12. Kolloquium Bauen in Boden und Fels - Januar 2020 445 Effect of free convection flow in water on frozen soil - an experimental and numerical study Mustafa Mustafa, MSc. University of Stuttgart, Institute for Geotechnical Engineering, Germany Dipl.-Ing Patrik Buhmann University of Stuttgart, Institute for Geotechnical Engineering, Germany Univ.-Prof Dr.-Ing. habil. Christian Moormann University of Stuttgart, Institute for Geotechnical Engineering, Germany Abstract The aim of this work is to investigate the effects of free convection flow of a free fluid during the melting of frozen porous media using the moving boundary method. The significance of this effect on the shape of the moving boundary as well as the temperature distribution in the melt region is shown. A physical experiment was conducted, where a frozen block of soil, insulated from 5 sides is lowered into a water tank and is allowed to thaw. The temperatures in the soil block were measured using pre-installed temperature sensors. First, a 1D numerical model was developed, where a constant heat flux was assumed and used as a boundary condition as a way to describe the free convection flow. A 2D model is used to better capture the free convection flow and the interaction between both solid and fluid domains. In both cases the frozen domains were considered to be porous media with their properties described by effective parameters. The model results are compared to the results of the experiments. 1. Introduction In recent years, the phase change model has received considerable attention due to its wide range of applications in various fields of engineering and environmental sciences, especially in artificial ground freezing which has been used for decades in engineering applications such as tunnel excavation, land slide stabilization, and in mining engineering, as well as in environmental engineering as a way of hazardous waste containment, however, the majority of these studies are mainly focused on the forced convection heat transfer process, due to its significant effects on the phase change phenomena, while the effects of natural convection flow in porous media have been previously investigated both experimentally and numerically, where one study by Weaver and Viskanta [1] performed melting experiments in cylindrical capsules filled with different sizes of glass and aluminium beads as the porous layer and water (and ice) as the phase change material. Another study by Beckermann [2] investigated the natural convection flow effects in a vertical rectangular enclosure partially filled with a vertical porous layer, with glass beads as the porous material (similar to a study by Weaver [4]) and gallium as the phase change material, and it was found that the natural convection flow considerably influences the shape and motion of the boundary during both melting and solidification experiments as the thawing front moved faster near the top where the temperature is higher due to the nature of free convection flow and slower near the bottom due to the liquid cooling as it descends. On the other hand, these studies were based on porous layers with a relatively high hydraulic conductivity which can cause significant temperature alterations and emphasise the effects of free convection flow, which can have a very different effect on porous media with a relatively lower hydraulic conductivity. Another study [3] investigated the effect of free convection flow during freezing of water, both numerically and experimentally, and the results are in good agreement for the initial time of the freezing process, however there are some discrepancies between the numerical prediction and the experiment became more significant. These studies [1, 2, 3] reported some anomalies when comparing the experimental and numerical results during solidification experiments due to a super cooling phenomena, where phase change occurred at temperatures much lower than the freezing temperatures of the liquid, which could not be modelled correctly, however, no super heating phenomena has been reported, therefore, this study will only consider the thawing part of the phase change process. 446 12. Kolloquium Bauen in Boden und Fels - Januar 2020 Effect of free convection flow in water on frozen soil - an experimental and numerical study The moving boundary method is another method that was implemented into the models of this study is, this method has been extensively investigated in the past, and Rattandecho [5], investigated the thermal behaviour of the freezing process in a water saturated porous media using the moving boundary method, and reported that the results of the model are in good agreement with the analytical solutions and experimental results. The objective of this study is to investigate the effect of free convection flow on the phase change process in frozen porous media using the moving boundary method in order to ensure a correct prediction of the location and time of the phase change from liquid to solid (and vice versa) within the material and the surrounding subsoil. This is done by developing two numerical models to simulate the effects of free convection flow and the interaction between both the solid and the fluid domains, then compare the simulation results to the results of an experiment, in an effort to shed some new light on natural convection and its effects on the phase change process in soils. 