Abstract: This experimental study present some micro-structural and thermo-hydrous properties of terra cotta. A tube, built from this porous material, bathes in a tank filled with water. The internal surface of the wall carries a water film which evaporates by forced convection of hot and dry air, allowing the storage of negative kilocalories. The form of the tube used is a factor that allows neglecting the variations of the thermo-physical properties of the air between the entry and the exit of the tube. A computer code (based on the equation of Darcy for fluid crossing saturated porous material, the equation of Laplace on pressures and the equation of heat in polar co-ordinates) was established to describe the thermo-hydrous behavior of the material. The code makes it possible to determine the zones of intrinsic permeability favorable to a continuous evaporation without draining off the water film and it also allows the calculation of the average temperature of bath.
INTRODUCTION
In a Sahelian Country, variations of temperature between dry air and the wet bulb can be higher than 15°C. Hot air destroys foodstuffs very quickly, but it constitutes a factor that is favorable to all processes of cooling by evaporation.
This study is undertaken within a project of construction of a heat exchanger of parallelepipedic form, containing a volume of water, crossed by porous tubes placed in quincunxes through which the hot and dry air can flow.
Figure 1 shows a diagram of the prototype under development. The porous tube allows the passage of a water flow, which, while evaporating, cools the great volume of water. Then the matter is to optimize the heat transfer between the hot and dry air and the volume of water to be cooled. In the first part of this study, we determine the hydraulic conductivity of the material by measuring the rate of water flow running out through the material under constant hydraulic load. The intrinsic permeability is deduced from the equation of Darcy. Then we determine the thermal conductivity of the material according to the water content (me/ms), by using the measures of heat fluxes on the two faces of the sample.
| Fig. 1: | Experimental prototype scheme |
Heat and mass transfer in porous media is a very complex phenomenon which has been the subject of a great number of theoretical and experimental studies. Previous concepts and research include capillary flow, diffusion theory, evaporation and condensation theory. Recently, more elaborate theories on simultaneous heat and mass transfer processes have been published by Nield and Bejan (1999), where the real discontinuous medium is converted to an equivalent fictitious continuous one. Transfers of heat and mass during evaporation are often described by the following empirical expressions:
| (1) |
In which coefficients a1, m, m1, n, n1 depend on the properties of the fluid and on the interval of variation of Reynolds numbers.
Kondjoyan and Daudin (1993) developed a method based on psychrometry for simultaneous measurements of heat and mass transfer coefficients in forced convection between air and a wet surface. This method which was tested on tubes, enables an easy evaluation of transfer coefficients.
Recently, Prabal et al. (2008) performed Computational Fluid Dynamics (CFD) simulations for laminar flow, convective heat and mass transfer between water surface and humid air flowing in a horizontal 3D rectangular duct, to validate experimental data. (A short pan of water forms the lower panel of the duct). Numerical data fall within the uncertainty range for most of the experimental data. Conrad and Carey (2007) used a low water pan which is not thick and remains on the lower surface of the channel. The numerical data obtained are in agreement with those obtained in experiments, with the same uncertainties of measures. The effects of buoyancy forces were found to be negligible. In this study, the average Nusselt and Sherwood numbers are calculated with:
| (2a) |
| (2b) |
where, Z is the co-ordinate in the direction of the air flow and Y is according to the vertical.
The assumption of Lewis makes it possible to obtain in the particular case of laminar boundary layer on a plane surface, the exact expression of the report/ratio:
which does not apply any more in the case of a turbulent flow and for a cylindrical geometry. The measures taken by Gilliland and Sherwood (1934) on the evaporation of a water film streaming along a cylindrical wall made it possible to establish a correlation giving the average of Sherwood number (Eq. 14a). This correlation is very close to that of Colburn in Saccadura (1982), (Eq. 15a), describing the convective transfers of heat under the same conditions. Studies of Berman (1961) on the determination of the Sh/Nu report/ratio for various configurations of flow, prove that for water temperatures ranging from 20 to 50°C, this report/ratio remains appreciably constant, close to 1, (Sh/Nu = 1.06). Over a long period, in variable mode, we cannot use average coefficients to suitably describe the coupling between the external transfer in the boundary layer and the internal transfer in the porous material. Then we re-compute at each step of time, heat and mass transfer coefficients that we introduce into the system of equations.
