07 - 7 Equations of State The properties of fluids can be...

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7 Equations of State The properties of fluids can be defined in two ways, either by the use of tabulated data (e.g. steam tables) or by state equations (e.g. perfect gas law). Both of these approaches have been developed by observation of the behaviour of fluids when they undergo simple processes. It has also been possible to model the behaviour of such fluids from ‘molecular’ models, e.g. the kinetic theory of gases. A number of models which describe the relationships between properties for single component fluids, or constant composition mixtures, will be developed here. 7.1 Ideal gas law The ideal and perfect gas laws can be developed from a number of simple experiments, or a simple molecular model. First the experimental approach will be considered. If a fixed mass of a single component fluid is contained in a closed system then two processes can be proposed: (i) the volume of the gas can be changed by varying the pressure, while maintaining the temperature constant; (ii) the volume of the system can be changed by varying the temperature, while maintaining the pressure constant. The first process is an isothermal one, and is the experiment proposed by Boyle to define Boyle’s law (also known as Mariotte’s law in France). The second process is an isobaric one and is the one used to define Charles’ law (also known as Gay-Lussac’s law in France). The process executed in (i) can be described mathematically as v = v( PIT (7.1) while the second one, process (ii), can be written v = v(T), Since these processes can be undergone independently then the relationship between the three properties is (7.3)
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122 Equations of state Equation (7.3) is a functional form of the equation ofstare of a single component fluid. It can be seen to obey the two-property rule, which states that any property of a single component fluid or constant composition mixture can be defined as a function of two independent properties. The actual mathematical relationship has to be found from experiment (or a simulation of the molecular properties of the gas molecules), and this can be derived by knowing that, if the property, v, is a continuous function of the other properties, p and T, as discussed in Chapter 6, then dv = ( $)T dp + ( $)p dT (7.4) Hence, if the partial derivatives (av/ap), and (av/aT), can be evaluated then the gas law will be defined. It is possible to evaluate the first derivative by a Boyle’s law experiment, and the second one by a Charles’ law experiment. It is found from Boyle’s law that pv =constant giving P =-- V Similarly, it is found from Charles’ law that V _- - constant T V ($)p =r Substituting eqns (7.6) and (7.8) into eqn gives V V = - - dp +- - dT P T which may be integrated to give = T (7.5) (7.7) (7.8) (7.9) (7.10) is known as the ideal gas law. This equation contains no information about the internal energy of the fluid, and does not define the specific heat capacities. If the specific heat capacities are not functions of temperature then the gas is said to obey the perfect gas law: if the specific heat capacities are functions of temperature (Le. the
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07 - 7 Equations of State The properties of fluids can be...

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