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2.3: Dimensionally Correct and Dimensionally Incorrect Equations

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  • In defining a unit of force on the basis of Equation \((2.2.2)\) we made use of what is known as the law of dimensional homogeneity. This law states that natural phenomena proceed with no regard for man-made units, thus the basic equations describing physical phenomena must be valid for all systems of units. It follows that each term in an equation based on the laws of physics must have the same units. This means that the units of \(\mathbf{f}\) in Equation \((2.2.2)\) must be the same as the units of \(d(m\mathbf{v})/dt\).

    It is this fact which leads us to the definition of a unit of force such as that given by Equation \((2.2.4)\). Equations that satisfy the law of dimensional homogeneity are sometime referred to as dimensionally correct in order to distinguish them from equations that are dimensionally incorrect. While the law of dimensional homogeneity requires that all terms in an equation have the same units, this constraint is often ignored in the construction of empirical10 equations found in engineering practice. For example, in the sixth edition of Perry’s Chemical Engineers’ Handbook11, we find an equation for the drop size produced by a certain type of atomizer that takes the form

    \[\bar{X}_{\mathrm{vs}} =\frac{1920\sqrt{\alpha } }{V_{a} \sqrt{\rho_{\ell } } } + 597\left(\frac{\mu }{\sqrt{\alpha \rho_{\ell } } } \right)^{0.45} \left(\frac{1000 Q_{L} }{Q_{a} } \right)^{1.5} \label{2-13}\]

    In order that this expression produce correct results, it is absolutely essential that the quantities in Equation \ref{2-13} be specified as follows:

    • \(\bar{X}_{\mathrm{vs}}\) = average drop diameter, \(\mu\mathrm{m}\) (a drop with the same volume-surface ratio as the total sum of all drops formed)
    • \(\alpha\) = surface tension, dyne/cm
    • \(\mu\) = liquid viscosity, \(P\)
    • \(V_{a}\) = relative velocity between air and liquid, ft/s
    • \(\rho_{\ell }\) = liquid density, \(g/cm^3\)
    • \(Q_{L}\) = liquid volumetric flow rate
    • \(Q_{a}\) = air volumetric flow rate

    When the numbers associated with these quantities are inserted into Equation \ref{2-13}, one obtains the average drop diameter, \(\bar{X}_{\mathrm{vs}}\), in micrometers. Such an equation must always be used with great care for any mistake in assigning the values to the terms on the right hand side will obviously lead to an undetectable error in \(\bar{X}_{\mathrm{vs}}\). In addition to being dimensionally incorrect, Equation \ref{2-13} is an empirical representation of the process of atomization. This means that the range of validity is limited by the range of values for the parameters used in the experimental study. For example, if the liquid density in Equation \ref{2-13} tends toward infinity, \(\rho_{\ell } \to \infty\), we surely do not expect that \(\bar{X}_{\mathrm{vs}}\) will tend to zero. This indicates that Equation \ref{2-13} is only valid for some finite range of densities. In addition, it is well known that when the relative velocity between air and liquid is zero (\(V_{a} =0\)) the drop size is not infinite as predicted by Equation , but instead it is on the order of the diameter of the atomizer jet. Once again, this indicates that Equation \ref{2-13} is only valid for some range of values of \(\bar{X}_{\mathrm{vs}}\) but the range is known only to those persons who examine the original experimental data. Clearly a dimensionally incorrect empiricism carries its own warning: Beware!

    The dimensionally incorrect result given by Equation \ref{2-13} can be used to construct a dimensionally correct equation by finding the units that should be associated with the coefficients 1920 and 597. For example, the correct form of the first term should be expressed as

    \[ \left( \begin{array}{c} \text{correct form needed} \\ \text{to produce micrometers} \end{array} \right) = \frac{1920 (\text{units}) \sqrt{\alpha } }{V_{a} \sqrt{\rho_{\ell } } } \label{2-14}\]

    in which the correct units are determined by

    \[(\text{units}) = (\text{micrometers}) \left[ \frac{(ft/s)\sqrt{g/cm^3} }{\sqrt{\mathrm{dyne}/cm} } \right]=10^{-4} \ cm \left[\frac{( ft/s)\sqrt{g/cm^3}}{\sqrt{ g/ s} } \right]=30.48\times 10^{-4} \sqrt{cm} \label{2-15}\]

    Thus the dimensionally correct form of the first term in Equation \ref{2-13} is given by

    \[\frac{1920\sqrt{\alpha } }{V_{a} \sqrt{\rho_{\ell } } } \Rightarrow \left(5.85\sqrt{ cm} \right)\frac{\sqrt{\alpha } }{V_{a} \sqrt{\rho_{\ell } } } \label{16}\]

    and it is an exercise for the student to determine the correct form of the second term in Equation \ref{2-13}.

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