A NEW APPROACH TO THE SOLIDIFICATION MODELING OF
CASTING PROCESSES
Dr. V. S. Lerner, University of California at Los Angeles, UCLA, CA, USA
Dr. Y. S. Lerner, University of Northern Iowa, Cedar Falls, Iowa, USA
ABSTRACT
A new systemic mathematical - informational approach based on Informational
Macrodynamics (IMD) is introduced for solidification modeling and optimization of the
casting processes. The IMD model takes into account the interrelated thermal, diffusion,
kinetic, hydrodynamic, and mechanical effects essential for the given casting process. The
optimum technological process parameters are found by simultaneous solution of problems
of identification and optimal control based on the extremum of the entropy functional,
evaluating a generalized assessment of the physicochemical properties of castings. For the
physical system considered, the IMD structures of the optimal model are connected with
controllable equations of Nonequilibrium Thermodynamics. This approach was applied to
horizontal continuous casting of ductile iron and results were compared with experimental
data. These comparisons have validated the accuracy and usefulness of the new approach.
INTRODUCTION
Traditional numerical modeling of the casting process is using mathematical models and
approaches that mostly describe a separated process's phenomena, or severely cross
phenomena, which simulate mold filling and solidification (1). These models need massive
computations, and are not suitable for operative process control. The proposed method is
based on Informational Macrodynamics (IMD) and is used here for a system informational
solidification modeling and optimization (2-4) of the casting process taking as an example
horizontal continuous casting (HCC). HCC is a relatively new, but promising method of
producing near net shape high quality cast products in ferrous materials (gray, ductile and NiResist irons and steel), as well as non-ferrous (aluminum and copper) alloys. In HCC, liquid
metal from the transfer ladle is poured into the metal receiver. A water-cooled graphite or
copper die is attached to the side of the receiver and bar is pulled out by an extraction system,
which controls stroke length and frequency. A special mechanism cuts and breaks the bars to
required lengths. The major advantage of this process is a high casting yield of 92-95%, since
it eliminates traditional feeder needs. Liquid metal in the receiver plays the role of a
preheated riser that continuously supplies liquid metal to feed the bar during solidification.
By maintaining an adequate balance between the metal chemistry, temperature, level in the
receiver, and drawing and cooling parameters, it is possible to produce defect free high
quality continuously cast bars (5,6).
THE BASIC IMD CONCEPTS
The IMD utilizes Lagrange's Mechanics of Uncertainly, Nonequilibrium Thermodynamics
(NT), Systems Information Theory and Probabilistic prognosis, uses a common information
language for computer modeling during a combined object's observation, control, and
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simulation. The IMD informational language represents the model's revealed uncertainty
directly via a corresponding information measure. That unifies and integrates different
physical models within IMD. IMD contains a formal mathematical model to describe the
transformation of random information processes (at microlevel) into the system dynamic
processes (at macrolevel). The observed process at microlevel is represented by a set of
random interactions connected via a Marcovian chain with a path integral evaluation. This
functional accumulates information contributions from the local functionals of the interacting
microlevel processes as a collective (macro) functional. Transformation from microlevel to
macrolevel is performed by using the informational form of variation principle (VP). The
dynamic macromodel follows from the solution of the VP minimax problem for information
form of the path functional (2-3). Microlevel stochastics transfer the irreversible
macrodynamics that extract order from microlevel randomness. Macromovement occurs
along segments of the initial n-dimensional extremals of the functional. These segments are
successively joining at discrete points (DP) effectively shortening the initial dimension and
ultimately leading to renovation of dynamic process. The main formal result is the LagrangeHamilton's equations for the conjugate informational macrocoordinates, which together with
the equation of the differential constraint (DC) describe the model's macrodynamic
regularities. DC reflects a ‘deterministic impact’ of microlevel stochastics on the Lagrange Hamilton's informational mechanics of uncertainties. In a chain of superimposing processes,
each subsequent process controls one or more the following chain's processes with a
possibility of changing the operator. In the IMD-NT macroequation:
n
dxi
 Li X i  ui , ui   Lij X j , Lii  1/ 2rii , rii  M [ xi2 ], i  j
(1)
dt
j 1
chain of superimposing processes ( i -1, i , i +1) is connected by mutual cross phenomena u,
which perform the control function depending on the state coordinate at ( i -1) DP interval:
dx
1
ui  2i xi (t ' )  2 i =2 xi 1 (t ' ) , Lii rii  i , vi (t ' ) =  2 xi (t ' ) . (2)
dt
The applying of needle controls:
v(t ' , l )  2 x(t ' , l )  2 x(t 'o, l ) , v(l ' , t )  2 x(l ' , t )  2 x(l 'o, t ) (3)
changes only the sign of the operator components without changing their absolute values.
The extremals of the entropy functional are the informational analogies of the solution for the
irreversible NT equations with the Onsager condition on the each of extremal segment.
