Optimized Design of Water Laminar Flow Sensor Tubes

Xiang Zheng Tu

POSIFA’s water laminar flow sensors use the thermal properties of water to measure the flow of water flowing in a tube. The water laminar flow sensors are structured to comprise of a thermal insulated base created in a silicon substrate, a long stripe polysilicon resistor as a heater disposed at the central region of the base, and two thermopiles with their hot junctions along the two sides of the heater and cold junctions disposed on the base surrounding region of the silicon substrate. A constant amount of heat is applied to the heater of the sensor. Some of this heat is lost to the flowing water. As flow rate increases, more heat is lost. The amount of heat lost is sensed by the thermopiles of the sensor. The output signal of the thermopiles is used to determine water flow rate based on the convection heat of the water flowing over the sensor.

A sensor chip is assumed to be mounted on the wall of a tube in such way that the water flowing perpendicularly to the long stripe polysilicon resistor and one thermopile is located up stream and the other thermopile is located down stream. The water flow is required to be completely developed which means the laminar flow can be considered as the relative motion of a set of concentric cylinders of fluid, the outside one fixed at the surface of the sensor chip and the others moving at increasing speeds as the centre of the tube is approached. The resulted forced convection heat transfer can be described by Newton’s Law of Cooling as

 Ǭ = hA(T_s – T_f )                               (1)

The rate of heat Ǭ transferred to the surrounding water is propotional to the sensor chip exposed area A, and the difference between the polysilicon resistor surface temperature T_s and the water free stream temperature T_w. The constant of proportionality h is termed the convection heat transfer coefficient, which is given by:

h = N_u κ_w / l                                      (2)

    

Where l is the characteristic length, Nu is the Nusselt number and κ_w is the thermal conductivity coefficient of water. l is the effective diameter of the tube which is defined as:

l = 4A/P                                             (3)

with A the flow cross sectional area, and P the perimeter, respectively. The Nusselt number has been calculated by Gianchandan et al as

N_u = 0.664 Re^1/2 Pr^1/3                          (4)

Where Re is Reynolds number and Pr is Prandtl number. Since Re= lv/υ and Pr = υ/α, the equation (4) can be expressed as

N_u = 0.664(lv/υ)^1/2(υ/α)^1/3                    (5)

Where v is the average velocity of water flow, υ is the kinematic viscosity of water and α is the thermal diffusivity of water.

 Replacing equations (2) and (5) into equation (1) results in the expression as

Ǭ = 0.664(lv/υ)^1/2(υ/α)^1/3A(T_s–T_f )/l     (6)                 

In this expression Ǭ can be measured by the thermopiles of the sensor as output voltage, v is required to be determined by the measured output voltage and all other parameters can be obtained from available physical and chemical data base.

A laminar water flow needs some length of tube to fully develop the velocity profile after passing through components like bends, valves, pumps, and turbines or similar. The entrance length can be expressed with the dimensionless Entrance Length Number as

El = le / d                                              (7)

Where

El = Entrance Length number

L_e = length to fully developed velocity profile (m, ft)

d = tube or duct diameter (m, ft)

The Entrance length number correlates with the Reynolds Number  and for laminar flow the relation can be expressed as:

El_laminar = 0.06 Re                                  (8)

The water flow is laminar when R_e < 2300. Reynolds Number and Entrance Length for one liter of water at approximately 20⁰ C flowing through tubes of different dimensions:

Note that the water viscosity varies with temperature. The kinematic viscosity of water at 20⁰C used to calculate the table above is 1.004 x10⁶m²/s.

At 0⁰C the kinematic viscosity is 1.787 x 10^-6m²/s the Reynolds values in the table above must be multiplicated with 1.004/1.787 = 0.56. At 100⁰C the kinematic viscosity is 0.29x10^-6m₂/s the values in the table above must be multiplicated with 1.004/0.29 =3.46.

As shown in the table above, the long entrance lengths are not accepted for the most applications of the POSIFA’s water laminar flow sensors, such as coffee makers and drinking water dispensers. In order to short the entrance lengths a laminar flow device (element) is required to place in the entrance region of the tube. The laminar flow device creates the flow of water to be laminar or restricts the water to be flow as laminar flow before flowing into the laminar flow developed region of the tube. The device typically utilizes a material which has randomly-arranged Capillaries for dividing the velocity components of the incoming fluid stream into smaller components. Some of the velocity components cancel each other thereby presenting a more uniform velocity profile, reducing the turbulence of the fluid, and allowing laminar flow at higher flow rates than would otherwise be possible.

The developed laminar flow region of the tube comprises a plurality of narrow passageways along the flow path. The flow sensor is incorporated one of the narrow passageways. It is proffered that the narrow passageways have two plane parallel surfaces where the width is much greater than the space between the plates than the characteristic dimension is equal to the distance between the plates. 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 tube cross-sectional area.

To facilitate measurement and control of larger flow rates, a bypass version of the water flow sensor was developed. The bypass flow sensor is comprised of a capillary sensor tube connected to the main flow tube as a shunt line. The sensor tube usually has inside diameter less than 3 mm so that the sensor is operated in the laminar flow region over its full operating range. The flow in the main tube is inferred by measuring the flow in a small bypass tube using the flow sensor. The main flow in the large tube can be estimated from the previously determined ratio of main flow to bypass flow. The diameter of the main tube can be as larger as 20 or 30 mm. In these cases the water flow will transition from laminar to turbulent flow. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction but the average velocity is still in flow direction.