Lecture23 Flowmeter Design. Contents of lecture Design of flowmeter Principles of flow measurement; i) Venturi and ii) Orifice meter and nozzle Relationship between flow rate and pressure drop Relation between pressure drop and mass flow rate Key Words: Fluid flow, Macroscopic Balance, Frictional Losses, Turbulent Flow, Venturimeter, Orifice Meter, Pitot Tube Design of flowmeters Efficient handling, utilization and disposal of fluids in engineering processes require knowledge of quantities of fluids flowing. Indirectly, this information can be obtained by stoichiometric calculations. However, precise and accurate measurements of flow quantities become essential to efficient operation. Most of the flow measuring devices for engineering purposes can be designed by using mechanical energy balance for the device. Principles of flow measurement It is known that pressure energy can be converted into kinetic energy and vice versa. Thus, if a restriction is placed in the flow passage, the fluid would be accelerated with the corresponding decrease in pressure head. There are three different ways in which this can be practiced. Venturi In the venturi, the cross sectional area of a flow passage decreases gradually in the direction of flow and attains a minimum cross section area at the throat and thereafter gradually increases further in the direction of flow. Figure shows a venturi meter.
Figure1: Design of a venturi meter Pressure taps are installed upstream the throat and at the throat, and the difference in pressure at these two locations is used to calculate velocity and the total rate of fluid flowing through a venturi. Due to the gradual decrease and increase in the cross section of the flow passage, the effect of frictional forces on decelerating the fluid velocity can be considered to be negligibly small. ii) Orifice meter and nozzle An orifice meter is of simple construction. A thin plate with a centrally located hole is inserted into the flow passage. The main path of flow through the orifice is same as that of venturi, but the flow contacts suddenly as fluid passes through the hole. The flow continues to contract a short distance downstream the hole as shown in the figure. Figure also shows the flow lines. It can be seen that the region of smallest cross section, known as vena contracta, is developed downstream the orifice. In the vena contracta the kinetic energy is maximum. Note that the minimum cross section in the orifice meter is not orifice diameter, but it is cross section at vena contracta. Accordingly pressure tap is to be installed at vena contracta as shown in the figure.
Figure 2: Design of orifice meter and flow lines. Note that the smallest cross section is downstream the orifice diameter. Nozzles are similar to orifices in general but the decrease in cross section area in the direction of flow is gradual when compared with an orifice in which it is abrupt. So that the losses due to friction are lower in nozzle than in the orifice. Relationship between flow rate and pressure drop We can consider flow of an incompressible fluid and apply mechanical energy balance at plane 1 and 2 and neglecting frictional losses. We get P P 0 (1) Equation of continuity for incompressible fluid, gives V V (2) Here V and V is average velocity at plane 1 and 2, and d and d are diameter at plane 1 and 2. By 1 and 2 we get, V P P / (3)
Note that velocity V according to eq. 3 corresponds to maximum velocity when effect of friction on flow is ignored. Pressure difference corresponding to V in eq.3 is the one which one would read at plane 2 in venturi and in orifice at vena contracta. It must be noted that the equation 3 is not specific to any flow measuring device; it is applicable to orifice, venturi meter, nozzle or any other. The equation relates velocity of the fluid to the pressure difference and diameter ratio. Now the cross section area at vena contracta is not known and hence d 2 at vena contracta is not known. Vena contracta is created due to the abrupt contraction as the fluid passes through an orifice. The cross sectional area at vena contracta would depend, among other factors, on shape of the orifice (circular, rectangular or square, etc.) and fluid dynamics. It can be determined experimentally. However, diameter d of the orifice is known. We introduce coefficient of discharge C D and replace d in eq. 3 by do we get. V C D P P /, V K P P / (4) Here λ and K flow coefficient C D 1λ. Note V in eq. 4 is V in case of venturi. Equations 3 and 4 can be applied to venturi, and nozzles as well once we know the value of. The discharge coefficient value is specific to the flow measuring device. C D Relation between pressure drop and mass flow rate For incompressible fluid, the mass flow rate is mk A.2ρ P P., (5) where A is the minimum cross section area of the flow passage. AA in case of orifice meter and AA in case of venturi. For flow of gases one has to take into account the compressibility factor or expansion factor Y so that mk Y A 2ρ P P. (6) where Y P P P P P P / / (7) Orifice plates are simplest and cheapest types of flow meters but they cause permanent pressure drop in the systems. The permanent pressure drop can be calculated from ΔP 1 λ P P In the venturi the flow passage is designed so that the friction is minimum. Permanent pressure drop can be taken to be equal to 10% of the measured pressure differential.