The aim of this study is to propose a simple model of storage of cold water in Sahelian rural zones.
STRUCTURAL AND THERMO-HYDROUS PROPERTIES OF THE MATERIAL
Density and specific surface: To measure the density of the ROU fireclay sample, (ROU abbreviation of the name of the site where the clay was taken), we introduce into a drying oven, a few well dried grams of this finely crushed sample (inferior to 40 μm) into a cell of 3.5 cm3. The unit is then placed in the helium pycnometer chamber.
Specific surface is an important parameter for the characterization of our porous solid. It makes it possible to be ensured of the quality of the material and the processes of transfer of water. Here, we use the BET method of study based on the theory of adsorption of water molecules. The results are shown in Table 1. The specific surface obtained favors a moderate transfer of water within the porous material.
Porosity and microstructure: A terra cotta sample which is previously weighed is used: M1. Then, it is placed in a desiccator in order to make the vacuum. Water penetrates the sample so that all the pores are filled. At the end of the immersion (30 mn), the sample mass M2 is measured. Finally, once out of the water the sample is weighed immediately: M3. The open porosity of the sample (ε = 32.3) is calculated according to the following relation:
| (3) |
| Table 1: | Specific surface and density of fireclay sample |
| Fig. 2: | SEM of sintered ROU clay |
Figure 2 shows a composite microstructure where coarse grains are embedded in an agglomerated clay matrix. But local heterogeneities are observed at the interfaces between the coarse grains and the matrix. They should have an effective role in the connectivity of the pores of the material.
Hydrous behavior through the average intrinsic permeability of the material: The method consists in studying the hydraulic conductivity of the material under a constant load in pressure.
First tests: During the first experiments (T = 25°C, 50 = RH = 60), tubes of 30 mm of diameter and 30 cm of height are used. Circular shaped samples were joined tightly to the lower part of each tube. Then, a quantity of water (350 cm3) was poured in the tubes. We measure the loss of mass according to time. We then numerically calculate the permeability of our samples using the Eq. 4.
| (4) |
The intrinsic permeability is deduced from the following formula:
| (5) |
We obtain an average value of 5.60 ×10-14 m2.
Second tests: During these tests, the moisture of the environment is controlled with the use of a NaCl brine which makes it possible to maintain a relative hygrometry of 75% between 20 and 40°C on the one hand and the use of a Mg brine (NO3)2 6H2O which makes it possible to maintain a relative hygrometry of 53% at 30°C (Fig. 3).
| Fig. 3: | Experimental device of permeability measurement |
| Table 2: | Intrinsic permeabilities |
That makes it possible to limit the variations of the rates of evaporation on the internal surface. We use a plane square surface of 15 cm of side. A constant water load of 10 cm is maintained on the top of the material. From the values of water flows at constant load, we deduce the hydraulic conductivity, then the intrinsic permeability. The numerical values of intrinsic permeability are consigned in Table 2.
We note a higher permeability for lower values of relative hygrometry of the air. We keep the value 2×10-15 m2 as average intrinsic permeability of our material because it is that which approaches more the conditions of saturation of the boundary layer.
Thermal behavior from conductivity
Thermal conductivity measured according to the water content: Fireclay
samples are cut out into squares of 3 cm side with variable thicknesses, in
order to ensure the parallelism of the two principal faces of the sample which
is then placed between two metal plates of which one emits a heat flow and the
other receives the outgoing flow crossing the sample. The values of these flows
are recorded every 15 min. Then the moisture of the samples is measured. Thermal
conductivity is then calculated by the Eq. 6.
| (6) |
where, φ (W.K -1) and e (m).
Figure 4 shows the thermal curve of conductivity of the ROU according to the water content. The curve shows that thermal conductivity follows a monotonous evolution to an ambient temperature of about 20°C. This is in agreement with the analysis of Krischer in Azizi et al. (1988). This analysis underlines a monotonous evolution for temperatures lower than 50°C with a growth of thermal conductivity according to moisture.
