Laminar flow meter

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Design and fabrication of laminar flow elements, Heilongjiang Provincial Intellectual Property Office, Ji Chongyuan. In fluid flow measurement, a type of laminar flow meter is attracting attention. A laminar flow meter is a device that converts the fluid under test, which is in a turbulent state, into a laminar state through a laminar flow element before measuring its flow rate; therefore, designing and manufacturing a high-performance laminar flow element is crucial for improving the quality of laminar flow meters. The structure of a laminar flow element is very simple; it consists of many small laminar flow tubes connected in parallel. For laminar flow meters designed to measure large flow rates, a huge number of such tubes are required. A high-quality laminar flow element should aim to reduce the number of tubes while still meeting the requirements for measuring flow rate. For this purpose, there are two factors to consider. The first is to improve the sensitivity of pressure difference measurement; reducing the maximum value of the differential pressure gauge allows for a larger allowable diameter of the laminar flow tube ； The second aspect is to select an appropriate cross-sectional shape for the laminar flow tube, as the cross-sectional shape of each such tube affects the Reynolds number of the fluid. For pipes with the same flow rate and cross-sectional area but different shapes, the Reynolds number of the fluid flowing through them varies; therefore, by choosing an appropriate cross-sectional shape for the laminar flow tube, it is possible to create laminar flow tubes, and by using these tubes to fabricate laminar flow elements, high-quality laminar flow elements can be obtained. 1. Characteristic length of the Reynolds number: Since the Reynolds number is a dimensionless quantity, it represents the ratio between the inertial force and the frictional force in fluid flow. In laminar flow, the fluid is subjected only to frictional forces, in addition to inertial forces. When the fluid flows through pipes of different shapes, although the flow rate remains the same, the varying frictional forces result in different Reynolds numbers. Research has shown that the product obtained by dividing the cross-sectional area of the pipe by the length of its sides is a very important parameter; for now, let’s call this characteristic length of the Reynolds number t. In the formula, S represents the cross-sectional area of the pipe ； b is the side length of the pipe cross-section. When fluid flows through two pipes with different cross-sectional shapes, if the cross-sectional areas of these two pipes are equal but the side lengths of their cross-sections are not, the Reynolds number of the fluid flowing through the pipe with the longer side lengths will be lower. In other words, when fluid flows through two pipes with equal cross-sectional areas but different Reynolds number characteristic lengths t, the Reynolds number of the fluid in the pipe with the smaller t value is lower, and the reverse is also true. Therefore, t actually represents an important parameter indicating the effect of pipe shapes on the Reynolds number. 2. Geometric shape of the laminar flow tube cross-section: A laminar flow element is composed of multiple laminar flow tubes connected in parallel. The geometric shapes of these tubes can vary, and different geometric shapes result in different characteristic lengths t for the Reynolds number. When the flow velocity of the fluid reaches a certain value, the Reynolds number of the fluid in the tube with the larger characteristic length t may already have reached the critical value, while that in the tube with the smaller t is still below the critical value. This means that a smaller t allows the fluid to achieve a higher critical flow rate in order to maintain a laminar state; therefore, when designing laminar flow elements, it is advisable to choose those with a smaller t for their tube cross-sections. When considering the geometric shape of pipe cross-sections, there are generally two main categories: convex pipes and concave pipes. Generally, the t-value of concave tubes is relatively low; however, laminar flow elements made of concave tubes have a low effective space utilization rate, making them unsuitable for laminar flow meters with high flow rates. They can be used in laminar flow meters with low flow rates, while in laminar flow meters designed for high flow rates, only convex laminar flow tubes are typically considered. Among convex tubes, there are circular tubes, polygonal tubes, square tubes, rectangular tubes, and triangular tubes, etc. Among these shapes, the circular tube has the highest t value, while the equilateral triangular tube has the lowest t value. The difference between them can be seen from the following