Glass batch calculation
Encyclopedia
Glass batch calculation or glass batching is used to determine the correct mix of raw materials (batch) for a glass
melt.
equation:
with NB and NG being the molarities 1-column matrices
of the batch and glass components respectively, and B being the batching matrix
. The symbol "T" stands for the matrix transpose
operation, "-1" indicates matrix inversion, and the sign "·" means the scalar product. From the molarities matrices N, percentage
s by weight (wt%) can easily be derived using the appropriate molar mass
es.
, 12 Na2O
, 10 CaO
, 5 Al2O3
, 1 K2O
, 2 MgO
, 3 B2O3
, and as raw materials are used sand
, trona
, lime
, albite
, orthoclase
, dolomite
, and borax
. The formulas and molar masses of the glass and batch components are listed in the following table:
The batching matrix B indicates the relation of the molarity in the batch (columns) and in the glass (rows). For example, the batch component SiO2 adds 1 mol SiO2 to the glass, therefore, the intersection of the first column and row shows "1". Trona adds 1.5 mol Na2O to the glass; albite adds 6 mol SiO2, 1 mol Na2O, and 1 mol Al2O3, and so on. For the example given above, the complete batching matrix is listed below. The molarity matrix NG of the glass is simply determined by dividing the desired wt% concentrations by the appropriate molar masses, e.g., for SiO2 67/60.0843 = 1.1151.
The resulting molarity matrix of the batch, NB, is given here. After multiplication with the appropriate molar masses of the batch ingredients one obtains the batch mass fraction matrix MB:
or 
The matrix MB, normalized to sum up to 100% as seen above, contains the final batch composition in wt%: 39.216 sand, 16.012 trona, 10.242 lime, 16.022 albite, 4.699 orthoclase, 7.276 dolomite, 6.533 borax. If this batch is melted to a glass, the desired composition given above is obtained. During glass melting, carbon dioxide
(from trona, lime, dolomite) and water
(from trona, borax) evaporate.
Another simple glass batch calculation can be found at the website of the University of Washington.
), the batch composition must be determined by optimization techniques
.
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...
melt.
Principle
The raw materials mixture for glass melting is termed "batch". The batch calculation is based on the common linear regressionLinear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
equation:
with NB and NG being the molarities 1-column matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
of the batch and glass components respectively, and B being the batching matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. The symbol "T" stands for the matrix transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
operation, "-1" indicates matrix inversion, and the sign "·" means the scalar product. From the molarities matrices N, percentage
Percentage
In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...
s by weight (wt%) can easily be derived using the appropriate molar mass
Molar mass
Molar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...
es.
Example calculation
An example batch calculation may be demonstrated here. The desired glass composition in wt% is: 67 SiO2Silicon dioxide
The chemical compound silicon dioxide, also known as silica , is an oxide of silicon with the chemical formula '. It has been known for its hardness since antiquity...
, 12 Na2O
Sodium oxide
Sodium oxide is a chemical compound with the formula Na2O. It is used in ceramics and glasses, though not in a raw form. Treatment with water affords sodium hydroxide....
, 10 CaO
Calcium oxide
Calcium oxide , commonly known as quicklime or burnt lime, is a widely used chemical compound. It is a white, caustic, alkaline crystalline solid at room temperature....
, 5 Al2O3
Aluminium oxide
Aluminium oxide is an amphoteric oxide with the chemical formula 23. It is commonly referred to as alumina, or corundum in its crystalline form, as well as many other names, reflecting its widespread occurrence in nature and industry...
, 1 K2O
Potassium oxide
Potassium oxide is an ionic compound of potassium and oxygen. This pale yellow solid, the simplest oxide of potassium, is a rarely encountered, highly reactive compound...
, 2 MgO
Magnesium oxide
Magnesium oxide , or magnesia, is a white hygroscopic solid mineral that occurs naturally as periclase and is a source of magnesium . It has an empirical formula of and consists of a lattice of Mg2+ ions and O2– ions held together by ionic bonds...
, 3 B2O3
Boron trioxide
Boron trioxide is one of the oxides of boron. It is a white, glassy solid with the formula B2O3. It is almost always found as the vitreous form; however, it can be crystallized after extensive annealing...
