c5bfc78b-db1c-4f07-8f54-cdc333b00773 49 3 12 decimal

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

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 :

Techniques to Thiele Continued fraction

Tanks Carlos for your immense contribution

Thiele Continued Fraction(s) have exceptional fitting qualities to many Engineering stuff,

mostly WASP [WaterSteamProperty]. Continued Fractions date back to Mesopotamians.

A huge book was published [Berlinski in French] ... never had it in hand.

Thiele works well on clean data sets, hard to handle on not too clean data sets.

Largely explored in Mathcad 11 [Theodore, Jean] ... Applied to T/C [Thermocouple],

Air atmospheric properties, WASP ...

]]>

Water ... °C(bar) PT 0.01 6.97 0.05 32.88 0.1 45.81 0.25 64.96 0.5 81.32 0.75 91.76 1 99.61 2.5 127.41 5 151.84 10 179.98 25 223.96 50 263.94 100311150 342.16 200 365.75 220 373.71 162 mat :

Xk PT 1 col : Yk PT 2 col : 12 mat XY PT . 14 black plot : 21 line

3 appVersion 0.98.6179 ≡ Timing 12 sec ≡ 21 mat

Thiele table of inverse differences i 1 Yk rows range φ i 1 el Yk i el : for j 2 Yk rows range i j Yk rows range φ i j el Xk j 1 - el Xk i el - φ j 1 - j 1 - el φ i j 1 - el - / : cont j : 21 line break try for for 21 line

With the Try / on error function, division by zero is avoided when the inverse differences of a column are all the same, and the loop is exited.

φ 11 mat Extend Thiele table ≡

cont 16

φ 11 mat 6.97 000000000000000 32.88 0.002 00000000000000 45.81 0.002 64.65 0000000000000 64.96 0.004 77.076 0.012 000000000000 81.32 0.007 89.168 0.016 58.921 00000000000 91.76 0.009 97.444 0.02 64.521 0.045 0000000000 99.61 0.011 103.908 0.023 69.099 0.049 55.806 000000000 127.41 0.021 128.068 0.038 87.303 0.07 67.764 0.125 00000000 151.84 0.034 150.452 0.057 105.469 0.097 81.682 0.155 85.783 0000000 179.98 0.058 177.051 0.088 128.279 0.137 100.187 0.203 96.961 0.447 000000 223.96 0.115 219.586 0.161 166.51 0.228 132.459 0.313 119.897 0.586 107.936 00000 263.94 0.195 258.818 0.257 203.126 0.343 164.926 0.449 146.783 0.738 137.736 0.839 0000311 0.329 305.342 0.415 247.529 0.528 205.527 0.661 181.974 0.988 166.568 1.279 113.576 000 342.16 0.447 336.261 0.552 277.406 0.684 233.344 0.839 206.636 1.2 186.044 1.6 131.336 2.815 00 365.75 0.557 359.704 0.678 300.181 0.827 254.71 1 225.702 1.394 200.769 1.885 143.38 3.355 92.593 0 373.71 0.6 367.619 0.726 307.883 0.882 261.942 1.062 232.133 1.469 205.522 1.998 146.64 3.629 85.992 3.03 - 1616 mat 11 mat

Assign the diagonal of the Matrix of the inverse differences i 1 cont range Yφ i el φ i i el 4 round : for

Rounding the values of the inverse differences for the purpose of presenting the equation to make it shorter

Yφ 6.97 0.0015 64.6502 0.0121 58.9206 0.0446 55.8065 0.1254 85.7831 0.4473 107.9361 0.8389 113.5761 2.8153 92.5928 3.0301 - 161 mat

Thiele Continued Fraction s P p size T i size 2 range i size ≡ T s i 1 - el eval p P i 1 - el eval - s i el eval / (+ : T T : 21 line T s i 1 - el eval p P i 1 - el eval - T / (+ : T T : 21 line if for s P p size T T : 21 line :

Thiele table above is constructed on the overal data set 'PT'. For shorter user range, reduce the range of the Thiele continued fraction by "size"

Thiele Size size 10 :

recast °C(bar) over p=bar p Tbar Yφ Xk p size T :

