modeling and simulation of metal melt filtration process

MODELING AND SIMULATION OF METAL MELT FILTRATION PROCESS Eric Werzner*, Miguel A. A. Mendes, Dimosthenis Trimis, Subhashis Ray Chair of Gas and Heat Technology, Institute of Thermal Engineering Technische Universität Bergakademie Freiberg Gustav-Zeuner-Str. 7 - 09596 Freiberg – Germany ABSTRACT Depth filtration of the melt is a common procedure for the production of metal parts, which have to meet highest requirements regarding strength and toughness. While typically employed ceramic foam filters yield high filtration efficiency for large inclusions, the removal rate for impurities with the size being less than 10 µm is still not satisfactory. In order to improve the filtration efficiency, a good knowledge about the effects of the filter geometry and various process conditions on the separation of inclusions is required. With respect to this, numerical simulations proved as a helpful tool, as they allow the variation of parameters and also the evaluation of the filtration efficiency in a straightforward manner compared to experimental studies. One way to determine the filtration efficiency in a fluid flow simulation is to track the movement of a sufficiently large number of suspended sample particles by solving the equation of motion for each particle on the basis of local fluid velocity. At velocities typical for melt filtration, the flow inside the filter structure is highly unsteady and characterized by a wide range of spatial and temporal scales, which are computationally expensive to resolve, especially for real structures, e.g. obtained from computed tomography scanning. In order to reduce the computational effort, two-equation turbulence models, such as k-ε, or Reynolds stress models can be employed. However, by virtue of their nature, they only provide statistical information about the unresolved flow scales in the form of distributions of turbulent kinetic energy or turbulent stresses. In order to consider these fluctuations, which greatly affect the dispersion of particles, the discrete random walk model (DRWM) is commonly used. Objective of the present study is to investigate, how the update time interval of the DRWM affects the prediction of the filtration efficiency. In the course of the present investigation, a direct numerical simulation of the threedimensional flow inside a periodic unit cell of a filter foam-like structure has been carried out. On the basis of distributions of mean and variance of the velocity components, the filtration efficiency was evaluated through a Lagrangian tracking of 100,000 particles, for which the velocity fluctuations were imposed using the DRWM. The filtration efficiency was evaluated for different DRWM update time intervals and different particle diameters and was found to largely depend on both quantities. KEYWORDS Metal Melt Filtration, Depth Filtration, Lattice Boltzmann Method, Particle Tracking, Turbulent Flow, Foam-like Structure

1. INTRODUCTION The mechanical properties of cast metals, especially fatigue life time and strength, are greatly determined by the cleanness of the melt during the casting process. A common procedure for increasing the purity of the liquid metal is to remove inclusions by filtration. Typically, ceramic foam filters are employed for this process, which facilitate the mechanism of depth filtration, where impurities are separated along the depth of the filter. While filtration using foam structures yields high filtration rates for large inclusions, the removal of inclusions with a size of less than 10 µm is not satisfactory in view of the posed requirements. In order to improve the filtration efficiency, a good understanding of the physical phenomena, which are involved in the filtration process, is needed. Due to the complex geometry of ceramic foam filters, experimental studies are usually limited to the measurement of integral quantities, such as filtration efficiency, pressure loss and flow rate. Further, parametric variations are often demanding to perform in an experiment. In view of this, numerical simulations have been proved as a helpful tool, as they allow the variation of parameters and also the evaluation of the filtration efficiency in a straightforward manner. Various studies, which make use of the numerical approach for investigation of the filtration process of liquid metals, can be found in the literature [1-5]. In those investigations, the filtration efficiency is usually determined by tracking a sufficiently large number of suspended sample particles solving the equation of motion for each particle on the basis of the local fluid velocity. At velocities typical for melt filtration, the flow inside the filter structure is highly unsteady and characterized by a wide range of spatial and temporal scales, which are computationally expensive to resolve, especially for real filter structures which may be obtained from computed tomography scanning. Several researchers reduced the computational effort by solving a two-dimensional problem [1, 2, 3] or by choice of a small representative synthetic element of a filter structure [1, 2, 4]. Instead of resolving the fluctuations, they can also be taken care of with two-equation [4, 5] or Reynolds stress models. However, by virtue of their nature, turbulence models only provide time-averaged information about the unresolved flow scales in form of distributions of turbulent kinetic energy or the turbulent stresses. In order to incorporate the random effect of those scales on the particle motion, the discrete random walk model (DRWM) is commonly used [4, 5]. In the DRWM, random velocity fluctuations are superimposed on the flow field solution, based on a normal-distributed random number, which is chosen according to the local turbulent kinetic energy or the Reynolds stresses. The random variable is updated with a time interval equal to the characteristic lifetime of the fluctuations, which corresponds to the integral time-scale of the flow. Although this quantity can be estimated from the turbulent kinetic energy and its dissipation rate, the accuracy of such estimations is limited and its significance for the correct prediction of filtration

