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Exergy costs

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The Exergy Replacement Cost of Natural Mineral Capital

Exergy is the minimum energy required to replenish a resource from its most degraded state or, in other words, to remake it from the reference environment via a reversible process. However, the real processes designed by humans are far from the ideal condition of reversibility and the exergy requirements to obtain a resource are always greater than those dictated by the second law. For this reason, we should not evaluate natural resources solely in terms of reversible processes, e.g. using solely Eq. 1 and Eq. 2, since this would ignore technological limits, which impose more costs from a physical point of view.

Therefore, we must include real physical unit costs in the thermodynamic evaluation of resources. These are defined as the relationship between the exergy invested in the real process of obtaining the resource and the exergy required if the process was reversible. It has a dimensionless value and measures the number of exergy units needed to obtain one unit of exergy of the product. Generally, the exergetic unit cost is tens or even hundreds of times greater than its exergetic content. The real thermodynamic value of a resource is determined by its exergy multiplied by the real physical unit cost of the process to obtain it, as in the following equation:

The following table shows the values of kch and kc for some minerals.

Table 2: Unit chemical exergy cost and unit concentration exergy cost for selected minerals. Source: Valero and Botero 2002.

Element Mineral Aver. conc. in Crust
xc (mass)
Ore grade
xm %mass
Real energy required GJ/ton bc
 kc Estimated  kch
Al Al2O3 8% 17 50 0,126 395,71 1
Ag Ag 0,05 ppm 0,01 ppm 1582 0,224 7046,49 1
Au Au 0,0018 ppm 0,0015 62245 0,152 408,74 1
Cu CuFeS 25 ppm 0,5 66,7 0,172 385,61 10
Fe Fe2O3 3,50% 50 1 0,022 43,98 1
Hg HgS 0,08 ppm 0,1 157 0,091 1706,87 10
Pb PbS 20 ppm 4 9,5 0,044 212,06 10
Sn SnO2 5,5 ppm 0,4 187 0,125 1493,02 1
Ti FeTiO3 0,30% 10 23 0,066 348,43 1
Zn ZnS 71,0 ppm 3,5 8,1 0,128 62,83 10

The Exergy Replacement Cost of the World’s Renewable Water Resources

In the case of water, its thermodynamic value has two basic components; its composition makes it useful for different human and agricultural activities and its potential energy can be used to perform mechanical work and generate electricity. These two conditions should be returned to the water from its more thermodynamically degraded state (the ocean in this case). Some authors have already proposed physical models to determine the thermodynamic value of a river, and to physically and economically calculate water resource values in a country or region. Nevertheless the models may not be very practical since they require a lot of informational inputs.

For this reason, we propose to thermodynamically evaluate the world's renewable water resources using the concept of Exergy Replacement Cost (ExRC). The first component of this replacement cost of water resources is the exergy needed to return the quality characteristics back to the water and is represented by the desalination exergy which is given by Eq. 3



  • bdes = Desalination exergy (J/mol).
  • v= molar volume.
  • TH2O =  Absolute temperature of water (K).
  • Rc = Recuperation ratio (N0/N1) (dimensionless). N1: number of moles of water at the input flow of the desalination plant (mol).
  • x1H2O = Molar fraction of the salt in water at the input flow of a desalination plant.

The second component is the minimum exergy needed to return the resource to its conditions of physical disequilibrium (or potential) with the chosen reference level (the ocean). This exergy can be calculated using the following equation:


  • bphys = Physical exergy (kW).
  • Q = Volumetric flow of water (m3/s).
  • h = Height (m).

The Exergy Abatement Costs

Although we are unable to replace fuels, we are spending them 3 or 4 times faster than they can be fixed by photosynthetic processes. This causes significant distortions in global material cycles but the demand is estimated to continue to increase by almost 2 % annually until 2020. In the light of our limited technologies and energy dependence, the least we can do is use these resources cleanly without adverse effects on ecosystems. We propose a new approximation to measure the environmental externality caused by the use of fossil fuels called the Exergy Abatement Cost (ExAC) defined as "the quantity of exergy needed to reduce the emissions of a specific contaminant to innocuous levels for the environment, using the best available technology".

The ExAC is a physical and anticipated way to internalize most external environmental effects. It can be used to decrease the exergy contained by any fossil fuel (the exergy needed to decrease its associated emissions), and express its remaining exergetic content as clean tonnes of oil equivalent, ctoe, instead of the traditional equivalent tonnes of oil, toe. The new unit of exergy content ctoe represents the fuel exergy that does not represent a risk for the environment.

 Clean fuel exergy can be calculated as:

  • CExR = Clean exergy [The value of the HHV is approximately equal to the fossil fuel exergy] contained in the specific fuel reserves (ctoe)
  • Ri = Fuel reserves  (t)
  • CEx= Clean exergy of the fuel (kJ/t)
  • HHVi   = High heating value of the ith fuel (kJ/t)
  • EFj = Emission factor of pollutant j (kg/t) (depends on the fuel type and the technology used to transform the chemical exergy)
  • ExACj = Exergy cost of reducing pollutant j (kJ/kg)

Both CEx and CExR are expressed in ctoe.

Created by aliciavd
Last modified 2015-09-10 01:51 PM
New book: Thanatia

Thanatia: The Destiny of the Earth's Mineral Resources

Thanatia: The Destiny of the Earth's Mineral Resources

A Thermodynamic Cradle-to-Cradle Assessment by (author): Antonio Valero Capilla (CIRCE — Universidad de Zaragoza, Spain), Alicia Valero Delgado (CIRCE — Universidad de Zaragoza, Spain)

Is Gaia becoming Thanatia, a resource exhausted planet? For how long can our high-tech society be sustained in the light of declining mineral ore grades, heavy dependence on un-recycled critical metals and accelerated material dispersion? These are all root causes of future disruptions that need to be addressed today.

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