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After studying the performance and characteristics of actual laminar flowmeters a new disposition for this type of sensors is proposed in such a way that the measurement errors introduced by the intrinsic nature of the device can be minimized. The preliminary study shows that the developing entry region introduces non-linearity effects in all these devices. These effects bring about not only errors, but also a change in the slope of the linear calibration respect of the Poiseuille relation. After a subsequent analysis on how these non-linearity errors can be reduced, a new disposition of this type of flowmeters is introduced. This device makes used of flow elements having pressure taps at three locations along its length and connected to three isolated chambers. In this way, the static pressure can be measured at three locations and contributed to by the pressure taps at the level of each chamber. Thus the linearization error is reduced with an additional advantage of producing a reduced pressure drop.

Flowmeters are devices of widespread use in many industrial processes that can use many different flows under many different conditions of pressure and temperature and can have many different requirements concerning cost, accuracy, safety, pressure losses, or materials compatibility, among others. A wide range of different types of flowmeters has been developed to satisfy the requirements in all cases regardless of these huge variations in fluid properties and circumstances [

Merging electronics into classical types of flowmeters has been quite common in the last decades as a means of increasing sensor accuracy, easing their use and/or facilitating their inclusion in monitoring or control systems. This trend started by just replacing mechanical or pneumatic based secondary devices by transducers allowing the translation of the physical quantity being measured into an analog or digital signal ready to be acquired by an electronic processor or a computer. In some cases this trend evolved lately towards the inclusion of some modifications in the original sensor design in order to obtain further advantages out of the electromechanical merger. This is the case of the work presented here, where it is shown that introducing some modifications on the standard design of a laminar flowmeter can lead to the enhancement of its characteristics after adding a simple auxiliary electronic board.

Laminar flowmeters are a well-known kind of differential pressure-based flow measurement devices mainly used for measuring low flow rates of gases and liquids [^{2}/4 is the pipe cross-sectional area. Under normal operating conditions and for values of the Reynolds number below 2,300, the flow remains laminar. The pressure drop (Δ

This equation establishes that for a laminar flow there is a linear relationship between flow rate and developed pressure drop; this linearity represents an advantageous characteristic of laminar flowmeters. A major drawback of this type of flowmeter, however, is its dependence on fluid viscosity, which in turn is mostly dependent on fluid temperature. Thus, any laminar flowmeter requires some form of temperature compensation to obtain precise measurements.

A laminar flow element can be constructed by various methods, but most commonly it consist of a set of capillary ducts whose length significantly exceed their inner diameter, and that are arranged in parallel. In this way the main flow is split among all of them obtaining, as a result, a reduced Reynolds number. To help in this reduction, very often the sum of the cross-sectional area of all capillaries is larger than the main pipe cross-sectional area.

There are three additional sources of pressure drop in this type of flowmeter that introduce nonlinearity and error to the capillary loss. These are: Inlet loss, exit loss, and capillary entrance loss. The inlet loss is produced by the effects of flow velocity changes when entering the pipe, as well as inlet edge effects on the flow [

In order to reduce the effects of this non-linear source of error, the Reynolds number inside the capillaries is commonly kept below 1,200 and the

It is customary to perform this analysis using non-dimensional variables. All lengths are normalized by the diameter ^{2}/2, obtaining a dimensionless pressure in the form: ^{2}/2). Using these dimensionless variables, from expressions 1 and 2 the following expression valid for fully developed Poiseuille flow is obtained:

A measure of the total pressure drop from the pipe inlet will include a term accounting for the fully developed flow plus the excess pressure drop

The term _{∞} in the developed region.

According to White [

There are two main ways to reduce this length; the first one is by reducing the pipe diameter and the second one by allowing a small level of non-linearity in the device response. The limitations to the first approach are related to the need of drilling holes on the pipe walls, while in the second case the required accuracy will be the constraint.

Shah [

Any laminar flowmeter, as in

In the case of measuring the pressure drop between two pressure taps on the pipe wall; one at position _{1} downstream the entrance and the other at the pipe end, the response curve of

In the present case the coordinate _{1} has been placed such that _{1}/_{1}, in addition its maximum value is now one sixth of the original one. On the other hand, the linear regression error has decreased by one order of magnitude.

