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Engineering LibreTexts

4.4: Measures of Velocity

  • Page ID
    44478
  • In the previous section we defined the mass average velocity according to

    \[\mathbf{v} = \sum_{B = 1}^{B = N}\omega_{B} \mathbf{v}_{B} , \quad \text{ mass average velocity} \label{47}\]

    and we noted that the total mass flux vector was given by

    \[\rho \mathbf{v} = \sum_{B = 1}^{B = N}\rho_{B} \mathbf{v}_{B} =\left\{\begin{array}{l} \text{ total mass} \\ \text{ flux vector} \end{array}\right\} \label{48}\]

    The mass average velocity and the species mass velocity are determined by the laws of mechanics for multicomponent systems5. In a course in fluid mechanics students will encounter the governing differential equation for the mass average velocity, \(\mathbf{v}\), and in a course in mass transfer students will encounter the governing differential equation for the species mass velocity, \(\mathbf{v}_A\). Throughout the chemical engineering literature, one also encounters the molar average velocity defined by

    \[\mathbf{v}^{*} =\sum_{B = 1}^{B = N}x_{B} \mathbf{v}_{B} , \quad \text{ molar average velocity} \label{49}\]

    Unlike the mass average velocity and the mass diffusion velocity, there is no governing differential equation for the molar average velocity. In the absence of a governing differential equation for the molar average velocity, it has historical value but limited practical value.

    In the previous section we used a decomposition of the species velocity of the form

    \[\mathbf{v}_{A} = \mathbf{v} + \mathbf{u}_{A} \label{50}\]

    so that the species mass flux vector could be expressed in terms of a convective flux and a diffusive flux according to

    \[\rho_{A} \mathbf{v}_{A} =\underbrace{ \rho_{A} \mathbf{v} }_{\text{mass convective flux}} + \underbrace{ \rho_{A} \mathbf{u}_{A} }_{\text{mass diffusive flux}} \label{51}\]

    When convective transport dominates, the species velocity, the mass average velocity and the molar average velocity are all essentially equal, i.e.,

    \[\mathbf{v}_{A} =\mathbf{v} \label{52}\]

    This is the situation that we encounter in our study of material balances and we will make use of this result repeatedly to determine the flux of species \(A\) at entrances and exits. While Equation \ref{52} is widely used to describe velocities at entrances and exits, one must be very careful about the general use of this approximation. If Equation \ref{52} were true for all species under all conditions, there would be no separation, no purification, no mixing, and no chemical reactions!

    Under certain circumstances we may want to use a total mole balance and this is obtained by summing Equation \((4.1.17)\) over all \(N\) species in order to obtain

    \[\frac{d}{dt} \int_{\mathscr{V}}\sum_{A = 1}^{A = N}c_{A} dV + \int_{\mathscr{A}}\sum_{A = 1}^{A = N}(c_{A} \mathbf{v}_{A} ) \cdot \mathbf{n} dA =\int_{\mathscr{V}}\sum_{A = 1}^{A = N}R_{A} dV \label{53}\]

    Use of the definitions given by Eqs. \((4.2.7)\) and \ref{50} allows us to write this result in the form

    \[\frac{d}{dt} \int_{\mathscr{V}}c dV + \int_{\mathscr{A}}c \mathbf{v} \cdot \mathbf{n} dA =\int_{\mathscr{V}}\sum_{A = 1}^{A = N}R_{A} dV - \int_{\mathscr{A}}\sum_{A = 1}^{A = N} (c_A \mathbf{u}_A ) \cdot \mathbf{n} dA \label{54}\]

    The first term on the right hand of this result represents the net rate of production of moles and this term need not be zero. Thus the overall mole balance is more complex than the overall mass balance given by Eq. \((4.2.18)\). The last term in Equation \ref{54} represents the diffusive fluxes at the surface of the control volume. This term can be neglected for the macroscopic balance problems that are treated in this text, but it will not be neglected in subsequent courses in mass transfer and reactor design.