Page 238 - Introduction to Computational Fluid Dynamics
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P1: IBE
                           CB908/Date
            0521853265c07
                        7.2 1D PROBLEMS FOR PURE SUBSTANCES
                        we must have    0 521 85326 5                              May 25, 2005  11:14 217
                                                   √                   √
                                              X i ∝  t    or    X i = C  t,                 (7.7)
                        where C can be determined from the interface condition (7.2). The transcendental
                        equation for determination of C thus becomes
                                      ρλC        T m − T w   K s          2
                                            =        √     √      exp (−C /4α s )
                                        2     erf(C/ 4α s )  πα s

                                                  T m − T sup  K l          2
                                              +        √      √     exp (−C /4α l ).        (7.8)
                                                erfc(C/ 4α l )  πα l
                           This transcendental equation shows that C = C (T m − T w , T m − T sup , K s , K l ,
                        α s ,α l ). Thus, C will be different for each initial and boundary condition and for each
                        specification of physical properties. The value of C must be iteratively determined
                        to calculate dX i (t)/dt from Equation 7.7 and hence to calculate the temperature as
                        a function of x and t from Equations 7.5 and 7.6. It can be shown that the system is
                        governed by a dimensionless number, called the Stefan number, which is defined as

                                                        C ps (T m − T w )
                                                   St =              .                      (7.9)
                                                              λ
                        The larger the value of St, the faster is the interface movement. A further point to
                        note is that, although the temperature profiles show discontinuity at the interface,
                        they are smooth within each phase and the variation of T with t at any x is also
                        continuous and smooth.

                        7.2.2 Simple Numerical Solution

                        It might appear that it is a straightforward matter to discretise Equation 7.4 to ob-
                        tain a numerical solution. However, there is a difficulty associated with predicting
                        continuous temperature histories when a numerical solution is obtained. To ap-
                        preciate the difficulty, we assume uniform and equal properties for both phases
                        (i.e., ρ s = ρ l = ρ, C ps = C pl = C p , and K s = K l = K). Thus, Equation 7.4 can be
                        written as
                                                               2
                                                       ∂     ∂ θ
                                                           =     ,                         (7.10)
                                                       ∂τ    ∂ X 2
                        where
                                           h − h s
                                        =          (dimensionless enthalpy),               (7.11)
                                              λ
                                           C p (T − T m )
                                       θ =               (dimensionless temperature),      (7.12)
                                                λ
                                           α t
                                       τ =      (dimensionless time),                      (7.13)
                                            L 2
                                            x
                                       X =     (dimensionless length).                     (7.14)
                                           L
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