Transcript nptel.ac.in
Advanced Transport Phenomena
Module 9 Lecture 40
Students Exercises: Numerical Questions (Modules 6-8)
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
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Students Exercises: Numerical Questions (Modules 6-8)
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NUMERICAL PROBLEMS
Module 6: Convective-diffusion mass transfer plays an important
role in the fabrication of small solid-state electronic devices,
which often start with the "chemical vapor deposition" (CVD) of
micron-thickness films of crystalline Si(s) grown on a sapphire
(Al2O3) substrate ("wafer").
a. Use the material studied in this course to make a preliminary
quantitative estimate of the average silicon deposition flux
(mg/min/cm2) to a flat,
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NUMERICAL PROBLEMS
growing wafer exposed to a parallel flow of SiH4(g)/H2 gas
mixture under the following conditions:
U =50 cm/s, T= 300 K,
L = 5 cm
Tw= 1273 K,
ySiH 4
p = 1 atm
, = 0.005 (mole fraction).
Assume that the Si-deposition flux is determined by SiH4(g)
("silane"-) vapor transport to the wafer surface, at which:
SiH4 g Si( s ) 2H2 g
That is, assume y, SiH4 ,w ySiH4 , but neglect the possible
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NUMERICAL PROBLEMS
complication of SiH4(g)-decomposition in the vapor phase
near the hot wafer. List and quantitatively defend each of
your important assumptions and property estimates.
Identify the "weakest links" in your estimates (for possible
future investigation). Ans. 405 mg/min/cm2.
b If the estimated equilibrium Si(g) vapor pressure over
Si(s) at this surface temperature is about 1.1 x 10-8 atm,
can the "physical" sublimation rate of silicon be neglected
under these CVD-reactor operating conditions?
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NUMERICAL PROBLEMS
c. Evaluate the natural convection/forced convection ratio
parameter
1/ 4
h
Gr
1/ 2
L
/ Re
and comment on the possible
importance of natural convection (caused by heat
transfer) in augmenting (or suppressing?) the masstransfer coefficient
Nu m,SiH4
if the heated wafer is
horizontal, and the gas mixture flows past above it.
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NUMERICAL PROBLEMS
Note: While the Stefan-mass flow ("suction") associated
with Si-deposition is negligible in this example (since
Si / SiH4 SiH4 1 the
"blowing" effect associated
with thermal (Soret) diffusion is appreciable (over 20%
reduction in
Nu m,SiH4 ) and should be taken into account
(Rosner (1980)).
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NUMERICAL PROBLEMS
Module 7: Application of the Mass/Energy Transfer
"Analogy"
and
Scale-Model
Theory.
Consider
the
following solution to the problem of scale-model testing
that involves heat-transfer measurements on a cluster of
2 – mm diameter rods in a water tunnel. This is based on
the feasibility of making dissolution (convective mass
transfer)-rate measurements on a cluster of 5-mm
diameter rods exposed to the flow of liquid water at 25°C.
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NUMERICAL PROBLEMS
a. Given the physical data below, systematically work out
how you would use benzoic-acid dissolution-rate data to
predict the heat-transfer performance of the full-scale
tube bundles, also circumventing the need for expensive
full-scale furnace tests. Defend the validity of each of
your underlying assumptions and itemize the
experimental precautions you would have to take. In your
opinion, does the present mass-transfer analogy
approach have any decisive advantages or drawbacks?
Basic data: Pure benzene carboxylic acid
(commonly called "benzoic acid") is a solid (mp 122°C,
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NUMERICAL PROBLEMS
density = 1.266 g/cm3) which is sparingly soluble in 25°C
water (0.0278 g-moles/liter H2O). Its diffusion coefficient
has been measured to be 1.00 x 10-5 cm2/s in 25°C water
(Table 2.6 of Sherwood, Pigford, and Wilke (1975)).
b. If the simultaneous role of natural convection cannot
conveniently be “simulated” in your mass-transfer model
experiment, is it possible to dispense with Grm- similarity
altogether?
