## Fachbereich Maschinenbau und Verfahrenstechnik

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- Coupling of Computational Fluid Dynamics and Population Balance Modelling for Liquid-Liquid Extraction (2010)
- The aim of this thesis was to link Computational Fluid Dynamics (CFD) and Population Balance Modelling (PBM) to gain a combined model for the prediction of counter-current liquid-liquid extraction columns. Parts of the doctoral thesis project were done in close cooperation with the Fraunhofer ITWM. Their in-house CFD code Finite Pointset Method (FPM) was further developed for two-phase simulations and used for the CFD-PBM coupling. The coupling and all simulations were also carried out in the commercial CFD code Fluent in parallel. For the solution methods of the PBM there was a close cooperation with Prof. Attarakih from the Al-Balqa Applied University in Amman, Jordan, who developed a new adaptive method, the Sectional Quadrature Method of Moments (SQMOM). At the beginning of the project, there was a lack of two-phase liquid-liquid CFD simulations and their experimental validation in literature. Therefore, stand-alone CFD simulations without PBM were carried out both in FPM and Fluent to test the predictivity of CFD for stirred liquid-liquid extraction columns. The simulations were validated by Particle Image Velocimetry (PIV) measurements. The two-phase PIV measurements were possible when using an iso-optical system, where the refractive indices of both liquid phases are identical. These investigations were done in segments of two Rotating Disc Contactors with 150mm and 450mm diameter to validate CFD at lab and at industrial scale. CFD results of the aqueous phase velocities, hold-up, droplet raising velocities and turbulent energy dissipation were compared to experimental data. The results show that CFD can predict most phenomena and there was an overall good agreement. In the next steps, different solution methods for the PBM, e.g. the SQMOM and the Quadrature Method of Moments (QMOM) were implemented, varied and tested in Fluent and FPM in a two-fluid model. In addition, different closures for coalescence and breakage were implemented to predict drop size distributions and Sauter mean diameters in the RDC DN150 column. These results show that a prediction of the droplet size distribution is possible, even when no adjustable parameters are used. A combined multi-fluid CFD-PBM model was developed by means of the SQMOM to overcome drawbacks of the two-fluid approach. Benefits of the multi-fluid approach could be shown, but the high computational load was also visible. Therefore, finally, the One Primary One Secondary Particle Method (OPOSPM), which is a very easy and efficient special case of the SQMOM, was introduced in CFD to simulate a full pilot plant column of the RDC DN150. The OPOSPM offers the possibility of a one equation model for the solution of the PBM in CFD. The predicted results for the mean droplet diameter and the dispersed phase hold up agree well with literature data. The results also show that the new CFD-PBM model is very efficient from computational point of view (two times less than the QMOM and five times less than the method of classes). The overall results give rise to the expectation that the coupled CFD-PBM model will lead to a better, faster and more cost-efficient layout of counter-current extraction columns in future.

- Point defects in piezoelectric materials – continuum mechanical modelling and numerical simulation (2010)
- The topic of this work is the continuum mechanic modelling of point defects in piezoelectric materials. Devices containing piezoelectric material and especially ferroelectrics require a high precision and are exposed to a high number of electrical and mechanical load cycles. As a result, the relevant material properties may decrease with increasing load cycles. This phenomenon is called electric fatigue. The transported ionic and electric charge carriers can interact with each other, as well as with structural elements (grain boundaries, inhomogeneities) or with material interfaces (domain walls). A reduced domain wall mobility also reduces the electromechanical coupling effect, which leads to the electric fatigue effect. The materials considered here are barium titanate and lead zirconate titanate (PZT), in which oxygen vacancies is the most mobile and most frequently appearing defect species. Intentionally introduced foreign atoms (dopants) can adjust the material properties according to their field of application by generating electric dipoles with the vacancies. Agglomerations of point defects can strongly influence the domain wall motion. The domain wall can be slowed down or even be stopped by the locally varying fields in the vicinity of the clusters. Accumulations of point defects can be detected at electrodes, pores or in the bulk of fatigued samples. The present thesis concentrates focuses on the self interaction behaviour of point defects in the bulk. A micro mechanical continuum model is used to show the qualitative and the quantitative interaction behaviour of defects in a static setup and during drift processes. The modelling neglects the ferroelectric switching mechanisms, but is applicable to every piezoelectric material. The underlying differential equations are solved by means of analytical (Green's functions) and numerical (Finite Differences with discrete Fourier Transform) methods, depending on the boundary conditions. The defects are introduced as localised Eigenstrains, as electric charges and as electric dipoles. The required defect parameters are obtained by comparisons