Equivalent thermal conductivity of the saturated porous medium: In a first approach, we notice in the literature that the equivalent thermal conductivity of a given porous material is intermediary between that of the solid phase and that of the liquid phase. The equivalent thermal conductivity of the porous saturated materials can be calculated theoretically by the following formula (law of composition).
| (7) |
Maxwell model can also be used:
| (8) |
In this model, the conductivity of the solid phase depends on the components forming the solid matrix. Its numerical value is approached by that of a dry and compact terra cotta brick.
λs = 1.004 W/m/K |
Then for saturated material we obtain:
| Fig. 4: | Thermal conductivity of the ROU: T = 20°C |
| • | By the law of composition:λm = 0.8112 W/m/K |
| • | By the formula of Maxwell:λm = 0.8694 W/m/K |
| • | In experiments:λm = 0.90000 W/m/K |
MODELISATION OF THE HYGROTHERMAL TRANSFERS IN FORCED CONVECTION
After determining the physicochemical characteristics of the material we propose to determine the ideal characteristics of a material allowing us to cool as efficiently as possible, a volume of water, using a porous tube whose internal surface is the seat of an evaporation in forced convection.
Formulation of the problem: We consider an initially dry porous tube (C = 8 cm, Di = 6 cm), of porosity equal to 0.323, whose internal surface of wall is in contact with a flow of hot and dry air and whose external surface bathes in a water vat with a square section of 18 cm of side. It is supposed that a small agitator homogenizes the temperature of water (Fig. 5). We can then define compactness for our module by the report/ratio:
The external surface of the wall is then subjected to a variable hydraulic load P(r, θ). The capillary invasion is made at the time when the internal surface becomes saturated. We then observe a very slow phenomenon of percolation. The measurements taken on different terra cotta by the CTTB (1998) and Perrin et al. (2004), show that the phenomenon of percolation occurs only when the water film approximately reaches 163 g of water m-2 of wall.
It is at this moment that we send the hot and dry air into the channel. Thereafter, there is a phenomenon of competition between on the one hand, the supplying of water to the water film by water infiltrating through the material and on the other hand, the flow evaporated in the environment.
| Fig. 5: | Layout of the studied system |
The internal surface of the wall which cools induces a transfer of heat from water towards the environment.
Assumptions:
| • | The porous material is homogeneous and constantly saturated with water |
| • | The density and the viscosity of the fluid are supposed to be constant |
| • | The porous environment can be treated like a continuum resulting from the simultaneous presence of two phases (solid and liquid in the present case). It is then modeled by a continuous, homogeneous isotropic and fictitious medium. The physicochemical characteristics of this medium (λm, (ρCp)m) can be obtained by the laws of composition |
| • | The local Reynolds number |
Equations of transfer: The equation of mass conservation in a saturated porous environment leads to the resolution of Laplace equation for pressure.
| (9) |
Momentum equations:
| (10a) |
| (10b) |
Heat equation within porous medium:
| (11) |
The average temperature of water is calculated starting from the following equation:
| (12) |
With
| (13a) |
| (13b) |
| (13c) |
| (13d) |
Heat and mass transfer coefficients: The average air velocity in the channel is 1 m sec-1. We obtain Reynolds numbers which are always higher than 2300, therefore, we deal with a turbulent flow. We use then the correlation of (Gilliland and Sherwood, 1934).
| (14a) |
| (14b) |
| (14c) |
The convective heat transfer coefficient between the air and the wall is evaluated starting from of Colburn, in Saccadura (1982) correlation in turbulent flow:
| (15a) |
| (15b) |
The heat transfer coefficient between the wall and water is obtained starting from the following correlation:
| (16a) |
| (16b) |
In which, Grashof number is calculated starting from the average temperature of the wall:
| (17) |
Initial conditions
t = 0;
Tf = T = Te = T0 | (18) |
Boundaries conditions
With
| (19) |
P = P0 | (20) |
| (21) |
P
= P0+ρg (h+r1 (1–sinθ)) | (22) |
Numerical resolution: Equation 9-11 associated at boundary conditions 19, 20, 21 and 22 are solved with an implicit numerical scheme and with the method of Gauss-Seidel.
