Transport Problems

1

May 2012

Problem 1

The shear forces in an incompressible liquid of constant viscosity are studied in a so-

called Couette cell: the liquid is filling the entire space between two concentric cylinders, which can be rotated around their axis independently.

Consider the case where only the inner cylinder is rotating (at constant angular velocity

v(r)/r, the flow is laminar, and effects from the top and bottom of the cell can be

ignored. See equations on the next page.

A) Explain why the velocity field takes the simple form v(r, q, z, t) = v(r) e

.

B) Using part A) and the Navier Stokes equation given on the equation sheet, write

down the simplified differential equation describing v(r). How many boundary

conditions do you need to solve this equation?

C) Assuming that there is no fluid slip at the cylinder walls, solve for the velocity profile.

D) Calculate the shear force F = A acting on the outer cylinder, where A is the

cylinder surface area. (A relation between stress and strain rate is given on the equation sheet.) If you could not solve (C), then use the following expression for the velocity:

v(r) = B r [ (R

2 / r)

2

-1] with some constant B.

Differential momentum balance (Navier-Stokes) for incompressible fluids:

in cylindrical coordinates:

vPgDt

vD

2

Transport Problems

2

Viscosity: Cartesian coord. (for )

cylindrical coord. or

Problem 3

The process of aerobic respiration is modeled in the lab using a glucose sphere, which is exposed to dry air at 300 K. Assuming that oxygen in the air reacts instantly at the surface of the sphere (reaction shown below), what is the rate of glucose consumption (g/min) and how much time (minutes) will be required to deplete the glucose sphere? Assume the diameter of the sphere is 1 cm, the molecular weight of glucose is 180 g/mol, and the density of glucose is 1.54 g/cm3. Assume that the diffusion coefficient for O2 in air at 300 K is 0.056 cm2/s. Also, assume pseudo steady state molecular diffusion.

C6H12O6 (s) + 6O2 (g) → 6CO2 (g) + 6H2O (I)

z

vv

r

vv

r

v

r

vv

t

v rz

rrr

r

2

2

2

22

2

2

211

z

vv

r

v

rrv

rrrr

Pg rr

rr

z

vv

r

vvv

r

v

r

vv

t

vz

rr

2

2

22

2

2

2111

z

vv

r

v

rrv

rrr

P

rg r

z

vv

v

r

v

r

vv

t

v zz

zzr

z

2

2

2

2

2

11

z

vv

rr

vr

rrz

Pg zzz

z

xx eyvv

)(

r

v

dr

drr

dr

dvzrz

dy

dvxyx

Transport Problems

3

Problem 5

Water at a temperature of 25° C and an average velocity of 0.5 m/s flows into a square

conduit, which has a height and width of 4 cm and length of 2 m. The surface

temperature of the conduit remains constant at Tsurface = 75° C and the Nusselt number

for this process is 700.

Properties of water at the relevant temperature: = 989 kg/m3, k = 0.640 W/mK, Cp =

4180 J/kgK, = 577x10-6 kg/ms, = 1.54x10-7 kg/ms, DAB = 2x10-8 m2/s.

A) Determine the outlet temperature of the water.

B) Determine the rate of heat transfer.

C) If the inside surface of the conduit is coated with calcium carbonate, which is slightly

soluble in water with Ca* = 0.05 mol/m3, determine the concentration of calcium

carbonate in the water exiting the channel (CAL).

Problem 7

A V-shaped tank has width w (into the paper) and is filled from the inlet pipe. A) If the tank is filled at a constant volumetric rate of Q, derive an expression for the speed of filling the tank (as a function of time). You may assume the tank is very large and the pipe is very small, so the filling is slow. B) Sketch the pressure as a function of time at the bottom of the tank. Assume the tank starts empty.

Q

h

x

y

θ

Transport Problems

4

January 2012

Problem 2

In one part of a modern paper machine a continuous belt of paper passes upward through a bath with a

coating solution at constant velocity v(paper) = v p and picks up a liquid film of constant thickness ,

constant viscosity μ, and constant density ρ. Although gravity leads to some downward draining motion of

the liquid relative to the paper, the upward motion of the paper prevents the liquid from running off

completely. We will assume a fully developed laminar flow at steady state, with zero pressure gradient

and zero shear stress at the free outer surface (x = ) of the film.