efficiency not well studied. Objective of the present study is therefore to investigate the sensitivity of filtration efficiency to the update time interval of the DRWM. In the course of the present investigation, a direct numerical simulation (DNS) of the three-dimensional flow inside a periodic unit cell of a filter foam-like structure has been carried out. On the basis of the distributions of the mean and variance of the velocity components, the filtration efficiency was evaluated through a Lagrangian tracking of 100,000 particles, for which the velocity fluctuations were imposed using the DRWM. The simulations were performed for different DRWM update time intervals and three different particle diameters. This article is structured as follows: In Section 2, the numerical approach used for simulation of the liquid metal flow and the particle tracking within the considered filter element are described in brief. Section 3 shows the results obtained from the present study and finally, section 4 provides the main conclusions as well as an outlook of this investigation. 2. MATHEMATICAL MODELING In the present study, the single-phase flow of liquid metal through an idealized filter structure, consisting of an infinite bundle of square bars, was considered numerically, as sketched in Figure 1(a). Nevertheless, it is important to note that the numerical model, described here, can be easily applied for the general case of flow through real filters. Figure 1(b) shows the computational domain, used for the fluid flow simulation, where the flow was assumed to be periodic and driven by an imposed pressure gradient in the x-direction. (a) Model representation of filter structure.

(b) Periodic unit cell chosen as domain of integration.

Figure 1: Idealized model of porous filter structure as suggested for example by Kuwahara et al. [13].

The trajectories of the metal inclusions and their adhesion to the filter walls were taken into account. However, instead of considering the particle tracking on the instantaneous 3D flow field, the inclusions were tracked on the 2D average flow field (averaged both in time and in the z-direction), where a random walk modeling approach was considered in order to account for the influence of the flow field

fluctuations on the particle trajectories. Additionally, the following important assumptions were also made in the present investigation: 1. The fluid is isothermal with constant viscosity and the flow is incompressible. 2. The flow is one-way coupled to the particles, where the particle motion is influenced by the fluid flow although the converse is not true. 3. The particles are spherical and the probability of agglomeration among them is neglected. 4. Simplified adhesion condition is considered, i.e., any particle that makes contact with the solid walls of the filter is trapped and remains attached. 5. Accumulation of the filtered particles does not modify the filter geometry. 2.1. Governing equations for fluid flow The transient flow field within the computational domain was obtained from the solution of the governing equations for mass and momentum, which were numerically solved using the LBM, see e.g. [6, 7]. Instead of directly solving the discretized governing equations for the macroscopic velocity and pressure, the LBM treats the fluid flow as movement of fictitious particles (distribution functions) undergoing consecutive streaming and collision processes over a discrete lattice. Based on this, the computational domain in Fig. 1(b) was discretized into a uniform lattice mesh, where each of the lattice nodes represents a voxel, made of either the solid or the fluid phase. Consequently, the micro-structure of the filter was represented by a finely-resolved staircase approximation, which can be regarded to be exact for the idealized filter considered in this study. A brief description of the LBM is presented here for the sake of completeness. In the LBM, the collision and streaming steps for the distribution function fα, in a particular streaming direction α, are given as [6]: collision

