Jorge A Laval
email: jorge.laval@ce.gatech.edu

Guillaume Costeseque

Bargavarama Chilukuri

The impact of source terms in the variational representation of traffic flow

Keywords: traffic flow, source terms, kinematic wave model

This paper revisits the variational theory of traffic flow, now under the presence of continuum lateral inflows and outflows to the freeway say Eulerian source terms. It is found that a VT solution can be easily exhibited only in Eulerian coordinates when source terms are exogenous meaning that they only depend on time and space, but not when they are a function of traffic conditions, as per a merge model. In discrete time, however, these dependencies become exogenous, which allowed us to propose improved numerical solution methods. In Lagrangian and vehicle number-space coordinates, VT solutions may not exist even if source terms are exogenous.

Introduction

The variational theory (VT) applied in traffic flow theory (Daganzo, 2005a,b;Claudel and Bayen, 2010a,b) was an important milestone. Previously, the only analytical solution to the kinematic wave model of [START_REF] Lighthill | On kinematic waves. I Flow movement in long rivers. II A theory of traffic flow on long crowded roads[END_REF]; [START_REF] Richards | Shockwaves on the highway[END_REF] was obtained thanks to the method of characteristics, which does not give a global solution in time as one needs to keep track of characteristics crossings (shocks and rarefaction waves) and impose entropy conditions to ensure uniqueness. This means that analytical solutions cannot be formulated except for very simple problems.

In contrast, VT makes use of the link between conservation laws and the Hamilton-Jacobi partial differential equation (HJ PDE), allowing the kinematic wave model to be solved using the Hopf-Lax formula [START_REF] Lax | Hyperbolic systems of conservation laws ii[END_REF][START_REF] Olejnik | Discontinuous solutions of non-linear differential equations[END_REF][START_REF] Hopf | On the right weak solution of the cauchy problem for a quasilinear equation of first order[END_REF], better known in transportation as Newell's minimum principle [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, I general theory, II queuing at freeway bottlenecks, III multi-destination flows[END_REF] whenever the fundamental diagram or Hamiltonian is piecewise linear. The big advantage is that this representation formula gives an analytical global solution in time that does not require explicit consideration of shocks and/or entropy conditions. Moreover, current approximation methods for the Macroscopic Fundamental Diagram (MFD) of urban networks rely on this approach [START_REF] Daganzo | An analytical approximation for the macroscopic fundamental diagram of urban traffic[END_REF][START_REF] Geroliminis | The effect of variability of urban systems characteristics in the network capacity[END_REF][START_REF] Leclercq | Estimating MFDs in simple networks with route choice[END_REF][START_REF] Laval | Stochastic approximations for the macroscopic fundamental diagram of urban networks[END_REF]. In addition, when the flow-density fundamental diagram is triangular (or piecewise linear more generally), numerical solutions become exact in the HJ framework, and this is not the case in the conservation law approach [START_REF] Leveque | Numerical methods for conservation laws[END_REF]. It has also been shown that the traffic flow problem cast in Lagrangian or vehicle number-space coordinates also accepts a VT solution, which can be used to obtain very efficient numerical solution methods [START_REF] Leclercq | The Lagrangian coordinates and what it means for first order traffic flow models[END_REF][START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF].

The traffic flow problem with source term is also of great interest; e.g., it can be used to approximate (i) long freeways with closely spaced entrances and exits, (ii) the effect of lanechanging activity on a single lane, or (iii) the effects of turning movements, trip generation and trip ends in the MFD. But the underlying assumption in the previous paragraphs is that vehicles are conserved. This begs the twofold question, are representation formulas for the VT solutions still valid, or even applicable, when there is a Eulerian source term? If not, can efficient numerical solution methods still be implemented? Recent developments in this area have not answered these questions as they are primarily concerned with discrete source terms (e.g., [START_REF] Daganzo | Singularities in kinematic wave theory: Solution properties, extended methods and duality revisited[END_REF]Costeseque and Lebacque, 2014a,b;[START_REF] Li | Modeling space-time inhomogeneities with the kinematic wave theory[END_REF]).

To answer these questions this paper is organized as follows. In section 2 we formulate the general problem and show that in general VT solutions are not applicable; but section 3 shows that they are when the source term is exogenous. Based on these results, section 4 presents numerical methods for the endogenous inflow problem that outperform existing methods. Section 5 briefly shows that in space-Lagrangian and time-Lagrangian coordinates VT solutions do not exist even if source terms are exogenous. A discussion of results and outlook is presented in section 6.

Problem Formulation

Consider a long homogeneous freeway corridor with a large number of entrances and exits such that the net lateral freeway inflow rate, φ, or inflow for short, can be treated as a continuum variable in time t≥ 0 and location x∈ R, and has units of veh/time-distance. The inflow is an endogenous variable consequence of the demand for travel, and could be captured by a function of the traffic states both in the freeway and the ramps. For simplicity, in this paper the inflow is assumed to be a function of the density, k(t, x), of the freeway only, i.e. φ = φ(k(t, x)), but also the exogenous case φ = φ(t, x) will be of interest.

In any case, the traffic flow problem analyzed in this paper is the following conservation law with source term:

k t + H(k) x = φ, k = g on Γ (1a) (1b)
where H is the fundamental diagram, g is the data defined on a boundary Γ, and variables in subscript represent partial derivatives. Now we define the function N(t, x) such that:

N x = -k say N(t, x) := +∞ x k(t, y)dy, (2)
and integrate (1) with respect to x to obtain its HJ form [START_REF] Evans | Partial Differential Equations[END_REF]:

N t -H(-N x ) = Φ, N = G on Γ, (3a) (3b)
where we have defined:

G(t, x) = Γ g(t, x)dΓ, (t, x) ∈ Γ, and Φ(t, x) = - x 0 φ(t, y)dy (4a) (4b)
where Φ is a potential function; the negative sign in (4b) follows from the traditional counting convention in traffic flow, where further downstream vehicles have lower vehicle numbers.

It is important to note that compared to the traditional case with zero inflow, under a continuum source term the interpretation of N changes: for fixed x it still defines a cumulative count curve due to (2) but its isometrics do no longer give vehicle trajectories. To see this we note that according to (3a) the flow q = H(k) is now:

q = N t -Φ, (5)
The time integration of (5) reveals that cumulative count curves are now given, up to an arbitrary constant, by Ñ(t, x) + t 0

x 0 φ(s, y)dsdy, where Ñ(t, x) := t 0 q(s, x)ds denotes the usual N-curve without source terms and the integral represents the net number of vehicles entering (φ > 0) or exiting (φ < 0) the road segment by time t and upstream of x.

The method to obtain the solution of (3) depends on the dependencies of the potential function. If the inflow function is allowed to depend on the traffic state in the freeway, i.e. φ = φ(k) then the potential function Φ depends non-locally on N, since:

Φ(t, x) = Φ(N, x) = - x 0 φ(-N x (t, y))dy, (6)
and therefore it is not obvious that (3) accepts a representation formula as a VT solution. To see this, we note that even in the simplest linear case:

φ(k) = a -bk, a, b ≥ 0, we get: Φ(N, x) = -ax + x 0 k(t, y) dy b = -ax -(N(t, x) -c)b, (7a) (7b)
where c= +∞ 0 k(t, y)dy = N(t, 0) is an arbitrary constant of integration. This means that (3) becomes the more general HJ equation N t -H(x, N, -N x ) = 0, where H is the Hamiltonian.

The Hamiltonian's N-dependency is what complicates matters. [START_REF] Barron | Hopf-Lax-type formula for u t + H(u, Du) = 0[END_REF]; [START_REF] Barron | Representation of viscosity solutions of Hamilton-Jacobi equations[END_REF] show that a Hopf-Lax type solution exists in this case only when (among other assumptions) a = 0 i.e. H(x, u, p) = Ĥ(u, p) where Ĥ does not depend on the space variable and H is homogeneous of degree one with respect to N x , which is of no use in traffic flow since in practice it means that the Hamiltonian has to be monotone linear. As explained in section 5, we argue that this N-dependency prevents formulating an equivalent VT problem (such as (8) below) that can be solved with variational methods.

As shown in the next section, it turns out that when inflows are exogenous a global VT solution can still be identified, albeit not in Hopf-Lax form because optimal paths are no longer straight lines.

Exogenous inflow

Setting of the VT problem

The results in this section are based on VT (Daganzo, 2005a), where the solution of the HJ equation N t -H(t, x, -N x ) = 0 is given by the solution of the following variational problem:

N(P ) = min B∈Γ P , ξ∈V BP f (B, ξ), with f (B, ξ) = G(B) + t t B R(s, ξ(s), ξ ′ (s)) ds (8a) ( 8b
)
where P is a generic point with coordinates (t, x), B ≡ (t B , y) is a point in the boundary Γ P , ξ is a member of the set of all valid paths between B and P denoted V BP , and ξ(t B ) = y; see Fig. 1a. The function R(•) gives the maximum passing rates along the observer and corresponds to the (concave) Legendre transform of H, i.e.,

R(t, x, v) = sup k H(t, x, k) -vk .
It is worth mentioning that in the simplest homogeneous case where H = H(k) the VT solution (8) becomes the Hopf-Lax formula:

N(P ) = min B∈Γ P G(B) + (t -t B )R x -y t -t B , (9)
where the minimization over ξ(t) is no longer necessary since characteristics become straight lines in this case. With a little abuse of notation, consider R(v) = sup k {H(k) -kv}. Unfortunately, even the presence of exogenous lateral inflows that vary in time or space make characteristics not to be straight lines and therefore a Hopf-Lax type solution cannot be devised. A VT solution, however, still exists as shown next.

We now formulate VT solution to account for source terms explicitly. In the remaining of the paper, we assume a triangular flow-density diagram. It may be defined by its free-flow speed u, wave speed -w (with w > 0) and jam density κ such that

H(k) = min {uk , w(κ -k)} for any k ∈ [0, κ]. (10)
It follows that the capacity is Q = κwu/(w + u) and the critical density

K = Q/u.
When the potential function is exogenous, i.e. Φ = Φ(t, x), the Hamiltonian can be written as the sum of the fundamental diagram and the potential function, i.e.:

H(t, x, k) = H(k) + Φ(t, x). In the case of a triangular fundamental diagram we have R(t, x, v) = Q -Kv + Φ(t, x) with v ∈ [-w, u],
and the function f (B, ξ) to be minimized reads:

f (B, ξ) = G(B) + (t -t B )Q -(x -y)K + t t B Φ(s, ξ(s)) ds =:J . ( 11
)
The J-integral in ( 11) is what separates this problem from the problems studied so far in traffic flow using VT principles. This integral in terms of φ is:

J = - t t B ξ(s) y φ(s, x) dxds, (12)
and represents the net number of vehicles leaving the area below the curve x = ξ(t), namely area A(ξ); see Fig. 1b. Therefore, minimizing J given y = ξ(t B ), can be interpreted as finding ξ(t) that maximizes the net number of vehicles entering A(ξ).

