plot(data,#,size,#)

e940aeb4-9cc3-4504-909e-fa1ed42d43ba 120 4 5 decimal

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

plot(data,#,size,#)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

DeCasteljau

Understanding SVG

Model from jmG Mathcad 8 Pro. made work in Smath by Dave Carpi

t = interpolation point in [0,1]
p = control points t p DeCasteljau De Casteljau SMath M 1 t - (p 1 row * t p 2 row * + : p rows 2 > i 2 p rows 1 - range M M 1 t - (p i row * t p i 1 + row * + stack : for t M DeCasteljau 21 line M if 31 line :

t = range of interpolation points in [0,1]
p = control points t p DeCasteljauRange M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 31 line :

ndp =  number of discretization points (> 1)
p = control points ndp p DeCasteljauNDP t 011 ndp 1 - / range : M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 41 line :

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

XY 100020201015 - 05050015 - 26 mat transpose :

a b c spec . 14 black 13 mat :

xy XY a b c spec pts :

p svg 1010015 - 15 - 023 mat 000005023 mat 0102050605023 mat 202020500023 mat 201010015 - 15 - 23 mat 15 mat transpose * :

svg 1 :

NDP 100 :

C 0200020000505025 mat transpose :

level 510151022 mat :

bin NDP p 1 el DeCasteljauNDP :

i 2 p length range bin bin NDP p i el DeCasteljauNDP stack : for

C bin level 3 1 sys

A 1 : a 0.025 : g 9.8067 : 13 mat

h time A a / 2 g / sqrt * 1 h - sqrt * :

0 time 18.06

x ellapsed 0 time x time - :

ratio level % 0.75 :

x c 10 x ≤ cases :

% ellapsed 9.03

x time x c * x ellapsed x c * % % ellapsed % 0 2 2 mat 0 18 . 14 red 1 5 mat 4 1 sys

X x time x ellapsed - 0 ≡ x 0.5 1 solve : 0.75

½ time ellapsed @ 0.75 discharge X ellapsed 9.032

Theoretical unary discharge . . .

The ratio A/a is typical to enable demo-graph. User will adapt to reflect installation,

i.e: consider a calibrated discharge ... demonstrated below for given 'h' and singular 'C'.

--------------------------------------

Experimental drainage . . . on referenced viscosity & mouthpiece.

Data segment represents the time required for discharging 10 rings of equal cylindrical height .

The pilot tank is cylindrical, it could be conical as well. Timing is sufficiently accurate for purpose.

This pilot experiment takes into account miscellaneous parameters [ mouthpiece discharge coefficient,

the viscosity of the fluid ... ]. From the collected data in this example, we can find a function

that will represent the timing for further analytical maths as they may apply.

Things happen in tank(s), mostly chemical processing. You can adapt the discharge by pressurizing.

]]> Genfit Functions ... plotG ... xd(L,H,N)

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 31 line :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Applicable parts are: Optimiz "numeric" XY f φ β βNlin X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx-> raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

Vector gradient ... Optimiz Symbolic x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

a 0.08942 - : b 0 : c 0.3 : x f b c x * + 1 a x * + / 8.01 + : 41 line

Segment 1 8.36 2 8.84 3 9.41 4 10.1 5116 12.1 7 13.8 8 16.3 92110 36.4 102 mat :

Time Segment o 5 black plot : x c 10 x ≤ (x 10.25 ≤ (& cases : 21 line

1 f 10 f 21 mat 8.34 36.37 21 mat

x f x c * Time 3 1 sys

Segment 2 col sum 147.31

XY 1 8.36 2 8.84 3 9.41 4 10.1 5116 12.1 7 13.8 8 16.3 92110 36.4 102 mat :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

homographic model function x β f 1 β 1 el x * + 1 β 2 el x * + / (0.77^β 3 el + :

0.77 better SSD than '1'

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

0.3 0.09 - 831 mat

<= Gradient Vector Function x β φ x β f 3 φ :

