transport phenomena problems
TRANSCRIPT
Some problems about transport phenomena (molecular and
convective behavior)
Ruben D. VargasWalter J. RosasAngel A. GalvisMayra P. Quiroz
Laura CalleWatson L. Vargas
Departamento de Ingeniería químicaUniversidad de los Andes, Bogotá 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
𝛿 ( 𝑧 , 𝑡 )=√ 𝜇𝜌 𝑔
𝑧𝑡
When time tends to infinite𝛿 ( 𝑧 , 𝑡 )=0
At the initial time
𝛿 ( 𝑧 , 𝑡 )=∞
Drainage of liquids
When time tends to infinite𝛿 ( 𝑧 , 𝑡 )=0
At the initial time
𝛿 ( 𝑧 , 𝑡 )=∞𝛿 ( 𝑧 , 𝑡 )=√ 𝜇
𝜌 𝑔𝑧𝑡
Drainage of liquids
Unsteady-state mass balance on a portion of the film between z and z + Δz to get:
Accumulation= in- out
Drainage of liquids
It’s dividing by
Lim Δz 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:
𝛿 ( 𝑧 , 𝑡 )=√ 𝜇𝜌 𝑔
𝑧𝑡
?
𝑓 ( 𝑧 ) h𝑑𝑑𝑡
+𝜌 𝑔𝜇
𝑓 2 ( 𝑧 )h2 (𝑡 ) 𝑑𝑓𝑑𝑧h (𝑡 )=0
Replacing
𝛿2
Drainage of liquids
𝑓 ( 𝑧 ) h𝑑𝑑𝑡
+𝜌 𝑔𝜇
𝑓 2 ( 𝑧 )h2 (𝑡 ) 𝑑𝑓𝑑𝑧h (𝑡 )=0
𝑓 ( 𝑧 ) h𝑑𝑑𝑡
+𝜌 𝑔𝜇
𝑓 2 ( 𝑧 )h3 (𝑡 ) 𝑑𝑓𝑑𝑧
=0
h𝑑𝑑𝑡h3(𝑡 )
=−𝜌 𝑔𝜇
𝑓 ( 𝑧)𝑑𝑓𝑑𝑧 ?
Drainage of liquidsSo we can solve h(t):
𝜙=−𝜌 𝑔𝜇
𝑓 (𝑧)𝑑𝑓𝑑𝑧
h𝑑𝑑𝑡h3(𝑡 )
=−𝜌 𝑔𝜇
𝑓 ( 𝑧)𝑑𝑓𝑑𝑧 ?
With a “beautiful” substitution!
Drainage of liquids
𝜙=−𝜌 𝑔𝜇
𝑓 (𝑧)𝑑𝑓𝑑𝑧
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=∞
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:
𝑘𝑘0
=(1+𝛽 )( 𝑇 −𝑇 0
𝑇1−𝑇 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
𝜙 (𝜂 )=1− 32
𝜂+12
𝜂3
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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