Numerical Modeling of One-Dimensional Hydrous Transfer in the Soil.

Transcription

1 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN Numerical Modeling of One-Dimensional Hydrous Transfer in the Soil. Aouattou Nabila, Saighi Mohamed University Mouloud Mammeri Tizi-Ouzou. Sciences Faculty; Deartment of Physics Algeria., Laboratory of Thermodynamics and Energetically Systems LTSE, Physics Faculty, University of Science and Technology Houari Boumediene, U.S.T.H.B, Bab Ezzouar 6, Algiers Algeria. ABSTRACT We roose, in this study, a mathematical model, which describes the hydrous transfer in the soil-lant-atmoshere systems. Indeed this roblem is resulted it from the combination of two rocesses essential to know the water run-off in the studied medium and the transfer of heat and mass. This model are develoed using an electric analogy, where hydrous transort in the ground and the lant are described in a couled way using an electric equivalent diagram, and the quadrule method. The balance equation; written and then numerically solved in the Lalace sace. The model is written in a FORTRAN rogram. For that, we resent the study of eriodic regime (day and year. The solution in the Lalace sace leads to the flow and otential evolution for long times. The return to real sace is done by the stehfest numerical method. Keywords: Modeling, hydrous, Heat Transfer; Mass Transfer; Electric analogy, Quadrioles. INTRODUCTION The heat and water exchanges between the atmoshere and the ground surface are imortant rocesses for environmental studies. Heat energy and water sulied to the ground surface as solar and long wave radiation and reciitation are redistributed to the ground and the atmoshere through the ground-surface rocesses, becoming the driving force of atmosheric henomena. Thus, the rocesses of heat and water exchanges at the ground surface lay an imortant role in determining the meteorological and climatologically conditions. Moreover, the rocesses of heat and water exchanges at the ground surface are essential comonents for ollutant movement. However, the behavior of heat and water in the soil lant-atmoshere systems, which can be found everywhere on the globe, is very comlicated and is still oorly understood. Many factors including meteorological, hydrological, and biological factors control the rocesses of heat and water exchanges in the system. The roerty of water that has solid, liquid, and vaor hases under the natural environment makes the rocesses more comlicated because hase change of water causes heat exchange.the mechanisms of water transfer between soil and atmoshere, articularly as influenced by lants, have been major themes in soil hysics (Hillel, 98, and many mathematical models describing these henomena have been roosed (Cambell, WATER TRANSPORT MODEL The ground water, the roots and the leaves are assumed to be at the average heights Z gw, Z R and Z L. Due to transiration (resulting from a heat and mass balance between the leaves and the surroundings the lant losses a water flux versus time T (t whereas due both to the rain and the evaoration, the soil surface a net water flux (t... The soil... Assumtions: The following hyothesis has been made: In a bare soil of deth e, transort is unidirectional in vertical (or z direction. Soil water diffusivity D and hydraulic conductivity k vary markedly with water content. However, D and K are assumed to be constant. Mass transfer is only in the liquid hase and the leaf water otential is considered to be the otential of the substomatal cells.. Water transort equations The equation of continuity if the water density is assumed constant combined with generalized Darcy's law for all isotroic medium, leads to: Volume, Issue 5, May 3 Page 463

2 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN K t ( Where the soil volumetric water of the soil, K is is the soil hydraulic conductivity and H the iezometric water head or hydraulic head. As indicated when develoing the various assumtions made, the hydraulic conductivity K is suosed here to be indeendent of the volumetric soil water content and hence of the sace variable. For unsaturated soil, the caillary caacity C is introduced d d If denotes the caillary water ressure ( C h P atm P g Where: P is the water ressure, P atm the atmosheric ressure, the water density and g the gravity intensity. The water ressure head H is given by H z (4 The caillary transort of water is given by the Richard's equation [] H ( K (5 t z z Assuming that the gas ressure is uniform in the soil and that is only a function of θerror! Bookmark not defined., combining equations (4 and (5 leads to: (3 d H d t H H ( K C (6 z z t The flow equation is then H D H (7 t K Where; D the soil water diffusivity. However, D and K are assumed to be constant in this study. C 3. Lalace transforms In the steady state, the resistive analogy is generally introduced in order to exress the linear relation between the otential variation and the rate of flow. In the transient case, the extension of this notion is a quadriole; a quadriole is built with three imedances, which hysically mean the simultaneity of resistive and caacitive effects in the medium. When the time tends to infinity (henomena aroach the steady state the limit of the quadriole is reresented in figure ( with R and R that exresses the resistance of the soil to the movement of water and C the caacity of storage of the soil. Mathematically when a Lalace transformation ( f ( s f ( t.ex( st. dt is alied, the artial differential equation of diffusion (7 becomes a differential equation of second order which solution is exactly known; using the boundary conditions one can write a linear relation between otential flux at the inut and the same at the outut. The quadriole exresses this relation. Let us oint out that the imedance is only significant in Lalace sace and the inverse transformation in temoral sace can only be made for the final solution (otential and/or rate of flow. Volume, Issue 5, May 3 Page 464