(a) Discuss why the velocity in the vertical liquid film has to have the following form:

(b) Using your considerations from part (A), simplify the z-component of the Navier-

Stokes equation

to obtain a tractable differential equation for vz ( x).

(c) State appropriate boundary conditions for the differential equation. How many

boundary conditions do you need to solve the equation?

(d) Solve for the flow profile vz ( x) .

Transport Problems

5

Problem 4

Water enters a 5-cm ID pipe (surface roughness unspecified) at 80 °C and at an average linear

velocity of 10 m/min. Under these conditions, a pressure drop of 100 N/m2 is recorded over a

pipe length of 8 m. If the surface of the pipe is maintained at 35 °C, estimate the water exit

temperature. Thermo-physical properties of water, as given below, may be assumed to be

constant within the range of temperatures involved.

Density () = 1000 kg/m3 Viscosity (µ) = 4.7x10

-4 Pa.s

Specific heat (Cp) = 4.2 kJ/kg-K Thermal conductivity (k) = 0.658 W/m-K

Prandtl Number (Pr) = 3.00

May 2011

Problem 3

Gaseous A diffuses to the surface of solid B and reacts to form gaseous C and solid D according

to the following reaction: A(g) + B(s) → 2C(g) + D(s). The reaction is diffusion controlled with

constant flux of A. Develop an expression to predict the thickness of the solid D layer as a

function of time. Clearly state all assumptions.

Problem 6

On an episode of the TV show 24, terrorists release nerve gas in the headquarters of CTU

(counter terrorism unit). To save CTU, the agents propose to turn on the air conditioning to vent

the nerve gas. Of course, the air conditioners aren’t functioning properly and a number of agents

die before they can get them to work. Had CTU recruited any chemical engineers as agents, they

would have realized that the nerve gas could also be cleared by activating the sprinkler system.

Suppose the volume of CTU headquarters is 10,000 cubic meters and the initial concentration of

gas is 100 ppm. The gas levels need to be below the lethal limit (3 ppm should be adequate) to

save the day. Assuming that the gas and liquid will be in equilibrium at 32C (p = 36.4 mm Hg)

once the sprinklers have done their job, how much water must the sprinkler system dispense for

our plan to work within an hour?

The properties of the nerve gas are as follows:

Mnerve gas = 140.09 g/mol

ρnerve gas = 1.0887 g/cm3

Hnerve gas = 4.43 × 10-3

atm/mol fraction

Transport Problems

6

Problem 7

Transport Problems

7

January 2011

Problem 2

A room is contaminated with 2 mol NH3 gas. The total volume of the room is 1300 m

3. The

temperature and pressure of the air in the room is 25°C and 1 atm. To clean the NH3, dry air

with a flow rate of 1000 m3/min is blown into the room. The airflow is sufficient to make the

room air composition spatially uniform so the outlet gas has the same composition as the gas in

the room. The temperature and pressure of input and output gas are the same as the room

temperature and pressure. Air and NH3 can be treated as ideal gases. At 0°C, the specific volume

of air is 22.4 m3/K-mol).

a) Write a differential NH3 balance as a function of time by letting N be the total moles of gas

in the room (it is constant), and x(t) the mole fraction of NH3 in the room. Also calculate the

initial mole fraction of NH3 (xt=0)

b) Calculate the NH3 fraction in the room after 10 minutes of air blowing.

Problem 6

The figure shows an “air-cushion” car, which is widely used to move heavy loads over smooth

surfaces. A blower forces air under pressure into a confined space under the car. This air

supports the car and its load. Air continually leaks out through a gap between the skirt of the car

and the floor. The blower provides this air. Assume that the

car and its payload have a mass of 2500 kg, the car is

circular with a diameter of 3 m, and the clearance between

the skirt and the floor is 0.5 mm.

a) What is the flowrate of air required?

b) Assuming 100 % efficiency, how much power is required for the blower?

h = 0.5 mm D = 3 m g = 9.81 ms-2

m = 2500 kg ρair=1.169 kg/m3

Transport Problems

8

Problem 5

Substance A is undergoing steady-state transfer from a gas phase into a liquid solvent. The

concentrations of A in both the gas and the liquid are dilute. The system is at a pressure P (in

kPa), and the partial pressure of A in the bulk gas is PAb. The bulk liquid-phase concentration of

A is CAb (mol/m3). Gas-liquid equilibrium is determined by a Henry’s constant HA (mol/m3/kPa).

