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&[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^* / * :

μ 1 :

A.1 π D.1 * : 0.157 m *

D.2 A.2 π D.2 * :

499.85 °C *

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

T ε D2 eq σ T 4^T.2 4^- (* 1 ε - ε A.1 * / 1 μ A.1 * / + 1 ε - ε π D2 * * / (+ / q' π D.1 2^* 4 / (* - :

This setup simulates the parametrization in the Mathcad solve blocks

ε .02 :

D2 60 mm * 100 mm * 1 m * 13 mat :

j 3 :

just to initialise D2 D2 j el :

about insensitive ... so fast MAXITERS 20 :

# 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 :

T.1 499.85 °C * :

T.2 1100 K * :

ε D2 fT T f T ε D2 eq : T f T.1 T.2 Bisection 21 line :

D2 60 mm * 100 mm * 1 m * 13 mat :

end low values ε D2 1 el fT ε D2 2 el fT ε D2 3 el fT 31 mat 932.9 K * 909.9 K * 875.8 K * 31 mat

c) For tube diameters of 60, 100, and 1000 mm and evacuated, compute and plot the surface temperatures as a function ofequivalent surface emissivities in the range from 0.1 to 1.0.

ε 0.01 1 0.0125 range : ε 60 mm * fT vectorize augment ε 0.01 1 0.0125 range : ε 100 mm * fT vectorize augment ε 0.01 1 0.0125 range : ε 1 m * fT vectorize augment 3 1 sys ε 0.01 1 0.0125 range : ε 60 mm * fT vectorize augment ε 0.01 1 0.0125 range : ε 100 mm * fT vectorize augment ε 0.01 1 0.0125 range : ε 1 m * fT vectorize augment 3 1 sys

a) Determine the center and surface temperatures of the rod:

0.2 60 mm * fT °C 519.0116

fT.mid q' D.1 2 / (2^* 4 k * / 0.2 60 mm * fT + : °C 519.22

D.o 60 mm * :

D.i 50 mm * :

T.i 0.2 50 mm * fT :

T.o 773 K * :

k.cond 0.0563 W m K * / * :

T ε D.o eq.cond 2 π * k.cond * T T.o - (* D.o D.i / ln / T ε D.o eq (+ :

ε D.o fT.cond T.cond f T ε D.o eq.cond : T.cond f T.i T.o Bisection 21 line :

b) Determine the center and surface temperatures of the rod if atmospheric air occupies the annular space:

0.2 D.o fT.cond °C 509.783

fT.cond_mid q' D.1 2 / (2^* 4 k * / 0.2 D.o fT.cond + : °C 509.9916