with atomistic methods (lattice statics). There are no standardised procedures available for the parameter identification. In this thesis, the mechanical parameter is obtained by a comparison of relaxation volumes of the atomic lattice and the continuum solution. Parameters for isotropic and anisotropic defect descriptions are identified. The strength of the electric defect is obtained by a comparison of the electric internal energies of atomistics and continuum. The appearing singularities are eliminated by taking only the energy difference of a infinite crystal and a periodic cell into account. Both identification processes are carried out for the cubic structure of barium titanate, which decouples the mechanical and the electrical problem. The defect interaction is analysed by means of configurational forces. The mechanical defect parameter generates a directional short-range attraction between defects. An electrical defect parameter produces the long-range Coulomb interaction, which predicts a repulsion of two similar charges. Additionally, an interaction with defect dipoles is taken into account. It is shown that a defect agglomeration is possible for any static defect configuration. Finally, defect drift is simulated using a thermodynamically motivated migration law based on configurational forces. In this context, the migration of point defects due to self interaction, and the influence of external fields is investigated.

- Multi-Scale Modeling and Simulation in Configurational Mechanics (2011)
- This thesis treats the extension of the classical computational homogenization scheme towards the multi-scale computation of material quantities like the Eshelby stresses and material forces. To this end, microscopic body forces are considered in the scale-transition, which may emerge due to inhomogeneities in the material. Regarding the determination of material quantities based on the underlying microscopic structure different approaches are compared by means of their virtual work consistency. In analogy to the homogenization of spatial quantities, this consistency is discussed within Hill-Mandel type conditions.

- Microstructural modeling of ferroelectric material behavior (2011)
- This thesis is concerned with the modeling of the domain structure evolution in ferroelectric materials. Both a sharp interface model, in which the driving force on a domain wall is used to postulate an evolution law, and a continuum phase field model are treated in a thermodynamically consistent framework. Within the phase field model, a Ginzburg-Landau type evolution law for the spontaneous polarization is derived. Numerical simulations (FEM) show the influence of various kinds of defects on the domain wall mobility in comparison with experimental findings. A macroscopic material law derived from the phase field model is used to calculate polarization yield surfaces for multiaxial loading conditions.

- On optimal control simulations for mechanical systems (2011)
- The primary objective of this work is the development of robust, accurate and efficient simulation methods for the optimal control of mechanical systems, in particular of constrained mechanical systems as they appear in the context of multibody dynamics. The focus is on the development of new numerical methods that meet the demand of structure preservation, i.e. the approximate numerical solution inherits certain characteristic properties from the real dynamical process. This task includes three main challenges. First of all, a kinematic description of multibody systems is required that treats rigid bodies and spatially discretised elastic structures in a uniform way and takes their interconnection by joints into account. This kinematic description must not be subject to singularities when the system performs large nonlinear dynamics. Here, a holonomically constrained formulation that completely circumvents the use of rotational parameters has proved to perform very well. The arising constrained equations of motion are suitable for an easy temporal discretisation in a structure preserving way. In the temporal discrete setting, the equations can be reduced to minimal dimension by elimination of the constraint forces. Structure preserving integration is the second important ingredient. Computational methods that are designed to inherit system specific characteristics – like consistency in energy, momentum maps or symplecticity – often show superior numerical performance regarding stability and accuracy compared to standard methods. In addition to that, they provide a more meaningful picture of the behaviour of the systems they approximate. The third step is to take the previ- ously addressed points into the context of optimal control, where differential equation and inequality constrained optimisation problems with boundary values arise. To obtain meaningful results from optimal control simulations, wherein energy expenditure or the control effort of a motion are often part of the optimisation goal, it is crucial to approxi- mate the underlying dynamics in a structure preserving way, i.e. in a way that does not numerically, thus artificially, dissipate energy and in which momentum maps change only and exactly according to the applied loads. The excellent numerical performance of the newly developed simulation method for optimal control problems is demonstrated by various examples dealing with robotic systems and a biomotion problem. Furthermore, the method is extended to uncertain systems where the goal is to minimise a probability of failure upper bound and to problems with contacts arising for example in bipedal walking.