2. Experimental setup The experiment consists of a rectangular coated plywood container with external dimensions of 43.3 cm x 70.3 cm x 57.8 cm in height, width, and depth respectively, and is lined with 9 cm of Styrofoam insulation which causes the container to have internal dimensions of 58.5 cm in height, 31.5 cm in width, and 52 cm in depth, the container is fitted with an external sensor used to measure the water temperature in the tank near the soil boundary, and an array of 9 temperature sensors at the centre of the container, these sensors are spaced at 5.8 cm across the depth of the container (Shown in figure 2) where the outer sensor is denoted by T2, and the sensor at the surface is denoted T1, and the internal sensors U8 - U1, the sensors are connected to a multi-channel data logger that logs one temperature measurement every 60 seconds. Figure 1 - Experiment setup The container is then filled with fully saturated sandy gravel soil with an average density of 1700 kg/ m 3 , and a porosity of approximately 30%, which was cooled down to a temperature of -19 °C in a climate controlled chamber, the temperature in the container was monitored via the installed sensors and when the soil reached the target temperature, it was then lowered into a temperature controlled water tank with a capacity of 2240 litres and dimensions of 2.0 m x 1.4 m x 0.8 m in length, width, and height respectively. The tank is filled with water at a temperature of 16 °C and the water temperature is monitored via the external sensor (seen in figure 2). The thawing process is then observed both visually and via the pre-installed sensors in the soil container. Figure 2 - Sensor arrangement The results of the experiment are shown in (Figure 3) where T2 is the outer sensor measuring the water temperature in the external tank, and it shows that the water temperature in the tank is decreasing due to the influence of the box, T1 shows the temperature on the surface of the box, and U8 through U1 represent the sensors embedded in the container. The thawing velocity is calculated using the equation (1), however, it should be noted that sensor T1 was excluded from the evaluation due to an anomaly at temperature = -15 °C which could be due to the unevenness of the soil surface, and the bottom sensor U1 is also excluded due to the fact that the box was lifted out of the tank before the thawing front could reach the sensor. Where l is the distance between the two sensors, and t is the time. The average thawing velocity was found to be around 0.57 cm/ min, however, it should be noted that a slight decrease in velocity was noticed towards the end 12. Kolloquium Bauen in Boden und Fels - Januar 2020 447 Effect of free convection flow in water on frozen soil - an experimental and numerical study Figure 3 - Experiment results of the experiment (Figure 4) which could be attributed to the fact that the water temperature in the tank was steadily decreasing due to the influence of the soil sample. It is also worth noting that the soil debris was manually removed from the container after thawing as it was acting as an insulation layer between the frozen soil and the free water which could have affected the thawing velocity. 3. Numerical setup Figure 4 - Experiment thawing velocity Developing an accurate numerical model can be challenging as some boundary conditions (i.e the exact water velocity in the tank, the material specifications, and the varying temperatures affecting the experiment setup) can be difficult to measure accurately, thus, assumptions for some these boundary conditions must be made. 3.1 1D Model The model consists of a 52 cm line representing the frozen soil (Figure 5), Since the frozen soil sample was fully submerged in the tank and is surrounded by water from all sides, a free convection flow is generated in the tank (represented by the arrows in Figure 5), that leads to temperature variability near the exposed surface, which cannot be properly modelled in 1D, thus, the heat flux is taken as a boundary condition and is assumed to be constant by keeping a constant heat flux gradient in order to account for the variability in temperature. Figure 5 - 1D model sketch 448 12. Kolloquium Bauen in Boden und Fels - Januar 2020 Effect of free convection flow in water on frozen soil - an experimental and numerical study Figure 6 - ID model boundary conditions The model boundary conditions are shown in (Figure 6) and the free water is represented by the heat flux gradient, which can be described by the equation: Where is the heat transfer coefficient, Nu is the Nusselt number and k is the hydraulic conductivity, and represents the length required to keep the heat flux constant. The Nusselt number can be obtained from equation (3), where Ra is the Rayleigh number, and Pr is the Prandtl number, and the Rayleigh number (equation 4) is the product of Prandtl and Grashof numbers which are obtained from equations (5) and (6). The heat flux gradient remains constant by introducing the moving boundary, where the free water domain moves with the thawing front, keeping the water domain length and, in turn, the heat flux gradient constant. The boundary velocity can be described by the equation: Where is the mesh velocity, is the Lagrange multiplier for temperature, is the soil density, and is the latent heat of fusion. While the heat flux equation can be described by Fourier’s law, and the heat conduction equation is described in equation 2.3. The phase change phenomena can be described by the equations: Figure 7 shows the results of the 1D model, where the X axis represent the model length in centimetres, and the y axis represents the temperature, while the lines represent the state of the soil sample each 10 minutes. Where is the heat capacity, u is velocity field, q is the heat flux, k is the thermal conductivity, and is the temperature gradient. 