Initial value problems

In the initial value problem (IVP) the boundary Γ is the line {t B = 0} ×R, so that the problem here is (3a) supplemented with:

N(0, x) = G(x), x ∈ R, (13)
where we assume that G ∈ C 2 (R). The candidate set for B, Γ P is reduced to B's x-coordinate, y, which is delimited by two points U = (0, x U ) and D = (0, x D ), where:

x U = x -ut, and

x D = x + wt, x U < y < x D (14a) (14b)
see Fig. 1a. The following subsections examine simplified versions of this problem that reveal considerable insight into the general solution.

Constant inflow

Consider the IVP with constant inflow problem:

φ(t, x) = a, (t, x) ∈ (0, +∞) × R, (15)
for some a ∈ R \ {0}. It follows from ( 12) that J = -aA(ξ) and therefore, for a fixed y = ξ(0) the minimum of ( 11) is obtained by a path that: (i) maximizes A(ξ) when a > 0; or (ii) minimizes A(ξ) when a < 0. These two optimum paths are the extreme paths in V BP that define its boundary; see "upper" and "lower" paths in Fig. 2a. This solution is a "bang-bang" solution, typical for this kind of optimal control problems. The reader can verify that the areas under these paths are given by A(y, x D ) and A(y, x U ), respectively, where we have defined:

A(y, x -) = 1 2 (x + x -)t -sign(a) (x --y) 2 u + w (16)
where sign(a) is the sign of a and x -is a placeholder for x D if a > 0 or x U if a < 0. Thus the optimization problem (8) has been reduced to a single variable, y

N(t, x) = min f (B, ξ) ⇔N(t, x) = min x U ≤y≤x D f (y), with f (y) := G(y) + tQ + (y -x)K -aA(y, x -) (17a) (17b)
whose first-and second-order conditions for a minimum y * read:

f ′ (y * ) = ψ (y * -x -) + G ′ (y * ) + K = 0, f ′′ (y * ) = ψ + G ′′ (y * ) > 0
, where:

ψ := |a| u + w , (18a) (18b) (18c)
where | • | denotes the absolute value. Notice that ψ > 0 as long as a = 0.

Of course, the optimal point y * needs to satisfy (14b):

x U ≤ y * ≤ x D .

Constant initial density

Here, in addition to φ(t, x) = a, we assume

g(x) = k 0 , ⇒ G(x) = -k 0 x -∞ < x < ∞ (19)
which implies that the function(17b) to minimize is a parabola:

f (y) = -c 0 -c 1 y + ψ 2 y 2 , ( 20
)
with constants c 0 = Kx U - ψ 2 x 2 --sign(a) (x -+ x) (u + w)t and c 1 = ψx --(K -k 0 )
, and extremum:

y * = x -- K -k 0 ψ . (21)
Notice that (18b) is always satisfied in this case and therefore y * is always a minimum and should be included so long as x U ≤ y * ≤ x D . Combining this with (21) gives that the final solution can be expressed as:

N(t, x) = f (y * ), t > (K -k 0 )/a > 0 min{f (x U ), f (x D )}, otherwise (22a) (22b)
Notice that (K -k 0 )/a represents the time it takes for the system to reach critical density, namely "time-to-capacity". This means that y * will be the optimal candidate only once a regime transition occurs. This will happen only if the time to capacity is positive, i.e. when sign(a) = sign(K -k 0 ) or more explicitly, when k 0 is under-critical and φ is an inflow (a positive) or when k 0 is over-critical and φ is an outflow (a negative).

Somewhat unexpectedly, we note that -f

x (x U ) = -f x (x D ) = -f x (y * ) = k 0 + at,
which means that the density is always given by the traveling wave k(t, x) = k 0 + at, which is also the solution of (1) in this case using the method of characteristics. One should also impose feasibility conditions for the density, i.e. 0 ≤ k ≤ κ, which gives:

k(t, x) =      0, t > -k 0 /a, a < 0 κ, t > (κ -k 0 )/a, a > 0 k 0 + at, otherwise (23a) (23b) (23c)
The flow can be obtained using ( 5), but it is equivalent and simpler to use q = H(k).

Extended Riemann problems

Riemann problems are the building blocks of Godunov-type numerical solution methods. As illustrated in Fig. 2b, in these problems one is interested in the value of N at x = x 0 ≥ ut, i.e. at point P = (t, x 0 ), with initial data typically given by the density at t = 0. This is a very special case for with the 'target' point P is located on the discontinuity of the initial data at x = x 0 . However the methodology below can be easily extended to the cases where P = (t, x) with x < x 0 or x > x 0 . Here, we extend the initial data to include the inflow:

(g(x), φ(x)) = (k U , a U ), x ≤ x 0 (k D , a D ), x > x 0 , (24a) (24b)
which in conjunction with (3) define an extended Riemann problem, or ERP for short. We assume that (a U , a D ) = (0, 0). Notice that now:

x U = x 0 -ut, and

x D = x 0 + wt. ( 25
)
For simplicity and without loss of generality we set G(x 0 ) = 0, which implies:

G(x) = (x 0 -x)k U , x ≤ x 0 (x 0 -x)k D , x > x 0 (26a) (26b)
It will be convenient to define:

η = a U /a D , ψ = a D u + w , θ = u/w. (27)
The J-integral in this case is a weighted average of the portion of A(ξ) upstream and downstream of x = x 0 , weighted by a U and a D , respectively. For instance, if ξ(t) > 0 for all t, we have J = -

t 0 ξ(t) 0 φ(x) dxdt = - t 0 (a U x 0 + a D (ξ(t) -x 0 ))dt = -(a U x 0 t + a D t 0 ξ(t) -x 0 dt),
where x 0 t is the portion of A(ξ) upstream of x = x 0 and t 0 ξ(t) -x 0 dt is the portion of A(ξ)

downstream of x = x 0 in this particular case.

It follows that the minimization of the J(ξ) can be achieved analogously to the previous section by considering the upper and lower paths from each candidate y = ξ(0). In addition, however, one has to include 2 middle "paths" that would reach and stay at x = x 0 until reaching P ; see Fig. 2b. To formalize, let j 1 , j 2 and j 3 be the value of J(ξ) when y ≥ x 0 along the upper, lower and middle paths, respectively; similarly for j 4 , j 5 and j 6 when y ≤ x 0 . Calculating the areas upstream and downstream of x = x 0 defined by each path, it can be shown that the j i 's can be obtained from:

2(j 1 (y) -J 0 )/ψ = -θt 2 w 2 (2η(θ + 1) + 1) + 2twy 0 + y 2 0 , 2(j 2 (y) -J 0 )/ψ = -ηθ(2θ + 1)t 2 w 2 + 2ηθtwy 0 + y 2 0 ((η -1)θ -1), 2(j 3 (y) -J 0 )/ψ = -(θ + 1) 2ηθt 2 w 2 + y 2 0 , 2(j 4 (y) -J 0 )/ψ = -θt 2 w 2 (2η(θ + 1) + 1) + 2twy 0 + y 2 0 (ηθ + η -1)/θ, 2(j 5 (y) -J 0 )/ψ = -η θ(2θ + 1)t 2 w 2 -2θtwy 0 + y 2 0 , 2(j 6 (y) -J 0 )/ψ = η(θ + 1) y 2 0 -2θ 2 t 2 w 2 /θ, (28a) (28b) (28c) (28d) (28e) (28f)
where y 0 = x 0 -y and J 0 is a constant of our problem that represents the net number of vehicles leaving the area upstream of x U , i.e. -a U x U t in this case. Notice that j 1 (x 0 ) = j 4 (x 0 ), j 2 (x 0 ) = j 5 (x 0 ) and j 3 (x 0 ) = j 6 (x 0 ), as expected. The function to minimize in this case can be written as:

f (y) = G(y) + tQ -(x 0 -y)K + J(y)
, where: (29) J(y) = min{j 1 (y), j 2 (y), j 3 (y)}, y > x 0 min{j 4 (y), j 5 (y), j 6 (y)},

y ≤ x 0 (30a) (30b)
The solution of (3) under these conditions can be reduced to the evaluation of f (y) at a small number of candidates. In addition to candidates y = x U and y = x D , we have to consider the discontinuity at y = x 0 and the possible minima produced by each of the components in (28). Since f (y) is piecewise quadratic, each one of these components has at most one minimum, namely y = y i , i = 1, . . . 6, which can be obtained by solving the first-order conditions f ′ (y i ) = 0 associated with each of the j i 's; i.e.:

y 1 = x D -(K -k D )/ψ, y 2 = x 0 + ηθx D -(K -k D )/ψ θ(η -1) -1 , y 3 = x 0 + K -k D (θ + 1)ψ , y 4 = x 0 + x U + θ(K -k U )/ψ 1 -η(1 + θ) , y 5 = x U + (K -k U )/(ηψ), y 6 = x 0 + θ (k U -K) η(θ + 1)ψ , (31a) (31b) (31c)
For the y i 's to be valid candidates they must meet the following conditions:

x 0 < y i < x D , i = 1, 2, 3 x U < y i < x 0 , i = 4, 5, 6 (32a) (32b)
which ensure that y i is on the boundary Γ P . With all, the sought solution can be expressed as:

N(t, x 0 ) = min y∈Y f (y), with: Y = {x U , x 0 , x D , y * 1 , . . . y * 6 } (33)
where y * i is y i if (32) is met and null otherwise. Notice that this solution method does not require imposing f ′′ (y * ) > 0 because maxima will be automatically discarded in the minimum operation.

The average flow q(t, x 0 ) during (0, t) can be obtained as follows:

q(t, x 0 ) = 1 t t 0 N t (s, x 0 ) -Φ(s, x 0 ) ds = 1 t (N(t, x 0 ) - t 0 Φ(s, x 0 ) ds) = 1 t (N(t, x 0 ) -(J 0 + x U a U )t) = N(t, x 0 )/t -J 0 -x U a U . (34a) (34b)
As shown in Section 4, this formula allows to formulate improved numerical solution methods for the endogenous problem.

We restrict the space to a segment of road [ξ, χ] with ξ < χ. In the initial and boundary value problem (IBVP) the boundary Γ is the set defined as

({t B = 0} × [ξ, χ]) ∪ ((0, +∞) × {x = ξ}) ∪ ((0, +∞) × {x = χ}) ,
so that the problem here is (3a) supplemented with:

     N(0, x) = G ini (x), on [ξ, χ], N(t, ξ) = G up (t),
on (0, +∞), N(t, χ) = G down (t), on (0, +∞).