β β 1 el 0.3 0.09 - 831 mat : iter 5 : i 1 : i iter ≤ β i 1 + el XY f φ β i el βNlin : i i 1 + : 21 line while β β iter el : 51 line :

β β :

Relative residuals δ i 1 XY rows range Δ i el 1 Y i el X i el β f / - eval : for X Δ augment 21 line :

β 0.4478 0.0932 - 6.958 31 mat

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 0.037

FIT U 010 0.01 range : i 1 U rows range fit i el U i el β f eval : for U fit augment 31 line :

1 β f 10 β f 21 mat 8.39 36.43 21 mat

data XY o 5 black plot :

x c 10 x ≤ (x 10 ≤ (& cases :

x image 0 β f x β f - 10 β f + :

X x β f x image - 0 ≡ x 910 solve : 9.1368

time ellapsed @ X discharge X β f 22.1936

Experimental data from in-situ metered timing of the free disharge [mouthpiece, viscosity ...] cylindrical tank.

10 100 10 10 - 2 2 mat FIT x image x c * data X X β f X 10 - 2 2 mat 6 1 sys

X 10 / 0.9137

1 X β f Y 10 el / - 0.3903

@ 91.4 % discharge 39% of time remains

SI units ρ.sph 8.0 : ρ.Liq 1.2 : R 0.005 : t 10 : h 1 : V h t / : g 9.8065 : 71 sys

Time Segment o 5 black plot :

xo 9 : (point of normalcy ≡

cp 1 :

a xo xo β f xo diff (* xo β f - : 44.497

in-situ, free-fall drainage of the cylindrical tank

x xo β f xo diff ( * a - x β f x c * xo 25 xo 10 - 2 2 mat Time 4 1 sys

Confirm cp ... ~ twice  cp H2O 0.01°C μ ρ.sph ρ.Liq - V / (g * 29 / * R 2^* 1000 * : cp 3.7

https://www.engineeringtoolbox.com/flow-liquid-water-tank-d_1753.html

[1] Segment [2] Average Height between Segment and Aperture (m) [3] Average Flow (m3/s) [4] Volume in Segment (m3) [5] Time to Drain Segment (s) ----------------------------------- 0 2.85 0.0359 0.3 8.36 1 2.55 0.034 0.3 8.84 2 2.25 0.0319 0.3 9.41 3 1.95 0.0297 0.3 10.1 4 1.65 0.0273 0.3 11 5 1.35 0.0247 0.3 12.1 6 1.05 0.0218 0.3 13.8 7 0.75 0.0184 0.3 16.3 8 0.45 0.0143 0.3 21 9 0.15 0.00823 0.3 36.4 3 147

image

V.jet 2 Z.surface Z.spout - g * sqrt ≡ (2 g * Δz * sqrt ≡

'C' discharge coefficient  [singular losses] Q Area.jet V.jet * ≡ (C Area.spout * V.jet * ≡

Unary Theoretical Torricelli Unary Singular 'C' Torricelli

A 1 : a 0.025 : g 9.8067 : 13 mat

A 1 : a 0.025 : g 9.8067 : C 0.5 : 14 mat

h time A a / 2 g / sqrt * 1 h - sqrt * :

h Time C A a / * 2 g / sqrt * 1 h - sqrt * :

0 time 18.06

0 Time 9.032

x ellapsed 0 time x time - :

x Ellapsed 0 Time x Time - :

x c 10 x ≤ cases :

x c 10 x ≤ cases :

Explain: for the unary time [1], time(0) and Time(0) represent the height 'h' in the drain tank

]]> x time x c * x ellapsed x c * 2 1 sys x Time x c * x Ellapsed x c * 2 1 sys

For a tank of given 'h' and discharge 'C' Torricelli is calibrated over 'C'

C x * time x c * C x * ellapsed x c * 2 1 sys

The maths of the drainage project

]]>

project Engineering

]]>

min 1 :

m 1 :

g 9.807 :

Torricelli constant wrt [min] Tor 2 g / sqrt 160 / * : 0.0075265

.... m ..... m ........ N/A H1 15 : H2 0 : 21 sys D 5 : d 0.05 : 21 sys Cd 0.8 : 13 mat

Time drain cylinder
-----
-----
Time drain cone Tcyl 1 Cd / D d / (2^* Tor * H1 sqrt * : Tcone 13 / (1 Cd / D d / (2^* Tor * H1 sqrt * * : 21 sys

Tcyl Tcone 21 mat min 364.4 121.5 21 mat

Cylindrical tank. . . . . . . . .

