Design and Analysis of a Thermal Flowmeter for Microfluidic Applications: A Study on Sensitivity at Low Flow Rates

Abstract

To address the challenge of precise flow rate measurement in microchannels, this research details the conceptualization and comprehensive evaluation of a thermal flowmeter which works on the principle of calorimetry for measuring small flow rates between 0.1 and 180 mL/h. The thermal flowmeter is composed of a silicone pipe, a heater, three platinum thermal sensors (T₁, T₂, T₃), and water as the working fluid. The flowmeter is strategically placed to monitor the complex thermodynamics between upstream and downstream flows. The analysis revealed a notable decay in the slope of the temperature differences beyond a flow rate of 40 mL/h, indicating the exceptional sensitivity of the device at lower flow rates and making it an ideal choice for medical applications. Parametric analysis was also carried out to place the sensors at optimized locations for better sensitivity.

1. Introduction

The measurement of physical variables has been one of the primary concerns in places where efficiency and accuracy are of utmost importance. Over the years, several physical variables, such as temperature, flow, pressure, speed, humidity, and pH, have been measured with different principles and in various dimensions. Owing to advances in science and technology, the miniaturization of devices has been achieved to meet the demands of the production industry and medical equipment. The nanometer technologies of the semiconductor industry have changed the perspective on miniaturization. These advances have led to the development of meters to measure physical variables at dimensions less than a millimeter. One such metering concept is called microfluidics, which measures the flow rate. Microfluidics is the branch that deals with the flow at a micro level in devices with length scales less than a millimeter [1,2,3]. The few listed applications of microfluidics are shown in Figure 1.

Microflow systems play crucial roles in a wide array of fields, including biotechnology, healthcare, sensor technology, and process control, where the precise mixing, flow regulation, or separation of different fluids and gasses is paramount [4]. In these contexts, maintaining precise delivery volumes or stable flow rates is critical for optimal performance. Additionally, modern medication emphasizes dose accuracy, economic efficiency, automated processing, and contamination-free techniques to achieve maximum patient comfort. Infusion therapy, encompassing treatments such as pain management, antibiotic delivery, hydration, transfusions, and nutritional support, is one of the fastest-growing methods for homecare medication [5,6,7,8].

Administering drugs intravenously by infusion poses significant challenges in critical scenarios since achieving consistent infusion flow rates is crucial. Inconsistent flow rates lead to dosing inaccuracies, resulting in harmful clinical outcomes. These clinical effects can be classified into two categories: underdosing, which leads to insufficient therapeutic effectiveness, or overdosing, which causes increased toxicity [9].

The sensing of flow at the micro level is performed with the help of various meters, such as a Coriolis flowmeter, a thermal flowmeter, and a pressure-based meter [10]. Among them, thermal flowmeters are widely used in variety of applications [11,12,13]. Thermal flow sensors operate by detecting changes in heat transfer caused by fluid flow, which are then converted into varying electrical signals, reflecting the sensor’s response to flow variations. To ensure maximum accuracy, it is necessary that these sensors are thermally isolated, allowing heat transfer to occur entirely because of flows such as those in cantilevers and pressure-difference-based sensors. Minimizing other heat transfer methods, such as through the substrate or electrical leads, is crucial, as they result in thermal losses that impair sensor performance. For a thermal flow sensor to function correctly, the fluid temperature must remain stable; any deviation necessitates the use of temperature compensation measures [14].

Several types of flowmeters have been developed via microelectromechanical system (MEMS) technology, capitalizing on thermal, Coriolis, ultrasonic, and mechanical flow effects [15,16,17,18,19]. A single heater enables a simple thermal flowmeter to function by regulating the heater temperature with consistent heating power or modulating the power accordingly to keep the heater temperature constant. However, one of the main challenges with thermal flowmeters is the slow response time (>1 min) [20]. Figure 2 illustrates the principle of calorimetric flow in a thermal flowmeter.

Calorimetric flow sensors, a type of thermal flow sensor, are highly sensitive in low-velocity ranges but tend to saturate at relatively high velocities due to the increasing prominence of the thermal boundary layer thickness [21]. Consequently, thermal flow sensors are commonly used in microfluidic devices that control low-velocity flow regimes and low Reynolds numbers, especially in biochemical applications. However, there has been an emergence of microfluidic systems designed to handle higher fluid pressures and flow rates, such as micro-steam turbines operating at up to 4.5 bars and 100 mg/s (with Reynolds numbers ranging from 100 to 2000) [22,23]. The present study aims to analyze the behavior of a calorimetric flow sensor at the micro level and to assess its sensitivity for use in medical applications.