RESULTS
Mass flow transferred according to the permeability of the material: Mass flows transferred grow in an exponential way when the intrinsic permeability is represented on a logarithmic scale (Fig. 6).
Temporal evolution of mass flow on the internal wall of the tube: Figure 7 shows that for values of Kp lower than 10-15 m2 the wall is drained at the end of 30 to 45 min. For Kp = 10-15 m2, the mass of liquid film decreases in the 1st h with a tendency to grow again beyond 3 h.
Figure 8 also shows, for an evaporation with 40°C, a more significant draining zone (Kp≤10-15 m2), a zone that is favorable to the evaporation of the liquid film without draining of the wall in the course of time (1.5×10-15≤Kp≤2×10-15 m2). In both cases of figures we notice that for higher permeability, the mass of water film increases very quickly and thus, these zones are not, with priori, adapted to our problem.
Influence of equivalent thermal conductivity on the change of temperature of the material: It is noticed naturally (Fig. 9) that the variations in temperature between the interface air-wall and the interface material-water attenuate when thermal conductivity is higher. These variations remain relatively weak when compared to the differences in selected thermal conductivities which are very high for terra cotta materials.
Evolution of the average temperature of the water according to the equivalent thermal conductivity of the porous material: The curves of Fig. 10 show a light fall in the average temperature of water according to the increase in equivalent thermal conductivity for several hours. Naturally, thermal resistance is all the more low as
| Fig. 6: | Water mass flow rate according to the permeability, in a controlled
environnement, Ta = 30°C, RH = 30%, Vair = 1 m sec-1 |
| Fig. 7: | Evolution of water film mass flow rate under evaporation, Ta = 30°C , RH = 30%, Vair = 1 m sec-1, Z0 = 0.05 m |
thermal conductivity is high. At the end of 6 h, a steady state of operation is practically reached and the heat gradients strongly attenuate. It is noticed on the other hand that the temperatures of the film vary little if we take into account the effect of thermal inertia brought by the great water mass to cool. The whole of these curves evolve towards the temperature of the wet bulb.
Evolution of the temperatures of the film and water mass according to the velocity of the air: The various air velocities in the channel, allow calculating the following Reynolds numbers:
| Re = 0.1 m sec-1; | Re = 7.5×103; | Re = 1.0 m sec-1 |
| Re =7.5× 104; | Re = 5.0 m sec-1; | Re = 3.75×105 |
| Fig. 8: | Evolution of water film mass flow rate under evaporation, Ta = 40°C , RH = 30%, Vair = 1 m sec-1, Z0 = 0.05 m |
| Fig. 9: | Temperature distribution into the material according to the
thickness, Ta = 30°C, RH = 30%, Vair = 1 m sec-1,
Z0 = 0.05 m, Kp = 2.10-15 m2 |
| Fig. 10: | Change of the film temperatures and the water mass: Ta = 30°C,
RH = 30%, Vair = 1 m sec-1, Z0 = 0.05 m,
Kp = 2.10-15 m2 |
| Fig. 11: | Temperatures evolutions of film and water mass according to
the air velocity: Ta = 30°C ; RH = 30%, Vair =
1 m sec-1, Z0 = 0.05 m, Kp = 2.10-15
m2 |
These values always lead to in turbulent flow, but large variations of behavior are observed. We observe on Fig. 11, with vair = 0.1 m sec-1, a drop in the temperature of the water mass of 8°C at the end of 6 h. For the same type of operation, with vair = 1 m sec-1 we lower the temperature of the water to 13.5°C. The increase in the Reynolds number induces an intensification of the thermal transfers.
DISCUSSION
The results obtained in this study, differ slightly from those of the previous study, because they are calculated with a higher degree of accuracy (a regular mesh was built with 240 nodes in the r direction, instead of 120 and 64 nodes in the θ direction, instead of 32). With this choice of the grid, we could determine the temporal evolution of mass flow on the internal wall of the tube.
As the goal is to cool as much as possible a mass of water while consuming the least possible, the material must have such permeability that the transferred flows are at least equal to the evaporated ones. In case it is the contrary, the wall of the tube is dried and this one heats by significant heat transfer with the flow of hot and dry air. The results of Fig. 7 and 8 confirm those obtained by Perrin et al. (2004) with permeabilities of the same order of magnitude.
