Optimization work was undertaken to advance a microfluidic network to a predetermined device scale. By hydraulic resistance set by the outermost channel to design the succession of microchannels widths for the multi-capillary network. This approach progressed previous work through a reverse analysis method. To determine succession resistance sequence at high flow rate optimized conditions. By fluidic distribution of flow between the longitudinal microchannels to remain uniform. To maximize lateral heat transfer to the fluid contained within a volume filled network in a material. In response to the challenges of surface high temperature through differing conductivity states by lower or increased absorption. This was undertaken using a theoretical approach of a leaf like model defined by electrical circuit analysis, will advance slot succession of microchannels for unified heat transport capture across a microfluidic platform. By extensional steady state flow for the required velocity profile without capillary flow breakup in network branching. Through systematic resistance networking to achieve a pressure gradient along the capillary channels that is uniform. This was determined through an optimized sequence of capillary widths, defined by setting a target resistance of one microchannel to set the resistance sequence network for all. Diversifying different sized channels is required to obtain the same velocity flow rate through slot network matrix architectures. This was achieved by simulation results validating the assumptions by setting flow rate at 1 ml/min at a constant temperature of 25 C. At this temperature, density and dynamic viscosity coefficient are equal to:

$$\rho =\frac{997\,{\rm{kg}}}{{m}^{3}}\,and\,\mu =8.9\times {10}^{-4}Ns/{m}^{2}$$

The volumetric flow rate, Q is

$$Q=\frac{1\,ml}{min}=\,\frac{1\times {10}^{-3}}{60}1/s=\frac{1\times {10}^{-3}\times {10}^{-3}}{60}{m}^{3}/s=1.66\dot{6}\times {10}^{-8}{m}^{3}/s$$

The aspect ratio of the channel is given by

$$\propto =\frac{h}{w}=\frac{2}{2.3}=0.8695$$

Po is 14.32808 for this channel aspect ratio.

The mean flow velocity in the channel \(\bar{{\mathfrak{u}}}\) is given by

$$\bar{{\mathscr{U}}}=\,\frac{Q}{A}=\frac{1.66\dot{6}\times {10}^{-8}}{0.0023\times 0.002}=3.62\dot{1}\times {10}^{-3}$$

The hydraulic diameter of the channels is defined as

$${D}_{h}=\,\frac{4\times area}{wetting\,perimeter}=\,\frac{4\times 0.0023\times 0.002}{2(0.0023+0.002)}=2.139\times {10}^{-3}m/s$$

The Rynolds number based on the hydraulic diameter of the channel is then defined by

$$Re=\frac{\rho \bar{{\mathscr{U}}}{D}_{h}}{\mu }=\frac{997\times 3.62\dot{1}\times {10}^{-3}\times 2.139\times {10}^{-3}}{8.9\times {10}^{-4}}=8.6764$$

To note Re < 2000 indicating that the flow is laminar, hence the flow rate could be increased to 100 ml/m and still be in a laminar state.

Since the flow is laminar Poiseuiille number can be used to calculate \(\bar{\tau }\) as a ratio of Poiseuille number and Rynolds number to evaluate the fanning factor \(f\)

$$\bar{\tau }=\frac{1}{2}\rho \,\overline{{\mu }^{2}}\,f=\,\frac{1}{2}\,\rho \overline{{\mu }^{2}}\,\frac{Po}{Re}\,\frac{1}{2}\times \,997\,\times {(3.62\dot{1}\times {10}^{-3})}^{2}\times \frac{14.32808}{8.6764}=0.01079N/{m}^{2}$$

The pressure drop for a fully developed flow rate along the central channel of length L will determine the pressure drop as a balance to average wall shear strength:

$${\rm{\Delta }}\rho =A=\bar{\tau }\times \,2(w+h)\times L\Rightarrow {\rm{\Delta }}\rho =\frac{\bar{\tau }\times 2(w+h)\times L}{A}$$

Pressure drop along the central channel for a flow rate of 1 ml/min

$${\rm{\Delta }}\rho =\frac{0.01079\times 2(0.0023+0.002)\times 0.186243}{0.0023\times 0.002}=3.7570\,N/{m}^{2}$$

Hydraulic resistance R of the central channel

$$R=\frac{{\rm{\Delta }}\rho }{Q}=\frac{3.7570}{1.66\dot{6}\times {10}^{-8}}=0.2255\times {10}^{8}\,kg{m}^{4}{s}^{-1}$$

Hence resistance evaluation will determine pressure drop at flow rates that can be defined by circuit resistance analysis.

$$R=\frac{{\rm{\Delta }}\rho }{{\rm{Q}}}\Rightarrow {\rm{\Delta }}\rho =Q\,R\,(cf.\,V=I\,R\,in\,electrical\,circuits)$$

Flow rate can be estimated from known applied pressure drop across the channel section

$$R=\frac{{\rm{\Delta }}{\rm{\rho }}}{{\rm{Q}}}\Rightarrow Q=\frac{{\rm{\Delta }}\rho }{{\rm{R}}}\,(cf.\,{\rm{{\rm I}}}=\frac{{\rm{V}}}{R}\,in\,electrical\,circuits)$$

