Measuring Pressure with a Multifluid Manometer
The water in a tank is pressurized by air, and the pressure is measured by a multifluid manometer as shown in Fig. 3–24. The tank is located on a mountain at an altitude of 1400 m where the atmospheric pressure is 85.6 kPa. Determine the air pressure in the tank if h_{1}=0.1 m , h_{2}=0.2 m , \text { and } h_{3}=0.35 m. Take the densities of water, oil, and mercury to be 1000 kg/m³, 850 kg/m³, and 13,600 kg/m³, respectively.
The pressure in a pressurized water tank is measured by a multifluid manometer. The air pressure in the tank is to be determined.
Assumption The air pressure in the tank is uniform (i.e., its variation with elevation is negligible due to its low density), and thus we can determine the pressure at the air–water interface.
Properties The densities of water, oil, and mercury are given to be 1000 kg/m³, 850 kg/m³, and 13,600 kg/m³, respectively.
Analysis Starting with the pressure at point 1 at the air–water interface, moving along the tube by adding or subtracting the 𝜌gh terms until we reach point 2, and setting the result equal to P_{\text {atm }} since the tube is open to the atmosphere gives
P_{1}+\rho_{\text {water }} g h_{1}+\rho_{\text {oil }} g h_{2}-\rho_{\text {mercury }} g h_{3}=P_{2}=P_{\text {atm }}
Solving for P_{\text {1 }} and substituting,
\begin{aligned}P_{1}=& P_{ atm }-\rho_{\text {water }} g h_{1}-\rho_{\text {oil }} g h_{2}+\rho_{\text {mercury }} g h_{3} \\=& P_{ atm }+g\left(\rho_{\text {mercury }} h_{3}-\rho_{\text {water }} h_{1}-\rho_{\text {oil }} h_{2}\right) \\=& 85.6 kPa +\left(9.81 m / s ^{2}\right)\left[\left(13,600 kg / m ^{3}\right)(0.35 m )-\left(1000 kg / m ^{3}\right)(0.1 m )\right.\\&\left.-\left(850 kg / m ^{3}\right)(0.2 m )\right]\left(\frac{1 N }{1 kg \cdot m / s ^{2}}\right)\left(\frac{1 kPa }{1000 N / m ^{2}}\right) \\=& 130 kPa\end{aligned}
Discussion Note that jumping horizontally from one tube to the next and realizing that pressure remains the same in the same fluid simplifies the analysis considerably. Also note that mercury is a toxic fluid, and mercury manometers and thermometers are being replaced by ones with safer fluids because of the risk of exposure to mercury vapor during an accident.