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4.2: Untitled Page 57

  • Page ID
    18189
  • Chapter 4

    v n. This follows from the fact that vn is negative over surfaces where mass is entering the control volume and v n is positive over surfaces where mass is leaving the control volume. In addition, one should remember that the control volume, V , in Eq. 4‐2 is arbitrary and this allows us to choose the control volume to suit our needs.

    Now we are ready to consider N‐component systems in which chemical reactions can take place, and in this case we need to make use of the two axioms for the mass of multicomponent systems. The first axiom deals with the mass of species A, and when this species can undergo chemical reactions we need to extend Eq. 4‐1 to the form given by

    (4‐3)

    In order to develop a precise mathematical representation of this axiom, we require the following quantities:

    (4‐4)

    (4‐5)

    (4‐6)

    Here it is important to understand that

    represents both the creation of species

    A

    r

    A (when

    is positive) and the consumption of species A (when is negative).

    A

    r

    A

    r

    In terms of these primitive quantities, we can make use of an arbitrary fixed control volume to express Eq. 4‐3 as

    Multicomponent systems

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    Axiom I:

    (4‐7)

    Within the volume, V , the total mass produced by chemical reactions must be zero.

    This is our second axiom that we express in words as

    Axiom II:

    (4‐8)

    and in terms of the definition given by Eq. 4‐6 this word statement takes the form (4‐9)

    The summation over all N molecular species can be interchanged with the volume integration in this representation of the second axiom, and this allows us to express Eq. 4‐9 as

    (4‐10)

    Since the volume V is arbitrary, the integrand must be zero and we extract the preferred form of the second axiom given by

    Axiom II:

    (4‐11)

    Here it should be clear that A

    r represents both the creation of species A (when A r

    is positive) and the consumption of species A (when is negative). In Eqs. 4‐7

    A

    r

    and 4‐11 we have used a mixed mode nomenclature to represent the chemical species, i.e., we have used both letters and numbers simultaneously.

    Traditionally, we use upper case Roman letters to designate various chemical species, thus the rates of production for species A, B and C are designated by A r ,

    and

    . When dealing with systems containing N different molecular species, B

    r

    C

    r

    we allow an indicator, such as A or D or G, to take on values from 1 to N in order to produce compact forms of the two axioms given by Eqs. 4‐7 and 4‐11. We

    99