- Dynamic Modelling and Simulation of (Pulsed and Stirred) Liquid-Liquid Extraction Columns using the Population Balance Equation (2012)
- The discrete nature of the dispersed phase (swarm of droplet) in stirred and pulsed liquid-liquid extraction columns makes its mathematical modelling of such complex system a tedious task. The dispersed phase is considered as a population of droplets distributed randomly with respect to their internal properties (such as: droplet size and solute concentration) at a specific location in space. Hence, the population balance equation has been emerged as a mathematical tool to model and describe such complex behaviour. However, the resulting model is too complicated. Accordingly, the analytical solution of such a mathematical model does not exist except for particular cases. Therefore, numerical solutions are resorted to in general. This is due to the inherent nonlinearities in the convective and diffusive terms as well as the appearance of many integrals in the source term. However, modelling and simulation of liquid extraction columns is not an easy task because of the discrete nature of the dispersed phase, which consist of population of droplets. The natural frame work for taking this into account is the population balance approach. In part of this doctoral thesis work, a rigours mathematical model based on the bivariate population balance frame work (the base of LLECMOD ‘‘Liquid-Liquid Extraction Column Module’’) for the steady state and dynamic simulation of pulsed (sieve plate and packed) liquid-liquid extraction columns is developed. The model simulates the coupled hydrodynamic and mass transfer for pulsed (packed and sieve plate) extraction columns. The model is programmed using visual digital FORTRAN and then integrated into the LLECMOD program. Within LLECMOD the user can simulate different types of extraction columns including stirred and pulsed ones. The basis of LLECMOD depends on stable robust numerical algorithms based on an extended version of a fixed pivot technique after Attarakih et al., 2003 (to take into account interphase solute transfer) and advanced computational fluid dynamics numerical methods. Experimental validated correlations are used for the estimation of the droplet terminal velocity in extraction columns based on single and swarm droplet experiments in laboratory scale devices. Additionally, recent published correlations for turbulent energy dissipation, droplet breakage and coalescence frequencies are discussed as been used in this version of LLECMOD. Moreover, coalescence model from literature derived from a stochastical description have been modified to fit the deterministic population model. As a case study, LLECMOD is used here to simulate the steady state performance of pulsed extraction columns under different operating conditions, which include pulsation intensity and volumetric flow rates are simulated. The effect of pulsation intensity (on the holdup, mean droplet diameter and solute concentration) is found to have more profound effect on systems of high interfacial tension. On the hand, the variation of volumetric flow rates have substantial effect on the holdup, mean droplet diameter and solute concentration profiles for chemical systems with low interfacial tension. Two chemical test systems recommended by the European Federation of Chemical Engineering (water-acetone (solute)-n-butyl acetate and water-acetone (solute)-toluene) and an industrial test system are used in the simulation. Model predictions are successfully validated against steady state and transient experimental data, where good agreements are achieved. The simulated results (holdup, mean droplet diameter and mass transfer profiles) compared to the experimental data show that LLECMOD is a powerful simulation tool, which can efficiently predict the dynamic and steady state performance of pulsed extraction columns. In other part of this doctoral thesis work, the steady state performance of extraction columns is studied taking into account the effect of dispersed phase inlet condition (light or heavy phase is dispersed) and the direction of mass transfer (from continuous to dispersed phase and vice versa) using the population balance framework. LLECMOD, a program that uses multivariate population balance models, is extended to take into account the direction of mass transfer and the dispersed phase inlet. As a case study, LLECMOD is used to simulate pilot plant RDC columns where the steady state mean flow properties (dispersed phase hold up and droplet mean diameter) and the solute concentration profiles are compared to the available experimental data. Three chemical systems were used: sulpholane–benzene–n-heptane, water–acetone–toluene and water–acetone–n-butyl acetate. The dispersed phase inlet and the direction of mass transfer as well as the chemical system physical properties are found to have profound effect on the steady state performance of the RDC column. For example, the mean droplet diameter is found to persist invariant when the heavy phase is dispersed and the extractor efficiency is higher when the direction of mass transfer is from the continuous to the dispersed phase. For the purpose of experimental validation, it is found that LLECMOD predictions are in good agreement with the available experimental data concerning the dispersed phase hold up, mean droplet diameter and solute concentration profiles in both phases. In a further part of this doctoral thesis, a mathematical model is developed for liquid extraction columns based on the multivariate population balance equation (PBE) and the primary secondary particle method (PSPM) introduced by Attarakih, 2010 (US Patent Application: 0100106467). It is extended to include the momentum balance for the dispersed phase. The advantage of momentum balance is to eliminate the need for often conflicting correlations used in estimating the terminal velocity of single and swarm of droplets. The resulting mathematical model is complex due to the integral nature of the population balance equation. To reduce the complexity of this model, while maintaining most of the information drawn from the continuous population balance equation, the concept of the PSPM is used. Based on the multivariate population balance equation and the PSPM a mathematical model is developed for any liquid extraction column. The secondary particle could be envisaged as a fluid particle carrying information about the distribution as it is evolved in space and time, in the meanwhile the primary particles carry the mean properties of the population such as total droplet concentration; mean droplet diameter dispersed phase hold up and so on. This information reflects the particle-particle interactions (breakage and coalescence) and transport (convection and diffusion). The developed model is discretized in space using a first-order upwind method, while semi-implicit first-order scheme in time is used to simulate a pilot plant RDC extraction column. Here the effect of the number of primary particles (classes) on the final predicted solution is investigated. Numerical results show that the solution converge fast even as the number of primary particle is increased. The terminal droplet velocity of the individual primary particle is found the most sensitive to the number of primary particles. Other mean population properties like the droplet mean diameter, mean hold up and the concentration profiles are also found to converge along the column height by increasing the number of primary particles. The predicted steady state profiles (droplet diameter, holdup and the concentration profiles) along a pilot RDC extraction column are compared to the experimental data where good agreement is achieved. In addition to this a robust rigorous mathematical model based on the bivariate population balance equation is developed to predict the steady state and dynamic behaviour of the interacting hydrodynamics and mass transfer in Kühni extraction columns. The developed model is extended to include the momentum balance for the calculation of the droplet velocity. The effects of step changes in the important input variables (such as volumetric flow rates, rotational speed, inlet solute concentrations etc.) on the output variables (dispersed phase holdup, mean droplet diameter and the concentration profiles) are investigated. The last topic of this doctoral thesis is developed to transient problems. The unsteady state analysis reveals the fact that the largest time constant (slowest response) is due to the mass transfer. On the contrary, the hydrodynamic response of the dispersed phase holdup is very fast when compared to the mass transfer due to the relative fast motion of the dispersed droplets with respect to the continuous phase. The dynamic behaviour of the dispersed and continuous phases shows a lag time that increases away from the feed points of both phases. Moreover, the solute concentration response shows a highly nonlinear behaviour due to both positive and negative step changes in the input variables. The simulation results are in good agreement with the experimental ones and show the usefulness of the model.

- Transient processes with hydrogels (2013)
- Hydrogels are known to be covalently or ionic cross-linked, hydrophilic three-dimensional polymer networks, which exist in our bodies in a biological gel form such as the vitreous humour that fills the interior of the eyes. Poly(N-isopropylacrylamide) (poly(NIPAAm)) hydrogels are attracting more interest in biomedical applications because, besides others, they exhibit a well-defined lower critical solution temperature (LCST) in water, around 31–34°C, which is close to the body temperature. This is considered to be of great interest in drug delivery, cell encapsulation, and tissue engineering applications. In this work, the poly(NIPAAm) hydrogel is synthesized by free radical polymerization. Hydrogel properties and the dimensional changes accompanied with the volume phase transition of the thermosensitive poly(NIPAAm) hydrogel were investigated in terms of Raman spectra, swelling ratio, and hydration. The thermal swelling/deswelling changes that occur at different equilibrium temperatures and different solutions (phenol, ethanol, propanol, and sodium chloride) based on Raman spectrum were investigated. In addition, Raman spectroscopy has been employed to evaluate the diffusion aspects of bovine serum albumin (BSA) and phenol through the poly(NIPAAm) network. The determination of the mutual diffusion coefficient, \(D_{mut}\) for hydrogels/solvent system was achieved successfully using Raman spectroscopy at different solute concentrations. Moreover, the mechanical properties of the hydrogel, which were investigated by uniaxial compression tests, were used to characterize the hydrogel and to determine the collective diffusion coefficient through the hydrogel. The solute release coupled with shrinking of the hydrogel particles was modelled with a bi-dimensional diffusion model with moving boundary conditions. The influence of the variable diffusion coefficient is observed and leads to a better description of the kinetic curve in the case of important deformation around the LCST. A good accordance between experimental and calculated data was obtained.

