# A Review of Passive Constant Flow Regulators for Microfluidic Applications
## 2 Passive Constant Flow Regulators--General Considerations  
### Working Principle  
The passive constant flow regulators considered in this review are related to microfluidics, which refers to the manipulation of fluids in channels or structures with dimensions of tens of micrometers [4]. By contrast to macroscopic fluids, viscosity is more important than inertia, and the Reynolds number (\(Re\)) that characterizes the ratio of inertial to viscous forces on the fluid is usually lower than 2000 in most microfluidic systems. The flow in such microstructures is in the laminar regime with highly predictable fluid dynamics. The estimation of the hydraulic resistances along the fluid pathway is done using the governing equations of laminar viscous fluid flow (see, e.g., References [5; 6]). A constant passive flow regulator is intended to deliver a constant flow rate, \(Q_{0}\), in a predefined pressure gradient range comprised between \(\Delta P_{\text{reg min}}\) and \(\Delta P_{\text{reg max}}\), as illustrated in Figure 2. Further increasing the pressure gradient between the inlet and the outlet can, depending on the regulator design and the target application, either stops the flow rate (passive shut-off feature) or generates a flow rate larger than \(Q_{0}\), the nominal flow rate. At a given temperature \(T\), the flow rate \(Q\) in the flow regulator is:  
\[Q=\frac{\Delta p}{R_{h}} \tag{1}\]  
where \(\Delta p=P_{\text{out}}-P_{\text{in}}\) is the pressure drop between the inlet and the outlet of the device and \(R_{h}\) is the hydraulic resistance of the device. According to Figure 2, the hydraulic resistance is constant and equal to \(R_{h0}=\frac{\Delta P_{\text{reg min}}}{Q_{0}}\) if \(\Delta p\leq\Delta P_{\text{reg min}}\). A further increase \(\delta p\) of the pressure gradient, with \(\delta p\leq\Delta P_{\text{reg max}}-\Delta P_{\text{reg min}}\), induces an increase of the hydraulic resistance, \(\delta R_{h}\). Using (1), we obtain:  
\[\Delta P_{\text{reg min}}+\delta p=(R_{h0}+\delta R_{h})Q_{0} \tag{2}\]  
and thus:  
\[\frac{\delta R_{h}}{\delta p}=\frac{1}{Q_{0}}=\text{{Constant}} \tag{3}\]  
Figure 1: Classification of the passive constant flow regulators dedicated to microfluidic applications. To maintain the flow rate constant in the gradient pressure range \(\Delta P_{\text{reg min}}\leq\Delta p\leq\Delta P_{\text{reg max}}\), the hydraulic resistance of the passive regulator shall linearly increase with the pressure gradient up to \(\Delta P_{\text{reg max}}\), where \(R_{h}\big{(}\Delta P_{\text{reg max}}\big{)}=\frac{\Delta P_{\text{reg max}}}{ \Delta_{0}}\). The change of slope in the \(Q(\Delta p)\) curve is representative of a nonlinear hydraulic resistor. At higher pressures, the regulator could further increase its hydraulic resistance until the flow is stopped or simply keep the hydraulic resistance constant (see Figure 2). A free-flow at high pressure can be obtained by fully by-passing the nonlinear resistor of the device (possible option for the piston spring valve [7]) or by adding a crack valve in parallel to the flow regulator (option implemented in the CRx Diamond Valve manufactured by Phoenix Biomedical Corp [8]). The \(Q(\Delta p)\) profile shown in Figure 2 is, therefore, representative of a three-stage valve:  
* Stage I--low-pressure stage, with low hydraulic resistance for \(\Delta p\leq\Delta P_{\text{reg min}}\)
* Stage II--flow regulation stage, for \(\Delta P_{\text{reg min}}\leq\Delta p\leq\Delta P_{\text{reg max}}\)
* Stage III--high-pressure stage, with a low or high hydraulic resistance for \(\Delta p>\Delta P_{\text{reg max}}\)  
The passive constant flow regulator is intended to be placed into a fluidic circuit. To keep the functioning point of the device onto the plateau of the \(Q\) (\(\Delta p\)) characteristic, it is recommended that the device exhibits the largest hydraulic resistance of the circuit. An alternative method to maintain the flow rate at \(Q_{0}\) consists of the control of the pressure at the inlet of a flow restrictor.