Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
A Generic Algorithm for Modelling Thermal Dynamics with Moving Boundary Using Comsol MultiphysicsTM Jiawei Mi and Patrick Grant Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH Email:
[email protected]
Abstract Droplets spray deposition is an important materials process in which materials in the form of particulates or droplets (a mixture of liquid, semisolid and solid) are projected onto an arbitrary collecting surface and consolidate to build up a thin-layer coating or a bulk preform. Droplets depositing at the collecting surface adds new materials and inputs new thermal energy continuously onto the already consolidated materials (named preform thereafter). Tracking the evolution of the sprayed surface (the free surface) of the preform and modelling the heat flow inside the preform is a great challenge. Fundamentally, this is a problem of tracking a moving boundary coupled with a thermal dynamics simulation. A multiphysics numerical model that captures all the important process physics of the spray forming process has been developed to simulate the spray forming process, including a submodel of shape dynamics developed using the MatlabTM and a submodel of thermal dynamics developed using the Comsol MultiphysicsTM to simulate the preform shape evolution and the internal heat flow. A generic algorithm of coupling the mass and enthalpy inputs at the sprayed surface has been developed to tackle the coupling of the shape and the thermal dynamics model at each time step simulation. It has been proven as a computationally efficient algorithm and can be used to handle the dynamic coupling of the mass and enthalpy inputs of any arbitrary distribution. Keywords: Modelling; Spray forming; Ni superalloys; Finite Element; Coupling
Introduction Spray forming is an inherent multiphysics process consisting of multi-length (micrometers to metres) and multi-time scale (microseconds to minutes) transport phenomena. The basic physics involved are: (1) the fragmentation of a continuous liquid stream into discrete droplets; (2) multiphase flow of the gas-droplet spray cone and non-linear heat transfer; (3) droplets deposition, splashing and re-deposition; and (4) preform solidification and microstructure evolution [1]. Over the last 30 years, many numerical models have been developed to simulate the preform shape evolution and heat flow during spray forming [2-8] and have helped to understand the correlation between the spray forming processing parameters and the resulting preform shape and the solidification behaviour inside. In the majority of these studies, the mass depositing at the preform top surface at each deposition cycle was calculated based on a predefined mass flux distribution profile. The corresponding enthalpy inputs was often calculated based on the deposited mass and a predefined uniform temperature or a measured temperature at a representative point on the preform top surface during experiments. This approach simplified the dynamic coupling of the mass and enthalpy inputs at each deposition cycle, but it often did not obey strictly the mass and energy conservation law because the droplets depositing at the preform top surface often did not have a uniform temperature.
1
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
A multiphysics numerical model [9] that captures all the important process physics of the spray forming process has been developed by the joint effort of the University of Oxford (UK), University of Bremen (Germany) and Inasmet (Spain) to simulate the spray forming process. Two important features of this multiphysics numerical model are that: (1) the mass and enthalpy distributions of the droplets spray were simulated all the way from the point of atomisation towards the points of droplets deposition, and (2) a generic algorithm was developed and implemented to deal with the dynamic coupling of the mass and enthalpy inputs at each deposition cycle by obeying strictly the mass and energy conservation law. It is a computationally efficient algorithm and can be used to handle the dynamic coupling of the mass and enthalpy of any arbitrary distribution. This paper describes in detail the development and implementation of this algorithm.
Melt atomisation, droplets spray and deposition Fig. 1 shows the gas atomisation of an IN718 alloy melt, the deposition of the resulting droplets spray at a sprayed surface to form a profiled ring preform and an instantaneous surface temperature of the resulting profiled ring preform during spray forming. The multiphysics numerical model [9] was used to calculate the size, mass and enthalpy of each individual droplet created during the atomisation process and then the discrete droplets were integrated over the whole range of the droplet size to form a continuous mass and enthalpy flux, acting as the inputs for the shape and thermal dynamics submodels [9]. Fig. 2 shows the mass deposition rate (converted from the mass flux) and the temperature distribution (converted from the enthalpy flux) of a scanning IN718 alloy droplets spray simulated using the multiphysics numerical model. The non-linear characteristics of the mass and temperature distribution over the scanning range and their dependence on the spray forming parameters (the spray distances and the atomiser scan operations) are clearly demonstrated. Fig. 1. High speed images, showing (a) the atomisation of an IN718 alloy melt, (b) the deposition of the resulting droplet spray at a sprayed surface to form a profiled ring preform, (c) one frame of a thermal imaging video, showing an instantaneous surface temperature of the resulting profiled ring preform during spray forming.
