User Manual
Contents
1. MFCS S N V Message comprising serial and version number mfcs purge on Message purge is switched on mfcs reset detected Message system has realized activation of reset signal button processor reset line Table 3 commands sent from the MFCS to the computer MFCS User Manual version 1 2 www fluigent com 22 Fluigent tel 33 0 15373 1551 fax 1552 MFCS e mail support fluigent com date SUGGESTIONS DYSFUNCTIONS Name Urgent yes O Suggestion O Society Dysfunctions O Device Version Software Version Menu or concerned function Error code if available DESCRIPTION please cite error messages operation before error operation to quit erl message add screen shots if possible Don t write in this field please MFCS User Manual version 1 2 www fluigent com 23 Fluigent SA 4 Avenue de l Observatoire 75006 Paris France Phone 0033 0 153731551 Fax 0033 0 153731552 web page http www fluigent com e mail supportmfcs fluigent com MFCS User Manual version 1 2 www fluigent com _ 242. For three channels p p2 ps and R1 R2 Rs joining in one point with pressure pc we find using the continuity equation the sum over all Qs going into one knot vanishes P3 R R R Q R R R Q w Pi Ba Pe P P P3 R R R P The corresponding flow in channel 1 as an example can be obtained with Ohm s law Q 0p1 pc0 R1 the other values are equivalent To obtain the corresponding formulas for n channels joining in one knot the latter formula can be generalized gt P i R n 1 2 R More complex geometries can be calculated using well known methods in analogy to electrical resistor networks pressure is equivalent to the electric potential flow is equivalent to the current and flow resistance of a channel plays the same role as the Ohm s resistance P MFCS User Manual version 1 2 www fluigent com 19 Appendix D Maintenance The MFCS is designed as a very robust device for usage in laboratory and industrial environment and easy to maintain For optimal accuracy calibration should be performed regularly typically every month or after the MFCS has been displaced in order to maintain optimal accuracy of the results and compensate for offset drift of the pressure sensors The housing can be cleaned with a moisty soft tissue iso propanol or window cleaning products Avoid aggressive organic solvents as acetone or chloroform or abrasive cleaning products Please use
3. 10s Calibration menu operations See Quick Start item 4 in page 6 Help menu The Help menu contains an ABOUT item informing about the contact address of the MFCS developers and about software and firmware version numbers 3 2 Graph display The graph window shows the measured pneumatic pressures as a multicolor graph The values can be recorded to a file using the SAVE TRACE option Pull down menus described above 3 3 Status indicator The MFCS status field bottom left shows the status of The USB connection The system state standby or active as selected by operation of the red and green buttons on the MFCS fornt panel Suspended operation due to input pressure exceeding the maximum input pressure as indicated in the technical specifications Purge activation as set by purge button see below MFCS User Manual version 1 2 www fluigent com 10 3 4 Purge Buttons Pushing the purge button bypasses the regulation and forwards the input pressure directly to the outlet 1 This operation is very useful for fast purging the system filling it with viscous buffers e g electrophoresis matrices removing dust bubbles etc IMPORTANT Never use this function if you are not sure that your microfluidic system will withstand the input pressure of the MFCS 3 5 Run Buttons Pushing the Direct control button mouse click alternates between the DIRECT CONTROL and the
4. Operation Irput pressure ndicator Pressure outputs 10 3 thread Pressure supply Powersupply USB connector On Off Switch Figure 2 Front and back sides of the MFCS 8C The front side consists of an input pressure display monitoring the value and stability of the external pressure supply The red button resets the system whereas the green button re starts it The output connectors are of 10 32UNF type The input pressure connector the USB and power 24Vdc connector as well as the on off switch are found on the back side 1 Install the MFCS in a clean and dry environment and avoid exposure to corrosive liquids and vapors or organic solvents which may attack the case Note that the MECS is a