All of the foregoing solution schemes for the Stefan problem given by (8), (9), (10) and (11) work wholly in terms of the temperature which necessitates explicitly tracking the location of the liquid/solid interface. In terms of difference computations this either requires a modified difference replacement near the interface (see the "direct" implicit formulation) or a more complicated set of equations are produced (see the variable grid, boundary immobilization and isotherm migration method). We now give an account of a novel application of a technique which has become known as the enthalpy
- 17
method. Some' of the work described in this section has appeared in the International Journal for Numerical Methods in Engineering [Wood, Ritchie and Bell (1981)].
It is convenient to define a total heat or "enthalpy"' function H(,T) [Rose (1960), Szekely and Themelis (1971), Atthey (1974)] which is a function of temperature and combines the sensible and latent heats of the material. The requirement of working entirely in terms of the temperature is removed as is the necessity of explicitly tracking the interface. As a result of this, standard finite difference schemes may be applied over the whole solution region with due regard being taken of any stability requirements associated with a particular scheme. By considering the definition of the enthalpy function H(T) it is a straightforward exercise to extract the interface location from the solution at any time. The flexibility of the method has been demonstrated by its simple application to a problem involving a "mushy" region in which the material melts or freezes over a temperature range [Voiler and Cross (1981)].
For the model problem previously defined and following Atthey (1974) we have the following definition for the enthalpy function.
, T = Tj T
H = I c d u + L , T > T .
^f r ^f r
where H = I c du, H = 1 c du + L and c is the specific heat
O'/ ^ 0
capacity of the material. These relations may be re-arranged to yield
T = (H - L) , H > cT. + L
Utilizing the non-dimensional change of variable E = H - cT^
with those given by equations (7), then (32) becomes
U = 0.0 , 0 E<C % - (33) U = E - Y , E >
where the latent heat parameter P( is given by (12). The non-dimensional enthalpy formulation is then (P( = 1.0)
^ = ^fu , 0 < X < X , f > 0.0 - (34) a t 3*2
U(x,0) = 0.0 , X > 0.0, t = 0.0
U(0,1) = 1.0 , X = 0.0, t > 0.0 - (35) U(x^,X) - 0.0 , X = x^, t > 0.0
with E(U) and U(x,4) related by (33). Standard difference schemes may now be applied to the whole solution region.
Despite the computational simplicity of explicit difference schemes a drawback when solving, for example, the heat conduction
2
problem is the stability restriction (r (=A//(Ax) ) 1/2 for the one-dimensional solution) which limits the size of the time step for a given mesh size. With Implicit difference replacements this restriction is theoretically lifted. However, it then becomes necessary to solve a system of equations at each time step, A method
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which combines both the computational simplicity of the explicit scheme and the large parameter range of the Implicit scheme was proposed by Gourlay (1970)» This scheme is given the name "Hopscotch" because of the way in which the solution progresses through the
space-time phase plane.
The odd-even hopscotch algor it Im combines explicit and implicit finite difference approximations at alternate mesh points in such a way that the overall scheme does not require the solution of a system of equations, that is it is overall explicit.
We construct a uniform grid (of N points) of mesh size Ax over the region 0 x x^. The explicit odd-even difference replacement for equation (34) is
e“+^ = e“ + - 2tl^ + U^_j) , 0,1,2,... -(3 6)
for points where m + i is EVEN and where E® = E(l6x,mAt) and - U(iAx,mAA). When m + i is ODD we use a three point implicit difference replacement which is infact explicit in the sense that the temperatures at the nodes (i+l)Ax and (i-l)Ax at the new time step have already been calculated using the explicit replacement (36). The following procedure is adopted. Initially a value E^^^ is calculated from
-m+1 = E* + ^ ym+l^ , m = 0,1,2,... - (37)
If E™”^^ x<: K then v< ^ and hence, from (33), = 0.0. If however
> K
it may be deduced that E™^ ^ > P( and+ 2roO
, m =0
,1
,2
,... - (38)(l+2r)
From (33) the corresponding temperature is given by jjm+l ^ gm+1 _ ^ ^ m = 0,1,2,
Equations (37) and (38) may be seen to be equivalent to one application of the implicit difference replacement (^ = 1.0)
gm+1 = gm + + U®^|) , m = 0,1,2,...