The kinetic operator is changing by jump at each DP time-space point (t', l' ):
x
x
L ( x(t ' , l ' ))  2b( x(t ' , l ' ))  M x (t , l ' ) [ x(t , l ) (t , l )*]  M x (t ' , l ) [ (t , l ) x(t , l )*] (4)
t
t
of the discrete control forming, where the following equality is true
Ii
I
(t j ' , l ' j )  g i (t j ' , l ' j )  g k (t j ' , l ' j )  k (t j ' , l ' j ) (5)
Xi
Xk
and g i , g k are the subsequent equalized components of the generalized transient conductivity.
The needle controls select them based on condition of the model controllability. The
1 x
generalized forces X= (2b)
are formed by the stochastics and applied macrocontrols,
t
which physically have a quantum (portion) character at the macrolevel interactions. The
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points of compensation diffusion and kinetics hold the chain connection. Sequence of the
chain dependable n-controllable conductivity's components can be reduced to one
controllable conductivity, for example, to electrical conductivity, which measuring is most
simple. The macrodynamic process is characterized by the discrete extremal intervals,
selected from Hamilton's solutions via DC with the time interval discretizations (DP),
determined by VP invariants. Hamilton's equations determine, in general, the reversible
dynamic solutions, and the DC equation (at the DP's) imposes the dynamic connection with
microlevel's stochastics. The DC connects macromodel's operator to statistical characteristics
of the random process, in general, in the form of the nonlinear correlations, which, in part,
lead to the precise and simple identification equations (2,3). The DC changes the structure
and value of dynamic macromodel operator at DPs, creating possibility of modeling jumpwise object's phenomena, connected with abrupt discontinuing or non-smoothness of
identified process. The character of specific object's stochastic data imposes the constraints
on the macrolevel's variation problem. Mathematically the constraints are determined by the
equations of Marcovian stochastics at microlevel, specifically by limitations on the structure
of shift and diffusions matrixes of Marcovian stochastic equation, or their widely used
nonMarcovian approximations. The information functional integrates these equations into the
dynamic constraint at the solution of considered minimax optimization problem (VP). The
VP solution uses Lagrange's method of eliminating constraints and Pontryagin's maximum
principle. The optimal control synthesis is solved using the corresponding Boltz's problem.
Mathematical condition on the constraints and the cost function is expressed by the
information analogue of function of action, which satisfies a joint solution both the HamiltonJacobi's and Kolmogorov's equations on some selected set of the extremal's field. This set
defines a natural boundary of the variation problem, and later on is selected constructively by
the synthesized optimal control functions (2). The applied variation techniques combine the
traditional ones with Kolmogorov's equations, which model the microlevel’s Marcrovian
stochastics and create a very specific variation minimax problem (2). The informational
dynamics and geometry of the macromodel are portaying both uncertainties (4). Information
Geometry (IG) visualizes the bound model's hierarchical information structure, space-time
locations, configurations, and shapes of interacting dynamic information flows as the
systemic categories. The IMD modeling selects the most informative data at DPs responsible
for the process phenomena. IMD combines object's observation and prognosis on the basis of
chosen performance criteria. Because of possibility of nonpredictable changing of object's
characteristics, the operations of object's identification and optimal control are joining in
time. It means, both object's identification and optimization problems can be solved
simultaneously by applying the same control strategy directed toward minimization
uncertainty between current observations and expected prognosis. Minimizing uncertainty
requires applying control that extracts maximum information compensating the uncertainty.
The successive joining of the piece-segments of model's extremals leads to a sequential
consolidation of object's processes (by shortening the initial dimension) and revealing the
object's hierarchical systemic structure during the optimal motion. The sequenced hierarchical
connections create the hierarchical information dynamic network (IN) of macronodes (Figure
1), formed by informational spectrum of macromodel's operator hio with the optimal path
through this network, identifiable in process of the object observation. Restoration of the
hierarchy of the identified IN's nodes brings a quantity measure of the level of systemic
organization, evaluated by the function of informational macrosystemic dynamic complexity
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(MC). The structurally stable macrodynamics (with a possible local instabilities) can create
the IN's macronodes formed by a unification of collective chaotic attractors. Chaotic regions
can generate the second level of stochastic behavior with a possibility of all kinds of chaotic
dynamic phenomena. Depending on the identified object's data, the model can reflect the
object's singularities and/or nonlinear behavior. The IMD model has 4-levels of hierarchy: 1)
statistical microlevel; 2) quantum dynamic level (at the DP locality); 3) dynamic (classical)
macrolevel as a result of selection of the initial macrostates and macrotrajectories; and, 4)
hierarchical informational dynamic network of macrostructures. This many-levels structure
