4.2. BRIEF OVERVIEW OF NUMERICAL METHODS FOR PHASE CHANGE PROBLEMS

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1 210 CHAPTER BRIEF OVERVIEW OF NUMERICAL METHODS FOR PHASE CHANGE PROBLEMS Consider the one-dimensional heat conduction problem ((1)-(4) of 4.1), but now regarding the material as a phase change material with a melt temperature T m, and take T init (x) < T m and T (t) > T m. Then melting of the material will occur, commencing at x = 0 at some time t init > 0, and we want to find t init, T(x, t) and X(t) such that ρc S T t = (k S T x ) x for 0 < x < l, 0 < t < t init and for X(t) <x < l, t > t init, ρc L T t = (k L T x ) x for 0 < x < X(t), t > t init, k S T x (0, t) = h[t (t) T(0, t)] for t < t init, k L T x (0, t) = h[t (t) T(0, t)] for t > t init, T x (l, t) = 0 for t >0, T(x,0) = T init (x), X(t init ) = 0, T(X(t), t) = T m, t > t init, ρ LX (t) = k L T x (X(t), t) + k S T x (X(t) +, t), t > t init. Clearly we have to separate the problem into two distinct problems: a pure heat conduction problem, until the face x = 0 reaches the melt temperature at time t = t init, and a two-phase Stefan problem after time t init. The numerical solution to the first problem was presented in 4.1, but that of the Stefan problem is much more difficult, due to the underlying geometric nonlinearity of the problem: the regions in which the two heat conduction equations are to hold change in time, and we have to compute the location of the interface x = X(t) concurrently. Several approaches have been devised with this aim, collectively referred to as front tracking schemes, because they attempt to explicitly track the interface using the Stefan condition. One approach is to fix the spatial step, x, but allow the time step, t n to float in such a way that the front always passes through a node (x j, t n ). An example of this approach is the method of [DOUGLAS-GALLIE]. Another approach is to fix the time step and allow the spatial step to float, in fact, use two distinct and time-varying space steps for the two phases. The isotherm migration method of J. Crank is of this type, [CRANK, 1981, 1984]. Yet another approach is based on the Landau transformation, which we have used in the perturbation method (see 3.3). By a change of variables ( 3.3.B), the regions representing the phases become fixed, the underlying geometric nonlinearity showing up algebraically now in the transformed equations. Then one solves the resulting system of nonlinear equations by some numerical method.

2 4.2 OVERVIEW OF NUMERICAL METHODS 211 All such approaches work well, more or less, for simple Stefan problems (that would arise e.g., in laboratory settings) in which we know what to expect, namely a single sharp front separating the two phases. It is not difficult to realize however, that such problems are not the rule in practice, particularly when time dependence of heat input/output or thermal cycling occur. For example, in Latent Heat Thermal Energy Storage, one must deal with cases of extreme thermal cycling, multiple fronts, disappearing phases and non-predictable behavior ( 1.3, 5.3). Internal heating is another source of difficulties, first documented by [ATTHEY], in which extended mushy zones may appear instead of sharp fronts. Constitutional supercooling of binary alloys results in similar effects which may not be ignored ( see [ALEXIADES-WILSON-SOLOMON, 1985] ). Simultaneous mass transfer by diffusion and/or convection complicate the phase change process to the point that we cannot guess a priori the qualitative picture in enough detail to even be able to formulate the problem in the classical fashion of a Stefan type problem with sharp front, etc. If so many complications can arise in 1-dimensional processes, what about 2- and 3-dimensional processes? Despite the difficulties, successful methods for 2-D hydrodynamic instability problems have been developed by Glimm and coworkers, [GLIMM]. Surveys of front tracking methods appear in [MEYER, 1978], [CRANK, 1981, 1984], [ALBRECHT-COLLATZ-HOFFMANN], etc. Such reasons make front-tracking schemes unviable as general simulation tools for modeling realistic phase-change processees. The only viable general approach is the so-called enthalpy method, precisely because it bypasses the explicit tracking of the interface. In this approach the jump condition (Stefan condition) is not forced on the solution, but it is obeyed automatically by it as a natural boundary condition (in the sense of the Calculus of Variations). Its theoretical basis is a formulation of the Stefan problem different than the classical one, the so-called weak or enthalpy formulation, described in 4.4. It is similar to the weak formulations commonly used in gas dynamics for shocks (see [HYMAN] for a brief overview). Another fixed-domain method (as opposed to front-tracking), based on variational inequalities [DUVAUT], [ODEN-KIKUCHI] reformulation of the Stefan problem and finite elements, lacks the direct physical interpretation of the enthalpy method and has not lived up to its initial promise for Stefan-type problems. It can be safely concluded today that the enthalpy method, to which we turn in the next section, discretized by (integrated) finite differences, is the most versatile, convenient, adaptable, and easily programmable numerical method available for phase change problems in 1, 2 or 3 space dimensions. We hasten to add however that it does not solve all the problems. Excluded are problems which we do not know how to formulate weakly due to their special interface conditions. Such is the case with supercooling problems, where the instability of the interface must be studied. A very successful computational approach for such problems is another fixed-domain type formulation, the socalled phase-field approach, under intense development lately, [CAGINALP, 1989, 1991], [KOBAYASHI].

