Small differences in gas pressures are commonly measured

Chapter 2, Problem 2.45

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Small differences in gas pressures are commonly measured with a micromanometer of the type illustrated in Fig. P2.45. This device consists of two large reservoirs each having a cross-sectional area \(A_r\) which are filled with a liquid having a specific weight \(\gamma_1\) and connected by a U-tube of cross-sectional area \(A_t\) containing a liquid of specific weight \(\gamma_2\). When a differential gas pressure \(p_1 - p_2\), is applied, a differential reading, h, develops. It is desired to have this reading sufficiently large (so that it can be easily read) for small pressure differentials. Determine the relationship between h and \(p_1 - p_2\) when the area ratio \(A_t/A_r\) is small, and show that the differential reading, h, can be magnified by making the difference in specific weights, \(\gamma_2 - \gamma_1\), small. Assume that initially (with \(p_1 = p_2\)) the fluid levels in the two reservoirs are equal.