(b) (c)
1610
0.6
(b) o
0.5
(a) Temperature ( K)
Deposition rate (m/s) ×10
-3
(a)
0.4 0.3 0.2 Scan angle ± 10.50
0.1 0
-0.1
-0.05 0 0.05 Scan range (m)
1600 1590
± 10.50
1570
0.1
Scan angle
1580
-0.1
-0.05 0 0.05 Scan range (m)
0.1
Fig. 2. The simulated: (a) mass deposition rate, and (b) droplets spray temperature at different spray distances for the spray forming of an IN718 ring. The lowest height profiles in each case represent the data at a spray distance of 0.34 m from the melt-delivery nozzle exit. The other profiles are at shorter spray distances, reducing in steps of 0.02 m. A sinusoidal scan profile was used with a scan angle of ±10.5º.
2
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
Shape and thermal dynamics modelling With the mass deposition flux as an input, the preform shape at each deposition cycle was calculated using the shape dynamics submodel [10]. By utilising the axisymmetry of the spray forming of a ring preform, the 3-D droplets deposition phenomena can be simulated by a 2-D approach in a cylindrical polar coordinates ( r, x ). Fig. 3 shows the calculated preform shape and the substrate at an arbitrary time step t in a cylindrical polar coordinates. A front tracking algorithm (a program coded using MatlabTM) in a fixed Cartesian grid was applied to simulate and track the preform shape evolution [10]. The heat equation for the preform is [10] :
∂T 1∂ ∂T ∂ ∂T ∂f ∂T = ρ C + L f l = K r + K r ∂t r ∂x ∂x ∂r ∂r ∂T ∂t C eff is the effective heat capacity, accounting for the latent heat release during solidification. The heat equation for the substrate is [10]:
ρ Ceff
(1)
∂T 1∂ ∂T ∂ ∂T = K sub r (2) + K sub r ∂t r ∂x ∂x ∂r ∂r The substrate was made of low carbon steel. During spray forming, the operating temperature of the substrate was controlled below the steel meting point and therefore solidification often did not occur in the substrate. A interfacial heat transfer scheme between the preform and the substrate was introduced to simulate the heat flow between them. On the preform and substrate free surfaces, the convective and radiation heat transfer were considered as detailed in [10]. The heat equations were solved using the commercial finite element software, Comsol MultiphysicsTM.
ρ sub C sub
r Ring preform Steel substrate
|
Substrate rotation axis
x
Fig. 3. The modelled IN718 ring preform shape and the substrate at an arbitrary time step t .
