XY from Carlos blue trace

10335d91-313c-4b45-9b7b-2e9521c63d3f 89 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.

1. Discrete solve export

D1 50 mm * :

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

<= Given:

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 export 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 :

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 :

2. A model function

β 2 el x * β 3 el + (x * β 4 el + β 1 el x 3^* + x 3^/

Model function x β f 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 771.1697737 4.3053446 0.0267923 - 0.0001042 41 mat :

110141 mat

4.325 - 4.372 0.031 - 041 mat

771.1697737 4.3053446 0.0267923 - 0.0001042 41 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 771.1697738 4.3053449 0.0267921 - 0.0001044 41 mat

x D x β f x diff :

λ 0.01 x ≤ (x 0.25 ≤ (* :

Relative residuals Residuals i 1 XY rows range Δ i el 1 Y i el X i el β f / - eval : for X 5000 Δ * augment 21 line :

data X Y . 12 black plotG :

Residuals Error

magnified 5000

SSD = "Sum Square Differences" β SSD Y i el X i el β f - (2^(i 1 n sum :

β SSD 1

data FIT 2 1 sys x D λ /

3. 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

4. Plot/solve on the canvas ... =>

Observe . . .

1. This section [4] is specific to Smath QuickPlot.

For convenience: unit system removed.

2. Here, we use °K as the solving variable.

You may re-use T but after Clear(T)=.

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D1 50 :

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

q 20 :

°K T2 773 :

Stefan Boltzmann constant: σ 5.670400 108 -^* :

A1 π D1 * : 157.08

Unknown μ 1 :

Setup to plot/solve on the canvas.
It solves for 407 pixels *3  = 1221
138 sec ... each sol in ~ 0.1 s x D2 sol σ °K 4^T2 4^- (* 1 x - x A1 * / 1 μ A1 * / + 1 x - x π * D2 * / (+ / q π D1 2^* 4 / (* - 0 ≡ °K 1 roots :

t0 1 time :

λ 0.01 x ≤ (x 0.5 ≤ (* :

user 0.0625 :

solutions [°K] over each D2
given => user 'x' User user 60 sol user 100 sol user 1000 sol 31 mat : 83482481031 mat

x 60 sol λ / x 100 sol λ / x 1000 sol λ / user User 1 el . 12 black 1 5 mat user User 2 el . 12 black 1 5 mat user User 3 el . 12 black 1 5 mat 6 1 sys

x 60 sol λ / x 100 sol λ / x 1000 sol λ / user User 1 el . 12 black 15 mat user User 2 el . 12 black 15 mat user User 3 el . 12 black 15 mat 61 sys

1 time t0 - 136 s *