enthalpy method how-to

8/12/2019 Enthalpy Method How-To

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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 Ther-

mal Energy Storage, one must deal with cases of extreme thermal cycling, multiplefronts, 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 super-

cooling 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 track-

ing of the interface. In this approach the jump condition (Stefan condition) is not

forcedon 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 formu-

lations 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 prob-

lems.

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 so-

called phase-field approach, under intense development lately, [CAGINALP,

1989, 1991], [KOBAYASHI].


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212 CHAPTER 4

4.3. 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

V E dV

dt =t+t

t

V q

. n

dS dt (1)

whereE= e is the energy density (per unit volume), and q . n

is the heat flux

into the volumeV across its boundaryV,n being the outgoing unit normal toV.The distinct advantage of this primitive form is that it is valid irrespectively of

phase, and even ifE andq

experience jumps, so it is actually more general than

the localized differential form

(2)Et+ divq

= 0 .

The two forms are equivalent for smoothE,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 solutionsobtained 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 time-

implicit schemes.

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 volumesVj and apply energy conservation, (1), to each control volume to

obtain a discrete heat balance. Note that this is the same discrete heat balance asfor plain heat conduction ( 4.1.B ), and we use it to update the enthalpy, Ej , of


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4.3 THE ENTHALPY METHOD IN ONE SPACE DIMENSION 213

each control volume. From the equation of state we know that Ej 0 ==>Vj issolid,Ej L ==>Vj is liquid, and 0 Vj is partially liquid andpartially 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 : j = EjL .

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 recovereda 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 of4.1, except now we assume that our slab 0x l is occupied by amaterial that changes phase at a melt temperatureTm . We assume that initially the

material is solid with

(3)T(x, 0) = Tinit(x) Tm , 0x l ,the facex = 0 is heated convectively byT(t)Tm :

(4)q(0, t) = h [ T(t)T(0, t) ] , t> 0 ,

and the facex =l is insulated :

(5)q(l, t) = 0 , t> 0 .

The energy conservation law in its integrated form (1) applied to the present one-

dimensional control volumes Vj= [xj12 ,xj+12 ] A (see 4.1.B ) becomestn+1

tn

t

A

xj+12

xj12

E(x, t)dx

dt = tn+1

tn

Axj+12

xj12

qx(x, t)dx dt. (6)

We seek numerical approximations to the temperature, energy and flux

obeying (3)-(6) withq= k Tx of course.This problem is formally identical to the heat conduction problem (1b,c),(8a)

of4.1. The two differ in that now the enthalpyE is the sum of sensible and latent

heat in the liquid, so that, instead of (6) of4.1.B, we have

E(x, t) =

T(x,t)

Tm

cS()d , T(x, t) Tm (liquid)(7)

The phases are described by


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214 CHAPTER 4

(8a)E(x, t) 0 => solid at (x, t)

(8b)0 interface at (x, t)

(8c)E(x, t) L => liquid at (x, t) .

Thus, only the relation between E andT 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

(9)cS , cL = constants .Then (7) becomes

E =

cS[ T Tm ] , TTm

(10)

or, solving forT,

T =

Tm+ E

cS

, E 0 ( solid )

Tm , 0


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The updating algorithm from any time tn to the nexttn+ tn proceeds as fol-lows: 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

Vj , defined as

nj =

0 , if Enj 0 ( solid )Enj

L, if 0 < Enj < L ( mushy )

1 , if LEnj (liquid)

. (13)

If 0 < nj < 1 the control volume is said to be mushy with liquid volume nj xj

and solid volume (1 nj )xj (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 atxj12 expressed as

Rj12 = xj1

2kj1

+xj

2kj

.

The effective conductivitykj 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

(14a)1

knj=

nj

kL(Tm)+

1njkS(Tm)

, j= 1, 2, . . . ,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 conduc-

tivities of the phases :

(14b)knj = njkL(Tm)+ (1

nj )kS(Tm) .

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]:

knj = kS(Tm)1+ 2/3( 1)

1+ (2/3 )( 1), = nj , =

kL(Tm)

kS(Tm). (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


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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,

(14d)knj = 12 (kS+ kL)

However, the best alternative, when applicable, is to employ the "Kirchoff trans-

formation" (see (7)4.4.D), to replace the temperatureT by the "Kirchoff tempera-ture"u. This can be used when the conductivity is a function of temperature only.

