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5. Energy Balance

5.1 Summary

5.2 Energy Requirements

The First Law of Thermodynamics also called the Law of Conservation of Energy states that, although energy may be converted from one form to another, energy cannot be created or destroyed. Whenever a quantity of one kind of energy disappears an exactly equivalent amount of the other kind must be produced.
Table 5.1 shows the different units of energy.

Table 5.1: Conversion of Energy Units.

The aluminum smelting process needs energy to keep the electrolytic cell at the reaction temperature and to produce aluminum. This required energy is furnished as electric energy. The Power (in kilo watts: kW) which is added to an electrolytic cell is written with

(5.1)

The energy (E) to produce one kilogram of aluminum (Specific Energy Consumption) is derived in the following way by using Equ. 2.15 for the electrolytic production of aluminum (EP_Al):

(5.2A)

(5.2B)

The difference between the added energy and the energy required to produce aluminum is lost as heat. This heat loss is of major concern since pot operation wants to keep its value at a minimum without disturbing optimal pot operation.
Discussing the energy balance one is speaking about values in energy per time (power in Watts or Kilowatts) or in energy while one kilogram of aluminum is produced (specific energy in kWh/kg). On the Cell Voltage Page you may select units for the energy balance: kWh, kW, kWh/kg and V. AlWeb and AlPrg calculate internally in Volts and converts the values into the wanted units. One the other hand the values for the enthalpies are given in kJ/mol.
The Tables 5-2 and 5-3 show how to convert these units:

Table 5.2: Conversion of Energy Balance Units.

Conversion of tension U (V), specific energy E_spec (kWh/kg), power P (kW) and enthalpy ΔH (kJ/mol). F is the Faraday Constant, I the electric current (kA) and M_Al the atomic weight of aluminum.

Putting values for the parameters into the relations one finds:

Table 5.3: Conversion of Energy Balance Units.

5.3 Enthalpies

Thermodynamic tables list energy values for a standard state (25 °C = 298 K) and for different temperatures. These values are called Enthalpies (J/mol) if the pressure is kept constant, normally at 1 atm.
To determine the energy necessary to produce aluminum two alternative ways are possible [Lit.]: the reaction takes place at the reference temperature, followed by heating the products to the reacting temperature (Figure 5.1)

5.1 Reaction Path for Aluminum Production.

The reaction takes place at room temperature and the reaction products are heated to the reacting temperature.

or the reactants are first heated to the reaction temperature and subsequently allowing the reaction to occur at the new temperature (Figure 5.2).

5.2 Reaction Path for Aluminum Production.

Contrary to Figure 5.1 the reactants are first heated to the reaction temperature and then the reaction takes place.

We follow the second alternative as [Lit. (p.19)] is doing. The Electrolysis Equation is rewritten (divided by 4η) to produce one mole of aluminum:

(5.3)

The enthalpies to heat the materials (ΔH) and the enthalpies of formation (ΔH_f) were taken from the JANAF Thermochemical Tables [Lit.] and a temperature dependence introduced by interpolating linearly between the temperatures 827 °C and 1027 °C (1100-1300 K).

(5.4)

According to Equation (5.3) on can write for the enthalpy of reaction (ΔH_react):

(5.5)

and for the total enthalpy (ΔH_total) i.e. the energy consumption that is necessary to produce aluminum:

(5.6)

In this derivation the following effects were neglected:
Technical alumina consists mainly of gamma alumina. Gamma alumina is transformed to alpha alumina when it comes into contact with the electrolyte. Since alpha alumina is more stable the heat of transformation is produced. However this effect is small about 0.02 kWh/kg. In addition to the electrolytic part some additional carbon is consumed. This anode excess consumption is mostly due to airburn e.g. the reaction of carbon anode with the ambient air. The excess carbon is mostly combusted in the cell; it does not occur in the electrolyte and will not contribute to the heat content of the bath.
According to the First Law of Thermodynamics the result of the consideration is independent of the reaction path. The state of the reactants and of the products is essential i.e. it does not matter how the chemical reaction takes place in the electrolytic cell. Therefore the heat of dissolution of alumina in the electrolyte or deviations from the ideal behavior must not be considered.

5.4 Heat Loss

The difference between the added energy and the energy required to produce aluminum is lost as heat. For the cell voltage that contributes to the heat loss the cell voltage (ΔU_cell) is taken reduced by the voltage drop that does not contribute to the cell heat balance in essence the voltage drop of the external bus bars (ΔU_ExtBus):

(5.7)

The heat loss (ΔQ) is then determined as the difference of the energy added to the electrolysis cell (ΔE_int) minus the energy (ΔH_total, Equ. 5.6 ) to produce aluminum:

(5.8)

AlPrg and AlWeb produce drawings similar to diagrams published by Haupin et al. [Lit.].

5.2 Haupin Diagram.

In this figure AlPrg and AlWeb draws on the left side the components of the energy balance (Equ. 5.8) and on the right side you find the components of the total energy (i.e. enthalpy) to produce aluminum. Red values mean energy producer and green values indicate energy consumers. The value for the decomposition enthalpy of alumina (ΔH_d(Al2O3)) is the nagative value of the enthalpy of formation (-ΔH_d(Al2O3)).

On the left side you find the components of the energy balance according to Equ. 5.8 and on the right side the values that contribute to the energy to produce aluminum (ΔH_total, Equ. 5.6). Red values mean energy producer like the formation of carbon dioxide (ΔH_f(CO2)). Green values indicate energy consumers like the decomposition of alumina where ΔH_d(Al2O3) = -ΔH_d(Al2O3) (enthalpy of alumina formation).