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# Transport Phenomena I CBE 30355

ND

GPA 3.57

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This 0 page Class Notes was uploaded by Nash Klein on Sunday November 1, 2015. The Class Notes belongs to CBE 30355 at University of Notre Dame taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/232707/cbe-30355-university-of-notre-dame in Biomolecular Engineering at University of Notre Dame.

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Date Created: 11/01/15

CBE 30355 Transport Phenomena I Final Exam December 17 2007 Closed Books and Notes Problem 1 20 points Parallel Plate Flow A controlled stress rheometer is depicted below It works by imposing some specified torque to the upper plate and then measuring the resulting angular velocity The viscosity of the uid between the rotating upper plate and stationary lower plate is proportional to the ratio of the torque and the resulting angular velocity a Solve for the relationship between the torque T and the angular velocity S2 in terms of the gap width h the plate radius R and the uid viscosity M For this system it is appropriate to assume low Reynolds number flow greatly simplifying the problem b Experimentally the measured gap width often has some error in it principally due to errors in the quotzeroquot as you examined in a problem earlier this semester If the actual gap width is h hmeas Ah explicitly show how you can determine the correct viscosity by measuring S2 for two different measured gap widths h1 and h2 at the same torque T You may find the following equations helpful 3VB 1 aVz Problem 2 20 points A Variant on Hero39s Fountain a Neglecting all frictional losses what is the height hf of the fountain water jet b Modify your answer by accounting for the head losses in the pipes and fittings Correlations for friction factors in pipes and fittings are given below You may take the total length of pipe to be 120 cm of 05 cm ID tubing You will probably need to do a couple of iterations to get the friction factor right c The result in part b leads to a rather unimpressive fountain What is the most practical way of fixing this and why does it work IL I Elli um ED um quotl 4D 1 sII V W 2 2 hL2K4 ff e 113 66Relt2100 ffz 4 3000ltRelt105 71 24010g10Ref 040 Regt3000 Fitting K value sudden contraction 045 sudden expansion 10 45 elbow 035 Problem 3 20 points Transient Boundary Layer Scaling You are asked to design an electrochemical probe capable of measuring both concentration and wall shear stress next to a wall The idea is that fluid flows along the surface of the wall containing some concentration c0 and then at the surface of the probe we electrochemically reduce the concentration to zero over the whole length L lnitially you develop a transient boundary layer independent of x and then at long times you develop a steady x dependent boundary layer independent of t The total reaction rate per unit width extension into the paper QW can thus be used to determine both c0 and 13W by examining these two limits a Scale the equations and boundary conditions at short time scales tc to determine how the transient boundary layer thickness 6t and QW depend on the parameters of the problem b Scale the equations and boundary conditions at long times to determine how the steady boundary layer thickness 6 and QW depend on the parameters of the problem in this limit c The transition between the short time and long time behavior occurs for some tc where the boundary layer thicknesses in a and b are of the same order What39s this tc d If our electrochemical measurement system requires a time of 1 second to determine a decent measure of the reaction rate the diffusivity is around 10396 cm2s and the wall shear rate is 100 s 1 what is the minimum length L of the probe What would be the characteristic boundary layer thickness under these conditions 6c 1W bc 6c 5c at ya 6x2 V L Q 6c D dx W y O Problem 4 30 points Pump Curves Additional Readings Short Answer The first seven questions refer to the pump curve on the last page 1 It is desired to pump 100 literssec from a pond to an elevation of 60 meters If we neglect all frictional losses say we use a really fat pipe is the pump HH150 recommended for the job 2 What is the RPM required to do the job 3 What is the mechanical work done by the pump g the fluid per unit time 4 What is the efficiency of the pump at the operating conditions 5 How far up the hill from the level of the pond can we put the pump Again neglect frictional losses Note 1atm 103 m water 6 Frictional losses always add to the required head What additional head losses can we tolerate before the pump is unable to achieve the required flow rate 7 It is proposed to use a 10cm diameter pipe for this system If we include just the losses due to the initial contraction and acceleration of the uid how does the answer to question 5 change 8 The displacement thickness is defined as 6 I 0 1 u d 0 U y Provide a brief physical interpretation of this quantity 9 Brie y describe one method for preventing stall on an aircraft wing 10 Give a physical description of the Reynolds stress eg where does it come from and how is it defined 11 In our class demonstration when we allowed a rod to settle in a viscous liquid Re 2 0 we observed that it didn39t rotate If