10335d91-313c-4b45-9b7b-2e9521c63d3f 42 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^* / * :

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.

L H N xd U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 0.01 : H 0.5 : N 100 : 13 mat

X L H N xd :

user 0.1 :

X rows 100

blue60 X X 60 mm * fT vectorize augment : red100 X X 100 mm * fT vectorize augment : green1000 X X 1 m * fT vectorize augment : 31 sys

USER user user 60 mm * fT augment user user 100 mm * fT augment user user 1 m * fT augment 31 sys :

plot blue60 red100 green1000 user 850 user 75022 mat 41 sys :

plot

USER 0.1 811.7485 K * 12 mat 0.1 805.4946 K * 12 mat 0.1 796.8138 K * 12 mat 31 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