cl203, autum n 2009 - iit bombayphenomena.pdf · cl203, autum n 2009 1 cl203: in tro duction to t...

CL203, Autumn 2009 1

CL203: Introduction to Transport PhenomenaMid-semseter exam (14:00-16:00, Sunday, 13 September, 2009)

Total Marks:20

1. Newtonian fluid (!, µ =constant) flows steadily through a channel (height 2h). In the middle of thechannel, an infinitely thin splitter plate is mounted. The channel walls move with a constant velocityU in the positive x1-direction. The two fluid streams separated by the plate are mixed at the end ofthe plate. At station [2], far away from station [1], a new velocity profile u1 = u1(x2) is developedthat 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 dependat station [2]. Justify your answer.

(b) Calculate the volume flow rate (per unit depth) Q at station [1]. The pressure gradient is zerohere.

(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 Navier-Stokes 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 disksof radius R. One measures the torgue Tz required to turn the upper disk at an angular velocity !.Develop the formula for deducing the viscosity from these measurements. Assume creeping flow.

(a) Postulate that for small values of ! the velocity profiles have the form ur = 0, uz = 0, andu! = rf(z). Postulate further that P = P(z, r). Write down the resulting simplified equationsof continuity and motion.

(b) From the "!component of the equation of motion, obtain a di"erential equation for f(z). Solvethe equation for f(z) and evaluate the constants of integration to determine u!(r, z).

(c) Determine the viscosity in terms of the measured torque, B, R and !.

Proceed using the Continuity equation and the Navier-Stokes equation (see reverse)[10 marks].

U

U[1] [2]

h

h

Splitter

plate

x1

x2

(a) Figure for problem 1

B

2R

z=0

z=B

!

(b) Figure for problem 2


CL203: Introduction to Transport PhenomenaMid-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 lengthL, and width W under the influence of a horizontal pressure gradient (po ! pL)/L. The fluid ratesare 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 andvelocity 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 ofthe combined momentum flux tensor (!) in any one of the phases.

(b) Use the given handout to replace components of ! and write the governing equation for bothphases 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 deter-mine the velocity profile in the two phases.

2. Liquid is present in the annular space between two vertical concentric cylinders of radius R1 andR2 (R2 > R1) that are rotating in opposite directions with angular velocities of magnitude "1 and"2. 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 noexternal torque would be required to hold this middle cylinder stationary. Use the handout for themomentum 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 a1:10 scale model placed in a pressurized wind tunnel. To minimize compressibility e#ects, the airspeed 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 themeasured force on the model is 4.5 N. Assume that the drag, D is a function of the density of air,!, viscosity of air, µ, speed of the airplane V and and characteristic length L. Further, while theviscosity is assumed to be una#ected by changes in pressure, the air pressure and density are relatedby ideal gas law.[5 marks]

Stationary wall

Stationary wall

2

1

x

z

2b

(a) Figure for problem 1

!2!1

R2

a

R1

(b) Figure for problem 2


CL203: Introduction to Transport PhenomenaWeekly 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.

x

y

Air

Stationary wall

y=h

Flowing Liquid

2. Two immiscible liquids, ! and ", flow in the x direction between two stationary walls. Liquid ! isthe 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

(a) True|False

Stationary wall

Stationary wall

!

"

y

x

(b) True|False

Stationary wall

Stationary wall

!

"

y

x

(c) True|False

Stationary wall

Stationary wall

!

"

y

x

(d) True|False


Roll No:



1. A solid sphere immersed in a stagnant fluid rotates about the z-axis. You are asked to postulatethat u! = u!(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, ! and " momentum balances.

x

y

z

!

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


Roll No:



1. A thin rectangular plate having a width w and a height h is located so that it is normal to a movingstream 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 !, respectively, and the velocity V of the fluid approaching theplate. Determine a suitable set of dimensionless groups to study this problem experimentally.


Roll No:



1. Two separate metal blocks of thickness d and 2d are kept at constant temperature T0 at the top anda constant heat flux q0 is provided at the bottom. Draw the steady state temperature profile for eachof 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. Whichmaterial has the higher thermal conductivity and Why ?


