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Graphical methods for 2 d heat transfer

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Graphical methods for 2 d heat transfer

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Graphical methods for 2 d heat transfer

  1. 1. NUMERICAL METHODS FOR 2-D HEAT TRANSFER
  2. 2. • Due to the increasing complexities encountered in the development of modern technology, analytical solutions usually are not available. • For these problems, numerical solutions obtained using high-speed computer are very useful, especially when the geometry of the object of interest is irregular, or the boundary conditions are nonlinear.
  3. 3.  Numerical methods are necessary to solve many practical problems in heat conduction that involve: – complex 2D and 3D geometries – complex boundary conditions – variable properties  An appropriate numerical method can produce a useful approximate solution to the temperature field T(x,y,z,t); the method must be – sufficiently accurate – stable – computationally efficient
  4. 4. General Features      A numerical method involves a discretization process, where the solution domain is divided into subdomains and nodes The PDE that describes heat conduction is replaced by a system of algebraic equations, one for each subdomain in terms of nodal temperatures A solution to the system of algebraic equations almost always requires the use of a computer As the number of nodes (or subdomains) increase, the numerical solution should approach the exact solution Numerical methods introduce error and the possibility of solution instability
  5. 5. Types of Numerical Methods 1. The Finite Difference Method (FDM) – subdomains are rectangular and nodes form a regular grid network – nodal values of temperature constitute the numerical solution; no interpolation functions are included – discretization equations can be derived from Taylor series expansions or from a control volume approach
  6. 6. 2. The Finite Element Method (FEM) – subdomain may be any polygon shape, even with curved sides; nodes can be placed anywhere in subdomain – numerical solution is written as a finite series sum of interpolation functions, which may be linear, quadratic, cubic, etc. – solution provides nodal temperatures and interpolation functions for each subdomain
  7. 7. • In heat transfer problems, the finite difference method is used more often and will be discussed here. • The finite difference method involves:  Establish nodal networks  Derive finite difference approximations for the governing equation at both interior and exterior nodal points  Develop a system of simultaneous algebraic nodal equations  Solve the system of equations using numerical schemes
  8. 8. The Nodal Networks The basic idea is to subdivide the area of interest into sub-volumes with the distance between adjacent nodes by Dx and Dy as shown. If the distance between points is small enough, the differential equation can be approximated locally by a set of finite difference equations. Each node now represents a small region where the nodal temperature is a measure of the average temperature of the region. Example: Dx m,n+1 m-1,n m,n m+1, n Dy m,n-1 x=mDx, y=nDy m+½,n m-½,n intermediate points
  9. 9. Finite Difference Approximation q 1 T Heat Diffusion Equation:  T   , k  t k where  = is the thermal diffusivity  C PV 2  No generation and steady state: q=0 and  0,  2T  0 t First, approximated the first order differentiation at intermediate points (m+1/2,n) & (m-1/2,n) T DT  x ( m 1/ 2,n ) Dx T DT  x ( m 1/ 2,n ) Dx ( m 1/ 2,n ) Tm 1,n  Tm ,n  Dx ( m 1/ 2,n ) Tm ,n  Tm 1,n  Dx
  10. 10. Finite Difference Approximation (cont.) Next, approximate the second order differentiation at m,n  2T x 2  2T x 2  m ,n m ,n T / x m 1/ 2,n  T / x m 1/ 2,n Dx Tm 1,n  Tm 1,n  2Tm ,n  ( Dx ) 2 Similarly, the approximation can be applied to the other dimension y  2T y 2 m ,n Tm ,n 1  Tm ,n 1  2Tm ,n  ( Dy ) 2
  11. 11. Finite Difference Approximation (cont.) Tm 1,n  Tm 1,n  2Tm ,n Tm ,n 1  Tm ,n 1  2Tm ,n   2T  2T    x 2  y 2   2 ( Dx ) ( Dy ) 2   m ,n To model the steady state, no generation heat equation: 2T  0 This approximation can be simplified by specify Dx=Dy and the nodal equation can be obtained as Tm 1,n  Tm 1,n  Tm ,n 1  Tm ,n 1  4Tm ,n  0 This equation approximates the nodal temperature distribution based on the heat equation. This approximation is improved when the distance between the adjacent nodal points is decreased: DT T DT T Since lim( Dx  0)  ,lim( Dy  0)  Dx x Dy y
