λ w w. (3) w 2 2 λ The obtained equation (4) may be presented in the stationary approximation as follows (5) with the boundary conditions: for = 0

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1 International Conference on Methods of Aerophysical Research, ICMAR 28 SIMULATION OF POROUS MATERIAL IMPREGNATION A.V. Fedorov, A.A. Zhilin Khristianovich Institute of Theoretical and Applied Mechanics SB RAS 639, Novosibirsk, Russia, Introduction. A wide use of porous materials of new generation in civil engineering and a number of other industry branches leads to a need in investigating the peculiarities of the mechanism of moisture interaction with porous materials. The work [1], in which the eperimental data on the moisture penetration into a porous material are presented and the dependence of the moisture diffusion coefficient on the moisture content in porous material is determined, is devoted to their study. In the present work, the mathematical modeling of the moisture migration process in a porous skeleton of solid material is carried out. The investigation of the pattern of a flow arising at a porous skeleton contact with water surface and the elucidation of the influence of initial parameters of the system the liquid + the solid on the liquid diffusion peculiarities are the main tasks. Physical-mathematical problem statement. Let us consider a vertically posed beam of porous material (the steam-cured aerocrete) whose side walls are moisture-insulated, and the lower face contacts with water surface. The upper plane is free and has a constant humidity. The impregnation process start corresponds to the moment of time when the beam dry face contacts with a humid medium (t = ). After the contact, a humidity wave propagates into the porous material. The problem is to determine the liquid flow pattern in the porous beam at subsequent (t > ) moments of time. To describe the process under study it is natural to invoke the nonstationary diffusion equation in the following form W t W = D( W ). (1) where D(W) is the diffusion coefficient; W is the humidity; t is the time; is the spatial coordinate. We augment equation (1) by the corresponding initial: for t = : =, W = W, < <l, W = and boundary conditions: for = : W = W, for = l: W/ =. (2) Self-similar approimation. Let us go over to new self-similar variables t and λ = t, in which equation (1) takes the form w λ w w t = + D( w). (3) t 2 λ λ λ The problem (2), (3) was solved numerically by the shooting method and involved two stages. The first stage is the testing. Assume that the diffusion coefficient is constant that is D(w) = const = D, then the diffusion equation may be written in self-similar variables (3) as 2 w λ w w t = + D. (4) 2 t 2 λ λ The obtained equation (4) may be presented in the stationary approimation as follows with the boundary conditions: for = λ λ w = w 2D w = w, for λ = + w =. (5) A.V. Fedorov, A.A. Zhilin, 28 1

2 Section V 1-6 D W.5 Fig. 1. Permeability vs. the diffusion coefficient. Fig. 2. Interpolation of eperimental data ( ) by the eponential function (solid line). The solution of equation (5) has the form w = w λ 1 Erf. (6) 2 D We will compare in the following the liquid flows in a porous body, which is characterized by the minimum and maimum values of the diffusion coefficient and with D = D(w). We consider for this purpose the behavior of the obtained solution versus the diffusion coefficient D. We show in Fig. 1 the solutions for three values of the diffusion coefficient D = 1 1-6, and m 2 /s. It is natural that as the diffusion coefficient reduces the permeability coefficient value decreases an, as a consequence, we have the liquid flow pattern presented in Fig. 1 when curve 2 lies above curve 1 corresponding to a lower value of the permeability coefficient. To improve the method of solving the boundary-value problem for the self-similar flow at D = const equation (5) was solved numerically by a shooting method. As the boundary conditions, the value of the function w() = w and the value of the first derivative w'() = w' were specified at the left boundary. The w value equaled.5 in all cases, and the function slope at point λ = was chosen from the criterion w(λ i ) < ε, where λ i is the finite value of the computational region. For the diffusion coefficient D = m 2 /s the w' value amounted to -.47, for D = it amounted to , and for D = it was equal to.473. The obtained solution is also plotted in Fig. 1. It turned out that the numerical solution coincides completely with the solution obtained analytically. The method of lines was applied for the numerical solution of the boundary-value problem for equation (4) with a subsequent solution of the arising system of ordinary differential equations using a method based on the backward differentiation formula. The numerical solution obtained by the method of lines is also shown in Fig. 1 for three values of diffusion coefficients. The numerical solution of the boundary-value problem for equation (4), which was obtained for large times of the process development, coincides completely with the analytical and numerical solution of stationary equation (5), which was obtained above. The second stage is the general case of a variable diffusion coefficient. Let us obtain the solution of the initial- and boundary-value problem for the original equation (3) for a variable value of the diffusion coefficient. Interpolation of eperimental data of [1] is used to determine the dependence of the diffusion coefficient on humidity. As can be seen (see Fig. 2) the eperimental data are fairly described with the aid of an eponential epression in the following form 8 w wp D( w) = a + cep, c where a = 1.5 1, c = 3 1, w =. 36 and c = p