Roll No:



1. A viscous fluid with temperature independent physical properties is in fully developed laminar flowbetween two flat surfaces placed a distance 2B apart. For z < 0 the fluid temperature is uniform atT = T1. For z > 0 heat is added at a constant, uniform flux qo at both walls. The velocity profile isgiven by, uz = umax(1! (x/B)2)

(a) Make a shell energy balance to obtain the di!erential equation for T (x, z). There is no need toinclude the energy flux component in the y direction. At this stage the energy balance shouldbe in terms of the energy flux vector. [3 marks]

(b) Substitute the terms in energy balance equation while noting that dH = CpdT + (1/!)dp. Atthis stage, the your di!erential 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 ofthe momentum balance in the z direction to obtain a reduced partial di!erential equation onlyin 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]

X=-B X=B

Z

X

Heating

element

Heating

element

Fully developed slit flow at z=0,

Inlet temperature is T1


Roll No:



1. Show that only one di!usivity is needed to describe the di!usional behavior of a binary mixture, i.e.,DAB = DBA


Roll No:CL203: Introduction to Transport Phenomena

Weekly Class Quiz - 10Total 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) tomake carbon dioxide, according to

O2 + 2CO → 2CO2 (1)

For this analysis, the reaction is assumed to occur instantaneously and irreversibly at the catalytic surface. Allvariations occur over a thin gas film of thickness d. The temperature and pressure are assumed to be constantthroughout 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 foreach of the fluxes. Your mass balance equation should be only in terms of the molar fluxes of each of thecomponents (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

CO2

CO

Z=0

Z=d

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 timet=0, the stopcock is opened. Write down how the molar fluxes of the two components are related during thisprocess. Note that this is an unsteady binary component problem.[3 marks]

Stopcock

Oxygen Nitrogen


Roll No:CL203: Introduction to Transport Phenomena

Weekly Class Quiz - 11Total 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. Att = 0, the surface at y = 0 is suddenly supplied with a constant heat flux, qo, and maintained at that flux fort > 0. The top surface is kept at To throughout.

(a) Write down the governing equation for the unsteady state one dimensional heat conduction problem. Thisshould 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 << td. Write the governing differentialin terms of heat flux (q) by differentiating the original equation with y.[2 marks]

(e) What is the corresponding boundary condition ?[2 marks]

(f) Using an analogy with a similar problem solved in class, can you guess the form of the similarity variablefor this problem ?[1 mark]

CL203, Fall 2008 1

CL203: Introduction to Transport PhenomenonEnd-semester exam (Total 50 marks)

Date: 17/11/2008

INSTRUCTIONS: State all assumptions clearly.

1. A wire of constant density ρ moves downward with uniform speed v into a liquid metal bath attemperature T0. It is desired to find the temperature profile T (z) in the metal wire. Assumethat T = T∞ at z = !, and that resistance to radial heat conduction is negligible. Assumefurther that the wire temperature is T = T0 at z = 0. Solve the problem for constant physicalproperties Cp and k. The equation of change for temperature is given in the vectorial formas:

ρCpDT

Dt= "# · q" τ : #v "

(∂lnρ

∂lnT

)

p

Dp

Dt

[8 marks]

2. A cold liquid film flowing down a vertical wall, as shown in figure (a), has a considerablecooling e!ect on the solid surface. Estimate the rate of heat transfer from the wall to thefluid for such short contact times that the fluid temperature changes appreciably only in theimmediate vicinity of the wall. [18 marks]

(a) Determine the velocity distribution in the falling film at steady state. Note that velocityvariation is only in the y direction.

ρDvDt

= "#p + µ#2v + ρb

(b) Deduce the energy equation for this situation by neglecting the conduction in the zdirection and any viscous heating e!ects. Your partial di!erential equation will be withrespect to z and y.

(c) Write the temperature boundary condition valid for short contact times only. Note thatthe z coordinate here is similar to the time coordinate in one dimensional time dependentproblem solved in the class. Further, the boundary condition at y = δ is replaced bythat at !.

(d) Next, use dimensionless variables, "(η) = (T " T0)/(T1 " T0) and η = y/ 3$

9βz, whereβ = µk/ρ2Cpgδ, and rewrite the di!erential equation in terms of " and η only.

(e) Write the corresponding boundary conditions.(f) Solve the above equation. Use the fact that #(4

3) =∫∞0 e−η3dη

(g) Determine the average heat flux over length, L of the plate and show that, qavg|y=0=

heff(T1 " T0).

3. Figure (b) shows a system in which a liquid, B, moves slowly upward through a slightly solubleporous plug of A. Then A slowly disappears by first order reaction after it has dissolved. Findthe steady state concentration profile cA(z), where z is the coordinate upward from the plug.Assume that the velocity profile is approximately flat across the tube. Assume further thatcA0 is the solubility of unreacted A in B. Neglect temperature e!ects associated with the heatof reaction and all variations in x and y directions. You are given [8 marks],

(∂cA

∂t+ 'v∗.#cA

)= DAB#2cA + RA (1)

Here, 'v∗ = v0ez, is a constant and assumed given.

CL203, Fall 2008 2

(a) Write down the reduced form of the governing equation for A.(b) What are the boundary conditions ?(c) Derive the expression of cA in terms of the known constants.

