Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System

In this work we will apply the three-dimensional mathematical modelling of fluid flow and heat transfer inside the furnaces based on the cylindrical coordinate system to describe the behavior of the transport phenomena. This modelling is constructed by using the mass, momentum, and energy conservation laws to achieve the continuity equation, the Navier-Stokes equations, and the energy conservation equation. Due to the moving boundary between the solid and melted materials inside of the furnaces we will impose the Stefan condition to describe the behavior of the free boundary between two phases. We will derive the variational formulation of the system of transport phenomena, then we will discrete the domain to complete the finite element method stages and we will obtain the system of nonlinear equations in 256 equations in 256 unknowns. To get the numerical solution of the large-scale system we will prepare a convenient mathematical work and gain some diagrams where they would be applicable in the design process of the furnaces shapes.


Introduction
Mathematical modeling of heat transfer is applied to investigate the environment of the furnaces and transport phenomena inside them, because it has the advantages of low cost and acceptable exactness. Quan-Sheng and You-Lan consider the two-dimensional Stefan problem in (1985) and they used the singularity-separating method to prepare the numerical solution. Ungan and Viskanta in (1986), (1987) investigated modeling of circulation and heat transfer in a glass melting tank in three-dimension. Henry and Stavros in (1996) prepared the convenient work about the mathematical modelling of solidification and melting process. Vuik, Segal, and Vermolen in (2000) studied about the discretization approach for a Stefan problem where they focused on interface reaction at the free boundary. Pilon, Zhao, and Viskanta in (2002), (2006) in their papers considered three-dimensional flow and researched about the behavior of thermal structures in glass melting furnaces by using the three-dimensional mathematical modeling. Their works were included theoretical and numerical sections where they applied sufficient boundary conditions. Sadov, Shivakumar, Firsov, Lui, and Thulasiram, provided an article about the mathematical model of ice melting on transmission lines in (2007). Choudhary, Venuturumilli, and Matthew in (2010) in their common paper introduce the mathematical modeling of flow and heat transfer phenomena in glass melting where it consists of delivery and forming processes, especially the turbulent conditions has been discussed in the paper for Newtonian and non-Newtonian fluids. Kambourova, and Zheleva had modelized and described temperature distributions ina tank of glass melting furnace in (2002).
The author studied the mathematical modeling of heat transfer and transport phenomena in (2016) in two-dimension with Stefan free boundary based on stream functions. due to the advantages of stream functions mathematical modeling has prepared and by invoking the finite element method the numerical solution of the transport phenomena derived, also we did the same work in three-dimension. In the current work we will apply the mathematical modeling in threedimension for the especial furnaces with cylindrical shapes, that they are called Garnissage tank. Due to their shapes we will illustrate the mathematical equations of transport phenomena in the cylindrical coordinate system in threedimension to use its symmetric properties.
The melting process in the Garnissage furnace starts when we impose the heat to the solid material by electrical boosters inside it, but the heat transferring by electrical boosters couldn't melt the whole materials and whenever the central parts are melted the parts far from the boosters is solid. In this case the boundary between the solid and liquid materials are moving during the process, then we will have free boundary problem. The Stefan condition is sufficient tool to describe the behavior of the free boundary and we will handle three-dimensional version of the Stefan condition in the mathematical modeling process.
We state the conservation equations in the cylindrical coordinate, so the Stefan condition, then we convert the system of equations into the weak formulation. For expressing the system in the variational formulation we will use convenient test functions with small support and then we will discrete the domain to follow the finite element method.
The approximate values of the variables would be applied instead of the variables, and finally we will get the system of equations that they are combination of linear and nonlinear equations, thus to solve the system numerically the Newton's method would be recommended.

Mathematical Modeling
We start the paper by reviewing the mathematical modeling of fluid flow in three-dimension. Cylindrical coordinate is applied for modeling because it provides the convenient environment to express the modeling, hence we can use the symmetric properties to introduce the equations. There are three different parts in the modeling process which we will introduce them.

Energy Equation
Deriving the weak formulation is second stage of the work after the mathematical modeling, then the variational version of the modeling has been obtained by imposing the convenient smooth test function with small support and using the integration by parts technique. Like the classical modeling the variational formulation has three sections:

Discretization of Domain
After weak formulation we discrete the domain Ω , and replace the approximate variables ( , ( , " ( , and # ( in the variational formulation to obtain the algebraic system of equations. Assume that * ( ⊂ , -Ω , which consists of test functions that they are piecewise continuous with the fix degree, and also suppose that where E = , E = , G = , # = , A = 1,2,3, … , 2 ℎ , must be determined. We insert the values of ( , ( , " ( , # ( ∈ * ( instead of , , ", #in the variational formulation of the heat transfer problem. In this case the heat transfer problem is restated as the problem of finding where they are satisfied in the system of equations which coefficients of the system are defined as Table 1. System Coefficients.
We need to determine the values of the coefficients to derive the numerical solution of system, then we must define the sufficient domain and discrete it to compute the relevant integrals, and at last we would found all coefficients and finally we will earn the numerical solution of the system for the interpretation of the fluid behavior in the furnaces.