  12. 12. A System of Algebraic Equations • The nodal equations derived previously are valid for all interior points satisfying the steady state, no generation heat equation. For each node, there is one such equation. For example: for nodal point m=3, n=4, the equation is T2,4 + T4,4 + T3,3 + T3,5 - 4T3,4 =0 T3,4=(1/4)(T2,4 + T4,4 + T3,3 + T3,5) • Derive one equation for each nodal point (including both interior and exterior points) in the system of interest. The result is a system of N algebraic equations for a total of N nodal points.
  13. 13. Matrix Form The system of equations: a11T1  a12T2   a1N TN  C1 a21T1  a22T2   a2 N TN  C2 a N 1T1  a N 2T2   a NN TN  CN A total of N algebraic equations for the N nodal points and the system can be expressed as a matrix formulation: [A][T]=[C]  a11 a12 a a22 21 where A=    aN 1 aN 2 a1N   T1   C1  T  C  a2 N   , T   2  ,C   2            aNN  TN  C N 
  14. 14. Numerical Solutions Matrix form: [A][T]=[C]. From linear algebra: [A]-1[A][T]=[A]-1[C], [T]=[A]-1[C] where [A]-1 is the inverse of matrix [A]. [T] is the solution vector. • Matrix inversion requires cumbersome numerical computations and is not efficient if the order of the matrix is high (>10). • For high order matrix, iterative methods are usually more efficient. The famous Jacobi & Gauss-Seidel iteration methods will be introduced in the following.
  15. 15. Iteration General algebraic equation for nodal point: i 1 a T j 1 ij j  aiiTi  N aT j i 1 ij j  Ci , (Example : a31T1  a32T2  a33T3   a1N TN  C1 , i  3) Rewrite the equation of the form: N aij ( k 1) Ci i 1 aij ( k ) (k ) Ti    T j   T j aii j 1 aii j i 1 aii Replace (k) by (k-1) for the Jacobi iteration • (k) - specify the level of the iteration, (k-1) means the present level and (k) represents the new level. • An initial guess (k=0) is needed to start the iteration. • By substituting iterated values at (k-1) into the equation, the new values at iteration (k) can be estimated • The iteration will be stopped when maxTi(k)-Ti(k-1), where  specifies a predetermined value of acceptable error
  16. 16. CASE STUDY Finite Volume Method Analysis of Heat Transfer in Multi-Block Grid During Solidification Eliseu Monteiro1, Regina Almeida2 and Abel Rouboa3 1CITAB/UTAD - Engineering Department of University of Tr´as-os-Montes e Alto Douro, Vila Real 2CIDMA/UA - Mathematical Department of University of Tr´as-os-Montes e Alto Douro, Vila Real 3CITAB/UTAD - Department of Mechanical Engineering and Applied Mechanics of University of Pennsylvania, Philadelphia, PA 1,2Portugal 3USA
  17. 17. The governing differential equation for the solidification problem may be written in the following conservative form ∂ (ρCPφ) ∂t = ∇· (k∇φ) + q˙ (1) where ∂(ρCPφ) ∂t represents the transient contribution to the conservative energy equation (φ temperature); ∇· (k∇φ) is the diffusive contribution to the energy equation and q˙ represents the energy released during the phase change.
  18. 18. undary conditions
  19. 19. Numerical solution method • Finite volume method
  20. 20. Finite difference method
  21. 21. Results and discussion
  22. 22. • Iterative performance of the three different numerical methods are also given.
  23. 23. Concluding remarks A multi-block grid generated by bilinear interpolation was successfully applied in combination with a generalized curvilinear coordinates system to a complex geometry in a casting solidification scenario. To model the phase change a simplified two dimensional mathematical model was used based on the energy differential equation. Two discretization methods: finite differences and finite volume were applied in order to determine, by comparison with experimental measurements, which works better in these conditions. For this reason a coarse grid was used. A good agreement between both discretization methods was obtained with a slight advantage for the finite volume method. This could be explained due to the use of more information by the finite volume method to compute each temperature value than the finite differences method. The multiblock grid in combination with a generalized curvilinear coordinates system has considerably advantages such as:
  24. 24. • better capacity to describe the contours through a lesser number of elements, which considerably reduces the computational time; • - any physical feature of the cast part or mold can be straightforwardly defined and obtained in a specific zone of the domain; • - the difficulty of the several virtual interfaces created by the geometry division are easily overcome by the continuity condition
  25. 25. THANK YOU

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