A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at 1000c, right side at 2000C. Upper part at 500 C, and lower at 1500C. If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes?

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Suresh Chand · Accepted Answer

Solution: Since height and weight of gird is same h = k = 0.8m The nodes are shown in Figure below: Now to get the temperature at the interior nodes we have to write Equation for all the combinations of i and j, i = 1, ......., m; j = 1, ......., n-1 For u1: u3 + u2 + 75 + 50 - 4u1 = 0    ----------(1) -4u1 + u2 + u3 = -125 For u2: u4 + u3 + 100 + 50 - 4u2 = 0   ----------(2) u1 - 5u2 + u4 = -150 for u3: u4 + u1 + 300 + 75 - 4u3 = 0   ----------(3) u1 + u4 - 4u3 = -375 For u4: u2 + u3 + 300 + 100 - 4u4 = 0   ----------(4) u2 + u3 - 4u4 = -400 Equations (1) to (4) represent a set of four simultaneously linear equations, which is given below: -4u1  + u2 + u3 = -125 u1 - 4u2 + u4 = -150 u1 + u4 - 4u3 = -375 u2 + u3 - 4u4 = -400 Now, (u_1 = frac{u_2 + u_3 + 125}{4}) (u_2 = frac{u_1 + u_4 + 150}{4}) (u_3 = frac{u_1 + u_4 + 375}{4}) (u_4 = frac{u_2 + u_3 + 400}{4}) Solving above equations by using Gauss-Seidel method with initial guess u2 = 0, u3 = 0, u4 = 0, we get Values → Iteration ↓ u1 u2 u3 u4 1 31.25 45.313 101.563 136.719 2 67.969 88.672 144.922 158.398 3 89.648 99.512 155.762 163.818 4 95.068 102.222 158.472 165.173 5 96.423 102.899 159.149 165.512 6 96.762 103.069 159.319 165.597 7 96.847 103.111 159.361 165.618 8 96.868 103.121 159.371 165.623 9 96.873 103.124 159.374 165.625 Thus, u1 = 96.873 u2 = 103.124 u3 = 159.374 u4 = 165.625

A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at \(100^{0}c\), right side at \(200^{0}c\). Upper part at \(50^{0}c\), and lower at \(150^{0}c\). If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes?

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A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at \(100^{0}c\), right side at \(200^{0}c\). Upper part at \(50^{0}c\), and lower at \(150^{0}c\). If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes?

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