midterm2_solution - PGE 322K-— TRANSPORT PHENOMENA Spring...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PGE 322K -— TRANSPORT PHENOMENA Spring 2008 EXAM 2 April 9, 2008 Except where noted, do all'calculations in SI units BEWARE OF UNNECESSARY INFORMATION. DO NOT SPEND TOO LONG ON ANY ONE PROBLEM. DO NOT LEAVE ANY PROBLEM BLANK! YOU CAN START ANWERS FROM EQUATIONS IN BSL, JUST GIVE THE EQUATION NUMBER T otal:100/120 pts Spring 2008 1. True or False questions. (25 points) a) In a fractured reservoir most of the fluid is in the porous rock, but most of the flow /[email protected]\) E conduits are fractures. b) When a shear thinning fluid is injected into a reservoir, its effective viscosity gets greater as it moves away from the well bore. «(Mg c) Metals with higher electric conductivity typically have higher thermal conductivity. (“Q C d) For two different substances with the same mass, the material with the higher heat capacity will heat up less when a certain amount of heat is added to it. 4 (20 g, e) Energy can is only transported in fluids through conduction and convection. (,{XKSE Spring 2008 x f E ‘7 z i i r 4 2. (40 pts) Consider a long wire of length L and radius R (R<<L). A current is applied through the wire, and heat is deposited in the wire with energy density per time of Se. The wire is in contact with the surrounding air at a temperature T=Ta, and the wire loses heat according to Newton’s law of cooling with heat transfer coefficient h. The temperature at the z = 0 end is held at T= To. The other end of the wire (z=L) is perfectly thermally insulated. You can make the estimation that the wire is much thinner than its length, and that the temperature in the wire is uniform radially (T=T(z)). The wire has properties k, A Cp , and p . The current through the wire is applied for long enough time so we can assume we are at steady-state. l l i l 5 2 t l i T(z=0) = T0 T(air) = Ta , . ___ z T(alr) Ta i? i é ; 9 i i a) Write down the boundary conditions in terms of T. Spring 2008 b) Set up a shell balance for the conductive heat flux in the Wire. From this, obtain a differential equation for the heat flux. You do not need to solve this differential equation. 3% 5C %1 L 9 j kfiKL. (go 1 kQT ‘TK\ c) Using what you know about heat conduction (i.e. the governing equation), and with the flux equation that you solved for above to obtain a differential equation for steady—state temperature T(z). You do not need to solve this differential equation. 1 - \Lfl E? (kit ‘ (FT 1 5 _~ slur-TA doc“ ’ 6’- Spring 2008 i E i i i E d) If the wire is sufficiently long; then for a sufficiently large 2 (i.e. far enough away from the z=0 end), the temperature will be independent of z. Derive the expression for the temperature in this region. _. "LT \ d»._, ._ A ”7:67“ " «DEL ’ :7 363% CT; 0 1 SU’) CT (EA ‘* SLR ._ sea M TAO Qua +\°L' e) Extra credit (10 pts): Calculate (the solution to this diff eq is in BSL, but with different variables) or estimate the distance 2 Where the temperature calculated in 1b is a good estimate (i.e. the z where T(z) is within a couple of percent of the difference between T0 and T(infinity)). This is may not be easy, only attempt if you have finished the rest of the exam). EQ, Q-OL Q Q as T'KR f»- a a} {5.4- TzTO @axo Q “xv-L. $00 ,. ilk/m 3: 501043»: T 1 “1% =— Q Q v99»! com (3’ LTOr’TDQ\ 4r TM Spring 2008 MT _ Akfi~TJ _ 5Q 1k ”“13 KR 3. (45 pts) During my post-doc at Stanford University, I needed to measure the permeability of a sand pack. The flow was provided by a pump with a constant flow rate. The brine (viscosity = 0.01 Pa 3; density = 1000 kg/m3) was pumped into and out of the sand pack through flexible tubing of 4mm diameter. The sand pack was cylindrical with a diameter of 8 cm, and a length of l m. A manometer was attached at the inlet, and the outlet of the tubing was such that it dripped into a volumetric flask. See figure below. The lengths of tubing between junctions and the absolute height of junctions (relative to the floor) are given in the following table. By measuring the water in the graduated cylinder versus time, you determine that the volumetric flow rate through the core is 6 ml/min (= 10'7 m3/sec). You can rip out this page for ease of reference during the calculations. B Pump 3 i g i s i l2 ”- ———_— __——— ————— __——- ————— -—__— Spring 2008 a) The tubing between B-F is basically a manometer. During the flow the water level at F is 1.6 m above the ground surface. Calculate the head difference (in Pa) of the brine between points F and E. <29? ‘. we... )tcTVt“ @E 'PigbrTv—x ','1’€-”\‘1L“ LL-LO Ha ,. Hg: 63 «we.» 2 omflgsfl Q DH =-» K? who Pa b) Now assume there is no head drop in the tubing due to the flow (the head drop between F and E is the head drop between C and D, or that the tubing is infinitely 3 conductive). Write down the equation that relates the flow through the core to the head drop; then calculate the permeability of the sandpack (core) in m2 and Darcy using this 2 E equation. If you are not confident in your head drop use AH=1x104 Pa in your E E i calculations. . :5 9 -3“, C/ ‘lXxo \c‘ “(5%“ 5;; - 75 2 K 5512‘. . 9 mm .’> Wu " L “L #- TTC‘lo/x\ AS-Qc-wfl’isxlb ”‘1 «A .. «I O Q L \Q Ci. 10 ‘——\—/ ,1 3—... M1 _.—. ___.. 1 ”k \L 3 A FLAH Sue’s “X93 0 Spring 2008 c) The assumption that tubing is infinitely conductive (no head drop) is, of course, wrong. By taking into account the tubing will the actual permeability of the sand pack be higher, lower, or the same as our initial estimate? Why? 30 Ltruafi, > KQST' d) We will now calculate the actual permeability. Use the actual size of the tubing (4 mm in diameter) and calculate the actual head at points B, C, and D relative to the head at point E. Assume laminar flow in the tubing. (Hint: Start at point F for points B and C, start at point B for point D). Hi5: Ht; CHO (Reno) 3 lHP‘HQ" iil<t0 9g ? (bi: 8 / I Tr a“ [A \ L LWM bHu‘ 3 "Q_%NCLX(O_3)L( —<\ 3 x a: if?» 2 if, '2: §xtc pa, (gmo’w— Z , _ 7. @ strut: Lécttm ,QM‘ZQM, '35 bu”; Sumo R 'L __.__,————-——-—-———""”" ‘3 Spring 2008 e) Is the assumption of laminar flow in the tubing correct? [JV LO . ‘5}, ’3 «g - ‘3 W ‘L M/ imo M/ (V) '- -—- Z 4’ 1 = ‘ 3 S TTRZ‘ BQLMO 3 ‘1 ( ’1 ‘ , (q =lhocmtmmo" "a: Elmo “:9 Lfimwfrfl , f) Now with the actual head drop between C and D (keep with your assumption of laminar flow in part (1 even if it is false), calculate the actual permeability in Darcy and m2 of the sand pack. . 3 ‘DHCO: AHce'bHoe,= 3ND 9:. (row swam 0.. ”\L 1 7;;- f" 23H l «J :9, mo‘w‘ «a g: D g) Assume the porosity of the pack is 0.33. Estimate the grain size of the sand. If you do not trust your answer above, use a permeability of 10D. 3 Z K1t.°rxi$°‘w¢.: L12 74,1, K V?) 5 ’lb 91 mass ( yfilw‘mul Cintw'd‘cuw : M : C/3\ f ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern