HEAT TRANSFER Problems with FEM solution VIJAYAVITHAL BONGALE DEPARTMENT OF MECHANICAL ENGINEERING MALNAD COLLEGE OF ENGINEERING HASSAN - 573 202. Mobile : 9448821954

Problem 1. For the composite wall idealized by the 1-D model shown in figure below, determine the interface temperatures. For element 1, let K1 = 5 W / m 0C, for element 2, K2 = 10 W / m 0C and for element 3, K3 = 15 W / m 0C. The left end has a constant temperature of 200 0C and the right end has a constant temperature of 600 0C. T= 200 0C 0.1 m 1 2 3 T= 600 0C A= 0.1 m2

Element conduction matrices are, Finite Element model: 1 2 3 4 0.1 m t1 t2 t3 t4 t1 = 200 0C , t4 = 600 0C , A = 0.1 m2, K1= 5 W / m 0C, K2= 10 W / m 0C , K3= 15 W / m 0C Element conduction matrices are,

And the global structure conduction matrix is,

Now the global equations are given by, That is Since the values of t1 and t4 are given, the equations are modified as follows

We have now two equations and two unknowns Solving we obtain, t3 = 527.3 0C and t2 = 418.2 0C

Problem 2: Heat transfer in 1D fin Calculate temperature distribution using FEM. 4 linear elements, 5 nodes

Element 1, 2, 3: Element 4:

For element 1, 2, 3 For element 4 ,

Heat source (Still unknown) t1= 80 0 C, four unknowns – eliminate Q Solving:

A composite wall consists of three materials as shown. Problem 3. A composite wall consists of three materials as shown. The outer temperature is T0 = 20 0C. Convection heat transfer takes place on the inner surface of the wall with T∞ = 800 0C and h = 25 W / m2 0C. Determine the temperature distribution in the wall. K1 K2 K3 0.3 m 0.15 m h, T∞ T0 = 20 0C K1 = 20 W/m 0C , K2 = 30 W/m 0C, K3 = 50 W/m 0C, h = 25 W/m2 0C , T∞ = 800 0C

Assume A = 1 m2 , the element conduction matrices are, Finite Element model: h , T∞ 1 2 3 4 0.3 m 0.15 m t1 t2 t3 t4 Element 1 Element 2 Element 3 Assume A = 1 m2 , the element conduction matrices are, For element 1, we must consider the convection from the free end (left end)

Global structure conduction matrix is,

Now the global equations will be in the form, And

Since h, T∞ and t4 are specified, the above equations are modified as, Therefore, Since h, T∞ and t4 are specified, the above equations are modified as,

We now obtain three sets of simultaneous equations in which we have three unknown temperatures. On solving, we get The temperatures are,

ONE DIMENSIONAL FINITE ELEMENT FORMULATION OF FIN A metallic fin, with thermal conductivity kxx = 360 W / m 0C, 0.1 cm thick, and 10 cm long, extends from a plane wall whose temperature is 235 0C. Determine the temperature distribution and amount of heat transferred from the fin to the air at 20 0C with h = 9 W / m2 0C. Take the width of fin to be 1 m. t W L 235 0C h = 9 W/m2 0C, T∞ = 20 0C L = 10 cm = 0.1 m t = 0.1 cm = 1x 10-3 m W = 1 m

Finite element model: h = 9 W/m2 0C, T∞ = 20 0C 235 0C 1 2 3 4 t1 t2 q = 0

Element stiffness (conduction) matrices written as, and

Global structure conduction matrix is,

The force matrix is given by,

The global equations are,

Applying the boundary conditions, t1 = 235 0C we get three sets of simultaneous equations with three unknowns.

The total heat loss in the fin, On solving The total heat loss in the fin, And Therefore,

Problem 5: The fin shown in figure is insulated on the perimeter. The left end has a constant temperature of 100 0C. A positive heat flux q*= 5000 W / m2 acts on the right end. Let Kxx = 6 W/ m 0C and cross sectional area A = 0.1 m2. Determine the temperatures at , , and L . Where L = 0.4 m. T= 100 0C L = 0.4 m q = 5000 W/ m2 Solution: Given: t1 = 100 0 C, Kxx = 6 W / m 0 C, A = 0.1 m2 , L1 = L2 = L3 = L4 = 0.1 m

Element conduction matrices are, Finite Element model: T= 100 0C L = 0.1 m q = 5000 W/ m2 t1 t2 t3 t4 t5 Element conduction matrices are,

Similarly we have

and the global structure conduction matrix is,

Since a positive heat flux q Since a positive heat flux q* = 5000 W / m2 acts on the right end , we will have to calculate force term at the end of the element 4. The global equations are,

Modified equations after imposing boundary conditions,

In expanded form we have, Solving,

Problem: Consider the circular heat transfer pin shown in the figure. The base of the pin is held at constant temperature of 100 0C ( i.e. boiling water ). The tip of the pin and its lateral surfaces undergo convection to a fluid at ambient temperature Ta. The convection coefficients for tip and lateral surfaces are equal. Given Kx = 380 W / m -0C , L = 8 cm, h = 2500 W / m2 -0C , d = 2 cm, Ta = 30 0C. Use a two finite element model with linear interpolation functions (i.e., a two-node element) to determine the nodal temperatures and the heat removal rate from the pin. Assume no internal heat generation. L h, Ta 100 0C

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