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The previous issue covered the principles of thermal mass flow meters and their controllers, conducted theoretical analysis based on these principles, and identified the fundamental issues that limit the improvement of the performance of thermal mass flow meters. In this issue, we will mainly explain the basic concepts of laminar flow gas mass flow meters and the principle of differential pressure flow meters. It is hoped that through the introduction to the concept of laminar flow meters and traditional differential pressure flow meters, everyone will understand that laminar flow meters are a special type of differential pressure flow meter. Its basic principle is, like that of traditional differential pressure flow meters, to measure flow rate through pressure difference, but the flow regime is different. 1. What are the basic concepts of the laminar flow principle? The principle of laminar flow is the product of a combination of ancient and modern theoretical techniques; why is that? Portrait of Evangelista Torricelli. Figure 1: The laminar flowmeter, also known as the laminar pressure difference flowmeter (Laminar Flowmeter, abbreviated as LFM), is a special type of pressure difference flowmeter. To distinguish it from traditional pressure difference flowmeters, China’s standards classify laminar flowmeters as a separate category. The development of differential pressure flow meters can be traced back to ancient hydraulic engineering and urban water supply systems. During the time of Caesar in ancient Rome, orifice plates (a type of differential pressure flow meter) were already used to measure the amount of drinking water supplied to residents. Around 1000 BC, ancient Egypt used weirs to measure the flow rate of the Nile River. China’s famous Dujiangyan water management project makes use of water level observations and flow volume measurements at the Baopingkou site. In the 17th century, Evangelista Torricelli (1608–1647), an Italian physicist and mathematician as shown in Figure 1, laid the theoretical foundations for differential pressure flow meters, which marked a milestone in flow measurement. The laminar flow principle mass flow meter is derived from the differential pressure flow meter. It inherits the advantages of traditional differential pressure flowmeters, while also effectively addressing their disadvantages. 2. What is the basic principle of traditional differential pressure flow meters? Principle of differential pressure flow meters (orifice flow meters) and pressure distribution diagram. Figure 2 shows that a differential pressure flow meter consists of a standard throttling element (such as a standard orifice), pressure lead pipes, and a differential pressure transmitter. Common standard throttling devices include standard orifice plate throttling devices, nozzle throttling devices, and Venturi tube throttling devices. Standard orifice throttling devices have been widely used due to their simplicity in manufacturing and installation, as proven by long-term practice. Rosemount’s orifice flow meters – Figure 3. Coriolis mass flow meters from Colombo – Figure 4. Traditional differential pressure flow meters; all of these devices operate on the basis of two laws: the conservation of mass of the fluid in a closed pipeline (as per the continuity equation) and the conservation of energy (as per Bernoulli’s equation). Assume that the cross-sections on both sides of the orifice plate are named Section 1 and Section 2, respectively. According to the laws of mass conservation and energy conservation, we have the following two formulas: Based on the law of mass conservation for fluids, the continuity equation is given by the following formula: Equation 1. For easier understanding, we assume that the fluid here is a viscous-free, incompressible fluid (an idealized fluid; viscous-free means there are no flow losses, and incompressible means the fluid density remains constant). In that case, Equation 1 can be rewritten as: Equation 2, where represents the fluid density at Section 1, represents the fluid density at Section 2, and refers to the density of this assumed incompressible fluid ； Represents the flow area at section 1, and represents the flow area at section 2 ； represents the average fluid flow velocity at section 1, and represents the average fluid flow velocity at section 2; the definition of these sections is shown in Figure 2. According to the law of energy conservation for fluids, and following Bernoulli’s equation, as shown in Equation 3. Similarly, for easier understanding, we assume that the fluid here is a viscous-free, incompressible fluid (an idealized fluid; viscous-free means there are no flow losses, and incompressible means the fluid density remains constant). In this case, Equation 3 can be rewritten as Equation 4. In these equations, represents the fluid pressure at Section 1, while represents the fluid pressure at Section 2; the locations of these sections are shown in Figure 2. By substituting Equation 2 into Equation 4 and simplifying, a relationship regarding can be obtained, as shown in the following equation: Equation 5 can be used to express the flow rate passing through a differential pressure flow meter as follows: By substituting Equation 5 into Equation 6 and simplifying, the following equation is obtained: Equation 7 It is easy to see from Equation 7 that the flow rate of a conventional differential pressure flow meter is proportional to the square root of the pressure difference (i.e., ). In other words, if the flow rate increases by 10 times, the pressure difference needs to increase by 100 times. This results in a nonlinear relationship between the pressure difference signal of conventional differential pressure flowmeters and the flow rate, which greatly increases the difficulty of calibration and reduces the accuracy and repeatability of the flowmeters. At the same time, as the flow rate decreases, the pressure difference drops much more rapidly. Therefore, the range ratios of traditional differential pressure flowmeters are not large, generally around 3 to 4 times (the measurement range of a single flowmeter is small). Let’s now summarize the advantages and disadvantages of traditional differential pressure flowmeters: 1 Advantages (1) The orifice plate flowmeter, which is the most widely used type, has a robust structure, stable and reliable performance, and a long service life ； (2) It has a wide range of applications, and to date no other type of flow meter can compare with it ； (3) The sensing element, transmitter, and display instrument are manufactured by different manufacturers, which facilitates economies of scale in production. 2 Disadvantages: (1) The measurement accuracy is generally low ； (2) Narrow range, generally only 3:1 to 4:1 ； (3) High requirements for on-site installation conditions, requiring long straight pipe sections ； (4) High pressure loss (referring to orifice plates, nozzles, etc.). Furthermore, traditional differential pressure flow meters operate in a turbulent state, and turbulence is a flow regime of fluids. In turbulent conditions, parameters such as the flow velocity and pressure of the fluid fluctuate irregularly, and this fluctuation further increases the difficulty of measuring with flow meters. On the other hand, the unevenness in velocity and pressure distributions across the flow channel cross-section also increases the measurement errors of traditional differential pressure flowmeters.