Heat Conduction in a cooling fin
ADVANCED TRANSPORT PROCESSES /
TRANSPORT PHENOMENACCB/CBB 30335. Mass TransportLesson 26:
Diffusion through a Spherical Stagnant Gas FilmCourse Outcomes
Semester May 2013CLO1Explain the theoretical aspect of momentum,
mass and energy transportCLO2Apply mathematical and numerical
methodology in analyzing momentum transfer problemCLO3Apply
mathematical and numerical methodology in analyzing heat transfer
problemCLO4Apply mathematical and numerical methodology in
analyzing mass transfer problemCLO5Analyze and solve transport
phenomena using Computational Fluid Dynamics (CFD) tools.23At the
end of the lesson the student should be able to Lesson 26. Solve
the problem of diffusion through a stagnant spherical gas film.
Lesson outcomesConsider vapor from the surface of a spherical
liquid droplet, A, diffuses through the stagnant gas film, B. It is
desired to determine the molar flux, the concentration profile and
the molar flow rate of A for constant temperature and pressure.
Diffusion through a stagnant spherical gas filmSolutionThe
problem can be solved by considering a hypothetical spherical
stagnant gas film around the droplet.4First, to get a general idea,
let me begin with the outline of my presentation.The background of
the problem and a brief overview of ozone photochemistry will be
introduced to gain familiarity with the subject.Then I will go over
some of the previous works that are based on statistical
methodologies.The environmental and meteorological data sets used
in this work will be discussed next. Here the time series of ozone
and its covariates are explored and the annual trends are
illustrated.The gist of this work lies on the application of
statistical analyses as an avenue to investigate O3 phenomena. In
particular, the results of O3 predictions based on linear
regression, kriging and stochastic simulation will be shown.Last
but not least, I will conclude by presenting the main findings of
this research and making some recommendations for future research
works.5Schematic DiagramAssumptionsB is stagnantNB=0Boundary
conditionsAt r=r1 xA=x A1At r=r2 xA=x A2
Diffusion through a stagnant spherical gas film6Mechanisms of
Mass Transfer
Combined mass transfer
(2)Since B is stagnant(1)Rearranging (2)
(3)Diffusion through a stagnant spherical gas film7Shell
Balance
(4)
Diffusion through a stagnant spherical gas filmTaking the shell
as thin as possible:
Dividing both sides of (4) by SZ
(5)(6)Integrating (6)
(7)Diffusion through a stagnant spherical gas film8First, to get
a general idea, let me begin with the outline of my
presentation.The background of the problem and a brief overview of
ozone photochemistry will be introduced to gain familiarity with
the subject.Then I will go over some of the previous works that are
based on statistical methodologies.The environmental and
meteorological data sets used in this work will be discussed next.
Here the time series of ozone and its covariates are explored and
the annual trends are illustrated.The gist of this work lies on the
application of statistical analyses as an avenue to investigate O3
phenomena. In particular, the results of O3 predictions based on
linear regression, kriging and stochastic simulation will be
shown.Last but not least, I will conclude by presenting the main
findings of this research and making some recommendations for
future research works.9Combining (7) and (3)
(9)(8)Diffusion through a stagnant spherical gas film10
Rearranging(9) we getSince c1=(1/r2) NAr from (7) , we get the
molar flux equation to be (10)(11)NB: Note that for gasses it is
common to give the partial and total pressure together with
temperature, rather than concentration in such cases (11) can be
changed as follows Diffusion through a stagnant spherical gas
film11 pA = partial pressure of A pB = Partial pressure of BP=
total pressure T= Temperature
using the above relations in (11) we get
(12)Diffusion through a stagnant spherical gas film12Integrating
(8) from r1 at composition xA1 to any r at composition
xA(13)Composition distribution/ partial pressure
distributionDividing (13) by (9) and rearranging
Simplifying (14) we get (15)
Diffusion through a stagnant spherical gas film(14)13
In terms of partial pressure(15)Diffusion through a stagnant
spherical gas filmThe total molar flow rate The flux at any radius
r is given by (11) (11)
14
In terms of pressure and temperature(16)(17)Diffusion through a
stagnant spherical gas filmMultiplying the flux (11) by the surface
area at r, i.e., 4r215Lesson 26. Solve the problem of diffusion
through a spherical stagnant gas film. Lesson outcomes15