2. Literature Survey

Miller et al., in 1982, were the first to create a liquid flow sensor for microfluidics, which was based on the principles of time-of-flight (TOF) technology [24]. During the 1990s, microfluidics advanced rapidly, giving rise to a broad array of techniques for determining flow rates. While PDMS-based microdevices have numerous benefits and have facilitated new applications, other methods have yet to move beyond the proof-of-concept phase. The different technologies for flow sensing are generally divided into two types on the basis of their measurement principles: active sensors and passive sensors [10]. Active sensors, which are designed to introduce energy into the liquid to detect disturbances, are divided into two main types: thermal and Coriolis flowmeters. Thermal flowmeters assess flow rates by applying heat to the liquid and measuring the transfer of heat. Within Coriolis sensors, the liquid’s flow path is excited, producing a force known as the Coriolis force. This force leads to an additional motion of the path, and the magnitude of this motion is measured to determine the flow rate. Overall, these techniques make up more than two thirds of the flow sensor technologies employed in microfluidic systems [25].

Passive sensors, which do not introduce energy to modify the fluid, cause the liquid to lose energy within the sensor body itself. These sensors are categorized on the basis of the transducing principle they employ. The first approach involves measuring the flow rate by detecting the sensor body’s movement or deformation resulting from fluid–structure interactions. This method capitalizes on the intricacies of fluid–structure interaction (FSI), where fluidic forces induce subtle mechanical deformations in the sensor’s structure, translating dynamic fluid properties into measurable mechanical responses with high precision. The second, more traditional method is the gravimetric technique, which tracks the mass of fluid over time. A more recent method for measuring the flow rate focuses on tracking the advancement of the meniscus front in a capillary at the exit of the microfluidic circuit, similar to the function of a macroscopic bubble flowmeter.

Generally, thermal flow sensors are active sensors consisting of two main components: a heater and a sensing element that detects variations in heat transfer between the heater and the flowing medium. The sensitivity of the sensor increases proportionally with the amount of heat delivered to the working fluid [10]. Both hot-wire anemometry (HWA) and hot-film anemometry (HFA) measure heat transfer from a heated surface to a moving fluid. In these systems, the heating element also functions as a sensor for heat loss. In the HWA technique, a resistive wire is positioned within the fluid stream, whereas the HFA method uses a thin resistive film sensor placed near the flow [11]. There are two operational modes for anemometer sensors: constant temperature and constant current. In constant-temperature mode, feedback circuits maintain a stable sensor temperature, necessitating increased power consumption as the flow rate increases. Conversely, in constant current mode, the heater temperature decreases as the flow rate increases, and this temperature variation is detected through resistance or voltage measurements [16]. Typically, HWA sensors are positioned away from the substrate, which allows for better heating uniformity and greater sensitivity, although they tend to be more delicate than HFA sensors [26].

The sensors, operating on calorimetric principles, measure the temperature difference induced by the fluid’s passage under a steady current supplied to the heater [27]. These sensors are more energy efficient than HWA sensors and can detect the flow direction with two sensors placed on each side of the heater. Despite their utility, calorimetric sensors are constrained by problems of nonlinearity and limited sensitivity. The readings become saturated after a certain flow rate, a phenomenon believed to stem from the limited heat exchange between the liquid and the surfaces of both the sensor and the heater filament [28,29].

3. Materials and Methods

The thermal micro-flowmeter designed in this study was modeled via Fusion 360 software, whereas the fluid dynamics simulations were conducted via Ansys Fluent 2023 R1. The design reference is taken from [30], where they made use of a calorimetric-principle-based thermal flowmeter for drug infusion system. The simulations proposed by them are configured in COMSOL Multiphysics 5.4 to analyze the workings of the thermal micro-flowmeter. In present study, the designed flowmeter consists of a 50 mm long silicone pipe, with an external diameter measuring 3 mm and an internal diameter of 2 mm, incorporating one microheater and three thermal sensors positioned appropriately to measure temperature variations. The primary materials used in the simulation include water as the working fluid, silicone for the pipe, copper alloy for the electrical board (PCB), and platinum for the thermal sensors and heating element. The computational fluid dynamics (CFD) simulations were performed via a pressure-based solver with steady time and absolute velocity formation settings under laminar flow conditions appropriate for low Reynolds numbers. The inlet flow rates varied between 0.1 mL/h and 180 mL/h, with the initial temperature set to 300 K to mimic room temperature conditions. The meshing process, which is critical for simulation accuracy, involves an element size of 0.0001 m. The results of the present study are verified by the simulation results of [30] along with a parametric study by changing the position of the sensors and heaters to find the optimum location for better sensitivity.

