Plot points module

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Plot points module

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

hatch

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x)
in x-range from x1 to x2 1 f 1 g x.1 x.2 n hatch dx x.2 x.1 - n / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

If you want to see what's in the collapsed areas, copy/paste in blank sheet. Opening a collapsed area dos NOT push down regions below. Thus, on re-collapsing again, it collapses the overlaping regions that were not out of the expanded "hatch".

Area of Circular Segment

In the Industry, area of circular segments aren't calculated from trigonometric formulas, i.e: from things that can't be measured. They were hand tabulated long time ago, before analog and digital computing machinery existed. The 'C' factor is found in Engineering Handbooks [Perry-Chilton ,,, TEMA ...etc]. For practical purposes [Mechanical, Instrumentation ...] 'C' is approximated exact to ± 0.00000 wrt to typical Perry, TEMA. 'C' is the C*π factor in formulas relating to segments where only 'h' between datum and level is measurable, C:=C(h) plotted below. Remark that 'C' auto ranges itself 0..1 Typical applications: mechanical inertia, partially filled pipes, volume in horizontal-cylindrical vessel, spherical tank, torus ...etc. In this document, we have neglected the "elliptical" caps [more details can be found in TEMA],,, TEMA = Tubular Exchange Manufacturer Association].

x C 1 π / 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

x C 1 1 sys

r 1 :

<= unit radius

x y1 x 1 + (2 r * x 1 + (- (* sqrt :

Unit Circle

x y2 x y1 - :

UnitArea 1 :

Level 33 :

% Level [XTR]

h Level 100 / :

h 0.33

2 H 2 col (* 1 -

<= formated to meet centered Unit 'r' circle

x datum 1 - :

2 0.5 x * 0.5 + C * 1 -

Area h C π 4 / * 2 r * (2^* :

Area UnitArea 0.9041

** Sanity check ** Full---------------- Empty -------- ½ filled ---------

π 4 / 2 r * (2^* 3.14159

π 4 / 02^* 0

π 8 / 2 r * (2^* 1.5708

1 C π 4 / * 2 r * (2^* 3.14159

0 C π 4 / * 2 r * (2^* 0

0.5 C π 4 / * 2 r * (2^* 1.5708

u f 0 u * 2 h * 1 - (+ (:

U x 12 h * 1 - (2^- sqrt : u x - x 0.9 x * - range : i 1 u rows range crest i el u i el f : idx i el u i el : 21 line for idx crest augment 41 line :

H u 01 0.1 range : i 1 u rows range H i el h u i el * : idx i el 0 i * : 21 line for idx H augment 31 line :

Operate the demo: as you change 'Level' [0..100%] the "hatching" & "XTR % read" will track 'Level'.

xtr 2 h * 1 - (0.175 + :

m 0.5 - xtr XTR % read 12 black 15 mat :

λevel U 1 col U 2 col . 14 black plotG :

XTR H 1 col 2 H 2 col (* 1 - . 14 orange plotG :

x.1 12 h * 1 - (2^- sqrt - :

x.2 12 h * 1 - (2^- sqrt :

<= hatching system

x f 0 x * 2 h * 1 - (+ (:

x g x y2 :

'C' π circular segment

x y1 x y2 2 0.5 x * 0.5 + C * 1 - x datum m λevel x f x g x.1 x.2 20 hatch 7 1 sys

x fnct mtp U 1 - 1 0.99 - range : j 1 U rows range vy j el U j el fnct : 11 line for U vy augment 31 line :

Discrete

Area 0.9041

Y1 x y1 mtp :

Y2 x y2 mtp :

Plot memo [system] x y1 Y1, discrete ≡ x y2 Y2, discrete ≡ 2 0.5 x * 0.5 + C * 1 - x datum m λevel x f x g x.1 x.2 20 hatch 81 sys

Technical note: If you don't find a XTR that linearises wrt C(h), you should be able to do it with an analog input device or at the central control system via some "Expert Extension" of the control software. Ancient float XTR had cam linearising mechanism.

Volumes in consistant units

V h C π 4 / * D 2^* L * ≡

horizontal tank

V h C 43 / * Π * r 3^* ≡

spherical tank

h measured level ≡

D L r 13 mat ID, Ir, Length ≡

Yet, I have not been able to reproduce Mathcad 11 random fill of the segment.

jmGiraud: 2015/09/15