purely pneumatic device there should not be any contact with liquids at any step of operation Plug the power cable into the power line and the outlet of the power supply to the 24V inlet of the MFCS rear panel Connect the USB cable to the dedicated computer and to the USB inlet of the MFCS rear panel Only one MFCS can be connected to one computer Switch the MFCS on The red front panel LED schould light up Press the green start button on the front panel to activate the system When the green LED lights up the system is ready for operation figure 2 Install MFCS Control software in order too keep the installation and compatibility as simple as possible the software consists of one single executable file mfcs VX XX X exe
5. SETTING mode In the DIRECT CONTROL mode button white and activated modification of the control slider buttons see below by the user are transmitted straight to the MFCS output In the SETTING mode button gray the new settings of the control slider buttons are stored but regulation still operates with the values displayed when the system was switched from the direct control to the setting mode The new values are taken into account only when the Direct control button is pressed again This permits to preselect pressure values and apply them simultaneously to the ouput channels Upon start the system is in setting mode Control slider buttons 4 8 slider buttons are employed to select independently the different output pressure levels The sliding bar range corresponds to the range of the pressure sensors The selected slider buttons can be controlled in different ways 1 Mouse or other pointing device 2 Cursor up and down keys finest possible step size 65536 steps over the whole range 3 Page up and down keys coarse step size 100 steps over the whole range 4 End and home keys jumps to 0 or to maximum pressure 5 Input numerical value in the Requested Pressure box auto mode or Valve Output box direct mode in the numeric block see below for auto and manual modes Pressure is automatically regulated to the requested pressure if the system is in direct control mo
6. allows to stop and reset the system immediately emergency stop and put it into the standby state In this state the security valve disconnects the input pressure source from the pneumatics of the MFCS and all outlets channels are closed The system starts up in the standby state The GREEN BUTTON allows to re start the system and to establish the USB communication with the connected computer The LEDs have following meaning 1 RED LED on standby status 2 GREEN LED on system operational 3 GREEN LED blinking purge activated 4 GREEN and RED LED blinking alternatively input exceeds limit value MECS User Manual version 1 2 www fluigent com 8 3 Software Description The MFCS Control software allows numerous operations as well as a basic programming capability for more complex operation we recommend the use of the optional MFCSmat stand alone scripting software The six labeled fields in the graphical window comprise the graph display top left the slider cursors top right the status bottom left the Run and Purge button bottom center the numeric block bottom right and configuration selector Pull Down Menus Graph Display Sliding Cursors lof sl I N Chan 1 Chan 2 Chan 3 Chan 4 I Ir Valve conn rs oo 0 l d FLUIGENT ee Ness ee cn Ce aS SS SS Pressure emra l m u ig oe fg i i RE l I l l i I I I I I I I l i eases ple i
7. the following form if you have problems or suggestions with your MFCS device or software and send it to us by fax or e mail MFCS User Manual version 1 2 www fluigent com 20 Appendix E Trace and Log files format MFCScontrol allows a complete real time recording of all settings and measured pressure values during operation 11 07 2005 15885 1 15886 15887 15888 15889 15890 15891 15892 O0 WO 00 00 17 33 03 424 0 426 424 0 426 386 0 395 23 71 0 426 485 0 395 462 0 426 402 0 426 333 0 395 146 100 100 123 100 100 146 123 O1 O1 OO oO 5267 MFCS S N 0003 V 000 011 20 20 20 20 20 20 20 560 477 454 583 583 500 500 Figure 7 an example of a Trace file The trace file format consists of a header line and a data block The header line is composed of the starting date and time serial number of the connected unit and version number of the MFCScontrol software The data block is composed of lines carrying a time stamp absolute time and the internal step number and the 4 8 pressure values N A means information lost e g transient transmission failure When using this option avoid the creation of oversized data files Adjust the storage frequency according to the needs see pull down menus 11 07 11 07 KT FS 110 11 07 11 07 1107 11 07 11 0 ILOs 11 07 E 07 11 07 Ely 07 200 200 200 200
