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A general mathematical modelling for heat and mass transfer in unsaturated porous media: an application to free evaporative cooling

Verallgemeinerte mathematische Modellierung des Wärme- und Stoffaustausches in nicht vollständig getränkten Medien

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Abstract

The present paper develops a general mathematical model with some improvements in mass, momentum and energy equations, which introduce more transport mechanisms to simulate simultaneous transfer of heat and mass in the porous media unsaturated with liquid. Numerical calculation results in two-dimension are obtained for the vertical packed bed with its right opening surface exposing to atmospherical environment. The calculating data can demonstrate the cooling effect of the water evaporation for the bed if it is used as a cooling wall of building for room air-conditioning in the hot and dry climate.

Zusammenfassung

In der vorliegenden Arbeit wird ein verallgemeinertes mathematisches Modell entwickelt, wobei hinsichtlich der Impuls-, der Kontinuitäts- und der Energiegleichungen einige Verbesserungen Eingang finden, die durch Hinzufügung weiterer Transportmechanismen eine Simulation des gleichzeitigen Wärme- und Stoffaustausches in nicht vollständig getränkten porösen Medien ermöglichen. Numerische Ergebnisse für ein zweidimensional betrachtetes, vertikal gepacktes Bett mit rechtsseitig offener, in Verbindung mit der Außenatmosphäre stehender Begrenzung, werden mitgeteilt. Die Berechnungsdaten belegen den Kühlungseffekt, der durch Wasserverdunstung aus dem Bett erzielbar ist, wenn dieses in heißen und trockenen Gebieten als Kühlwand ausgebildet wird, um eine Raumklimatisierung zu bewirken.

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Abbreviations

A :

area, m2

c :

specific heat, J/(Kg·K)

D l :

diffusivity of water in porous materials, m2/s

D v :

molecular diffusivity of vapor in air, m2/s

D Tv :

diffusivity defined in Eq. (11), m2/(s·K)

D lv :

diffusivity defined in Eq. (12), m2/s

g :

acceleration of gravity, m/s2

h :

convective heat transfer coefficient, W/(m2·K)

h m :

convective mass transfer coefficient, m/s

H :

horizontal width inx direction of bed, m

k m :

apparent thermal conductivity, W/(m·K)

K g :

infiltrating conductivity of gas mixture, m/s

K l :

hydraulic conductivity of water, m/s

L :

vertical height iny direction of bed, m

\(\dot m\) :

mass rate of phase change per unit volume, Kg/(m3·s)

m :

mean vapor quantity in porous bed per unit volume, Kg/m3

\(\vec n\) :

normal vector

p :

pressure, Pa

\(\vec q_v \) :

vapor diffusion flux, kg/(m2·s)

q r :

solar radiation, W/m2

RH, h :

relative humidity in ambient air, or in gas mixture, %

S :

liquid saturation, %

t :

time, s

T :

temperature, K (°C)

u :

velocity component inx-direction, m/s

v :

velocity component iny-direction, m/s

\(\vec V,\vec v\) :

velocity vectors, m/s

V :

averaging volume, m3

V t :

volume of porous packed bed, m3

\(\vec w\) :

velocity vector of gas-liquid interface, m/s

α:

tortuosity factor in Eq. (2)

β:

thermal expansion coefficient, 1/K

γ:

latent heat, J/Kg

ε:

phase content, %

ν:

kinematic viscosity, m2/s

ρ:

density, Kg/m3

υ:

mass flow factor in Eq. (2)

ϕ:

porosity, %

Ψ:

water pressure potential, m

Φ:

actual value of certain physical quantity

a :

air, ambient

cw :

cooling wall

g :

gas mixture

gl :

gas-liquid interface

i :

inside surface

l :

liquid, water

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Authors and Affiliations

  1. Department of Power Engineering, Huazhong University of Science and Technology, 430074, Wuhan, PR China

    W. Liu

  2. Department of Mechanical Engineering, Ehime University, 790, Matsuyama, Japan

    S. W. Peng &  K. Mizukami

Authors
  1. W. Liu
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  2. S. W. Peng
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  3. K. Mizukami
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Additional information

This project is supported by the National Natural Science Foundation of PR China.

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Liu, W., Peng, S.W. & Mizukami, K. A general mathematical modelling for heat and mass transfer in unsaturated porous media: an application to free evaporative cooling. Heat and Mass Transfer 31, 49–55 (1995). https://doi.org/10.1007/BF02537421

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  • DOI: https://doi.org/10.1007/BF02537421

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