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HeatTransfer-I-Section-5

# 23 transient conducon soluons for semiinnite solid

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Unformatted text preview: 5.1 or approximated using the following: Plane Wall 1.5708 ζ1 = 2.139 Ⱥ 0.4675 Ⱥ Ⱥ1 + 1.5708 / BiL Ⱥ Ⱥ Ⱥ 2.4048 ζ1 = 2.238 Ⱥ 0.4468 Ⱥ Ⱥ1 + 2.4048 / 2 BiR Ⱥ Ⱥ Ⱥ 3.14159 ζ1 = 2.314 Ⱥ 0.4322 Ⱥ Ⱥ1 + 3.14159 / 3BiR Ⱥ Ⱥ Ⱥ € ) ( Sphere ) ( Cylinder ( ) Transient Conduc/on •  Some comments on these solu/ons: –  We only need one term when Fo > 0.2 for all values of Bi. –  If Fo < 0.2 addi/onal terms are needed and hence addi/onal eigenvalues. –  Addi/onal eigenvalues can be approximated by adding Pi = 3.14159… to the previous eigenvalue. –  For very small Fo numbers, many terms are required. –  We may resort to the semi ­inﬁnite region problems for certain cases for a more eﬃcient solu/on. We will see this criteria later. –  Finally when Bi < 0.1, the solu/ons reduce to the lumped capacitance approxima/on given earlier for all Fo. 20 Example 3 •  For Bi = 1, calculate and compare the eigenvalues from Table 5.1 with the simple formulas given earlier. 21 Example 4 •  Show that when Bi < 0.1 we obtain the same solu/on from the single term approxima/on for a plane wall as that returned by lumped capacitance. 22 Example 5 •  A very long A316 stainless steel ingot of 1...
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