3 212 CHAPTER THE ENTHALPY METHOD IN ONE SPACE DIMENSION 4.3.A Introduction The, so called, enthalpy or weak solution approach is based on the fact that the energy conservation law, expressed in terms of energy (enthalpy) and temperature, together with the equation of state contain all the physical information needed to determine the evolution of the phases. It turns out that, for the purpose of obtaining numerical schemes, the most appropriate and convenient way to state energy conservation is the primitive integral heat balance over arbitrary volumes and time-intervals, from which all other formulations can be obtained, namely t+ t t t EdV V dt = t+ t t q. n ds dt (1) where E = ρe is the energy density (per unit volume), and q. n is the heat flux into the volume V across its boundary V, n being the outgoing unit normal to V. The distinct advantage of this primitive form is that it is valid irrespectively of phase, and even if E and q experience jumps, so it is actually more general than the localized differential form (2) E t + divq = 0. The two forms are equivalent for smooth E, q, thanks to the Divergence Theorem ( 1.2 ). In the presence of a phase-change, the partial differential equation (2) can only be interpreted in the classical pointwise sense inside each phase separately, and then conservation across the interface must be imposed explicitly as an additional interface (Stefan) condition, making front-tracking necessary. Alternatively, the PDE (2) may be interpreted in a generalized (weak) sense globally, as described in detail in 4.4. It turns out that the numerical solutions obtained via the enthalpy method approximate this weak solution, as we shall show in 4.5. In this section, we describe the enthalpy method for Stefan problems in one space dimension, and its numerical implementation via time-explicit or timeimplicit schemes. V 4.3.B The enthalpy method The idea of the enthalpy approach is very simple, direct, and physical. We partition the volume occupied by the phase-change material into a finite number of control volumes V j and apply energy conservation, (1), to each control volume to obtain a discrete heat balance. Note that this is the same discrete heat balance as for plain heat conduction ( 4.1.B ), and we use it to update the enthalpy, E j,of

