ABSTRACT
Consider the constant pressure specific heat of air at high temperatures presented in Table 4.12, where T is the temperature (K) and Cp is the specific heat (JIgm-K). The exact values, approximate values from a least squares quadratic polynomial approximation, and the percent error are also presented in the table. Determine a least squares quadratic polynomial approximation for this set of data:
For this problem, Eq. (4.158) becomes
5a + b L Ti + C L Tl = L Cp.i a L Ti + b L Tl + C L T? = L TiCp.i a L Tl + b L T? + C L Ti4 = L TlCp.i
Evaluating the summations and substituting into Eq. (4.160) gives
10 x 103a + 22.5 x 106b + 55 x 109c = 12.5413 X 103
22.5 x 106a +55 x 109b+ 142.125 x 1012c =288.5186 x 106
Solving for a, b, and c by Gauss elimination yields
Cp = 0.965460 + 0.211197 x 1O-3T - 0.0339143 x 1O-6 T2
(4.159)
(4.160a) (4. 160b) (4.160c)
(4.16Ia) (4.16Ib) (4.16Ic)
(4.162)
Substituting the initial values of T into Eq. (4.162) gives the results presented in Table 4.12. Figure 4.12 presents the exact data and the least squares quadratic polynomial approximation. The quadratic polynomial is a reasonable approximation of the discrete data.