A single channel will define resistance succession sequence of multi microchannels are equal to each other. This can be determined by iterative method to follow the above procedure, except for the target channel was determined from the outmost channel working inwards. This advances previous work that was determined by the central channel to define the sequence of channel geometries. The optimized sequence of multi microchannels widths starting from the outer most channel working inward is 3.00 mm, 2.804 mm, 2.613 mm, 2.445 mm and 2.300 mm with a constant microchannel depth of 2 mm. This method sets the footprint of the system that is constrained to a pre-determined size starting from the outermost channel in this case R4 3 mm. By design of the succession of microchannel widths for R3, R2, R1, R0. If undertaken in a reverse analysis the outmost channel width is unknown and the width of the device cannot be known in advance.

The flow rates through the microchannels are very close to the optimized condition. From the results the flow rates in R1 to R4 have a maximum error of 0.4% from the desired mass flow rate and within 1.3% of R0. This demonstrates the resistance model to reduce maximum potential pressure drop is a valid approach, Table 1.0. The results indicated flow rate through the central channel is still slightly too small. Observations of CFD-derived manifold resistances suggested a slightly wider central channel 2.329 mm compared to 2.300 mm. This would redistribute some of the flow into the central channel away from the outer channels. However using a theoretical resistance model tends to have lower flow rate in the central channel. By over-predicted manifold resistance, into the network circuit that was Rm1 in this case. In the CFD model, Rm1 is lower and thus more flow is diverted towards the outer channels. Velocity flow rates are determined by pressures at the inlet and outlet of the longitudinal manifolds and this can be derived from CFD simulations as an expression of:

$$R{m}_{4}=\frac{{\rm{\Delta }}{\rho }_{4}}{Q{m}_{4}}=\frac{{\rho }_{inlet3}-{\rho }_{outlet4}}{Q{m}_{4}}=\frac{(3.80210-3.68215)}{1.766750251\times {10}^{-08}}=6.789301\times {10}^{6}kg{m}^{-4}{s}^{-1}$$

Table 1 CFD simulations at flow rates of 0.9 ml/min. Full size table

Flow rates through the manifold are denoted by Qm. The flow rates in the manifold taper inlet and outlet are found by the summing the flows rates in the longitudinal channels by;

$$\begin{array}{c}Q{m}_{4}={Q}_{4}\\ Q{m}_{3}={Q}_{4}+{Q}_{3}\\ Q{m}_{2}={Q}_{4}+{Q}_{3}+{Q}_{2}\\ Q{m}_{1}={Q}_{4}+{Q}_{3}+{Q}_{2}+{Q}_{1}\end{array}$$

The parabolic velocity profile using CFD derived theoretical manifold resistances are very close to the optimized condition. This procedure of parallel resistance networking as an analytical method, to reduced total potential pressure drop and maximized flow rate is effective. To measure and understand the nature of this approach the microfluidic network geometry was run against differing velocity flow rate profiles to investigate flow rate impact. The assumption at lower volumetric flow rates would produce steady state parabolic shape in capillary channel distribution. Using higher extensional flow may start to break up capillary flow through non-linear and turbulence effects at the inlet and outlet manifolds. CFD simulations was undertaken by modulating volumetric flow rates set at 0.9 ml/min, 9.0 ml/min and 90.0 ml/min. CFD Tables 1, 2 and 3 of results to measure pressure driven steady state flow as mass flow rate.

Table 2 CFD simulations at flow rates 9.0 ml/min Full size table

Table 3 CFD simulations at flow rates 90 ml/min. Full size table

9.0 ml/min CFD analysis is reflective of the iterative method flow rate = 1.0 ml/min in each channel. The results indicate even at a flow rate of 90 ml/min, the distribution of the flow between the longitudinal channels remains almost uniform. By knowledge of hydraulic resistances allows us to calculate the pressure drop across the device for any given flow rate, without having to perform a CFD simulation (delta p0 = Q0 x R0). Once we know the pressure drop across the device, the method to compute the power required to overcome the viscous forces in the fluid is determined through power = delta P 0 x Total flow rate, Q. CFD results of simulated flow rate velocities 9.0 ml/min and 90.0 ml/min analysis to reduce maximum pressure drop, Figs 1, 2.

Figure 1 Flow velocity distribution at flow rate of 9.0 ml/min. Full size image

Figure 2 Flow velocity distribution at flow rate 90.0 ml/min. Full size image

Using planar extensional flows in maximizing flow rates with reduced total potential pressure drop simulates a leaf model. By energy conservation as the sum of pressure drop in a closed loop capillary network to achieve zero. Through uniform laminar flow at fluidic input channel node to achieve steady state pressure for a required velocity, without shortcuts pathways. This process method of pressure equalization in diminishing flow variation in a leaf like rule order. Using minimum pressure, minimum power output flow to achieve minimum energy loss for energy capture and storage.