represents a basic analytical model's form, which is concurrently updating, validating, and
progressively refined. This structure can capture the essential object's regularities and ignore
the irrelevant aspects, with different level of abstractions, including the automatic modeling
both sooth and jump- wise phenomena. A combination of the identification, optimization, and
disclosure object's hierarchy reveals the object's systemic functions and regularities in process
of functioning. It leads to constructive engineering solutions.
hio
Figure 1. Structure of the space-time Informational Network of the superimposing processes,
defined by the initial information spectrum hio , which are cooperating along the hyperbola.
The IMD specifics allow us modeling the object's complex phenomena characterized by a
variety of superimposition processes. The informational IN network of superimposing
processes models the chain of their sequential cooperations by the speed and space's
interactions. Both the IMD and NT equations are used to connect the traditional physical
approach with IMD's modeling formalism. The virtual INs can be used for mutual
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communications with possibility of connecting to WEB-based modeling. This communication
can involve human activities in decision-making and the corresponding model's adjustments
using artificial intelligence approach. Using the IN's formal logic and its information
language for description the IMD model and IG operations (4), brings unification for IMD
algorithms with an opportunity of formal application to wide diversity of complex objects.
The formal set of possible IMD models are classified in terms of their MC-complexity, which
depends on the basic identifiable model's parameters. The identification of these basic object's
parameters leads to choosing the particular information model's computational code
depending on the object's complexity (that includes the size of object's computation).
This approach is directed toward the solution of scalability problem for information model
with decreasing its cost. Basic IMD information computation software is simpler comparing
with traditional massive numerical computations, and can work on regular PC with Java
language code. Unified IMD approach brings standardization of module's functions with no
software changes and no degradation in efficiency.
THE HCC SOLIDIFICATION OPTIMAL MODEL
Solidification is created by the HCC’s system of superimposing processes is shown on Table
1, where phase coordinates ( xnt k , xnl k ) is presented as the derivatives and integrals of
corresponding ( n  k  1) coordinates, associated with physical quantities: temperature  ,
rate of solidification vc , concentration  C, rate of mass transfer v m , thermomechanical
stresses  , strains  , density  , and pressure p . The number of total variables n is
previously unknown. The thermoconductivity (T) is the basic phenomenon generating
thermostresses (TS), solidification (S), the mass transfer M, initiated by effective diffusion
(D), the phase transformation (PT), the motion of the liquid phase transfer under the
influence of the hydrodynamic forces HT. Both the IMD and NT equations are used to
connect the traditional physical approach with IMD modeling formalism.
For example, to describe heat conduction (T) and solidification (S) the Fourier and Stefan
equations are used. Diffusion (D) is described by equations of Fick's first and second laws,
associated with the laws of conservation. The equations of thermomechanical stresses (TS)
are determined by the bar temperature distribution in time and space, e.g., by solution of
equations T, and the equations, which describe the phase conversions (PT), associated with
the change in concentration of carbon C= C-C1 in the liquid (C) and solid (C1) phases. To
describe the hydrodynamics (H) the Navier-Stokes equation is used, etc. As an optimization
criterion for the systemic macromodel, the entropy functional of quality is employed, the
minimal value of which determines the maximum order of the bar’s structure. This criterion,
determined by the conditions of the extremum of nonequilibrium entropy functional for the
process (Table), emerges as a general indicator of bar structural uniformity on the micro- and
macrolevels. The order on the microlevel implies structural uniformity of the metallic matrix
through the cross section of the bar with uniform distribution of graphite nodules, and the
uniform distribution of chemical elements through the cross section of the grain boundaries
without segregation. The order on the macrolevel corresponds to production of a bar with a
uniform macrostructure without internal defects, with fulfillment of the conditions of
directional solidification and compensation for shrinkage. As a result of the optimization
problem solution, the regularity of change in the kinetic operator is characterized by
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successive reduction with time in its intrinsic values, which are equated at specified DP
points. The corresponding dynamic processes illustrates the connections between T, S, D, PT
processes in the two-phase zone. The ranged processes in optimal model are characterized by
three basic parameters: the number of equations n, the variability  , and the space k
parameters, which are common for the macromodel.
Table 1. Scheme of the main interrelated physical phenomena
Physical phenomena and their
notations
Phase-space
coordinates
l
x
Phase-time coordinates
xt
xnl 1
2. Heat conductivity (T) (initial
phenomenon)
xnl
3. Solidification (S) under
temperature gradient
xnl 1
 (l) ~