(

) 35
For obvious compatibility reasons, we request these conditions to satisfy

G ini (ξ) = G up (0) and G ini (χ) = G down (0).
We also consider a a = 0 and k 0 ∈ [0, κ] such that the inflow rate is given by

ϕ(t, x) = a, for any (t, x) ∈ [0, +∞) × [ξ, χ]
and the initial density

g ini (x) = k 0 , for any x ∈ [ξ, χ].
According to the position of point P = (t, x) with t > 0 and x ∈ (ξ, χ), we have the following cases to distinguish (see also [START_REF] Jin | Continuous formulations and analytical properties of the link transmission model[END_REF])

(t U , x U ) =        0, x - t u , if x ≥ ξ + t u , t - x -ξ u , ξ , else,
and

(t D , x D ) =        0, x + t w , if x ≤ χ - t w , t + x + χ w , χ , else,
that define 4 regions (see Figure 6)

Region Set of points Upstream point Downstream point I : (t, x) χ - t w ≥ x ≥ ξ + t u (t U , x U ) = 0, x - t u (t D , x D ) = 0, x + t w II : (t, x) x ≤ min ξ + t u , χ - t w (t U , x U ) = t - x -ξ u , ξ (t D , x D ) = 0, x + t w III : (t, x) x ≥ max ξ + t u , χ - t w (t U , x U ) = 0, x - t u (t D , x D ) = t + x + χ w , χ IV : (t, x) ξ + t u ≥ x ≥ χ - t w (t U , x U ) = t - x -ξ u , ξ (t D , x D ) = t + x + χ w , χ
Then in Region I, everything is exactly the same than in previous Subsection 3.2 while in the other cases, we will compare initial data G ini coming from (t D , x D ) with upstream data G up coming from (t U , x U ) (Region II), initial data G ini from (t U , x U ) with downstream data G down from (t D , x D ) (Region III) and finally upstream data G up from (t U , x U ) and downstream data G down from (t D , x D ) (Region IV). Everything can be done exactly in the same way and finally the idea is to use the inf-morphism from Aubin et al. ( 2011)

N(t, x) = min {N ini (t, x) , N up (t, x) , N down (t, x)}
where N ini , N up and N down denote the partial solutions obtained by considering respectively initial, upstream and downstream conditions. The interested reader is also referred to [START_REF] Mazaré | Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model[END_REF].

It is noteworthy that additional constraints could be also addressed in the same framework.

In particular, any internal boundary condition could be added as long as this condition is exogenous and do not depend on the traffic state. Such internal boundary condition can represent for instance a moving bottleneck constraint as discussed in Claudel and Bayen (2010b); [START_REF] Mazaré | Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model[END_REF].

Numerical solution methods

In the following two subsections we formulate two numerical solution methods to find the global solution of the endogenous inflow problem in conservation law form (1), ( 7) and in HJ form (3), ( 7), respectively. The basic idea is that in discrete time, if the (endogenous) inflows are computed using the traffic states from the previous time step using an explicit-in-time numerical scheme, then they become exogenous for the current time step and therefore the VT solution may be applied.

Godunov's method

This method has been traditionally used to solve the traffic problem in conservation law form without inflows, and constitutes the basis of the well known Cell Transmission (CT) model [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory[END_REF] assuming the triangular Hamiltonian (10). In this method, time and space are discretized in increments ∆t and ∆x = u∆t, respectively, and we let:

k j i = k(j∆t, i∆x) (36)
be the numerical approximation of the density. The update scheme is the following discrete approximation of the conservation law (1):

k j+1 i -k j i ∆t + q j i+1 -q j i ∆x = φ(k j i ) ( 37
)
The key to Godunov's method is the computation of the flow into cell i, q j i , which are obtained by solving Riemann problems. Traditionally, inflows have been considered explicitly only in the update scheme (37) but not in the solution of the Riemann problems [START_REF] Laval | Continuum approximation for congestion dynamics along freeway corridors[END_REF]. This implies that the computation of q j i corresponds to the original CT rule (also equivalent to the minimum formula between upstream demand and downstream supply [START_REF] Lebacque | The godunov scheme and what it means for first order traffic flow models[END_REF]):

q j i = min{Q, uk j i , (κ -k j i+1 )w}, (CT rule) (38)
Here, we will compare the CT rule with the ERP rule (34b), i.e. the flow based on extended Riemann problems. Both methods are first-order accurate since both are Godunov-type methods, and therefore the rate of convergence of both methods should be similar and roughly proportional to the mesh size. The main difference is in the magnitude of the error, where the ERP rule should be more accurate because the impacts of inflows are explicitly considered in the solution of Riemann problems. We illustrate this with the following example.

Example. Consider an empty freeway at t = 0 subject to an inflow linear in both x and k; i.e.:

g(x) = 0, φ(k) = ax -buk, a, b > 0. (39a) (39b)
Notice that [START_REF] Laval | Continuum approximation for congestion dynamics along freeway corridors[END_REF] showed that linear inflow functions arise in the continuum approximation of the Newell-Daganzo merge model [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, I general theory, II queuing at freeway bottlenecks, III multi-destination flows[END_REF][START_REF] Daganzo | The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory[END_REF], which accounts for the interactions between freeway and on-ramp demands. The particular coefficients of the linear function depend upon the state of the freeway and on-ramps. In particular, (39b) corresponds to the case where both are in free-flow, ax represent the inflow demand rate at x and b the exit probability per unit distance. Using the method of characteristics [START_REF] Laval | Continuum approximation for congestion dynamics along freeway corridors[END_REF] showed that the solution of ( 1), ( 39) is :

k(t, x) = a b 2 u bx -1 + (1 -b(x -tu))e -btu (40)
provided k(t, x) ≤ K. To get an idea of the solution for all densities, Fig. 3 shows the numerical ERP solution with ∆t = 1 s, with parameters given in its caption. Notice that similarly to [START_REF] Laval | Continuum approximation for congestion dynamics along freeway corridors[END_REF] we chose θ = 1 (say u = w) to avoid the numerical errors intrinsic to Godunov's method and to focus on those caused by the treatment of the inflow. that the main difference, as expected, is in the flow estimates, particularly at t = 0 where the CT rule predicts zero flow since the freeway is empty and it does not consider inflows in its calculations.

To assess the accuracy of each method, part (c) of Fig. 4 shows the density root-meansquared error (RMSE) of each method with respect to (40) for varying ∆t and only until t = 2.35 s, when the density exceeds the critical density K; see Fig. 4a. It becomes apparent that both methods converge to the right solution as ∆t → 0, but the accuracy of the proposed method outperforms the existing method by a factor of two for all values of ∆t.

Fig. 4d shows the optimal candidate that minimizes f (y) at each time step of the numerical method, which is an element of the set Y, for the ERP rule, and of {x U , x 0 , x D } for the CT rule. It can be seen that both methods coincide except for the time step where the density approaches the critical density, in which case the proposed method finds the more accurate optimal candidate y * 1 . Daganzo (2005b) introduced time-space networks to solve the traffic problem without inflows in variational form using shortest paths. Notice that this is an application of Bellman's dynamic programming principle. Each link i in these "variational networks" is defined by its: (i) slope v i : wave speed, (ii) cost c i : maximum number of vehicles that can pass, (iii) time length τ i , and (iv) distance length δ i = τ i v i ; see inset in Fig. 5.

Variational networks

Since the fundamental diagram is assumed triangular and the freeway homogeneous, there are only three wave speeds to be considered, u, -w and 0, and the corresponding passing rates are given by

L(v i ) =      wκ, v i = -w Q, v i = 0 0, v i = u (41a) (41b) (41c)
Let J i be the contribution of the J-integral in the cost of each link i. It corresponds to the (negative of) integral of φ(t, x) over the shaded region in Fig. 5, S i , and can be approximated by:

J i = -τ i j∈S i δ j a j , (42a)
where a j is the inflow associated with link j and j ∈ S i means all links that "touch" area S i . Finally, the cost to be used in each link becomes:

c i = L(v i )τ i + J i . (43)
The advantage of this method is that it is free of numerical errors (when inflows are exogenous) but it may be cumbersome to implement unless θ is an integer. In that case, as illustrated in Fig. 5 for θ = u w = 2, the location of nodes align on a grid pattern. This allows defining a conventional grid with cell size ∆t, ∆x where inflows may be assumed constant. The other disadvantage is that merge models are typically expressed in terms of flows or densities rather than N values, and therefore an additional computational layer has to be added.

Other coordinates

As pointed out in [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF] there are two additional coordinate systems that provide alternative solution methods of the traffic flow problems without inflows. In space-Lagrangian coordinates the quantity of interest is X(t, n), the position of vehicle n at time t; in time-Lagrangian coordinates one is interested in T (n, x), the time vehicle n crosses location x.

These representations correspond to the same surface in the three-dimensional space of vehicle number, time and distance, but expressed with respect to a different coordinate system. We briefly analyze these two alternatives when considering Eulerian source terms and conclude that even if a HJ equation is still valid in both cases, one cannot expect to get a VT representation formula even when inflows φ(t, x) are exogenous.

Space-Lagrangian coordinates: X-models

Let s(t, n) be the spacing of vehicle n at time t. To derive the X-model we multiply the Nmodel (3a) by s to get sN t -sH(k) = sΦ(t, x). Noting that s = -X n and sN t = -X n N t = X t , this can be rewritten as:

X t -V (-X n ) = -X n Φ(t, X), (44)
where V (s) = sH(1/s) is this spacing-speed fundamental diagram. We conclude that ( 44) is still a HJ PDE but does not admit a VT solution due to the term involving X. The corresponding conservation law can be obtained by taking the partial derivative with respect to n of (44):

s t + V (s) n = -φ(t, X)s 2 -Φ(t, X)s n , (45)
Notice that van Wageningen-Kessels et al. ( 2013) identified (45) but without the term Φ(t, X)s n using a different approach, and used it to formulate a numerical solution method in the case of discrete inflows.

Time-Lagrangian coordinates: T-models

Let r = T x and h = 1/q be the pace and the headway of vehicle n at location x, and let F (r) be the fundamental diagram in this case, i.e. h = F (r). Here, the T-model is simply F (r) = 1/q, where q is given by q = N t -Φ(t, x), per (5). Noting that N t = 1/T n this can be rewritten as:

T n - F (T x ) 1 + Φ(T, x)F (T x ) = 0, (46)
which, again, is still a HJ PDE but does not admit a VT solution due to the term involving T . The corresponding conservation law can be obtained by taking the partial derivative with respect to x of ( 46):

r n - F (r) x + F (r) 2 φ(T, x) (1 + Φ(T, x)F (r)) 2 = 0. (47)
To summarize, it becomes apparent that in Lagrangian and vehicle number-space coordinates the solution to our problem does not accept VT solutions and therefore becomes more difficult to solve.