8. 200 200 200 200 200 200 200 200 200 200 5 2 5 5 1 AAA D 2 Dil 5 1 5 1 Sc SS Pe III I JJJ 227 21 27 2273 27 2273 2273 i27 27 2 24 e257 2 ELE 2 8 139 43 AU 45 AU 45 AU 38 BEGIN OF LOG 43 AU 43 AU 43 AU 43 AU 43 AU 43 AU 46 AU 46 AU 46 AU SR e TO Chan TO Chan TO Chan TO Chan TO Chan TO Chan TO Chan TO Chan TO Chan TO Chan TO Chan LO Chan END OF LOG 2196 2221 2246 2283 2332 2381 2406 2715 2727 2739 2739 2139 Figure 8 an example of a Log file MFCS User Manual version 1 2 www fluigent com 21 ALPHA Chan X YYYYY Order set a value to YYYYY in Channel X AUTO Chan X YYYYY Order set value YYYYY in Channel X in feed back mode BEGIN OF LOG Order start logging END OF LOG Order stop logging MANUAL Chan X YYYYY Order set value YYYYY in Channel X in manual mode PURGE OFF Order open purge valve PURGE ON Order close purge valve USB OFF Order close USB channel USB ON Order open USB channel ZERO Chan X YYYYY Order set zero value YYYYYon channel X Table 2 commands sent from the computer to the MFCS mfcs exceeding input pressure Message input pressure exceeds threshold value mfcs operating Message system is operating normally mfes purge off Message purge is switched off
9. PR d FLUIGENT MicroFluidics Control System User Manual MECS software 2006 Fluigent MFCS User Manual version 1 2 www fluigent com MFCS User Manual version 1 2 www fluigent com Contents 1 REPO CU GION ae ea 4 1 1 Required Computer Hard and Software ss 5 1 2 Package C ntentSs s usasraassaannann nn nnnnannnnnn nenn ann ann ann nenn NEES ann anna nun anne 5 2 Initial Setup Operations ess 6 Zele o TT e GE 6 2 2 Front Panel DeSCriptiOn isseictciccccccsccstveicicceccesntsatecteacsaccuceseccceeng sencnaderereeavaacaneecedee 8 3 Software Description hennenunn anne 9 31 TOUR Le Ou TN 10 3 2 Graph diSplAY uses ereen on ere eNEEENENNEChEEN ee na 10 3 3 Status indiCator sssssssssssscceesasnsnsteennncnenasnenseseann gennnenenteesenpaneencentes 10 34 Purge BUONS so s AIEEREIFERERFFERATRRERSCHRERARAET 11 3 5 RUN Te de D EE 11 3 6 Automation Of Protocols wisssicisccces cacccsesnsccccecensnsectesiecceccasceseneessececcaccnsneseedeeesuens 14 Appendix A Technical Specifications MFCS eenn 15 Appendix B Unit Conversion Table 17 Appendix C Hydrodynamic Resistance of Rectangular Channels and Networks un ee 18 Appendix D Maintenance inner 20 Appendix E Trace and Log files format 21 MECS User Manual version 1 2 www fluigent com 3 1 Introduction The MFCS is a compact high precision pneumatic pressure controller designed for reliable flow control in Lab on a Chip applications i e in microch
10. Please copy it from the CD delivered with the MFCS or from www fluigent com Please also copy the user manual in pdf format in the same directory as the MFCS Control program allowing direct access to the manual from the MFCS Start MFCS by clicking on the MFCS Control icon Calibration please verify that no pressure input is connected to the MFCS and that output connectors of the MFCS are all open to a free quiet atmosphere After a heating up time of approx 10 minutes select in the set up pull down menu the compensate offset item and follow the indications figure 3 Calibration is required only once for first operation However it is recommended to recalibrate the MFCS once a month or after moving or servicing the device MFCS User Manual version 1 2 www fluigent com 6 MFCS Last Average value value value Chan 1 318 318 318 El Setup Help Set sensors zero Chan 2 366 366 366 Chan3 327 327 327 v Purge warning Chan4 307 307 307 d FLUIGENT Figure 3 The calibration window is opened if compensate offset has been selected The first column displays the current pressure value the second column shows the currently programmed value and the last column averages the MFCS value over 1 second Pushing the Save to MFCS button writes the new values into the MFCS processors 5 Connect a pneumatic pressure source with only dry dust and oil free non corrosive gases to the inlet at the rear of the MFCS Use any elastic tu
