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The Hall–Héroult process is the major industrial process for smelting aluminium. It involves dissolving aluminium oxide (alumina) (obtained most often from bauxite , aluminium 's chief ore, through the Bayer process ) in molten cryolite and electrolyzing the molten salt bath, typically in a purpose-built cell.
The process produces a quantity of fluoride waste: perfluorocarbons and hydrogen fluoride as gases, and sodium and aluminium fluorides and unused cryolite as particulates. This can be as small as 0.5 kg per tonne of aluminium in the best plants in 2007, up to 4 kg per tonne of aluminium in older designs in 1974.
The Hall–Heroult process produces aluminium with a purity of above 99%. Further purification can be done by the Hoopes process. This process involves the electrolysis of molten aluminium with a sodium, barium, and aluminium fluoride electrolyte. The resulting aluminium has a purity of 99.99%. [46] [143]
The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all the equations or inequalities. If the solution set is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities.
Molten cryolite is used as a solvent for aluminium oxide (Al 2 O 3) in the Hall–Héroult process, used in the refining of aluminium. It decreases the melting point of aluminium oxide from 2000–2500 °C to 900–1000 °C, and increases its conductivity [ 16 ] thus making the extraction of aluminium more economical.
In mathematics, the solution set of a system of equations or inequality is the set of all its solutions, that is the values that satisfy all equations and inequalities. [1] Also, the solution set or the truth set of a statement or a predicate is the set of all values that satisfy it. If there is no solution, the solution set is the empty set. [2]
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions.
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the ...