In particular, for constantkS,kL the "Kirchoff temperature" is

u =

kS[ T Tm ]0

kL [ T Tm ]

if T < Tm

if T = Tmif T > Tm

. (15)

Thenq k Tx= ux , so the discrete flux is simplyqj12= ( uj1uj ) /x . Tocompare this with (12e), resubstitute u in terms ofT: uj= kj [ Tj Tm ], andwrite it as

qj12 = Tj1Tm

Rj1

+TmTj

Rj

, Rj : = x/kj . (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 thenTj= Tm and thenode is not contributing to conduction. We see that in this prescription each node

has its own resistivity Rj= x/kj , which may be found conveniently from

(17)Rj= x { j/kL+ (1j) /kS} ,

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 con-

sistent with the mushy nodes being treated as isothermal.

Note that such issues arise only whenkL(Tm) andkS(Tm) are substantially dif-

ferent, in which case the sensitivity of the solution to the choice of effective con-ductivity 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 computa-

tion at all, this being an essential advantage of the enthalpy method. If the prob-

lem being modeled admits a sharp interface, then the enthalpy scheme ought to

produce asingle mushy node at each time step. If at timetn the mushy node is the

m-th node, then a good approximation to the interface locationX(tn) is given by

(18)Xn : = xm12+ nmxm .

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


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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

Ifc S=cS(T),cL=cL(T), then the equation of state is the nonlinear rela-tion (7), and findingT fromE is not quite as simple as in the constant cS,cL

case. If E 0 we need to find T from the equation E=T

Tm

cS()d, forwhich a Newton-Raphson method may be employed. Alternatively, we may

rewrite the equation asdE

dT= cS(T), or

dT

dE=

1

cS(T), which is an ODE with

initial condition T= Tm for E= 0. Similarly, if E L, then we may

solve the equationE=T

Tm cL()d+ L via a Newton-Raphson method, or

solve the ODEdT

dE=

1

cL(T)with initial condition T= Tm for E= 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

ci(T) = Ai+ B iT+Ci

T2, i= S,L , (19)

withT in degrees Kelvin and Ai, Bi, Ci 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,Tnj .

4.3.C A time-explicit scheme

Choosing = 0 in (12), the fluxes are evaluated at the old time tn and weassume that up to time tn+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

T0j = Tinit(xj) , j= 1, 2, . . . ,M, (20)

which in turn determine the enthalpies E

0

j , j= 1, 2, . . . ,M, via (10). Assume thatwe have found enthalpies, temperatures and phase-states ( j ) through then-th


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218 CHAPTER 4

time step. From (13) we find the liquid fractions,nj , hence the phase of each node

and from (12f) the (mean) temperatures. If only one node is mushy the interface

location at timetn is given by (18).

Now we compute the conductivities from (14), the resistances and fluxes from

(12b,c,e), with= 0, and thenEn+1j is found from (12d), j= 1, 2, . . . ,M. Note thatfor the boundary control volumes ( j= 2,M 1 ) we may use the analog to theimplicit 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 timetn+1 is complete.

The stability condition is the same as in the pure conduction case, namely

tn 1

2

(min x)2

(maxn), (21)

where min x=j=1,2,...,M

min xj and

maxn = max

kL(Tnj )

cj,

kS(Tnj )

cj, j= 1, 2, . . . ,M

.

It is good practice to take a number slightly smaller than 12 in order to avoid stabil-

ity problems arising from roundoff. The value of maxn can be computed at each

time step tn and then we can use tn=1

2

(min x)2

maxnas the next time step. As

this quantity may become impractically small, it is good programming practice to

halt the computation iftn becomes smaller than a prescribed minimum t, andcarefully 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 nanosecondpulses 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 of4.3.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 dimension-less variables appears in (14)-(16) of2.1.

To retain direct physical meaning, we implement the enthalpy scheme on the

original formulation (1)-(4) of2.1, which can be made identical to the dimension-

less formulation ((19)-(23), 2.1) by choosing

Tm= 0 . = cL= kL= 1 , TL= 1 , L= 1St

. (22)


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We simulate melting in the slab 0x 1 for two extreme values of the Stefannumber: St= . 1 andSt= 5; the corresponding transcendental roots are found tobe= . 22 and= 1. 06.

To exhibit the convergence of the algorithm, we discretize the slab 0x 1withM= 10, 20 and 40 uniform subintervals. Since=kL/cL= 1 (from (22)),

the corresponding time steps will be (see (21))t 12

( 110

)2, 12

( 120

)2 and 12

( 140

)2,

showing the severe limitation on the time step imposed by the stability criterion.

Table 4.3.1 shows temperatures at several locations x, at time t= 3 for theproblem withSt= 0. 1 and at timet= . 12 whenSt= 5. At these times the meltfronts have not reached x= . 8 yet, so there has been no backface influence. Thesecond column shows the exact (Neumann) temperatures found as in 2.1. The

numerically computed temperatures withM= 10, 20 and 40 nodes are listed in theother 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 forSt= 0. 1 and 7, 9 and24 seconds forSt= 5, respectively forM= 10, 20 and 40 illustrating the dramaticslowdown the finer mesh causes.

