Claims:We claim the following from our invention,

Claim:
1. A system and method was developed for monitoring temperature profile of slab and sphere.
a) For a given Initial temperature distribution and diffusion coefficient in the original phase, time taken for phase change at a given point increases with β.
b) For a given β and k, time taken to reach a particular temperature at any given point increases with Initial temperature distribution.
c) For a given β and V, cooling is faster in the second phase (Initial phase) for higher diffusion coefficient in the original phase.
d) Communication, critical thinking, collaboration and creativity stages occupies the higher level in the model which requires higher order teaching skills (HOTS)
2. As mentioned in claim 1, higher β implies longer time for phase change at any point and time taken for phase change does not depend on diffusion coefficient in the original phase
3. According to claim 1, in the enthalpy formulation, after each time step spatial grid conversion of enthalpy to temperature is required. , Description:Field of Invention
The present invention relates to, heat flow with phase change, for monitoring the temperature profile of slab and sphere. The invention also relates to keep track of shape of the knee at the phase change temperature.
Background of the invention
Temperatures in the two phases are u and v at any time and at any point, is known as interface (front) of the two phases. The basic difficulty to solve one dimensional two phase problems is there is no domain for the space variable. Very few methods have been developed for the two phase problem. Series expansion of the variables gives only the qualitative behavior of the solution (Scott W Mccue et al. [2001], proc. Royal Soc. A, 464, p 2055-2076). The integral transform method was used to obtain the interface as an integral (Dewynne J.N. et al. [1986], Acta Mechanica, 58, p 201-228).
The other method is with enthalpy formulation. This method has a serious limitation of ignoring the Stefan condition at the point of phase change and gives a discontinuity in the enthalpy (V. Voller et al. [1980], Int. J. Heat Mass Transfer, 24, p 545-556). To obtain smooth solution, the latent heat is distributed heuristically into neighboring points. As step sizes h and are constant, the points of the front are obtained (Douglas Jr Jim et al. [1955], Duke Math. J., 22, p 557-571). Further, this method cannot be used if the temperature of the phase change is not a constant but a known function such as (Crank. J.[1984], Oxford Science Publications, p 252-253 ).
One of the recent patent work focuses on two-phase heat transfer loop applicable in spacecraft electronics thermal management (US20160047605A1). A Moving boundary as vapor permeates through the substrate is another active area of recent patents (US4195055). The general principle of compressed-air energy storage is the theme for number of patents (US20140000251A1). Finally increasing the efficiency of energy storage through control system or by changing the chemical or physical property of heat-exchange liquid is discussed many US patents (US20110296822A1).
In this invention, slab problem as well as sphere problem solved for various parametric combinations. Package developed for the two phase problem can be used for the one phase problem by taking . The only results available to us on a freezing of a one phase sphere (Davis G.B et al. [1982], IMAJ. Appl. Math., 29, p 99-111). Research findings on one dimensional one phase, two phase and multi phase moving boundary problems have received considerable importance as patent because of its applications in industrial processes, such as solidification of steel and chemical reactions, metal processing, solidification of castings, environmental engineering, thermal energy storage system in a space station, melting of ice, growth of metallic crystals, feezing and thawing of foods, ice formation on pipe surface, and medical science; where particular cancer cells may be destroyed under extremely cold temperatures (cryosurgery).
The objective of this invention is to develop the front tracking method to determine the points on the moving boundary along with temperature profiles and solution to the two phase Moving boundary problem with quadratic polynomial approximation to the front.
Summary of the invention
All the other methods assume the initial temperature of the material as a constant. In the present innovative invention, realistic profile was considered. The nonlinear condition on the interface can be easily handled by our method as bisection does work for nonlinear function as well.
In the enthalpy formulation, after each time step at every point of the spatial grid conversion of enthalpy to temperature done. Stefan condition is derived using the latent heat of phase change and whereas Front Tracking Method is built with the methodology in a natural way.
Brief description of Drawing
In the figures which are illustrate exemplary embodiments of the invention.
Figure 1 Moving boundary with fixed space step and variable time step
Figure 2 Temperature profile of (a) slab and (b) sphere at x = 0.25, 0.50 and 0.75
Detailed description of the invention
In the prior art proposed methods may solve very limited class of problems. The method developed in our invention covers a wide range of variations, like the presence of heat source (or sink) or the phase change temperature is a function of the position on the front. The two phase sphere problem can be handled by the same methodology as for the slab with minor changes. Shape of the knee at the phase change temperature could be discussed because of the ease with which problem could be solved for various physical properties combinations.
The effect of discontinuity in the initial and boundary data of the parabolic problems (Pearson C.E. [1965], Math. Comp. 19, p 572-576), shows that whatever space and time step sizes are taken, the numerical solution is not very accurate. Indeed, difference between the solutions obtained with two different step sizes monotonically reduces and goes to zero. Discontinuity is a common phenomenon in Stefan problems and our computational experiments confirmed the same.
By applying Green’s theorem for the equation over the region

where C is the closed curve enclosing the said region.

The first two integrals are evaluated using Trapezoidal rule. Applying Green’s theorem over the region

Using of Crank-Nicholson scheme is not possible at the first interior point G, as a point outside the domain of occurs. To enable us to use this stable and more accurate scheme, we first write down the fully implicit scheme at G giving us

Few observations on the behavior of profile against the parameters β, k and V are shown in the figure 1 and 2. For given V (Initial temperature distribution) and k (diffusion coefficient in the original phase), profile for a given x is higher than the profile for a lower x for all values of Stefan number. Accordingly, the time taken for phase change at a given point increases with β. This is as expected since higher β implies longer time for phase change at any point. For given β and k, the time taken to reach a particular temperature at any given point increases with V. This phenomenon is natural, as higher initial temperature implies they need for longer time for cooling. For given β and V, the profiles do not change significantly for any given ‘x’ up to the point of phase change. Cooling is faster in the second phase (Initial phase) for higher k. Thus, Knee gets sharper. This also means the time taken for phase change does not depend on k significantly. Knee gets flatter and flatter in the case of profiles with increasing x. For small values of β, the duration of phase change is short. Thus, for small values of parameters one needs to take much smaller h (large N) to observe the knee as it gets bypassed.
3 Claims & 2 Figures