4 4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 213 each control volume. From the equation of state we know that E j 0 ==> V j is solid, E j ρ L ==> V j is liquid, and 0 < E j < ρ L ==> V j is partially liquid and partially solid, so we call it "mushy". A mushy cell contains an interface and the fraction of the cell occupied by liquid is naturally given by the value of the liquid fraction : λ j = E j ρ L. Note that in this scheme the phases are determined by the enthalpy alone, with no mentioning of interface location(s). It is a volume-tracking scheme, as opposed to front-tracking. Since the front location may be recovered a posteriori from the values of the enthalpy, it may be characterized as a front-capturing scheme, similar in spirit to shock-capturing schemes of gas dynamics (see [HYMAN] for an overview of various types of schemes). Let us see how the method works in detail, by considering the heat conduction problem of 4.1, except now we assume that our slab 0 x l is occupied by a material that changes phase at a melt temperature T m. We assume that initially the material is solid with T(x,0) = T init (x) T m, 0 x l, (3) the face x = 0 is heated convectively by T (t) T m : q(0, t) = h [ T (t) T(0, t)], t >0, and the face x = l is insulated : q(l, t) = 0, t >0. (5) The energy conservation law in its integrated form (1) applied to the present onedimensional control volumes V j = [ x j 1 2, x j+1 2 ] A (see 4.1.B ) becomes t n+1 t n A t x j+1 2 E(x, t)dx dt = A q x (x, t) dx dt. (6) x j 1 2 We seek numerical approximations to the temperature, energy and flux obeying (3)-(6) with q = kt x of course. This problem is formally identical to the heat conduction problem (1b,c),(8a) of 4.1. The two differ in that now the enthalpy E is the sum of sensible and latent heat in the liquid, so that, instead of (6) of 4.1.B, we have t n+1 t n x j+1 2 x j 1 2 (4) E(x, t) = T(x,t) ρc S(τ )dτ, T(x, t) <T m (solid) T m T(x,t) ρc L (τ )dτ + ρ L, T(x, t) >T m (liquid) T m (7) The phases are described by

5 214 CHAPTER 4 E(x, t) 0 => solid at (x, t) (8a) 0 < E(x, t) < ρ L => interface at (x, t) (8b) E(x, t) ρ L => liquid at (x, t). (8c) Thus, only the relation between E and T is now different than in 4.1, while the discretization of (6) is still (12) of 4.1.B. The flexibility and generality of this approach will be further illustrated in 4.3.H where even a wall layer will be incorporated into the global scheme by adjusting the energy. Consider the case in which c S, c L = constants. (9) Then (7) becomes E = ρc S [ T T m ], T < T m ρc L [ T T m ] + ρ L, T > T m (10) or, solving for T, T = T m + E ρc S, E 0 ( solid ) T m, 0 < E < ρ L ( interface ) T m + E ρ L, ρc L E ρ L ( liquid ) Proceeding with the discretization of fluxes and boundary conditions as in 4.1, we arrive at the following discrete problem. initial values: T 0 (12a) j = T init (x j ), j = 1,..., M boundary condition at x = 0: boundary condition at x = l : interior values: E n+1 j where and q n+θ j 1 2 = T j n+θ T j 1 n+θ R j 1 2 T n j = 1 = T 1 n+θ T n+θ 2 1 h + R1 2 q n+θ q n+θ M+ 1 2 = 0, = E n j + t n q n+θ x j j 1 2 q j+ n+θ 1 2,, R1 2 = 1 2 x k 1 j = 1,..., M (11) (12b) (12c) (12d) 1 2 x 1 j 1 2 x j (12e) with R j 1 2 = +, j = 2,..., M, k j 1 k j T m + En j, ρc S E n j 0 ( solid ) T m, 0 < E n j < ρ L ( interface ) T m + En j ρ L, ρc L E n j ρ L ( liquid ) (12f)

6 4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 215 The updating algorithm from any time t n to the next t n + t n proceeds as follows: Knowing the enthalpy, temperature and phase (see below) of each control volume, we compute the resistances and fluxes, which are then used to update the enthalpies, which in turn yield new temperatures and phase states. The most convenient phase-indicator is the liquid fraction of a control volume V j, defined as λ n j = 0, if E n j 0 ( solid ) E n j ρ L, if 0 < En j < ρ L ( mushy ) 1, if ρ L E n j (liquid). (13) If 0 < λ n j < 1 the control volume is said to be mushy with liquid volume λ n j x j and solid volume (1 λ n j ) x j (per unit cross sectional area). The definitions of resistances and fluxes between control volumes are identical to their definitions in 4.1, with the resistance at x j 1 2 expressed as R j 1 2 = x j 1 + x j. 2 k j 1 2 k j The effective conductivity k j of a mushy control volume depends on the structure of the phase-change front, and it is not always clear how to choose it, especially in 2 or 3 dimensional situations. Some alternative choices are: Sharp front(s): A control volume containing a sharp front consists of layers of solid and liquid in a "serial" arrangement, for which the effective resistivity is the sum of the resistivities of the layers. With the layer thicknesses determined from the solid and liquid fractions, we have 1 λ n j (14a) k n = j k L (T m ) + 1 λ n j, j = 1, 2,..., M. k S (T m ) Columnar front: A front consisting of columns of solid and liquid constitutes a "parallel" arrangement, so the effective conductivity is the sum of the conductivities of the phases : k n j = λ n j k L (T m ) + (1 λ n j ) k S (T m ). (14b) Amorphous mixture of solid and liquid: The inter-phase region may be a random mixture of solid and liquid. In this case one may use the following formula which interpolates the previous two cases [CHEMICAL ENGINEERING GUIDE, p.242]: k n 1 + λ 2/3 (κ 1) j = k S (T m ) 1 + (λ 2/3 λ)(κ 1), λ = λ n j, κ = k L(T m ) k S (T m ). (14c) In most situations we do not know which of the above cases is relevant. In 2 or 3 space dimensions even a sharp front will generally not be moving in the

