The main vector of development in ZSoil v2024 has been on simulating heat exchangers in 3D. This new development is based on the original work by H.J.G. Diersch et al. (2011) Finite element modeling of borehole heat exchanger system. Part 1. Fundamentals. Computers & Geosciences 37: 1122-1135.
HEAT EXCHANGERS
A heat exchanger is a system or a device which is used to transfer heat between a source and working fluid that are at different temperatures. Heat exchangers are used in both heating and cooling processes. In geotechnics they are widely used as borehole heat exchangers (BHEs) or heat exchanger piles. Integrating heat exchanger pipes with structural foundations in one system creates a new renewable solution providing heating and cooling for the built environment.
Heat exchanger pipes attached to the reinforcement cage of the pile
The heat exchangers in ZSoil v2024 can be modeled using 1D elements, including explicit or implicit grout modelling, embedded within the continuum elements mesh in an arbitrary way, using non-local constraints to avoid any mesh dependency (for detail see new non-local approach FE for modelling piles and barrettes).
Available thermal analyses
To solve a complex thermo-hydro-mechanical soil-structure interaction boundary value problem in which heat exchangers play a role in the design procedure, the following two strategies are proposed:
1. Weakly coupled hydro-thermal formulation which consists of a two-stage procedure where fluid flow is first computed and is then mapped onto a thermal simulation (Fig.1). In the first stage, the standard fluid flow problem is solved. In the second, the thermal analysis is carried out taking into account the following phenomena that are computed for a two-phase medium :
-
- partial saturation
- advection
- effective thermal conductivity
- and volumetric heat capacity
The implemented mapping algorithm makes it possible to efficiently transfer fluid flow to the thermal model even if the two meshes or time step increments are different.
Fig.1 Weakly coupled hydro-thermal analysis strategy consisting of mapping results from a “flow” project to a thermal analysis
Fig.2 Weakly coupled hydro-thermal analysis strategy: (left) associating of previously simulated “flow” model to a thermal (heat) one, (right) new fluid thermal properties to be defined in a two-phase analysis (advection term can be locally neglected)
2. Weakly coupled thermo-hydro-mechanical formulation (known from previous ZSoil versions) in which the temperature field is mapped onto the hydro-mechanical FE mesh, and the temperature field which results from heat exchangers is mapped onto the macro pile elements.
Fig.3 Weakly coupled thermo-hydro-mechanical analysis consisting of associating the results from a previously simulated “heat” project to a mechanical or hydro-mechanical analysis.
Modelling of heat exchangers
Two general approaches to model heat exchanger elements are brought to you with ZSoil v2024.
1. Heat exchanger with EXPLICIT GROUT
The heat exchanger element models the pipe and refrigerant only while the grout should be defined using standard continuum elements. This option makes it possible to model complex pipe systems that can be embedded in any user-designed geometry of the grouting material.
Fig.4 An example of modelling of a pile exchanger by means of the heat exchanger elements with EXPLICIT GROUT
2. Heat exchanger with IMPLICIT GROUT
The heat exchanger element consists of pipes, refrigerant and grout and all thermal resistances are computed based on analytical models. In this group four types of typical geometries have been developed which are shown in Fig.5. Therefore, this option can be applied to model heat exchangers that are embedded in the circular section of the grouting material.
Fig.5 The four types of heat exchanger elements with IMPLICIT GROUT
The heat exchanger can be defined in the material dialog window at the end for the material type list:
Fig.6 Heat exchanger in the material formulation list
Material properties for each type of the heat exchanger can be defined under HEAT property group.
Fig.7 Dialog window for borehole heat exchanger
Heat exchangers can be embedded within the soil mesh in an arbitrary manner using the concept of non-local constraints. This concept removes mesh dependency when 1D heat exchangers elements are linked with the 3D mesh representing continuum. This effect is illustrated in the following figures which are taken from a benchmark that deals with a transient heat transfer problem for a CXA heat exchanger. The first image illustrates an initial 3D mesh while the following ones show the mesh refinement based on the mesh tying option which makes it possible to ensure continuity of results between coarse and fine mesh zones.
Fig.8 Initial FE mesh size for benchmarking mesh dependency
Fig.9 Mesh refinement from 1m element size to 0.5, 0.25, 0.125 and 0.0625m
Fig.10 Refrigerant temperature time history obtained at the outlet for the laminar flow case, conforming soil mesh, with the inner zone grid size he = 1m
(exchanger soil nodes coincide with the soil mesh nodes)
Fig.11 Refrigerant temperature time history obtained at the outlet for the laminar flow case, conforming soil mesh, with the inner zone grid size he = 1m
(exchanger soil nodes are linked to the soil mesh using non-local constraints)
Fig.12 Refrigerant temperature time history obtained at the outlet for the laminar flow case, locally refined soil mesh, with the inner zone grid size he = 0.0625m
(exchanger soil nodes coincide with the soil mesh nodes)
Fig.13 Refrigerant temperature time history obtained at the outlet for the laminar flow case, locally refined soil mesh, with the inner zone grid size he = 0.0625m
(exchanger soil nodes linked using nonlocal constraints to the soil mesh)
To minimize mesh dependency, local mesh refinement and application of non-local constraints are advised. For further information refer to ZSoil’s report dedicated to heat exchangers.
PARTIALLY SATURATED CONTINUUM
In the ZSoil v2024, the modified Bishop’s effective stress principle which was introduced in ZSoil v23.50, becomes the default formulation for hydro-mechanical analyses, and it relies on the full form of the van Genuchten Soil Water Retention Curve (SWRC).