11. annels nanochannels and capillaries It allows the generation of stable pulsation free flows and dynamic coupling as well as automation of complex flow patterns in microchannel networks It involves state of the art electronic and pneumatic technology and user friendly software based on FLUIGENT s proprietary FASTAB technology Different versions and configurations are available 4 or 8 channels Figure 1 versions equipped with 4 different pressure ranges 25 69 345 and 1035 mbar Standard 10 32 UNF thread output connectors allows connection with numerous adapter systems Figure 1 Top Left 8 channels MFCS Bottom left 8 channels MFCS with pneumatic connections Right Typical experimental microfluidic work station consisting in the MFCS a chip holder and an epi fluorescence microscope computer not shown The MFCS is a high precision device solving many of the problems encountered in microfluidics such as long equilibration times hysteresis unwanted hydrostatic flows difficulties in controlling simultaneously several interconnected channels flow pulsing The MFCS provides extremely precise pressure regulation with small increments 16 bit and an short time constant stabilized by a pressure feed back loop Since the MFCS is pressure based and pressure is an intensive variable it retains its accuracy independently of the scale of the controlled microchannels network making it particularly powerful for small scale microflui
12. bes of about 4 6 mm ID 6 Please use a regulator adjusted to the pressure prescribed for your own version of the apparatus Table 1 prescribed pressure maximum Pressure MFCS 25 mBar 500 mBar 600 mBar MFCS 70 mBar 500 mBar 600 mBar MFCS 350 mBar 800 mBar 900 mBar MFCS 1000 mBar 1300 mBar 1400 mBar Table 1 prescribed and maximum pressure inlet for the MFCS devices N B when you open the four channels simultaneously the initial pressure is decreasing from 10 to 15 depending on the air flow of the pressure source 7 Connect your application to the MFCS and start your experiment The outlets are provided with 10 32 UNF type threaded connectors 8 Have success with your MFCS experiment IMPORTANT No liquids schould enter the MFCS Liquid backflow into the MFCS will void guarantee To avoid liquid backflow the Fluiwell also being a reservoir or a similar protection system is recommended If sample and reagent reservoirs are integrated on the chip the fluiwell can also be used just as intermediate liquid trap 1 The 10 32 thread can also receive connectors with European M5 threads even though we don t guarantee for tightness and reliability of those connections MFCS User Manual version 1 2 www fluigent com 7 2 2 Front Panel Description The front panel carries an input monitor display red and green buttons red and green LEDs and 4 8 pressure outlets figure 2 The RED BUTTON
13. de Direct control button white and the channel in auto mode see below Control coupling parallel coupling It is often useful to couple the pressures in different channels e g to increase the global pressure in a network while keeping the flows in the different channels fixed parallel coupling or to adjust the relative flow rate in two channels without changing the overall combined flow anti parallel coupling This can be achieved by pushing the down arrow on the right side of the coupling window in the Numeric Block frame This calls a menu A A B B Selecting A on channel 1 and A on channel 2 locks the two slider buttons in a parallel way they move together in such a way that the difference remains constant When the shift key is pressed during slider motion the coupling is suspended and the concerned slider button MFCS User Manual version 1 2 www fluigent com 11 can be moved independently of the other slider buttons again It is further possible to couple more than 2 channels A and B allows to couple independently two groups of channels Control coupling antiparallel coupling The anti parallel coupling is achieved by setting one channel to A B and a second to A B Then increasing decreasing of one channel value results into the decreasing increasing of the other while keeping the algebric mean value constant This is useful for example for modifying the relative flow in two channels