In Table 4.3.2 we compare the exact and numerical interface locations at a few

sample times from the same runs as above.

In Figures 4.3.1 and 4.3.4, the computed temperature history (melting curve) at

a fixed location is compared with the exact (Neumann) solution whenStSt= 0. 1 andStSt= 5 respectively. The staircase shape is characteristic of enthalpy methodsand it is much more pronounced forM= 10 nodes (Fig. 4.3.1(a), 4.3.4(a)) than for

Table 4.3.1: Exact and Computed Temperature Profiles

For St= 0. 1 at timet= 3. 0 For St= 5 at timet= . 12

x Texact M= 10 M= 20 M= 40 x Texact M= 10 M= 20 M= 40

0 1.0000 1.0000 1.0000 1.0000 0 1.000 1.000 1.000 1.000

.1 .8667 .8667 .8677 .8672 .1 .8132 .8144 .8135 .8133

.2 .7336 .7333 .7359 .7346 .2 .6341 .6360 .6346 .6342

.3 .6010 .6000 .6051 .6026 .3 .4692 .4711 .4698 .4693

.4 .4691 .4666 .4755 .4712 .4 .3236 .3254 .3243 .3238

.5 .3380 .3333 .3473 .3405 .5 .2003 .2035 .2015 .2004

.6 .2080 .2000 .2203 .2105 .6 .1001 .1076 .1017 .1002

.7 .0793 .0667 .0942 .0809 .7 .0220 .0331 .0193 .0241

.8 0.0 0.0 0.0 0.0 .8 0.0 0.0 0.0 0.0

.9 0.0 0.0 0.0 0.0 .9 0.0 0.0 0.0 0.0

1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0


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220 CHAPTER 4

Table 4.3.2: Exact and Computed Interface Locations

For St= 0. 1 For St= 5

time Xexact M= 10 M= 20 M= 40 time Xexact M= 10 M= 20 M= 40

0 0. 0. 0. 0. 0 0. 0. 0. 0.

.5 .3111 .3089 .3101 .3108 .02 .2998 .2995 .3008 .3004

.0 .4400 .4375 .4401 .4399 .04 .4240 .4237 .4243 .4251

1.5 .5389 .5369 .5390 .5388 .06 .5193 .5227 .5197 .5206

2.0 .6223 .6207 .6218 .6225 .08 .5996 .6082 .6023 .6006

.10 .6704 .6958 .6724 .6722

Neumann solution

numerical with M = 10 nodes

Figure 4.3.1(a). Temperature history atx = . 3, withM= 10 nodes,compared with the exact solution, StSt= 0. 1 .

M= 40 nodes (Fig. 4.3.1(b), 4.3.4(b)). This is due to the fact that while the inter-face lies anywhere inside a particular mesh interval, the temperature of that inter-

val is held atTm, 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 tothe 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, usingM= 80 nodes the computed and exact solu-tions would be indistinguishable graphically. Figures 4.3.2 and 4.3.5 show that the

interface location computed with onlyM= 10 andM= 20 nodes, forStSt= 0. 1 andStSt= 5 respectively, agrees well with the exact interface. Finally, temperature pro-files with M= 10 and M= 40 nodes are shown in Figures 4.3.3(a),(b) (at timet= 2.,StSt= 0. 1) and Figures 4.3.6(a),(b) (at timet= . 1,StSt= 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.


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Neumann solution

Figure 4.3.1(b). Temperature history atx = . 3, withM= 40 nodescompared with the exact solution, StSt= 0. 1 .

Neumann solution


Figure 4.3.2. Melt front location withM= 10 nodes,compared with the exact solution, StSt= 0. 1 .


Neumann solution

Figure 4.3.3(a). Temperature profile at timet= 2., withM= 10 nodes,compared with the exact solution, StSt= 0. 1 .


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222 CHAPTER 4

Neumann solution


Figure 4.3.3(b). Temperature profile at timet= 2., withM= 40 nodes,compared with the exact solution, StSt= 0. 1 .


Neumann solution

Figure 4.3.4(a). Temperature history atx = . 3, withM= 10 nodes,compared with the exact solution, StSt= 5 .


Neumann solution

Figure 4.3.4(b). Temperature history atx = . 3, withM= 40 nodes,compared with the exact solution, StSt= 5 .


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Neumann solution


Figure 4.3.5. Melt front location withM= 20 nodes,compared with the exact solution, StSt= 5 .

Neumann solution


Figure 4.3.6(a). Temperature profile at timet= . 1, withM= 10 nodes,compared with the exact solution, StSt= 5 .

Neumann solution


Figure 4.3.6(b). Temperature profile at timet= . 1, withM= 40 nodes,compared with the exact solution, StSt= 5 .

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