Department of Materials Science and Engineering
Massachusetts Institute of Technology
3.044 Materials Processing- Spring 2011
Exam II
Wednesday, April27, 2011
The Rules:
1) No books allowed; no computers allowed; etc.
2) A simple calculator is allowed
3) One hand written 3x5 index card may be prepared as a crutch .
4) Complete 5 out ofthe 6 problems. If you do more than 5 problems, I will grade the first
5 that are not crossed out.
5) Make sure that you READ THE QUESTIONS CAREFULLY
6) Supplementary materials are attached to the end of the test ( eqns ., etc.)
7) WRITE YOUR NAME HERE:
s 0 l u -rt s
0 ('J
Problem #1: Swimming for Their Lives (the particles, not the students)
In continuous casting, one major problem is that small
inclusion particles can be caught in the casting. Let's
analyze this problem , for small inclusions of AI 20 3 (which
are actually quite common in steel).
. - + - --me lt
Particle 'A'
The key consideration is that the alumina particles float in
molten steel. For such a particle in the melt region at point
'A', the particle is (a) trying to float to the top due to the
buoyancy force (F = ~p·g·V , with V the particle volume),
(b) but being propelled downward at the casting velocity Vc
(= I m/min) at the same time.
so lid metal
Part A:
\
Write down a reasonable and simple condition that decides whether the particle will eventually
escape at the top surface, or be carried down and get incorporated in the metal in the sump.
.
I 4'
~ Vel OL. 4-/ {!r0 jNI hv::>yc,J;t c! 1"3, () (~ #,...... /i."'-"- 1-0. Jo (;) Y1 ,vd k(_ J
"
v
__./~
v
v(_
>
+tat -
Part B:
~
--
_____./
ftq_( w ,?l fla-..+
~......
oJI-
Let's calculate the velocity at which the particle would rise. Write down the force balance for
particle 'A', and indicate with what velocity it rises. (For this part of the problem, ignore the Vc
effect.)
Fdr(.__j::: :S1i;v
Vfla-t
cL
L:>y· ~ 37t(;)3 ~
'
.
-
' I vn
~
Part C:
v~Jo<-+ d
3-rrI
2_
ov ·l\ ·d
V~lo ...f ::
J .1 (e_vr._,~J2
;~ jJ
Some particles will escape, and some will not- calculate which size particles will escape and
which will not. You may need the information at the back ofthe test, and the viscosity ofthe
meltis7·10-3 Pa ·s.
./
5-e + v+lo.-+ :: Vc
6o\VL
,._;
for d_
Icvs~r d -Pio . . t
S v1--l l-tr J
5t)t
~
J _,-
-
'h.
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;v
.o
\b r-"ts
bJ~
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L-bo
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Prohlcm #2: A Question you arc in
H
Super Position to Answer
It i-. Slllllctimes possihk tu usc the principle or-.upcrpnsition for fluid now probkm .... (01 course
thi" is no '-'Urprise. seeing the pervasive analogies between lluw and di!Tusinn.)
l3clow an.: SL)fllC common situations in nuid now. Exp lain whether or not you can usc
supcrpn..,ition to sulvc J'or the flow J'i cld in each case. If superposition is possible, explain what
geometries/solutions are "-Uperposed. II it is nol pnssible. explain why nul.
Part 13) Gravity drives !low down a
l.
s
fk_
b--'+
/,
r~.::ctangular chanlll:l open to the air on top.
w~-..l h
0"'..
1/t--=--
)
~
~
~
d:t
-h (.I
c)
6
ue
( <-4"1,
+ us~
'51Vfl>S ,' h'J"'.
l'at·t.C) A cylinder or asp~.::ct ratio I is filled with water. c losed, and spun on its axi". lcuding tu
rotaltllllal !low or the liquid inside .
F!
t
~
J Yl f.-,_
I
CJW
A-Jl
I')
bou!rldt-i,'e 5
So
0"1(7
c...fe_
I,A·rv.... -for
So-L+.
6)"'
5~
1'>--
lT
+t re:
cy {,\ul v
~ -fvo
i)
t?•
C
•<.J"\.
r~ =
0
~~.),(h..)
-fwtJ-h)L~
(r ~~ +m rff.. )
Problem #3: The Students Face Down a Real Phase Diagram
c c. l-; C--s (. q__
•
<::.><.
I
/
I
Alo m u: Perce nt Ca bo l\
You are gtven a rod
o
10
~ ao IV Jo
co
~o
eo
1800
of titanium that
ot
contains I at% cobalt,
~
\,
and asked to purify it.
1600
''
Looking at the Ti-Co
L
' '
''
phase diagram, you
'
liOO i
reason that this alloy
•
''
~
'
'\
could be zone refined, '=
exactly as discussed
'\
... 1200
.,
in your 3.044 class.
''
You place the rod into .:
TIC.
{It!)
!,-"' 1000
an induction coil and
pull it through slowly,
locally melting and
BOO
resolidifying the rod
as the hot zone passes
over it.
......... . .... - ...
GOO
.
~
~
0..
.. ..
I .
I
..
---,--L-.....J,..---'UI.~U-..-.....,.....--'-"-
10
0
10
!1(1
Yic•l i! h: Pe r ce n t
T1
80
10
100
80
Co l><~ lt
Co
At the end of one
pass, you analyze the leading end of the rod- this is the first part of the rod to be melted and
resolidified.