The hot and dry season of Sahelian countries, is characterized by temperatures varying between 30 and 40°C with a relative humidity of the air that is generally lower than 30%. The mass flow on the internal wall is then due to phenomenon of competition between, on the one hand, the flow of water film through the material and on the other hand, the flow evaporated in the air.
Curve 2 of Fig. 7 and curve 4 and 5 of Fig. 8 present minima. That is explained by the attenuation of the gradient of water vapor partial pressure between air and water film, since this one cools in the course of time and consequently a reduction in the evaporated flow.
Intensification of thermal transfer occurs when increasing Reynolds number. But this one always leads to temperatures that are higher or equal to the temperature of the wet bulb. These results were also observed by Boukadidia and Nasrallah (2001) during water evaporation in laminar flow inside rectangular channel. In conclusion the evolution of the various temperatures within present system is guided primarily by convective transfers. The transfers by conduction have only one secondary influence.
CONCLUSION
We have determined through experiments the permeability and the equivalent thermal conductivity of a terra cotta material in order to model the hygrothermal exchanges through this material. This work constitutes a first stage towards the dimensioning of a heat exchanger for the storage of negative kilocalories in water mass. This study shows that:
| • | The permeability of the material constitutes data that are difficult to determine. It can vary in a significant range of values for the same material according to the selected procedure |
| • | The thermal measures of conductivity obtained in experiments are confirmed by the theoretical approaches |
| • | The choice of the intrinsic permeability of the material is capital data in so far as it makes it possible to be ensured of the permanence of a liquid film on the wall during evaporation. The permeability of the studied material seems a little high in comparison with the objectives that are laid down (Fig. 7, 8) |
| • | An increase in the thermal conductivity of the material improves only very little the thermal transfers compared to the transfers by convective evaporation |
ACKNOWLEDGMENT
Authors (team: 7226), thank IRD, AIRES-SUD development for their support.
NOMENCLATURES
| C | : | Compactness (m2 m-3) |
| CP | : | Specific heat to constant pressure (J/kg/K) |
| D | : | Mass diffusivity (m2 sec-1) |
| Di | : | Diameter interns (m) |
| C | : | External diameter (m) |
| g | : | Acceleration of gravity (m sec-2) |
| Gr | : | Grashof number |
| H | : | Height (m) |
| ha | : | Air heat transfer coefficient (W/m2/K) |
| hm | : | Mass transfer coefficient (m sec-1) |
| K | : | Hydraulic conductivity (m sec-1) |
| Kp | : | Intrinsic permeability (m2) |
| L | : | Length of the tube (m) |
| : | Evaporation latent heat at temperature T (J kg-1) | |
| Me | : | Mass water (kg) |
| : | Mass flow rate (kg sec-1) | |
| : | Average nusselt number | |
| P | : | Pressure (Pa) |
| Pr | : | Prandtl number (υ/α) |
| R | : | Radius of the tube (m) |
| Re | : | Reynolds number |
| Rv | : | Perfect gases constant relating to water vapor (J/kg/K/mole) |
| Sc | : | Schmidt number |
| Si | : | Internal surface of the tube (m2) |
S0 |
: | External surface of the tube (m2) |
| Sh | : | Sherwood number |
| T | : | Temperature (°C) |
| U | : | Radial velocity (m sec-1) |
| V | : | Tangential velocity (m sec-1) |
| VT | : | Total volume of water |
Greek symbols
| αm | : | Thermal diffusivity (m2 sec-1) |
| ε | : | Porosity |
| θ | : | Angular co-ordinate |
| λ | : | Thermal conductivity (W/m/K) |
| μ | : | Dynamic viscosity (kg/m/sec) |
| ρ | : | Density (kg m-3) |
| υ | : | Kinematic viscosity (m2 sec-1) |
Indices
| a | : | Air |
| D | : | diameter |
| e | : | Water |
| f | : | Interface |
| i | : | Internal |
| o | : | External |
| s | : | Saturation |