4. Formulation

The thermal microflow sensor (TµFS) employs three sensors located before and after a heating element to detect temperature changes. The observed temperature difference (ΔT) between these sensors is then used to calculate the flow rate.

In a quasistatic situation, the heat balance equation governing the incoming and outgoing heat leads to a differential equation that connects temperature (T) with the fluid’s flow direction (x). This relationship is articulated in Equation (1).

$$\mathrm{D}{\displaystyle \frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{x}}^2}}-\mathrm{F}{\displaystyle \frac{\partial \mathrm{T}}{\partial \mathrm{x}}}-\mathrm{g}\mathrm{D}=0$$

(1)

where D represents the fluid’s thermal diffusivity and where F indicates the velocity of the fluid. The parameter l_z refers to the length of the flow channel, and g is defined as l_z⁻². The calculation of heat flux (Q) involves the method outlined in Equation (2):

$$\mathrm{Q}=-{\displaystyle \frac{\mathrm{K}\mathrm{d}\mathrm{T}\left(\mathrm{x}\right)}{\mathrm{d}\mathrm{x}}}$$

(2)

where K denotes the thermal conductivity, dT(x) is the temperature difference within the microheater, and dx is the length of the microheater through which the heat flux (Q) is produced. At each measurement position i, convection heat transfer by the water flow can be simply depicted as in Equation (3), where ${\dot{\mathrm{q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v},\mathrm{i}\to \mathrm{i}+1}$ denotes the rate of convective heat transfer between consecutive points i and i + 1.

$${\dot{\mathrm{q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v},\mathrm{i}\to \mathrm{i}+1}=\dot{\mathrm{m}} {\mathrm{C}}_{\mathrm{p}} ({\mathrm{T}}_{\mathrm{i}+1}-{\mathrm{T}}_{\mathrm{i}})$$

(3)

Here, $\dot{\mathrm{m}}$ refers to the mass flow rate, whereas C_p denotes the fluid’s specific heat capacity. On the basis of the calorimetric principle, F is strongly associated with the temperature variations between configurations T₁–T₂ and T₁–T₃, T₂–T₃.

5. Methodology

Figure 3a,b represent the thermal micro-flowmeter. The thermal micro-flowmeter along the silicone pipe measures 50 mm in length. The outer diameter of the pipe is set to 3 mm, and the inner diameter of the pipe is set to 2 mm. The TµFS is composed of one microheater and three thermal sensors: the reference thermal sensor (T₁), the heater thermal sensor (T₂), and the microheating element and outlet thermal sensor (T₃). The microheater, along with the thermal sensors, is modeled with dimensions of 1.65 mm (L) × 0.85 mm (T) × 0.85 mm (W).

A convective heat transfer coefficient of 5 W/m²K is added during the simulation to the walls of the microheater, heater thermal sensor, outlet thermal sensor, and PCB. The wall of reference thermal sensor T₁ has been given an adiabatic condition.

CFD simulation was performed on the model. The type was set to pressure-based, with a steady time and absolute velocity. Named selections were set for the fluid, inlet, outlet, wall, microheater, and thermal sensors. Figure 4a,b show the mesh applied to the model. The number of nodes generated for the meshing is 601,025, with 3,294,219 elements. The element quality ranged from a minimum of 0.224 to a maximum of 0.99, with an average value of 0.844. The skewness values varied from a minimum of 8.232 × 10⁻⁵ to a maximum of 0.799, with an average value of 0.216.

Laminar flow was set for the fluid flow throughout the pipe as per the analysis of the Reynolds number of the simulated mass flow rates. The inlet was set to the mass flow rate, with the initial temperature set to be the same as room temperature. A pressure-based outlet was selected for the outlet boundary condition. At the beginning of the simulation, the model’s initial temperature was set to 300 K.

The heat flux through the microheater was set to 11,000 W/m². Parametric analyses were carried out with inlet flow rates ranging between 0 mL/h and 180 mL/h with the other parameters held constant. The reported definitions of the area-weighted average were set individually for all three thermal sensors for the analysis of the change in temperature during the simulation. TµFS had a lower detection limit for a flow rate of approximately 0.1 mL/h. Figure 5a,b show the flow contours corresponding to flow rates of 1 mL per hour and 10 mL per hour, respectively.

Figure 5a depicts the thermal distribution profile at a low flow rate of 1 mL/h, where the temperature gradient along the silicone pipe is more gradual. Owing to the reduced velocity of the fluid, the heat generated by the microheater disperses more uniformly throughout the fluid. This slower flow rate enhances the fluid’s capacity to absorb heat, resulting in a pronounced temperature differential between the sensors located upstream and downstream of the heater (T₁, T₂, and T₃). This particular arrangement underscores the flowmeter’s heightened sensitivity to low flow rates, enabling it to detect even the most subtle thermal variations essential for accurate flow measurements in microfluidic contexts.

