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5. Numerical Results

5.1. Vaporization Test Problem

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 [13] . The thermo-physical properties are listed below in Table 1.

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 Figure 3. In this figure, both vapor/liquid and liquid/solid moving boundaries are plotted both on the same figure. The liquid/solid moving boundary is in the left side, while the other one is on the right. From this figure, one can clearly determine the melting, vapor and the time at which the process ends. These times are determined and summarized in Table 2, at two different time steps. The absolute errors between the results due to the present method compared with the finite elements method are also presented in the same table. As we see, the absolute errors decrease by decreasing the time step. On the other hand by decreasing the time step, the number of iterations increases. The number of iterations are shown in Table 2, corresponding to the two time steps used. It is clear that the increase in the number of iterations is not so much but the accuracy improved to

Figure 2. Flow chart for THREEPHASE subroutine.

Figure 3. Moving boundaries location for vaporization test problem.

Table 1. Numerical data for vaporization test problem.

Table 2. Comparison between melting, vaporization and end time.

nearly the half.

5.2. Ablation Test Problem-1

A solid medium initially at uniform temperature, , the boundary exposed to two different cases of input heat flux, constant, and linear, respectively. The domain in the present problem is still fixed while appearing two moving interfaces, solid-liquid and vapor (gas)-liquid (Ablation surface). The results due to the present algorithm are shown in Figures 4-6, respectively and the results are compared with the results due to the source and sink method [14] . Figure 4 shows the movement of solid-liquid due to constant input heat flux, and the resulting ablation surface due to the same input heat flux is shown in Figure 5. The same results due to linear case are plotted on the same plot as shown in Figure 6. From the computations and figures, it is found that the solid-liquid interface has the same behavior in both constant and linear cases of input heat flux that is concave upward. In case of linear heat flux input this concavity becomes more apparent than the constant case. In the contrary, the vapor (gas)/liquid interface behaves concave downward but in linear case this concavity increases.

5.3. Ablation Test Problem-2

This problem is for a long enough solid mold initially at a uniform temperature. The surface exposed to two different high input heat flux, constant and linear and the vapor is removed as soon as it formed- ablation problem-. Two specific heat flux boundary condition are chosen in the present computations, namely, and the following values used in the calculation [15] (Table 3).

The ablation thickness due to the present method compared with the corresponding from the heat balance integral method is shown in Figure 7. It is clear that the ablation thickness is concave downward in both case of

Figure 4. Solid/liquid due to Constant heat flux-problem-1.

Figure 5. Ablated surface due to constant heat flux-problem-1.

Figure 6. Solid/liquid and Ablated surface due to linear heat flux-problem-1.

Figure 7. Ablation thickness for ablation test problem-2.

Table 3. Thermo-physical parameters.

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.

6. Conclusion

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.

Cite this paper

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

References

  1. 1. Meaad, N.M.I. (2014) Mesh-Less Methods and Its Application to Free and Moving Boundary Problems. M.Sc. Thesis, Zagazig University, Zagazig. Supervision S. G. Ahmed.

  2. 2. Crank, J. (1984) Free and Moving Boundary Problems. Clarendon Press, Oxford.

  3. 3. Mohamd, N.A.E. (2002) Boundary Elements Methods and Its Non-Linear Analysis of Hydrofoils. Supervisors: Prof. S. G. Ahmed and Prof. A. F. Abd-Elgawad. Zagazig University, Zagazig.

  4. 4. Nofal, T.A. and Ahmed, S.G. (2015) An Integral Method to Solve Phase-Change Problems with/without Mushy Zones. Journal of Applied Mathematics and Mathematics, II, 152-157.

  5. 5. Ahmed, S.G. and Mekey, M.L. (2010) A Collocation and Cartesian Grid Methods Using New Radial Basis Function to Solve Class of Partial Differential Equations. International Journal of Computer Mathematics, 87, 1349-1362.

  6. 6. Meshrif, S.A. and Ahmed, S.G. (2007) Approximate Integral Method Applied to Ablation Problem in a Finite Slab. International Symposium on Recent Advances in Mathematics and Its Applications (ISRAMA 2007), Culcutta, 15-17 December 2007, 1-12.

  7. 7. Zerroukat, M. and Chatwin, C.R. (1994) Computational Moving and Boundary Problems. Research Studies Press Ltd., Baldock.

  8. 8. Ruan, Y. and Zabaras, N. (1991) An Inverse Finite Element Technique to Determine The change of Phase Interface Location in Two-Dimensional Melting Problem. Communications in Applied Numerical Methods, 7, 325-338.
    http://dx.doi.org/10.1002/cnm.1630070411

  9. 9. Mohamed, W.A. (2012) Advanced Mathematical Analysis for Phase Change Problems with and without Mushy Zones. Supervisors: Prof. S. G. Ahmed and Prof. M. E. Mohamed. Zagazig University, Zagazig.

  10. 10. Liu, W., Chen, Y., Jun, S., Chen, J.-S., Belytschko, T., Pan, C., Uras, R. and Chang, C. (1996) Overview and Application of the Reproducing Kernel Particle Methods. Archive of Computational Methods in Engineering: State of the Art Reviews, 3, 3-80.
    http://dx.doi.org/10.1007/BF02736130

  11. 11. Ahmed, S.G. (2005) A New Algorithm for Moving Boundary Problems Subject to Periodic Boundary Conditions. International Journal of Numerical Methods for Heat and Fluid Flow, 16, 18-27.

  12. 12. Abd-El Fatah, W.M. and Ahmed, S.G. (2006) Boundary Integral Formulation for Binary Alloys from a Cooling Solid Wall. International Journal of Computational and Applied Mathematics, 5, 687-696.

  13. 13. Ahmed, S.G. (2003) A New Algorithm for Front Tracking of Ablation Problem in Unbounded Domain. Ain Shams University, Egypt.

  14. 14. Mehdi, A. and Hsieh, C.K. (1994) Solution of Ablation and Combination of Ablation and Stefan Problems by a Source and Sink Method. Numerical Heat Transfer, Part A, 26, 67-86.
    http://dx.doi.org/10.1080/10407789408955981

  15. 15. Ahmed, S.G. (2003) A Numerical Solution for Nonlinear Heat Flow Problem Using Boundary Integral Method. 6th International Conference of Computer Methods and Experimental Measurements for Surface Treatment, Crete, March 2003.

Nomenclature

: Temperature

: Fundamental solution

: Thermal diffusivity

: Conductivity

: Density

: Input heat flux

: Liquid-solid interface

: Vapor-liquid interface

: Truncated long enough boundary length

: Latent heat

Subscript

: Solid

: Liquid

: Initial

: Vapor

: Melting

NOTES

*Corresponding author.

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