ternary mixtures (english version)

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ternary mixtures (english version)

Beitragvon Sculpteur » 28.08.2022 18:47

DISCLAIMER:
Despite careful examination, the author does not assume any guarantees or liability for the information, calculations and logic contexts presented in this script.
Despite careful examination, the author has no knowledge of whether the topic discussed in this script has ever been discussed and published in the same or a different way and is grateful for any further information.

Determining the amount of possible mixing ratios in ternary mixtures

Abstract
This article deals with the possiblities of determining the complete amount of mixing ratios in (static) ternary mixtures. It is shown, that the majority of a mixture gives the summation key for determinming the number of possible mixing ratios.

1. Possible amounts of mixing ratios in static ternary mixtures
The complete number of possible mixing ratios in a (static; e.g. alloy) ternary mixture is determined by the majority of a specific static mixture.
The majority in a static ternary mixture denotes the largest possible proportion of one of three components.
Static ternary mixtures are mixtures in which each of the 3 components used is always present (in a specific; from a minimum up to the by a majority of a specific static mixture defined proportion.
Since each component in a static ternary mixture must be contained with a minimum (defined by the graduation of a mixture), the distribution of a ternary mixture in the majority is given by the value of the majority, because by x = first component; y = second component and z = third component it follows x + y + z = 100\% or Mixture 100\% = xyz.
The majority of a static ternary mixture is determined by the staggering of a specific mixture, i.e. the specification of the minimum proportion increment (e.g. 10 increments or 1 increments, e.g. per thousand increments or parts per million):

1.1 Example:
With a staggering in 10\% increments, the majority of the mixture is 8 : 1 : 1,
because 80\% + 10\% + 10\% = 100\%.
Formula for determining the majority of a static ternary mixture
The formula for determining the majority of a static mixture follows from the proportions of the components to each other, it is (for example in percent) xyz% - ( y% - z%) = largest possible proportion in a mixture by x = largest proportion; y and z = proportion of minimum stakes.
When we calculate arithmetic, we can calculate for example (when the graduation of a mixture is for example 10% , the minimum stakes arithmetic is 1 (one), because:

1 = 10%
x = 10 - 2
y = 1
z = 1

xyz = 10 (or 100%)
x = 8 (or 80%)
y = 1 (or 10%)
z = 1 (or 10%)

it follows, that the majority of a static ternary mixture with graduation of 10%-steps is 8 : 1 : 1.

2. Determining amount of possible mixing rations in static ternary mixtures via summation key
The total number of all possible mixing ratios in a static ternary mixture is determined via the majority: The numerical value of the majority of a static ternary mixture corresponds to the summation key for determining the number of possible mixing ratios of a mixture.
The summation key, derived from the majority of a static ternary mixture, specifies the degree to which the natural numbers, starting from their origin 1, must be summed up together; Example:

SUM = SIGMA

In the case of a static ternary mixture with a majority of 8 : 1 : 1, the summation key of the mixture is 8 and the total of SUM(1...8) is 36 because 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

2.1 Example
a) A static ternary mixture with a graduation in 1\%-steps has SUM(1...98) possible mixing ratios, because the majority of such mixture is 100 - 2 = 98 and the summation key of the mixture is 98. It follows, that there are SUM(12...98) = 4.851 possible mixing ratios for such mixture, because
(1 + 2 + 3 + ... + 96 + 97 + 98) = 4.851.

b) A static ternary mixture with a graduation in 0.1%-steps with a majority of 998 : 1 : 1 with a summation-key of 998 has the possible mixing ratios of
SUM(1...998) = (1 + 2 + 3 + ... + 996 + 997 + 998) = 498.501.