The mass transfer coefficients in the gas and liquid phases are kp (mol/m2/s/kPa) and kc (m/s)

respectively.

a) Derive the steady-state molar flux NA (mol/m2/s) in terms of only the quantities given

above.

b) SO2 is being absorbed into water at 2 atm. The bulk gas mole fraction of SO2 is 0.085, and

the bulk liquid mole fraction is 0.001. The system is at 50°C, at which temperature the

Henry’s constant is 8 mol/m3/kPa. The mass transfer coefficients are 5 x 10

-5 m/s (liquid) and

0.01 mol/m2/s/kPa (gas). The molar density of the liquid is 55,000 mol/m

3. What is the molar

flux of SO2 ?

c) For the same bulk conditions, it is proposed to control the molar flux by controlling the gas

flow velocity. Is this a good idea? Why, or why not?

d) What phenomenon needs to be corrected for in the case that the concentrations are NOT

dilute?

Transport Problems

9

Problem 8

A counter-current packed bed absorption column is used to reduce the levels of a

pollutant gas stream emitted to the atmosphere from a mole fraction of yin = 0.03 to yout =

0.0002. The pollutant gas stream enters the column from the bottom at a rate of G = 10.0

m3/min at the atmospheric pressure, P = 1.013 x 10

5 Pa. Pure water (molecular weight

18.0 g/mol) enters the top of the column at a flow rate of L = 20.0 kg/min. The 1.5 m

diameter column is packed with Raschig rings. You can make the following assumptions:

The pollutant gas follows Henry’s law with the Henry’s constant of H = 1.50×105

Pa.

Raschig rings provide an interfacial surface area-to-volume ratio of a = 220 m2/m

3

The overall mass transfer coefficient based on the gas phase driving force is KG =

60 mol/m2/h

The column operates at T = 40 oC.

The gas constant is R = 8.314 m3Pa/K/mol.

Ignore the volume reduction of the gas stream due to the absorption.

Ignore the pressure drop of the gas.

Based on the design requirements above, obtain the following values:

a) Mole fraction of the pollutant in the exiting water stream, xout.

b) Total height of packing required to achieve the desired separation.

Hints:

The height of a transfer function unit is

(m), where U is the total molar velocity of the

gas stream in mol/m2/h. The number of transfer unit is

, where is

the log-mean driving force difference,

. Here, yin* and

yout* are equilibrium mole fraction with the liquid that contacts the gas at the inlet and

outlet, respectively.

Transport Problems

10

May 2010

Problem 1

Problem 4

Transport Problems

11

January 2010

Problem 3

Transport Problems

12

Problem 5

Problem 7

Transport Problems

13

May 2009

Problem 1

Problem 6

Transport Problems

14

Problem 5

Transport Problems

15

January 2009

Problem 2

A spherical water droplet is suspended in stagnant dry air (i.e., velocity of droplet and air is zero, and

the air has initially no water vapor). If the initial diameter of the droplet is Di, compute the time, tevap,

it takes for the droplet to completely evaporate. Assume that the saturation vapor concentration of

water, Cs, (which is much smaller than the concentration of air molecules) and water vapor

diffusivity in air, Daw, are constant and known for the conditions of the experiment. Recall that Sh =

2 for such conditions.

Problem 7

Transport Problems

16

Problem 5

Transport Problems

17

May 2008 Problem 3

You are to measure the viscosity of a liquid. You have a relatively large reservoir with an outlet

tube on the bottom of the reservoir. The tube has a diameter of 0.18 cm, is 0.55 m long, and goes

straight downward. The reservoir has a cross-sectional area of 0.1 m2. You have 8 liters of the

fluid. By measuring the rate the fluid level falls in the reservoir, you measure the draining rate of

the fluid to be 0.273 cm3/s. What is the kinematic viscosity of the fluid? How would you get the

dynamic viscosity of the fluid? (i.e. what other equipment do you need?) You may neglect the

resistance in the constriction and the velocity of the fluid in the tube.

