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Miniaturized thermal flow sensors have opened the doors for a large variety of new applications due to their small size, high sensitivity and low power consumption. Theoretically, very small detection limits of air velocity of some micrometers per second are achievable. However, the superimposed free convection is the main obstacle which prevents reaching these expected limits. Furthermore, experimental investigations are an additional challenge since it is difficult to generate very low flows. In this paper, we introduce a physical method, capable of generating very low flow values in the mixed convection region. Additionally, we present the sensor characteristic curves at the zero flow case and in the mixed convection region. Results show that the estimated minimum detectable air velocity by the presented method is 0.8 mm/s. The equivalent air velocity to the noise level of the sensor at the zero flow case is about 0.13 mm/s.

The minimum detectable flow (MDF) becomes a crucial feature when flow sensors are used in very low-flow applications, such as gas detection and accurate supply of gases in some medical applications [

MDF is basically influenced by natural (free) convection and thermal noise in the case of thermal flow sensors. Natural convection is a complex mechanism in which the fluid motion is generated by density differences in the fluid due to temperature gradients [

Flow is characterized mainly by Reynolds number (Re). Re is a dimensionless number used in fluid mechanics to study the flow; it represents the ratio of inertial forces to viscous forces in the fluid. Reynolds number characterizes different flow regimes,

Free convection is characterized by Grashof number (Gr) which expresses the ratio between buoyancy forces due to spatial variation in fluid density (caused by temperature differences) to viscous forces acting on the fluid. It is given as:
_{s} and T_{∞} are temperatures of the surface and the surrounding fluid, respectively; L is the characteristic length and υ is the kinematic viscosity of the fluid. Free convection on a surface depends on several parameters such as geometry, orientation, variation of temperature on the surface and thermo-physical properties of the fluid. For a vertical plate position, the plate is aligned with the gravitational vector, and the buoyancy force induces fluid motion in the upward (or downward) direction. However, if the hot plate is horizontal, as in our case, the buoyancy force is normal to the surface and the resulting fluid motion is in the vertical direction. When the temperature difference (T_{s} − T_{∞}) rises, the surrounding air starts to move and the heat losses rise quickly. However, when the convective flow is established, the heat transfer rises slightly with increasing temperature difference [

The ratio Gr/Re^{2} defines the importance of natural convection in respect to a forced convection. This ratio of the buoyancy forces and the inertial forces is expressed as:
^{2} ≪ 1, forced convection is negligible when Gr/Re^{2} ≫ 1, and both are significant when Gr/Re^{2} ≅ 1. In the strict sense, a free convection flow is induced by buoyancy forces, if there is no well-defined forced convection velocity and Gr/Re^{2} = ∞ [

Van Putten ^{2}) equals to [0.3–0.8] for a horizontal hot plate. The method used to generate velocities in the mixed convection region is based on a vertical piston controlled by a computer. It moves back and forth in order to generate the airflow in two opposite directions. A hardware clock in the engine control unit measures the number of rotations of the engine that moving the piston. Velocities achieved by this method ranged from 1 to 25 mm/s, 1 mm/s was clearly detected whereas velocities below 0.5 mm/s could not be generated in a reliable way. Cubacku

The focus of this paper is directed towards the minimum detectable air velocity by calometric thermal flow sensor. Thermal noise and free convection are considered as basic parameters which affect MDF. After a short description of the used sensor, we present a simple experimental method provide very low flow rates. It allows obtaining the characteristic curve in the mixed convection region, up to 20 mm/s. This value is the velocity for the upper limit of the mixed convection region determined by the ratio Gr/Re^{2}. Then, a statistical study for sensor output at zero flow is done. From the characteristic curve and noise level at zero flow, we calculate the minimum detectable velocity by this method and the corresponding velocity to noise level at zero flow.

The investigated thermal flow sensor is based on silicon as substrate material; it consists of a heater and two symmetric thermopiles embedded in silicon nitride membrane as shown in ^{2} with a thickness of 600 nm. The distance between heater and both thermopiles is 20 μm. More information about the fabrication process of the sensor can be found in [^{2}. The sensor is operated by a constant power circuit which provides constant power to the heater during the measurement. Response time of the sensor is related to the velocity and geometry of the membrane. It decreases from about 5 ms in the stagnant flow case to 1.5 ms in the case of 44 m/s as air velocity [

In order to measure the sensor MDF, we built a physical method which generates very small flow rates. The method principle is based on weighing the mass changes of one bottle partially filled with water during its discharge into another bottle. Mass readings were taken in time steps of 2.5 s in order to calculate the mass flow. Water flow between the two bottles occurs by means of a small pipe. This method is shown in

The accuracy of the calculated flow velocity depends basically on the accuracy of the balance. The balance accuracy is 0.1 mg. The mass flow is calculated as successive discrete values. Each one represents the mean flow between two successive weighing operations separated by 2.5 s. Thus we have 600 flow values. The accuracy of velocity values (is calculated by substituting the corresponding values of pipe section area and water density) resulted from using the balance is about 0.03 mm/s. Additionally, the relative errors generated by using the mean velocity value between each two successive weighing operations are less than 5%.

