Analysis of the range ratio for differential pressure flow measurement

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　In process control, differential pressure-type flow meters are commonly used. Although there are various types of them, such as advanced orifice plates, ordinary orifice plates, elbow flow meters, and venturi tubes, their principle is the same: they all rely on Bernoulli’s equation. Manufacturer promotional materials often include information on the range ratio of such flow meters, such as 10:1, 20:1, or even 40:1, 100:1. This means that even when the measured flow rate is as low as 1/40 or 1/100 of the maximum flow rate, the accuracy of the orifice plate is still maintained. So what is the truth? How large can the range ratio be? The orifice plate is used as an example for analysis below. 　　　　1 Parameter analysis affecting the range ratio of orifice plates　　　The flow calculation formula for differential pressure instruments is　　　　　　 (1)　　　It can be seen from this formula that flow rate is related to factors such as the discharge coefficient C, the opening ratio β, the expansion coefficient ε, the differential pressure, and density ρ; therefore, the range ratio of orifice plates is also influenced by these parameters. To discuss the range ratio issue, we first need to examine the various parameters involved. 　　　　1.1 Outflow coefficient C The outflow coefficient is the ratio of the actual flow rate to the theoretical flow rate. Since the early 1990s, numerous experiments have been conducted in Western laboratories, a large amount of experimental data has been collected, and international standards have been established. As a result, the orifice plate is, to date, one of the flow meters among all types of flow meters that requires only accurate geometric dimensions to ensure measurement precision, without the need for calibration using actual flow rates. Moreover, since the parts of the orifice plate that come into contact with the fluid are essentially purely mechanical components capable of withstanding high temperatures and pressures, this enables its wide range of applications. In particular, the advent of advanced replaceable orifice plates **has expanded its range of applications. 　　　　Ignoring other factors, the uncertainty of the discharge coefficient for orifice plates is generally 0.6%. 　　　　Table 1 shows the variation of the discharge coefficient for orifice plates with angular inlet conditions as defined in ISO5167, as a function of flow rate. 　　　　The Reynolds number is used in the table. In the same medium within the same pipe, the Reynolds number is proportional to the flow rate. Taking β=0.5, Reynolds number=3′105 and Reynolds number=3′104 as examples, when the flow rate changes by a factor of 10, the discharge coefficient varies from 0.6037 to 0.6096; the error is significant. It can be seen that the discharge coefficient changes considerably with variations in flow rate – a 10-fold change in flow rate results in a change in the discharge coefficient of nearly 1%. Many people now believe that accuracy can be maintained over a wide range as long as the density is compensated for, which is unscientific. Using the discharge coefficient obtained from the calculation of the orifice across the entire flow range will inevitably lead to significant errors. Therefore, it is essential to compensate for the outflow coefficient. 　　　　Now, the formula for the outflow coefficient is given in ISO5167. The formula for pressure extraction at the corner joint is　　　　　　　　　　　　where ReD is a variable that changes with flow rate. Integrating this formula into a computer essentially solves the problem of compensating for the outflow coefficient. 　　　　Formulas for other pressure measurement methods can also be entered into the computer according to standards for compensation purposes. 　　　　1.2 Opening ratio β      The orifice plate ratio is the ratio of the opening area of the orifice plate to the inner diameter of the pipe. Once the orifice plate is completed, its opening ratio is already determined. However, since pipes and orifice plates are often made of different materials with varying coefficients of expansion, the orifice ratio still changes to some extent as pressure and temperature change. When calculating orifice plates, the orifice size is typically determined based on the operating temperature and pressure specified by the process, and then converted to the orifice size ratio at 1 atmosphere and 20°C. Although the relationship between it and the flow rate range is not very strong, if the operating temperature and operating pressure are not provided accurately, it will directly affect the accuracy of the calculations; of course, the flow rate ratio will also be affected. As long as the pressure loss requirements are met, it is recommended to set the β value at around 0.5 to 0.6. 　　　　1.3 Coefficient of expansion The coefficient of expansion also changes with the differential pressure, that is, it varies with the flow rate. If the same expansion coefficient ε is used for the entire range of flow rates, large errors will occur. Now, ISO5167 provides a formula for the expansion coefficient. With the development of computers, similar to compensation for the outflow coefficient, it is only by incorporating this formula into the orifice plate calculation formulas used in control systems that it is possible to **improve measurement accuracy**. 　　　　1.4 Differential Pressure Value As is well known, the accuracy and range of a differential pressure transmitter have a significant impact on those of a orifice plate. Therefore, in recent years, manufacturers have been continuously improving the accuracy and range ratio of differential pressure transmitters. The following analysis takes the commonly used ROSEMENT3051CD2-5 differential pressure transmitter as an example. 　　　　