Unit conversion: 1000 liters = 1 m3

2

2

22

2

2

2211

z

vv

r

v

rrv

rrrg

r

P

z

vv

r

vv

r

v

r

vv

t

v rrrr

rz

rrr

r

2

2

22

2

2

2111

z

vv

r

v

rrv

rrrg

P

rz

vv

r

vvv

r

v

r

vv

t

v rz

rr

2

2

2

2

2

11

z

vv

rr

vr

rrg

z

P

z

vv

v

r

v

r

vv

t

v zzzz

zz

zzr

z

Problem 5

In a coal-fired power plant, pulverized coal is burned in air at a constant temperature and

pressure of 1450 K and 1 atm, respectively. The coal may be modeled approximately as a

carbon sphere with a radius of Ro. The mole fraction of oxygen infinitely far away from the

coal particle is XO2∞ = 0.21. The oxygen diffuses to the surface of the particle with a constant

diffusion coefficient, DA, where it reacts by the first order surface reaction equal to –ksC

a. Neglecting changes in the radius of the pellet, (i) obtain the appropriate differential equation

and (ii) specify appropriate boundary conditions for this situation to enable determining the

radial concentration distribution of oxygen and carbon dioxide assuming complete

combustion.

b. Use your result from part (a) to derive an expression for the mole fraction profile, XO2(r) as

a function of distance from the center of the coal particle. Your results contain ks, DA, R

and XO2∞, but not the value XO2 (Ro), since this is not known explicitly.

Transport Problems

18

Problem 6 A 100 gallon tank is initially filled with fresh water (density = w = 8.33 lb/gal). A stream of salt water

containing 1.92 lb/gal of salt flows at a fixed rate of 2 gal/min. The density of this incoming solution is

71.8 lb/ft3. The solution, kept uniform by stirring, flows out at a fixed rate of 19.2 lb/min. How many

pounds of salt will there be in the tank at the end of 1 hour and 40 min? There are 7.48 gallons in 1 ft3.

2 gal/min

CS=1.92 lb/gal salt

= 71.8 lb/ft3

19.2 lb/min

Transport Problems

19

Problem 8

A square metal pan (12 x 12”, 0.5” deep) is exposed to saturated steam at 1 atm and 100°C. The bottom

of the pan is in good thermal contact with cooling water at 70°C, which keeps the temperature of the pan

constant. The cooling water tubes and outside surface of the pan are well insulated, so that condensation

occurs only on the inside of the pan.

A) Sketch a qualitative graph that presents the condensation rate m (kg/s) in the pan as a function of

time for three different pan orientations: horizontal (θ = 0°), vertical (θ = 90°), inclined (θ = 45°). For

each case, an empty pan without condensate is exposed to the humid air at time t = 0.

B) For the horizontal pan position (θ = 0°), the condensate slowly fills the pan and the thickness of the

condensate layer is . Derive an expression for the rate d dt at which the

pan fills up as a function of the layer thickness , the temperatures T∞ and Ts, and physical

properties selected from the list below. You may assume that natural convection does not

occur in the condensate layer.

Do not calculate a numerical answer!

Physical properties of potential interest:

Density ρ, viscosity , surface tension , dielectric constant r,

heat capacity cp, latent heat of vaporization hfg, thermal conductivity k.

Transport Problems

20

January 2008

Problem 2

Molecule B is to be produced from reactant A via the reaction 10A B. This reaction is carried out

inside a catalytic membrane of thickness L (m). The membrane permits flow of A through it with

practically zero mass transfer resistance, whereas it is impermeable to the large product molecule B. The

support side of the membrane is maintained at concentration )/( 30 mmolCA of reactant A, whereas the

permeate side is maintained at perfect vacuum. Reactant A diffuses through the membrane from the

support side and reacts inside the membrane to form B, which diffuses out and is recovered at the

permeate end. The diffusivities of the two species in the membrane are AD and )/( 2 smDB . The reaction

is first-order in A: AA CsmreactedAmolr )//( 3 , where is the rate constant. The membrane reactor is

operated at steady-state conditions.

