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4.6: Alternate Flow Rates

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  • There are a number of relations between species flow rates and total flow rates that are routinely used in solving macroscopic mass or mole balance problems provided that either the velocity profile is flat or the concentration profile is flat. For example, we can always write Equation \((4.5.5)\) in the form

    \[\dot{M}_{A} = \int_{A_{exit} }c_{A} \mathbf{v}\cdot \mathbf{n} dA = \int_{A_{exit} }\frac{c_{A} }{c} c \mathbf{v}\cdot \mathbf{n} dA = \int_{A_{exit} }x_{A} c \mathbf{v}\cdot \mathbf{n} dA \label{69}\]

    If either \(c \mathbf{v}\cdot \mathbf{n}\) or \(x_{A}\) is constant over the area of the exit, we can express this result as

    \[\dot{M}_{A} = \langle x_{A} \rangle \dot{M} \label{70}\]

    where \(\dot{M}\) is the total molar flow rate defined by

    \[\dot{M} = \sum_{A = 1}^{A = N}\dot{M}_{A} \label{71}\]

    If the individual molar flow rates are known and one desires to determine the area averaged mole fraction at an entrance or an exit, it is given by

    \[\langle x_{A} \rangle = \dot{M}_{A} \left/ \sum_{B = 1}^{B = N}\dot{M}_{B} \right. \label{72}\]

    provided that either \(c \mathbf{v}\cdot \mathbf{n}\) or \(x_{A}\) is constant over the area of the entrance or the exit. It will be left as an exercise for the student to show that similar relations exist between mass fractions and mass flow rates. For example, a form analogous to Equation \ref{70} is given by

    \[\dot{m}_{A} = \langle \omega_{A} \rangle \dot{m} \label{73}\]

    and the mass fraction at an entrance or an exit can be expressed as

    \[\langle \omega_{A} \rangle = \dot{m}_{A} \left/ \sum_{G = 1}^{G = N}\dot{m}_{G} \right. \label{74}\]

    One must keep in mind that the results given by Eqs. \ref{70} through \ref{74} are only valid when either the concentration (density) is constant or the molar (mass) flux is constant over the entrance or exit.

    When neither of these simplifications is valid, we express Equation \ref{69} as

    \[\dot{M}_{A} = \int_{A_{ exit} }x_{A} c \mathbf{v}\cdot \mathbf{n} dA = \langle x_{A} \rangle_{b} \dot{M} \label{76}\]

    where \(\langle x_{A} \rangle_{b}\) is the cup mixed mole fraction of species \(A\). The definition of the mole fraction requires that

    \[\sum_{A = 1}^{A = N}x_{A} = 1 \label{77}\]

    and it will be left as an exercise for the student to show that

    \[\sum_{A = 1}^{A = N}\langle x_{A} \rangle_{b} = 1 \label{78}\]

    This type of constraint on the mole fractions (and mass fractions) applies at every entrance and exit and it often represents an important equation in the set of equations that are used to solve macroscopic mass balance problems.

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