~    fα ( x , t + δt ) = fα ( x , t ) + Ωα ( x , t ) + δt Fα

(1)

streaming

  ~  f α ( x + cα δt ,t + δt ) = f α ( x ,t + δt )

(2)

where 𝑥⃗, t, δx and δt are the spatial location, time, lattice constant and time step, respectively. A D3Q19 lattice model was considered (see Figure 2) for the present simulation and therefore, the distribution functions were only allowed to stream on the lattice with a discrete set of velocities 𝑐⃗𝛼 . The collision operator Ωα in equation (1) represents the rate of change of fα resulting from collision and was modeled using the incompressible Multi Relaxation Time (MRT) formulation [8], where the fluid viscosity was adjusted through the relaxation frequency ω. The source term δtFα in

equation (1) is the rate of change of fα due to the component of pressure gradient in the α direction, in which Fα is given as [9]: Fα =

wα ∂P   eα ⋅ cα c s2 ∂x

(3)

where 𝑒⃗𝛼 is the Cartesian coordinate vector in x-direction, wα are the weighting factors of the D3Q19 lattice model [7] and 𝜕𝑃/𝜕𝑥 is the imposed pressure gradient in the x-direction. Regarding the boundary conditions, the LBM implicitly imposes the periodic condition on the boundaries of the computational domain through the streaming step. Moreover, the no-slip condition at the solid walls was imposed by applying the half-way bounce back treatment on the solid lattice nodes adjacent to the fluid region [6]. It is important to note that, since no modeling approach was considered here in order to resolve turbulence, it Figure 2: D3Q19 lattice stencil with lattice  was assumed that all the turbulent velocities ( c 0 = 0 ). scales of the flow are resolved, which implies that a fine lattice mesh is required for the domain discretization. In the incompressible limit, the LBM equations revert into the macroscopic governing equations for the mass and momentum and therefore, the macroscopic flow velocity and pressure are retrieved as: 18   P( x , t ) = cs2 ∑ fα ( x , t )

(4)

18     u f ( x , t ) = ∑ cα fα ( x , t )

(5)

α =0

α =0

where 𝑐𝑠 = 𝛿𝑥/𝛿𝑡/√3 is the lattice sound speed for the D3Q19 lattice model. The corresponding fluid viscosity, derived from equations (1) and (2), is related to the relaxation frequency by [6]:  1 1 2 − c s δt ω 2

νf =

(6)

The described LBM was implemented in an in-house Fortran code that enables parallel computation by using the Message Passing Interface (MPI) protocol. The code uses a domain parallelization approach, by dividing the computational domain into a number of parallelepipedic sub-domains of nearly equal volume, which depends on the number of processors used for the specific computation. Although the coding is relatively simple and straightforward, the developed code was thoroughly tested by solving several conventional fluid flow problems. The results for these test cases are, however, not presented in the paper for sake of brevity. 2.2. Governing equations for particle motion The velocity and position of the inclusions within the filter can be obtained from the equations of motion using a Lagrangian approach. The velocity 𝑢 �⃗𝑝 of a particle moving in the fluid is calculated from the balance of forces, according to Newton’s second law, as:

mp

 du p dt

 = FD

(7)

where 𝑚𝑝 = 𝜋𝜌𝑝 𝑑𝑝3 /6 is the mass of the spherical particle, which is a function of its density 𝜌𝑝 and diameter dp. Moreover, 𝐹⃗𝐷 is the drag force acting on the particle and

the term on the left-hand side of Eq. (7) represents the inertial force. It may be pointed out here that the influence of other external forces on the particle motion, besides drag, was neglected in the present study. This is justified by the fact that, in general, the buoyancy and drag are the dominant forces acting on the particles, although the lift force may play an important role for very small particles moving in regions of high velocity gradients [4, 10]. Nevertheless, buoyancy force does not play any role in the present study since the density of the particles and the fluid was assumed to be equal. The drag force is calculated, using the quasi-steady drag assumption, as:  18ν f f S ρ f   − FD = m p u u f p d p2 ρ p