(1) Develop the differential equation for the concentration of A ( AC ) at any location x in the membrane.

(2) Solve the differential equation to obtain the concentration profile of A.

Hint: try a solution of the form kxkxA eCeCxC 21)( , where kCC ,, 21 are constants.

(3) Obtain the production rate of B per unit membrane surface area (mol B produced/m2/s) at the permeate

side. Your answer should contain only physical parameters from the set 0,,,, ABA CLDD .

Problem 3

A tank is used to heat oil by saturated steam, which is condensing in steam coils at 40.0 psia. Oil flows in

and out of the tank at a rate of 1018.0 lbm/h. The tank, which is perfectly mixed by a stirrer, contains

5000 lbm of oil initially at 60°F. The temperature of the inflowing oil is also 60°F. The rate of heat

transfer from the steam to the oil is given by Newton’s heating law,

Q = U(Tsteam - Toil)

Where Q is the rate of heat transfer in Btu/h and U is an overall heat transfer coefficient. Calculate the

time in hours it will take for the discharge from the tank to rise from 60°F to 90°F and the maximum

temperature that can be achieved in the tank.

Additional data:

Power of stirrer motor: 1.0 hp; 75% of this power is delivered to the oil.

U= 291 Btu/(H∙°F)

Cp,oil = 0.5 Btu/(lbm∙°F)

1 hp = 2547 Btu/h

Water saturation temperature at 40 psia is 130.7°C = 267.2°F

Transport Problems

21

Problem 5

The mass transfer of liquid A via evaporation into an inert carrier gas B must be characterized for a

practical application in which both diffusion and convection are hypothesized to be of importance. In

order to quantify this hypothesis, two laboratory experiments are performed under the same conditions:

1) Evaporation of liquid A from the bottom of a narrow cylindrical tube. Pure carrier gas B is blown

over the top of the tube at velocity v∞. The distance from the liquid surface to the top of the tube is

L = 25cm at the start of the experiment (left figure).

2) Evaporation of liquid A from a flat pan with diameter d = 10cm. Pure carrier gas B is blown across

the pan at velocity v∞ (right figure).

An analytical balance is used to monitor the rate of evaporation. When (pseudo) steady state is reached,

the molar flux of A (i.e., moles per time per unit area) from the pan is found to be 10 times larger than

from the tube.

A) Derive an equation for the molar flux of A from the narrow tube.

B) Calculate the Nusselt number: Nu = kc d / DAB, which defines the relative importance of

convective and diffusive mass transfer at the gas-liquid interface.

Additional information:

The evaporation process takes place at T = 293K and P = 1 atm.

The vapor pressure of A, Pvap,A, is much smaller than 1 atm.

The generalized diffusion equation for (pseudo) steady state diffusion:

0AN , where 0AN is the molar flux

Fick’s rate equation for a binary gas (A diffusing in B):

A AB A A A BN c D y y N N ,

where ABD is the diffusion coefficient, ctot the total molar concentration and yA the molar fraction of A.

The expression for convective mass transfer: NA = kc A

d

tot

Transport Problems

22

Problem 8

You are asked to evaluate a proposed gas sensor to detect combustible gas mixtures. The sensor concept

is shown below. The gas mixture, air with a little combustible gas, flows over a small cylinder whose

surface is covered with catalyst. When any combustible compound reaches the surface it reacts rapidly

and completely with oxygen in the air releasing heat. The temperature of the cylinder, which may be

assumed constant (that is the Biot number

k

hL is much less than one) is then correlated with the

concentration of combustible gas in air.

a) Explain the Chilton-Colburn analogy, DH jj , where

3

23

2

ScandPr

cD

p

H

kj

C

hj

b) Using the Chilton-Colburn analogy, develop a relationship among the concentration in the gas, the heat

of combustion, physical properties, and the temperature difference between the cylinder and the bulk gas.

c) If the minimum temperature difference that can be measured reliably is 0.1 °C, what is the detection

limit for CO. Physico-chemical properties are given below.

