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4.5: Molar Flow Rates at Entrances and Exits

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    44479
  • Here we direct our attention to the macroscopic mole balance for a fixed control volume

    \[\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{\mathscr{V}}R_{A} dV , \quad A=1,{ 2, .... }N \label{55}\]

    with the intention of evaluating \(c_{A} \mathbf{v}_{A} \cdot \mathbf{n}\) at entrances and exits. The area integral of the molar flux, \(c_{A} \mathbf{v}_{A} \cdot \mathbf{n}\), can be represented as

    \[\int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{A_{ e} }c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA + \int_{A_{i} }c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA \label{56}\]

    where \(A_{ e}\) represents the entrances and exits at which convection dominates and \(A_{i}\) represents an interfacial area over which diffusive fluxes may dominate. In this text, our primary interest is the study of control volumes having entrances and exits at which convective transport is much more important than diffusive transport, and we have illustrated this type of control volume in Figure \(\PageIndex{1}\). There the entrances and exits for both the water and the air are at the top and bottom of the column, while the surface of the control volume that coincides with the liquid-solid interface represents an impermeable boundary at which \(c_{A} \mathbf{v}_{A} \cdot \mathbf{n}=0\). For systems of this type, we express Equation \ref{56} as

    \[\int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{A_{ e} }c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{A_{entrances}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA + \int_{A_{ exits} }c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA \label{57}\]

    In order to simplify our discussion about the flux at entrances and exits, we direct our attention to an exit and express the molar flow rate at that exit as

    \[\dot{M}_{A} =\int_{A_{ exit} }c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA \label{58}\]

    On the basis of the discussion in Sec. 4.2, we assume that the diffusive flux is negligible (\(\mathbf{v}_{A} \cdot \mathbf{n} = \mathbf{v}\cdot\mathbf{n}\)) so that the above result takes the form

    \[\dot{M}_{A} =\int_{A_{ exit} }c_{A} \mathbf{v}\cdot\mathbf{n} dA \label{59}\]

    It is possible that both \(c_{A}\) and \(\mathbf{v}\) vary across the exit and a detailed evaluation of the area integral is required in order to determine the molar flow rate of species \(A\). In general this is not the case; however, it is very important to be aware of this possibility.

    image
    Figure \(\PageIndex{1}\): Entrances and exits at which convection dominates

    Average concentrations

    In Sec. 3.3.1 we defined a volume average density and we use the same definition here for the volume average concentration given by

    \[\langle c_{A} \rangle =\frac{1}{\mathscr{V}} \int_{\mathscr{V}}c_{A} dV , \quad \text{ volume average concentration} \label{60}\]

    At entrances and exits, we often work with the "bulk concentration” or "cup mixed concentration” that was defined earlier in Sec. 3.3.1. For the concentration, \(c_{A}\), we repeat the definition according to

    \[\langle c_{A} \rangle_{b} =\frac{1}{Q_{exit} } \int_{A_{ exit} }c_{A} \mathbf{v}\cdot\mathbf{n} dA =\frac{\int_{A_{ exit} }c_{A} \mathbf{v}\cdot\mathbf{n} dA }{\int_{A_{ exit} }\mathbf{v}\cdot\mathbf{n} dA } \label{61}\]

    In terms of the bulk concentration the molar flow rate given by Equation \ref{59} can be expressed as

    \[\dot{M}_{A} =\langle c_{A} \rangle_{b} Q_{exit} \label{62}\]

    in which it is understood that \(\dot{M}_{A}\) and \(\langle c_{A} \rangle_{b}\) represent the molar flow rate and concentration at the exit. In addition to the bulk or cup-mixed concentration, one may encounter the area average concentration denoted by \(\langle c_{A} \rangle\) and defined at an exit according to

    \[\langle c_{A} \rangle =\frac{1}{A_{ exit} } \int_{A_{ exit} }c_{A} dA \label{63}\]

    If the concentration is constant over \(A_{exit}\), the area average concentration is equal to this constant value, i.e.

    \[\langle c_{A} \rangle =c_{A} , \quad \text{when }c_{A} \text{ is constant} \label{64}\]

    We often refer to this condition as a "flat” concentration profile, and for this case we have

    \[\langle c_{A} \rangle_{b} =\langle c_{A} \rangle =c_{A} , \quad \text{ flat concentration profile} \label{65}\]

    Under these circumstances the molar flow rate takes the form

    \[\dot{M}_{A} =c_{A} Q_{exit} , \quad \text{ flat concentration profile} \label{66}\]

    The conditions for which \(c_{A}\) can be treated as a constant over an exit or an entrance are likely to occur in many practical applications.

    When the flow is turbulent, there are rapid velocity fluctuations about the mean or time-averaged velocity. The velocity fluctuations tend to create uniform velocity profiles and they play a crucial role in the transport of mass orthogonal to the direction of the mean flow. The contribution of turbulent fluctuations to mass transport parallel to the direction of the mean flow can normally be neglected and we will do so in our treatment of macroscopic mass balances. In a subsequent courses on fluid mechanics and mass transfer, the influence of turbulence will be examined more carefully. In our treatment, we will make use of the reasonable approximation that the turbulent velocity profile is flat and this means that \(\mathbf{v}\cdot\mathbf{n}\) is constant over \(A_{ exit}\). Both turbulent and laminar velocity profiles are illustrated in Figure \(\PageIndex{2}\) and there we

    image
    Figure \(\PageIndex{2}\): Laminar and turbulent velocity profiles for flow in a tube

    see that the velocity for turbulent flow is nearly constant over a major portion of the flow field. If we make the "flat velocity profile” assumption, we can express Equation \ref{61} as

    \[\langle c_{A} \rangle_{b} = \frac{\int_{A_{ exit} }c_{A} \mathbf{v}\cdot\mathbf{n} dA }{\int_{A_{ exit} }\mathbf{v}\cdot\mathbf{n} dA } = \frac{\int_{A_{ exit} }c_{A} dA }{\int_{A_{ exit} }dA } \frac{\mathbf{v}\cdot\mathbf{n}}{\mathbf{v}\cdot\mathbf{n}} = \frac{1}{A_{ exit} } \int_{A_{ exit} }c_{A} dA = \langle c_{A} \rangle \label{67}\]

    For this case, the molar flow rate at the exit takes the form

    \[\dot{M}_{A} =\langle c_{A} \rangle Q_{exit} , \quad \text{ flat velocity profile} \label{68}\]

    To summarize, we note that Equation \ref{62} is an exact representation of the molar flow rate in terms of the bulk concentration and the volumetric flow rate. When the concentration profile can be approximated as flat, the molar flow rate can be represented in terms of the constant concentration and the volumetric flow rate as indicated by Equation \ref{66}. When the velocity profile can be approximated as flat, the molar flow rate can be represented in terms of the area average concentration and the volumetric flow rate as indicated by Equation \ref{68}. If one is working with the species mass balance given by Equation \((4.1.7)\), the development represented by Eqs. \ref{55} through \ref{68} can be applied simply by replacing \(c_{A}\) with \(\rho_{A}\).

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