Transport+Phenomena(1)

1

CL203, Autumn 2009

CL203: Introduction to Transport Phenomena Mid-semseter exam (14:00-16:00, Sunday, 13 September, 2009) Total Marks:20 1. Newtonian fluid (ρ, µ =constant) flows steadily through a channel (height 2h). In the middle of the channel, an infinitely thin splitter plate is mounted. The channel walls move with a constant velocity U in the positive x1 -direction. The two fluid streams separated by the plate are mixed at the end of the plate. At station [2], far away from station [1], a new velocity profile u1 = u1 (x2 ) is developed that does not change anymore with x1 . The body forces can be neglected. (a) Using the equation of motion, on what spatial coordinates does the pressure gradient depend at station [2]. Justify your answer. (b) Calculate the volume flow rate (per unit depth) Q at station [1]. The pressure gradient is zero here. (c) Obtain the velocity profile u1 = u1 (x2 ) at station [2]. Note that the volume flow rate at station [1] and [2] should be the same. (d) What is the pressure gradient at [2] ? There is no need to perform a shell balance. Proceed using the Continuity equation and the NavierStokes equation (see reverse) [10 marks]. 2. A fluid, whose viscosity is to be measured, is placed in the gap of thickness B between the two disks of radius R. One measures the torgue Tz required to turn the upper disk at an angular velocity Ω. Develop the formula for deducing the viscosity from these measurements. Assume creeping flow. (a) Postulate that for small values of Ω the velocity profiles have the form ur = 0, uz = 0, and uθ = rf (z). Postulate further that P = P(z, r). Write down the resulting simplified equations of continuity and motion. (b) From the θ−component of the equation of motion, obtain a differential equation for f (z). Solve the equation for f (z) and evaluate the constants of integration to determine uθ (r, z). (c) Determine the viscosity in terms of the measured torque, B, R and Ω. Proceed using the Continuity equation and the Navier-Stokes equation (see reverse)[10 marks].

U x2

Splitter plate

h x1

z=B

h

! B

z=0

U

[1]

[2]

(a) Figure for problem 1

2R

(b) Figure for problem 2

1

CL203, Autumn 2008

CL203: Introduction to Transport Phenomena Mid-semseter exam (14:30-16:30, Saturday, 6 September, 2008) Total Marks:20 1. Two immiscible, incompressible liquids are flowing in the z-direction in a horizontal thin slit of length L, and width W under the influence of a horizontal pressure gradient (po − pL )/L. The fluid rates are adjusted so that the slit is half filled with fluid 1 (more dense phase) and remaining with fluid 2. The interface between the phases remains flat. The goal is to determine the momentum fluxes and velocity distributions. Assume that flow is fully developed and has reached steady state. [8 marks] (a) Choose a thin element of length L and width W and perform z-momentum balance in terms of the combined momentum flux tensor (Φ) in any one of the phases. (b) Use the given handout to replace components of Φ and write the governing equation for both phases in terms of the shear stress. Postulate that p = p(z), uz = uz (x), and ux = uy = 0. (c) Substitute the Newton’s law of viscosity and apply the relevant boundary conditions to determine the velocity profile in the two phases. 2. Liquid is present in the annular space between two vertical concentric cylinders of radius R1 and R2 (R2 > R1 ) that are rotating in opposite directions with angular velocities of magnitude Ω1 and Ω2 . We would like place a thin circular hollow cylinder of negligible wall thickness and radius a (R2 > a > R1 ) concentric and in between with the two cylinders. Determine the value of a when no external torque would be required to hold this middle cylinder stationary. Use the handout for the momentum balance equations [7 marks]. 3. The drag on an airplane cruising at 400 km/h in standard air is to be determined from tests on a 1:10 scale model placed in a pressurized wind tunnel. To minimize compressibility effects, the air speed in the wind tunnel is also to be 400 km/h. Determine the required air pressure in the tunnel (assuming the same air temperature for model and prototype), and the drag on the prototype if the measured force on the model is 4.5 N. Assume that the drag, D is a function of the density of air, ρ, viscosity of air, µ, speed of the airplane V and and characteristic length L. Further, while the viscosity is assumed to be unaffected by changes in pressure, the air pressure and density are related by ideal gas law.[5 marks]

Stationary wall x

2 2b

z

R1

1

Stationary wall

(a) Figure for problem 1

!1

a

R2

(b) Figure for problem 2

!2

1

CL203, Autumn 2008

CL203: Introduction to Transport Phenomena Weekly Class Quiz - 2 Total Marks:10 Time alloted: 10 minutes Date: 14/08/2008 1. Write the boundary conditions at the air-liquid and liquid-solid interface for the following situation. Air

y=h

Flowing Liquid

y x

Stationary wall

2. Two immiscible liquids, α and β, flow in the x direction between two stationary walls. Liquid α is the lighter of the two. State which of following velocity profiles are possible (True/False) and why ? Be brief and to the point in your response. No marks for guess work. Stationary wall

Stationary wall

!

!

"

y x

"

y x

Stationary wall

(a) True|False

(b) True|False

Stationary wall

Stationary wall

!

!

"

y x

Stationary wall

"

y

Stationary wall

(c) True|False

x

Stationary wall

(d) True|False

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 4 Total Marks:10 Time alloted: 10 minutes Date: 28/08/2008 1. A solid sphere immersed in a stagnant fluid rotates about the z-axis. You are asked to postulate that uφ = uφ (r) while other velocity components are zero. Gravity acts in the negative z direction. Assume steady state. (a) Using the given handout, write down the reduced set of r, θ and φ momentum balances.