3. Basic principle of scrambling in ternary mixtures
The basic scrambling principle operating in a static ternary mixtures is:
(which here develops according to a specific logic, used by the author)

[component I] ; [component II] ; [component III] ; [pos.]
x, y, z[I]
x, z, y [II]
y, x, z [III]
y, z, x [IV]
z, x, y [V]
z, y, x [VI]

This scrambling principle applies to every possible batch in static ternary mixtures. This basic principle of scrambling is expressed in a complete listing of all the possibilities of a specific static ternary mixture - depending on the way of listing and the associated structure-forming principle, more or less complexly nested.
complete list of mixing ratios with a majority of 4 : 1 : 1 by a graduation of 16.666...%

component I : component II : component III [pos.]
1 : 1 : 4 [I]
1 : 2 : 3 [II]
1 : 3 : 2 [III]
1 : 4 : 1 [IV]
2 : 1 : 3 [V]
2 : 2 : 2 [VI]
2 : 3 : 1 [VII]
3 : 1 : 2 [VIII]
3 : 2 : 1 [IX]
4 : 1 : 1 [X]

4. Overview of examples of numbers of possible mixing ratios for specific majorities
[graduation] ; [majority (arithmetic)] ; [majority (percent)] ; [SUM] ; [mixtures (ratios)]
[10,00%] ; [8 : 1 : 1] ; [80% : 10% : 10%] ; [SUM(1...8)] ; [36]
[5,00%] ; [18 : 1 : 1] ; [90% : 10% : 10%] ; [SUM(1...18)] ; [4.851]
[1,00%] ; [98 : 1 : 1] ; [98% : 1% : 1%] ; [SUM(1...98)] ; [498.501]

5. Row of Triangular numbers for determining mixture-varieties of static ternary mixtures
From the calculation method, it follows that the summation of the total number of possible mixtures always comes from the series of triangular numbers inwhich is
{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... }.
In the result we can use the row of triangular numbers to simple determine the amounts of static ternary mixtures fundamental.
It follows, that we can simple calculate the amount of possible mixing rations of each specific majority in strictly order from origin majority 1 : 1 : 1.
(for this purpose it´s more usefull to flip the way of presentations of majorities for example from
4 : 1 : 1 to 1 : 1 : 4.)

6. mixture-varieties of static ternary mixtures

[majority (arithmetic)] ; [graduaty (percent)] ; [SUM] ; [ratios (mixtures) (row of triangular numbers)] ; [pos]
[1 : 1 : 1] ; [100.00%] ; [SUM(1...1)] ; [1] ;[1]
[1 : 1 : 2] ; [25.00%] ; [SUM(1...2)] ; [3] ; [2]
[1 : 1 : 3] ; [20.00%] ; [SUM(1...3)] ; [6] ;[3]
[1 : 1 : 4] ; [16.66...%] ; [SUM(1...4)] ; [10] ;[4]
[1 : 1 : 5] ; [14.28...%] ; [SUM(1...5)] ; [15] ; [5]
[1 : 1 : 6] ; [12.50%] ; [SUM(1...6)] ; [21] ; [6]
[1 : 1 : 7] ; [11.11...%] ; [SUM(1...7)] ; [28] ; [7]
[1 : 1 : 8] ; [10,00%] ; [SUM(1...8)] ; [36] ; [8]
[1 : 1 : 9] ; [9.09...%] ; [SUM(1...9)] ; [45] ; [9]
[1 : 1 : 10] ; [8.33%] ; [SUM(1...10)] ; [55] ; [10]


SOURCES

(For the sake of simplicity, the author used Wikipedia sources in this script.)

Seite „Dampf-Flüssigkeit-Gleichgewicht“. In: Wikipedia – Die freie Enzyklopädie. Bearbeitungsstand: 14. Juli 2020, 11:03 UTC. URL: https://de.wikipedia.org/w/index.php?ti ... =201866759 (Abgerufen: 15. August 2022, 08:52 UTC)

https://www.dlr.de/mp/Portaldata/22/Res ... eich-k.pdf

Seite „Dreieckszahl“. In: Wikipedia – Die freie Enzyklopädie. Bearbeitungsstand: 4. April 2022, 16:17 UTC. URL: https://de.wikipedia.org/w/index.php?ti ... =221784081 (Abgerufen: 15. August 2022, 08:51 UTC)

Hoppe, 2022:
viewtopic.php?f=138&t=6667
Ein Physiker-Witz: "Was ist der Unterschied zwischen Doppelmoral und Dopplermoral?"
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