Conversely, Figure 5b presents the thermal distribution at a higher flow rate of 10 mL/h, where a steeper temperature gradient emerges. The increased velocity of the fluid shortens the time available for effective heat transfer from the heater to the fluid, leading to a less distinct temperature differential between the sensors. The rapid movement of the fluid facilitates faster heat dissipation, thereby diminishing the device’s sensitivity at elevated flow rates. This phenomenon is indicative of a reduced thermal boundary layer thickness and a consequent decline in heat transfer efficiency.

6. Results

Figure 6a shows the simulation graph of the three individual sensors, T₁, T₂, and T₃, for flow rates varying between 0 and 180 mL/h. When the heater is activated, the temperature of all the sensors initially increases from 0.1 mL/h. The sensor T₁ output exhibits a baseline temperature, reflecting environmental factors such as heat and the starting temperatures of the liquid or air. Sensor T₂ measures variations in heat loss upstream during fluid flow and temperature changes when the microheater is unable to preserve the initial temperature. Sensor T₃ monitors a temperature rise for lower flow rates and decreases progressively as the flow rate increases.

Figure 6b shows the changes in the simulated temperature differences resulting from the combinations of the three temperature sensors: T₃–T₂, T₂–T₁, and T₃–T₁. The sensitivity of the flowmeter is elevated at lower flow rates and reduced at higher flow rates at the micro level. Hence, this flowmeter can be used for measuring the flow at a micro level of 0–40 mL/h for better sensitivity. It can be used for applications such as drug delivery, where the greatest sensitivity and accuracy are needed. The results of the present study are in accordance with the work of D. Lee et al. [30], as shown in Figure 7, with the error falling in the band of 10–15%. The meshing considered in their work is different from that in the current work, which has a considerable impact on simulation results, and also the simulation platform used is different, leading to the mentioned deviation.

A parametric study was carried out by moving the T₁ sensor 2 mm to the left, and its influence on the temperature profiles was observed as shown in Figure 8a–c. Also, the T₃ sensor was moved a further 2 mm to the right, and temperature profiles were observed as shown in Figure 9a,b. From this analysis, it was observed that there was the smallest change in the sensitivity observed when varying the location of the sensors. The changes account for the temperature differences becoming constant at 45 mL/h for T₁ variation and 50 mL/h for T₃ variation. Therefore, this parametric study helped to find the sensitivity ranges for different positions.

7. Discussion

The simulation demonstrates a thermal microflow sensor (TµFS) that offers high sensitivity and reliability, a significant advancement in the field of microfluidics. This innovative device employs three strategically placed temperature sensors and a microheater designed to capture flow rates as minimal as 0.1 mL/h to 180 mL/h accurately. The TµFS operates by monitoring the complex thermodynamics between the points preceding and following the sensor. The thermal sensors at various points on the TµFS record the differences in these thermodynamics, providing a comprehensive picture of the flow dynamics within the system.

The temperature differences between the three sensors, T₁, T₂, and T₃, were used to assess the flow rate. This method of estimation ensures a high degree of accuracy in flow rate detection. A unique feature of the TµFS is its increased sensitivity, which is achieved through a mixture of heat dissipation and heat diffusion. This enhanced sensitivity allows for more precise flow rate detection, making TµFS a highly effective tool in microfluidic applications. Interestingly, the temperature increases at a flow rate of 0.1 mL/h, which is influenced by the thermal conductivity of the silicone material in the TµFS. The increase in temperature is a significant element of the sensor’s design and performance. The initial flow rate of the fluid was set to a minimum of 0.1 mL/h. With a steady increase in the flow rate to 180 mL/h, the TµFS effectively records a stable temperature change corresponding to the higher liquid flow rates. This stable temperature change, even at higher flow rates, further attests to the reliability of the TµFS. The estimated minimum flow rate detection limit for TµFS is approximately 0.1 mL/h. The combination of this low detection limit and the sensor’s high sensitivity of up to 40 mL/h makes TµFS an ideal choice for medical applications where precision and reliability are paramount. Since we are dealing with very low flow rates, this means the lowest chance of fluids becoming corrupted.

In conclusion, the proposed thermal microflow sensor system, as demonstrated by the simulation, offers significant advancements in the field of microfluidics. Its high sensitivity, reliability, and ability to detect minute flow rates make it an invaluable tool, particularly in medical applications where precision is crucial. The successful development and implementation of TµFS underscore the potential of microfluidic technologies in revolutionizing various fields, including healthcare and diagnostics.