One way of avoiding these non-linearity effects could be achieved by placing the first pressure tap at a distance larger than the entrance length, corresponding to the maximum Reynolds number allowed. As we can see now, this approach will result in very long devices. In fact, using the limit value of Re = 2,000, and as according to White [_{L} defined as the point where _{L}/(Re_{L}/

As shown in the previous section, entry effects are always present in the performance of any laminar flowmeter having the classical disposition shown in

Taking these considerations into account, the modifications on the flowmeter design proposed here consider the non-linearity errors and try to reduce them to a required level. This will be achieved by placing several pressure taps along the flow element pipes, which require a pipe diameter of few millimetres. The

Following the proposed approach, and in order to produce correct measurements, three pressure taps are placed along one or several of the laminar flow element tubes. The tubes are arranged as if they are a part of a “shell and tube” heat exchanger but having two extra intermediate walls, resulting in three chambers along the shell, as sketched in

The idea is to use the pressure difference between chambers 1 and 3 to measure low flow rates and switch to 2 and 3 at higher flow rates, at a point where entrance effects begin producing significant non-linearities at position 1. The distance from the pressure taps in chamber 3 to the pipe end can be just of about two pipe diameters, as this distance is enough to avoid exit losses. The distance between pressure taps in chambers 2 and 3 must be such as to produce a pressure drop equal to the full span of the pressure transducer used when the maximum allowed Reynolds number is achieved. The distance between the inlet and the pressure taps at position 2 must be such that at the maximum allowed value for the Reynolds number, the entry effects at this position result in an error level below the value imposed in the design. The pressure taps in chamber 1 should be placed in a position such that when the pressure drop between 1 and 3 reaches the full range of the transducer, the entry effects at position 1 introduce an error below the maximum allowed.

Another advantage of the proposed setup is that the full range of the pressure transducers is used to measure half of the range of the flow rate, thus the errors referred to the transducer full scale are now divided by two when referring to the flowmeter’s full scale. Nevertheless, the only way of making a functional instrument based on this concept will be to introduce an electronic unit able to execute adequate switching between both halves of the full range and give a correct single output. This aspect of the device is clarified in the next section.

Notice that in the example presented above, the maximum allowed value for the Reynolds number has been chosen. Reducing this value or reducing the diameter of the flow element pipes will produce a proportional reduction on the maximum non-linear error, in the same way that they do for conventional laminar flowmeters. Both reductions will have an effect of decreasing the maximum flow rate per pipe, thus the number of pipes needed for a given total flow rate should be increased.

Prior to defining the electronic unit and its functions, it is necessary to decide the disposition of the pressure transducers. One possibility is to permanently connect two pressure transducers; one between pressure connectors 1 and 3 and the second one between 2 and 3. In this case the first pressure transducer should be able to support the overpressure suffered at the high flow rate range, where the other transducer is performing the task. Another possibility is to use a single pressure transducer having its low-pressure port directly connected to pressure connector 3 and an additional electrovalve in charge of switching connectors 1 or 2 to the high-pressure port.

The electric diagram for the device is presented in

The PCU can have some linearization programs implemented. In this case the programs should take into account the non-linearities introduced by the differential pressure transducer in addition to those of the LFE. However, maintaining the LFE non-linearity error below a given value will be enough in most applications. One interesting point here is that the response curve of practically any pressure transducer has a negative second derivative, that is, its slope diminishes as the pressure goes higher, which is the opposite of the LFE. Consequently the combined non linearity errors will try to compensate each other and taking the higher of both values as the non-linearity error of the assembly would be a conservative approach.

After studying the performance of actual laminar flowmeters, it is found that the developing entry region introduces non-linearity effects even in cases when the

A new arrangement of this type of flowmeters, with flow elements that include pressure taps along their pipes, has been introduced. The proposed setup splits the full flow rate range into two parts, reducing in this way the linearization error, with the additional advantage of producing a reduced pressure drop and a better use of the pressure sensor range.

This proposal accounts for the non-linearity error of the device and permits limiting its level at design, in order to pair it with the accuracy of the flowmeter and the rest of its components. A fully linear device is also proposed at the cost of increasing its length.

A general description of the device, as well as a discussion on the setup, its associated transducers, pneumatic connections and electric scheme is presented. The fields of application of the proposed flowmeter remain the same as for conventional laminar flowmeters,

This work has been partially supported by the Galician Government under projects 07DPI166E and 08TIC035E.

A typical laminar flowmeter.

Pressure drop development along a pipe.

Flowmeter response and two possible linearization paths.

Linearization errors corresponding to

Nonlinearity errors achieved by the linear approximations.

Flowmeter response after introducing a pressure tap (solid line) and its linear Poiseuille model (dashed line).

Linearization errors corresponding to

Proposed flowmeter arrangement.

Response of the proposed flowmeter.

Linearization errors of the proposed flowmeter.

Pressure transducers and pneumatic connections disposition.

The flowmeter’s electrical block diagram.