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Analysis of heating a three-dimensional porous bed utilizing the two energy equation model

Berechnung der Aufheizung eines dreidimensionalen porösen Bettes unter Verwendung eines auf zwei Energiegleichungen basierenden Modells

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Abstract

In this paper heating a three-dimensional porous packed bed by a non-thermal equilibrium flow of incompressible fluid is analytically investigated. A two energy equation model is employed to simulate the temperature difference between the fluid and solid phases. Using the perturbation technique, an analytical solution for the problem is obtained. It is shown that the temperature difference between the fluid and solid phases forms a wave localized in space and propagating from the fluid inlet boundary in the direction of the flow. The amplitude of the wave decreases while the wave propagates downstream.

Zusammenfassung

In der Arbeit wird die Aufheizung eines dreidimensionalen porösen Bettes über eine nichtthermische Gleichgewichtsströmung eines inkompressiblen Fluids analytisch untersucht, und zwar mit Hilfe eines Modells, das auf zwei Energiegleichungen fußt und die Temperaturdifferenz zwischen flüssiger und fester Phase zu simulieren gestattet. Unter Einsatz der Störungsrechnung läßt sich eine analytische Lösung des Problems finden. Es wird gezeigt, daß die Temperaturdifferenz zwischen Flüssigkeit und Festphase räumlich lokalisierte Wellen bildet, die sich vom Ort des Fluideintritts in Strömungsrichtung ausbreiten. Die Amplitude der Wellen nimmt während deren stromabwärts erfolgender Ausbreitung ab.

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Abbreviations

a sf :

specific surface area common to solid and fluid phases (m2/m3)

c p :

specific heat at constant pressure (J kg−1K−1)

d :

particle diameter (m)

h w :

heat transfer coefficient at the porous bed walls (W m−2K−1)

h sf :

fluid-to-particle heat transfer coefficient between solid and fluid phases (W m−2K−1)

L i :

length of the packed bed (inx i -direction, m)

Nu fs :

fluid-to-solid Nusselt number (=h sf d/λ f )

Pr :

Prandtl number (=μ(c p ) f /λ f )

R i :

dimensionless length of the packed bed (inx i -direction)

Re p :

particle Reynolds number (=<ρ f >f vd/μ)

t :

time (s)

T :

temperature (K)

T in :

inlet temperature of the fluid phase (K)

T 0 :

initial temperature of the packed bed (K)

u :

dimensionless temperature (=1−Θ)

ν :

velocity of the fluid phase (m s−1)

x 1,x 2,x 3 :

Cartesian coordinates (m)

α :

dimensionless heat transfer coefficient at the porous bed walls

β :

constant

δ :

dimensionless small parameter

ɛ :

porosity

λ :

thermal conductivity (W m−1K−1)

μ :

absolute viscosity of the fluid (kg m−1s−1)

Θ:

dimensionless temperature

ΔΘ:

dimensionless difference between temperatures of the fluid and solid phases

ρ :

density (kg m−3)

τ :

dimensionless time

ξ i :

dimensionless coordinates

eff:

effective property

feff:

effective property for fluid

in:

inlet

f :

fluid

0:

initial

s :

solid

seff:

effective property for solid

\(\langle \Phi \rangle = \frac{1}{V}\mathop \smallint \limits_{V_\Psi } \Phi dV\) :

local volume average of a quantity Φ

\(\langle \Phi \rangle ^\Psi = \frac{1}{{V_\Psi }}\mathop \smallint \limits_{V_\Psi } \Phi dV\) :

intrinsic local volume average of a quantity Φ associated with phase Ψ

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Authors and Affiliations

  1. Institute of Fluid Mechanics and Heat Transfer, Technical University Vienna, Wiedner Hauptstrasse 7/322, A-1040, Vienna, Austria

    A. V. Kuznetsov

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  1. A. V. Kuznetsov
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Additional information

The results presented in this paper were obtained while the author was a Research Fellow of the AvHumboldt Foundation (Germany) at Ruhr-University Bochum. The support of the Christian Doppler Laboratory for Continuous Solidification Processes is also appreciated.

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Kuznetsov, A.V. Analysis of heating a three-dimensional porous bed utilizing the two energy equation model. Heat and Mass Transfer 31, 173–177 (1996). https://doi.org/10.1007/BF02333316

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  • DOI: https://doi.org/10.1007/BF02333316

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