(

)

where 𝑓𝑆 is the Stokes correction factor according to the correlation of White [11]:

fS = 1+

Re p

4 1 + Re p

+

Re p

(8)

(9)

60

where 𝑅𝑒𝑝 = �𝑢 �⃗𝑓 − 𝑢 �⃗𝑝 �𝑑𝑝 /𝜈𝑓 is the particle Reynolds number. It is important to note that the drag force, given by Eq. (8), depends on the flow velocity 𝑢 �⃗𝑓 at the particle position. Consequently, in order to obtain the particle velocity, Eq. (7) must be solved using the flow field solution previously calculated, assuming that the flow is one-way

coupled, as mentioned before. Moreover, for purpose of particle tracking, the original 3D unsteady flow field is not directly taken into account. Instead, its 2D time statistics (averaged over the z-direction) are considered in order to reduce the dimensionality of the filtration analysis and hence the computational effort. Therefore, 𝑢 �⃗𝑓 = (𝑢, 𝑣) required by Eqs. (8) and (9) is estimated based on the mean of the velocity field ���� ′ 2 , ���� 〈𝑢 �⃗𝑓 〉 = (𝑢�, 𝑣̅ ) as well as on its variance ����⃗ 𝜎 2 = (𝑢 𝑣′2 ) using the DRWM given as [4]: u = u + u' 2 ξ

(10)

v = v + v' 2 ξ

(11)

where 𝜉 is a standard normal distributed variable. It is important to mention here that 𝜉 is updated with a time interval Δ𝜏, which should be representative for the dominant time scales of the flow field fluctuations. Once the particle velocity 𝑢 �⃗𝑝 is known, the particle position 𝑥⃗𝑝 is calculated by integrating the following relation:

 dx p dt

 = up

(12)

For every particle injected in the domain, Eqs. (7) and (12) are numerically solved, using an explicit Euler scheme, in order to find the particle trajectory. Since the LBM provides a discrete field for the flow velocity statistics, the calculation of 𝑢 �⃗𝑓 , from Eqs. (10) and (11), at the particle position is performed with a double linear interpolation between the four closest nodes of the lattice. Moreover, although the flow field in the computational domain is periodic, the same is not considered for the particle trajectory. Therefore, the present approach is able to predict how many computational domains the particle crossed in each direction before it gets trapped on the filter walls and consequently, the filtration efficiency can be calculated as function of the filter thickness. Additionally, a particle is assumed to make contact with the filter wall when the distance between its center and the wall is smaller than its radius 𝑑𝑝 /2. The algorithm described here for the calculation of the particle motion, has been implemented in an in-house FORTRAN code. 3. RESULTS AND DISCUSSION In a first part of this section, the numerical details of the simulated case will be described, followed by the results of the fluid flow DNS. A second part presents and discusses the results of the filtration efficiency obtained for different update time intervals of the DRWM.

3.1. Details of the numerical simulation The computational domain of the idealized filter, as shown in Fig. 1(b), was discretized into a uniform lattice mesh with a resolution of 𝐿⁄𝛿𝑥 = 512 amounting to a total number of 16.7M nodes. A small time step size of 𝛿𝑡 = 9.77 ∙ 10−7 s was chosen in order to satisfy the assumption of incompressible flow. During the simulation, the pressure gradient was continuously adjusted by a controller in order to assure that 𝑅𝑒 = 400, where 𝑅𝑒 = 𝐷𝑢𝑎𝑣 𝜑⁄𝜈𝑓 with 𝑢𝑎𝑣 denoting the average interstitial velocity in the x-direction and 𝜑 = 1 − 4(𝐷⁄𝐿)2 being the porosity of the filter. The simulation was set up according to typical process conditions for the filtration of metal melt inside ceramic foam filters, i.e. 𝐿 = 6.4 mm, 𝐷 = 1.0 mm and 𝜈𝑓 = 6 ∙ 10−7 m²/s.