inertial effects are important it does Why is the behavior different in this case 12 For a shear stress of 16 dynescm2 in the turbulent flow of water through a pipe about how rough does the pipe wall have to be before it influences the flow 13 Why do dimpled golf balls and fuzzy tennis balls have less drag than their smooth counterparts One sentence please 14 Provide two interpretations of u 15 Provide two interpretations of pu M 391 1W L WHPJ LUW Pump Cesxng SB PUN Sucnpnvaer SB PUN AwSeperenpnTenK SB PUN Beenng Bree et Pump Shaft 431 Stewn ess Stee mpeHer 316 Stewn ess Stee eerp etes 316 Stewn ess Stee Mecnemce Seek Human Swhcon N W S 8 PW nwm HEMP Pump Descnpnpn Smg e Stage Mute type 3 b eded My Open Centn uge ump Sucnpn Henge 200 mm Dehver y Henge 150 mm Npmne Cesxng anckness 18 mm Sands Hendhng Swze 38 mm Uperenng Speed M N MUUrpm MM QUUUrpm Mexwmum Head 98 m M 2 Mn 1 7 Mexwmum Cepecwty WEULsec wum 1250 mm Hexgnt 1700 mm Length 2750 mm Dry Wexgnt 1850 kg Fue Upnsumptmn 15 5 m d QHE SEXKS CBE 30355 TRANSPORT PHENOMENA I TTh 1100 Second Hour Exam Closed Books and Notes Problem 1 30 points Scaling Lubrication Theory Consider a disk of radius R located a distance H above a plane as depicted below Fluid of viscosity u is pulnped into the center of this geometry at a rate Q volulne thne and ows outward radially This ow results in a pressure gradient and hence develops a high pressure underneath the disk applying an upward force on it In this problem you are to determine this force You may simplify the problem by looking at the lubrication limit H R ltlt 1 a j39 a Using a mass balance determine the characteristic magnitude of the radial velocity b Via scaling analysis of the equations of motion determine the characteristic magnitude of the pressure in the lubrication limit c How does the magnitude of the force depend on the paraIneters of the problem e g get the force to within an unknown but hopefully order one constant d Solve for the velocity distribution pressure distribution and force to get this constant The following equations may be helpful note the vast majority of these terms may be neglected a 1 Vr 1 Dve DvZ Dr E0 tlH pDvr Dvr Eavr DvrDp vr vZ 7 Dt Dr 139 be 139 Dz Dr 3 23 3 DV 3 M i l7r vr inzrlzir 7 p gr Dr 139 Dr r be r quotbe 02 p DvZJr rDvZJrvieD ZDvZ E Dt Dr 139 D6 Dz Dz 1 D DvZ 1 DzvZ D2vZ if 7 H 139 Dr I Dr r2 362 322 p gz Problem 2 30 points Two Dilnensional Flows Consider the twodilnensional geometry depicted below A very viscous uid is draining out of the bottom of a tank through a narrow slit at a rate Q W per unit extension into the paper l39n2 a Set up the problem as completely as possible with all boundary conditions clearly determined b Using the boundary conditions as a guide determine the form of the streamfunction and develop the corresponding ODE and boundary conditions Hint remember the relationship between the net ow rate through any curve connecting two streamlines and the value of the streaInfunction on the streamlines c Solve for the streamfunction Make maximuln use of symlnetry Remember that in the cylindrical geometry we have 31p 131p vs ar Vr was and Problem 3 10 points Short Answer Identify the following equations and state under what conditions they are valid aV41p0 quot0147 b E70 boil 0 C quotbx dE4zp0 Problem 4 10 pts Dimensional Analysis You are assigned the task of estimating the drag on a 300 meter long aircraft carrier moving at 20 m s roughly 40 knots or 40 nautical miles per hour You choose to do this using a towing tank with a 1 meter long model a The dilnensionless drag depends on two key dimensionless parameters what are they and what is their approxilnate magnitude for the fullsized ship Hint If you can39t remember them you can easily get them by appropriate scaling of the Navier Stokes equations b Using the concept of approximate rather than strict dynamic similarity determine the velocity at which the model should be towed to best approxilnate the ow pattern of the fullsized carrier What are the values of the i1nportant dimensionless paraIneters for the model c What is the ratio of the drag on the carrier to that measured for the model CBE 30355 TRANSPORT PHENOMENA I TTh 1100 Second Hour Exam Closed Books and Notes Problem 1 45 points Scaling Unidirectional Flow In this problem we are exalnining the drainage of a tube of radius R and length L initially filled with a uid of density p and Viscosity u as depicted below The uid is initially at rest eg we have plugged the end of the tube with a finger and at t 0 we release the fluid to drain out of the tube We want to calculate the drainage time Td In the limit R L ltlt 1 we may assulne unidirectional ow in the axial direction We also ignore any surface tension effects turbulence film drainage issues or other nastiness A V A R gt L ht PM g r r V V a Apply the Buckingham H theorem to this problem to determine how Td depends on the various parameters of the problem Try to pick reasonably physical dilnensionless groups there are many correct answers here but some are obviously better than others b Using the unidirectional ow approximation write down the differential equation governing the uid velocity hint unsteady in general keeping only the nonzero terms Also write down the relation between the velocity and the change in height with time dh dt of the