DAB=1x10-5

m2/s Cp=1005 J/kg-K p=10

5 Pa

-5 Pa-s

3 T=298 K

comb=-3x105 J/mol Pr=0.7281

T=25 C

Air with small amount of

combustible gas,

atmospheric pressure

v

Transport Problems

23

May 2007

Problem 2 I2 (A) is being sublimed into still air (B) from a small iodine spherical particle of outside diameter, do, at

110 mm Hg and 90 °C, where the vapor pressure of iodine is 22 mm Hg. Neglect changes in do, in

your derivations and assume that YA* is known at a radial distance equal to r* = 100ro .

a. Derive a differential equation for molar flux through a spherical shell in the gas phase surrounding the

particle. Substitute for the flux expression that accounts for the non-zero contribution due to bulk flow

since the gas phase iodine mole fraction YA >> 0. This should provide a differential equation in terms

of gas phase mole fraction of A.

b. Indicate appropriate boundary conditions to allow solving for both the flux and YA(r) profile as a

function of distance from the surface of the sphere at ro = do/2, but you do not need to solve for the

NAr(r) or YA(r) profiles.

c. Assuming one has solved the above equations for NAr(r) or YA(r) profiles, so that they are known,

indicate how to estimate the instantaneous rate of sublimation of the particle (gm/sec) if physical

properties of solid I2 [ g/cm, Mw g/mol and DAB are all known].

do

I2

I2 in air

I2 in air r

Transport Problems

24

Problem 4

Consider the process for mixing orange juice concentrate and cranberry concentrate to produce cranberry-

orange juice. Assume that the tank is well mixed. The purpose of this problem is to set up the mass

balances and solve for the unknown functions P, xJ(t) and xC(t). Initially (t≤0) the tank is fed only pure

water (W=50 kg/h) at steady state (C0=J0=0), and hence the mass in the tank and the outlet is pure water.

The tank is filled with M = 1000 kg water initially and a level-control device keeps the overall mass in the

tank constant. For t >0, the J and C streams are opened to start feeding juices to the mixer at 25 kg/h each

Problem 8

Dry air enters a rectangular cavity. The cavity walls are maintained wet and at a constant temperature,

so that the water vapor pressure at the walls is uniform throughout the cavity. Assume that the flow

velocity profile is uniform, equal to U, throughout the cavity. Derive an approximate expression for the

length le required for the water vapor concentration to develop to a steady state profile, as a function of

key transport parameters. If you can, express the result in terms of the Reynolds and Schmidt numbers.

Also assume that air has a constant density, ρ, and viscosity, μ, and that the water vapor diffusivity in

air, Dw, is constant. Recall that the timescale of diffusion is 2L / Δ , where L is the lengthscale over

which diffusion takes place and Δ is the appropriate diffusivity.

M=1000

kg

W =50 (water, kg/h)

P (kg/h)

xJ(t)

xC(t)

J = 25 (orange juice

concentrate) (kg/h)

xJ,J = 0.95

25 (cranberry juice

concentrate) (kg/h)

xC,C = 0.9

xW,C=0.10

Transport Problems

25

January 2007

Problem 2 In the following diagram, fluid flows out of the storage tank (with a large diameter) into the small pipe

(diameter of the pipe is 0.3 cm). If the fluid has a constant density and viscosity is flowing out at a rate of

1 cm3/s, what is the kinematic viscosity of the fluid? In fact, this is a common way to determine the fluid

viscosity experimentally.

Here are some equations that you may find useful.

0)(

CVCS

dVt

dAnv

FdVv

tdAnvv

CVCS

)(

CVCSdVe

tdAnve

dt

W

dt

Q

)(

where ugyv

e 2

2

2

2

2

22

1

1

2

11

22

Pvgy

Pvgy

2

32

D

v

dx

dP avg

0

v

t

PgDt

vD

0 v

vPgDt

vD 2

h2=60 cm y

Patm

Patm

h1=10 cm

Transport Problems

26

Problem 6

Consider the problem of diffusion of reactant in a sphere of porous catalyst. The radium of the sphere is

R. We may assume that concentration varies only in radial direction. The concentration of the external

fluid phase is uniform everywhere and remains constant at Cex

. The reactant inside the sphere is being

consumed through the first order kinetics of r=kC.