l
 (t) ~

~ vc
l
xnt 1

~ C
t
xnt 2
 2   C
~
t 2
t
xnl 2
 2   vc
~
l 2
l
5. Mass transfer (M)
xnl 3
 2 C
~ vm
 l2
6. Phase transformation (PT)
causing stresses and pressure
  
~
~
 l t  t
 vm  3
l
~ 3
xn 4
l
t
8. Hydrodynamic mechanism
(H) developing a pressure

l
xnt
4. Effective diffusion (D) under
heat flow
7. Hydrodynamic
transformations in liquidsolid state (HT)
xnt 1  (t)
  dl ~  (l)
1. Thermomechanical stresses
(TS)
xnl 5
 vm
~ p(l)
 l2
xnt 3
t
n 4
x
 2 C
 t2
 vm
 2
~ pdl ~ 2
t 
t
xnt 5
p(t)
xnt 6
 p
t
2
 p
l
xnl 6
The solution of optimization problem ensures a maximum value of the functional of microand macrostructural ordering.
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Figure 2 illustrates the diagram of computation of the IMD optimal model's
process x(t)  xi (t,l,u(ti )) using a given space image with distributed information
S per cross-section F .
i (t i )
S
S
F
F
x[t,l,u(t)]
n
k
invar
li
i (t i )
x i(t)
ti
ti
v(t i)
u (t i )
u (t i )
i (t i )
x i(t)
Figure 2. The diagram illustrating the sequence of the IMD optimal model's process
computations.
The algorithm calculates the models' basic parameters ( n ,  , k ), which define the model's
invariants INVAR, time ti and space li discrete intervals, eigenvalues  i (ti ) of differential
model's operator, MC-function, speeds CT ,C , simulates the inner vi (ti ) and output ui (ti )
optimal controls. The model’s specific Hamiltonian hv e is used to calculate the optimal
hydrodynamics and basic three parameters of new model.
OPTIMAL HYDRODYNAMIC CONDITIONS FOR THE IMD MODEL.
CALCULATION THE MODEL’S PARAMETERS
The systemic solidification macromodel (Table) includes the hydrodynamic components (H,
HT) that interact with motion of the liquid and solid phases participating in HCC. Maximal
macrostructure order and obtaining a homogenous bar without defects can be reached by
minimizing hv e that defines the optimal function of the entropy production  =  (t , l ) . The
problem consists of finding the optimal functions for hydrodynamic variables: the pressure
p , the velocity  of the liquid phase movement, which are defined by the given function
 =  (1,  ) . Let us consider the movement of a non-compressed Newton's liquid within a
cylindrical channel, described by the equations:
t
zo

