CBE Transport Phenomena I Final Exam. December 19, 2013

Transcription

1 CBE Tanspot Phenomena I Final Exam Decembe 9, 203 Closed Books and Notes Poblem. (20 points) Scaling analysis of bounday laye flows. A popula method fo measuing instantaneous wall shea stesses in tubulent flows is the use of an electochemical pobe. The fluid is doped with some eactant at concentation c 0 which decomposes electochemically at an electode flush with the wall. The eaction is essentially instantaneous, so the concentation of eactant at the electode is zeo, and the eaction ate is detemined by the ate with which the stuff diffuses to the wall. By measuing the cuent, we can detemine the eaction ate and hence the integated mass flux to the electode. The mass flux gets elated to the shea stess because diffusive tanspot of mass is so slow elative to momentum tanspot that the mass tansfe bounday laye essentially samples the linea shea flow ight at the wall. We thus have the following poblem: c = c 0 y x c = 0 L u ~ " # ~ c = D # 2 c ; u ~ ~ ˆ e ~ x " w µ y N y = 0 = "D # c L ; I = W " N # y y = 0 dx 0 y = 0 a. Using scaling analysis, detemine how the cuent pe unit width I/W scales with the paametes of the poblem (e.g.,! w, D (the diffusion coefficient), µ, c 0, and L). b. Estimate the time esolution of such a pobe (e.g., how long would it take fo the signal to appoach steady state in the pesence of shea tansients). Choose L = mm, D = 0-6 cm 2 /s,! w = 00 dynes/cm 2, and the popeties of wate. c. The cuent is only elated to the shea stess if the bounday laye appoximation holds. In geneal, we would like to make the pobe as shot as possible to get the best time esolution possible. What is the shotest length the pobe can be?

2 Poblem 2. (20 pts) Scaling/Unidiectional flows: Conside a hoizontal staw of length L and adius a containing a liquid with viscosity µ and density " as depicted below. The liquid is initially at est. At time t=0 we stat to blow the liquid out of the staw by applying a constant pessue diffeential #p. The length of the staw filled with fluid at any time t is given by h. In this poblem we wish to detemine the time T d equied to empty the staw in the high Re limit - e.g., how long does it take fo h to each zeo. p = p0 +!p h(t) p = p0 2a L a. Using the unidiectional flow appoximation, wite down the diffeential equation govening the fluid velocity (hint: unsteady, in geneal), keeping only the non-zeo tems. Also wite down the elation between the velocity and the change in length with time dh/dt of the column of fluid in the staw of length L. Wite down all elevant bounday conditions and initial conditions. b. Scale the equations fo HIGH Reynolds numbes, and detemine the (unknown) chaacteistic blowout time t c in this limit. What bounday/initial condition(s) have to be thown out in this limit, and what dimensionless goup of paametes has to be small? c. Solve the poblem to obtain the dimensionless non-linea second ode ODE which govens the evolution of the liquid slug length in this limit, togethe with initial conditions. This equation is tivial to solve numeically using matlab, of couse, (the dimensionless blowout time comes out to be a nice O() value of (π/2)^.5) but don t do it hee! The following equations may be helpful: % " #u z ' #t +u & #u z # + u $ #u z #$ +u z #u z #z ( ) " " u + "u # "# +"u z "z = 0 ( * = + # p ) #z + µ, # % # #u ( z ' * + # 2 u 2 z & # ) 2 #$ +# u /. z + " g - 2 #z 2 z 0

3 & " #u ( #t +u ' #u # + u $ #u 2 #$ % u $ +u z #u #z ) + = % # p, * # + µ # &. # ( -. ' ( ) # # u ) + + # 2 u * 2 #$ % 2 #u #$ +# u / #z " g Poblem 3. (20 points) Scaling/Lubication: A common way to poduce a thin film on a suface is to spin coat it - take a disk of adius R, deposit some viscous liquid of kinematic viscosity $ on the suface, and then otate it with some angula velocity %. Centifugal foces cause the fluid laye to thin out ove time, leaving a final thickness h f << R at the end of the pocess. Hee we analyze this pocedue. a). Fo thin fluid layes the otational velocity is just u & = % ove the entie film (e.g., it moves with the otational velocity of the disk). Using this, detemine the diffeential equations and bounday conditions which goven the adial and vetical velocity distibution in the film, and the equation fo the change in film thickness ove time. (Hint: How does the change in thickness with time elate to u z at z = h(t)?) It is appopiate to use h f, the final thickness, as the vetical length scale. b). Using scaling analysis, detemine how long the spin coating pocess takes as a function of the paametes of the poblem to within the usual unknown O() constant. c). Explicitly solve fo the velocity distibution and the height as a function of time to get the O() constant. You may take the initial height to be H whee H/h f >>.

4 Poblem 4. Hee we detemine the flow ate of a siphon used to dain the tank depicted below. Wate fills a tank (open at the top to the atmosphee) to the height shown, and the siphon consists of 6 metes of 4cm ID pipe (it is filled with wate too!). 7 m 8 m m a). Neglecting all fictional losses, what is the flow ate and what height (elative to the siphon outlet) would the spay fom the siphon outlet each? Give you answe in lites/s fo the flow ate and metes fo the spay height. b). Modify you answe to the flow ate and spay height by accounting fo the head losses in the pipes and fittings. Coelations fo fiction factos in pipes and fittings ae given below. You will pobably need to do a couple of iteations to get the fiction facto ight - don t foget the domain of validity of the coelations! It helps to stat the iteative calculation of the velocity with a easonable guess fo f f since you know the othe paametes. h L = u 2 2 g! K + 4 f L u 2 D 2 g ff = 6 Re ; Re < 200 f f! Re 4 4 ; 3000 < Re < 0 5 f f = 4.0 log 0 Re f f 0.40 ; Re > 3000 Fitting K value sudden contaction 0.45 sudden expansion.0 90 elbow 0.7

5 Poblem 5). (20 points) Pump Cuves / Additional Readings / Shot Answe: The fist five questions efe to the pump cuve below:. It is desied to pump 33 lites/sec of wate fom a pond to an elevation of 5 metes. If we neglect all fictional losses (say we use a eally fat pipe!) is the pump CP80i ecommended fo the job? 2. What is the RPM equied to do the job? 3. What is the efficiency of the pump at the opeating conditions? 4. What vetical distance elative to the level of the pond can we put the pump at? (Again, neglect fictional losses) 5. If we ae pumping hot wate at 60 0 C (vapo pessue of 50mmHg vs. that at 25 0 C of 23.7mmHg), and the tank is in Denve (630mmHg), what is the vetical distance? (OK, a hint: mecuy has a density of 3.56 times that of wate )

6 6. At the Mole Hole (a gift stoe on the East Race) I saw an inteesting vaiant on a Galileo s Themomete. Font and back pictues of it ae epoduced below. It is filled with a fluid, pobably wate, and one end of a chain is attached to the wall, and the othe to the floating disk (you can see this moe clealy fom the back side pictue). Biefly explain how the chain makes the themomete wok. 7. Give a physical desciption of the Reynolds stess (e.g., whee does it come fom, and how is it defined?). 8. Why do fisbees have idges? Biefly, please! 9. Fo a shea stess of 25 dynes/cm2 in the tubulent flow of wate though a pipe, about how ough does the pipe wall have to be befoe it influences the flow? 0. Explain the key diffeence between quasi-paallel (e.g., what you get fo lubication flows o the Blasius poblem) and unidiectional flows.