]]>

The cone parameters β Tcyl H1 sqrt / (0.5 21 mat : 94.0818 0.5 21 mat

[min] to drain 'h' h β tcyl β 1 el h β 2 el^* :

'h' drained in [t min] t β hcyl t β 1 el / ln β 2 el / exp :

drain  [m] from top H1 drain 8 :

tt 115.5 :

drain Vidange H1 drain - :

Time T Tcyl tt - :

Tcyl Vidange β tcyl - min 115.5

hh H1 T (β hcyl - : m 8

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

xh 12 :

xt 120 :

Th Tcyl H1 xh - (β tcyl - :

hT H1 Tcyl xt - (β hcyl - :

Th min 201.4

hT 8.253

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

h sol Tcyl h β tcyl - :

t Φsol H1 Tcyl t - (β hcyl - :

x h 10 x ≤ (x H1 ≤ (& cases :

x Φ 10 x ≤ (x Tcyl ≤ (& cases :

Solution: given 't' find => 'h' ----- [min] to drain 'h' -----

x - H1 + sol x h * H1 500 H1 100 - 2 2 mat 2 1 sys

Reciprocal: find 'h' <= given 't' ------ [h] drained in 't' ------

x Φsol x Φ * Tcyl 20 Tcyl 10 - 2 2 mat 2 1 sys

Normal cone. . . . 1/3 cone . . . .

]]>

The cone parameters β Tcone H1 2.5^/ (2.5 21 mat : 0.1394 2.5 21 mat

time to drain 'h' h β t β 1 el h β 2 el^* :

'h' drained in [t min] t β h t β 1 el / ln β 2 el / exp :

drain  [m] from top H1 drain 6.262 :

tt 90 :

drain Vidange H1 drain - :

Time T Tcone tt - :

Tcone Vidange β t - min 90

hh H1 T (β h - : m 6.262

Horizontal-Cylindrical tank. . . . . . Unary_Sphere . . .

]]>

x C 0.31831 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

L 1 : D 1 : d 0.025 : Cd 0.8 : α 1 : 15 mat

x Time α L Cd / * D d / (2^* Tor * 1 h sqrt / h 0 x C int * ≡

Options/Calculations/Int_acc 5000 ... sanity x time α L Cd / * D d / (2^* Tor * 1 h sqrt / h 0.00001 x C int eval * :

MCD 8.1
MCD 21.3
MCD 30.1 0.125 time 0.5 time 1 time 31 mat 8.3 24.3 36.7 31 mat

lowest limit wrt 1/sqrt(h)

x f α L Cd / * D d / (2^* Tor * 2 x C sqrt * * :

Y 15 :

solving level for given time [Y] X x f Y - 0 ≡ x 01 solve : 0.2965

X f 15

Y x f x c * X 40 X 10 - 2 2 mat 3 1 sys

[194, 222, 241]

C(x) exact 5 decimals TEMA, Perry-Chilton-Kirpatrik x C 0.31831 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

Unary discharge time of the sphere

2 x C sqrt * 2 1 x C sqrt - ( * 2 1 sys

@ ½ time, [1-0.3 = 0.7] 70% level  remains to drain
f(x) has NO explicit inverse, only Maple RootOf X 2 x C sqrt * 21 x C sqrt - (* - 0 ≡ x 01 solve : 0.298

2 X C sqrt * 1

x C 0.31831 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

x sol t C x - 0 ≡ t 01 solve :

X 0.75 sol : 0.702

X C 0.75

x C x sol 2 1 sys