Coupling of dynamic mass and enthalpy inputs A generic algorithm of coupling the mass and enthalpy inputs at the sprayed surface when new materials were deposited was developed and implemented to tackle the coupling of the shape and the thermal dynamics model at each time step calculation. The algorithm is described in detail in Fig. 4 and 5. Fig. 4 shows the algorithm of dynamic meshing and coupling of mass and enthalpy inputs during spray forming. At the start of a simulation loop of the time step t , a fixed-curve (the red curve shown in Fig. 4 and its physical position did not change in this loop) was set below the preform top surface to divide the preform meshes into two functional parts: (1) the fixed-meshes – the meshes below the fixed-curve and their physical positions did not change in this loop as shown in Fig 4(a)
3
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
and (b); and (2) the stretched-meshes – the meshes above the fixed-curve and their mesh points could be stretched to reach the corresponding positions of the new preform top surface if new materials were deposited as shown in Fig 4(b). The fixed-meshes is used to transfer the mesh data (temperature in this case) of the preform at t to the corresponding mesh points of the new preform at t + ∆t . The stretched-meshes is used to accommodate the newly deposited mass and couple the enthalpy inputs. Fig. 5 shows a procedure of how to couple the mass and enthalpy inputs when new materials are added on top of the preform top surface. Fig. 5(a) shows three mesh nodes T1 , T2 and
T3 (also denoting the temperatures) and an associated infinitesimal area in the stretched-meshes. This infinitesimal area is actually the area bounded by a-b-e-f in the corresponding infinitesimal volume shown in a cylindrical polar coordinates of Fig. 5(b). Assuming the three mesh nodes have the radial distance of r1 , r 2 and r3 respectively. T1 and T2 were known and they were the temperatures of the mesh nodes on the fixed-curve and on the preform top surface output from the simulation at the time step t , before the new materials were deposited. After the new materials having a temperature of Tarr were deposited and the mesh point of T2 was stretched to T3 , the temperature of the mesh node T3 on the new preform top surface became an unknown. To continue the simulation to the next time step t + ∆t , a procedure was developed to calculate T3 using the mass and energy conservation law. The enthalpy of the infinitesimal volume bounded by a-b-c-d, V f shown in Fig 5(b) is: T + T2 T + T2 1 = ρ δx ⋅ φ ⋅ (r2 ) 2 − (r1 )2 ⋅ C f ⋅ 1 H f = M f ⋅ C f ⋅ 1 (3) 2 2 2 where H f , M f and C f are the enthalpy, mass and heat capacity of V f respectively. ρ is assumed a constant and C f is calculated using the average temperature of T1 and T2 [10]. The enthalpy of the infinitesimal volume bounded by c-d-e-f, Vs shown in Fig 5(b) is: 1 H s = M s ⋅ C s ⋅ Tarr = ρ δx ⋅ φ ⋅ (r3 ) 2 − (r2 ) 2 ⋅ C s ⋅ Tarr (4) 2 C s is calculated using the arrival temperature of the newly deposited materials Tarr [10]. Therefore, the total enthalpy of the infinitesimal volume bounded by a-b-e-f, shown in Fig 5(b) is: T + T3 1 2 2 H all = H f + H s = ρ δx ⋅ φ ⋅ (r3 ) − (r1 ) ⋅ C all ⋅ 1 (5) 2 2 C all can also be calculated using the weighted mixture of the heat capacity at T1 and T3 [10]: T ⋅ C (T1 ) + T3 ⋅ C (T3 ) C all = 1 T1 + T3
(6)
Substitute the right-hand side terms of Eqs. (3), (4) and (6) into (5) and make a rearrangement, T3 can be calculated:
(
T3 =
)
(
)
T + T2 2 2 2 2 (r2 ) − (r1 )2 C f 1 + (r3 ) − (r2 ) C s Tarr 2
((r ) 3
2
2
)
− (r1 ) C (T3 )
4
T ⋅ C (T1 ) − 1 C (T3 )
(7)
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
(a) Step 1
Mesh the preform and substrate at the time step, t . The fixed-meshes and stretched-meshes are defined.
Step 2
Mesh the new preform at t + ∆t using the fixed-meshes at t . Stretch the stretched-meshes at t to reach the new preform top surface at t + ∆t . Copy the mesh data of the fixedmeshes at t to that at t + ∆t . Couple the mass and enthalpy inputs for the stretched-meshes at t + ∆t . Model preform heat flow at t + ∆t
Step 3
Step 4
(b)
Step 5
Step 6
(c)
Step 7 Step 8
Create a new fixed-meshes for the new preform at t + ∆t . Interpolate the mesh data from step 6 to the new fixed-meshes.
Fig. 4. The algorithm of dynamic meshing and coupling of mass and enthalpy inputs during spray forming. (a) At the start of a simulation loop of the time step t , a fixed-curve (the red curve) is set below the preform top surface to divide the preform meshes into two parts: (1) the fixed-meshes and (2) the stretched-meshes; (b) at the next time step t + ∆t , using the fixedmeshes at t to mesh the new preform while stretching the stretched-meshes at t to reach the new preform top surface to accommodate the new mass and enthalpy inputs at t + ∆t ; (c) creating a new fixed-meshes for the new preform at t + ∆t for the interpolation of the mesh data from Fig. 4(b) to (c) after the heat flow calculation at t + ∆t .
