1D Verification Freezing and Thawing

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1 1 Introduction 1D Verification Freezing and Thawing An important class of problems in geotechnical engineering involves phase change. Typical examples include ground freezing for the construction of mine shafts and tunnels, analysis and design of thermosyphons in permafrost regions, and the effect of climate and surface disturbance on the thermal regime in the ground. A phase change problem involves the liberation or adsorption of thermal energy in the zone under-going the change of state. Numerical analysis of phase change problems is very challenging because of the vastly different material properties on each side of the phase change front. In particular, the latent heat of fusion is included in the specific heat capacity term for the soil undergoing phase change. The latent heat of fusion for water is approximately two orders-of-magnitude greater than the average heat capacity of a fully saturated soil. As a result, phase change analysis is prone to numerical oscillation. The objective of this example is to benchmark TEMP/W to a well-established closed-form solution for the freezing and thawing of liquid water. GeoStudio feature highlights include: 1. Comparing TEMP/W to closed-form solutions for freezing and thawing; 2. Establishing initial conditions using an activation temperature; and, 3. Demonstrating the robustness of the TEMP/W under-relaxation scheme for controlling numerical oscillation. 2 Numerical Experiments and Closed-Form Solutions The numerical experiment considers two cases: freezing and thawing of pure liquid water. Analysis of liquid water was selected over soil because it is more numerically demanding. Soil comprises both solids and liquid water, so the amount of heat that must be liberated or adsorbed during freezing and thawing is lower than the water-only scenario. Moreover, pore-water in soil does not freeze at a single temperature due to the generation of cryogenic suction. This phenomenon is manifest in the unfrozen water content function of a soil, which implies that the latent heat is released incrementally over a range of temperatures. The model domain consists of a 10 m column of elements having a global element size of 0.2 m. The spatial discretization has been refined in the upper 3 m of the profile due to the sharpness of the phase change front in this zone. The element size in the vertical direction was set to 1 cm by clicking on Draw Mesh Properties, selecting the left edge of the domain, and generating elements with a length of 1 cm (Figure 1). TEMP Example File: Verification Freezing and Thawing (pdf)(gsz) Page 1 of 6

2 Figure 1 - Mesh setting for Line 1 in upper 3 m Figure 2 presents the material properties used for the both the freezing and thawing cases. Note that the insitu volumetric water content has been set to 1.0 in order to model only water. Two materials have been created in this project for the sole purpose of activating the regions at different temperatures. The activation temperature has been set to + 2 C and 2 C for the freezing and thawing cases, respectively. The activation feature is a convenient technique for establishing the initial conditions in a geo-thermal analysis, although any of the other options available in GeoStudio could have been used. Figure 2 - Material properties and activation temperature TEMP Example File: Verification Freezing and Thawing (pdf)(gsz) Page 2 of 6

3 Elev. (m) GEO-SLOPE International Ltd, Calgary, Alberta, Canada The latent heat of fusion for water is set to J/m 3 in the project settings (Set Units and Scale). For the freezing case, the top and bottom boundary conditions are set to 2 C and + 2 C, respectively, while the boundary conditions are opposite for the thawing case. The duration of both analyses is 100 days and the analysis is solved using ten time steps of 10 days each. Closed-form analytical solutions to the phase change problem, which is commonly called the Stefan Problem in the literature, are presented by Carslaw and Jaeger (1986). These solutions were developed by Franz Neumann in the 1860 s. The Neumann equations were developed for a semi-infinite region initially at a constant temperature. The boundary temperature at x = 0 is either increased or decreased above/below the phase change temperature, while the far-field boundary (x ) is held constant at the initial temperature. The equations are not presented herein because the solution is non-linear and requires a mathematic root finding scheme. 3 Results and Discussion Figure 3 and Figure 4 present the freezing and thawing temperature profiles, respectively, calculated using TEMP/W and the analytical solutions for days 10, 50, and 100. TEMP/W matches the closed-form Neumann solution with the exception of the first time step. Both the location of the phase change front and the shape of the temperature profile are in-keeping with the analytical solution days 50 days 100 days Neumann's Solution (10 days) Neumann's Solution (50 days) Neumann's Solution (100 days) Temp. (Deg. C) Figure 3 - Results for freezing case TEMP Example File: Verification Freezing and Thawing (pdf)(gsz) Page 3 of 6

4 Figure 4 - Results for thawing case It is interesting to note that the location of the phase change front on the first time step matches the Neumann solution very accurately; however, the shape of the temperature profile is inconsistent with the analytical solution. This is due to non-convergence caused by numerical oscillation. The first time step is particularly difficult for this class of problem because of the sharp inflection at the phase change front. Figure 5 presents a plot of iteration count verses elapsed time for the freezing case. The maximum number of iterations was set to 100 in the convergence tab under KeyIn Analyses. Only the first time step reached the max iteration count, indicating that it did not converge. Regardless, the location of the phase change front is accurate because of the robust under-relaxation (UR) scheme implemented in TEMP/W. The UR scheme ensures that the iterative solution moves in a controlled and incremental manner toward the final solution. It also dampens the numerical oscillation as the solution is approached. Unfortunately, even the slightest oscillation near the actual answer prevents the convergence criteria from being met. Additional effort in adjusting the spatial and temporal discretization and under-relaxation rate is simply not warranted given the accuracy of the solution. TEMP Example File: Verification Freezing and Thawing (pdf)(gsz) Page 4 of 6

5 Iteration C ount GEO-SLOPE International Ltd, Calgary, Alberta, Canada Iteration Count vs Time Time (days) Figure 5 - Iteration count verses time. Figure 6 presents a plot of freezing front depth verses time. TEMP/W compares very well with the closed-form solution. The slight discrepancy is due solely to the spatial resolution of the TEMP/W analysis. An impractically fine level of discretization would be required to determine the exact location of the phase change front. Figure 6 - Comparison between TEMP/W and Neumann solution for freezing front depth TEMP Example File: Verification Freezing and Thawing (pdf)(gsz) Page 5 of 6

6 4 Summary and Conclusions In this example, TEMP/W is benchmarked against the Neumann closed-form analytical solution for one the most challenging phase change problems available: freezing and thawing of pure water. TEMP/W produces accurate results that are in-keeping with the analytical solution. The robust under-relaxation scheme implemented in the software dampens numerical oscillations and promotes convergence for steep freeze/thaw fronts. TEMP Example File: Verification Freezing and Thawing (pdf)(gsz) Page 6 of 6