transport phenomena problems

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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