3 International Conference on Methods of Aerophysical Research, ICMAR 28 w.6.5 a) b) λ **.5.55 w = λ * λ i λ Fig. 3. Influence of the initial humidity on permeability, a) for low values of w, b) for high values of w. In the stationary approimation, equation (3) was solved by the shooting method, its solution for w =.5 is shown in Fig. 1 and in Fig. 3a for different w (in a larger scale). The obtained solution represents the humidity profile in self-similar variables. It can be subdivided into three intervals. The first interval is characterized by a humidity increase from to + to w * for λ *. We replace the point at + with the λ i value, the humidity value at which corresponds to a deviation of the humidity profile from the zero value by the quantity ε = 1 1-4, the λ i values are presented for different initial humidity values in the table (Table 1). The point λ * is marked in Figs. 1 and 3а. Thus, the complete width of the humidity profile in self-similar variables amounts to λ i = The second interval is the interval from point λ = λ * to λ = λ ** (point λ ** is shown in Figs. 1 and 3a). This interval is determined by the wave width after Prandtl (l Pr = 5.621). In the interval under consideration, an abrupt increase in humidity is observed, and the inflection point also lies therein, after which the humidity profile changes from the concave one to a conve one. In this region, there is also the maimum gradient, it corresponds to the coordinate λ ma, the humidity value w ma and to its derivative w' ma. For the initial humidity w =.5 these values correspond to 28.6,.25149, and.8925, respectively (see Table 1). The third interval of the humidity profile starts at λ = λ ** 25.8, and terminates at λ =. In the given interval, the humidity profile has a conve shape and smoothly increases from w ** and adheres to the initial humidity value w specified at the left boundary. The solution of the initial- and boundary-value problem (3), (2) was also carried out in the nonstationary approimation, having the purpose of obtaining a self-similar flow by the pseudounsteady method. Figure 4 shows the humidity profiles obtained at the moments of time t = 1, 1.5, 2, 5, 1, and 1. Note that the humidity profile obtained at the time value 1 coincides with the one obtained above in the stationary approimation. At a further increase in t up to 1 and higher the humidity profile remained invariable that is the same as for t = 1 (see Fig. 4). Thus, we have shown that as t there is a passage to the self-similar liquid flow in a porous sample, this flow type proves to be stable. Initial humidity influence. Let us investigate the initial humidity influence on the flow behavior in the sample. Figure 3 shows the humidity distribution along the self-similar variable λ under the initial humidity variation w varying from.5 to.95 with step.5. At small values of w from to.15 the humidity profile in self-similar variables has the shape of a concave monotonously increasing function in the entire interval of λ values from λ = λ i to. The maimum humidity gradient (w' ma ) lies at the left boundary, thus, the second and third regions do not eist in the interval of w under consideration (see Fig. 3а). Consequently, the impregnation wave front is not realized for such parameters. Within increasing initial humidity w from.15 to 5, a shift of the maimum humidity gradient from the left boundary towards the higher λ values is observed (see Table 1), thus, the second region formation occurs. Starting from w = 5, the first region appears. At a subsequent increase in the initial humidity, a shift of the (λ ma ) location of the maimum humidity gradient towards higher λ values is observed (see Table 1). This is related to the beginning of the phase of an intense growth of the diffusion coefficient depending on humidity (see Fig. 2). 3