Part A:
From the phase diagram, you estimate that the alloy composition at the end of the bar should be
about 0.6 at% after one pass through the zone refiner. Explain this estimate.
1<. ~
13
/ 2..,_
(!
L
.y,'("j1
.-\?ra~
f k. . .se
IIIA _Jnrn (
~v"'-Tt:.-11._
h
d.;-4-~l.:..~
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COVj05t-<ho,
}t_ C0
1't-
? 1
()_(,
Part B:
Ito
You are shocked to find, however, that the end of the bar is quite substantially purer than you
first suspected it might be! The composition is ~0.1 at% Co. Explain why the rod is so pure at
this end, and why specifically it is at a value near 0.1 at% Co. (Note: your measurements are
made after the rod is cooled back down to room temperature) .
c)
/1
-~
f
~ 5"eco~'~...J [ kc...s e
~ Dl ·kvt s tc.~" Vh--ko ._
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--
Problem #4: Which Driving Force is Doing the Driving?
In class, a long time ago, we looked at the problem of glass fiber
production. We explicitly looked at the required cooling rate to
keep a new fiber of diameter 20 11m, drawn from the fluid , from
breaking up into droplets before it became rigid . A schematic of
the process in shown at the right.
Molten glass
T ~ ~L
lcm
Now, back when we looked at this problem, we knew heat
transfer but not fluid flow . Let 's talk about fluid flow this time
around.
Information: the material is glass at 900° C, 11 = I 02 Pa ·s, p =
2500 kg/m3, y = 0.5 J/m2 .
•
Part A) There are two driving forces for flow of the molten glass through the aperture. Name
them , and determine which is dominant.
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@ ~r~v , ~··(·
tl
8
J';J(+!.S)vfe_
F.p -;:
looJ( +'ort.e. ·
FJ -: : j _j
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b_P - J>Cjh «-- I""'- ~
L ~
L-
c.~
..t-'
~ore t~orh . -+~
Part B) There is one driving force for the breakup of the stream into drop lets : capillary stress, a
2y/r. Compare it to the driving forces from Part A and convince yourse lf (and me) that it is
significant in this process. Pay attention to units here.
=
()=
c_o~ <I~
·Jo
ckJ~d-eA3k<_ /e . . ~il 5c . . /e ·
~ - ?:> 0 '<I o q IJ/,..._J
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-
$u.rfc__ce
..J--e.., .s ,~."- 1 s
v~1
Co~.eff,~)
~r~
Problem #5: Stretchy or not so Stretchy?
Polyethylene is a very nice
material to form into shapes with
vi scous forming operations. At
the right are data for the rheology
of polyethylene. NOTE: These
data are plotted on
LOGARITHM IC SCALES . The
slope tells you about the strain
rate sensitivity.
Part A) For a fixed forming rate
(a shear rate of, say, I 0 s-1) , hi gher
temperatures lead to more stable
flow . Provide an estimate of the
tensile ductility at 300° C vs. 120°
N
e lr:i
u
.....
41)
cu
c:
>-
~
en
en
1.1.1
0:
I-
en
0:
ct
1.1.1
:X:
en
10 1
c.
10
103
2
SHEAR RATE (sec- 1 )
Flow curves at various temperatures for a low density
polyethylene.
[Reproduced from R. Mendelson, Polymer
Eng. Sci., ~..
lo~
235
(1968)].
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·-c_::
slo~e
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+-
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fh:=
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6
3~() I
5
y'L
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jz_o"c
~f ~ z_oa%
3l)t/ c
2·-+ ~ 1-(6o /o
Part B) As a process designer, you would like to operate at the lowest possible temperature and
the highest possible rate. Explain briefly how, with the help of this data, you would identify an
optimum processing condition? What data would you need to do such an analysis?
~rork ti.e_ pc..r·~ 5~o~~~ we WoJ)d ltc_x,e ~ .dc'S,Y~J s+r?t,-~ _/ ~+·ftt~
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In
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c.Jaove
fvih J,e-hi-t-5 tk 5'/ ~~e
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Pmhlcm #6: Amazing Brazing
..
...
Copper
L31li/e allo\
-
I
/
_,.~- I
13
A
The prncess or hra;.ing i-. C'.Selltia ll y like ca-.ting: il melt is placed between two nhjn·ts to he
joined : tile melt then solidifie-; _ holding the uhjcch tugethcr . These pil'lure-. show cro-.s--.cctiun-.
llf hr:11.e j11illts het\\cent\\·u copper piece-. (\\.hich arc on the top and hntlonl) .
I !ere arc three variables in this process: (i) braLe al loy composition, (ii) thickness of the hr:11.c
region. (iii) temperature or the copper sheets
Part.'\) lr ( i) <IIlLi (ii) arc exactly the same between the twu cases A and
I) ,
L'xplain how (iii) must
he dirkrcnt het\Vl'ell thl'lll . Provide tln l' '>elllencc or physical explanation .
.
•
tt1 .
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• I
Part B) If (ii) ami (iii) arc exactly the \arne between the two cases A a11d 13. exp l <~in how (i) lllll'>l
he dil"lcrelll llL'tWL:ell them. Pruvide lllle seil!CilCC or physical explanatinn .
ftvt. ~ .
u.ulereo o l.~
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3.044 Materials Processing
Spring 2013
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