plot(data,a,b,c)

e303ccaa-090b-4640-8887-fbc8204b49a8 5 3 12 decimal

&[DATE] &[TIME] - &[FILENAME]

plot(data,a,b,c)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

0. Water masse volumique kg/m^3/°C

t0 1 time :

Thiele(X,Y,Δ)

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

"data" source: M.P. WUKALOWITSCH [1958] Russian, German, English, French

X Y 0 Thiele generate Thiele vector ≡ X Y 1 Thiele generate Thiele table ≡ 21 line

step 0 => reduce the data Y

WUKALOWITSCH [1958] data 10 999.65 50 988.01 100 958.35 150 917.01 200 864.67 250 798.89 300 712.14 325 654.32 350 574.69 360 527.84 370 450.03 374322122 mat :

0 999.8 4100010 999.7 20 998.2 30 995.7 40 992.2 50 988.1 60 983.2 70 977.8 80 971.8 90 965.3 100 958.4 122 mat

X data 1 col : Y data 2 col : 21 line

X kg m 3^* ≡ Y °C ≡ 21 line

reduce i 1 data rows range R i el 1 999.9 Y i el - / : for R 21 line :

An expansion tank is required on all closed loop boiler systems. As a boiler heats the system water, the water will expand in volume. Water cannot be compressed but air can, so an expansion tank uses air (or an inert gas like nitrogen) that acts like a cushion or spring against this force.

step 1 => Thiele vector of the "reduced"

X reduce 0 Thiele 10450 10.214777 - 100 4.025304 - 150 6406.546109 200 0.023623 250 392323.244204 - 300 0.000572 325 930691.116845 - 350 0.000314 360 76987.173805 - 370 0.000284 - 374 35984.735995 - 122 mat

TK 10450 10.214777 - 100 4.025304 - 150 6406.546109 200 0.023623 250 3.923232 105^* - 300 0.000572 325 9.306911 105^* - 350 0.000314 360 76987.173789 - 370 0.000284 - 374 35984.735997 - 122 mat :

reduce 4 0.0841 0.02407 0.01206 0.00739 0.00497 0.00348 0.00289 0.00235 0.00212 0.00182 0.00148 121 mat

step 2 => Thiele continued fraction of the "reduced"

T TK 1 col : ρ TK 2 col : 12 mat

r(t)

t r ρ 1 el x T 1 el - ρ 2 el x T 2 el - ρ 3 el x T 3 el - ρ 4 el x T 4 el - ρ 5 el x T 5 el - ρ 6 el x T 6 el - ρ 7 el x T 7 el - ρ 8 el x T 8 el - ρ 9 el x T 9 el - ρ 10 el x T 10 el + ρ 11 el x T 11 el + ρ 12 el / + / + / + / + / + / + / + / + / + / + / + :

step 3 => Recast/Reverse Engineer the "reduced"

recast as function of 'x' x kgm3 999.9 1 x r / - :

Optimiz => numeric W t 0.01 3741 range : i 1 t rows range ρ i el t i el kgm3 eval : 11 line for t ρ augment 31 line :

Notes: 1. the triple point of water 273.16 K, 0.01 °C 2. max density @ approx: 4 °C 3. "surfusion" water droplet may be as low as - 10°C 4. any & all kg/m^3 formulas are approximation. 5. this Thiele is of exceptional quality [Jean Giraud]. 6. Thiele is exact @ the support points

step 4 =>Plot discrete, Interpolate kg/m^3 vs °C

XY data . 10 black plot :

Water masse volumique ... kg/m^3 vs °C critical range

322 W XY 3 1 sys

kg m 3^/

kg_m³ 1 :

Interpolate in-situ 100 kgm3 200 kgm3 325 kgm3 31 mat kg_m³ 958.35 864.67 654.34 31 mat

100 kgm3 200 kgm3 325 kgm3 31 mat kg_m³ 958.35 864.672 654.341 31 mat

1 time t0 - 2 s *

x kgm3 x 0.9 - kgm3

Optimiz => numeric v t 0.01 3741 range : i 1 t rows range specificvolume i el t i el kgm3 eval (1 -^: 11 line for t specificvolume augment 31 line :

support pts specific vol xy i 1 Y rows range spec i el 1 Y i el / : for X spec augment 21 line :

xx xy 1 col : yy xy 2 col : 21 line

uv xy . 10 black plot :

uv v 2 1 sys

1 time t0 - 3 s *

jmGiraud:revisited 2016/06/02

VUKALOWITSCH [1958]

WUKALOWITSCH [1958] Water/Steam data 10 999.65 0.009398 50 988.01 0.08306 100 958.35 0.5977 150 917.01 2.547 200 864.67 7.862 250 798.89 19.98 300 712.14 46.21 325 654.32 70.57 350 574.69 113.6 360 527.84 144370 450.03 203374322288123 mat :

rho-Steam

ρs 0.009398 543.020824849 0.128192581 - 651.141760016 - 1.682165726 117.019936457 19.46537573 - 302.928543896 - 0.064612944 270.341818192 6.346153635 - 1.040581502 121 mat :

t ρSteam ρs 1 el t 10 - (ρs 2 el t 50 - (ρs 3 el t 100 - (ρs 4 el t 150 - (ρs 5 el t 200 - (ρs 6 el t 250 - (ρs 7 el t 300 - (ρs 8 el t 325 - (ρs 9 el t 350 - (ρs 10 el t 360 - (ρs 11 el t 370 - (ρs 12 el / + / + / + / + / + / + / + / + / + / + / + :

X data 1 col : Y data 3 col : 12 mat XY X Y augment : ρSteam XY . 12 black plot : 31 line

v 5 x ≤ (x 374 ≤ (* :

374 ρSteam kg_m³ 288

ρSteam x ρSteam v / 2 1 sys