This transmitter has an accuracy of 0.065% within a range of 10:1, and its accuracy decreases beyond that range. 　　　　As can be seen from equation (1), the flow rate is proportional to the square root of the differential pressure; therefore, when the flow ratio of the differential pressure transmitter is 10:1, the flow range ratio is only 1:3.6:1. And a gain ratio of 3.16:1 is far from sufficient to meet the requirements on-site. 　　　　Below, calculations are carried out using a higher flow rate point of 1/10 as an example. The flow rate is 1/10 of the higher flow rate, and the pressure difference is 1/100 of the higher pressure difference. Therefore, the accuracy of the differential pressure transmitter calculated using the above formula at that point is 0.515%. If the larger differential pressure corresponding to the rated flow rate is 100 kPa, then the differential pressure at that point should be 1 kPa. Therefore, 1 ± 1*0.515% = 1 ± 0.00515, meaning the differential pressure at that point ranges from 0.99485 to 1.00515 kPa. After taking the square root, this becomes 0.9974 to 1.00257, indicating that the flow rate is 99.74% to 100.26% of the actual flow rate, with an error of 0.26%. Together with the influence of parameters such as the discharge coefficient and expansion coefficient, it becomes very difficult to ensure a measurement accuracy of 1% or even 0.5% at a flow rate of 1/10Qmax. Interested readers can calculate that when the flow rate is 1/20 of the higher value, the error of the differential pressure transmitter at that point is already well above 1%. So, if a flow ratio of 10:1 can still barely ensure accuracy, then it becomes very difficult to maintain accuracy at a ratio of 20:1. If a differential pressure transmitter with a low range is used, its accuracy and range ratio are both lower than those of medium differential pressure gauges; as a result, the error in its output changes more significantly as the flow rate varies. No further explanation is needed here. 　　　　1.5 Density ρ      The variation in density has a crucial impact on flow rate, especially in the case of flow measurement involving gases or vapors. With the development of computer technology, compensating for density has become a very easy task; it is straightforward to enter into a computer the equations for gas laws, tables of vapor density, or formulas for liquid density as a function of temperature and pressure. Therefore, although density is crucial, as long as the relevant compensation methods are implemented properly, its impact on the range ratio is relatively small. However, not every change in temperature and pressure within a certain range can be compensated for by changes in density. 　　　　In addition, issues such as the installation location, installation concentricity, straight pipe sections, resistance elements, temperature drift of the transmitter, and zero drift all have a significant impact on flow rate. It will naturally also affect the range ratio. 　　　　　　　　2 Several methods for expanding the range ratio of orifice plates Wide-range orifice plates remain an issue that requires further exploration; relying solely on density compensation cannot ensure measurement accuracy over a wide range. If, in addition to density compensation, compensation is also applied to the other parameters discussed above, and the installation conditions on site meet the standard requirements, with a flow ratio within 10:1, the accuracy of the entire orifice plate flow meter can generally reach 1.0%–1.5%. If the range ratio has already exceeded 10:1, the author recommends choosing another type of flow meter based on the properties of the medium, or opting for other methods to increase the range ratio of the orifice plate. 　　　　The methods for expanding the range ratio of orifice plates generally fall into three categories: 1) Measuring by combining multiple high-flow channels in parallel ； This method is generally used in calibrating equipment. Different pipelines have different flow ranges; to achieve a higher flow rate, it is sufficient to use the pipeline whose capacity is equal to the required flow rate. 2) By changing the β value through the replacement of the orifice plate, measurements can be taken; this allows for the replacement of the orifice plate flow meter, commonly known as an orifice plate valve ； 3) Use a orifice flow meter in parallel with differential pressure gauges of different ranges for measurement. In practical applications, one orifice plate flow meter is used in parallel with three differential pressure transmitters: one for low differential pressure, one for medium differential pressure, and one for high differential pressure. All three transmitters operate simultaneously, sending their readings to the calculation unit at the same time. The computing unit uses a program to filter the three differential pressure values; it also determines the parameter, the flow coefficient C, and the expansibility coefficient ε. Temperature and pressure are still compensated in real time. This approach enables the system uncertainty of the flow measurement to be kept at ±1.2%, with a range ratio that can be increased to 15:1. 　　　　3 Conclusions     In general, the differential pressure flow meter remains a flow measurement instrument with great potential for development. By continuously exploring new methods to address its shortcomings, it can be made more widely and accurately applied.