p
=  
, z
,  2
(6)
r o

z
ro
with the boundary conditions:

r
 (0, R )   ( R ) , R  ,  ( ,1)   ( ) =0;
=-2g(  ).
(7)
R
ro
The tangent tension on the liquid-solid phase boundary is given by the equation:

f  a1
, a ~  ,    .
(8)
R 1
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The equations are written in relative (non dimensional variables) as the time  , the radius R
and the one of space coordinates l  z , and with the absolute variable: t as a time,  ,  as
the kinematic and dynamic viscosities, r , ro are the current and fixed channel radiuses
accordingly, zo is the axis coordinate,  is the density.
The entropy production is connected with the tangent tension f by the relation:
 ( R,  )  a 2 f 2 , a 2 ~ 
(9)
where it is assumed that the parameters a , a1 ,  ,  are given and fixed.
The entropy production for the optimal systemic model is defined by the equation:
 (1, )  b 2 ot (1, )
(10)
where
 ot (l , ) =  t ( ) is the given function of time ( in the IMD equations) , and b is the
constant coefficient. Function g (  ) (7) can be expressed via  using (10) in the form:

b
b2 t = a 2 f 2 = a 2 a12 ( ) 2 , g  
(11)
( t )1/ 2 .
R
2a1 a
p
 f at the given  t , we obtain
By solving the equations (6-9) for an arbitrary function z
the following integral equation with regard to f:
 
b
f ( ) exp(1  2k (   )) d = C ( t )1 / 2 , C  
(12)


a
k 1 0
t
2
where  k are the eigenfunctions of equation (6). The solution gets representation:
f  f ,
k
k

f
k
( ) exp( 2k (   )) d =
0
C
( t ) 1 / 2 .
k
2
(13)
After substituting (13) into (6-9), the field of velocities    ( R, ) ) can be found.
Let us consider the concrete results. Suppose  is given by the following equations:
 =  o  ,  =  (  o ,  o , z),  o =  o (