Discussion

We have shown in this paper that VT solutions to the traffic flow problem exist only in Eulerian coordinates when inflows are exogenous. In all other cases the Hamiltonian is a non-local function of the independent variable, and the corresponding variational may not be possible to formulate. Even in the simplest endogenous linear case (7) the reader can appreciate the mathematical difficulties: using Φ = Φ(s, N(s, ξ(s))) in (8a)-( 11) turns the problem implicit in N, and therefore it is no longer a VT problem.

Improved numerical solution methods for the endogenous case were derived by taking advantage of this insight. In other fields, it appears that solving the extended Riemann problems explicitly considering the inflows has not been possible, and the only alternative has been to use high-resolution Riemann solvers [START_REF] Schroll | Finite difference schemes for scalar conservation laws with source terms[END_REF][START_REF] Leveque | Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm[END_REF]. We have shown that this is not the case in traffic flow, and that the ERP method presented here is indeed more accurate.

A streamlined version of the ERP method could be envisioned that drastically improves computation times with minimal impact in the quality of the solution. To see this, recall that Fig. 4d showed that the candidate y * 1 was optimal only during the time step where the density approaches the critical density. Based on the discussion following eqn. (41) it is reasonable to conjecture that candidates y * 1 , y * 2 . . . y * 6 in (33) would be optimal only when a transition takes place. Therefore, the streamlined method would consider only the reduced set Y = {x U , x 0 , x D } in (33), which would induce an error only in the time step where the transition occurs. This error can be made arbitrarily small by decreasing ∆t. Notice that this does not mean that the continuum solution (33) can be streamlined in this way; this is only possible in discrete time

where N-values are updated at each time step.

The implications of our findings in the context of MFD analytical approximation methods considering turns as a continuum inflow are not encouraging. This is because inflows in this case would have to be endogenous for the method to be meaningful, and in such case we have seen that there is no VT solution. This implies that the method of cuts-the only method used so far to provide analytical MFD approximations-is no longer applicable. At the same time, however, a stochastic extension of the method of cuts proved successful in approximating a real-life MFD [START_REF] Laval | Stochastic approximations for the macroscopic fundamental diagram of urban networks[END_REF]. This would indicate that, at least in the context of the MFD, VT solutions still provide good approximations. Research in this topic is ongoing.

Fig. 4

 4 Fig. 4 compares the CT-rule with the ERP-rule numerical solution (with ∆t = 40 s) for this example vis-a-vis the exact solution (40). Parts (a) and (b) show the time evolution of the density and flow, respectively, at x = 14 km obtained with each method. It can be seen

Figure 1 :

 1 Figure 1: Illustration of key definitions in VT: (a) the initial value problem; (b) the area of the integration to obtain the J-integral.

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 23456 Figure 2: Possible paths to minimize the J-integral: (a) constant initial density; (b) extended Riemann problems.

Acknowledgements

This research was supported by NSF Grants 1055694 and 1301057. The author is grateful for the comments of two anonymous reviewers, which greatly improved the quality of this paper.


J Pal
email: j.pal@hzdr.de

A Cramer

S Franke

S Eckert

R Nauber

N Thieme

L Büttner

J Czarske

Experimental study of the flow driven by combined AC magnetic fields using a novel ultrasound array system

Keywords: Ultrasound Sensor Array, Flow Field Measurements, Magnetohydrodynamics, Electromagnetic Stirring

Ultrasound velocity measurements were performed in a liquid metal flow (GaInSn) inside a cubic vessel. The flow was driven by the combined action of a rotating (RMF) and a travelling magnetic field (TMF) leading to an inherent threedimensional electromagnetic force distribution in the fluid. As a result flow structures develop which are essentially determined by the frequency difference and the strength relation of both fields. The driving force is stationary for identical frequencies whereas it undergoes a slow temporal modulation in case of slightly varying frequencies. Beside the magnetic field parameter the alignment of the TMF's to the fluid volume's axis plays a crucial role. A novel ultrasound array system was used in the present study to measure two velocity components in two perpendicular planes (2d-2c) simultaneously. This contribution reports on the influence of axial alignment on the flow, and on the flow dynamics resulting from slightly different field frequencies of the RMF and the TMF.

Introduction

Electromagnetic processing of materials is well established in industrial applications such as steel casting or crystal growth. Both static and alternating magnetic fields may be employed in order to optimize the stirring process by means of Lorentz forces. The trend for the last years has been to combine and tailor the action of different magnetic field types. A prerequisite for such studies are suitable bulk flow measurement instruments. In opaque fluids the UDV (ultrasound Doppler velocimetry) technique provides a great potential [1,2]. In a previous work [3] ultrasound velocity measurements in a liquid metal flow were performed in order to study the combined action of an RMF and a TMF. Only one traversable transducer was used in this study. In [4] the flow driven by a TMF was the object of investigation. Few transducers were used to study the sensitivity of the flow with respect to a shift between the axes of the liquid metal column and of the TMF. The present work is devoted to the combination of the previous studies using a novel ultrasound array system. As we shall see, this system allows to measure simultaneously both velocity components in an entire plane. Furthermore, it is demonstrated that the use of ultrasound arrays extends the capabilities for flow mapping significantly.

The measurement system and the experimental setup

For a detailed description of the entire system including the electronics we refer to [5,6], thus, only some relevant information will be reviewed in this section. The measuring principle is based on the well-known pulsed-wave Doppler technique [2]. A short ultrasound pulse is thereby emitted by an ultrasound transducer into the fluid under investigation. The wave propagates through the fluid and is scattered on small particles such as oxides inherently present inside the medium. Assumed the particles follow the fluid without slip, the scattered wave contains information about the position and the velocity of these particles. Thus, a 1d-1c (1 dimension and 1 component) velocity profile can be obtained along the sound propagation direction. Instead of single transducers, the system used in the present work utilizes ultrasound line arrays to gain a 1c velocity measurement in the spanned plane. Each array consists of 25 piezo-transducer elements with a size of 2.5×5 mm 2 and a gap of 0.2 mm between them. Hence, a measurement field of 67 mm width is covered by such an array. During operation, two adjacent elements are always driven together acting as a single square transducer of 5 mm edge length. In order to scan the entire field covered by one array in a high temporal resolution without cross-talk between the transducer, their operation is parallelized by a special time division multiplex scheme [6]. A frame rate of up to 30 Hz is possible per array. To determine two or more velocity components multiple arrays have to be utilized in a suitable arrangement. Hence, the arrangement as shown in the picture in Fig. 1 is able to deliver the required information to reconstruct the 2d-2c velocity field in an entire plane. Due to technological reasons given by the ultrasound array, a cubic vessel was chosen as the object of the present investigation. The inner dimensions of 67×67×67 mm 3 match the field size of the transducer array. Four arrays were used in a crossed-plane configuration as shown in Fig. 2. The arrays spanning the horizontal half-height plane capture the primary swirling flow, whereas the arrays of the vertical plane measure the secondary meridional flow inside the vessel. The cube was filled with the eutectic alloy GaInSn. For the purpose to drive the electromagnetically-induced flow in the melt, the vessel was installed inside the MULTIMAG (MULTIpurpose MAGnetic) research facility [7]. MULTIMAG offers a variety of magnetic field types and linear superpositions thereof since it was built without any ferromagnetic yokes. The RMF used here was produced with six rectangular solenoids arranged hexagonally and the TMF with six linearly arranged solenoids in the vertical direction. To allow a precise positioning with respect to the axis of the TMF coil system, the vessel with the assembled transducer arrays was placed on a non-magnetic two-axes crossbar.

The electromagnetically driven flow

Applying an AC magnetic field with strength B to a cylindrical volume with height H and radius R of liquid metal induces an eddy current j within the fluid and the interaction of the current with the magnetic field that produced it leads to an electromagnetic Lorentz force f L =jB acting on the fluid as a volume force. The here used RMF and TMF cause, operating separately, an axisymmetric base flow. Although we use a cubic vessel, the fundamental concept in a cylinder is broadcasted for the present work. Without going into details, we refer to [3,4] and references therein for this purpose, the forces may be expressed as

2 4 2 2 , ) , (    R B Ta e z r s r Ta R R R    f (1)   2 5 2 3 2 4 , , , 2     kR B F f f f F R T T r z T   f (2)
Apart from material and geometry characteristics the above Taylor number Ta and forcing parameter F are determined by the strength and frequency of the applied magnetic field and represent a dimensionless measure for the driving action of the respective magnetic field. Assumed that axisymmetric condition prevails between the axes of the liquid metal column and that of the magnetic field, the Lorentz force in Eq. ( 2) has a sole axial component. However, this force exhibits non-vanishing components f r and f  if a certain displacement between the axis is present. Beside the respective body forces an additional interaction term occurs in the Lorentz force in the case of a combined RMF/TMF [3]:

  ) ( ) sin( 4 , z T R T R I R T T R T R I T R L kr kz t r B B e e f B f B j f f f f f f                      (3)
If the frequency difference  =  R - T exceeds a certain value determined by inertia, the impact of the interaction term on the flow vanishes and one is concerned with the linear superposition of the respective forces. Thus, for equal frequencies,  the interaction force has only components in the and z-directions and depends on these components. It becomes static but helical and therewith 3d. As soon as the frequencies differ slightly, up to  ~ u/R, the force is slowly modulated in time and its influence on the fluid flow decreases accordingly. All measurements presented in the next section were done for Ta =210 6 and F=210 6 at the technical relevant power line frequency of =250 Hz, or in the case of the combined fields in its vicinity.

Results

The first step covers the axisymmetric case of the TMF. Thus, the unshifted "zero"-position in the TMF was deduced in a recursive procedure by recording the flow and repositioning the vessel until the best possible axisymmetry was reached. The meridional and horizontal flow maps in the Fig. 4: The same as in Fig. 3 for a shift of 2 mm between the TMF's and vessel's axes.

As can be seen from the vector fields, a measurement in the direct vicinity of the transducers of both velocity components was not possible. This is due to stationary echoes which were caused by multiple reflections inside the vessel wall. These stationary echoes impede the measurements because they interfere with the echoes from the flow. Since a cubic vessel was used, the inward oriented radial flow shown on the right side of Fig. 3 is not so evenly distributed as it would be expected in a cylindrical vessel. Nevertheless, the axisymmetric orientation becomes obvious. Displacing the vessel 2 mm away from the center axis of the TMF, the original toroidal structure in the meridional plane is completely destroyed (see Fig. Whereas for =0° in the meridional plane the rotation sense of the flow is seemingly counter clockwise it changed completely to a clockwise one for =180°. It is emphasized that not the flow itself is changed but rather the prevailing static structure was rotated inside the fixed laboratory measurement frame. Thus, varying  between 0° and 360° allows a precise positioning of the whole structure. The flow maps in Fig. 6 show that the azimuthal rotation sense is not affected by .

