hter 5. Direct Method of Interoltion More Exmles hemicl Engineering Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t 6. The secific het of wter is given s function of time in Tble. Tble Secific het of wter s function of temerture. Temerture, T Secific het, J kg 48 4 479 5 8 499 47 Determine the vlue of the secific het t T 6 using the direct method of interoltion nd first order olynomil. 5..
5.. hter 5. Figure Secific het of wter vs. temerture. Solution For first order olynomil interoltion (lso clled liner interoltion), we choose the secific het given by T T y x, y f x x, y x Figure Liner interoltion.
Direct Method of Interoltion More Exmles: hemicl Engineering 5.. Since we wnt to find the secific het t T 6, nd we re using first order olynomil, we need to choose the two dt oints tht re closest to T 6 tht lso brcket T 6 to evlute it. The two oints re T 5 nd T 8. Then T 5, T T 8, T 499 gives 5 5 8 8 499 Writing the equtions in mtrix form, we hve 5 8 499 Solving the bove two equtions gives 46.5.4 Hence T T 46.5.4T, 5 T 8 At T 6, 6 46.5.46 J 489.9 kg Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t 6. The secific het of wter is given s function of time in Tble. Tble Secific het of wter s function of temerture. Temerture, T Secific het, J kg 48 4 479 5 8 499 47
5..4 hter 5. Determine the vlue of the secific het t T 6 using the direct method of interoltion nd second order olynomil. Find the bsolute reltive roximte error for the second order olynomil roximtion. Solution For second order olynomil interoltion (lso clled qudrtic interoltion), we choose the secific het given by T T T y x, y x, y f x x, y x Figure Qudrtic interoltion. Since we wnt to find the secific het t T 6, nd we re using second order olynomil, we need to choose the three dt oints tht re closest to T 6 tht lso brcket T 6 to evlute it. The three oints re T 4, T 5, nd T 8. Then T 4, T 479 T 5, T T 8, T 499 gives 4 4 4 479 5 5 5 8 8 8 499 Writing the three equtions in mtrix form, we hve 4 764 479 5 74 8 674 499
Direct Method of Interoltion More Exmles: hemicl Engineering 5..5 Solving the bove three equtions gives 45..67 6.6667 Hence T 45..67T 6.6667 T, 4 T 8 At T 6, 6 45..676 6.6667 6 J 49. kg The bsolute reltive roximte error obtined between the results from the first nd second order olynomil is 49. 489.9 49..6% Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t 6. The secific het of wter is given s function of time in Tble. Tble Secific het of wter s function of temerture. Temerture, T Secific het, J kg 48 4 479 5 8 499 47 Determine the vlue of the secific het t T 6 using the direct method of interoltion nd third order olynomil. Find the bsolute reltive roximte error for the third order olynomil roximtion. Solution For third order olynomil interoltion (lso clled cubic interoltion), we choose the secific het given by T T T T
5..6 hter 5. y x, y x, y f x x, y x, y x Figure 4 ubic interoltion. Since we wnt to find the secific het t T 6, nd we re using third order olynomil, we need to choose the four dt oints closest to T 6 tht lso brcket T 6 to evlute it. The four oints re T 4, T 5, T 8 nd T. (hoosing the four oints s T, T 4, T 5 nd T 8 is eqully vlid.) Then T 4, T 479 T 5, T T 8, T 499 T, T 47 gives 4 4 4 4 479 5 5 5 5 8 8 8 8 499 47 Writing the four equtions in mtrix form, we hve 4 4 764 7.488 479 5 5 74.46 5 8 674 5.57 499 6 47 Solving the bove four equtions gives
Direct Method of Interoltion More Exmles: hemicl Engineering 5..7 Hence 478. 4.477.67.849 4 T T T T 478. 4.477T.67T T J 49. kg The bsolute reltive roximte error nd third order olynomil is 49. 49. 49..795%.849 6 478. 4.477 6.67 6.849 4 6 4 T, 4 T obtined between the results from the second INTERPOLATION Toic Direct Method of Interoltion Summry Exmles of direct method of interoltion. Mjor hemicl Engineering Authors Autr Kw Dte November, 9 Web Site htt://numericlmethods.eng.usf.edu