Velocity Flow Meters – Pitot/Pitot-Static Tubes

Pitot tubes (also called pitot-static tubes) are an inexpensive and convenient way to measure  velocity  at  a  point  in  a  fluid.  They  are  used  widely  in  airflow measurements in ventilation and HVAC applications.  Definitions for three types of  pressures  or  pressure  measurements are  given  below.  Understanding these definitions aids in understanding the pitot tube equation.  Static pressure, dynamic pressure and total pressure are defined below and illustrated in Figure 5.

Static pressure is the fluid pressure relative to the surrounding atmospheric pressure. It is measured through a flat opening which is in parallel with the fluid flow as shown with the first U-tube manometer in Figure 5.

Stagnation pressure is the fluid pressure relative to the surrounding atmospheric pressure. It is measured through a flat opening which is perpendicular to and facing into the direction of fluid flow as shown with the second U-tube manometer in Figure 5. This is also called the total pressure.

Dynamic  pressure  is  the  fluid  pressure  relative  to  the  static  pressure.  It  is measured through a flat opening which is perpendicular to and facing into the direction of fluid flow as shown with the third U-tube manometer in Figure 5. This is also called the velocity pressure.

Figure 5. Various Pressure Measurements

Static pressure is typically represented by the symbol, p.   Dynamic pressure is equal to ½ ρV2.    Stagnation pressure, represented here by Pstag, is equal to static pressure plus dynamic pressure plus the pressure due to the height of the measuring point above some reference plane, as shown in the following equation.

Where the parameters are as follows: Pstag   =  stagnation pressure, lb/ft2

P     =  static  pressure, lb/ft2

ρ   =   density of fluid, slugs/ft3

γ  =   specific weight of fluid, lb/ft3

h  =  height above a specified reference plane, ft

V  =   average velocity of fluid, ft/sec

(V =  Q/A = volumetric flow rate/cross-sectional area normal to flow)

For pitot tube measurements, the reference plane can be taken at the height of the pitot tube measurement so that h = 0.   Then, stagnation pressure minus static pressure is equal to dynamic pressure:

The pressure difference, Pstag   -  P, can be measured directly with a pitot tube such as the third U-tube in Figure 5, or more simply with a pitot tube like the one shown in Figure 6, which has two concentric tubes.   The inner tube has a stagnation pressure opening and the outer tube has a static pressure opening parallel to the fluid flow direction.  The pressure difference is equal to the dynamic pressure (½ ρV2) and can be used to calculate the fluid velocity for known fluid density, ρ.

A consistent set of units is:  pressure in lb/ft2, density in slugs/ft3, and velocity in ft/sec.

For use with a pitot tube, Equation (8) will typically be used to calculate the velocity of the fluid. Setting (Pstag  – P)   =   ∆P, and solving for V results in the following equation:

In order to use Equation (9) to calculate fluid velocity from pitot tube measurements, it is necessary to obtain a value for the density of the flowing fluid at its temperature and pressure.  For a liquid, a value for density can typically be obtained from a table similar to Table 1 in this course.  Such tables are available in handbooks and fluid mechanics or thermodynamics textbooks. Pitot tubes are used more commonly to measure gas flow, for example, air flow in HVAC ducts. Density of a gas varies considerably with both temperature and pressure. A convenient way to obtain a value for density for a gas, at a known temperature and pressure, is through the use of the Ideal Gas Law.

The Ideal Gas Law, as used to calculate density of a gas, is as follows:

Where:        ρ  =  density of the gas at pressure, P, and temperature, T, slugs/ft3

MW = molecular weight of the gas, slugs/slug-mole (The average molecular weight typically used for air is 29.)

P =  absolute pressure of the gas, psia

T  =  absolute temperature of the gas, oR  (oF + 459.67 = oR) R =  Ideal Gas Law constant, 345.23 psia-ft3/slug-mole-oR

But, if this is the Ideal Gas Law, how can we use it to find the density of real gases?  Well, the Ideal Gas Law is a very good approximation for many real gases over a wide range of temperatures and pressures.  It does not work well for very high pressures or  very low temperatures (approaching the  critical  temperature and/or critical pressure for the gas), but for many practical situations, the Ideal Gas Law gives quite accurate values for the densities of gases.

Example #5: Estimate the density of air at 16 psia and 85 oF.

Solution: Convert 85 oF to oR:     85 oF =  85 + 459.67 oR  =  544.67 oR Substituting values for P, T, R, and MW into Equation 11 gives:

ρ  =  (29)[16/(345.23)(544.67)] =   0.002468 slugs/ft3

Flow Nozzle Meter:  

The flow nozzle meter is simpler and less expensive than a venturi meter, but not quite as simple as an orifice meter.  It consists of a relatively short nozzle, typically held in place between pipe flanges as shown in Figure 4.