4. Suppose helium gas is contained in a pyrex tube of inner radius, R1 and outer radius, R2.Obtain an expression for the rate at which helium will leak out of the tube at steady state.You are given the di!usivity of helium through pyrex, DHe, the concentration of helium atR1, c1 and that at R2 being c2. Assume that x1, x2 << 1 and therefore neglect the convectionterms. [8 marks]

(a) Use shell balance to obtain the di!erential equation governing the mass transport ofhelium.

(b) Write the boundary conditions(c) Solve to obtain the concentration distribution and the rate at which helium is lost over

a length L of the tube.

5. An open circular tank 8 m in diameter contains benzene at 22 oC exposed to the atmosphere insuch a manner that the liquid is covered with a stagnant air film estimated to be 5 mm thick.The concentration of benzene beyond the stagnant film is negligible. The vapor pressure ofbenzene at 22 oC is 100 mm Hg. If benzene is worth Rs 20/Kg, what is the value of the lossof benzene from this tank in rupees per day ? The specific gravity of benzene is 0.88 and thedi!usivity of benzene in air is 0.096 cm2/s [8 marks]

(a) Figure for problem 2 (b) Figure for problem 3

CL203, Fall 2007 1

CL203: Introduction to Transport PhenomenonEnd-semester Exam (Total 50 marks)

Date: 28/11/2007 Time: 2:30 PM

INSTRUCTIONS: No queries will be answered during the exam. In case of doubt, make appropriate assumptions.

1. The steady-state, one-dimensonal temperature distribution in a composite wall, made of slabs of three di!erentsolid materials, is shown in figure 1. Each material has a constant, but di!erent, thermal conductivity k. Theheat fluxes q within each material are also indicated in the figure. [10 marks]

(a) What is the relative magnitude of qB and qC ?

(b) What is the relative magnitude of qA and qB at position 2 ?

(c) What is the relative magnitude of kB and kC ?

(d) What is the relative magnitude of kA and kB ?

(e) Sketch a plot of q versus x labeling the positions 1,2,3, and 4 and showing qA, qB and qC .

(f) What is likely to be the left of position x ? What else might be there ?

(g) The region to the right of slab c is a fluid with heat transfer coe"cient h and temperature far from position4 of T∞. Write an expression relating surface temperature T4, T∞, kc, and dT

dx in slab c.

(Source: From MIT website)

2. An open circular tank 8 m in diameter contains benzene at 22 oC exposed to the atmosphere in such a mannerthat the liquid is covered with a stagnant air film estimated to be 5 mm thick. The concentration of benzenebeyond the stagnant film is negligible. The vapor pressure of benzene at 22 oC is 100 mm Hg. If benzene isworth Rs 20/Kg, what is the value of the loss of benzene from this tank in rupees per day ? The specific gravityof benzene is 0.88 and the di!usivity of benzene in air is 0.096 cm2s−1 [10 marks] (Source: From Cussler’s bookon Diffusion)

3. Consider a layer of bacteria contained between two semipermeable membranes that allow the passage of achemical solute S, but do not allow the passage of bacteria. The movement of the bacteria B is described witha flux equation roughly parallel to a di!usion equation,

jB = −DodB

dz+ χB

dS

dz

where Do and χ are constant transport coe"cients. In other words, the bacterial flux is a!ected by S, althoughthe bacteria neither produce or consume S. If the concentrations of S are maintanied at So and 0 at the upper(z = h) and lower (z = 0) surfaces of the bacterial suspensions, determine S(z) and B(z). Since the bacteria arecontained, we have 1

h

∫ h0 B(z)dz = N , where N is a constant. Neglect convective e!ects. [10 marks] (Source:

From Cussler’s book on Diffusion)

4. A thin sheet of fused-silica (glass) of thickness h separates helium gas at temperature T1 from the outside airwhich is at an higher temperature To and carries negligible helium. The mole fraction of helium on the insideis xA1. It is well known that helium di!uses through the silica where the coe"cient of di!usivity, DAB isknown. In this situation both mass and energy transfer occur simultaneously across the sheet of silica. Butyour friend tells you that you have to stop the heat transfer across the helium. Further, she recommends thatyou change the inside temperature so as to match the outside temperature to achieve this objective. Do youagree with her ? If not, can this objective be achieved by some other means. If so, determine that particularvalue of T1 at which the energy transfer is completely eliminated. Assume the over-all thermal conductivity of

CL203, Fall 2007 2

silica-helium system, k to be a constant. Further, assume ideal gas behaviour and uniform pressure. Also, thephysical properties are assummed constant. The partial molar enthalpy is given as Hα = Cαp(T − Tref ). [10marks]. (Source: Self)