Tarr
T3
T3 Vs
T2
c
δx
T1
r3
b
r2 T1 (b)
d
Vf a
δx r1
(a)
f
e
T2
φ x
Fig. 5. (a) Three mesh nodes having the temperature of T1 , T2 and T3 respectively in the stretched-meshes with an associated infinitesimal area (the shaded area) having the width of δx . (b) In a cylindrical polar coordinates ( r , φ, x ), this infinitesimal area is actually the area marked by a-b-e-f. The three mesh nodes have the radial distances of r1 , r2 and r3 respectively.
5
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
By the calculation of Eq. (7), the new temperature of the new mesh point on the new preform top surface was calculated based strictly on the mass and energy conservation law. Numerically, the mass and the enthalpy carried by the depositing materials were added exactly into the top layer of the preform without interfering the mesh data of the fixed-meshes. Therefore the mass and energy conservation from the droplets spray to the depositing layer on the top surface of the preform at each deposition cycle were maintained and simulation could proceed to next loop. The heat flow submodel was validated and calibrated via numerous thermal measurement using an in-situ thermocouple datalogger, a pyrometer and a thermal imaging camera, ensuring sensible boundary conditions to be adopted in the heat flow model [10].
(a)
(b)
Fig. 6. The simulated cross sections and the internal heat flow of the spray formed ring and the substrate using the data shown in Fig. 2 as the inputs at the spraying times of: (a) 167 s, (b) 250 s and (c) the end of spray forming. The modelled alloy liquid fraction contours were also superimposed on the simulated ring preforms. (d) A cross section of the corresponding spray formed IN718 alloy ring using exactly the same spray forming parameters, showing the final ring shape and a uniform macrostructure.
0.5 0.4
0.0
0.0
0.5 0.4
0.5 0.4
(c)
0.0
(d)
Fig. 6 shows three frames of the simulated ring cross sections, the heat flow, the superimposed liquid fraction and a cross section of a real ring. The difference in term of area between the simulated final ring shape and measured one from the real ring was less than ~6%. The simulated top surface temperature was agreed with the measured one using a pyrometer [10]. Therefore, the simulation was in a good agreement with the experiment. This multiphysics numerical model has been intensively used in helping the optimisation of the spray forming IN718 alloy rings and proven as a powerful tool for modelling the spray forming processing.
Summary 1. A generic algorithm has been developed and implemented in a multiphysics numerical model of the spray forming process to couple the dynamic mass the enthalpy inputs at each simulation loop. 2. This algorithm has been proven as a computationally efficient algorithm and can be used to tackle the coupling of dynamic mass and enthalpy inputs of any arbitrary distribution.
Acknowledgement The support of the European Commission under the contract G4RD-CT2002-00762 is gratefully acknowledged. References [1] Grant PS. Prog Mater Sci 1995;39 (4–5):497. [2] Mathur P, Apelian D, Lawley A. Acta Metall 1988;37(2):429.
6
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Birmingham
[3] Gutierrez-Miravete E, Lavernia EJ, Trapaga GM, Szekely J, Grant NJ. Metall Trans A 1989; (20):71. [4] Pryds NH, Hattel JH, Pedersen TB, Thorborg J. Acta Mater 2002;50:4075. [5] Mi J, Shi Z, Grant PS. Materials Science Forum 2005;475-479:2807. [6] Shi Z, Mi J, Grant PS. Materials Science Forum 2005;475-479:2803. [7] Cui C, Fritsching U, Schulz A, Li Q. Acta Mater 2005;53:2765. [8] Cui C, Fritsching U, Schulz A, Li Q. Acta Mater 2005;53:2775. [9] Mi J, Grant PS, Fritsching U, Belkessam O, Garmendia I, Landaberea A. “Multiphysics Modelling of the Spray Forming Process”, accepted by Mater. Sci. Eng. A. [10] Mi J, Grant PS. “Modelling the Shape and Thermal Dynamics of Spray Forming Ni Superalloys”, submitted to Acta Materialia.
7
