HEAT TRANSFER FINITE ELEMENT FORMULATION VIJAYAVITHAL BONGALE DEPARTMENT OF MECHANICAL ENGINEERING MALNAD COLLEGE OF ENGINEERING HASSAN - 573 202. Mobile : 9448821954 17/08/2014 N I T CALICUT

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What is Finite Element Method? FEM is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. 17/08/2014 N I T CALICUT

Applications of FEM: Equilibrium problems or time independent problems. e.g. i) To find displacement distribution and stress distribution for a mechanical or thermal loading in solid mechanics. ii) To find pressure, velocity, temperature, and density distributions of equilibrium problems in fluid mechanics. Eigenvalue problems of solid and fluid mechanics. e.g. i) Determination of natural frequencies and modes of vibration of solids and fluids. ii) Stability of structures and the stability of laminar flows. 3.Time-dependent or propagation problems of continuum mechanics. e.g. This category is composed of the problems that results when the time dimension is added to the problems of the first two categories. 17/08/2014 N I T CALICUT

Similarities that exists between various types of engineering problems: 1. Solid Bar under Axial Load 17/08/2014 N I T CALICUT

2. One – dimensional Heat Transfer 3. One dimensional fluid flow 17/08/2014 N I T CALICUT

How the Finite Element Method Works Discretize the continuum: Divide the continuum or solution region into elements. Select interpolation functions: Assign nodes to each element and then choose the interpolation function to represent the variation of the field variable over the element. Find the Element Properties: Determine the matrix equations expressing the properties of the individual elements. For this one of the three approaches can be used. i) The direct approach ii) The variational approach or iii) the weighted residuals approach. 17/08/2014 N I T CALICUT

Assemble the Element Properties to Obtain the System equations: Combine the matrix equations expressing the behavior of the elements and form the matrix equations expressing the behavior of the entire system. Impose the Boundary Conditions: Before the system equations are ready for solution they must be modified to account for the boundary conditions of the problem. Solve the System Equations: Solve the System Equations to obtain the unknown nodal values like displacement, temperature etc. Make Additional Computations If Desired: From displacements calculate element strains and stresses, from temperatures calculate heat fluxes if required. 17/08/2014 N I T CALICUT

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General Steps to be followed while solving a problem on heat transfer by using FEM: Discretize and select the element type Choose a temperature function Define the temperature gradient / temperature and heat flux/ temperature gradient relationships Derive the element conduction matrix and equations by using either variational approach or by using Galerkin’s approach Assemble the element equations to obtain the global equations and introduce boundary conditions Solve for the nodal temperatures Solve for the element temperature gradients and heat fluxes 17/08/2014 N I T CALICUT

ONE DIMENSIONAL FINITE ELEMENT FORMULATION USING VARIATIONAL APPROACH Step 1.Select element type. Step 2.Choose a temperature function. t1 t2 1 x One –D element L 2 t1 t2 T = N1t1+ N2 t2 1 2 Temperature Variation along the length of element 17/08/2014 N I T CALICUT

We choose, T (x) = N1 t1 + N2 t2 --------------------------- (1) N1 & N2 are shape functions given by, In matrix form and 17/08/2014 N I T CALICUT

Temperature gradient matrix is given by, Step 3. Define the temperature gradient / temperature and heat flux/ temperature gradient relationships Temperature gradient matrix is given by, Where matrix B is, Where matrix B is, The heat flux /temperature gradient relationship is, The heat flux /temperature gradient relationship is, 17/08/2014 N I T CALICUT

The material property matrix is, Step 4. Derive the element conduction matrix and equations Consider the following equations q x = 0 L T=TB S1 Insulated L T=TB S1 q x = + q*x S2 With T =TB on surface S1, 17/08/2014 N I T CALICUT

is the heat flow / unit area Where, Q is heat generated / unit volume is the heat flow / unit area is positive when heat is flowing into body is negative when heat is flowing out of the body is Zero on an insulated boundary Consider With the first boundary condition of above equation and /or second boundary condition and /or loss of heat by convection from the ends of 1-D body, we have Insulated h T 2 17/08/2014 N I T CALICUT