, and as raw materials are used sand
Sand
Sand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...
, trona
Trona
Trona ; Na3•2H2O is an evaporite mineral. It is mined as the primary source of sodium carbonate in the United States, where it has replaced the Solvay process used in most of the rest of the world for sodium carbonate production.- Etymology :The word "trona" comes to English by way of either...
, lime
Limestone
Limestone is a sedimentary rock composed largely of the minerals calcite and aragonite, which are different crystal forms of calcium carbonate . Many limestones are composed from skeletal fragments of marine organisms such as coral or foraminifera....
, albite
Albite
Albite is a plagioclase feldspar mineral. It is the sodium endmember of the plagioclase solid solution series. As such it represents a plagioclase with less than 10% anorthite content. The pure albite endmember has the formula NaAlSi3O8. It is a tectosilicate. Its color is usually pure white, hence...
, orthoclase
Orthoclase
Orthoclase is an important tectosilicate mineral which forms igneous rock. The name is from the Greek for "straight fracture," because its two cleavage planes are at right angles to each other. Alternate names are alkali feldspar and potassium feldspar...
, dolomite
Dolomite
Dolomite is a carbonate mineral composed of calcium magnesium carbonate CaMg2. The term is also used to describe the sedimentary carbonate rock dolostone....
, and borax
Borax
Borax, also known as sodium borate, sodium tetraborate, or disodium tetraborate, is an important boron compound, a mineral, and a salt of boric acid. It is usually a white powder consisting of soft colorless crystals that dissolve easily in water.Borax has a wide variety of uses...
. The formulas and molar masses of the glass and batch components are listed in the following table:
| Formula of glass component | Desired concentration of glass component, wt% | Molar mass of glass component, g/mol | Batch component | Formula of batch component | Molar mass of batch component, g/mol |
|---|---|---|---|---|---|
| SiO2 | 67 | 60.0843 | Sand | SiO2 | 60.0843 |
| Na2O | 12 | 61.9789 | Trona | Na3H(CO3)2*2H2O | 226.0262 |
| CaO | 10 | 56.0774 | Lime | CaCO3 | 100.0872 |
| Al2O3 | 5 | 101.9613 | Albite | Na2O*Al2O3*6SiO2 | 524.4460 |
| K2O | 1 | 94.1960 | Orthoclase | K2O*Al2O3*6SiO2 | 556.6631 |
| MgO | 2 | 40.3044 | Dolomite | MgCa(CO3)2 | 184.4014 |
| B2O3 | 3 | 69.6202 | Borax | Na2B4O7*10H2O | 381.3721 |
The batching matrix B indicates the relation of the molarity in the batch (columns) and in the glass (rows). For example, the batch component SiO2 adds 1 mol SiO2 to the glass, therefore, the intersection of the first column and row shows "1". Trona adds 1.5 mol Na2O to the glass; albite adds 6 mol SiO2, 1 mol Na2O, and 1 mol Al2O3, and so on. For the example given above, the complete batching matrix is listed below. The molarity matrix NG of the glass is simply determined by dividing the desired wt% concentrations by the appropriate molar masses, e.g., for SiO2 67/60.0843 = 1.1151.
The resulting molarity matrix of the batch, NB, is given here. After multiplication with the appropriate molar masses of the batch ingredients one obtains the batch mass fraction matrix MB:
or The matrix MB, normalized to sum up to 100% as seen above, contains the final batch composition in wt%: 39.216 sand, 16.012 trona, 10.242 lime, 16.022 albite, 4.699 orthoclase, 7.276 dolomite, 6.533 borax. If this batch is melted to a glass, the desired composition given above is obtained. During glass melting, carbon dioxide
Carbon dioxide
Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...
(from trona, lime, dolomite) and water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
(from trona, borax) evaporate.
Another simple glass batch calculation can be found at the website of the University of Washington.
Advanced batch calculation by optimization
If the number of glass and batch components is not equal, if it is impossible to exactly obtain the desired glass composition using the selected batch ingredients, or if the matrix equation is not soluble for other reasons (e.g., correlationCorrelation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
), the batch composition must be determined by optimization techniques
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
.