XY from Carlos blue trace

10335d91-313c-4b45-9b7b-2e9521c63d3f 49 3 5 auto

&[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:

D1 50 mm * :

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

q 20 kW m 3^/ * :

°K T2 773 K * :

Stefan Boltzmann constant: σ 5.670400 108 -^* W m 2^K 4^* / * :

A1 π D1 * : 0.157 m *

unknown NOT used k 15 W m K * / * :

Unknown μ 1 :

L H N T x 0 : dx H L - N 1 - / : 12 mat i 1 N range x i el L dx i * dx - (+ : for j 13 range i 1 x rows range T i j el q π D1 2^* 4 / (* 1 x i el - x i el A1 * / 1 μ A1 * / + 1 x i el - x i el π * D2 j el * / (+ (* σ / T2 (4^+ 4 nthroot : for 11 line for x T K / augment 41 line :

Sloution => Carlos Adapted => Jean

User: range/mesh T 0.01 1100 T :

Round OFF a matrix for convenient sub-process M n Mround i 1 M rows range j 2 M cols range m i j 1 - el M i j el n round : for for M 1 col m augment 21 line :

collect T 1 col T 2 col T 3 col T 4 col augment : Export collect 0 Mround : 21 line

Export 0.01 10381004952 0.02 933910876 0.03 888871846 0.04 86384982944 mat

XY from Carlos blue trace

XY 0.01 1038.173 0.02 932.927 0.03 888.355 0.04 863.295 0.05 847.144 0.06 835.84 0.07 827.476 0.08 821.033 0.09 815.914 0.1 811.748 0.11 808.292 0.12 805.377 0.13 802.886 0.14 800.732 0.15 798.851 0.16 797.195 0.17 795.724 0.18 794.41 0.19 793.229 0.2 792.162 0.21 791.192 0.22 790.307 0.23 789.497 0.24 788.752 0.25 788.065 0.26 787.429 0.27 786.839 0.28 786.289 0.29 785.777 0.3 785.298 0.31 784.849 0.32 784.427 0.33 784.03 0.34 783.656 0.35 783.303 0.36 782.969 0.37 782.653 0.38 782.353 0.39 782.068 0.4 781.797 0.41 781.539 0.42 781.293 0.43 781.059 0.44 780.834 0.45 780.62 0.46 780.415 0.47 780.218 0.48 780.029 0.49 779.848 0.5 779.674 0.51 779.507 0.52 779.346 0.53 779.191 0.54 779.042 0.55 778.898 0.56 778.759 0.57 778.625 0.58 778.495 0.59 778.37 0.6 778.249 0.61 778.132 0.62 778.019 0.63 777.909 0.64 777.802 0.65 777.699 0.66 777.599 0.67 777.502 0.68 777.407 0.69 777.316 0.7 777.226 0.71 777.14 0.72 777.056 0.73 776.974 0.74 776.894 0.75 776.816 0.76 776.741 0.77 776.667 0.78 776.595 0.79 776.525 0.8 776.457 0.81 776.39 0.82 776.325 0.83 776.262 0.84 776.2 0.85 776.139 0.86 776.08 0.87 776.022 0.88 775.966 0.89 775.911 0.9 775.857 0.91 775.804 0.92 775.752 0.93 775.701 0.94 775.652 0.95 775.603 0.96 775.556 0.97 775.509 0.98 775.464 0.99 775.419 1 775.375 1002 mat :

Genfit Utilities

vx vy char size clr plotG n vx length : plot vx 1 el vy 1 el char size clr augment : i 2 n range plot plot vx i el vy i el char size clr augment stack : for plot 41 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx-> raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Vector gradient ... Optimiz Symbolic x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

Minimize model "ConjugateGradient" wrt parameters [β's]

Model function x β f 0 β 1 el β 2 el x 1 -^* + β 3 el x 2 -^* + β 4 el x 3 -^* + + :

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

<= ConjugateGradient Vector Fnct x β φ x β f 4 φ :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

β 4.3251307 - 4.3714704 0.0312143 - 0.0001421 41 mat :

110141 mat

4.325 - 4.372 0.031 - 041 mat

Genfit coeff's of fit β XY f φ β Minimize :

re-inject to iterate manually

FIT range, populate U 0.01 1 0.011 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

β 771.1697737 4.3053446 0.0267923 - 0.0001042 41 mat

x D x β f x diff :

λ 0.01 x ≤ (x 0.25 ≤ (* :

data X Y . 12 black plotG :

data FIT 2 1 sys x D λ /

Maple Analytical Solution

T D2 Sol σ T 4^T2 4^- (* 1 x - x A1 * / 1 μ A1 * / + 1 x - x π * D2 * / (+ / q π D1 2^* 4 / (* - 0 ≡ T solve maple :

x D2 Analytical T D2 Sol 1 el :

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

x 60 mm * Analytical x 100 mm * Analytical x 1 m * Analytical 3 1 sys