4 Section V Table 1. w λ i λ ma w' ma w ma l Pr The following peculiarities are observed as w increases from.5 to.95. Firstly, as the initial humidity increases the predecessor region gradually reduces down to the step size along the selfsimilar variable for w =.75, and then vanishes completely (see Figs. 3a and 3b). Secondly, there is a gradual reduction of the wave width after Prandtl, and for w =.7 the formation of an internal discontinuity is observed for λ ma = and the value w ma = At a further increase in the initial humidity up to w =.75 and so on, the discontinuity amplitude increases, and the etension of the second region that is the wave width reduces (l Pr ). Finally, a considerable nearly two-fold increase in the etension of the third region during each step w =.5 is observed. This means that a wave mechanism of the impregnation is organized when an abrupt increase in saturation occurs in a region of small etension. Then the humidity relaation proceeds up to the boundary value in the etended zone. Hence, at large values of the initial humidity, the humidity profile in self-similar variables has an etended smoothly increasing conve form starting from the jump in which the humidity increases from zero to w ** (Fig. 3b). Solution in the region of physical variables. Let us go over to the posed problem solution in the plane of physical variables and t. The solution of the initial- and boundary-value problem (1), (2) was carried out numerically by the above-tested method..5 w t = (1) 1 5 λ Fig. 4. Obtaining of a stationary humidity profile by the pseudo-unsteady method in the self-similar approimation. 4

5 International Conference on Methods of Aerophysical Research, ICMAR 28 W a).2 t = W b) W.6 t = c) t = Fig. 5. Propagation of the humidity wave a) W =, b) W =.5, c) W =.7. Humidity wave propagation. Figure 5 shows the distribution of humidity profiles for moments of time 1 1 with step 1, 1 1 with step 1, and 1 1 with step 1 for three values of the initial humidity: W = (Fig. 5 а), W =.5 (Fig. 5 b), and W =.7 (Fig. 5 c), and Table 2 presents the values of main parameters of the humidity profiles. Analyzing the humidity profiles for W = one can note that as the time increases an intense growth of the filtration impregnation occurs that is i >> ma. With increasing t the humidity profile slope becomes more slanted, and the maimum humidity gradient value becomes less that is W' ma, and as a consequence of this, an increase in the width after Prandtl of the humidity wave width occurs. 5

6 Section V Table 2. W t i ma W' ma W ma L Pr As W increases up to.5 the picture of the humidity profile propagation changes considerably, in particular, the propagation velocity of the filtration predecessor remains nearly the same as for the above-considered case of W =, and the profile inflection point (the wave center) has accelerated and amounted to about 5 % of the predecessor velocity. A nearly two-fold narrowing of the wave width after Prandtl in comparison with the case of W = has also occurred (see Table 2). A further increase in the initial humidity up to.7 leads to a change in the humidity profile form. In particular, the humidity profile has a small filtration region, an abrupt spasmodic growth, and a subsequent etended increase up to the value at the left boundary, which is, however, small in its amplitude. As the moisture propagates into the porous material the humidity profile slope increases, but the humidity wave width after Prandtl is nearly 8 times lower than for W =.5 and by 16 times lower than for W =. Note that the propagation velocity of the humidity wave has increased considerably, by a factor of nearly 1 as compared to the case of W =.5. As is seen in Fig. 5c the main alteration in the humidity profile occurs in the third region. Its etent rapidly increases and begins to take practically the entire width of the humidity profile. Note that as the time increases the propagation velocity of the humidity profile slows down, so in the time interval from 1 to 1 the velocity decreases by 1 times. And a similar trend is observed for W =.5 and. One more peculiarity is the location of the humidity maimum gradient point or the inflection over the profile height at W = W ma.22, which amounts to 73 %; at W =.5 W ma.25, it amounts to 5 %, and for W =.7 W ma.25, it amounts to 35 %. Thus, the inflection point shifts along the profile from the very top (1 %) downwards and stops near the value W ma.26 or about 3 % of the entire profile height for W =.9. 6