Conclusions

The flow driven by a TMF and a combined RMF/TMF was studied experimentally. A new ultrasound array system was used, thus allowing to measure simultaneously both the complex flow map in different planes and also the transient flow. In contrast to previous works, the results demonstrate that a more complex study of the spatio-temporal behavior of the flow is now possible without the restrictions of using either one traversed single sensor or few of them at different positions. The present UDV array system offers great potential for the field of MHD and of the interaction between electromagnetic fields and conductive fluids.

Fig. 1 :

 1 Fig. 1: A picture of the the ultrasound linear array and below it a schema illustrating a crossed arrangement for the reconstruction of a 2d-2c velocity field.

Fig. 2 :

 2 Fig. 2: The measurement configuration used here.Each of the planes is at the half height to the vessels edge, respectively.

  Fig.'s 3 and 4 illustrate this axisymmetric case.

Fig. 3 :

 3 Fig. 3: TMF flow maps under axisymmetric conditions. On the left side the meridional and to the right the horizontal plane, respectively. The arrows indicate the flow direction and the length correlates with the calculated velocity magnitude shown in the color bar.

Fig. 5 :

 5 Fig. 5: Vertical flow maps measured with a phase difference of 0°, 90°, and 180° between the RMF and the TMF.

Fig. 6 :

 6 Fig.6: The same as in Fig.5but in the horizontal measurement plane.

  The large-scale flow cell begins to rotate by non-vanishing frequency difference , which becomes obvious by analyzing the local flow velocity at one fixed position. The diagrams in Fig.7show the vertical velocity component near to the rim of the vessel as function of the time and the frequency difference. The change in the flow direction is clearly observed for not too large  (on the left in Fig.7) as it was already argued in context with Eq. (3).

Fig. 7 :

 7 Fig. 7: The vertical velocity component near to the vessels rim for slightly different frequencies (left diagram) between the TMF and the RMF. For a higher difference (right diagram) the velocity large scale oscillations disappear.

Acknowledgment

Financial support of this research by the German Helmholtz Association in the frame of the Helmholtz-Alliance LIMTECH is gratefully acknowledged.


C Kratzsch

R Schwarze

G Gerbeth

S Eckert

K Timmel
email: k.timmel@hzdr.de

B Willers

Modelling of the

Keywords: Continuous casting of steel, electro-magnetic flow control, liquid metal models, numerical simulation, scale adaptive simulations

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The persistent effort to achieve a better product quality and higher productivity of the continuous casting of steel implies the high importance of powerful capabilities to control the flow in tundish and mold, and the initial solidification in the mold. Numerous sophisticated numerical simulations concerned with the metal flow during the casting process need a fundamental experimental validation. The use of water models has the advantage to save expenses and to be able to apply a number of well-proofed measuring methods. However, a generalization of these results to liquid metal flows has to be considered as questionable because the realistic values of flow parameters (Re, Pr, Gr, Ha, etc.) are difficult to meet. In many cases, for instance liquid metal flows with strong temperature gradients, with an additional gaseous phase or under the influence of electromagnetic fields, the flow phenomena cannot reasonably be modelled by means of water experiments. The application of electromagnetic fields provides a considerable potential to control the fluid flow in the mold and to influence the solidification in the strand. First strategies for EM applications in steel casting were mainly guided by simplified pictures of the magnetic field impact on the global flow field. Many numerical investigations have been reported until now to improve the understanding of the magnetic field influence on the mold flow (see for instance [2][3][4][5]). However, the problem has to be considered as challenging because of the complex geometry, the highly turbulent flow, and specific peculiarities occurring in case of MHD turbulence. Obviously, a validation of the numerical predictions by liquid metal experiments is indispensable. However, related experimental studies are rather scarce until now. Several plant trials were carried out [6,7] to test the efficiencies of electromagnetic brakes in the real casting process. Because of the lack of suitable measuring techniques for liquid steel at 1500 °C such trials cannot provide any reliable knowledge about the magnetic field effect on the flow in the mold. First model experiments employing simplified mercury models have been reported by Japanese [8,9] and French [START_REF] Etay | PAMIR conference, Proc[END_REF] groups. With our work we want to continue the strategy of cold metal models. The main value of such cold metal laboratory experiments consists in the capabilities to obtain quantitative flow measurements with a reasonable spatial and temporal resolution. New ultrasonic or electromagnetic techniques for measuring the velocity in liquid metal flows came up during the last decade allowing for a satisfying characterization of flow quantities in the considered temperature range up to 400°C [START_REF] Wondrak | [END_REF].

Experimental facilities

The experimental program of the LIMMCAST facilities at HZDR aims to model the essential features of the flow field in the continuous casting of steel. Basically, these are the flow fields in the tundish, in the submerged entry nozzle (SEN) and in the mold cavity, the complex two-phase flow in SEN and mold due to argon injection as well as the effect of electromagnetic actuators at the mold. For this purpose, there have been build three experimental facilities dealing with the modelling of the continuous casting process. The X-LIMMCAST facility is a liquid metal loop in particular designed for the visualization of the liquid metal-gas two-phase flow by means of X-ray imaging. It is operating with the model fluid GaInSn at room temperature. The liquid metal is circulated by an electromagnetic pump working with rotating permanent magnets. An electromagnetic flow meter is used to monitor the flow rate of the liquid metal. A photograph and a schematic drawing of the X-LIMMCAST facility are displayed in Fig. 1. A photograph showing an overall view of the big LIMMCAST facility is displayed in Fig. 2. All components are made of stainless steel, including the tundish, the SEN and the mold. The low melting point alloy Sn60Bi40 is used as model liquid. The liquidus temperature of 170°C allows for an operation of the facility in a temperature range between 200 and 350°C. The investigations will be explicitly focused on the behavior of the isothermal melt flow. Argon gas bubbles can be injected with tunable flow rates through the stopper rod into the SEN resulting in a two-phase flow inside the nozzle and the mold. Pipe connections with flanges are realized at various locations within the loop allowing in principal a replacement of the particular components, which gives us a flexibility to modify the flow geometries for miscellaneous requirements. Fig. 3 shows the small-scale setup Mini-LIMMCAST, which is operated again with the eutectic alloy GaInSn as model fluid. The experiments at this setup allow for flow measurements at room temperature and gained in valuable experiences for the detailed design of the larger LIMMCAST facility during its build-up. The previous experiments at Mini-LIMMCAST were performed in a discontinuous mode, i.e. after filling the tundish with the melt the stopper rod was lifted to drain the fluid into the mold. The setup has been improved during the last months and can now be run in a continuous operation. The turbulence closure is made through a k-ω SST model [12]. The magnetohydrodynamic effects are considered by a one-way coupled approach (inductionless approximation):

Numerical model

𝒇 𝐿 ̅̅̅ = 𝒋̅ × 𝑩 0 (3) 𝒋̅ = 𝜎 ( -∇𝜓 ̅ + 𝒖 ̅ × 𝑩 0 ) (4) ∇ 2 𝜓 ̅ = ∇ • ( 𝒖 ̅ × 𝑩 0 ) (5)
Meaning of the symbols: uvelocity ρdensity η -dynamic viscosity p -Pressure f L -Lorentz-Force j -el. current density B 0 -magn. flux density ψ -el. potential σ -el. conductivity

The magnetic field influence inside the MHD turbulence model is not yet included in the current model of OpenFOAM. As shown earlier by Miao et al. [13] in his numerical simulations using the CFX solver, the phenomenon of large-scale oscillations of the jet can also be reproduced by a turbulence model without anisotropy parameter, an MHD extension proposed by Widlund et al. [14]. This is only true for the case of electrically insulated walls at moderate Ha numbers. However, a turbulence model without anisotropy parameter can produce wrong results for the magnetic field effect on the mold flow in the case of conducting mold walls [13]. Therefore, we restrict our analysis here to this specific situation, where the mold walls are considered to be electrically insulating.

Experimental & simulation results

Previous experiments showed a dramatic influence of a static magnetic field on the liquid metal flow in the mold [1]. Low frequency oscillations of the jet flow were detected. To investigate the influence of such a static magnetic field on the free surface, the position of the liquid metal level was scanned with a line laser system. Due to spatial restrictions, the system was not able to measure the elevation in the whole mold width and was restricted to the part near the narrow mold wall. Fig. 4 depicts the free surface at one time instance for the reference case without magnetic field. The surface is rather smooth without disturbances. When the static magnetic field is applied in the same way as in the previous experiments [1], the surface becomes visibly more turbulent. Fig. 5 shows a snapshot of the free surface under the influence of a static magnetic field with electrically isolating boundary conditions on the mold walls (Magnetic flux density B = 310 mT, Hartmann-number Ha = 417 and Magnetic Interaction Parameter N = 75). This snapshot shows a typical bulge close to the narrow wall. The bulge is not static and can switch between the mold halves or even disappear for some time. It is likely that the bulge and the site shifting are related to the low frequency jet oscillations. The jet is there temporarily deflected upwards resulting in a strong upwards flow near the narrow wall [13]. This strong upward flow might be responsible for the deflection of the free surface. Experiments for closer look on the connection between jet flow and free surface shape are ongoing.

The mold flow was addressed also by numerical simulations. In conjunction with the experiments, the simulations identify a low frequency oscillation of about 0.3 Hz in case of an applied static magnetic field with isolating mold walls (see Fig. 6). It is clearly seen from the snapshots of the velocity distribution in the middle plane of the mold, that a reciprocal bending of the jets is present. The typical two roll structure collapses in the lower mold and an oscillation manifests itself. An evaluation of the velocity signal in the frequency domain yields a distinct peak in the experiments as well as in the CFD simulations.

2 )Fig. 1 (

 21 Fig. 1 (top, left):X-LIMMCAST -small scale acrylic glass model for investigation of liquid metal-gas two-phase flow in the mold and SEN by X-ray imaging.

Fig. 2 (

 2 Fig. 2 (top, right): LIMMCAST -large scale (250 °C) liquid metal experiment for continuous flow measurement in tundish, SEN and mold.

Fig. 3 (

 3 Fig. 3 (bottom, left): Mini-LIMMCAST -small scale acrylic glass model for experimental flow investigations at room temperature using GaInSn.

Fig. 4 :

 4 Fig. 4: Smooth mold level at one time instance without electromagnetic interference.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Snapshot of mold level under the influence of a static magnetic field, which creates a wave. (B = 310 mT, Ha = 417, N = 75)

Acknowledgment

The authors acknowledge the financial support from the German Helmholtz Association in the framework of the Helmholtz Alliance LIMTECH.

