3.5 Bernoulli principle

In hydrodynamics, Bernoulli principle and derived equations are a special form of energy conservation equation. More specifically, these equations are special forms of general fluid flow energy conservation equations, which were first explained mathematically by Leonhard Euler in 1757. The principle is actually a set of related equations, whose form varies with different flow types. The basic Bernoulli equation for stable incompressible flow is as follows:

This equation applies to fluids moving along a "streamline" (defined as "s"), which is a continuous path through which the fluid moves. All changes of the fluid occur in the streamline, and there is no flow in or out of the streamline. This concept is applied to the flow meter. The fluid in the flowmeter enters the pipeline. Now the pipeline is the streamline. This one-dimensional model is sufficient to describe the flow field in the tube for steady-state expansion flow.

Bernoulli's equation can be used as a transfer function between input (flow and fluid state) and output (differential pressure) of differential pressure flow. The advantage of Bernoulli equation is simple and clear. It is a feasible method of fluid flow measurement in the field of engineering.

In the process of flow measurement, it must be considered that the fluid flow is in "steady state", that is, the speed rate or state of the measurement operation will not change significantly. However, these conditions may be limited. In fact, most fluid systems are designed to operate in a steady state with small changes to prevent excessive pressure transients or vibration in the system.

In order to achieve energy balance, it is assumed that the system is not heated and the system is not working or the system is not working. In fact, both the pump and the fan are working on the fluid - otherwise there is no need for them. However, when the system boundary is around the flowmeter, this approximate treatment can eliminate the system energy term well. In the application form of fluid flow, Bernoulli equation reflects the energy difference between two points of fluid flow. In this description, because of the energy conversion between potential energy (pressure) and kinetic energy (velocity), point 1 shows high pressure; point 2 shows low pressure

Since the flow coefficient of the multi primary element varies with the Reynolds number, this test was performed on a series of Reynolds numbers to determine the CD and re curves, or flowmeter characteristics. For cross-section flowmeter, the curve of different specific pressure can also be determined. This data summary describes the flow coefficient characteristics for a large area ratio or beta for various possible flow states. According to the range of measured parameters, hundreds of data points will be generated.

Gas expansion coefficient (Y1)

The coefficient of gas expansion is also derived from laboratory tests in which a gaseous fluid, usually air, can be used to produce a known flow rate. The reason why it is not applicable to the actual flow state is that when the gas flows through the limiting position, the pressure drops, resulting in gas expansion and density reduction, so ρ 1 ≠ ρ 2. When the density decreases, the velocity will be slightly higher than the predicted value calculated by the theoretical discharge equation.

Known:

Y1 can be determined. For example, laboratory tests of the coefficient of gas expansion determined that 90 lbs should be collected according to the theoretical formula and 54.1 lbs in practice. 54.1 LBS is the mass flow rate including flow coefficient and gas expansion coefficient.

Figure 3.5. E - flow laboratory apparatus used to determine a gas flow expansion data point.

CD = 0.607 due to the same pipe and specific pressure used in this test.

Now you can calculate Y1.

Since the calculated flow depends on CD and Y1, and these two coefficients depend on the flow, it is necessary to recalculate the flow, then calculate the new CD and Y1 values, and repeat the operation until the difference between the subsequent calculation results becomes smaller.