REFRIGERANTS Industry’s great equations


Dick Powell, technical consultant with Refrigerant Solutions takes a look at the scientific theory of refrigeration.


I


n a fascinating book, edited by Graham Farmelo, various contributors wax lyrical on the ‘great equations of modern science’, each encapsulating a fundamental theory in few symbols.


Not surprisingly, one chapter features Einstein’s well-known E=mc2


-0.5R gab , which emerged


from his Theory of Special Relativity, and another his less well-known Rab


= -8pGTab ,


representing gravity as the curvature of space- time induced by mass from the Theory of General Relativity.


More relevant to refrigeration, science journalist Aisling Irwin, cites the fundamental chemical equations of Rowland and Molina summarising the destruction of stratospheric ozone: CF2


Cl2 +UV radiation = CF2 ClO+O=Cl+O2


But, despite their profound effect on our industry, they are obviously not fundamental to it.


Farmelo’s book does not include what I consider to be the great equations of refrigeration. I suggest these are to be found in Lars Onsager’s work on non-equilibrium thermodynamics.


Basic thermodynamic theory – as taught to engineers, physicists and chemists at college – might be better termed ‘thermostatics’, since it applies to processes at equilibrium, or that take place infinitely slowly. Real-world refrigeration is somewhat faster.


Fortunately, even simple cycle calculations, based on classical thermodynamics, plus a few fudge factors, do provide useful insight into the operation of actual machines. But if we wish to be more technically correct, we need non- equilibrium thermodynamics.


Onsager’s starting point was the relationship between the Peltier and Seebeck effects of thermoelectricity, which he expressed in the form of two linked equations:


Cl+Cl; Cl+O3


(1) J1 (2) J2 where J1


and J2


= L11 = L21


.X1 .X1


+ L12 + L22


.X2 .X2 is the heat ‘flux’; X1


is the electric charge ‘flux’ (current) and X2


are the


thermodynamic driving forces; and Lxy constants.


The terms L11 .X1 and L22 .X2 are represent the


thermodynamically irreversible processes occurring in the system equivalent, for example, associated with Ohm’s Law and Fourier’s law of thermal conduction respectively. The term L21


.X1 represents the Peltier effect


– that is, the heat flow transferred reversibly from the cold to the hot junction induced by the portion of the electric charge, L12


.X2


reversibly through the circuit. Onsager rigorously proved that L12


=ClO+O2 ; =L21 (3)


of the motor/compressor. Obviously, for efficient operation L11


and L22 (= L21 should be minimised and L12 ) should be maximised.


For me, Equations (1), (2), and (3) elegantly summarise the phenomenon of refrigeration and can truly be regarded as its ‘great equations’. They link holistically the inefficiencies in the total refrigeration system, notably from the motor, compressor and insulation, with the effectiveness of the refrigerating process to convert electricity into useful cooling.


Furthermore, the equations can represent other refrigerating systems that are not driven by electricity.


, flowing , known


as the ‘reciprocity relationship’, which Kelvin had previously postulated based on experimental results.


Onsager also generalised the equations to represent the induced flow of one property in a system by the applied flow of another property. In the refrigeration industry we are especially interested in the flow of heat caused by the flow of electric charge, not only via the Peltier effect, but more importantly by an electric motor/ compressor combination.


The heat flow equation (2) describes mathematically the cooling of refrigerated volume. The heat flow L22


.X2


An obvious example is the absorption refrigerator where J1


in equation (1) represents


the flow of heat from a high temperature source – such as a gas flame, solar thermal collector, or steam – driving the refrigeration process of Equation (2).


is the heat flow


into the refrigerator from its surroundings, for example through the insulation and by addition of warm items, while L21


.X1 represents the heat


removed as a result of the flow of electric charge by the action of the motor/compressor. To maintain a constant temperature inside the refrigerator these values must be equal – that is to say, J2


is zero. In the electric charge flow equation (1), L12 .X2


represents the flow of electricity actually required to deliver the cooling effect of the refrigeration circuit, while L11


.X1 represents the inefficiencies


For his work on non-equilibrium thermodynamics Onsager received the 1968 Nobel Chemistry Prize. He died in 1976 the USA having become a naturalised citizen in 1945. On his tombstone is inscribed his academic post at Yale University, ‘J Willard Gibbs Professor’, plus simply ‘Nobel Laureate’ to which his children added an asterisk after the ‘Nobel Laureate’, and ‘*etc’ in the bottom right hand corner when his wife’s name was added following her death in 1991.


Do we need to raise a memorial to Onsager’s work? The answer is ‘no’ – one already exists on which is engraved his reciprocity relationship in generalised form (Lij


=Lji ), and was erected


by his alma mater, the Norwegian Institute of Technology, on its Trondheim campus, now part of the Norwegian University of Science and Technology. Certainly a most fitting location, considering the University’s renowned contribution to refrigeration technology, especially the pioneering work of Gustav Lorentzen on the modern super-critical CO2


cycle. www.acr-news.com August 2016 53


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