Summary

For physical modeling of the continuous casting process, it is essential to use liquid metals for conducting the experiments when regarding special effects, e.g. in case of two-phase flows or magnetic fields. To provide an experimental tool for these cases, the three experimental facilities LIMMCAST, Mini-LIMMCAST and X-LIMMCAST were built up at HZDR with slightly different scopes. This paper shortly presented all three experimental setups and exemplarily one measurement example. The measurement results of the liquid metal experiments represent a valuable data base for the validation of numerical models, e.g. [13,[15][16][17]. Additionally, a numerical model is proposed that gives the possibility to study the flow quantities in detail. It was shown that the numerical data are in very good accordance with experimental results. In future it will be further developed in direction to multiphase flows. So, the advantage of combined experimental and numerical work was demonstrated.


C F Fang

G X Liu

Y M Wang

L Zhang

L G Meng

X G Zhang

The Role of Electromagnetic Stirring in Preparation of TiB 2 /AZ31 Composite under Coupled-field

Keywords: Electromagnetic stirring, magnesium matrix composites, Ultrasonic filed, Microstructure Acting volume of ultrasonic vibration Ultrasonic probe Crucible Electromagnetic stirrer Rotating electromagnetic field Traveling electromagnetic field Spiral electromagnetic field Mg melt

In the present study, results of water simulation visually show that the acting volume of ultrasonic vibration is pretty limited and just locates around the ultrasonic probe. Considering that, the idea that controls melt flow by electromagnetic field was proposed to indirectly enlarge the acting volume of ultrasonic vibration. Therefore, the comparative trials of TiB 2 /AZ31 composites prepared under ultrasonic field coupled with rotating, traveling and spiral electromagnetic field respectively were conducted. The results show that the spiral electromagnetic stirring can not only effectively enlarge acting volume of ultrasonic vibration indirectly by controlling the composite melt flow, but also significantly improve the utilization rate of TiB 2 reinforcement particles.

Introduction

As for the same reinforcement particles, properties of magnesium matrix will be further enhanced with decreasing particle size [1][2]. However, because of the large surface free energy, fine particles (<5μm) are easily agglomerated into large clusters. Significant efforts have been taken out to distribute the reinforcement particles, including mechanical stirring [3], secondary deformation [4] and ultrasonic treatment [5]. But the results are less than satisfactory. Not only that, the particle clusters are tend to deposit, which leads to a low utilization rate of reinforcement particles [5]. As a result, the uniform distribution and ideal utilization rate of such fine particles have been a challenge.

In the present work, the acting volume of ultrasonic vibration was investigated by water simulation experiments.

The idea was proposed that electromagnetic fields were used to make melt flow through the zone of ultrasonic probe and then indirectly enlarge the ultrasonic vibration effect. Accordingly, in-situ TiB 2 /AZ31 composites were prepared under ultrasonic field coupled with three different types of electromagnetic fields. Microstructures of TiB 2 /AZ31 composites, including particle clusters size, particle distribution and utilization rate, were investigated to evaluate the effects of three different types of electromagnetic stirring.

Experimental procedure 1.1 Water simulation about acting volume of ultrasonic vibration

Pure water was selected as the media to study the acting volume of ultrasonic vibration. The frequency and the power of the ultrasonic vibration were 20 kHz and 330W, respectively. As shown in Fig. 1a, ultrasonic probe was dipped into the pure water served in a beaker, and a digital camera was used to record the shape of sound beam below the radiation face of the probe. To further investigate the weakening trend of ultrasonic energy, an aluminum foil (16μm in thickness) was placed below the probe (Fig. 1b). The distance from radiation face to aluminum foil was 0mm, 5mm, 15mm and 20mm, respectively. When the ultrasonic probe worked for 5s, the aluminum foil was removed and its morphology was observed to analyze the penetration capacity of ultrasonic vibration. Nominal composition and preparation process of Al-Ti-B preform have been mentioned in our previous studies [6]. The AZ31 magnesium alloy was first melt in a steel crucible. At 780℃, about 4wt% Al-Ti-B preform was added into the melt. 15 min later after the completion of SHS, the composite melt was mechanical stirred (MS) for 10min at 620℃ to preliminarily break up the SHS products. Subsequently, the composite melt was reheated to about 700℃. While the alloy melt is stabilized at this temperature, an ultrasonic field (power and frequency were 600 W and 20 kHz) coupled with three different types of electromagnetic field (rotating electromagnetic field (RMF), traveling electromagnetic field (TMF) and spiral electromagnetic field (SMF), respectively) was applied to modify particle dispersion in the composite melt. Current and frequency of the three types of electromagnetic fields were all 120 A and 10 Hz. Schematic illustration of the coupled-field treatment on the composite melt was shown in Fig. 2. After having been coupled-field treated for 20min, the remaining composite melt was reheated up to 720℃ and cast into a steel mold. For comparison, AZ31 alloy and TiB 2 /AZ31 composite treated only by ultrasonic field (UF) were also made. Material designations and preparing condition were shown in table 1. Samples for optical microstructure observation were firstly homogenized at 415℃ for 20 h. Then they were mechanically polished and etched in the acetic picral (2.1 g picric acid + 5 ml acetic acid + 5 ml water + 35 ml ethanol).

In present study, the area percentage of TiB 2 in optical micrographs calculated by Image J software was used to show the utilization rate of reinforcement particles. The average grain size of each sample was obtained by using the mean linear intercept method.

Results and discussion

Acting volume of ultrasonic vibration

It is believed that the strong impact of power ultrasound can overcome the Van der Waals force and the capillary force between fine particles in clusters, and thereby break nanoparticle clusters [5,7]. Results of extraction experiments in our previous study [8] also showed that the ultrasonic vibration could break TiB 2 particle clusters, but the utilization rate of them was low. The low utilization rate should be ascribed to the limited acting volume of ultrasonic vibration [8,9].

The acting volume of ultrasonic vibration can be intuitively observed by water simulations. As can be seen from Fig. 3, the energy of ultrasonic vibration is emitted from the radiation face of the ultrasonic probe. The cross-sectional area of the sound beam reduces gradually when it is far away from the probe, which visually indicates a limitation of ultrasonic vibration. Fig. 4 shows the photographs of penetration of ultrasonic vibration on the aluminum foils. As the distance from foil to probe is less than 5mm, the foil is fully penetrated (Fig. 4a,b). The more far away from the probe, the weaker penetration of ultrasonic vibration becomes (Fig. 4c). Moreover, when the distance is further increased to 20mm, the foil can't be penetrated and just some pits are found on the foil surface, as shown in Fig. 4d.

Effects of the coupled-field types on the distribution and utilization rate of TiB 2 particles

The optical microstructures of the as-homogenized AZ31 alloy and the TiB 2 /AZ31 composites prepared with different external fields are shown in Fig. 5. Fig. 6 shows the average grain size of the matrix and utilization rate of TiB 2 reinforcement particles. Under the effects of TiB 2 particles and UF treatment, sample MU showed a finer microstructure than AZ31 matrix. However, in addition to lots of small size TiB 2 particles, some pretty large TiB 2 particle clusters which not deposited during casting process were kept in the matrix (Fig. 5 b). And the utilization rate of TiB 2 particle is very low (6.4%) (Fig. 6). As RMF and TMF introduced respectively, size of reinforcement particle clusters was decreased significantly and the particles distributed more uniformly in the matrix (Fig. 5 c andd). But there was little improvement in grain refinement of the matrix compared with sample MU. What's more, utilization rates of TiB 2 particles in sample MUR and MUT were just increased to 9.8% and 11.1%, respectively. Although the size of TiB 2 particle clusters was not further decreased in sample MUS, there was an obvious grain refinement (65μm) and a significant increase in TiB 2 particle utilization rate (23.1%). electromagnetic fields. With the introduction of RMF, a cycle flow in the cross section of the melt can be realized and the melt will be forced toward the crucible wall because of centrifugal force. Instead, there is a vertically circulation as TMF introduced. In both cases, there are relatively low chances for melt unit to pass through the acting volume of ultrasonic vibration and the problem that large TiB 2 particle clusters deposited on the bottom of crucible was not well solved. As a result, there was not a significant increase in particle utilization rate in sample MUR and MUT. And this unobvious improvement in particle utilization rate contributed little to grain refinement. On the contrary, after the introduction of SMF, a good mixed flow consisting of a vertically cycle and a horizontally turning will be produced in the melt. This mixed flow can effectively control the melt unit to pass through the acting volume of ultrasonic vibration and indirectly enlarge the acting volume of ultrasonic vibration. Thus, more deposited TiB 2 particle clusters will be ultrasonically processed and then easily kept in the composite matrix. At last, a homogenous microstructure with small size particle clusters and high particle utilization rate was obtained in sample MUS.

Conclusion

(1) Water simulation visually shows that acting volume of ultrasonic vibration is pretty limited and just located around the ultrasonic probe.

(2) The introduction of electromagnetic stirring can effectively decrease the size of particle clusters and improve the utilization rate of particles by enlarging the acting volume of ultrasonic vibration. Among the three types of electromagnetic fields, the spiral electromagnetic stirring has best effects.

Fig. 1 . 1 . 2

 112 Fig. 1. Schematic illustration of water simulation Fig. 2. Schematic illustration of the coupled-field treatment on about acting volume of ultrasonic vibration the molten TiB 2 /AZ31 composites 1.2 Preparation and characterization of TiB 2 /AZ31 composite Mg-3Al-Zn magnesium alloy was used as the matrix in the present study. In-situ TiB 2 particles was synthesized via the self-propagating high-temperature synthesis reaction (SHS) by adding Al-Ti-B preform into the molten matrix alloy.

Fig. 3 .

 3 Fig. 3. Shape of sound beam below the Fig. 4. Penetration of aluminum foil by ultrasonic vibration for different radiation face of the probe distances (a) 0mm; (b) 5mm; (c) 10mm; (d) 20mm

Fig. 5 .

 5 Fig. 5. Optical micrographs of (a) AZ31, (b) MU, (c) MUR, (d) MUT and (e) MUS after homogenizing treatmentThe acoustic streaming effect produced by ultrasonic vibration is very weak and can't fully stir the whole melt. In addition, the particle clusters were easily to aggregate together and deposit on the bottom of the crucible because of the density discrepancy between reinforcement particles and the matrix melt. Consequently, the size of particle clusters was pretty large and utilization rate was low in sample MU. After additionally introducing different types of electromagnetic fields, the size of TiB 2 particle cluster was decreased and the particle utilization rate was improved, which should be mainly ascribed to the change of melt flow. Fig.7illustrates the macroscopic melt flow under different kinds of

Fig. 6 .

 6 Fig. 6. Average grain size of the matrix and utilization Fig. 7. Schematic illustration of the melt flow with different rate of TiB 2 reinforcement particles types of electromagnetic fields

Table 1 .

 1 Material designations used in the present work

	Designation	Al-Ti-B (wt%)	Processing conditions
	AZ31	0	MS
	MU	4	MS + UF
	MUR	4	MS + UF + RMF
	MUT	4	MS + UF + TMF
	MUS	4	MS + UF + SMF

Acknowledgement

The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (No.51374047) and the Foundation of Liaoning Educational Committee of China (No. L2013031).


