Chapter for Thermodynamics and the Destruction of Resources, B. R. Bakshi, T. G. Gutowski, D. P. Sekulic, Camb. U. Press 1. INTRODUCTION In this chapter, we will develop several models for the materials recycling process. The focus will be on the separation of materials from a mixture. This problem can be modeled using the principles of thermodynamics, particularly the concept of mixing entropy, as well as by using some of the results from information theory. In doing this calculation we will find from a thermodynamic point of view, that the theoretical minimum work required to separate a mixture is identical to the work lost upon spontaneous mixing of the chemical components. In other words, the development in this chapter in conjunction with the results from previous chapters will allow us to track both the degradation in materials values as they are used and dispersed in society, as well as the improvement and gain as materials are restored to their original values. Of course this restoration does not come for free, and so we will also look at the losses and inefficiencies involved in materials recycling. This approach allows us to look at the complete materials cycle as they move through society and to evaluate the gains and losses at each step. The chapter starts with the development of the needed thermodynamics concepts and then moves on to the application of these ideas. This chapter also introduces an alternative way of looking at the recycling problem using information theory. 2. THE THERMODYAMICS OF MATERIALS SEPARATION The basic separation problem can be illustrated by considering the separation of a molecular mixture into its pure components. This result will then be developed for the special case of an ideal mixture. Ideal mixtures include ideal gas mixtures and ideal solutions, but not necessarily many of the material separation situations that occur in recycling, material extraction and material purification. These cases may deviate from ideal mixtures because of specific interactions between dissimilar molecules, such as volume effects and heat effects, or because the mixtures are not actually molecular mixtures. Never the less, the ideal mixture result can provide guidance, for example, by suggesting concentration scaling effects that could apply to many situations including non-ideal processes. We will show that these results have useful applications in the fields of resource accounting and industrial ecology. An introduction to the thermodynamics of mixing and separation can be found in the text by Çengel and Boles [2006]. The development here follows the material found in Chapters 1 and 2 in this book and the work of Gyftopoulos and Beretta [2005]. Consider the open system shown in Fig. 1. A mixture denoted by “12” at temperature T0 and pressure p0 enters on the left and the pure components “1” and “2”, also at T0 and p0, exit on the right. Each stream has enthalpy “H” (measured in Joules, J) and entropy “S” (measured in J/K) which will be denoted by their subscripts. The system has a work input W and can exchange heat Q with the surroundings at temperature To. One can then write the rate balance equations (shown by the dot over each variable that changes with Materials Separation and Recycling Timothy G. Gutowski2 time) for constituents, energy and entropy as given below. Fig 1 An ideal separation process out,iin,isys,iNNdtdN!!!= i = 1,2 (1) 2112HHHWQdtdEinout!!!!!!!++!= (2) irroutSSSSTQdtdS!!!!!+!!+!=21120 (3) Where Ni are in moles, and “Sirr“ is the entropy production associated with irreversibilities in the system. This term allows us to write (3) as a balance even though entropy is not conserved. Assuming steady state and eliminating the heat rate Q dot between (2) and (3) yields an expression for the work rate for separation. irrooinST)S)SS((T)H)HH((W!!!!!!!!+!+!!+=12211221 (4) The mass balance is implied because the terms in (4) are all extensive. This result can also be written using the molar intensive forms of the thermodynamic properties (denoted by lower case font) as, irrmixmixinST)sTh(NW!!!0012+!"!"= (5) or, irrmixinSTgNW!!!012+!"=# (6) Where Δhmix = ( h12 – x1h1 –x2 h2) and Δsmix = (s12 –x1s1 – x2s2), and x1 and x2 are mole fractions, (N1 / N12 and N2 / N12 respectively), with N12 = N1 + N2. And recognizing the term within the brackets in (5) as the intensive form of the Gibbs Free Energy of Mixing at the so-called restricted dead state ( T0 , p0 ) i.e., Δg*mix = Δhmix –T0 Δsmix. We can now obtain an expression for the minimum work of separation per mole of mixture by letting 0=irrS! This gives the minimum rate of work as Pure Component 2 Pure Component 1 Mixture 12 inW!outQ!3 mixgNW!"#=12min!! (7) and the minimum work as mixgNWw!"#==12minmin!! (8) That is, the minimum work of separation is negative the Gibbs Free Energy of mixing. When two substances spontaneously mix, the Gibbs Free Energy of mixing is negative. So the minimum work required to separate these is the positive value of Δg*mix. Losses in the system i.e., 0>irrS!, will make the work required even larger. If we consider the reverse problem, the one of mixing two pure streams, this could be accomplished without any work input provided Δg*mix < 0 (A common enough occurrence for many systems). Now the irreversible loss upon mixing is mixirrgNST!"#=120!! (9) That is, the irreversible loss upon spontaneous mixing is the same as the minimum work for separation. Compare with (7). (Note the sign change when you write the material flows in the opposite direction.) For an ideal mixture the enthalpy of mixing is zero, i.e., Δhmix= 0. Hence the minimum work for separation becomes wmin = T0Δsmix. (10) The mixing entropy Δsmix for non-interaction particles, can be calculated from the case of mixing ideal