7 International Conference on Methods of Aerophysical Research, ICMAR 28.5 W, w t = 1 l = 1 1 1, λ Fig. 6. Comparison of computational results obtained in the plane of physical variables (solid line) and the self-similar approimation (dotted line). For the initial humidity W =.5 the test series of computations was eecuted whose results are presented in Fig. 6. The solid line shows here the humidity profiles obtained at the solution of problem (1), (2) in the physical plane of the motion at the moments of time t = 1, 1, 1 and the computational region etension l = 1, 1, and l = 1, respectively. A feature of the mapping is the alteration of scale of the abscissa ais because at a passage from l = 1 to 1, and then to 1 the ais scale increases by one order of magnitude. The thick dashed line shows the computed results obtained at the solution of the initial- and boundary-value problem (3) and (2) in selfsimilar variables for the moments of time t = 1 and t = 1 (1) (these lines were presented above in Fig. 4 in the form of solid lines). Note that relation l t equals 1 for all three cases. The solution obtained in the plane of physical variables is seen to coincide with the solution obtained in the self-similar approimation. W = 9 mm Fig. 7. Humidity distribution over the section. t, h 7

8 Section V, mm W Table 3. Moisture distribution over the section of porous material. Let us investigate the dynamics of moisture variation in sections 1, 2,, 9 mm in the beam of a porous material whose length l = 1 mm. Figure 7 shows the moisture distribution in the sections under consideration versus time. The humidity profile in each section may be subdivided into three intervals. The first interval (I) is characterized by a zero humidity value, it starts from the time of the eperiment beginning and terminates by the time when the humidity wave reaches the section under consideration. In the second interval (II), the humidity increases smoothly up to W К. The third interval (III) is characterized by a constant humidity value as the first interval, but the constant humidity value equals W К. The finite humidity values in the sections for different values of the initial humidity are presented in Table. 3. Note the features arising at the initial humidity variation. As the section under consideration shifts from the plane of contact with water into the beam depth an increase in interval I is observed. When W =.1 the interval II of the passage from W to W К, is the most etended and amounts to about 1 hours, and it is characterized by the minimum gradient. The W К value varies considerably depending on the section, so at a passage from section 1 to 9 mm the variation of W К makes 8 %. As W increases from.1 to.4 an insignificant reduction of the transition region II occurs, and the variation of the W К values remains practically within the same ranges. At W =.5 the transition region II is narrowing and amounts to about 1 hours, and the range of the W К variation amounts to about 25 %. An increase in W from.5 to.9 is characterized by an intense narrowing of interval II from hours to minutes and seconds. The slope gradient increases considerably and reaches a discontinuous one, and the range of the W К values is narrowing. At W =.9 the humidity transition (II) from W to W К becomes very narrow and spasmodic and amounts to about 5 seconds, and the deviation of W К varies by no more than.5 % depending on the section. Mathematical model verification. A comparison of the results of numerical computations eecuted by the proposed mathematical model with eperimental data obtained in the work [1] has been carried out. As is seen in Fig. 7, the proposed mathematical model adequately describes the eperimental data (cross sections = 4 and 6 mm). Conclusions. 1. A diffusion physical and mathematical model of the capillary impregnation of a porous material, which is based on the obtained analytical epression for the diffusion coefficient describing fairly the eperimental data, has been proposed and verified. 2. The mathematical technology for solving the one-dimensional nonstationary problems of impregnation has been developed and tested on a self-similar solution of the corresponding boundary-value impregnation problem. 3. It has turned out that at large times of the impregnation process development, the liquid flow in the sample is described by a solution of the self-similar type. Its stability is shown. 8

9 International Conference on Methods of Aerophysical Research, ICMAR Depending on the initial humidity at the sample boundary it proved to be possible to give a classification of the moisture diffusion in the form of: a smooth impregnation wave, a diffusion impregnation wave followed by a predecessor ahead of the transitional structure and a diffusion relaation zone towards the boundary value of saturation, a wave with an abrupt change of the profile in the wave head. The work was made under a partial financial support of the Russian Foundation for Basic Research (grant No ), the grant of the Russian Federation President MK REFERENCES 1. Nisovtsev M.I., Stankus S.V., Sterlyagov A.N., Terekhov V.I. and Khairulin R.A. Eperimental determination of moisture diffusion coefficients in porous materials at a capillary and sorption moistening // J. Eng. Phys. 25. Vol. 78, No. 1. P