r
2
) 1 ,  o =  o (
o

r 2o
) 1 ,  o =  o (  ) (14)
where  o ,  o =  o  are determined by the parameter  of the optimal model at given ro and
 . Then the sought function f '   acquires the form:
|  | r 2  (t )
1/ 2
1/ 2
  25.1 ( o o
- [ (0)] )
(15)
1 / 2 + [ ( t )]
4 25.1 [ (t )]
From that at  (0) =0,  (t ) =  (t),  o =-2.29,  o =-1.1456,  =0.5 we get the solution
3.3 (t )   (t )
 o  25.1 (
).
(16)
[ (t )]1 / 2
The pressure changing should be applied to the solidified iron bars by alliterating the metal
level within the iron receiver. In the casting process with applied pressure, for fulfillment the
directional solidification, the external pressure as a control function should satisfy the
relation for  o . The feeding and the solidification processes in systemic model are mutually
372
interconnected such a way that the speed of the liquid metal movement is coordinated with
changing density of the solidified bar. The found optimal impulse feeding law
C ( ln )=C [ ln ( t n , l T , t c , t )] satisfies these conditions, where t n is the starting moment of
changing linear feeding velocity C (t); t k is the fixed time interval, t is the current time
interval, l T is the space interval of the impulse feeding controls; t c is the time interval of
stopping the feeding impulses; Cl is the average solidification speeds.
The linear bar size increment satisfies the function l n  (C  C l )t n 1 , where the component
t
l
C  T (1  c ) , t = t k - t n characterizes the average speed of feeding. The difference ( ln )
t
tk
defines the space distribution of metal that is necessary to compensate a volumetric
contraction at l  l n , and during the time interval t n 1 . At the real conditions of the
gravitational feeding, the impulse cycle takes place, when the continuous speed feeding Cl is
equal to the average solidification speed, and the bar high is equal ln .
The time of solid skin formation, required to start bar’s extraction or draw, is governed by the
conditions needed to control the elastic deformations with compensation for shrinkage. The
optimal drawing speed changes in a discontinuous pattern and is characterized by a cycle
frequency: drawing-pause. The implemented algorithm allows the computation of the
optimal value of casting speed, the drawing and pause intervals, the cast iron temperature in
the receiver, the cast bar temperature, the variation in the flow rate and water temperature in
the die, and the parameters of die design and secondary cooling conditions.
CALCULATION OF CHEMICAL COMPOSITION AND MICROSTRUCTURE
PREDICTION
Each cross section of continuously cast bar is characterized by its own solidification pipe with
the zone of effective diffusion. The concentration of chemical components in this zone
depends on the solidification time. With increasing cross section of the bar, solidification time
and the time of effective diffusion increase resulting in structural microheterogeneity. To
ensure the same concentration of chemical elements in the model bar, as in the actual bar, the
model controls the concentration gradient at the boundary of the solidification zone and its
distribution within this zone. The problem is solved by selecting the chemical composition of
the bar as a function of the initial distribution coefficient. Knowing the change in
concentration of chemical elements during solidification of the base bar, the optimal chemical
composition of the given bar may compute. The model also calculates amount of alloying
elements, for example Sn, Cu, Mo, which are required under real solidification conditions to
control the initial distribution coefficient within the zone of effective diffusion and as-cast
microstructure. The model calculates a phase transition in the presence of k points of
nonequilibrium, particularly, the nodule count (NC), when (n-4+k), or two (n-2+k) elements
are precipitated. The radius R of the bar corresponds to the model length of the solidification
pipe, which is characterized (at the fixed angle at its vertex) by the segment of the IN’s helix
of the cone l (n+1)), where the number of elements
2 2
(Ln+1/Ln+k-1)  2(n-2-k)(l+  )
(17)
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is found. NC per unit of cross-sectional area S=  R , precipitated at the k-th point, is:
NC=4  /S (n-2+k) 2 (Ln+1/Ln+k-1) 2 (l+  2 ) 2
(18)
where n and  are the models' basic parameters, L(n+1)=L(n,  ) is the model length of the
solidification cone; Ln+k-1=L (n,  ) is the model length of the effective diffusion zone
calculated on computer. In the optimal model, graphitization occurs in the spatial interval
L (n-4) with the maximum NC (L (n-4)) nodule count. The computerized method calculates
the NC at any point along the entire cross section of the bar with a sufficient accuracy (7).
2
CONCLUSIONS
A new information approach to the simulation and control of the solidification process using
the IMD has been described. This new computation techniques and software have been
applied to the horizontal continuous casting of ductile iron. Comparisons with the results of
actual test runs have validated the accuracy and usefulness of this new approach.
ACKNOWLEDGMENTS
The authors would like to thank Professor J.T. Berry of Mississippi State University for
helpful suggestions rendered during the preparation of this paper.
REFERENCES
1. Godsell B: ‘Continuous Casting of Ductile Iron: A Numerical Approach’, AFS
Transactions, 1987, 613-616.
2. Lerner V: ‘Mathematical Foundation of Information Macrodynamics’, International
Journal ‘Systems Analysis-Modeling-Simulation’, Vol. 26, 1996, 119-184.
3. Lerner V: ‘ Informational Macrodynamics: System Analysis Modeling & Simulation
Methodologies’, Proceedings of the 1998 Winter Simulation Conference, Washington,
December 1998, 125-133.
4. Lerner V., Talyanker M: ’Informational Geometry for Biology and Environmental
Applications’, Proceedings of the 2000 Western MultiConference, San Diego,
January 2000,79-84.
5. Lebau C: ‘Properties of Continuous Cast Austempered Ductile Iron Bar,’ 2nd International
Conference on Austempered Ductile Iron: Your means to improve Performance,
Productivity and Cost, Rackham School, University of Michigan, Ann Arbor, MI, ASME,
March 17-19, 1985, 215-226.
6. Lerner Y., Griffin G: “Developments in Continuous Casting of Gray and Ductile Iron’,
Modern Casting, November 1997, pp.41-44.
7. Lerner Y: ‘Continuous Casting of Ductile Iron. Solidification, Microstructure, and
Properties,’ 50th Electric Furnace Conference Proceedings, ISS, V. 50, Atlanta, USA,
December 13-15, 1992, 331-340.
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