Question

The constant-pressure specific heat of air at different temperatures are tabulated in the following table: T(K) 1.0 CP (kJ/kg.K) 1.1410 1.1573 1.1 1.2 1.1722 1.3 1.1858 1.1982 1.4 1.5 1.2095 a) Use a third order interpolation polynomial to find Cp at T = 1.45 K. Show complete solution. b) If you want to develop a model that passes through all these data points using the Newton's difference method, explain briefly without calculations how you will do it. Also, how can you test the sensitivity of your developed model?

Answer 1

Let the Polynomial be

where x = Temp, P = Cp

P(2) 03 + 2r- + 11 + do

Now by substituting the given values we find the Constants $a_3,a_2,a_1,a_0$

a3 + a2 + au + do = 1.1410

a: (1.1) + a2(1.1)2 + a1(1.1) + do = 1.1573

By solving the equations we get a = 0.0167, b = -0.125, c= 0.3703, d = 0.879

Now for Cp at T=1.45K

For Newtons method we should Construt the differences table first.

Something similar to.

ху хо УО Ayo дуо (= x + b) Ауі Уа (ex + 2A) Ау. Уа ду, = (х + 3 ) Ауа У. а (х + 4А) У. = (х + 5 )

Then Using all the points we should construct a 6 degree polynomial(as we have 6 points).

Please Ask for any further clarifications..