I Kaldre

A Bojarevics

Directional solidification of metallic alloys under pulsed transverse electromagnetic force

Keywords: Directional solidification, alternating electromagnetic force, transverse magnetic field

Studies of solidification of metallic alloys are usually aimed to understand the formation of solidified microstructure depending on various solidification parameters. Melt flow is one of the factors which change heat and mass transfer conditions, thus it has influence on homogeneity and microstructure of the solidified sample. This work analyses directional solidification of metallic alloys under static transverse magnetic field and simultaneous electric current pulses through the sample. Different type of current pulses is used to create alternating Lorentz force in direction perpendicular to growth direction thus time varying forced convection along the solidification front is created. Changes in microstructure and component distribution of Pb-Sn alloys under such interaction are observed and described in this work.

Introduction

Solidification of metallic alloys under electromagnetic influence has been widely studied recently. Previous experimental results demonstrate that applied magnetic fields may have different influence on solidified metal microstructure and component distribution within the ingot [1][2][3][4]. It is well known that an alternating magnetic field refines grain structure and increases composition homogeneity [5]. Static magnetic field is used to damp unwanted melt convection and to influence grain structure by means of thermoelectromagnetic convection around dendrites. Static transverse magnetic field during directional solidification creates crucible scale macrosegregation [6]. Static magnetic field and imposed DC current combination allows us to achieve segregation which is similar than segregation caused by TEMC, but in this case it is easier to control the segregation pattern and magnitude by changing the current value and direction [7].

The aim of this work is to verify if time varying transverse electromagnetic force can cause macrosegregation of binary metallic alloy and to investigate its impact on grain structure of directionally solidified alloy under this electromagnetic interaction. Growth velocity is chosen to be sufficiently high to avoid profound macrosegregation by thermoelectric effect as demonstrated by the previous works [2] and also so that growth circumstances are closer to real solidification conditions. Typical grain size of Sn-10%wt.Pb alloy directionally solidified at 50 µm/s velocity is about 100-200 μm [8] so we chose current pulse length to be sufficiently long so that solidified length during one pulse is larger than grain size.

Experiment

Experimental setup consists of temperature gradient furnace with resistive heater and a water cooled sample holder. Alloy is prepared from high purity tin and lead and poured into alumina tube. This sample is then directionally solidified in temperature gradient furnace at a constant pulling velocity of 50 µm/s. Solidified samples are then examined by optical microscopy to visualize grain structure and component distribution in the sample. DC magnetic field is provided by permanent NdFeB magnet system, while pulsed electric current is provided by programmable current source. Experimental setup and permanent magnet system are depicted in Fig. 1. Permanent magnet system creates 0.5 T static transverse magnetic field.

Results and discussion

Electromagnetic interaction created by magnetic field and pulsed electric current refines grain structure and prevents radial segregation of the alloy. By varying force pulse magnitude and frequency it is possible to achieve different modifications of material structure. Microstructure and macrosegregation of different alloys are analysed by optical microscopy and x-ray fluorescence spectroscopy. Horizontal cross section of the directionally solidified 5 mm diameter Sn-10%wt.Pb alloy is shown in Fig. 2. Fig. 2 (a) shows the cross section of solidified alloy under 0.5 T transverse magnetic field, it can be seen that segregation is visible perpendicular to magnetic field direction as a consequences of thermoelectromagnetic force and electromagnetic damping. Fig. 2(b) shows the solidified sample under 0.5 T magnetic field and applied electric current along the sample axis. It can be seen that grain size is reduced while macrosegregation across the diameter perpendicular to magnetic field direction can still be observed. In Fig. 2(c) sample solidified under different current regime is given and similar conclusions can be made. Pulsed electromagnetic interaction is interesting because it can be more effectively used for lower conductivity materials because integral Joule heat is significantly smaller. Pulsed interaction is also one of the perspective methods which is considered for dielectric particle dispersion in metals.

Fig 1 .

 1 Fig 1. a) Sketch of the experimental setup; b) permanent magnet system (view from the top)

Fig 2 .Fig 3 .

 23 Fig 2. Cross section of directionally solidified 5 mm diameter Sn-10%wt.Pb samples: a) under 0.5 T static field; b) under 0.5 T static field and J1 current regime; c) under 0.5 T static field and J2 current regime

Acknowledgement

Financial support from LIMTECH alliance and Latvian Science Council are gratefully acknowledged.


W Wang

T M Wang
email: tmwang@dlut.edu.cn

C L Zou

H J Kang

Z Q Cao

T J Li

Influence of direct current on the ageing behavior of a CuCrZr alloy

Keywords: direct current, ageing, precipitate, mechanical properties, CuCrZr alloy

In recent decades, much work has been done on the effect of electric current on processing and properties of metals. And influence of an electric current on the solid state phase transformation in metals is complicated: an electric current can either enhance or retard the precipitation rate, depending on the alloy composition, the current density and the frequency of pulsed dc current. In this study, we imposed direct current during the ageing process of a CuCrZr alloy and studied the influence of current on the ageing behavior. With the imposition of current, the ultimate strength of CuCrZr alloys is improved, and the improvement in strength can be explained by finer precipitates modified by current. The effect of direct current pulses with the frequency of 1,000 Hz is most significant. The morphology of precipitates changes from spherical to cylinder with the imposition of current.

Introduction

Continuous efforts have been made to obtain materials that exhibit both excellent mechanical strength and high electrical conductivity for high performance applications, such as integrated circuit lead frames and high speed railway contact wire [1][2][3]. As an age-hardenable alloy, the excellent mechanical properties of Cu-Cr-Zr alloys are attributed to the precipitation and particle-dispersion strengthening mechanism [4,5]. In recent decades, much work has been done on the effect of electric current on processing and properties of metals. Ageing process can be promoted by current [6,7], and electric current pulses have a remarkable effect on recovery and recrystallization [8,9]. The effects of electric current pulses on the ageing process of deformed copper alloys may be interesting. So far, very little work on this subject has been reported. In this paper, we focus on the influence of current pulses on the aging behaviors of a hot rolled-quenched Cu-Cr-Zr alloy during thermomechanical treatment. Different current were imposed and their influence on mechanical strength was evaluated by tensile test. Scanning electron microscope (SEM), optical microscope (OM) and transmission electron microscope (TEM) were applied to characterize the evolution of the microstructures, upon which detailed discussion of the mechanism was based.

Experimental procedure

Alloy with a composition of Cu-0.6 wt.% Cr-0.3 wt.% Zr was melted in a vacuum-induction melting furnace with electrolytic copper, pure chromium and copper-zirconium master alloy (Cu-50 wt.% Cr alloy)., and then casted in a graphite mold with a size of 40 mm × 200 mm. The ingot was cut into a size of 30 mm × 30 mm × 120 mm before hot rolled at 850 ℃to a thickness of 15 mm. The hot rolled plate was solution treated at 960 ℃ for 1 h,followed by quickly quenching in cold water. The solution treated plate was then planed on both sides to remove surface defects and rolled with 85% reduction at room temperature. The cold rolled plates were cut into small plates with a size of 20 mm × 50 mm × 2 mm. The schematic diagram of ageing process is illustrated in Fig. 1. The samples were gripped on both sides to conduct current and Ar was used as a productive atmosphere. To modify the effect of joule heat, a thermal couple was fitted to the surface of the sample to accurately measure the temperature. The cold rolled pieces were aged with and without current. In this paper, direct current and a single-phase d.c. current pulse with the frequency of 50 Hz and 1,000 Hz were employed. The density of d.c. pulses was fixed below 100 A/cm2. Due to the low frequency of current, the influence of skin effect can be ignored in the present study. The aging treatment was carried out in a tube furnace and current pulses were imposed from the beginning of heating.

Results and discussion

Fig. 2 shows the optical micrographs of the Cu-Cr-Zr alloys under different state. In the as-cast state, typical coarse columnar grains are observed, which can be attributed to the low cooling rate of iron mold. The grains were elongated by hot rolling (Fig. 2 (b)) and grew into equiaxed grains during solution treatment ( Fig. 2 (c) ). However, there're still undissolved eutectic Cr particles due to its limited solubility in copper. The grains were severely elongated during cold rolled (Fig. 2 (d)) and the deformed structure remains upon ageing at 450 ℃ for 4 h, which indicates that work hardening is still effective. After ageing at 600 ℃ for 4 h, equiaxed grains appear again, indicating the onset of recrystallization . Table 1 shows the mechanical properties of CuCrZr alloys under different conditions. With the imposition of current, all the ultimate strength of aged CuCrZr alloys is improved with the exception of Sample 5. Comparing Sample 2&3 with Sample 4&5, it goes without saying that the effect of pulses is more obvious than direct current. Comparing Sample 2 with Sample 3, it seems the effect of higher frequency (1,000 Hz) is more obvious than a lower one (50 Hz).

Comparing Sample 4 with Sample 5, the invariant ultimate strength of sample 5 can be attributed to the low density of current. As for the elongation of CuCrZr alloys, imposition of current shows little difference. For Cu-Cr system alloys under peak ageing and over ageing conditions, the strengthening is dominated by Orowan mechanism [10], the Orowan stress (TOS) can be expressed as

          0 2 ln ) 1 ( 2 8 . 0 r r L Gb T OS   (1)
Where G is the shear modulus of matrix, b is the Burgers vector of dislocations, L is the inter-precipitate spacing, is the Poisson's ratio, r is radius of precipitates and r0 is the inner cutoff radius of dislocations, the value of which can be taken as equal to b. It can be inferred from Equation ( 1) that the Orowan stress is mainly controlled by the radius of precipitates r and the inter-precipitate spacing L. As shown in Fig. 3, the precipitates are clearly refined with the imposition of current, thus the radius of precipitates r is reduced. Finer precipitates would result in higher density of precipitates in the matrix, consequently reduce the inter-precipitate spacing L. Therefore, strengthening of Orowan mechanism is enhanced. Fig. 4 shows the SEM micrographs of CuCrZr alloys. Apart from much finer precipitates in the matrix, the morphology of precipitates is changed by current. The precipitates change from spherical to cylinder. To confirm this change in morphology, CuCrZr alloys were aged at 450 ℃ for 4 h with a current density of 100 A/cm 2 . From the results shown in Fig. 5, we can clearly see the cylinder precipitates formed during ageing with current. The precipitates' aspect ratio of sample aged under direct current is the highest, while that of sample aged under positive pulses of 1,000 Hz is the lowest. This result confirms with the effect of current on mechanical strength, as spherical precipitates are more effective than cylinder precipitates in Orawan strengthening. The changes in morphology with the imposition of current may be partly explained by the electronwind force. For substitutional solutes like Cu-Cr-Zr alloys, aging process is a diffusion-controlled phase transformation. Therefore, it is expected that the electromigration plays a key role in influence of may be important when an electric current is introduced. According to the Nernst-Einstein equation, the atomic drift flux of atoms that occurs under the influence of an imposed electric current is given by

               j e Z x x N kT kT D N i i i i   * ln (2)
In the equation above, the term kT∂lnN i /∂x is determined by the chemical potential gradient, Ω∂σ/∂x is determined by the internal stress produced by precipitates and Z * eρj is determined by electronwind force. With the imposition of current, the mobility of atoms is accelerated by electronwind force. The accelerated mobility along the direction of current may promote the growth of precipitates along the specific directions, thus changes the morphology of precipitates. In substitutional solute alloy systems, the effect of an electric current is complex [12]. Further study is needed to clarify the mechanism of influence of current on the ageing process.

