^{1}

^{1}

^{*}

The present paper deals with very important practical problems of wide range of applications. The main target of the present paper is to track all moving boundaries that appear throughout the whole process when dealing with multi-moving boundary problems continuously with time up to the end of the process with high accuracy and minimum number of iterations. A new numerical iterative scheme based the boundary integral equation method is developed to track the moving boundaries as well as compute all unknowns in the problem. Three practical applications, one for vaporization and two for ablation were solved and their results were compared with finite element, heat balance integral and the source and sink results and a good agreement were obtained.

The differential equation in a certain domain satisfying some given conditions is referred to as boundary-value problem. If one or more of the boundaries are not known and moving with time, the problem then is referred to as moving boundary problem [

A semi-infinite solid initially at uniform temperature with the following constraints, there is no sub cooling, no mushy zone, solid and liquid phases have equal and constant properties and finally no convection. The mathematical formulation consists of three different stages as follow:

Heating stage

As

Liquid-solid stage

Gas-liquid-solid stage

Starting by the weighted residual statement for diffusion equation as follows:

In Equation (18)

In which,

Integrating equation (18) twice by parts leads to:

For one-dimensional problems, Equation (20) takes the following form:

In Equation (21)

Making use of Equations (22) and (23) into Equation (20), the last one takes the following form:

For any point the integral equation takes the following form:

Equation (25) after the discretization of time takes the following form:

Assume that the potential u and the flux q are constant within each time step therefore Equation (26) can be written as:

In Equation (27), we have two different time integrals, they are:

The domain integral

Now, the final form for the integral equation corresponding to the diffusion equation defined over fixed domain will be:

Similar procedure can be carried out taking into consideration the moving boundaries and their normal velocities, therefore and according to [

In Equation (33),

in our case study, this velocity may be

In the present paper, a generalized numerical algorithm and code for multi-moving boundary problem are developed, using visual Fortran 6.6. The main code consists of main program and three subroutines. The flow chart describing the main parts of the proposed algorithm is shown in

In this subroutine, a single phase bounded by two fixed boundaries are solved, using Equation (32) and the output will be the time at which melting starts and the corresponding location of the first moving boundary, separating the liquid and the solid,

1) Input data, mold length

2) Apply the boundary integral equation, given by Equation (32) at the end points of the domain of interest to

estimate

3) Check

time of melting and the corresponding position for the moving boundary separating liquid and solid. If no update both the spatial and time variables

This subroutine concerns mainly with two phases each phase will be bounded by one fixed and the second is moving are solved and the output will be the time at which vapor starts and the corresponding location of the first and second moving boundaries

1) The input unit here is the output from one-phase subroutine.

2) Solve solid phase subjected to melting temperature at the moving boundary and initial temperature at the

fixed end to get

3) By knowing

4) Check

appearing with the new position of the moving boundary separating liquid/solid and the starting position of the second moving boundary separating vapor(gas)/liquid, if no update the moving boundary separating liquid/solid.

This subroutine is for three phases, the first one is bounded by two boundaries, one fixed and one moving, the second phase is bounded by two moving boundaries, and the third phase is also bounded by two boundaries one fixed and one moving. The output of this subroutine is to track all moving boundaries, determination all unknowns in all phases up to the end of the process with minimum number of iterations and high prescribed accuracy. The flow chart of this subroutine is shown in

In this section, three test problems are solved to check the validity of the proposed algorithm and the high accuracy expected. The first example is a multi-moving boundary problem [

The solid material herein is of a low thermal conductivity and so the vaporization occurs before the moving boundary separating liquid and solid reaches the adiabatic boundary. The result due to the present algorithm is shown in

Parameter | Definition | Numerical data |
---|---|---|

Specific heat for solid | 4.944 | |

Specific heat for liquid | 4.944 | |

Thermal conductivity for solid | 0.259 | |

Thermal conductivity for liquid | 0.259 | |

Melting temperature | 1454 | |

Vaporization temperature | 3000 | |

Latent heat for melting | 2160 | |

Latent heat for vaporization | 37,200 | |

Initial temperature | 27 | |

Input heat flux at the boundary | 2500 | |

Density | 1 |

Time step | Type of time | FE | Present | Absolute error | Average number of iterations |
---|---|---|---|---|---|

0.36150 | 0.36158 | 12 - 16 | |||

1.63034 | 1.63041 | ||||

9.75464 | 9.75468 | ||||

0.32767 | 0.32771 | 20 - 24 | |||

1.63446 | 1.63449 | ||||

9.38719 | 9.38721 |

nearly the half.

A solid medium initially at uniform temperature,

This problem is for a long enough solid mold initially at a uniform temperature. The surface

The ablation thickness due to the present method compared with the corresponding from the heat balance integral method is shown in

Symbol | Physical meanning | Numerical value |
---|---|---|

Thermal diffusivity | ||

Characteristic time | ||

characteristic length | ||

Reference heat flux | ||

Temperature difference | ||

Inverse Stefan number | 1 | |

Reference time | 100 sec |

input heat flux and at the same time, the ablation thickness in case of linear heat flux is higher than that for constant case. There is a good agreement between the two methods in both cases.

The importance of the present paper comes from its dealing with practical applications of wide range of our daily life. The boundary integral equation method is not so new but it is used as a mathematical tool due to its simplicity in use. Based on this method, a generalized numerical algorithm and computer code are developed to solve such applications. It is found from the computations that the developed algorithm and the code are so simple to handle and that an acceptable accuracy is obtained. Also by decreasing the time step, and a little bit increase of iterations, the absolute errors are decreased to the half nearly. Finally, the algorithm and subsequently the code can be easily modified to cover higher dimensional problems with acceptable accuracy, which can be improved by decreasing the time step, but on the other hand, stability should be achieved.

Kawther K.Al-Swat,Said G.Ahmed, (2015) A New Iterative Method for Multi-Moving Boundary Problems Based Boundary Integral Method. Journal of Applied Mathematics and Physics,03,1126-1137. doi: 10.4236/jamp.2015.39140