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Thermodynamics filled in class notes_Part_92

# Thermodynamics filled in class notes_Part_92 - 191 5.6 SOME...

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Unformatted text preview: 191 5.6. SOME CONSERVATION AND EVOLUTION EQUATIONS concentration (kmole/m 3 ) 0.002 ρ N2 0.0015 0.001 0.0005 ρ N 0.001 0.002 0.003 0.004 t (s) 0.005 Figure 5.5: ρN2 (t) and ρN (t) in isobaric, isothermal nitrogen dissociation problem. 5.6 Some conservation and evolution equations Here a few useful global conservation and evolution equations are presented for some key properties. Only some cases are considered, and one could develop more relations for other scenarios. 5.6.1 Total mass conservation: isochoric reaction One can easily show that the isochoric reaction rate model, Eq. (5.279), satisﬁes the principle of mixture mass conservation. Begin with Eq. (5.279) in a compact form, using the deﬁnition of the reaction rate r , Eq. (5.281) and perform the following operations: d dt dρi = νi r, dt ρci = νi r, Mi (5.390) (5.391) d (ρci ) = νi Mi r, dt L d Ml φli r, (ρci ) = νi dt l=1 (5.392) (5.393) =M i d (ρci ) = dt L l=1 Ml φliνi r, (5.394) CC BY-NC-ND. 18 November 2011, J. M. Powers. 192 CHAPTER 5. THERMOCHEMISTRY OF A SINGLE REACTION N N d (ρci ) = dt i=1 d ρ dt N i=1 L i=1 l=1 ci = L Ml φli νi r, (5.395) Ml φli νi r, (5.396) N l=1 i=1 =1 dρ =r dt L l=1 N Ml φli νi . (5.397) i=1 =0 Therefore, we get dρ = 0. dt Note the term 5.6.2 N i=1 (5.398) φli νi = 0 because of stoichiometry, Eq. (5.24). Element mass conservation: isochoric reaction Through a similar series of operations, one can show that the mass of each element, l = 1, . . . , L, in conserved in this reaction, which is chemical, not nuclear. Once again, begin with Eq. (5.281) and perform a set of operations, dρi = νi r, dt dρ φli i = φli νi r, dt N i=1 d dt (5.399) l = 1, . . . , L, d l = 1, . . . , L, (φli ρi ) = rφli νi , dt N d (φli ρi ) = rφliνi , l = 1, . . . , L, dt i=1 (5.400) (5.401) (5.402) N N φli ρi =r φli νi , l = 1, . . . , L, (5.403) i=1 i=1 =0 d dt N φli ρi = 0, i=1 CC BY-NC-ND. 18 November 2011, J. M. Powers. l = 1, . . . , L. (5.404) ...
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