# Response time of the U-tube manometer

The manometer will not immediately sense pressure changes because a flow has to be generated in the pipe connecting the measuring point and the manometer to compensate for the change in density of the gas inside the tube and the manometer as well for the displacement of the manometer liquid (the first cause of delay is negligible in measurements performed in liquids).

If there is a sudden change in pressure, ±dp, a very slow viscous flow (the Poiseuille flow) is generated in the tube connecting the pressure tap to the manometer; the average speed in the section is proportional to the pressure difference between the tap and the manometer:  – 1

V= — V

2 max

where m is the dynamic viscosity of air, and £ and d are the length and the diameter of the tube.    The air flow rate passing through the tube produces a change in the mass in the manometer due to both a change in density and a variation in the volume available (Vol) caused by the displacement of the liquid:

Since the displacement of the liquid surface in the tube connected to the tap is half of Ah, by using Equation (1.1), is it obtained:

dVol = Adh = Ad{Ah = A dp dt dt 2dt 2g p£ dt

where A is the area of the section of the manometer tube. In isothermal conditions, the change in density is related to the change in pressure : dp _ p dp dt p dt

Substituting Equations (1.3), (1.5) and (1.6) into Equation (1.4) yields : pn – put d = A + Vol I d

128p£ ^ 2gp£ p J dt

The pressure in the manometer varies with speed decreasing gradually as the pressure difference between inlet and manometer decreases, therefore the manometer adapts only asymptotically to the new pressure, p + dp.

We define the time constant, t*, as the time the manometer would take to adapt to the change in pressure dp if the rate of change of pressure

were maintained equal to the initial value obtained by putting Pin – pout = dp in Equation (1.7):

Sp 128 pi f A Vol ^

T* = dp/dt = ndA [2g + ~J (1.8)

For normal-sized gages the time constant varies from 1 to 10 ms (frequency range 0.1 – r – 1 kHz). For this reason, and the difficulty of registering pressure fluctuations by hand, no hydrostatic manometers are suitable for unsteady pressure measurements.