A thermocouple with its reference junction at has a Seebeck coefficient fitted over by

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

A thermocouple with its reference junction at

0\,{}^\circ\mathrm{C}

has a Seebeck coefficient fitted over

0 \le T \le 1000\,{}^\circ\mathrm{C}

S(T) = 40/(1 + T/800)\ \mu\mathrm{V\,K^{-1}}

, and its output voltage is

E(T) = \int_0^TS(T')\,\mathrm{d}T'

. The closed form

E(T) = 32000\ln(1 + T/800)\ \mu\mathrm{V}

may be used as the reference value in checks, but the rules below may use only values of

S

. Errors are measured as value minus exact value. ((a)) Compute

E(500\,{}^\circ\mathrm{C})

with the three-node Gauss-Legendre rule and with the three-node Clenshaw-Curtis rule (nodes

\cos(k\pi/2)

, mapped to

[0,500]

), and give the error of each. ((b)) Let

Q_1

be each value from part (a) and

Q_2

the sum of the same rule on

[0,250]

and

[250,500]

. Estimate the error of

Q_2

for each rule a posteriori, using the divisor that its local error requires. Compare each estimate with the true error of

Q_2

, count the function values needed for

Q_1

and

Q_2

together in each rule, and state the assumption behind the estimates. ((c)) Tabulate

E

at the five Chebyshev points

T_j = 500(1 + \cos(j\pi/4))\ {}^\circ\mathrm{C}

j = 0, \dots, 4

, and evaluate the barycentric interpolant at

300\,{}^\circ\mathrm{C}

. Give its error. ((d)) A measurement gives

E = 12000\ \mu\mathrm{V}

. Solve

E(T) = 12000\ \mu\mathrm{V}

by Newton's method from

T_0 = 400\,{}^\circ\mathrm{C}

, using

S

as the derivative of

E

. Give

T_1

and

T_2

, and the order observed from the errors of

T_0

T_1

and

T_2

. ((e)) The voltage is uncertain by

\pm5\ \mu\mathrm{V}

. Find the resulting uncertainty in

T

from the sensitivity of the inversion, and say which of the errors in parts (a) and (c) can be neglected beside it.

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