Minimize the following functional : (Analogous to the potential energy functional Π) Where, 17/08/2014 N I T CALICUT

Important: q* and h on the same surface cannot be specified simultaneously because they cannot occur on the same surface We now have the functional given by Consider the first term, 17/08/2014 N I T CALICUT

Similarily, fourth term gives, Second term gives, Third term gives, Similarily, fourth term gives, Substituting equations (11),(12),(13) and (14) in equation (10) we obtain 17/08/2014 N I T CALICUT

Equation (15) has to be minimized with respect to and equated to zero 17/08/2014 N I T CALICUT

The above equation is of the form, On simplifying, The above equation is of the form, 17/08/2014 N I T CALICUT

Element Conduction matrix is Where, Element Conduction matrix is The first term is conduction part of K and second term represent convection part of K And the force matrices have been defined by, 17/08/2014 N I T CALICUT

Consider the conduction part, Substituting for B, D and dV in the above equation, 17/08/2014 N I T CALICUT

Therefore, element conduction matrix is, The convection part is, Therefore, element conduction matrix is, 17/08/2014 N I T CALICUT

The force matrix terms will be, 17/08/2014 N I T CALICUT

Convection force from the end of the element By adding we get, Convection force from the end of the element 1 2 h T00 17/08/2014 N I T CALICUT

N1 = 0 and N2 = 1 at right end and We have an additional convection term contribution to the stiffness matrix and is, N1 = 0 and N2 = 1 at right end and The convection force from the free end 17/08/2014 N I T CALICUT

The global structure conduction matrix is Step 5. Assemble the element equations to obtain the global equations and introduce boundary conditions. The global structure conduction matrix is The global force matrix is and global equations are 17/08/2014 N I T CALICUT

Step 6. Solve for the nodal temperatures Step 7. Solve for the element temperature gradients and heat fluxes. 17/08/2014 N I T CALICUT

ONE DIMENSIONAL FINITE ELEMENT FORMULATION GALERKIN’S APPROACH Equation representing one dimensional formulation of conduction with convection is given by, Element – 1-D linear 2 noded element with temperature function T (x) = N1 t1 + N2 t2 --------------------------- (2) Where, 17/08/2014 N I T CALICUT

The residual equations for the equation (I) are Where i = 1,2 Integrating the first term of equation (4) by parts and rearranging, 17/08/2014 N I T CALICUT

Integration by parts: 17/08/2014 N I T CALICUT

Substituting for T , we get 17/08/2014 N I T CALICUT

Let x1= 0 and x2 = L, then, The above equations are of the form, And 17/08/2014 N I T CALICUT

If one substitute for N1 & N2 in equation (9) and solve, then, The forcing function vectors on the right hand side of the equation (7) are given by 17/08/2014 N I T CALICUT

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Two Dimensional Finite Element Formulation: 1-d elements are lines 2-d elements are either triangles, quadrilaterals, or a mixture as shown Label the nodes so that the difference between two nodes on any element is minimized. 17/08/2014 N I T CALICUT

Three noded triangular element: Assume (Choose) a Temperature Function: 17/08/2014 N I T CALICUT

Define Temperature Gradient Relationships Analogous to strain matrix: {g}=[B]{t} 17/08/2014 N I T CALICUT

[B] is derivative of [N]: Heat flux/ temperature gradient relationship is: 17/08/2014 N I T CALICUT

Derive the element conduction matrix and equations: Where, If thickness is assumed constant and all terms of integrand constant, then the conduction portion of the total stiffness matrix is, 17/08/2014 N I T CALICUT

Now the convection portion of the total stiffness matrix is, Consider the side between nodes i and j of the element subjected to convection, then, Nm = 0 along side i-j 17/08/2014 N I T CALICUT

Where Li-j is the length of side i-j We Obtain Where Li-j is the length of side i-j Force Matrices: 17/08/2014 N I T CALICUT

Can be found in a same manner by simply replacing The integral Can be found in a same manner by simply replacing 17/08/2014 N I T CALICUT

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Thank You 17/08/2014 N I T CALICUT