Flow resistance

Determining the necessary pressure to achieve the desired flow through a microfluidic circuit can be hard or even fail-prone for the tested system if it is purely done by heuristic methods. Thus, some simple calculations using the flow resistor concept may help you save time and resources.

Introduction to microfluidics and flow resistance

Microfluidics, the science of the very small fluids, is becoming more and more popular. It was born in the nineties following the creation of the first microelectronics systems [1]. After the first small gas detector was invented making use of the silicon manufacturing techniques, it rapidly showed that these microfluidic systems could work as a solution to perform biological and chemical analyses, among other applications, with limited sample quantities. Nowadays, the applications of these systems go further, from advanced materials manufacturing [2] to even the creation of organs-in-a-chip [3].

In this context, the concept of flow resistance, also called hydrodynamic resistance or just microfluidic resistance, corresponds to the opposition that a fluidic element, like a pipe, offers to a flow through itself. Each element of a microfluidic circuit offers some resistance to the flow which is translated into a drop of pressure. Thus, if you know how to calculate the total flow resistance of your system, you can easily calculate the necessary pressure of your pump to achieve the desired flow rate in your system  by using the Hagen-Poiseuille equation, the fluidic analogous Ohm’s law for electrical resistance:

Δ P = QR_H  

Where Δ P is the pressure difference, or drop, between two points of the system, Q is the flow rate and R_H is the hydraulic resistance.

The concept of flow resistance through examples

Flow resistors in a fluidic circuit are the analogous elements to electrical resistors in an electrical circuit. In these analogous circuits, the pressure [mbar] corresponds to the voltage [V], the flow rate [µL/min] to the electrical current [A] and the flow resistance [mbar·min/µL] (or hydrodynamic resistance) to the electrical resistance [Ω]. An example can be shown for the sake of simplicity.

Here, the output of each part of the circuit (pipe, channel, sensor, etc.) is connected to the input of the following one. Thus, the total flow resistance of the whole circuit is simply the sum of all the individual resistances of the elements.

R_HT=R_H1+R_H2+R_H3+R_H4+R_H5

Now, in the next example, that you can see in the figure 3, two analogous schematics of a slightly more complex system are shown: a) the microfluidic example circuit and b) is its electrical analogous circuit.

The total resistance of the cascade circuit can be calculated in the usual way as it is done with the electrical circuits, considering it different in case of serial or parallel association. Thus, the total flow resistance can be calculated as before but using the flow resistors values:

Where the two first flow resistors are given in parallel and the third flow resistor is in series with them. Then, for a given voltage/pressure drop, the resultant current/flow can be calculated using the analogous Ohm’s law previously introduced by rearranging the terms:

Notice that here the pressure of the two inlets is the same because we are assuming that they are connected to the same pressure pump outlet. 

This circuit could represent a simple and common microfluidic T-mixer. As it can be seen, in this example there is no need for a complex microfluidic simulation with expensive software and rather it can be calculated by hand.

Theoretical calculation

As you have seen, the problem is reduced to find the values of the hydraulic resistances. In [4], it can be seen that, under some circumstances, it can be demonstrated that the pressure drop associated with any section of microchannel only depends on the flow rate, the dimensions of the channel and the dynamic viscosity of the fluid itself. For the most thtypical channel cross sections, the flow resistor expressions are also given in [5]. These are, respectively: circle, rectangle and square shapes. 

Where:

• r is the radius of the circle.

• 𝜇 is the dynamic viscosity.

• L, h, w the length, height and width of the channel.

By simply looking at the flow resistance expressions it can be observed that its value will be higher the smaller the section is and the longer the channel is. It is also directly proportional to the viscosity of the fluid. You can check how these parameters change the final resistance value by using the resistance calculator that Elvesys created for you.

Limitations of theoretical flow resistance calculation

The circumstances, or assumptions, under which this theoretical model represents the reality are the following:

• Small Reynolds number, i.e. laminar flow regimen through the flow resistor.
• Incompressible fluid, which is a good assumption in most of the cases.
• Unidirectional flow.
• Steady flow along the channel.
• Small fluid mass per distance unit, so gravity is negligible.

