10335d91-313c-4b45-9b7b-2e9521c63d3f 18 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

A long uniform rod of 50-mm diameter with a thermal conductivity of 15 W/m-K is heated internally by volumetric energy generation of 20 kW/m3. The rod is positioned coaxially within a larger circular tube of 60-mm diameter whose surface is maintained at 500 degC. The annular region between the rod and the tube is evacuated, and their surfaces are diffuse and gray with an emissivity of 0.2. a) Determine the center and surface temperatures of the rod. b) Determine the center and surface temperatures of the rod if atmospheric air occupies the annular space. c) For tube diameters of 60, 100, and 1000 mm and for both the evacuated and atmospheric conditions, compute and plot the center and surface temperatures as a function of equivalent surface emissivities in the range from 0.1 to 1.0.

Given:

k 15 W m K * / * :

q' 20 kW m 3^/ * :

D.1 50 mm * :

T.2 773 K * :

T.2 °C 499.85

Stefan Boltzmann constant:

σ 5.670400 108 -^* W m 2^K 4^* / * :

F.12 1 :

A.1 π D.1 * : 0.157 m *

D.2 A.2 π D.2 * :

T 300 K * :

Numerical methods can't handle equals without some symbolic help. Use rhs-lhs = 0 instead.

T ε D.2 eq σ T 4^T.2 4^- (* 1 ε - ε A.1 * / 1 F.12 A.1 * / + 1 ε - ε D.2 A.2 * / (+ / q' π D.1 2^* 4 / (* - :

This setup simulates the parametrization in the Mathcad solve blocks

ε .1 :

D.2 60 mm * :

1 f x a b Bisection k 1 MAXITERS range c a b + 2 / : c f 0 ≡ break a f c f * 0 > a c : b c : if if 21 line for c 21 line :

MAXITERS 50 :

T.1 500 K * :

T.1 ε D.2 eq 189.1835 N s / * -

T.2 900 K * :

T.2 ε D.2 eq 112.9434 N s / *

ε D.2 fT T f T ε D.2 eq : T f T.1 T.2 Bisection 21 line :

ε D.2 fT 811.748 K *

ε 0.1 1 0.11 range : ε D.2 fT vectorize augment

Seems that Newton methods never found a good interval for the roots. Bisection never fails! If the interval T1..T2 isn't enough good for the entire range of ε can use a recursive search of interval values.