3 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN V V max M Figure b: Limit of the quadriole when t figure: water volume versus water ressure of the lant We assume for simlicity that at t =, the soil is at hydrostatic equilibrium (that is to say H = H i = is constant elsewhere. If H L( H H i et q L( q designs the Lalace transforms of the ressure head and the volumetric water flux ( q ks H, the linear relation between H q z, and H, q at two different sections z = z and z = z of the soil is written with e = z z [] : H A A H (8 q A A q With: A A ch( le ; sh l e A (. ; A ( ( kl sh le ; kl Where; l and is the Lalace variable. D Thus, in the unsteady state a soil layer of thickness e can be associated with a quadriole. As this quadriole is assive, it can be reresented by a T - association of imedances (see figure b with [3]: A Z Z ; Z 3 A A D. t D. t It is interesting to find the behaviours for these imedances when t ( : e e Fourier number in relation to mass transfer in Lalace sace at Pe D In this case the quadriole aroximation is given by: e Z Z k (Equivalent to a resistance. Z 3 (Equivalent to a caacity ec 4. PLANT h According to Katerji's model [4] there are several resistance mechanisms to water transort inside a lant. But the most imortant ones are the resistance between the soil and the roots R r and the stomatic resistance of the leaves R f. More the lant is able to accumulate water [4, 5]. If we admit a linear relation between the accumulated water volume V in the lant and water ressure of the lant, then: P V V (9 max M Volume, Issue 5, May 3 Page 465

4 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN Where V max is the accumulated water volume when the lant is fully wet ( = and M < the ressure corresonding to definitive withering of the lant (figure. Therefore, the water flux q from the lant is given by: dv V d q C d ( max ( ( dt M dt dt The lant acts as a caacitance C which is suosed to be connected to the main water flux by a resistance r (figure.. Lalace transformation in the time t from equation ( gives i ( q C( CH Where i is the initial value of. For simlicity we will assume here that the system is initially at equilibrium. 5. The Atmoshere The main factors of atmoshere influencing the behaviour of water flow towards lant are: the net solar radiation, the rains, the humidity and temerature of air, and the wind seed. 6. Balance equation 6.. Radiation Balance The main energy source on earth surface is from solar origin. The net radiation R N is the balance of radiative energy received by the covered vegetal. 4 R a R R T ( N ( G S C S F Where R C is the atmosheric radiation and R G the total radiation, which is comound of the direct and diffuse radiations [6]. 6. Energetic Balance A art of net radiation serves to vaorize the water available on the surface of the covered vegetal and give a flux of latent heat (LE. The rest is dissiated by convection as heat in the air (sensitive heat H C or stored in the soil (as a conductive heat flux S. If the art of net radiation R N used for hotosynthesis by the vegetation and the conductive flux in the soil S are neglected, the equation of the energy balance can be written: R LE (3 N H C The electrical equivalent schema soil-lant-atmoshere system reresented in Lalace sace is comound of three quadrioles figure (3. Figure 3: schematic diagram of soil-lant- atmoshere system. 7. Results and Discussion The system of the established equation from this electrical reresentation see (figure 3 has been given as follows: Volume, Issue 5, May 3 Page 466

5 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN abc ab ab H = - ( E + v + f abc abc abc H = H f + q r C H q = - C H f = - rc q a c E c H v f a b c E c H f v H a b a a H r = a H v Where: ab abc abc ch ( e ; ab ch ( e th ( e sh ( e ; abc ch ( e th ( e sh ( e ch ( e sh ( e ch ( e ch ( e C ( rc th ( e th ( e sh ( e C ( r R ( rc sh ( e Volume, Issue 5, May 3 Page 467 ; ; abc sh ( e C ( r R ( rc sh ( e th ( e ch ( e. R P ab sh ( e ; ab ch ( e C ( rc sh ( e th ( e The resolution of the system of equations obtained in the Lalace sace is numerically given; have allowed us to know the evolution of all hysical magnitudes (flow and otential intervening in rocess of exchange at different levels of lant or soil. The return to real sace is done by the Stehfest numerical method [6]. A art of results obtained has been illustrated in figures 4 and 5; they reresent the evolution curves of the otential and density flow of water with resect to time for different arameters characterizing the soil-lant-atmoshere. The hysical roerties of the soil; (relative or balance humidity, hydraulic diffusivity and conductivity of soil, the deth of ground water and the water flux arriving on earth surface from the rain or the artificial irrigation. Different magnitudes in relation with the lant (stomatal resistance, lant-reservoir caacity... roosed in secialized literature have been tested too. The results obtained are in good agreement with the exerimental results obtained by various authors. For a mean flow density q =, 8mm.day - at the soil surface, a mean transiration T=8, 47 mmday-, a volumetric water content of soil = 35% and a water at.5m deth (where the soil water being at atmosheric ressure for a diurnal cycle (T = 4h. Curves relating to different otentials are resented in fig. 4a, the leaves otentials and the tank-lant otentials decrease very fast during the ten first hours. This is confirmed by the density flux evolution shown in fig. 4b. This brutal dro of tank-lant otentials shows that the tank-lant will be solicited first. It becomes raidly emty. The roots flow issuing from the humid surrounding soil will feed the lants and will become the main water sring satisfying the climatic request, because the resonse times of soil are much longer than those of lant. The two curves are confused along the day; a low difference between them, exlains the low value of flow coming from the tank-lant. A otentials (H f and that of the tank-lant (H P : can follow know a similar variation. This brutal dro of tank-lant otentials shows that the tank-lant will be solicited first. The Flow coming from the tank-lant is definitely more imortant, its contribution to satisfy the climatic request is considerable. Because of the diurnal cycle of the climatic demand inducing water utake, ch ( e R ; ;