Conclusions

1. With the imposition of current, the ultimate strength of CuCrZr alloys is improved. The improvement is most significant when aged under positive pulses with the frequency of 1,000 Hz. 2. The improvement in strength can be explained by finer precipitates modified by current. 3. The morphology of precipitates changes from spherical to cylinder with the imposition of current.

Fig. 1

 1 Fig. 1 Schematic diagram of ageing process

Fig. 2

 2 Fig.2 Optical micrographs of the CuCrZr alloys under different state: (a) as cast (b) hot rolled (c) solution-treated (d) cold rolled (e) aged at 450 °C for 4 h and (f) aged at 600 °C for 4 h

Fig. 3 Fig. 3

 33 Fig.3illustrates the TEM image of CuCrZr alloy aged at 450 °C for 4 h with and without current pulses. When aged at

Fig. 4

 4 Fig.4 SEM micrographs of CuCrZr alloys (a) ageing at 450 ℃ for 4 h (b) ageing at 450 ℃ for 4 h with positive pulses of 40 A/cm 2 and (c) marked square area of (b)

Table 1

 1 Mechanical properties of CuCrZr alloys under different conditions Number Applied current Frequency/Hz density/(A/cm 2 )

	UTS/(MPa) Elongation/%

Acknowledgment

The authors gratefully acknowledge the supports of National Natural Science Foundation of China (Nos. 51274054, U1 332115, 51271042, 51375070), the Key grant Project of Chinese Ministry of Education (No. 313011). The authors wish to thank all the staff members of the BL14B1 beamline of SSRF for corresponding experiments and discussion.


A Bojarevics

T Beinerts

I Kaldre

I Grants

G Gerbeth

Metal matrix nanocomposite production using contactless electromagnetic interaction

Keywords: Metal matrix nano-composites, ODS steel, cavitation in liquid metals

Metal matrix nano composites (MMNC) are a class of composite materials where nano and micro scale dielectric particles are evenly distributed in metal matrix. Such materials may have qualitatively different properties and thus specific applications in technology and nuclear industry.

This work analyses the possibility to mix and disperse nano size particles into molten metal using intense contactless electromagnetic interaction. Superimposed AC and DC magnetic fields create oscillating pressure in the liquid metal. If magnitude of this pressure is sufficiently high, acoustic cavitation in the liquid metal can be achieved. Cavitation bubbles are expanding and collapsing very rapidly causing intense microscale jets which can overcome Van der Waals forces and disperse added particles.

Introduction

Evenly distributed dielectric particles in metal matrix can significantly improve some of metal properties, like mechanical strength, thermal properties and radiation absorption properties [1][2][3][4][5]. Particles added to liquid metal tend to form agglomerates due to Van der Waals forces and electromagnetic separation between metal and particles or their ensembles [6]. Dispersion of these agglomerates is a technical challenge which limits the production of these materials in larger quantities. One of the best known material of such type is oxide dispersion strengthened (ODS) steel, which consists of steel matrix and dispersed yttria or other nanoparticles. Typically the particle concentration is from 0.2 to 0.7 weight percent. ODS steel is used in nuclear applications and other ODS alloys are used in turbine blades and high temperature heat exchanger tubing [7,8]. Particle doped aluminum has increased mechanical strength and better extrudability can be achieved for some aluminum grades by adding TiB and Zr 2 O particles [9,10].

Currently most of the MMNC are produced by powder metallurgy when metal powder is mixed with particles and then sintered together by electric arc. To ensure good mixing of powders it takes lot of milling time which increases the cost of the final material. Material produced by this method has limited quality high trapped gas concentration. For aluminum and other low temperature alloys mechanical sound generation by sonotrode works well and there are several works demonstrating that in this way it is possible to achieve improved material quality, reduce porosity and improve particle distribution [11]. In case of steel this method does not work and even niobium coated sonotrode decays rapidly in liquid steel. Several works report different approaches to mix particles into liquid metals by electromagnetic methods. It is experimentally demonstrated that funnel type flow can be used to stir particles into liquid metal even if their densities are very different [12]. Immersed AC coil can also be used to stir the liquid metal and distribute particles into metal [13]. Nevertheless this type of mixing in many cases is insufficient to disperse particle agglomerates and does not break existing nanoparticle agglomerates. Electromagnetically induced sound waves in liquid metals are recently considered as one of the ways to disperse particles, if sound intensity is sufficiently high [14][15][16] It is well known that cavitation phenomena can occur in liquids if pressure change is sufficiently high. Small bubbles filled with saturated vapor appear and collapse in liquid in very short time, thus local thermodynamic quantities can reach very high values. It is known if certain speed is reached in duct flow it may lead to rapid increase of channel wall decay due to cavitation. Cavitation has also been investigated in liquid metals by several authors; however results are ambiguous due to rapid speed and small size of the bubbles. In pure metal to reach cavitation threshold liquid metal should be stretched by negative pressure comparable to the tensile strength of the liquid. In reality cavitation starts earlier and bubbles are starting to form trapped gas bubbles and impurities, thus cavitation thresholds can be very different for same material [17]. Most common way to detect cavitation is to register the sound spectrum from the liquid region. Typical cavitation spectrum contains sub-harmonics and ultra-harmonics of the main frequency f which is used to excite the cavitation. Thus main lines in frequency spectrum to acoustically detect cavitation are 1/2f and 3/2f [START_REF] Brennen | Cavitation and bubble dynamics[END_REF].

Experimental setup

Experimental setup is designed to allow simultaneous application of both AC and DC magnetic fields to the sample. DC magnetic field is created by permanent magnet system while AC field is applied by high frequency inductor. A schematic cross section of the initial experimental setup and the stably semi-levitated pure liquid copper region are shown on the Fig. 1(a). The AC field frequency range used in experiments was from 9 to 18 kHz, maximum AC field induction amplitude up to 0.09 T. Induction melted copper droplet is shown in Fig. 1(b) assumes meniscus shape as a consequence of electromagnetic forces. The DC magnetic field with maximum induction of 0.35 T was delivered by a permanent magnet assembly, permitting to apply quite uniform field over the sample region in the direction range from axial to horizontal. During processing of the liquid sample surface oscillations are observed, and their intensity and morphology are very different depending on magnetic field [START_REF] Bojarevičs | Magnetohydrodynamics[END_REF]. It was find out that the best method how to prepare initial sample is mixing together metal and powder and particles and then pressing it in hydraulic press. Such sample contains particle chunks within the size range 20-200 µm. This sample is melted by AC inductor under static magnetic field. To allow applying larger electromagnetic interaction sample is intensively cooled from bottom to prevent overheating. Maximum electromagnetic interaction is maintained by 10 minutes and then reduced to value which allow liquid metal to solidify. It was find out that it is important to maintain electromagnetic interaction until metal is completely solid to prevent particle segregation and reagglomeration.

Results and discussion

Results show that for different metal and particles combination required interaction intensity can be very different. Contactless electromagnetic method for metal matrix nanocomposite production can be feasible for some combinations. Particle dispersion and influence on material structure are examined using optical microscopy, scanning electron microscopy and X-ray fluorescence spectroscopy. Most promising experimental results in terms of particle dispersion are achieved for aluminum matrix and SiC particles, and Fe or FeCr steel matrix and and TiN particles. For these pairs of materials mutual solubility is minimal even in high temperatures as shown in Fig. 2. Some other combination like Yttria and Fe-12%wt.Cr turned out to be chemically incompatible in molten state at 1500 o C. 3(a) shows SEM image of produced composite material. Initial particle content is 2%. wt, which agrees well with the observed concentration. It can be seen that particles are not fully dispersed in all cases but achieved dispersion is very good compared to initial sample where particle conglomerate sizes were up to 200 μm. Al and SiC were chosen as a test combination because this is one of the composite material with practical applications and it has lower melting temperature. Previous study shows that SiC particle reinforced aluminum has improved mechanical properties and wear resistance [20][START_REF] Ozben | [END_REF]. Compatibility and efficiency of this method for iron alloys were tested by melting pure iron and 800 nm TiN particles. Experiment showed good stability of the particles, melted sample cross section is shown in Fig. 4. Fig. 4(a) shows that typical particle size is slightly larger than particle size, indicating that some particles are still not fully dispersed. Fig. 4(b) shows secondary electron image revealing element composition in the sample in larger scale. Here it can be seen that particles tend to concentrate on grain boundaries, which appear as linear structures in some cases. Particle concentration primarily at the grain boundaries were expected and it is investigated that this is one of the mechanism, which increases material mechanical properties by preventing sliding along grain boundaries [22].

Conclusions

Proposed electromagnetic contactless method to mix and disperse dielectric particles in liquid metal showed promising results for some metal-particle pairs in laboratory scale tests. Experiments show that different metals and particles behave in a very different way. Not all material pairs are chemically compatible in high temperatures thus this method can be applicable for metals and particles with low solubility and good mutual wetting. Most promising results we obtained are for Al/SiC, Fe/TiN and Cu/SiC combinations, but not in all cases particles were fully dispersed. Obtained Al/SiC composite shows good dispersion of particle conglomerates of various sizes. For better dispersion electromagnetic interaction intensity should be increased above the cavitation threshold and maintained for the longer time during treatment and solidification phase. In this work acoustic spectrum was not measured to determine whereas cavitation is achieved but our previous studies show that with slightly higher magnetic fields cavitation frequencies can be detected in liquid aluminum [23].

Fig 1 :

 1 Fig 1: (a) Schematics of the setup; (b) semi-levitated copper in axial 10 kHz AC field and DC field

Fig 2 :

 2 Fig 2: a) Al-SiC phase diagram; b) Fe-TiN phase diagram

Fig 3 :

 3 Fig 3: Al-SiC sample solidified under combined AC/DC electromagnetic interaction. a) SEM image; b) XRF image

Fig 4 :

 4 Fig 4: Fe-TiN sample solidified under AC/DC electromagnetic interaction. a) SEM image; b) back scattered electron image showing element distribution (white is Ti rich areas)

Acknowledgment

Financial support from LIMTECH alliance and Latvian Science Council are gratefully acknowledged.