Unknown Coefficients
We continue the process by computing the coefficients in the transport system, for this objective we need to divide the domain Ω to the mesh cubes, the Figure 1 shows the Ai`−mesh cube where we will construct the test functions on Ai`−mesh cube. To define the test functions we divide the Ai`−mesh cube into the 24 tetrahedrons as Figure 2. Now we are looking for the continuous piecewise linear function 8 = 8 j, k, @ where it equals to 1 in the origin node ofAi` mesh cube, and 0 in the other nodes.
Suppose that 8 j, k, @ 5 Vj fk l@ for example to the point j, k, ? in the first tetrahedron 8 mSj = , k X , @ U 1 8 WSj =n , k Xo , @ U 0 8 YSj =n , k X , @ U 0 8 ZSj = , k X , @ n U 0 then we will have We get the solution of the linear system as then we derive the test function 8 j, k, @ 1 3 j j = @ @ 1 We perform the same approach for all of the other tetrahedrons and we demonstrate them as Parts 1, 2 Parts 5, 6 8 j, k, @ 1 3 S k k X @ @ U 1 We are in the position to compute the coefficients of the continuity equation 5 = and V = for the system (13) We compute the values of integrals in different tetrahedrons separately and derive the final value of 5 == as j = + ℎ 6j = ! − 8j = j = + ℎ ! + 3 j = + ℎ 9 and by the similar operation we gain 5 =,=oS; n;o U = 1 72ℎ 12j = 9 − 6j = ! ℎ + 40j = ℎ ! − 3ℎ 9 − 1 6ℎ ! j = v + 3j = ! ℎ ! + 2j = ℎ 9 − 2ℎ v ln b1 + ℎ j = d, 5 =,=nS; n;o U = 1 72ℎ 12j = 9 + 6j = ! ℎ + 40j = ℎ ! + 3ℎ 9 + 1 Also for the coefficient V = in (13)  after inserting the values in the system (13) we will reach the coefficients matrix G, ƒ with 64 rows and 128 columns, where Now we have obtained a linear system of equations that it has 128 variables E L , E , … , E OE• ; E L , E , … , E OE• within 64 equations. We will return to the system (13) in the next sections again, thus we focus on the Navier-Stokes system (14) to earn the remainder coefficients.
We continue the process by determining the coefficients of the Navier-Stokes system (14), (15). We start by specifying the coefficient W = , and as we done before we suppose the Ai`−mesh cube that it was divided to 24 tetrahedrons, then after integration process in the whole parts we gain  (14) and we earn the coefficient matrix We continue the process of finding the coefficients of the Navier-Stokes system (14)   then we will get the coefficient matrix We note that the last coefficient \ = in the system (14) is Now we reach to the nonlinear system in 64 variables E L , E , … , E OE• in 64 equations, then we repeat precisely the same process for the second part of the Navier-Stokes equations (15), and again we get the second nonlinear system, that it has 64 variables E L , E , … , E OE• within 64 equations.

Numerical Solution
As we have shown in the section (4) equation (13) may be exhibited as a linear system in 64 equations in 128 variables, where we got the coefficients matrix G, ƒ Also we compute the _ =X by applying the Mathematica codes, and at last we derive # as  We enter the values of # into the relation (12) to simulate the function # ( , and we will show the result in the Figure 5.
Numerical solution of the transport phenomena and its mathematical simulation, where they have been gotten in the current work, are suitable tools to describe the environment of furnaces and fluid flow inside them. Also they help to designers to determine the optimal position of the electrical boosters. In particular designers analyze the mathematical simulation to decide about the optimized style of the furnaces, then we hope this work is useful for them.

Summary and Conclusion
In this work we performed the process to get the mathematical modeling of heat transfer in the Garnissage furnace in three dimension in the cylindrical coordinate system. The cylindrical coordinate system has chosen for the modeling process because of its symmetric advantages, then we applied the physical conservation laws, that is the mass, momentum, and energy conservation laws, to achieve the continuity, Navier-Stokes, and heat equation. To modeling the free boundary between the solid and liquid phase we used the three dimensional version of Stefan condition.
When we derived the mathematical modeling of the transport phenomena immediately we started to rearrange the modeling to the weak formulation by handling the sufficient test functions. In this part we divided the domain by cubes and every cubic parts had 24 tetrahedrons. Then we defined the test functions on the tetrahedrons and we got the coefficients after solving the integrals on the mentioned tetrahedrons, and we completed the finite element technic by constructing the system of equations with 128 variables within 64 linear equations and 64 nonlinear equations. Newton's method has been used to achieve the numerical solution of the system and we simulated the numerical solution of the heat transfer system where that was applicable for furnace designers.