cea881c4-a2ea-4c77-a0b7-76b99383e4af 34 4 5 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 :

Data 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 : XY PT . 14 black plot : 31 line

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 6.97 000000 32.88 0.0015 00000 45.81 0.0023 64.6502 0000 64.96 0.0041 77.0761 0.0121 000 81.32 0.0066 89.1681 0.0163 58.9206 00 91.76 0.0087 97.4437 0.0198 64.5207 0.0446 0 99.61 0.0107 103.9078 0.0229 69.0993 0.0491 55.8065 127.41 0.0207 128.0685 0.0378 87.3027 0.0705 67.7642 151.84 0.0344 150.452 0.0571 105.4693 0.0967 81.6822 179.98 0.0577 177.0509 0.0881 128.2791 0.137 100.1869 223.96 0.1152 219.5862 0.1607 166.5098 0.2277 132.4587 263.94 0.1945 258.8183 0.257 203.1257 0.3433 164.9259 311 0.3289 305.3417 0.4151 247.5293 0.5275 205.527 342.16 0.4475 336.2607 0.5519 277.4065 0.6843 233.3444 365.75 0.5574 359.7045 0.6775 300.1815 0.8269 254.7095 373.71 0.5999 367.6194 0.7258 307.8832 0.8817 261.9423 167 mat

cont 16

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 ≡

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 °C *

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 green 1 5 mat Tlogpts XY 3 1 sys

Procedure to J-Fraction . . .

. . . is a procedure used to reduce the Thiele native Continued Fraction. It originates from the very first IBM Main Frame

prior to the more efficient Padé rational fraction. The J_Frac is native in Mathcad/Mathsoft.

In Smath, you have to construct from maple(convert(f(x),confrac,x)) . . . There is a possible pitfall . . . Explain:

In some application(s), the Maple expansion is much too long, some β will not evaluate, The technique to rescue the red

is to isolate the 'K' coefficients and round off to less digits.

]]>

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 1 x β 3 el + β 8 el 1 x β 4 el + β 9 el 1 x β 5 el + β 10 el 1 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 + / - / - / - / - :

sanity check 1.333 J_Frac 1.667 J_Frac 2 J_Frac 31 mat 108.2 114.8 120.5 31 mat

CREATE a WILD continued fraction on length 'j' of support points 'X'

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 :

Optimiz NONE ... Export the RHS member 10 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 / + / + / + / + / + / + / + / + / +

X Xk 1 size 1 - (range el : K Yφ 1 size range el : 21 line

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

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 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 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 / + / + / + / + / + / + / + / + / + :

This example produces two branches. Show these two branches distinctively.

x Th 1 X 1 el x ≤ (x 8 ≤ (& cases :

x ThLHS 1 X 1 el 7 - x ≤ (x 2 - ≤ (& cases :

x Thiele x Th * x Thiele x ThLHS * 2 1 sys

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

x Th 1 0.01 x ≤ (x 8 ≤ (& cases :

Export J_Fraction ...  apply more maths with caution.
Integration of  J_Frac(x) will fail in the low range,
native Smath Simpson integrator is inadequate, simply.
For the low range region, apply Carlos "Adaptive". 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 + / - / - / - / - :

x J_Frac x 48 int 634

x J_Frac x diff x Th * 0.5 x J_Frac x Th * ( * 2 1 sys