The discharging curve in the previous experiments is exponential as shown in ^{3}, and the section area of the air channel is 3 mm^{2}. The resulting curve of velocity (v) as function of time has similar behavior of the sensor output voltage difference (∆U) as function of time. In order to obtain the direct relationship between the sensor output voltage difference and air velocity, we modeled the both curves by using MATLAB based program. The resultant fitted curves for the air velocity (in mm/s) and the sensor output voltage difference (in mV), are given in the following expressions, respectively:

We can obtain the characteristic curve of the sensor in the mixed convention region by eliminating time between the above two equations, which give:

This equation assumes the linear relationship in the mixed convection region. Sensor sensitivity (S) is defined as the derivative of the output voltage difference with respect to the airflow velocity, as in the following equation [

The sensitivity is then 0.017 V/m/s. The experimental data between ∆U and v are plotted in

The relative error by this method is less than 20% for the range from 0 to 5 mm/s; it decreases significantly afterwards to less than 10% for the range 5 to 20 mm/s. The larger relative errors for small velocities are due to the high significance of the free convection in heat transfer in addition to the instability of the balance at low values which is another reason for these relatively high errors. By extracting the maximum deviation from the fitting line we found that the maximum error is about 0.03 mV. The corresponding error in velocity according to

Noises on the sensor signals are caused by the sensor itself and by the measurement system. At zero flow, new experiments were done to evaluate noise level. The noise of the sensor is mainly caused by thermopiles noises and natural convection. In this case, we can examine the pure natural convection together with the thermal noise as there is no defined forced convection and

Firstly, the thermopile noise is basically the temperature noise and the thermal noise. The temperature noise is caused by temperature fluctuations in the surrounding atmosphere. We assume that this noise has negligible effect on our calculations as all our measurements have been performed at room temperature 20 to 22 °C. Meanwhile the thermal noise or the Johnson noise is an electrical noise source caused by random motion of electrical charges in the material. The Johnson noise is determined by the following equation [_{B} is the Boltzmann's constant; T_{ext} is the external temperature; R_{e} is the electrical serial resistance and ∆f is the frequency bandwidth. With k_{B} = 1.38066 × 10^{−23} J/K; T_{ext} = 323 K; R_{e} = 200 K; ∆f = 1 Hz. The thermal noise of the sensor is then 0.06 μV.

Secondly, the main noise source of the measurement system is that of the Analogue to Digital Convertor (ADC). Since the thermopiles signals are analogue they are converted into digital by ADC with reference voltage of 400 mV and resolution of 16 digits. The root-mean-square quantization noise (N) is obtained from the following equation [

This value represents the theoretical limit for the minimum detectable flow velocity for the studied sensor.

These results show that thermal flow sensors are capable to detect very low air velocities by optimizing the noise sources. Firstly, the thermal noise of the thermopile is very small as it gives a detection limit of 0.9 μm/s for temperature resolution of 0.1 mK. Secondly, the natural convection can be minimized by either reducing the characteristic length such as by using narrow and deep air channels, or by reducing the temperature difference between the heater and the surrounding air. The first solution requires reducing the sensor dimensions whereas the second solution will decrease the sensor sensitivity and the measuring range. Thirdly, the noise arising from the measurement system can be reduced by optimizing the choices of the circuit elements such as ADC with higher resolution. Moreover, the promising results of using microchannels realized by microfluidic structures in providing very accurate measurements for very low flow rates, especially for liquids, motivate us to use such structure for air as flow as well.

We have introduced in this paper a physical test method which is capable of generating very low flow values in the mixed convection region from 0 to 20 mm/s. We found that the characteristic curve is linear in this region and the sensor sensitivity is about 0.017 V/m/s. The estimated minimum detectable velocity obtained by the presented method is 0.8 mm/s. Equivalent velocity to the noise level at zero flow is about 0.13 mm/s.

The authors would like to thank the International Graduate School for Dynamics in Logistics (IGS) at the University of Bremen for supporting this work.

The authors declare no conflict of interest.

^{2}≫1 by silicon anemometry

Representation of natural, mixed, and forced convections around thermal flow sensor.

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Setup for generating very small flow rates. The flow is identified by measuring the water flow rate between two closed bottles.

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Sensor output voltage difference (∆U) as function of air velocity (v) in the mixed convection region.

Sensor's output voltage difference

Representation of the detection limit of the flow sensor.