derivation: Let the diameter of the circular tube’s cross-section be d; then its cross-sectional area is..., and its side length is..., so.. ； Let there be another equilateral triangular tube, whose cross-section is an equilateral triangle with side length a; then its cross-sectional area, and its side length is, so we have ； At that time, it was: d=0.7425a, and. This shows that using an equilateral triangular tube as a laminar flow tube results in a smaller t value compared to using a circular tube, but this does not mean that it is the only option to choose; there is another cross-sectional shape available with an even smaller t value, namely an acute triangular tube. 3. Design of laminar flow elements using right-angled triangular tubes as laminar flow pipes 3.1 Reynolds number characteristic length t of the right-angled triangular tube: The right-angled triangle referred to here is one that is formed by dividing an equilateral triangle along its height; thus, the right-angled triangle in question has side lengths of /2 and . For an acute right triangle, its perimeter is: , its cross-sectional area is: , and thus the characteristic length for the Reynolds number is: . Compared to a circular pipe with the same cross-sectional area, it can be shown that a=1.9047d. By comparing the Reynolds number characteristic length of a right-angled triangle with that of a circular pipe having the same cross-sectional area, it can be shown that: . It can be seen from this that laminar flow pipes composed of pipes with right-angled triangular cross-sections can have a larger pipe diameter when achieving laminar flow. 3.2, Determination of design parameters. The parameter to be determined here is the side length a of the right-angled triangle. Before determining this parameter, it is first necessary to specify the dynamic viscosity and density of the fluid being measured; thereafter, the range of the differential pressure gauge used is determined to determine the maximum pressure difference that can be measured. In addition, there are two parameters that must be given in advance, namely the length l of the laminar flow tube and the wall thickness of the laminar flow tube, which are known values. When fluid flows through a laminar flow pipe in the shape of a right-angled triangle, the formula for calculating the Reynolds number is as follows: where Q is the flow rate of the fluid (), ρ is the density of the fluid (), a is the length of the hypotenuse of the right-angled triangle (in cm), and μ is the dynamic viscosity of the fluid (). When the fluid flows laminarly through such a pipe, the formula for calculating the flow rate is:, where ΔP represents the pressure difference across the two ends of the pipe, measured in units of pressure. Since it is a laminar flow meter, the maximum flow rate that can be measured should correspond to the critical Reynolds number; that is, . According to the flow formula, it can be obtained that: ; by equating this with the previous expression, we get: , that is: ; from this, the diameter of the right-angled triangle designed can be determined. 3.3, Structural design of laminar flow elements: Laminar flow elements are composed of laminar flow tubes connected in parallel. When the cross-section of the laminar flow tube is an equilateral triangle, a layer-by-layer circular structure can be employed. From the perspective of the cross-section of the laminar flow element, it consists of numerous circles with common centers; the first layer may contain 12 equilateral triangular tubes, the second layer may contain 48 such tubes, and the nth layer contains n equilateral triangular tubes. Thus, the total number of triangles in the n concentric circles is n×… The outer diameter of these concentric circles corresponds to the inner diameter of the laminar flow element. Therefore, the maximum flow rate that can be measured by the laminar flow element composed of n concentric circles is: . 3.4, Illustration: The flow rate measurement of air at room temperature is used to illustrate this design. An equilateral triangular tube is selected as the laminar flow tube to form the laminar flow element, a micro-differential pressure gauge ranging from 0 to 30 is used to measure the differential pressure, and the length of the laminar flow tube is set at 10 cm. The length of the hypotenuse of the cross-section of this equilateral triangular tube is therefore 0.526 cm, while its maximum flow rate is 11.29. Since the laminar flow tube section designed uses a right-angled triangular shape, its Reynolds number characteristic length is small, which allows for a larger diameter of the laminar flow tube section. As a result, the laminar flow elements can be manufactured as injection-molded parts, thereby **reducing manufacturing costs and accelerating production speeds, allowing laminar flow meters to be produced in bulk.