Xk 0.01 0.05 0.1 0.25 0.5 0.75 1 2.5 5102550100150200220161 mat

Yφ 6.97 0.0015 64.6502 0.0121 58.9206 0.0446 55.8065 0.1254 85.7831 0.4473 107.9361 0.8389 113.5761 2.8153 92.5928 3.0301 - 161 mat

Data [Xk,Yφ] represent the overall Thiele. @ 'size 10', the last Thiele interpolant is shown on this overall data. At this point, the expansion Tbar(p)can be exported to Mathcad up to MCD 11. Smath and Mathcad native kernel do take the continued fraction. Other softwares [OriginLab 7.5, Kalculator, Excel ...] will have to be scripted bottom to top.

size 10 ≡

... so, the remaining of the demo will be according to the deprecated convergent 10

The size 10 continued fraction. p Tbar 6.97 p 0.01 - 0.0015 p 0.05 - 64.6502 p 0.1 - 0.0121 p 0.25 - 58.9206 p 0.5 - 0.0446 p 0.75 - 55.8065 p 1 - 0.1254 p 2.5 - 85.7831 p 5 - 0.4473 / + / + / + / + / + / + / + / + / +

Optimiz => numeric Tbar p 0.01 size 0.025 range : i 1 p rows range u i el p i el : ρ i el p i el Tbar eval : 21 line for u ρ augment 31 line :

Domestic pressure cooker 1.333 Tbar 1.667 Tbar 2 Tbar 31 mat 108.2 114.8 120.5 31 mat

108.2 114.8 120.5 31 mat

Tbar 1 99.6 + 20 red 1 5 mat XY 3 1 sys

p 0.01 10100 logpts :

low high N logpts linear-log populate step high low / log10 N / : i 1 N range x i el low 10 step i *^* : for x 41 line :

Optimiz => numeric Tlogpts i 1 p rows range u i el p i el : ρ i el p i el Tbar eval : 21 line for u ρ augment 21 line :

Tlogpts Tlogpts . 8 red plot :

Water saturation temperature °C <= p[bar]

1 99.6 + 40 blue 1 5 mat Tlogpts XY 3 1 sys

Procedure to J_Fraction

x f 6.97 x 0.01 - 0.0015 x 0.05 - 64.6502 x 0.1 - 0.0121 x 0.25 - 58.9206 x 0.5 - 0.0446 x 0.75 - 55.8065 x 1 - 0.1254 x 2.5 - 85.7831 x 5 - 0.4473 / + / + / + / + / + / + / + / + / + :

x f confrac x convert maple 10000 x * 6309 / 848507889931339803481000 / + 111710845198715706562677889125560080814500000000000 x 30288851625378698409623538760337148057735239186117934869635196740085050000000 / + 301208571564049986373806642390415626548940453056774685938741209411596239656467505712014632758460358357121527837384748160500000000000000 x 1851099814171466887511629987804627476496402173083836603754851396666802737923637514437906734798475110666750391668368908271214565215593918649493473823793755312540133658418546748086372344394780000000 / + 14377719008148800741917822072627254369670641674911262516837576537112912406469604428468956378380703233466284272291863601304105928457730240180399117454182829428617522163782382921623515824470936662510072847583655429580752093514176979048633462732815238436407024144027470320400000000000000 x 2931982996350764432489852693624348299476869318844755228772795966794032646204832786537694243611160165198550454513354234133540460460158271376178620037523618614462617464186765147478106391069181407656933529379824830500270528261140348254672795868166396056688113019162286285179791636426847042387657374595050280888904869473005693520186420277151837004845786750303700000000 / + 51762217446351539484846067223502594864658197699915375954300058123536509145907988452776297681702009061479098997007805837026423144348617351990509862294410039322526881383111981330610694624287224393483409815969041392045455358332354097811421311605320388533207368166992912542617703683054890588408574966820195304364115275676651132007285398353447112899762032333959190878313578251329717576731135489958496872690589512887885061278975096122343171089925777411430201914062500000000 x 1756563092824708338901837168873439719893584469472576160694275722830141869267013748020069414132999162692308675750510034768750956500686934495809647132240543607128233310760447653311312927632433886986381468208636484314421340738112175000000 / + (* / - (* / - (* / - (* / -