(a) By performing a shell balance, derive a PDE that describes the dynamics of the

concentration profile C(r,t).

(b) Suggest appropriate boundary condition. Assume that the mass transfer at the interface

(between the fluid and catalyst surface) is very fast compared to the diffusion process.

(c) Non-dimensionalize the equation and the boundary conditions using u=C / Cex

and s=r/R.

Note: The volume and surface area of a sphere of radius R is (4/3) πR3 and 4πR

2 respectively. Use Dr

as the diffusion coefficient.

Problem 7 A small spherical particle is released into the atmosphere, and it falls through the air toward the ground.

Treat the atmosphere as an ideal gas with pressure equal to100 kPa absolute and 298 K. The drag (both

friction and form) on the particle is given by Stokes law,

RVFD 6 ,

where µ is the viscosity, R the particle radius, and V , the velocity.

A) Use a force balance to develop an equation that relates the terminal velocity,

V to physical

properties of the air, the density of the particle, gravitational constant, and particle size.

The figure below shows the terminal velocity based on this Stokes relationship, as well as experimental

data. It is seen that at both small and large diameters departures from the predicted values exist. For large

diameter particles the Stokes relation over predicts the velocity. This is due to a transition from laminar to

turbulent flow.

B) Using the data below, at approximately what Reynolds number does this occur?

C) In contrast, for very small particles the Stokes relationship under predicts velocity. Propose an

explanation.

Transport Problems

27

May 2006

Problem 2 A capillary viscometer is commonly used to measure viscosity of a fluid. The viscous or frictional losses

(expressed as “head loss” in meters) in a smooth tube are given by the Hagen-Poiseuille relationship

4

8(m)

Rg

QLhL

Where Q is the volumetric flowrate; R the radius of the capillary, and L the

length of the capillary tube.

If h is the distance from the surface of the fluid to the

end of the capillary, use the Bernoulli equation (general energy balance) to

show that

LQ

g

VhgR

8

2

2

24

V2 is the velocity of the fluid exiting the capillary. What assumptions have

gone into this analysis? Under what conditions might this method provide an

inaccurate measurement of the viscosity?

Problem 3

The effective diffusion coefficient of component A in a cylindrical layer of material, B, is equal to DAB,

and this coefficient is a known constant. The density of the layer may be assumed to be constant and

equal to .

The mass fraction of A at the inner surface of the pipe at r= r0 is known to be wA0, and is greater than the

mass fraction of A the outer surface at r = r1, which is wA1. Both mass fractions, wA0 and wA1 , are

known. Steady state diffusion conditions prevail, and there is no reaction within the pipe wall.

a. Consider the case in which the mass fractions are fairly high and flux includes

contributions due to bulk flow. This case arises due to the observation of the flux, nAr(r), relative to a

fixed frame of reference. For a section of the pipe that has a length, L, determine an expression for

the :

i. mass flux of A through the inside wall at r = r0 .

ii. total rate of mass flow of A at r = r0 in the radial direction assuming no end effects.

b. For the special limiting case in which wA0 and wA1 << 1, so contributions due to bulk flow, which

arises due to the observation of the flux, n Ar, relative to a fixed frame of reference are negligible

determine:

i. the mass flux of A as a function of r.

ii. the fraction profile in the radial direction as a function of r, wA(r).

L

h

Transport Problems

28

Problem 6

Fluid flows down a vertical surface. The flow is driven by gravity and the liquid has density ρ and

viscosity µ.

A. Show by shell balance that g

dx

dxz

. Substitute in the shear stress/velocity relation to derive

an equation relating gravity and velocity.

B. Show that Navier-Stokes equation reduces to the same equation.

C. Assume pseudo steady state, the air has a negligible viscosity, and the liquid film has a thickness

. Calculate the velocity profile. (Define the wall to be x=0.)