12. Kolloquium Bauen in Boden und Fels - Januar 2020 449 Effect of free convection flow in water on frozen soil - an experimental and numerical study Figure 7 - 1D model results 3.2 2D Model The two dimensional model unlike the one dimensional model is capable of modelling the free convection flow in both x and y axis which provides a higher level of accuracy, and eliminates the need for a constant heat flux gradient such as the one discussed in section (3.1). The model consists of two domains, free water and frozen porous media, separated by a moving boundary, the boundary conditions shown in (Figure 8), where the water domain has a temperature boundary condition of 16 °C and the frozen domain at -19 °C, while the moving boundary is set to a temperature of 0 °C to represent the thawing front. Due to the moving boundary method, the free water domain expands and the soil domain shrinks, which assumes that the soil “instantly” turns to water as it goes above the temperature threshold. Figure 8 - 2D model boundary conditions The governing equations for heat conduction for the 2D model are described by the following equations: Mass: Momentum: Energy: Where is the density, T is the Temperature, P is the pressure, is the velocity vector, with and being the x and y components of velocity. For a more detailed description refer to a study by Alexiades [6]. The results of the 2D model are shown in (Figure 9) show a significant difference in thawing velocity between the top and the bottom of the soil container, which can be attributed to the free convection flow in the tank, represented by the black arrows in Figure 9, causing the colder water particles to sink to the bottom thus reducing the thawing velocity in the adjacent soil The thawing velocity was monitored at 5 different points along the moving boundary (Figure 10), the maximum velocity was recorded at the top of the container at 0.45 cm/ min, the thawing velocity shows a steady decline towards the bottom of the container where the minimum velocity was recorded at 0.11 cm/ min. Figure 9 - Free Convection effects on the moving boundary Figure 10 - Monitoring points Point Velocity [cm/ min] M1 0.45 M2 0.24 M3 0.17 M4 0.14 M5 0.11 Table 1 - 2D model thawing velocities 450 12. Kolloquium Bauen in Boden und Fels - Januar 2020 Effect of free convection flow in water on frozen soil - an experimental and numerical study 4. Results and discussion The one dimensional model shows that thawing velocity is around 0.32 cm/ min (Table 2), this velocity is also kept constant due to the constant heat flux. While the thawing velocity in the 2D model is a little more trivial to simulate, since the shape of the moving boundary and in turn, the thawing velocity is significantly altered by the free convection flow, the velocity at the top is significantly higher than the velocity at the bottom of the container due to temperature variations. Experiment 1D 2D Min 0.48 0.32 0.11 Max 0.64 0.32 0.45 Average 0.57 0.32 0.19 Table 2 - Thawing velocity [cm/ min] The temperature difference between the experiment, and the models can be attributed to the soil parameters, these parameters were assumed since they couldn’t be accurately measured due to the heterogeneity of the soil sample, another factor which can have a significant influence on the results is the flow variability (both vertical and horizontal flow) and the tank size, since only a small portion of the water directly adjacent to the soil sample was simulated in the 2D model which does not accurately describe the flow and temperature variability in the tank during the experiment, where the relatively small volume of water simulated is heavily influenced by the temperature of the soil sample, moreover, it is clear that due to the free convection flow, the colder water particles settle at the bottom of the tank which explains the slow thawing velocity, which can significantly influence the average thawing velocity in the container. 5. Conclusion An experiment has been conducted to investigate the effect of free convection flow on a frozen soil sample, two models were developed. A one dimensional model where a constant heat flux was used to describe the effect of the free convection flow. A second model was developed to better describe the free convection flow and implementing the moving boundary method, this moving boundary utilizes an interface that separates the two domains and moves in the direction of the heat fluxes. It was found that the free convection flow can have significant effects on both the shape of the moving boundary as well as the thawing velocity. The model results were presented, and the factors influencing the results were also explained in detail. Further investigation is required to further develop the model and to investigate whether a 3D model will be required to describe the phenomena more accurately. References [1] Weaver J. A.; Viskanta R.: Melting of frozen, porous media contained in a horizontal or a vertical, cylindrical capsule. Int. J. Heat Mass Transfer. Vol. 29. NO.12, pp.1943-1951,1986. [2] Beckermann C.; Ramadhyani S.; Viskanta R.: Natural Convection flow and heat transfer between a fluid layer and a porous layer inside a rectangular enclosure. Journal of heat transfer. Vol. 109, pp.363, 1987. [3] Giangi M.; Stella F.; Kowalewski A.: Phase change problems with free convection: fixed grid numerical simulation, Computing and Visualization in Science Vol. 2, pp. 123-130, 1999 [4] Weaver K. A.; Solid liquid phase change heat transfer in porous media. MSME Thesis, Purdue University, 1985. [5] Rattanadecho P.; Wongwises S.: Moving boundary moving mesh analysis of freezing process in water saturated porous media using a combined transfinite interpolation and PDE mapping methods; Journal of heat transfer Vol. 130 pp.012601-1 : 012601-10 , 2008. [6] Alexiades V.; Hannoun N.: Tin melting: Effect of grid size and scheme on the numerical solution, , Electronic Journal of differential equations, pp. 55-69, 2003.