^{1}

^{2}

^{*}

^{1}

^{1}

^{2}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper introduces a simple theoretical model for the response time of thermal flow sensors. Response time is defined here as the time needed by the sensor output signal to reach 63.2% of amplitude due to a change of fluid flow. This model uses the finite-difference method to solve the heat transfer equations, taking into consideration the transient conduction and convection between the sensor membrane and the surrounding fluid. Program results agree with experimental measurements and explain the response time dependence on the velocity and the sensor geometry. Values of the response time vary from about 5 ms in the case of stagnant flow to 1.5 ms for a flow velocity of 44 m/s.

Micro-machined thermal flow sensors are used for many applications, especially ones that need a fast response time such as medical and automotive applications. For example, in the medical field, the respiration disturbances related to some cardiovascular diseases are a supplementary risk for the cardiovascular system. They require urgent diagnostic assessment and consistent therapeutic measures. Thermal flow sensors satisfy such specific requirements of high dynamic flow range and fast response time in controlling the patient's respiration [

Although there is no established method for the response time measurements due to the difficulties of realizing generated defined fluidic steps [

This modeling program uses the numerical analysis approach in order to solve heat transfer equations within the sensor membrane and the surrounding fluid. A cross section in the membrane and the air flow channel represents a two dimensional body with uniform thickness in the z-direction. We assume that there is no temperature gradient in that direction. By choosing appropriate spacing, the body is divided into a network of nodes. Each node is characterized by a single nodal point at its center as shown in

The formula for the evaluation of temperature in each node based on the explicit Finite-Difference conduction equation [_{mn}_{n}_{k},_{mn}_{c,mn}_{n}

The wall heat transfer coefficient is given by:
^{2}·K for the stagnant air case, but when we have flow the previous conditions are no longer applicable. The Nusselt number is calculated in this case according the following equation [

Firstly, a one-dimensional model is established to obtain the thermal response time of the flow sensor. A cross section of the sensor membrane is divided into 100 nodes as shown in

Two sensor configurations are considered regarding the membrane geometry: TS20 and TS50. They have the same membrane area 1 × 1 mm^{2}, but differ in the distance d between the heater and the hot contact of the thermopiles as shown in

The response time of the configuration sensor TS20 is 5 ms in the stagnant air case. However, in the case when a flow exists, the time constant decreases to

Model results also explain the dependence of the response time on the distance between the thermopiles and the heater. The larger the distance, the higher the response time because heat travels further, as shown in

Secondly, another dimension (2D) is added to the previous model, mainly for the steady state case, to be able to model the thermopiles output as a function of flow velocities. The new dimension consists of virtual sublayers of the air flowing over the membrane through the air channel as depicted in

For each time step, the conduction equations are applied to all nodes, and then each virtual air node moves one step in the direction of the flow to replace the next node. This means we can assume for the sake of simplicity that all layers move at the same speed, which is not true in reality as the velocity should follow a parabolic pattern. Boundary conditions were taken by considering that heat transfer occurs only in the membrane and the air channel; other surroundings keep the same ambient temperature.

This 2D model is used to provide sensor output signals in the steady state case. In this case the velocity was set to values in the range from 0 to 70 m/s. The up- and downstream thermopile signals are extracted, their difference is calculated and the three of them are plotted as a function of the velocity. At zero flow, the two thermopiles give the same signals. When a flow exists, the difference between the two signals is detected, which increases with the increment of the air velocity, as expected [

A simple theoretical model for response time of thermal flow sensors was presented. Model results agree with experimental measurements and explain the dependence of the thermal response time on sensor geometry and air velocity.

Nodal representation of two dimensions body (1, 2, 3 and 4 are the four adjacent nodes to node

Schematic representation of the one dimensional model where a cross section in the membrane is divided into 100 nodes.

Temporal changes of thermal flow sensor signal (thermopile) for different values of air velocities according to the theoretical model. For this 1D model, the direction of the flow is not taken into account, and therefore the both thermopiles have exactly the same temperature.

Comparison between experimental and model results for flow sensor (TS20) response time. Experimental results from measurements performed by Sosna

Comparison in response time between two sensor configurations TS20 and TS50. Dots are model results for some discrete values of velocity and lines are the splines interpolation.

Comparison between model and experimental sensor output signals: up- and downstream thermopiles TP1 and TP2 and their differences as functions of air velocity. Solid lines are for model results and dotted lines are for experimental results.

Properties of constituent's elements.

_{p} |
^{3}] | ||
---|---|---|---|

Titanium | 20 | 530 | 4,500 |

Tungsten | 177 | 130 | 19,300 |

Poly-silicon | 34 | 710 | 2,300 |

Silicon Nitride | 4 | 750 | 3,100 |

Air | 0.03 | 1,006 | 1.18 |