Measuring Pressure — Question 7

Question 7

An inclined well manometer has two moving levels

A well manometer contains liquid of density $800\,\mathrm{kg/m^3}$. A narrow tube of area $a$ slopes at $30^\circ$ above horizontal and opens to atmosphere. The well area is $A=20a$. Increasing gas pressure over the well moves the tube meniscus a distance $s=0.100\,\mathrm{m}$ up the tube. Initially both surfaces were at the same elevation. Use $g=10\,\mathrm{m/s^2}$.

1. Find the well-surface fall and the gas gauge pressure.

2. For an along-tube reading uncertainty of $\pm1.0\,\mathrm{mm}$, find the corresponding pressure bound when other quantities are exact. Why does ignoring well motion bias the result?

Question 7 - Solution

An inclined well manometer has two moving levels

(a) Derivation and calculation

$$Ad=as\Rightarrow\boxed{d=(a/A)s=.100/20=.00500\,\mathrm m}.$$$$h=s\sin30^\circ+d=.0500+.00500=.0550\,\mathrm m.$$$$\boxed{p_g=\rho gh=(800)(10)(.0550)=440\,\mathrm{Pa}}.$$

(b) Result and interpretation

$$p_g=\rho gs(\sin\theta+a/A).$$$$\boxed{|\delta p|\le\rho g(\sin\theta+a/A)|\delta s|=800(10)(.50+.05)(.001)=4.4\,\mathrm{Pa}}.$$$$p_{\mathrm{omit\ well}}=400\,\mathrm{Pa}.$$Ignoring the well fall omits 40 Pa of actual vertical head.

Physical check

The bound is exact for the stated linear calibration with all other quantities fixed.