Full form of the van Genuchten soil water retention curve
The current upgrade offers a full form of the van Genuchten’s law including 3 parameters. It is expressed in the following form:
The m parameter depends on the n value (m = 1 – 1/n), and n no longer needs to be fixed to value 2 as it was in the previous versions of ZSoil (versions < 23.50). Note that here n is a measure of the pore-size distribution and not the porosity.
In order to help the user to define the three parameters Sr, alpha and n, a dedicated calculator is proposed in the material window under group Flow. Based on basic geotechnical data i.e. eo, d60, d10 for coarse grained, and eo , LL (liquid limit) for fine grained soils, a best fit of the full van Genuchten model to the Modified Kovacs model can be obtained. The new user interface for Flow properties is illustrated in Fig.14, whereas the van Genuchten’s parameters estimator is shown in Fig.15. It makes it possible to find van Genuchten’s parameters and visualize a comparison with the Modified Kovacs model.
In addition, the effective suction pressure which appears in the Bishop’s effective stress principle, is displayed in order to anticipate the resulting maximal apparent cohesion value. The effective suction pressure may appear in a partially-saturated medium. Parameter alpha is the only one which affects the maximal value of the apparent cohesion. In order to keep back compatibility with previous ZSoil versions, the user may set n=2 and modify Sr and alpha values.
Modified Bishop’s effective stress principle
The current upgrade introduces a modified formulation for the Bishop’s effective stress principle. The new formulation makes it possible to maintain control over the resulting apparent cohesion. It takes following form:
where the modified effective saturation is expressed as follows:
In addition, the Biot’s coefficient (only the elastic part) has been introduced to make numerical analyses in rock geomaterials more realistic.
In order to keep back compatibility with previous ZSoil versions (<23.50), the user can always use the previous form of the Bishop’s principle. This can be done under the new menu item Control/two-phase formulations (see Fig.14). All older projects by default will use standard saturation ratio in the Bishop’s principle while in every new project the modified effective one will be set as default. It is important to note that the modified effective saturation preserves monotonic and asymptotic behavior of the resulting apparent cohesion with increasing suction.
Fig.14 The new user interface for setting seepage-related properties
Fig.15 Calculator for SWRC parameters based on the best fit of van Gencuhten’s curve to the Modified Kovacs model
Fig.16 Defining Bishop’s principle formulation under Control/two-phase formulations menu
Saturation state profiling
In addition, using the Initial State Profile tool which is included in the Virtual Lab, it is now possible for the user to inspect the profiles of initial state variables that control the new formulations. It makes it possible to test the sensitivity of each parameter describing material behavior (see Fig.17) in terms of:
• Total stress (gravity analysis)
• Saturation (van Genuchten’s model for the soil water retention curve), e.g. Fig.18
• Effective stress (Bishop’s principle),
• Overconsolidation (preconsolidation state depending on mechanical constitutive model)
• Permeability (Irmay or Mualem model for permeability of partially-saturated medium), e.g. Fig.18
Fig.17 Definition of parameters in the initial state profile tool in Virtual Lab
Fig.18 Visualization of partial-saturation effects in the initial state profile tool in the Virtual Lab
Ko cut off in the Hardening Soil model for large OCR values
In the Hardening Soil model the initial Ko coefficient can automatically be calculated based on a given OCR or POP (preoverburden pressure) profile using a commonly recognized formula
For certain cases at shallow depths, the OCR values computed from POP definition may lead to very high Ko values. Starting from the version 23.50, the user may set the upper limit for the Ko coefficient for computing the initial in situ stresses (see Fig.19). Note that the default upper limit is defined by the passive earth pressure coefficient Ko.
Fig.19 Setting the initial Ko value for the Hardening Soil model including the user-defined upper limit for K0
Fig.20 Testing of initial Ko setup using Initial state profiler in the Virtual Lab
PREPROCESSING
The new ZSoil v2024 brings a few brand new tools which aim at improving FE model preparation and data verification :
- bulk edit of material properties in the grid layout (Fig.20) for an efficient material data inspection
- inspection grid for verifying association of existence functions to any type of element in a FE model (Fig.21 and Fig.22)
- inspection grid for verifying association of load functions to any type of element in a FE model (Fig.23 and Fig.24)
- import of color lines from CAD drawings and coloring geometrical objects (Fig.25) for better geometrical visibility, selection and control
- customized model verification in terms of data and geometrical consistency (Fig.26)
Fig.20 Bulk edit of material properties in the grid layout
Fig.21 Inspecting all the defined existence functions and the corresponding associated types of elements
Fig.22 Inspecting all existence functions associated to finite elements – “shell one layer” in this example
Fig.23 Inspecting all the defined load functions and the corresponding associated to types of elements
Fig.24 Inspecting all load functions associated to the defined surface loads
Fig 25. Importing color lines from CAD drawings and coloring geometrical objects
Fig.26 Customizing model verification in terms of data and geometrical consistency
VIRTUAL LAB
The Virtual Lab v2024 provides the user with new correlations with a special reference to soil characteristics defining the full form of the van Genuchten soil water retention curve model (see above):
- particle diameters corresponding to 60% finer (d60)
- particle diameters corresponding to 10% finer (d10)
which are used to identify the van Genuchten model parameters:
- measure of the pore-size distribution n
- saturation constant alpha
All these properties can be determined in the Automatic or Interactive mode for parameter selection.
Fig.27 Determination of particle diameters corresponding to 60% finer with the aid of the Virtual Lab
Fig.28 Determination of parameter n (measure of the pore-size distribution) using the Virtual Lab