column of uid in the tube of length L Write down all relevant boundary conditions and initial conditions c Scale the equations for HIGH Reynolds numbers and determine the unknown characteristic drainage time tC in this limit What boundary initial conditions have to be thrown out in this limit and what dimensionless group of parameters has to be small d Solve for the actual drainage time in the high Reynolds number limit e g solve the dimensionless equations obtained in part c to get the numerical value e Scale the equations for LOW Reynolds numbers and determine the new unknown characteristic drainage time tC in this limit What boundary initial conditions have to be thrown out in this limit and what dimensionless group of parameters has to be small f Solve for the actual drainage time in the low Reynolds number limit eg solve the dimensionless equations obtained in part e to get the numerical value g If we let R 05cm L 2 ft and we have the properties of water which limit is more appropriate Be quantitative The following equations may be helpful note the vast majority of these terms may be neglected 13rvrlaVZiO r Dr r 36 DZ 7 a VrDVZViea VZDVZ E at Dr 139 36 DZ Dz 2 2 13 sz 1 sz sz if 7 Mir Dr I Dr r2 362 322 W Problem 2 15 points Short Answer Multiple Choice a 10 pts Brie y identify the physical mechanism described by each of the following terms 2 V6 3 pr vv 4 r6 p r Duii SE70 Dui Dui 3P bui FtpujaX j39 itla 931 6 10 each term 9 2 X j b 5 pts Multiple Choice 1 It is proposed to study the ight dynamics of a fruit y by building a10cm scale model What dimensionless number do you have to keep constant to preserve dynamic similarity A Reynolds Number B Prandtl Number C Froude Number D Weissenberg Number 2 In ship modelling we try to preserve which dimensionless number A Reynolds Number B Prandtl Number C Froude Number D Weissenberg Number 3 Which if any of the following can be discontinuous at a uid uid interface A Shear rate B Heat ux C Mass ux D Velocity 4 You are struck by a desire to solve the equations of motion for the followin systems For which of the following would you use the viscous scaling give all that apply A A golf ball in ight B Mixing corn syrup C The USS Enterprise D A glucose blood monitor 5 Crooke39s Radiometer works because of A The momentum of light B Thermal transpiration C Maxwell said it should D Hot gas on the black face of the vane CBE 30355 TRANSPORT PHENOMENA I First Hour Exam 10207 This test is closed books and closed notes Problem 1 15 pts A mass M is sitting on a plate as depicted below balanced on a vertical jet of water If the volumetric ow rate of the jet is Q and the crosssectional area of the nozzle is A we can do a force balance on the plate Don39t forget that the jet slows down and spreads out as it travels upwards For a jet like this Bernoulli39s equation resulting from a mechanical energy balance in the absence of friction provides a good relationship between velocity and height 1 2 1 2 59u1P1pgh1 puz Pzpgh2 Given all this calculate the steadystate height H of the plate and determine the minilnuln jet ow rate Q necessary to support the plate M Problem 2 10 pts The lower atmosphere may be regarded as adiabatic thus we have P pY where Y 14 is the adiabatic exponent of a diatomic gas Given that the density of the atmosphere at sea level is 12 kgm3 the pressure at sea level is 101X10 5 Nmz and the acceleration due to gravity is 98 m s2 solve the equation governing hydrostatic pressure variation and calculate the altitude where the density drops to half the value at sea level Problem 3 10 pts Starting from the arbitrary stationary control volume depicted below derive the continuity equation for a compressible uid 5D Problem 4 10 pts In this problem we design a hydrometer a simple device for measuring the density of a liquid usually used in deterimining the salt concentration A simplified version is depicted below and consists of a bulb of volume Vb and density pb and a column of length LC cross sectional area AC and density pC Given this determine the relation between the height h the column sticks up out of the water as a function of psw the density of the salt water solution we are interested in and the other parameters in the problem What is the range of densities we can measure using this device Psw Vb Pb Problem 5 15 pts Index notation Additional Readings Multimedia CD questions 1 1 pt Flow visualization using the timelapsed photography of a tracer particle technique reveals which of the below for unsteady ows A Pathlines B Streaklines C Streamlines D All of the above 2 1 pt Flow visualization using the dye release technique dye ejected from a fixed array of nozzles reveals which of the below for steady flows A Pathlines B Streaklines C Streamlines D All of the above 3 2 pts Provide two interpretations of u 4 2 pts Provide two interpretations of pu 5 3 pts Using index notation prove that the divergence of the vorticity is zero 6 2 pts Write down the continuity equation for an incompressible uid using index notation 7 2 pt Using index notation write down the representation of the symmetric part of the general matrix 8 2 pts Match up the kinemch viscosities of the following materials 1 Water A 650 cSt 2 Air B 170 cSt 3 Glycerine C 10 cSt 4 Mercury D 0118 cSt

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