- Numerical and Analytical Investigation of a Phase Field Model for Fracture (2013)
- This thesis is concerned with a phase field model for brittle fracture. The high potential of phase field modeling in computational fracture mechanics lies in the generality of the approach and the straightforward numerical implementation, combined with a good accuracy of the results in the sense of continuum fracture mechanics. However, despite the convenient numerical application of phase field fracture models, a detailed understanding of the physical properties is crucial for a correct interpretation of the numerical results. Therefore, the driving mechanisms of crack propagation and nucleation in the proposed phase field fracture model are explored by a thorough numerical and analytical investigation in this work.

- Computational Fluid Dynamics Aided Design of Stirred Liquid-Liquid Extraction Columns (2013)
- Aim of this work was the extension and development of a coupled Computational Fluid Dynamics (CFD) and population balance model (PBM) solver to enable a simulation aided design of stirred liquid-liquid extraction columns. The principle idea is to develop a new design methodology based on a CFD-PBM approach and verify it with existing data and correlations. On this basis, the separation performance in any apparatus geometry should be possible to predict without any experimental input. Reliable “experiments in silico” (computer calculations) should give the engineer a valuable and user-friendly tool for early design studies at minimal costs. The layout of extraction columns is currently based on experimental investigations from miniplant to pilot plant and a scale-up to the industrial scale. The hydrodynamic properties can be varied by geometrical adjustments of the stirrer diameter, the stirrer height, the free cross sectional area of the stator, the compartment height as well as the positioning and the size of additional baffles. The key parameter for the liquid–liquid extraction is the yield which is mainly determined at the in- and outlets of the column. Local phenomena as the swirl structure are influenced by geometry changes. However, these local phenomena are generally neglected in state-of-the are design methodologies due to the complex required measurement techniques. A geometrical optimization of the column therefore still results in costs for validation experiments as assembly and operation of the column, which can be reduced by numerical investigations. The still mainly in academics used simulation based layout of counter-current extraction columns is based at the beginning of this work on one dimensional simulations of extraction columns and first three dimensional simulations. The one dimensional simulations are based on experimental derived, geometrical dependent correlations for the axial backmixing (axial dispersion), the hold-up, the phase fraction, the droplet sedimentation and the energy dissipation. A combination of these models with droplet population balance modeling resulted in a description of the complex droplet-droplet interactions (droplet size) along the column height. The three dimensional CFD simulations give local information about the flow field (velocity, swirl structure) based on the used numerical mesh corresponding to the real geometry. A coupling of CFD with population balance modeling further provides information about the local droplet size. A backcoupling of the droplet size with the CFD (drag model) results in an enhancement of the local hydrodynamics (e.g. hold-up, dispersed phase velocity). CFD provided local information about the axial dispersion coefficient of simple geometrical design (e.g. Rotating Disc Contactor (RDC) column). First simulations of the RDC column using a two dimensional rotational geometry combined with population balance modeling were performed and gave local information about the droplet size for different boundary conditions (rotational speed, different column sizes). In this work, two different column types were simulated using an extended OpenSource CFD code. The first was the RDC column, which were mainly used for code development due to its simple geometry. The Kühni DN32 column is equipped with a six-baffled stirring device and flat baffles for disturbing the flow and requires a full three dimensional description. This column type was mainly used for experimental validation of the simulations due to the low required volumetric flow rate. The Kühni DN60 column is similar to the Kühni DN32 column with slight changes to the stirring device (4-baffles) and was used for scale up investigations. For the experimental validation of the hydrodynamics, laser based measurement techniques as Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) were used. A good agreement between the experimental derived values for velocity, hold-up and energy dissipation, experimentally derived correlations from literature and the simulations with a modified Euler-Euler based OpenSource CFD code could be found. The experimental derived axial dispersion coefficient was further compared to Euler-Lagrange simulations. The experimental derived correlations for the Kühni DN32 in literature fit to the simulated values. Also the axial dispersion coefficient for the dispersed phase satisfied a correlation from literature. However, due to the complexity of the dispersed phase axial dispersion coefficient measurement, the available correlations gave no distinct agreement to each other. A coupling of the modified Euler-Euler OpenSource CFD code was done with a one group population balance model. The implementation was validated