In order to evaluate the filtration efficiency, a number of 𝑁𝑡𝑜𝑡𝑎𝑙 = 100,000 sample particles were randomly injected with the local fluid velocity along the line 𝑥 = 𝐿/4 of the computational domain. As described in section 2, the particle tracking was performed on the basis of the time-averaged flow field, while the DRWM was used to model the velocity fluctuations under consideration of the velocity variance. The simulation was carried out for a duration of ~ 22 melt residence times, which is long enough to assure that all unfiltered particles passed at least 10 computational domains. In order to study the effect of the DRWM time scale on the filtration efficiency, the simulation was repeated for different values of ∆𝜏 ∗ = ∆𝜏⁄𝛿𝑡, ranging from 10 up to 10,000. Further, different particle populations with a diameter of 1, 10 and 100 µm, representing typical sizes of metal melt inclusions, were considered. For each case, the filtration efficiency is obtained as

η=

N trapped

(13)

N total

where 𝑁𝑡𝑟𝑎𝑝𝑝𝑒𝑑 stands for the number of particles, which either adhered to the filter surface or remained trapped inside recirculation regions at the end of the simulation. 3.2. Solution for velocity ���� ′ 2, The statistical information of the velocity field, namely mean (𝑢�, 𝑣̅ ) and variance (𝑢 ���� ′ 2 ) of the velocity components in x- and y- direction, were obtained as cumulative 𝑣 moving averages, evaluated over more than 40 melt residence times. Since the geometry does not vary in the z-direction, the computed distributions were additionally averaged along this coordinate. The resulting two-dimensional fields of the mean and variance of velocity are displayed in Figure 3. As clearly visible, the strongest velocity fluctuations are found at the side-faces of the square bar for the xcomponent and in the region of vortex shedding for the y-component, respectively. For the present study, the integral time scale of the flow was estimated from the autocorrelation of a velocity time series, recorded at the point 𝑃(0.25𝐿, 0.375𝐿, 0.125𝐿) in the computational domain shown in Figure 1.

u

v

u'2

v' 2

Figure 3: 2D velocity field statistics used for the particle tracking. Quantities were obtained by timeaveraging over ~40 melt residence times and additional averaging over the z-dimension.

The decay of the normalized autocorrelation coefficient 𝑟, defined as [12] r (τ ) =

1

T u′2

T

∫ u ′(t )u ′(t + τ )dt

(14)

0

is displayed in Figure 4. The characteristic time scale ∆𝜏 for the DRWM is evaluated from the integral time scale 𝐼 of the flow, which is obtained as the integral over the correlation coefficient [12] tb

I = ∫ r (τ )dτ

(15)

0

where two zero-crossing times were chosen as the upper integration bound 𝑡𝑏 . Finally, ∆𝜏 is obtained as ∆τ = 2 I

(16)

The time scales, estimated in this study, were found to be of the order of ∆𝜏 ∗ ≈ 1000, corresponding to ~1 ms in physical units. However, the values obtained for the individual velocity components vary by almost a factor of two, with the x-component showing the longest times, as also visible from the slower decay in Figure 4. Since the DRWM employed here assumes isotropy of the fluctuation time scale, the effects of this variation are not taken into account. It may be further noted, that the integral time scale is also expected to vary in space.