5. A liquid is in the annular space between two vertical cylinders of radii κR and R, and the liquid is open to theatmosphere at the top. Show that when the inner cylinder rotates with an angular velocity #, and the outercylinder is held fixed, the free liquid surface has the shape

zR − z =12g

(κ2R#1− κ2

)2

(ξ−2 + 4 lnξ − ξ2))

in which zR is the height of the liquid at the outer-cylinder wall, and ξ = r/R. The Navier Stokes equation inthe cylindrical coordinates is as follows:

ρ

(∂ur

∂t+ ur

∂ur

∂r+

uθ

r

∂ur

∂θ+ uz

∂ur

∂z− u2

θ

r

)= −∂p

∂r+ µ

[1r

∂

∂r

(r∂ur

∂r

)+

1r2

∂2ur

∂θ2+

∂2ur

∂z2− ur

r2− 2

r2

∂uθ

∂θ

]+ ρgr

ρ

(∂uθ

∂t+ ur

∂uθ

∂r+

uθ

r

∂uθ

∂θ+ uz

∂uθ

∂z+

uruθ

r

)= −1

r

∂p

∂θ+ µ

[1r

∂

∂r

(r∂uθ

∂r

)+

1r2

∂2uθ

∂θ2+

∂2uθ

∂z2+

2r2

∂ur

∂θ− uθ

r2

]+ ρgθ

ρ

(∂uz

∂t+ ur

∂uz

∂r+

uθ

r

∂uz

∂θ+ uz

∂uz

∂z

)= −∂p

∂z+ µ

[1r

∂

∂r

(r∂uz

∂r

)+

1r2

∂2uz

∂θ2+

∂2uz

∂z2

]+ ρgz

while the continuity equation is given by,

1r

∂

∂r

(rur

)+

1r

∂uθ

∂θ+

∂uz

∂z= 0.

[10 marks](Source: From BSL)

T

xx=0

qA qB qC

1 2 3 4

A B C

Figure 1: For Problem 1

CL203, Fall 2007 1

CL203: Introduction to Transport PhenomenonMidsemester Exam (Total 20 marks)

Date: 13/9/2007 Time: 2:30 PM

INSTRUCTIONS: State all assumptions clearly.

1. A Newtonian liquid is ejected out into air of a cylindrical tube of circular cross-section ofradius R in the form of a laminar jet. A steady state flow is assumed. At section (1), the flowis laminar and the velocity profile follows the Hagen-Poiseuille profile (derived in class), i.e.

uz(r)UCL

= 1−(

r

R

)2

(1)

where UCL is the centerline velocity. Upon exiting, the liquid jet contracts and the velocityprofile becomes flat some distance away from the exit (say, at (2)) with no further contraction,

uz = u∗ for r ≤ R∗ (2)

The goal of this problem is to find the precise numerical value of R∗/R subject to the as-sumption of negligible gravity and surface tension effects. [5 marks]

2. Consider the steady axial flow of a Newtonian liquid in an annular region between two coaxialcylinders of radii κR and R as shown in figure 2. The fluid is moving upward in the annularregion, that is in the direction opposed to gravity. The goal of this problem is to determinethe velocity profile and the mass flow rate (liquid density is ρ).

(a) Perform a shell balance of z−momentum around a cylindrical element while postulating,vz = vz(r), v! = 0, vr = 0, and p = p(z). At this stage, the differential equation shouldbe in the form of stresses, pressure and body forces.

(b) Replace the pressure with the modified pressure that includes effects of both the pressureand gravity. Integrate the equation to obtain the shear stress in terms of the modifiedpressure gradient, radial coordinate and the constant of integration.

(c) Now substitute the Newton’s law of viscosity, τrz = −µ(dvz/dr), and integrate oncemore. Apply the requisite boundary conditions to obtain the velocity profile.

(d) Determine the mass flow rate. [7 marks]

3. A fluid flows in the positive x-direction through a long flat duct of length L, width W , andthickness B, where L >> W >> B. The duct has porous walls at y = 0 and y = B, so thata constant cross flow can be maintained, with vy = vo, a constant everywhere. Flows of thistype are important in connection with separation processes. The goal of this problem is todetermine the velocity profile, vx(y) ?

(a) State the postulate. Starting from the x-momentum equation (given with this paper) forconstant density and viscosity liquid, show clearly the terms that survive on applyingthe postulate.

(b) Using the no-slip boundary condition at the walls for vx, show that the velocity profilefor the system is given by

vx =(P0 − PL)B2

µL

1A

(y

B− eAy/B − 1

eA − 1

)

(3)

in which A = Bvoρ/µ [8 marks]

CL203, Fall 2007 2

(1)

R*R

(2)

Figure 1: For problem 1

kRR

z

r

Flow directionFlow direction


yx

L


cl203, autum n 2009 - iit bombayphenomena.pdf · cl203, autum n 2009 1 cl203: in tro duction to t...

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