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Some problems about transport phenomena (molecular and
convective behavior)
Ruben D. VargasWalter J. RosasAngel A. GalvisMayra P. Quiroz
Laura CalleWatson L. Vargas
Departamento de Ingenierรญa quรญmicaUniversidad de los Andes, Bogotรก D.C. , Colombia
Outline Introduction
Drainage of liquids
Transient diffusion in a permeable tube with open ends
Heating of a semi-infinite slab with variable thermal conductivity
Conclusions
Introduction
Drainage of liquids
J.J. van Rossum, Appl. Sci. Research, A7, 121-144(1958)V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, N.J. (1962)
Wall of containing vessel
Initial level of liquid
Liquid level moving downward with speed s
x
y
Drainage of liquids
Wall of containing vessel
Initial level of liquid
Liquid level moving downward with speed s
x
y
๐ฟ ( ๐ง , ๐ก )=โ ๐๐ ๐
๐ง๐ก
When time tends to infinite๐ฟ ( ๐ง , ๐ก )=0
At the initial time
๐ฟ ( ๐ง , ๐ก )=โ
Drainage of liquids
When time tends to infinite๐ฟ ( ๐ง , ๐ก )=0
At the initial time
๐ฟ ( ๐ง , ๐ก )=โ๐ฟ ( ๐ง , ๐ก )=โ ๐
๐ ๐๐ง๐ก
Drainage of liquids
Unsteady-state mass balance on a portion of the film between z and z + ฮz to get:
Accumulation= in- out
Drainage of liquids
Itโs dividing by
Lim ฮz 0
Drainage of liquids
With the following assumption:
We obtain:
Drainage of liquidsTaking the terms to one side of the equation
Supposing that viscosity and density remains constant
We can obtain this first order differential equation:
Drainage of liquids
Is clear:
So,
We need solve this equation:
๐ฟ ( ๐ง , ๐ก )=โ ๐๐ ๐
๐ง๐ก
?
๐ ( ๐ง ) h๐๐๐ก
+๐ ๐๐
๐ 2 ( ๐ง )h2 (๐ก ) ๐๐๐๐งh (๐ก )=0
Replacing
๐ฟ2
Drainage of liquids
๐ ( ๐ง ) h๐๐๐ก
+๐ ๐๐
๐ 2 ( ๐ง )h2 (๐ก ) ๐๐๐๐งh (๐ก )=0
๐ ( ๐ง ) h๐๐๐ก
+๐ ๐๐
๐ 2 ( ๐ง )h3 (๐ก ) ๐๐๐๐ง
=0
h๐๐๐กh3(๐ก )
=โ๐ ๐๐
๐ ( ๐ง)๐๐๐๐ง ?
Drainage of liquidsSo we can solve h(t):
๐=โ๐ ๐๐
๐ (๐ง)๐๐๐๐ง
h๐๐๐กh3(๐ก )
=โ๐ ๐๐
๐ ( ๐ง)๐๐๐๐ง ?
With a โbeautifulโ substitution!
Drainage of liquids
๐=โ๐ ๐๐
๐ (๐ง)๐๐๐๐ง
From :
Solving to f(z):
This equation can be write as:
Is possible to arrange the terms and integrate
Drainage of liquids
In summary:
We obtain:
Heating of a semi-infinite slab with variable thermal conductivity
x
y
y=0; T1
y=โ
The surface at y = 0 is suddenly raised to temperature T 1 and maintained at that temperature for t > 0. Find the time-dependent temperature profiles T(y,t) Thermal conductivity varies with temperature as follows:
๐๐0
=(1+๐ฝ )( ๐ โ๐ 0
๐1โ๐ 0)
Heating of a semi-infinite slab with variable thermal conductivity
Dimensionless heat conduction equation:
Heating of a semi-infinite slab with variable thermal conductivity
Replacing , we can obtain:
Heating of a semi-infinite slab with variable thermal conductivity
Heating of a semi-infinite slab with variable thermal conductivity
Heating of a semi-infinite slab with variable thermal conductivity
Heating of a semi-infinite slab with variable thermal conductivity
๐ (๐ )=1โ 32
๐+12
๐3
Heating of a semi-infinite slab with variable thermal conductivity
Heating of a semi-infinite slab with variable thermal conductivity
Using uniqueness
Heating of a semi-infinite slab with variable thermal conductivity
Heating of a semi-infinite slab with variable thermal conductivity
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