6 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN the model reroduces greater dryness around the roots during high demand (3 h than during low demand (9 h. The same calculations were done for a ground water located at.5m deth. Then we comare the evolution of the curves relatively to otentials fig 5a and to density flow fig. 5b for the two volumetric water content of soil = 35% and 5%. The fig. 6(a, b, shows a variations of flow and otential for a eriod of one year (T = month at deth z =. 5m; with the same soil moisture = 35% q qa E otential(bars - -5 H H Hf Hr flux density(mm/j time(h time(h (a Figure4 (a:- Potential variation for = 35% at.5m deth.5m deth (b Figure4 (b:- flow density variation for =35% at 5 otentiel (bars H 35% H r 35% H 5% H r 5% Flux (mm/j 5 5 q(35% qa(35% q(5% qa(5% Time (h Time (h (a (b Figure5 (a, b: Comarison of otential and density flow evolution at the.5m deth for = 35%, 5%. (a (b Volume, Issue 5, May 3 Page 468

7 International Journal of Alication or Innovation in Engineering & Management (IJAIEM Volume, Issue 5, May 3 ISSN Figure6 (a Potential variation for = 35% at.5m deth; Figure6 (b flow density variation for =35%. 8. CONCLUSION We develoed, in this study a simlified mathematical model to describe the transfer in the soil-lant-atmoshere continuum system (SPAC. In this global aroach of comlicated and multifarious roblems, the assumtions considered reviously have allowed to obtain a simle and quick solution. These results have allowed us to rove the imortance of taking into account the lant-reservoir, which aears during a water stress eriod. On the other hand, for short times case, the foliar resistance becomes a significant arameter and its influence on flux and the otentials is imortant in the first hours. We were interested thereafter, in the influence of a few numbers of arameters such: water content of the ground, deth of the ground, foliar resistance on the conditions of aearance of the hydrous stress. This study allows the ossibility to analyze the sensitivity of each arameter on the evolutions of the otentials and flows at various levels of the ground and the lant. References [.] Richards L.A., Caillary conduction of liquids in orous media, Physics ( (93. [.] A.DEGIOVANNI, conduction dans un mur multicouche avec sources: extension de la notion de quadriôle. [3.] Degiovanni, A. Conduction. Techniques de l ingénieur. Traité généralités, 989, A 53, 4. [4.] Katerji N. et. al "Modèle dynamique de circulation de l'eau dans la lante: vérification sur une culture de omme de terre dans la région arisienne". Etude sur les transferts d'eau dans le système sol-lante-atmoshère. INRA (988. [5.] Carlson T.N. and Lynn B. "The effects of lant water storage on transiration and radiometric surface temerature".agricultural and forest meteorology.57, 7-8 (99. [6.] Chourd P. et al. "Bilan thermique d'une maison solaire, méthode de calcul raide". EDF Editions Eyrolles. (977. [7.] Stehfest H., "Remark on algorithm 368". Numerical inversion of Lalace transforms, A.C.M., 53. (97 [8.] Celia M.A., Bouloutas E.T., Zarba R.L., A general mass conservative numerical solution for the unsaturated flow equation, Water Researches. Res. 6, (99. [9.] Hillel D, Alications of soil hysics. Academic Press, New York, 9-96 (98. [.] M. Saighi., C. M; Un modèle simlifié de transfert de chaleur et de masse dans le système sol-lante-atmoshère. Int. J. Heat Mass Transfer. 4, N, (998. [.] D. Hillel, L'eau et le sol. Princies et rocessus hysiques. Vander, Louvain (984. Volume, Issue 5, May 3 Page 469