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Transport in fluid mixtureit’s a take home examI will send you the notes when the question is assign it.ONLY BID IT IF YOU CAN HANDLE IT
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Columbia University – Chemical Engineering
CHEN E4112 Y Spring 2024
Transport in Fluids/Mixtures
Take Home Midterm
Professor C.J. Durning
March 20, 1:00 PM – March 22, 5:00 PM EST
Information/Instructions.
Solve only TWO problems; work on a third problem will not receive
credit.
Submit your answers through the course website in .pdf or .docx format
with your name/uni cearly indicated in the …le title by the due date/time;
a word processed document is preferred.
Prepare your own solutions; collaboration and/or use of AI tools are not
allowed. Documents will be screened for duplication/plagiarism/use of AI
tools.
Unless otherwise indicated, symbols in what follows have their usual meaning (e.g.
means the ‡uid density, etc.)
You may use results from references if needed, provided you properly cite
the source.
1
1. CONSTITUTIVE DEVELOPMENT FOR RIGID MEDIA INCLUDING A NON-LOCAL EFFECT: Consider an isotropic rigid
media with the density
assumed to be a …xed constant. Unlike the case
discussed in class, we will consider that the media’s local state depends on
high order gradients of the temperature as a way of exploring non-local
e¤ects. Recall for rigid media treated in class, the primary …elds were
and T . Here we consider
to be a …xed constant so that the only
primary variable is T (r; t), governed by the energy balance
@ ^
U +r q=0
@t
and subject to the second law
@ ^
S+r s
@t
0
^ ; q;S;
^ s are the non-equilibrium (local valued) speci…c internal
where U
energy, conductive energy ‡ux, speci…c entropy, and conductive entropy
‡ux, respectively. The temperature …eld T appears implicity in the
above; for example, the local internal energy must depend on the local
temperature. To ensure that T has the usual signi…cance at equilibrium,
we include the relation
^ T S^
A^ = U
where A^ is the local speci…c free energy. This de…nes the local temperature.
(a) Determine the degrees of freedom (DOF) in the above framework
(b) The DOF can be satis…ed by the constitutive laws
^
A^ = A^ (:::) ; S^ = S(:::)
q = q(:::) ; s = s(:::)
where (:::) are frame indi¤erent arguments derived from T (r; t)
assuming that only the current state matters. Recall, to realize the
feature of “local action”, we included rT among (:::). To model
a non-local e¤ect one could include higher order gradients of T , for
example
(:::) = T; rT; rr2 T
Consider the set of “linearized” (i.e. “weak gradient”) constitutive
laws in this class of materials:
A^ =
S^ =
A^ (T )
^ )
S(T
q
=
kq1 (T )rT
kq3 (T )rr2 T
s
=
ks1 (T )rT
ks3 (T )rr2 T
2
(note: this is NOT the most general representation, but a possible,
relatively simple one). Adopt a weak assumption that A^ = A^ (T )
has the same functional relationship with T
as at equilibrium;
^
@A
^
= S . One can then apply Liu’s theorum
consequently
@T
to determine the constraints on the above constitutive functions imposed by the second law. Recall one …rst creates an “augmented”
inequality by adding the balance laws as constraints, each multiplied
by a Lagrange multiplier. In this case:
@ ^
S+r s
@t
T
@ ^
U +r q
@t
0
Then inserting the constitutive laws, the de…nition of T and applying
the chain rule leads to
!#
”
@ A^ ^
@T
@ S^
T
T
+S
1
T
@T
@T
@t
T
+
+rT
T
kq1
rkq1
ks1 r2 T +
T
ks3 r2 r2 T
kq3
rks1 + rr2 T
T
rkq3
rks3
0
In the context of the class discussion of Liu’s theorum, what are the
X in this case? According to Lui’s Theorum, which terms in this
inequality have to be set to zero? State why this is so, and show the
consequences of setting these terms to zero are
T
=
1
1
; s= q
T
T
(c) Show that after the conditions in part b). are determined, the following inequality still needs to be satis…ed
h
i
kq1
kq3
2
2
(rT ) + 2 r2 (rT )
2
T
T
0
Explain the additional conditions needed to satisfy this inequality.
3
2. DIRECTIONAL SOLIDIFICATION OF A LIQUID NEAR THE
MELTING POINT. Controlled solidi…cation of liquids is of great importance in many applications. We want to model the solidi…cation of a
+
liquid …lm just above its melting point (say at Tm
) moving at constant
speed V along the x direction past a stationary heat sink with a …xed
temperature T0 < Tm located at x = 0. At steady state a solid-liquid
interface appears at some …xed distance x =
> 0 ahead of the heat
+
sink. For x >
the media is liquid at temperature Tm
while for
0 0) is
insulated. Viscous dissipation causes heating of the ‡uid and therefore a
temperature increase in the insulated section. Model the steady temperature distribution T (r; z) in the far downstream portion of the tube for
large P e, where “far downstream” is de…ned by
z0 =
z
RP e
O 100
with R being the tube radiius.
(a) Apply the FCET model to this situation to …nd a di¤erential equation
and associated boundary conditions governing T (r; z) for 0 < r < R
and
1 < z < 1. Its helpful to remember that the velocity
distribution for an isothermal, incompressible Newtonian ‡uid in fully
developed steady laminar ‡ow in a tube is
v
=
((0; 0; vz (r))
vz (r)
=
vmax 1
r
R
2
(b) A natural scaling of the independent variables for the far downstream
behavior is
r
z
r0 =
; z0 =
R
RP e
To scale the temperature properly use
T0 =
T
T0
T
with
T determined by balancing the downstream energy convection term with the energy dissipation term. What
T results?
Show that the resulting scaled energy eqution reduces to
1
for P e >> 1.
equation?
r02
@T 0
1 @
‘ 0 0
@z 0
r @r
r0
@T 0
@r0
+ 4r02
What are reasonable boundary conditions for this
(c) Use the method indicated on p 163 in V&O, or that in Deen Example
10.5-2, to …nd the downstream temperature distribution
5
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