Solução da Série para Condução Transitória em Parede Plana - Problema 5.43

PROBLEM 543 KNOWN Series solution Eq 542 for transient conduction in a plane wall with convection FIND Midplane x0 and surface x1 temperatures for Fo01 and 1 using Bi01 1 and 10 with only the first four eigenvalues Based upon these results discuss the validity of the approximate solutions Eqs 543 and 544 SCHEMATIC ASSUMPTIONS 1 Onedimensional transient conduction 2 Constant properties ANALYSIS The series solution Eq 542a is of the form 2 n n n n 1 C exp Fo cos x where the eigenvalues n and the constants Cn are from Eqs 539b and 539c n n n n n n tan Bi C 4sin 2 sin 2 The eigenvalues are tabulated in Appendix B3 note however that 1 and C1 are available from Table 51 The values of n and Cn used to evaluate are as follows Bi 1 C1 2 C2 3 C3 4 C4 01 03111 10160 31731 00197 62991 00050 94354 00022 1 08603 11191 34256 01517 64373 00466 95293 00217 10 14289 12620 43058 03934 72281 02104 102003 01309 Using n and Cn values the terms of designated as 1 2 3 4 and are as follows Fo01 Bi01 Bi10 Bi10 x 0 1 0 1 0 1 1 10062 09579 10393 06778 10289 01455 2 00072 00072 00469 00450 00616 00244 3 00001 00001 00007 00007 00011 00006 4 299107 300107 247106 246107 396106 283106 09991 09652 09931 07235 09684 01705 Continued PROBLEM 543 Cont Fo1 Bi01 Bi10 Bi10 x 0 1 0 1 0 1 1 09223 08780 05339 03482 01638 00232 2 835107 835107 122105 117106 349109 138109 3 7041020 4701020 4301024 4 4771042 7931042 8521047 09223 08780 05339 03482 01638 00232 The tabulated results for x Bi Fo demonstrate that for Fo1 the first eigenvalue is sufficient to accurately represent the series However for Fo01 three eigenvalues are required for accurate representation A more detailed analysis would show that a practical criterion for representation of the series solution by one eigenvalue is Fo 02 For these situations the approximate solutions Eqs 543 and 544 are appropriate For the midplane x0 the first two eigenvalues for Fo02 are Fo02 x0 Bi 01 10 10 1 09965 09651 08389 2 000226 00145 00096 09939 09506 08293 Error 026 153 116 The percentage error shown in the last row of the above table is due to the effect of the second term For Bi 01 neglecting the second term provides an error of 026 For Bi 1 the error is 153 Hence we conclude that the approximate series solutions with only one eigenvalue provides systematically high results but by less than 15 for the Biot number range from 01 to 10