, assuming isothermal, incompressible flow. Obtain an expression for the tube radius for which critical damping occurs. Neglect the density of the gas above the manometer liquid.

(b) A sphere of radius R is falling in creeping flow through a stationary fluid of viscosity M. with a terminal velocity . At what horizontal distance from the sphere in a plane perpendicular to the direction of fall does the velocity of the fluid fall to 0.01 times the terminal velocity of the sphere ?

(c) A semi-infinite body of liquid with constant p and n is bounded on one side by a flat surface (the xz -plane). Initially, the fluid and the solid surface are at rest, but at time t = 0 the solid, surface is set in motion in the positive x – direction with a velocity V. Obtain the expression for the velocity as a function of y and t. There is no pressure gradient or gravity force in the x-direction and the flow is assumed to be laminar.

4. Attempt any two parts of the following :

(a) A stainless steel sphere (k = 15 W !m K)

having a diameter of 40 mm is exposed to convection environment at 20°C,

in the sphere at the rate of 1 MW/m3. Calculate the steady state temperature for the centre of the sphere.

(b) In a steady state forced convection, the following differential equation is obtained oh = 15W I m K . Heat is generated uniformly for large r\ the solution is of the following from where CQ is a constant. Solve the differntial equation for the following boundary conditions £, = 0 (§) = finiteB® ,^ -^r=11-Tl= J (l-52)®(5. T|)d5 0

(c) A cylindrical tank capable of holding 28.3m3 of liquid is equipped with an agitator having sufficient power to keep the liquid contents at uniform temperature. Heat is transferred to the contents by means of a coil arranged in such a way that area available for heat transfer is proportional to the quantity of liquid in the tank. This heating coil consists of 10 turns, 1.22 m in diameter, of 0.0254 m OD tubing. Water at 20°C is fed into this tank at a rate of 0.15 kg/s, starting with no water in the tank at tune t = 0 Steam at 105°C is contained within the heating coil and the overall heat transfer coefficient is 570 W/m2K. What is the temperature of water when the tank is filled ?

5. Attempt any four parts of the following :

(a) Discuss the effect of temperature and pressure on diffusivity of binary mixture of gases.

(b)   A dimerization reaction 2A —» A2 is carried out is a catalytic reactor consisting of spherical catalyst particles. Each catalyst particle is surrounded by a stagnant gas film through which A has to diffuse is order to arrive at the catalyst surface. At the surface of the catalyst dimerization reaction occursinstantaneously and that the product A2 thendiffuses back out through the gas film to the main turbulent gas stream composed of A and A2. Derive an expression for the local rate of conversion from A to A2 when theeffective gas film thickness 8 and main gas stream compositions are known. The gas film may be considered to the isothermal.

(c) Deduce the expression for enhancement factor for diffusion of a gas through a liquid phase which is accompanied with a slow first order chemical reaction.

(d) For an equimolar counter diffusion from a sphere to a surrounding stationary infinitemedium, the mass flux of the diffusingcomponent A at the interface is given byNAi = ^<CAi-CAb)where DA is the diffusivity, R is the radius of the sphere, CAi and CAb are the molarconcentration of A at the interface and at a point for away from the sphere. Show that the Sherwood number, based on the diameter of sphere is equal to 2.

(e) State and explain Fick’s law of diffusion.

(f)  What are various analogies between heat and mass transfer ?