The odd-even scheme is particularly sensitive to discontinuous initial data [Bell and Ritchie (1980)] and so the numerical algorithn is initialised at t = 0.5 using the analytic solution. The initial enthalpy distribution may be determined from application of (33). To estimate the initial value of E(U) at the point(s) adjacent to the interface we suppose that each grid point is located at the right hand side of a segment of length Ax. The starting value of E(U) at this point is given as a fraction of the segment that is liquefied at the starting time. The results, discussed in section 1.4, are presented (for the case ^ = 1.0) at 1 = 1.0 for r = 0.4 (tables 1.6(a) and 1.6(b)) and for r = 2.0 (tables 1.7(a) and 1.7(b)). For both cases it is evident that the algorithm converges as the difference mesh is refined (see appendix 1.5 for the FORTRAN code).
From the enthalpy solution obtained it is possible to determine the location of the interface by reversing the procedure described earlier for estimating E(U) in the solid region. The only non-zero value of E^ in the solid region is that which is adjacent to the
— 21 —
Table 1.6(a)
Enthalpy Formulation ("RIGHT" definition) : Values of the temperature distribution as predicted by the numerical and analytic solutions at
X ~ 1.0 for r = 0.4
Explicit Hopscotch technique
method Analytic
A x = 0 .05 X Ax=0.05 Ax=0.025 A x = 0 .0125 solution
1.0
0.0
1.0
1.0
1.0
1.0
0.909990 0.1 0.910050 0.909516 0.909261 0.909998 0.820413 0.2 0.820533 0.818479 0.818972 0.818450 0.731695 0.3 0.731879 0.730331 0.729578 0.728802 0.644259 0.4 0.644514 0.642502 0.641508 0.640489 0.558527 0.5 0.558864 0.556396 0.555174 0.553293 0 . 474928 0.6 0 . 4753 5 5 0 . 472386 0 . 470961 0 . 469 494 0.393893 0.7 0.394409 0.390804 0.389225 0.387560 0.315808 0.8 0.316390 0.311970 0.310295 0.308442 0.240923 0.9 0.241517 0.236284 0.234462 0.232425 0.169213 1.0 0.169738 0.164269 0.161880 0.159752 0.100258 1.1 0.100622 0.096255 0.092341 0.090621 0.033194 1.2 0.033325 0.031630 0.026596 0.025189 Table 1.6(b)Predicted location of the interface at
t -
1.0 (Hopscotch) A x S(f) % Error 0.05 0.025 0.0125 1.230576 1.234900 1.237786 -0.7700 -0.4213 -0.1886 Analytic Solution 1.240125Table 1.7(a)
Enthalpy Formulation ("RIGHT" definition) ; Values of the temperature distribution as predicted by the numerical and analytic solutions at
t -
1.0 for r ~ 2.0Explicit Hopscotch technique
method Analytic
6x=0.05 X Ax=0.05 Ax=0.025 Ax=0.0125 solution
0.0
1.0
1.0
1.0
1.0
0.1 0.911779 0.910648 0.909686 0.908998 0.2 0.824173 0.821750 0.819826 0.818450 0.3 0.737687 0.733756 .0.730869 0.728802 0.4 0.652361 0.647106 0.643247 0.640489 Method 0.5 0.568777 0.562222 0.557376 0.553923 Unstable 0.6 0.487106 0.479485 0.473642 0.469494 0.7 0.407489 0.399209 0.392400 0.387560 0.8 0.329939 0.321584 0.313975 0.308442 0.9 0.254316 0.246622 0.238697 0.232 425 1.0 0.185876 0.176137 0.166838 0.159752 1.1 0.124912 0.109440 0.098347 0.090621 1.2 0.061244 0.046207 0.032452 0.025189 Table 1.7(b)Predicted location of the interface at
i. -
1.0 (Hopscotch)A X S(t) % Error 0.05 0.025 0.0125 1.265715 1.259068 1.247850 2.0635 1.5275 0.6229 Analytic Solution 1.240125
23 -
interface. This value represents the fraction of the segment of size
hx.
that is liquefied. The estimates produced in this way are presented in tables 1.6(b) and 1.7(b) at A - 1.0 for r = 0.4 and r = 2.0 respectively.An alternative algorithm for estimating the enthalpy is given by Crank (1975)2 in which each grid point is considered to be at the centre of a segment of length Ax. Thus, for the point x “ iAx, the interface moves through values from (i-0.5)Ax to (i+0.5)Ax as the enthalpy moves through the values 0.0 to 1.0. The results, discussed in section 1.4, are presented in tables 1.8(a) and 1.8(b) and tables 1.9(a) and 1.9(b) respectively for the above values of the parameters and r. Again convergence is evident.