Team Member: Yilun Ma Kewei Qian Yiqing Xu Yunan Wang
Date of experiment: Jan,28th,2014 Date of submission of report: Feb, 11th, 2014
1. Introduction: In our second experiment, we do the tasks about venturi tube. In this experiment, we need to use Bernoulli’s equation because it’s one of the foundational analytic tools in Fluid Mechanics. The main goals of this experiment for students are to gain more understanding of measurements of pressure and to show how useful about the Bernoulli’s equation.
2. Objectives:
(1) This experiment needs to demonstrate availability of the Bernoulli’s equation for the flow of a fluid with a Venturi tube;
(2) This experiment also needs to help students gain deeper understanding about pressure measurement.
3. Theory: The Bernoulli’s equation is one of the foundational analytic-al tools in Fluid Mechanics. For an inviscid, incompressible and steady-state flow of density, acceleration due to gravity g, static pressure at a cross section P, mean velocity at that cross-section V, and elevation z from a reference datum, the Bernoulli’s equation may be expressed as: P/ρg +V2/2g +z = h (2.1)
The parameter h in equation (2.1) represents the total (stagnation) head, which is a constant. The terms on the left hand side of equation (2.1) are respectively the static pressure, dynamic pressure and elevation heads. Equation (2.1) provides a reasonable basis for explaining certain phenomena of fluid flow through pipes. It shows why for a pipe of varying cross-section, the pressure may be lower at points where the mean velocity is higher, and vice versa.
Figure 2.1: A schematic of a Venturi tube having a flow of fluid (in the direction of the arrows shown), and with manometers reading pressure measurements at points (1) and (2).
For the present consideration, our focus will be limited to a Venturi tube, as shown in figure 2.1. Using the above-mentioned assumptions, equation (2.1) may therefore be expanded to cover two different locations along two points in the tube, denoted by subscripts 1 and 2. Because the center-line of the cross-sections lies on the same elevation, the elevation head terms (z) can be neglected such that equation (2.1) becomes:
P1/ ρg + V12/ 2g = P2/ ρg + V22/ 2g = h (2.2)
Thus, Bernoulli’s equation reduces to that of the sum of the static pressure head (hstatic = P/ (ρg) ) and the dynamic pressure head (hdyn = V2 / (2g) ) respectively at a cross-section. The mean velocity at a given cross-section can be evaluated from the total pressure head (h) and the static pressure head (hstatic) at that cross-section as:
V= √2g( h- P/ ρg) = √2g( h- hstatic) (2.3)
If the cross-sectional areas A1 and A2 at 1 and 2 respectively are circular with respective diameters of D1 and D2, then the continuity equation for the incompressible flow at the two cross-sections is given by:
ρA1V1 = ρA2V2 (2.4)
After some manipulation of equation(2.4), the following relation between the mean velocities, areas, and diameter at the two locations result V2/V1 = A1/A2 =(D1/D2)2 (2.5)
Thus, from equation(2.2) and (2.4), the static pressure at 2 is found to be P2 = P1 + ρ·(V1)2 ·[1-(D1/D2)4] /2 (2.6)
The static pressure head at any given location x with respect to a given reference location 1 may be given by h(x) = h1+(v1)2 [1- (D1/D2)]4/ 2g (2.7) where D(x) is the diameter at x, and h1 is the static pressure head at 1.
(Reference:Lab2 Manual, MECH 2262;Chapter2 : VENTURI TUBE EXPERIMENT)
