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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
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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 steadystate. l
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, . ___
z T(alr) Ta i? i
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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
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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
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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, QOL Q Q as T'KR f» a a} {5.4 TzTO @axo
Q “xvL. $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 postdoc 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
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———_—
__———
—————
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—__— Spring 2008 a) The tubing between BF 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.
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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
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\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“
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‘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 ...
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 Spring '08
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