Navier-Stokes: vPgDt

vD 2

Cartesian coordinates:

2

2

2

2

2

2

z

v

y

v

x

vg

x

P

z

vv

y

vv

x

vv

t

v xxx

x

x

z

x

y

x

x

x

2

2

2

2

2

2

z

v

y

v

x

vg

y

P

z

vv

y

vv

x

vv

t

v yyy

y

y

z

y

y

y

x

y

2

2

2

2

2

2

z

v

y

v

x

vg

z

P

z

vv

y

vv

x

vv

t

v zzzz

zz

zy

zx

z

wall liquid air

x

z

g

Transport Problems

29

January 2006

Problem 3

A polymer of constant density and viscosity is processed through an annular die. The inner

radius of the die is r1 and the outer is r2. For steady, laminar flow, ignoring entrance/exit effects

and neglecting gravity, determine the velocity profile in the annular space between r1 and r2.

a) Using the continuity equation and assuming that the only nonzero velocity component is in the

z direction, show that. ).(rvv zz

b) From the Navier-Stokes equation simplify the momentum balance for the z direction to an

ordinary differential equation. What are the boundary conditions and what is the physical

significance of these boundary conditions?

c) Integrate to determine the velocity profile. You don’t need to evaluate the constants.

The divergence in cylindrical coordinates is

Navier-Stokes equation for constant density and viscosity

Transport Problems

30

Problem 5 To treat severed (damaged) nerves, a new biomedical material is used that releases growth factors (GF)

that promote healing of the nerve ends. The growth factor is bound to a gel contained within a tube which

is placed on each end of the severed nerve (see figure below). The success of the treatment depends on the

“bound” growth factor (bGF) being present at sufficiently high concentrations throughout the treatment

period. Unfortunately, “bound” growth factor (bGF) can dissociate to “free” growth factor (fGF), which

diffuses away from the gel into the body through both nerve endings. It is known that:

i) the dissociation rate for liberating bGF is proportional to the dissociation rate constant kD, and

the concentration of bGF Cb. Also

ii) the association rate for the binding of fGF is proportional to the association rate constant kA,

the concentration of fGF Cf and the concentration of available binding sites in the gel - Cb.

Assume that L, kD, kA, the diffusion coefficient of fGF in the gel DfGF, the initial concentrations of bGF,

fGF and the total concentration of free and occupied binding sites in the gel , are known.

a) Write governing equations that describe the concentrations of bGF and fGF as functions of time and

position in the guide tube. Suggest initial and boundary conditions as well. Assume axial diffusion only.

b) Assuming that bGF and fGF are always in equilibrium, and simplifying the equations you wrote down

in part a, what is the characteristic timescale for depleting fGF at the center of the tube guide?

L

x

x

x

x

Nerve Guide

Tube (NGT)

Gel containing bound

Growth Factor (GF)

One part of the

damaged nerve

The other part of the

damaged nerve

Transport Problems

31

Problem 7

Consider the co-current heat exchanger pictured above. Assume that the outer tube is insulated, and that

the temperature of each fluid is uniform in the radial direction due to mixing. Neglect conduction in the

fluid. U is the heat transfer coefficient between the inner and outer fluid. Wn represents the mass or

molar flow of the fluid in stream n.

1) Derive the energy balance on fluid 1 for a small element of width .

2) Do the same for fluid 2.

3) Let go to zero and write these equations as differential equations.

4) The solution of this coupled system of equations takes the form

2211

22

11

11

)(

)(

pp

Cx

Cx

CWCWdUC

eBAxT

eBAxT

Verify that these general equations could satisfy your energy balances from part 3.

d

T1(x)

T2(x)

T1*,W1

T2*, W2

Cp1

Cp2

x

Transport Problems

32

May 2005

Problem 3

Problem 4

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33

Problem 5

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34

Problem 7

January 2005

Problem 4

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35

Problem 5

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36

Problem 6

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May 2004

Problem 5

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38

Problem 6

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39

January 2004

Problem 5

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40

Problem 6

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41

Problem 8

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42

May 2003

Problem 6

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43

Problem 8

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44

January 2003

Problem 4

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45

Problem 8

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46

May 2002

Problem 3

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47

Problem 4

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48

Problem 7

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49

Problem 8

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50

January 2002

Problem 2

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51

Problem 4

May 2001

Problem 1

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52

Problem 7

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53

Problem 8

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54

January 2001

Problem 3

Problem 5

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55

Problem 8

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56

January 2000

Problem 1

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57

Problem 8