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NUMERICAL PROBLEMS
Examine this possibility on the basis of available
experimental data for an isolated cylinder under the
simultaneous
influence
of
forced
convection
and
("opposed") natural convection. (In that case how much
is
Nu h ,
decreased below its "pure" forced convection
counterpart) when
1/ 4
h
Gr
1/ 2
/ Re
is as large as expected
in our "prototype"?)
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NUMERICAL PROBLEMS
c. Why in Part (a) above, is it reasonable to test with Rem =
Rep, assuming the dependence of Nu on Pr (or Sc) is
"known" (from isolated cylinder data)? Would it be
reasonable to also dispense with the “set-up rule”: Rem =
Rep by assuming that the Re-dependence of Nu (tube
bundle) is the same as the Re-dependence of
Nu (isolated cylinder)? Are any experimental data
available which would allow you to compare the Redependence of Nu for both an isolated cylinder and, say,
two cylinders "in tandem"?
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NUMERICAL PROBLEMS
d. In a previous solution, no mention was made of the
possible importance of simultaneous natural ("free")
convection in the full-scale furnace application.
Investigate this possibility by first calculating the
magnitude of Grh and then forming the important ratio
1/ 4
1/ 2 (governing the relative importance of free
Grh / Re
and forced convection). Is this ratio small enough to
consider natural convection negligible, or will it be
necessary to also do model testing at the same Grashof
number as for the full-scale prototype?
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NUMERICAL PROBLEMS
In the benzoic-acid dissolution model (mass-transfer
analog) experiment considered in this exercise, what is
the value of (Grm)m given the fact that the density
difference between benzoic-acid-saturated water and
pure water is only 1.42 x 10-3 g/cm3 at 25°C. To force
(Grm)m =(Grh)p what value of dm would be needed, and
what water velocity, Um would now be needed to preserve
Rem= Rep? Would the Pr and/or Sc-dependence of
be
known under theseNumore complicated (combined forced
convection and natural convection) conditions?
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NUMERICAL PROBLEMS
Module 8: By stating and where possible, numerically
evaluating
the
relevant
dimensionless
ratios,
quantitatively defend the following approximations made
in solving
the illustrative problems of heat transfer and ash fouling
in a pulverized-coal combustor
a. Continuum approximation,
b. Neglect of viscous dissipation,
c. Neglect of natural convection
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NUMERICAL PROBLEMS
With additional information it would also be possible to
defend (or relax) the:
d. neglect of mainstream turbulence
e. neglect of effects of neighboring heat-exchanger tubes
f. neglect of deposition on the cylinder by “eddy impaction”
g. neglect of radiative energy gain (from gases and
suspended ash)
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NUMERICAL PROBLEMS
8.2 Suppose, for structural-design reasons, you wanted to
estimate the drag force per unit length of the heatexchanger tube. Could this also be done based on the
Nu h Re,Pr - relation using the so-called "analogy"
between momentum and energy transfer?
8.3 In estimating the submicron particle (ash) mass-transfer
rate we exploited the (a) mass-energy transfer "analogy"
(i.e., Nu h Re,Pr and Nu m Re,Sc - are the same functions).
Can this analogy be derived from "dimensional analysis"?
Similitude analysis? (b) proportionality Nu h ~ Pr1/ 3 for large
Prandtl numbers. Can this proportionality be derived from
"dimensional analysis"? Similitude analysis?
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NUMERICAL PROBLEMS
8.4 Verify the effective "capture efficiency" figures quoted in
the course for submicron particle collection by the heatexchanger tube under the conditions treated above. How
does it compare to that estimated above for inertial
capture of 20 m m particles?
8.5 Show that the particle Stokes' number can be
re-expressed in terms of the prevailing values of the Mach,
Reynolds', and Schmidt numbers in accordance with:
2( Ma )2
Stk
( mg / mp )( Sc )p (Re)'
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NUMERICAL PROBLEMS
where cp / cv and mg / mp is the ratio of individual
gas molecule-to-particle mass. Can analogous
momentum nonequilibrium phenomena be observed for
mixtures containing trace amounts of a high-molecularweight vapor (e.g., WF6(g)) in a low-molecular-weight
carrier gas (e.g., H2(g)) under continuum flow
conditions (see, e.g., Fernandez de la Mora, J. (1982))?
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