10335d91-313c-4b45-9b7b-2e9521c63d3f 24 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:

D.1 50 mm * :

k 15 W m K * / * :

q' 20 kW m 3^/ * :

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 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.

D.2 0.06 0.1 131 mat m * :

To m 1 ε rows range n 1 D.2 rows range To m n el ε m el D.2 n el fT : for 11 line for To 21 line :

D.2 transpose 0.06 m * 0.1 m * m 13 mat

To K 811.7483 805.4946 796.8138 808.292 802.6031 794.7259 805.3774 800.1695 792.9734 802.8862 798.0927 791.4813 800.7324 796.2997 790.1957 798.8515 794.7359 789.0764 797.1948 793.3599 788.093 795.7243 792.1399 787.2223 794.4104 791.0506 786.446 793.2292 790.0723 785.7493 792.1617 789.1885 785.1208 791.192 788.3864 784.5508 790.3074 787.6551 784.0315 789.4971 786.9856 783.5565 788.7521 786.3704 783.1203 788.0649 785.8031 782.7184 787.429 785.2784 782.3468 786.8387 784.7916 782.0023 786.2894 784.3388 781.682 785.777 783.9164 781.3834 785.2979 783.5217 781.1044 784.8488 783.1518 780.8432 784.4271 782.8046 780.598 233 mat

ε 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 231 mat