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Filling the segment project

1. Prepare the project

2. Construct the project

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

Density of random N 500 : (Density of random ≡ r 1 : (Unit circle ≡ 21 line

h 0.5 :

level 0 h ≤ (h 2 ≤ (& (≡

xmax 1 : xmin 1 - : 21 line

ymax 1 - h + (: ymin 1 - : 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

On large number of points, pts(data,spec) plots about 12 times faster than Smath plotG(,,,,,). This module is clean/short.

x y1 x 1 + (2 r * x 1 + (- (* sqrt : x y2 x y1 - : 21 line

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x y1 + 0 ≥ (y x y1 - 0 ≤ (* :

t0 1 time :

A routine that finds an array of N random 
pairs in the critical region. Pts the routine is a compressor it maintains density 'N' within the variable region i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 51 line :

a b c spec . 6 #A0A0A0 13 mat :

pts Pts a b c spec pts :

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

1 time t0 - 8 s *

Mesh the datum level U x 11 - h + (2^- sqrt : u x - x 0.95 x * - range : i 1 u rows range datum i el u i el f : idx i el u i el : 21 line for idx datum augment 41 line :

3. View the project

plot x y1 x y2 pts λ 41 sys :

a b c spec . 8 black 13 mat :

4. Technical end use

λ U a b c spec pts :

The project runs on the UnitCircle ...

Partially filled

plot

level 0 h ≤ (h 2 ≤ (& (≡

... it must be converted wrt XTR [level transmitter 0..100%] and the C(x) function [0 .. 1]

Technical 'h'

h 0.5 h * :

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

x cπ 2 0.5 x * 0.5 + C * 1 - :

h C 0.196

cπ x 1 - 1 0.995 - range : i 1 x rows range c i el x i el cπ : for x c augment 31 line :

References: the C(x) formula agrees within the 5 D tables in: 1. Chemical Engineer's Handbook [Perry, Chilton, Kirkpatrik] 2. TEMA [Tubular Exchanger Manufacturers Association]

Explain ...

Figure:: Partially filled

x y1 x y2 pts λ cπ 5 1 sys

To compute the area of the segment, "solving" is not an option. The C(x) is pretty old, accurate to 5 decimals+. It is a factor to 'π' in formulas involving the measure of 'h', for instance: the sole possible measurement in partially filled pipe, horizontal tank, spherical tank as well as partially filled elliptical bassin ... In the "Figure", we have superposed this curve over the partially filled circle. For experimental 'h', the area of the segment will be: C(h)*π*(D^2)/4. Observe that in the figure, the graph of C(x) has been shifted left/down to ease the pictorial understanding of the project.

h 0.25 : h Area 2 r * (2^π 4 / * h Cπ * : 21 line

Recast the C(x) function x Cπ 1 - x ≤ (x 1 ≤ (& x C invalid if :

C(x) graph and Cπ coefficient

x Cπ 0 h h Cπ + 20 red 1 5 mat 3 1 sys

================

h h Cπ h Area 31 mat 0.25 0.196 0.614 31 mat

Plot arguments Figure x y1 x y2 pts λ cπ 51 sys ≡

jmGiraud: 2015_12_29

x f 1 x 2^- sqrt ± : x C 1 π / 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* : x cπ 2 0.5 x * 0.5 + C * 1 - : 31 sys

x f x cπ 2 1 sys

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