β 100001 * 6309 / 848507889931339803481000 / 30288851625378698409623538760337148057735239186117934869635196740085050000000 / 1851099814171466887511629987804627476496402173083836603754851396666802737923637514437906734798475110666750391668368908271214565215593918649493473823793755312540133658418546748086372344394780000000 / 2931982996350764432489852693624348299476869318844755228772795966794032646204832786537694243611160165198550454513354234133540460460158271376178620037523618614462617464186765147478106391069181407656933529379824830500270528261140348254672795868166396056688113019162286285179791636426847042387657374595050280888904869473005693520186420277151837004845786750303700000000 / 1756563092824708338901837168873439719893584469472576160694275722830141869267013748020069414132999162692308675750510034768750956500686934495809647132240543607128233310760447653311312927632433886986381468208636484314421340738112175000000 / 111710845198715706562677889125560080814500000000000 / 301208571564049986373806642390415626548940453056774685938741209411596239656467505712014632758460358357121527837384748160500000000000000 / 14377719008148800741917822072627254369670641674911262516837576537112912406469604428468956378380703233466284272291863601304105928457730240180399117454182829428617522163782382921623515824470936662510072847583655429580752093514176979048633462732815238436407024144027470320400000000000000 / 51762217446351539484846067223502594864658197699915375954300058123536509145907988452776297681702009061479098997007805837026423144348617351990509862294410039322526881383111981330610694624287224393483409815969041392045455358332354097811421311605320388533207368166992912542617703683054890588408574966820195304364115275676651132007285398353447112899762032333959190878313578251329717576731135489958496872690589512887885061278975096122343171089925777411430201914062500000000 / 101 mat :

β 1.585 213.1743 8.5952 1.7354 0.322 0.0368 889.7003 4.8273 0.1577 0.0036 101 mat

Raw J_Fraction β 1 el x * β 2 el + β 7 el x β 3 el + β 8 el x β 4 el + β 9 el x β 5 el + β 10 el x β 6 el + / - / - / - / -

β 1.585 213.1743 8.5952 1.7354 0.322 0.0368 889.7003 4.8273 0.1577 0.0036 101 mat :

Convenient formula for Engineering document

Export J_Fraction x J_Frac β 1 el x * β 2 el + β 7 el x β 3 el + β 8 el x β 4 el + β 9 el x β 5 el + β 10 el x β 6 el + / - / - / - / - :

Continued on length 'j'  of support points 'X' j K X x Thiele Optimiz NONE j <= rows(k) N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 71 line :

CREATE a WILD continued fraction on length 'j' of support points 'X'. This will construct a "Conventional Continued Fraction", compatible to various sources, some ancient not obsolete and unique of the kind of application.

°C 1 :

Pressurized cooker: °C/bar 1.333 J_Frac 1.667 J_Frac 2 J_Frac 31 mat °C 108.2 114.8 120.5 31 mat

size 10

Optimiz NONE ... Export the RHS member size K X x Thiele K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el x X 6 el - K 7 el x X 7 el - K 8 el x X 8 el - K 9 el x X 9 el - K 10 el / + / + / + / + / + / + / + / + / +

recover the support points
....
recover Thiele Convergents 'K' X Xk 1 size 1 - (range el : K Yφ 1 size range el : 21 sys

X 0.01 0.05 0.1 0.25 0.5 0.75 1 2.5 591 mat :

K 6.97 0.0015 64.6502 0.0121 58.9206 0.0446 55.8065 0.1254 85.7831 0.4473 101 mat :

x Cfr K 1 el x X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el x X 6 el - K 7 el x X 7 el - K 8 el x X 8 el - K 9 el x X 9 el - K 10 el / + / + / + / + / + / + / + / + / + :

x c 10 x ≤ (x 10 ≤ (& cases :

XY PT . 14 black plot :

Water °C(bar) saturation

XY x Cfr x c * 2 1 sys

°C 1 :

Home pressurized cooker 1.333 Cfr 1.667 Cfr 21 mat °C 10811521 mat