7 216 CHAPTER 4 direction of one of the axes, so the choice is not clear. A simple expedient is to take the average of the solid and liquid conductivities, k n j = 1 2 (k S + k L ) (14d) However, the best alternative, when applicable, is to employ the "Kirchoff transformation" (see (7) 4.4.D), to replace the temperature T by the "Kirchoff temperature" u. This can be used when the conductivity is a function of temperature only. In particular, for constant k S, k L the "Kirchoff temperature" is u = k S [ T T m ] 0 k L [ T T m ] if T < T m if T = T m. (15) if T > T m Then q kt x = u x, so the discrete flux is simply q j 1 2 = ( u j 1 u j )/ x. To compare this with (12e), resubstitute u in terms of T : u j = k j [ T j T m ], and write it as q j 1 2 = T j 1 T m Rˆ j 1 + T m T j Rˆ j, Rˆ j : = x / k j. (16) Thus the flux neatly splits to a sum of two terms, one for each of the two adjacent nodes. Note that if one of the nodes, say node j, is mushy then T j = T m and the node is not contributing to conduction. We see that in this prescription each node has its own resistivity Rˆ j = x / k j, which may be found conveniently from Rˆ j = x { λ j / k L + (1 λ j )/k S }, (17) the value for mushy nodes being irrelevant since they are not contributing to the flux. No averaging of values is used, so this prescription results in the highest effective conductivity. It is the best choice for the enthalpy scheme since it is consistent with the mushy nodes being treated as isothermal. Note that such issues arise only when k L (T m ) and k S (T m ) are substantially different, in which case the sensitivity of the solution to the choice of effective conductivity should be examined by comparing the results from the various choices. (14a) yields the lowest effective conductivity and (16)-(17) yields the highest, so these two pretty much bracket the system behavior. We emphasize again that the interface location is not involved in the computation at all, this being an essential advantage of the enthalpy method. If the problem being modeled admits a sharp interface, then the enthalpy scheme ought to produce a single mushy node at each time step. If at time t n the mushy node is the m-th node, then a good approximation to the interface location X(t n ) is given by X n : = x m λ n m x m. (18) REMARK 1. Missing the phase-change The latent heat effect is only felt in mushy nodes, so each control volume should pass through the mushy state before changing phase. The algorithm