Figure 4. Flow Nozzle Meter Parameters

For a flow nozzle meter, the exit diameter of the nozzle, d, is used for D2 (A2 = An), and the discharge coefficient is typically called a nozzle coefficient, Cn, resulting in the following equation:

Due to the smoother contraction of the flow, flow nozzle coefficients are significantly higher than orifice coefficients.   They are not, however, as high as venturi coefficients.  Flow nozzle coefficients are typically in the range of 0.94 to

0.99.   There are several different standard flow nozzle designs.   Information on pressure tap placement and calibration should be provided by the meter manufacturer.

Orifice Meter:  

The orifice meter is the simplest of the three differential pressure flow meters.  It consists of a circular plate with a hole in the middle, typically held in place between pipe flanges as shown in Figure 2.

For an orifice meter, the diameter of the orifice, d, is used for D2 (A2 = Ao), and the discharge coefficient is typically called an orifice coefficient, Co, resulting in the following equation for an orifice meter:

The preferred locations of the pressure taps for an orifice meter have undergone change  over  time.  Previously  the  downstream pressure  tap  was  preferentially located at the vena-contracta, the minimum jet area, which occurs downstream of the orifice plate as shown in Figure 2. For a vena-contracta tap, the tap location depended upon the orifice hole size.   This link between the tap location and the orifice size made it difficult to change orifice plates with different hole sizes in a given meter in order to alter the range of measurement.  In 1991, the ISO-5167 international standard came out, in which three types of differential measuring taps were identified for orifice meters as illustrated in Figure 3 below. In ISO-5167, the distance of the pressure taps from the orifice plate is specified as a fixed distance or as a function of the pipe diameter, rather than the orifice diameter as shown in Figure 3.

In ISO-5167, an equation for the orifice coefficient, Co, is given as a function of β, Reynolds Number, and L1 & L2, the respective distances of the pressure taps from the orifice plate as shown in Figures 2 and 3.  This equation (indicated below) can

be  used  for  an  orifice  meter  with  any  of  the  three  standard  pressure  tap configurations.

Figure 3. ISO standard orifice meter pressure tap locations

The ISO-5167 equation for Co  (see Reference #3 at the end of this course) is as follows:

Co   =  0.5959 +  0.0312 β2.1   - 0.1840 β8   +  0.0029 β2.5(106/Re)0.75

+ 0.0900(L1/D)[β4/(1 - β4)]  - 0.0337 (L2/D) β3                     (5)

Where:     Co   =  orifice coefficient, as defined in equation (4), dimensionless

L1   =  pressure tap distance from upstream face of the plate, inches

L2   =  pressure tap distance from downstream face of the plate, inches

D =  pipe diameter, inches

β  =  ratio of orifice diameter to pipe diameter  =  d/D, dimensionless

Re  =  Reynolds number =  DV/ν  =  DVρ/µ,  dimensionless (D in ft) V =  average velocity of fluid in pipe =  Q/(πD2/4),  ft/sec (D in ft)

ν  =  kinematic viscosity of the flowing fluid, ft2/sec

ρ  =  density of the flowing fluid, slugs/ft3

µ =  dynamic viscosity of the flowing fluid, lb-sec/ft2

As shown in Figure 3:  L1  = L2  = 0 for corner taps;  L1  = L2  = 1 inch for flange taps;  L1 =  D  and  L2 =  D/2 for D – D/2 taps.   Equation (5) is not intended for use with any other arbitrary values for L1 and L2.

There are minimum allowable values of Reynolds number for use in equation (5) as follows.  For flange taps and (D – D/2) taps, Reynolds number must be greater than 1260β2D.  For corner taps, Reynolds number must be greater than 10,000 if β is greater than 0.45, and Reynolds number must be greater than 5,000 if β is less than 0.45.

Fluid properties (ν or ρ and µ) are needed in order to use equation (5).  Tables or graphs with values of ν,  ρ,  and µ for water and other fluids over a range of temperatures are available in many handbooks or textbooks (such as fluid mechanics or thermodynamics) as for example in Reference #1 at the end of this course. Table 1 shows density and viscosity for water at temperatures from 32o F to

70o F.

Table 1. Density and Viscosity of Water

Example #2:  What is the Reynolds number for water at 50oF, flowing at 0.35 cfs through a 4 inch diameter pipe?

Solution:  Calculate V from V = Q/A = Q/(πD2/4) = 0.35/[π(4/12)2/4] = 4.01 ft/s. From Table 1:  ν  =  1.407 x 10-5 ft2/s.  From the problem statement:  D = 4/12 ft. Substituting into the expression for Re = (4/12)(4.01)/(1.407 x 10-5):

Re = 9.50 x 104