## 2 Theoretical Fundaments: Laminar Flowmeters  
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.  
A laminar flowmeter is composed of a laminar flow element and a differential pressure gauge or transducer. The laminar flow element ensures that the flow passing through the flowmeter is in laminar condition. In this condition the viscous forces, generated by internal friction of the fluid, overcome the inertial forces and therefore the dominant mechanism for resistance to fluid motion is friction against the surrounding walls [8].  
The pressure drop \((\Delta p)\) caused by fluid friction between two points separated by a distance \(\Delta x\) along a pipe in a laminar flow regime is quantifiable, and can be expressed by the Hagen-Poiseuille equation (Equation (1)) [9].  
\[\Delta p=\frac{128\mu Q\Delta x}{\pi D^{4}}, \tag{1}\]  
where \(D\) is the pipe internal diameter, \(\mu\) the fluid's viscosity and \(Q\) the volumetric flow rate.  
Equation (1) establishes that for a laminar flow there is a linear relationship between the flow rate and developed pressure drop. 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 [4]. In Figure 1 there is a scheme of a longitudinal and a transversal cut of a laminar flowmeter.  
The assumptions of the Hagen-Poiseuille equation (Equation (1)) [10] are:  
1. The flow has negligible kinetic energy,
2. The flow is laminar and steady,
3. The capillary is straight and has a uniform, circular cross section,
4. The fluid is incompressible and its density is constant,
5. The fluid is Newtonian,
6. The temperature of the fluid is constant and viscous heating is negligible,
7. There is no slip at the wall of the capillary.  
This is a simplistic model.  
In real situations, it is natural that some of these conditions cannot be met and therefore errors arise in the flow measurement. The practical models used by experimentalists must correct errors arising from a breach of the conditions listed. According to the practical model adopted in this work, each laminar flowmeter has its calibration curve that is generated using a quadratic Equation (2) [11]. This equation is only valid for temperature and pressure at standard conditions (at sea level: \(T=15\,^{\circ}\mathrm{C}\) and \(p=1013,25\) mbar).  
\[Q=\left[B\times\Delta p+C\times(\Delta p)^{2}\right], \tag{2}\]  
where \(B\) and \(C\) are calibration constants.  
The actual volumetric flowrate, \(Q_{a}\), can be obtained by measuring the differential pressure (\(\Delta p\)) across the laminar flow element and the temperature (\(T\)) of the inlet fluid. This flow rate can be obtained by the Equation (3).  
\[Q_{a}=\left[B\times\Delta p+C\times(\Delta p)^{2}\right]\ast\frac{\mu_{std}}{ \mu_{a}}, \tag{3}\]  
where \(\mu_{a}\) is the actual viscosity of the fluid and \(\mu_{std}\) the viscosity of the fluid at standard temperature.  
The word \"standard\" when associated with flow rate, means that the flow rate has been normalized to an assigned standard pressure and temperature [7]. If the standard volumetric flowrate is desired it can be obtained by multiplying the actual volumetric flowrate by the pressure and temperature ratios:  
_standard temperature \((T_{std})/\)actual temperature \((T_{a})\),_  
and _actual pressure \((p_{a})/\)standard pressure \((p_{std})\)_.  
This way, the standard volumetric flowrate can be obtained by Equation (4).  
\[Qstd=\left[B\times\Delta p+C\times(\Delta p)^{2}\right]\times\frac{\mu_{std}}{ \mu_{a}}\times\frac{T_{std}}{T_{a}}\times\frac{p_{a}}{p_{std}}. \tag{4}\]