8 4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 217 described here is "robust" in this respect, so skipping this transition indicates a bug in the code, or too large a time-step. Some other implementations may not be so robust and one must be careful not to miss the transition. Especially prone are algorithms that track the temperature and account for the latent heat via a source term. REMARK 2. Temperature dependent heat capacities If c S = c S (T), c L = c L (T), then the equation of state is the nonlinear relation (7), and finding T from E is not quite as simple as in the constant c S, c L case. If E 0 we need to find T from the equation E = ρc S(τ )dτ,for T m which a Newton-Raphson method may be employed. Alternatively, we may rewrite the equation as de dt = ρc S(T), or dt de = 1, which is an ODE with ρc S (T) initial condition T = T m for E = 0. Similarly, if E ρ L, then we may solve the equation E = T ρc L(τ )dτ + ρ L via a Newton-Raphson method, or T m solve the ODE dt de = 1 ρc L (T) with initial condition T = T m for E = ρ L. Any ODE solver can be used for this purpose, e.g. forward Euler, backward Euler, Runge-Kutta, etc. Actually, the temperature dependence of heat capacities is commonly expressed in the form c i (T) = A i + B i T + C i, i = S, L, (19) T 2 with T in degrees Kelvin and A i, B i, C i given constants. Then the integrals expressing the sensible heat can be computed analytically, and the resulting algebraic equations may be solved very effectively via a Newton-Raphson method. Note that this needs to be done for each node at each time step, adding considerably to the expense of the computation. A reasonable starting value is the temperature at the previous time step, T n j. T 4.3.C A time-explicit scheme Choosing θ = 0 in (12), the fluxes are evaluated at the old time t n and we assume that up to time t n+1 the process is driven by these fluxes. The explicit scheme proceeds as follows. Initially, the phase and temperature of each control volume are known with T 0 j = T init (x j ), j = 1, 2,..., M, (20) which in turn determine the enthalpies E 0 j, j = 1, 2,..., M, via (10). Assume that we have found enthalpies, temperatures and phase-states ( λ j ) through the n-th

9 218 CHAPTER 4 time step. From (13) we find the liquid fractions, λ n j, hence the phase of each node and from (12f) the (mean) temperatures. If only one node is mushy the interface location at time t n is given by (18). Now we compute the conductivities from (14), the resistances and fluxes from (12b,c,e), with θ = 0, and then E n+1 j is found from (12d), j = 1, 2,..., M. Note that for the boundary control volumes ( j = 2, M 1 ) we may use the analog to the implicit relation (43) of 4.1, in order to guarantee the Maximum Principle and still have the CFL condition guarantee no growth of errors. Thus, the updating of enthalpies to time t n+1 is complete. The stability condition is the same as in the pure conduction case, namely where min x = min x j and j=1,2,...,m t n 1 2 (min x) 2 (max α n ), (21) max α n = max k L (T n j ) ρc j, k S (T n j ) ρc j, j = 1, 2,..., M. It is good practice to take a number slightly smaller than 1 2 in order to avoid stability problems arising from roundoff. The value of max α n can be computed at each time step t n and then we can use t n = 1 (min x) 2 as the next time step. As 2 max α n this quantity may become impractically small, it is good programming practice to halt the computation if t n becomes smaller than a prescribed minimum t, and carefully examine what caused it to become so small. The great advantage of the time-explicit scheme lies in its simplicity and the ease with which it can be programmed. In situations where the time-step must be small for physical reasons (to capture rapidly moving fronts or resolve rapid changes in data, for example), the stability requirement may not impose undue restrictions, and the explicit scheme may turn out to be as efficient as implicit schemes. The extreme case of laser annealing with picosecond or nanosecond pulses is such a situation [ALEXIADES et al, 1985a]. 4.3.D Performance of the explicit scheme on a one-phase problem To see how the scheme of 4.3.C performs, we test it on the simplest problem with known exact solution, namely the one-phase Stefan Problem with constant imposed temperature at x = 0. The Neumann (similarity) solution in dimensionless variables appears in (14)-(16) of 2.1. To retain direct physical meaning, we implement the enthalpy scheme on the original formulation (1)-(4) of 2.1, which can be made identical to the dimensionless formulation ((19)-(23), 2.1) by choosing T m = 0. ρ = c L = k L = 1, T L = 1, L = 1 St. (22)