to the analytical solution of the population balance equation for constant breakage and coalescence kernels. A further validation of the population balance transport equation was done by comparing the results of a five compartment section to the results of the commercial CFD code FLUENT using the Quadrature Method of Moments (QMOM). For the simulation of the droplet-droplet interactions in liquid-liquid extraction columns, several breakage and coalescence models are available in the literature. The models were compared to each other using the one-group population balance model in Matlab which allows the determination of the minimum stable droplet diameter at a certain energy dissipation. Based on this representation, it was possible to determine the parameters for a specific breakage and coalescence model combination which allowed the simulation of a Kühni miniplant column at different rotational speeds. The resulting simulated droplet size was in very good agreement to the experimental derived droplet size from literature. Several column designs of the DN32 were investigated by changing the compartment height and the axial stirrer position. It could be shown that a decrease of the stirrer position increases the phase fraction inside the compartment. At the same time, the droplet size decreases inside the compartment, which allows a higher mass transfer due to a higher available interfacial area. However, the shifting results in an expected earlier flooding of the column due to a compressed flow structure underneath the stirring device. In a next step, the code was further extended by mass transfer equations based on the two-film theory. Mass transfer coefficient models for the dispersed and continuous phase were investigated for the RDC column design. A first mass transfer simulation of a full miniplant column was done. The change in concentration was accounted by the mixture density, viscosity and interfacial tension in dependence of the concentration, which affects the calculation of the droplet size. The results of the column simulation were compared to own experimental data of the column. It could be shown that the concentration profile along the column height can be predicted by the presented CFD/population balance/mass transfer code. The droplet size decreases corresponding to the interfacial tension along the column height. Compared to the experimental derived droplet size at the outlet, the simulation is in good agreement. Besides the occurrence of a mono dispersed droplet size, high breakage may lead to the generation of small satellite droplets and coalescence underneath the stator leads to larger droplets inside the column and hence to a change of the hold-up and of the flooding point. A multi-phase code was extended by the Sectional Quadrature Method of Moment (SQMOM) allowing a modeling of the droplet interactions of bimodal droplet interactions or multimodal distributions. The implementations were in good agreement to the analytical solution. In addition, the simulation of an RDC column section showed the different distribution of the smaller droplets and larger droplets. The smaller droplets tend to follow the continuous phase flow structure and show a higher distribution of inside the column. The larger droplets tend to rise directly through the column and show only a low influence to the continuous phase flow. The current results strengthen the use of CFD for the layout of liquid-liquid extraction columns in future. The coupling of CFD/PBM and mass transfer using an OpenSource CFD code allows the investigation of computational intensive column designs (e.g. pilot plant columns). Furthermore the coupled code enhances the accuracy of the hydrodynamics simulations and leads to a better understanding of counter-current liquid-liquid extraction columns. The gained correlation were finally used as an input for one dimensional mass transfer simulations, where a perfect fit of the concentration profiles at varied boundary conditions could be obtained. By using the multi-scale approach, the computational time for mass transfer simulations could be reduced to minutes. In future, with increasing computational power, a further extend of the multiphase CFD/SQMOM model including mass transfer equation will provide an efficient tool to model multimodal and multivariate systems as bubble column reactors.

- Analysis of Off-Road Tire-Soil Interaction through Analytical and Finite Element Methods (2013)
- Tire-soil interaction is important for the performance of off-road vehicles and the soil compaction in the agricultural field. With an analytical model, which is integrated in multibody-simulation software, and a Finite Element model, the forces and moments generated on the tire-soil contact patch were studied to analyze the tire performance. Simulations with these two models for different tire operating conditions were performed to evaluate the mechanical behaviors of an excavator tire. For the FE model validation a single wheel tester connected to an excavator arm was designed. Field tests were carried out to examine the tire vertical stiffness, the contact pressure on the tire – hard ground interface, the longitudinal/vertical force and the compaction of the sandy clay from the test field under specified operating conditions. The simulation and experimental results were compared to evaluate the model quality. The Magic Formula was used to fit the curves of longitudinal and lateral forces. A simplified tire-soil interaction model based on the fitted Magic Formula could be established and further applied to the simulation of vehicle-soil interaction.