Properties for Design of Composite Structures. Neil McCartney

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СКАЧАТЬ upper U With caret Over partial-differential p EndFraction equals minus StartFraction partial-differential ModifyingAbove upper F With caret Over partial-differential upper V EndFraction comma upper V equals StartFraction partial-differential ModifyingAbove upper G With caret Over partial-differential p EndFraction comma upper S equals minus StartFraction partial-differential ModifyingAbove upper F With caret Over partial-differential upper T EndFraction equals minus StartFraction partial-differential ModifyingAbove upper G With caret Over partial-differential upper T EndFraction comma 2nd Row mu Subscript k Baseline equals StartFraction partial-differential ModifyingAbove upper U With caret Over partial-differential upper M Subscript k Baseline EndFraction equals StartFraction partial-differential ModifyingAbove upper F With caret Over partial-differential upper M Subscript k Baseline EndFraction equals StartFraction partial-differential ModifyingAbove upper G With caret Over partial-differential upper M Subscript k Baseline EndFraction comma k equals 1 comma ellipsis comma n period EndLayout"/>(2.56)

      The total mass of the system denoted by M is given by

      (2.57)

      Any extensive state variable P depends directly on the total M of the system and is such that for any value of the dimensionless parameter λ

      (2.58)

      On applying this principle to the functions introduced in (2.54)

(2.59)

      It follows that the functions U^,F^ or G^ are homogeneous of order unity in the extensive state variables. Provided that λ is independent of the state variables, on differentiating relations (2.59) with respect to λ, and then setting λ=1, it follows on using (2.56) that

(2.60)

The quantities U, F, G, S, V and {Mk} are the extensive variables of uniform multi-component systems used in thermodynamics. The intensive variables, which do not depend on the total mass M of the system, are the temperature T, pressure p and the chemical potentials μk,k=1,…,n. The thermodynamic relations may only be used for uniform systems which are in a state of mechanical and thermal equilibrium. Their use needs to be extended to non-uniform states where gradients of state variables are encountered, and to dynamic states where the various species are subject to motion. In this book, diffusion mechanisms will be neglected, which means that all species at any given location in the material will move with the same velocity v.

      2.9.2 Local Thermodynamic Relations

      The extensive quantities are not used directly in simulations, as local expressions of these thermodynamic quantities defined per mole, per unit mass or per unit volume need to be defined. The approach to be taken here is to define thermodynamic variables per unit mass by introducing the state variables υ,ψ ,φ ,Ω and η using the relations

(2.61)

      implying that

(2.62)

      where υ,ψ​ and φ are the specific internal, specific Helmholtz and specific Gibbs energies, respectively, Ω is the volume per unit mass (i.e. 1/ρ where ρ is the mass density), whereas η and ωk,k=1,…,n are, respectively, the specific entropy and mass fractions for the n species. Substitution of (2.61) into (2.54) then leads to

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