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We use the parallel between the 27 component conservation Eq. 1-1) and Eq. 3-1) to shorten the following development. By analogy to the procedure in Sec. 2-2, the accumulation term in Eq. 3-2) where the total energy includes internal, kinetic, and potential energy. 3-3) where U is an overall internal energy (total energy/total mass), and ρ is the overall 2 represents total density given by Eq. 2-24). In Eq. 3-3) the term 1/ 2 ρ v ( ) kinetic energy per unit bulk volume and − ρ gDz total potential energy per unit bulk volume with reference to the depth below some horizontal plane.
The equation of continuity is ⎞ ⎛ NP ⎞ ∂ ⎛ NP ⎜ φ ∑ ρ j S j + (1 − φ ) ρ s ⎟ + ∇i⎜ ∑ ρ j u j ⎟ = 0 . 2-33) can be written totally in terms of pressure and saturation derivatives using Eqs. 2-24); this equation is a form of the “pressure” equation. 2-33) is strictly correct only when the velocity is the massaveraged velocity, not volume-averaged. 2-3 ENERGY BALANCE EQUATIONS For steam, hot water injection, and in situ combustion—some of the most important EOR and remediation processes—the temperature changes with both space and time.
That is, the entropy change with internal energy is inversely proportional to temperature when the volume is fixed and will be greatest at temperatures near absolute zero. Similar logic leads to the relationship that for constant internal energy ( ∂S ∂V )U = P T . Because all properties 37 in these relationships are state functions (independent of the path), they can be combined to give the macroscopic definition of entropy as dS = ( ∂S ∂U )V dU + ( ∂S ∂V )U dV = (1 T ) dU + ( P / T ) dV . Again, there is nothing magical about this definition except that like Boltzmann’s microscopic definition it gives a value for entropy that is always positive and increasing monotonically towards equilibrium.