10 4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 219 We simulate melting in the slab 0 x 1 for two extreme values of the Stefan number: St =.1 and St = 5; the corresponding transcendental roots are found to be λ =.22andλ = To exhibit the convergence of the algorithm, we discretize the slab 0 x 1 with M = 10, 20 and 40 uniform subintervals. Since α = k L /ρc L = 1 (from (22)), the corresponding time steps will be (see (21)) t 1 2 ( 1 10 )2, 1 2 ( 1 20 )2 and 1 2 ( 1 40 )2, showing the severe limitation on the time step imposed by the stability criterion. Table shows temperatures at several locations x, at time t = 3 for the problem with St = 0. 1 and at time t =. 12 when St = 5. At these times the melt fronts have not reached x =. 8 yet, so there has been no backface influence. The second column shows the exact (Neumann) temperatures found as in 2.1. The numerically computed temperatures with M = 10, 20 and 40 nodes are listed in the other columns (linearly interpolated from nodal values for M = 20 and M = 40). Observe the progressive convergence to the exact solution as M increases. It is also interesting to compare the code execution run times for the three mesh sizes: on an IBM-PC/XT, they were 12.5, 41.5 and 260 seconds for St = 0. 1 and 7, 9 and 24 seconds for St = 5, respectively for M = 10, 20 and 40 illustrating the dramatic slowdown the finer mesh causes. In Table we compare the exact and numerical interface locations at a few sample times from the same runs as above. In Figures and 4.3.4, the computed temperature history (melting curve) at a fixed location is compared with the exact (Neumann) solution when St = 0. 1 and St = 5 respectively. The staircase shape is characteristic of enthalpy methods and it is much more pronounced for M = 10 nodes (Fig (a), 4.3.4(a)) than for Table 4.3.1: Exact and Computed Temperature Profiles For St = 0. 1 at time t = 3. 0 For St = 5 at time t =.12 x T exact M = 10 M = 20 M = 40 x T exact M = 10 M = 20 M =

11 220 CHAPTER 4 Table 4.3.2: Exact and Computed Interface Locations For St = 0. 1 For St = 5 time X exact M = 10 M = 20 M = 40 time X exact M = 10 M = 20 M = numerical with M = 10 nodes Figure 4.3.1(a). Temperature history at x =. 3, with M = 10 nodes, compared with the exact solution, St = M = 40 nodes (Fig (b), 4.3.4(b)). This is due to the fact that while the interface lies anywhere inside a particular mesh interval, the temperature of that interval is held at T m, so the temperature in the rest of the slab relaxes to a steady state corresponding to a fixed isotherm through that node. When the interface moves to the next mesh interval, the temperature adjusts rapidly and then relaxes to a new steady state. It follows that the duration of each step is strictly a function of the time the interface remains in each mesh interval, and therefore, the finer the mesh the shorter the steps. Indeed, using M = 80 nodes the computed and exact solutions would be indistinguishable graphically. Figures and show that the interface location computed with only M = 10 and M = 20 nodes, for St = 0. 1 and St = 5 respectively, agrees well with the exact interface. Finally, temperature profiles with M = 10 and M = 40 nodes are shown in Figures 4.3.3(a),(b) (at time t = 2., St = 0. 1) and Figures 4.3.6(a),(b) (at time t =. 1, St = 5). These figures bare out the fact that while numerical methods for phase-change problems can easily capture interface locations and even temperature profiles, the errors show up vividly in temperature history plots.

12 4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 221 numerical with M = 40 nodes Figure 4.3.1(b). Temperature history at x =. 3, with M = 40 nodes compared with the exact solution, St = numerical with M = 10 nodes Figure Melt front location with M = 10 nodes, compared with the exact solution, St = numerical with M = 10 nodes Figure 4.3.3(a). Temperature profile at time t = 2., with M = 10 nodes, compared with the exact solution, St = 0. 1.

13 222 CHAPTER 4 numerical with M = 40 nodes Figure 4.3.3(b). Temperature profile at time t = 2., with M = 40 nodes, compared with the exact solution, St = numerical with M = 10 nodes Figure 4.3.4(a). Temperature history at x =. 3, with M = 10 nodes, compared with the exact solution, St = 5. numerical with M = 40 nodes Figure 4.3.4(b). Temperature history at x =. 3, with M = 40 nodes, compared with the exact solution, St = 5.

14 4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 223 numerical with M = 20 nodes Figure Melt front location with M = 20 nodes, compared with the exact solution, St = 5. numerical with M = 10 nodes Figure 4.3.6(a). Temperature profile at time t =. 1, with M = 10 nodes, compared with the exact solution, St = 5. numerical with M = 40 nodes Figure 4.3.6(b). Temperature profile at time t =. 1, with M = 40 nodes, compared with the exact solution, St = 5.