In addition to that, it is observed that in parallel channel arrangement, the fluid does not always make a predictable flow distribution among the branches, so there may be situations for which the theoretical set of equations will not represent the reality as expected. 

Finally, the theoretical expression for calculating the flow resistance of rectangular channels is an approximation that works better when the channel width is very large compared to its height. This way, the more of these assumptions that are not fulfilled, the less realistic this model will behave compared to your real system.

Estimation from simulation

The previous approach for the calculation of microfluidic resistance, pressure drop and flow rate is not applicable to microfluidic circuits with complex shapes or when the above assumptions cannot be made.That’s why traditionally fluidic simulations are run using FEM (Finite Element Method) software like COMSOL [7], OpenFOAM [8] or SimScale [9], to name a few, which are usually expensive and can be really computing consuming.

To perform these simulations, you usually need to create a 3D model of your microfluidic channels which will be converted into a mesh model. These mesh models are an approximation of the geometry made of polygons which are used to split the simulated volumes into small pieces where properties can be calculated. In addition, it is necessary to define some initial conditions for the simulation, like the inlet and outlet pressures, as well as to set several parameters. One common result returned by the fluidic simulation is the velocity profile of the liquid flowing through the microfluidic channels, from which the channel’s resistance can be deduced. You can see some simulation results in the Figure 4.

This can be used, for instance, in order to improve the manufacturing techniques for producing flow resistance, as it is necessary to find a proper way to measure the flow. However, flow sensors have some limitations when working with extremely low flow values, thus, it can be convenient to run accurate simulations and to compare the data from the sensor and from the simulations in order to reduce the uncertainty of their values. 

Experimental measurements

As a validation step or alternative to simulations in the design process of a microfluidic circuit, some experiments can be made to measure flow resistance. In that aim, two magnitudes of the circuit must be measured: pressure and flow rate. To do it, it is necessary to place two pressure sensors, one before and another after the flow resistor, as well as a flow sensor before the resistor. 

Once you have those measurements, it is as simple as again using the analogous Ohm’s law to get the flow resistor value between the pressure sensors:

Being P₁ the pressure measured by the sensor before the flow resistor (or microfluidic element), P₂ the pressure after it, and Q_m the measured flow.

Again, for some of those microfluidic circuits, the simulation and experimental data can be compared with the expected values from the theoretical models. In case the similarity is enough for the application, further study of the system, like the variation of some parameters in order to check how the system behaves, can be achieved in a faster way using the electrical-analogous equations.

Conclusions on flow resistance

Flow resistance is clearly one of the most important concepts to which you must pay attention when designing a microfluidic circuit so that you can adjust the operation conditions of your system through the Hagen-Poiseuille equation. The main methods for the estimation of the flow resistance of a fluidic component that have been presented are the following:

• Theoretical models: when possible, they are the simplest solution.
• FEM simulations: for accurate results and complex shapes or just when the previous models are not valid.
• Heuristic methods: to validate the computed models or measure resistance of fluidic components which are too hard to model. 

As well as flow resistors, there exists the analogy of flow capacitors, in which some fluid is stored in an elastic or volumetric element where the pressure is stored as potential energy. As it happens with the electrical circuits, these elements can be used to produce delays in the flow response and, together with flow inductive elements, which store kinetic energy, can be used to create fluidic “RLC” circuits [6].

Finally, it can be emphasized that a proper understanding of microfluidic resistance has the potential of allowing a multitude of new applications and design procedures. If a system is created with the necessary constraints for the flow resistors and other analogous elements to properly represent the actual behavior, it would be possible to simulate it in real time, allowing the control of complex microfluidic circuits like the actuators of a soft robot [6]. Additionally, this understanding and new design techniques could lead to a highly automated creation of microfluidic circuits in the same way digital systems are produced today as a combination of small predefined blocks [10].

For more microfluidic reviews, you can have a look here: «Microfluidics reviews». The photos in this article come from the Elveflow® data bank, Wikipedia or elsewhere if precised.

Article written by Juan Sandubete López and Alex McMillan.

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