3.3. Effect of DRWM time scale In order to evaluate how accurately the DRWM can recover the spectral composition of the velocity fluctuations in foam-like structures, velocity time series were generated using the DRWM for different ∆𝜏 ∗ , having the same variance as found for the xcomponent of velocity at the considered point P(0.25L, 0.375L, 0.125L). Subsequently, the power spectral density was computed for the synthetic signals as well as for the velocity signal, recorded at P. Figure 5 shows the comparison of the resulting spectra. Up to the frequency of the random number generation 𝑓 = 1/∆𝜏 all signals exhibit a flat distribution of power spectral density. At frequencies larger than 𝑓, where only the higher-order components of the square wave-shaped signals are present, the PSD decreases with constant slope. However, as a consequence of the fact that the integral over the spectra must equal the prescribed variance, the distribution along the frequency axis varies significantly. For small values of Δ𝜏 ∗ , the variance of low-frequency fluctuations is underestimated in comparison to the DNS result, while high-frequency components become too pronounced. In contrast to this, a value of Δ𝜏 ∗ = 10,000 results in an overprediction of the fluctuations with low frequency with respect to the data obtained from the DNS. The effect of these differences can be observed from Figure 6, which displays the variation of filtration efficiency with respect to the dimensionless filter length, measured by the number of computational domains, crossed by particles of different sizes. As expected, for almost all cases the filtration efficiency grows with the filter depth, following an exponential law as also reported in the literature [14]. Also, the filtration efficiency increases with the size of the particles since large particles are more prone to filtration due to the mechanisms of inertial impaction and direct interception. Further, the DRWM time scale strongly affects the dispersion of the particles and hence also the filtration efficiency. At large values of Δ𝜏 ∗ the particles are exposed to the randomly generated fluctuation velocity for a longer time, which causes them to depart more from the time-averaged streamlines. In this way, the chance of collision with a filter wall is increased. As one approaches very small values of Δ𝜏 ∗ , the intensity of fluctuations at higher frequency increases Figure 4: Variation of the autocorrelation at the cost of the low-frequency range (see coefficient for the individual components of Figure 5). Since the large particles cannot velocity at a point, located in the adapt to the fast changes of the flow field, computational domain at x = 0.25L, y = 0.375L and z = 0.125L. they are only influenced by the low-

frequency fluctuations, which have very low energy. Consequently, the particles are almost not dispersed and follow the streamlines, which gives them low chance to collide with the obstacle once the initial filtration stage has passed. This results in a constant value of filtration efficiency as shown in Figure 6 for the case of Δ𝜏 ∗ = 10 and 𝑑𝑝 = 100µm. Finally, it may be noted that the filtration efficiency for the particles of size 𝑑𝑝 = 1µm decreases as Δ𝜏 ∗ goes from 10 to 100, which apparently contradicts the previous finding. However, at Δ𝜏 ∗ = 10 the particle movement caused by the velocity fluctuations is so small that many particles remain trapped in the recirculation regions of the flow, once they entered them, as a look on the final particle distribution revealed. For the particles with a diameter of 𝑑𝑝 = 1µm, this effect contributes to almost 50 % of the filtration efficiency.

∆τ * = 10

∆τ * = 100

∆τ * = 1000

∆τ * = 10000

Figure 5: Comparison of power spectral density for the time series of x-component of velocity obtained from the DRWM and the DNS for different DRWM time scales ∆𝜏 ∗ .

∆τ * = 10

∆τ * = 100

∆τ * = 1000

∆τ * = 10000

Figure 6: Variation of filtration efficiency 𝜂 along the depth of the filter 𝑥 ⁄𝐿 for different particle diameters 𝑑𝑝 and values of the DRWM time scale ∆𝜏 ∗ .

4. CONCLUSIONS In the course of the present study, a DNS of the three-dimensional flow inside a periodic unit cell of a synthetic filter structure has been carried out. On the basis of distributions of mean and variance of the velocity components, the filtration efficiency was evaluated through a Lagrangian tracking of particles of different size, for which the velocity fluctuations were imposed using the DRWM. In order to evaluate, how the choice of the model time scale Δ𝜏 affects the filtration efficiency, four different orders of magnitude of Δ𝜏 were considered. Power spectra of velocity time series,

generated using the DRWM, were computed and compared against the spectra obtained from the DNS. Finally, the effect on the filtration efficiency was analyzed. The results show that the filtration efficiency 𝜂 is significantly influenced by the choice of the DRWM time scale Δ𝜏. For all sizes of particles considered here, 𝜂 is observed to increase with Δ𝜏 due to the enhanced particle dispersion. Exceptions are only found for very low Δ𝜏. In order to test the accuracy of the DRWM for the prediction of filtration efficiency, it should be compared to a particle tracking using the detailed transient velocity field. Further investigations could be carried out regarding an improvement of the DRWM applied here. This may include extension of the DRWM to facilitate anisotropy and local variation of Δ𝜏, as well as the use of random distributions for Δ𝜏.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the German Research Foundation (DFG) for supporting the Collaborative Research Center CRC 920, subproject B02. REFERENCES [1]

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