### “U”-tube liquid filled manometers

**Fig. 1**shows the schematic of this manometer. A glass tube

**, bended to the “U”-shape, is filled by one half of its volume with liquid**

*1***(water, mercury). This tube is placed vertically, and a scale**

*2***(usually in**

*3**mm*) is attached along its height. Pressures P

_{1}and P2 are supplied to legs of the tube, and levels of liquid in the legs change their position.When static balance between a measuring pressure and the head of the liquid column is reached, this pressure can be evaluated according to the equation:

**(3)**

_{1}

**and**

**P**

_{2}- pressures supplied to the legs of the manometer,

_{1}

**and**

**h**

_{2}- deviations of liquid levels from the zero point of the scale in two legs of themanometer,

m;H= h

_{1 + }h

_{2}

**- total length of the liquid column corresponded to the measuring differentialpressure,**

**- density of liquid filled the “U”-tube,**

^{3};g

_{loc}

**- local gravitational acceleration, m/s**

^{2}.

_{1}

**and h2 are not equal. As the result of such reading the error introduced during pressure measurement will be reduced. When this type of manometer is used for pressure measurements three cases may take place:**

_{1 }is above atmospheric pressure, P

_{2 }= P

_{atm}. In this case the manometer measures the difference between absolute and atmospheric pressures: P

_{1 = }P

_{g }= ρg

_{loc}(h

_{1}+h

_{2})

**2).**P is below atmospheric pressure, P

_{1 }

**=**P

_{atm}. In this case manometer measures the difference between atmospheric and absolute pressures: P

_{2 = }P

_{vac }

**=**ρg

_{loc}(h

_{1}+h

_{2})

**.**

**3).**In this case the equation (3) refers to measurements of differential pressures.

**Fig. 2**gives examples how operator should make readings when using “U”-tube manometer with various liquids. We should always read a surface of the meniscus in its centre. In the case with water - in the bottom, and in the case with mercury - in the top of the meniscus. But in everyday industrial measurements the first two corrections (gravitational and thermal) are not always used, whereas the last one (the meniscus correction) must always be taken into account.

Hagen-Poiseuille equation applies for a laminar flow of liquid in the tube according

**(13)**

.

__In this equation:__

a - acceleration of liquid in the tube,

m/s

^{2};A - cross-sectional area of the tube,

m

^{2};D - internal diameter of the tube,

m;F

_{P}- displacement pressure force,

N;F

_{f}- frictional force for the laminar flow,

N;F

_{g}- gravitational restoring force,

N;g - gravitational acceleration,

m/s

^{2};h - variation of the height of the liquid column in one leg of the manometer,

m;L - total length of the liquid column in the tube,

m;m - mass of liquid in the tube,

kg;∆P - difference of pressures supplied to both legs of the manometer,

Pa;P

_{g}- pressure developed by the gravitational force acting on the liquid columnin the tube, in other words this is liquid pressure head,

Pa;P

_{1,}P

_{2}- pressures supplied to both legs of the manometer,

Pa;∆Q - variation of the volumetric flowrate of the liquid in the tube duringdisplacement of liquid,

m

^{3}/s;∆V - variation of the liquid volume displaced,

m

^{3};∆t - time during which ∆V occurred,

s;ρ - density of the liquid in the tube (density of gas above liquid is negligible),

kg/m

^{3}η - dynamic viscosity of the liquid in the tube Pa x s.

Let:

**(14)**

**(15)**

**(16)**

where,

τ - characteristic time of the system, s;

ζ - damping factor, dimensionless value;

K

_{p}- steady state, or static, or simply gain of the system, m

^{2}* s

^{2 }/kg.

Substitute (14)-(16) into (13) and use variables in deviation form:

**(17)**

Apply Laplace transform to (17):

**(18)**

**(19)**

**(20)**

**(21)**

**ζ**

__1. At__

__= 1 we have__*a critically damped response*.**(22)**

**(23)**

^{3}- density of the liquid (mercury) in the manometer;g = 9.80665, m/s

^{2}- standard gravitational acceleration.

_{p}= 3.767*10

^{-6}, m

^{2}*s

^{2}/kgFinally, we’ve got an expression for a critically damped response:

*Figure 3.**Critically damped response for a “U”-tube manometer.*

radian frequency of oscillations: ,

*Figure 4.**Underdamped response of a “U”-tube manometer*

*Figure 5.**Overdamped response of a “U”-tube manometer.*

Article Source:: Dr. Alexander Badalyan, University of South Australia