Low Detectable Range Thermopile Flow Sensors

 Xiang Zheng Tu

Thermal flow sensors are inevitably influenced by natural convection. An operating thermal flow sensor is a hot source surrounding by air. The air receives heat from the sensor, becomes less dense and rises. The cooler air then moves to replace it. This cooler air is then heated and the process continues, forming the convection current around the sensor. The temperature of the sensor will changes due to a part heat of the sensor is carried away by the convection current. Therefore the temperature change will add to the signal of sensor as a part of offset of the sensor.

But for thermopile flow sensors the natural convection can be reduced as low as being ignore. The installation of the thermopile flow sensors in a tube is commonly required to have their hot surface facing down to the Earth’s surface. In this case the heat transferred from the sensor chip by natural convection is the lowest. This is not surprising, since the hot air is “trapped” under the sensor chip and can not move away from the sensor chip easily. As a result, the cooler air in the vicinity of the sensor chip will have difficulty reaching the sensor chip, which results in a reduced rate of heat transfer.

Even the hot surface of the sensor intersects the Earth’s surface at a specific angle, the formed natural convection current is small and the influence still can be ignore. The reason can be explained as follows. Natural convection is characterized by Grashof number G_r which expresses the ratio between buoyancy forces due to spatial variation in air density to viscous forces acting on air. It is given as:

Gr = gβ(T_sensor - T_air )L³ /  ע² ,     (1)

where g is the acceleration due to gravity; β is the volumetric thermal expansion coefficient; T_s and T_air are temperature of the heater of the sensor and the surrounding air, respectively; L is the characteristic length and the ע is the kinematic viscosity of the surrounding air. With g = 9.8 m/s², β = 3.25x10^-3 /k,   ע = 1.65 x 10^-5, T_s - T_air = 21.9 ^oC, and assuming the characteristic dimension L = 1x10^-3 m, we have

Gr = gβ(T_s - T_air )L³ /  ע² = 2.56     (2)

As well know, the ratio Gr/Re₂ defines the importance of natural convection in respect to a forced convection. The R_e is Rayleigh Number which represents forced convection and can be expressed as R_e = vL/ ע. The thermopile flow sensors provided by POSIFA Microsystems have been widely used for air mass flow meters. The characteristic length of the house tube of the air mass flow meter is 6mm. If assuming the velocity of air flow is 0.1mm/s, the Re is calculated to be 3.64 and the ratio Gr/Re₂ to be 0.19. This is lower than the lower bound of Gr/Re2 =0.3, which means that compare with the forced convection the contribution of the natural convection to the heat transfer can be negligible. (E. M. Sparrow, R. Eichhorn, and J. L. Gregg, “Combined forced and free convection in a boundary layer flow,” Phys. Fluids, vol. 2, no. 3, pp. 319–328, 1959.)

In addition to natural convection, the noises of the thermopile flow sensors are also limiting the detectable flow velocity. The noises of the thermopile flow sensors are basically the temperature noise and the thermal noise. The temperature noise is caused by temperature fluctuations in the surrounding atmosphere. This noise can be cancelled because the thermopile flow sensors are responsive to the temperature difference between a hot place and a cold place instead of a temperature. The thermal noise or the Johnson noise is an electrical noise source caused by random motion of electrical charges in the material. It is determined by the following equation:

V_noise = (4k_bT_eR∆f)^1/2       (3)

where k_B is the Boltzmann’s constant; T_e is the absolute temperature in kelvin ; R is the electrical serial resistance and ∆f is the frequency bandwidth.

Thermal noise calculation can be carried out online. The following results are given by http://www.sengpielaudio.com/calculator-noise.htm.

  

In the calculation T = 45.9 ⁰c representing the operating temperature of the thermopile flow sensor, R = 150 kΩ representing the serial resistance of a thermopile of the sensor. The bandwidth is calculated using equation ∆f = f_cutoff – 20 Hz, where f_cutoff is determined by measuring the response time of the sensor, as shown in the following figure.

It can be obtained from Fig.1 that the response time of the sensor is 0.74ms corresponding cutoff frequency 1351 Hz. With these parameters the thermal noise of the sensor is 1.88µV.

In conclusion, the thermopile flow sensors provided by POSIFA Microsystems can detect as low as 0.1mm/s velocity for air mass flow meter applications. This is resulted by the inherence low Grashof number of the sensors, which is down to 2.56. The temperature noise of the sensor can be canceled since the sensor is responsive to the temperature difference instead of temperature. The thermal noise or the Johnson noise is calculated to be 1.88µV, which is very low and is suitable for medical applications.  Furthermore the cutoff frequency of the sensors is up to 1331Hz, which is required by several medical instruments such as spirometers.