! z

y

x

(b) What spatial coordinates should the pressure be dependent on and why? No marks for guesses.

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 5 Total Marks:10 Time alloted: 10 minutes Date: 4/09/2008 1. A thin rectangular plate having a width w and a height h is located so that it is normal to a moving stream of fluid. Assume the drag, D, that the fluid exerts on the plate is a function of w and h, the fluid viscosity and density, µ and ρ, respectively, and the velocity V of the fluid approaching the plate. Determine a suitable set of dimensionless groups to study this problem experimentally.

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 6 Total Marks:10 Time alloted: 10 minutes Date: 25/10/2008 1. Two separate metal blocks of thickness d and 2d are kept at constant temperature T0 at the top and a constant heat flux q0 is provided at the bottom. Draw the steady state temperature profile for each of the blocks. The temperature at the bottom plate and the slope should be indicated.

2. At steady state the temperature profiles in a laminated system appear as shown in figure. Which material has the higher thermal conductivity and Why ?

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 7 Total Marks:20 Time alloted: 30 minutes Date: 08/10/2008 1. A viscous fluid with temperature independent physical properties is in fully developed laminar flow between two flat surfaces placed a distance 2B apart. For z < 0 the fluid temperature is uniform at T = T1 . For z > 0 heat is added at a constant, uniform flux qo at both walls. The velocity profile is given by, uz = umax (1 − (x/B)2 )

(a) Make a shell energy balance to obtain the differential equation for T (x, z). There is no need to include the energy flux component in the y direction. At this stage the energy balance should be in terms of the energy flux vector. [3 marks] ˆ = Cˆp dT + (1/ρ)dp. At (b) Substitute the terms in energy balance equation while noting that dH this stage, the your differential equation should contain derivatives of T, uz , and p. [ 6 marks] (c) Next, discard the viscous dissipation and the axial heat conduction term. Further, make use of the momentum balance in the z direction to obtain a reduced partial differential equation only in T . [6 marks] (d) List all the boundary conditions. [3 mark] (e) For temperature variations in regions far from the entrance, write down the integral boundary condition that will replace the boundary condition at z = 0. [2 mark]

Heating element

Heating element

Z X

X=-B

X=B

Fully developed slit flow at z=0, Inlet temperature is T1

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 9 Total Marks:10 Time alloted: 10 minutes Date: 16/10/2008 1. Show that only one diffusivity is needed to describe the diffusional behavior of a binary mixture, i.e., DAB = DBA

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 10 Total Marks:10 Time alloted: 10 minutes Date: 6/11/2008 1. Figure 1 shows schematically how oxygen and carbon monoxide combine at a catalytic surface (palladium) to make carbon dioxide, according to O2 + 2CO → 2CO2 (1) For this analysis, the reaction is assumed to occur instantaneously and irreversibly at the catalytic surface. All variations occur over a thin gas film of thickness d. The temperature and pressure are assumed to be constant throughout the gas film. Note that this is a three component mass transfer problem [7 marks]. (a) Using the above assumptions, write down the final steady state mass balance differential equation for each of the fluxes. Your mass balance equation should be only in terms of the molar fluxes of each of the components (i.e. there is no need to substitute the Maxwell-Stefan equation). (b) How are each of the fluxes related ? (c) What is the concentration of O2 and CO at the catalyst surface ? O2 CO Z=d CO2 Z=0

Catalyst

2. A two bulb apparatus containing pure oxygen in the left bulb and nitrogen in the right bulb is shown in fig 2. The stopcock is placed in the middle. The entire gas system is at constant temperature and pressure. At time t=0, the stopcock is opened. Write down how the molar fluxes of the two components are related during this process. Note that this is an unsteady binary component problem.[3 marks]

Stopcock

Oxygen

Nitrogen

1

CL203, Autumn 2008

Roll No: CL203: Introduction to Transport Phenomena Weekly Class Quiz - 11 Total Marks:10 Time alloted: 10 minutes Date: 11/11/2008 1. A solid metallic block occupying space between y = 0 and y = d is kept at an initial temperature of To . At t = 0, the surface at y = 0 is suddenly supplied with a constant heat flux, qo , and maintained at that flux for t > 0. The top surface is kept at To throughout. (a) Write down the governing equation for the unsteady state one dimensional heat conduction problem. This should be a differential equation in terms of T .[1 marks]

(b) Write down the initial and boundary condition in terms of T . [2 marks]

(c) What is the thermal diffusion timescale, td ?[2 marks]

(d) We are interested in the temperature profile at short times, i.e., t W >> B. The duct has porous walls at y = 0 and y = B, so that a constant cross flow can be maintained, with vy = vo , a constant everywhere. Flows of this type are important in connection with separation processes. The goal of this problem is to determine the velocity profile, vx (y) ? (a) State the postulate. Starting from the x-momentum equation (given with this paper) for constant density and viscosity liquid, show clearly the terms that survive on applying the postulate. (b) Using the no-slip boundary condition at the walls for vx , show that the velocity profile for the system is given by (P0 − PL )B 2 1 vx = µL A in which A = Bvo ρ/µ [8 marks]

#

y eAy/B − 1 − B eA − 1

$

(3)

2

CL203, Fall 2007

R

R*

(2) (1)

Figure 1: For problem 1

kR R

Flow direction

Flow direction

z r

Figure 2: For problem 2

y x

L

Figure 3: For problem 3