14. dics and nanofluidics The range of potential applications in biology includes e g single molecule or single cells manipulations kinetic studies flow addressing and generation of micro gradients automated and reproducible manipulation of buffers samples and separation matrices in analytical applications and bio assays Potential applications in Chemistry and Physics involve e g fundamental microfluidics nanofluidics studies of instabilities and mixing in micro channels combinatorial chemistry chemistry of explosives back pressure local small scale synthesis of toxic compounds microfluidic scale chemical synthesis emulsions crystallization MFCS User Manual version 1 2 www fluigent com 4 1 1 Required Computer Hard and Software USB 1 1 no additional driver required Screen resolution 1024x768 Minimum P II 500 MHz if no other applications are running in parallel Operating system default configuration of Windows 98SE Windows2000 Windows XP 1 2 Package Contents The MFCS package comprises following items MFCS unit Power supply power line cable USB cable Pneumatic connection kit User manual MFCS Control software on CD Fluiwell if required If any parts are missing or damaged please contact your local dealer or Fluigent immediately MFCS User Manual version 1 2 www fluigent com 5 2 Initial Setup Operations 2 1 Quick start Rese
15. lor Manual Auto Two different pressure generation modes can be chosen The manual manual and the automatic mode auto The normal operation mode is auto In this mode the FASTAB technology provides constant pressure values The requested pressure is set and MFCS User Manual version 1 2 www fluigent com 12 the MFCS dynamically maintains the pressure at the set requested value irrespective of the changes occurring in the microfluidic network In the manual mode the slider button s position or numerical setting in the valve output box is directly converted into the electrical control current to the regulator valve important note that in this case it does not represent a pressure value The use of the manual mode is discouraged since it does not benefit from the powerful features of the auto mode for dynamical compensation of the intrinsic nonlinear characteristics hysteresis and limited reproducibility of the solenoid valves Only special applications may require to switch to manual mode e g if the connected system is highly instable if other control variables than pressure have to be stabilized or for unraveling anomalous operation of complex networks Feed Back Coefficient auto mode only The MFCScontrol software provides optimal pressure and flow regulation in a transparent way to the user in a wide range of situations covering the majority of microfluidic applications in particular those involving extreme
16. ly low pressures and or volume flow rates ul to pl min Optimal operation of the MFCS for extremely high or low pressure flow rates may require an adjustment of the reactivity of the feed back loop This can be achieved independently for each channel by changing the setting in the feed back coefficient box The factory setting 50 should be appropriate for most applications Too small coefficients e g 5 for the 25mbar version result in a slower reaction high values gt 200 can destabilize the system in particular when large pressure changes are requested abruptly The maximal value is 255 The optimal response is obtained just below the critical damping situation between a monotonous return to equilibrium and oscillatory overshooting Please note that for high pressure versions of the MFCS at the factory setting initial value operation is oscillation free for small pressure changes but shows overshooting and transient oscillations for drastic pressure changes If this overshooting is a nuisance to your application it can be suppressed by decreasing the feedback coefficient at the expense of the speed of regulation for small pressure changes A working compromise has to be found depending on the specific application The optimal value depends on several features of the connected system volume and elasticity of tubing hydrodynamic resistance and geometry of the microfluidic channels and on the time constant of the phenomena under in
17. m in a convenient way protocols and automate efficiently complex experiments Further coupling with other software is possible MFCS User Manual version 1 2 www fluigent com 14 Appendix A Technical Specifications MFCS Characteristics Input pressure range Output pressure precision Min output pressure step Max output pressure Operation temperature range Input voltage range Weight Power consumption Characteristics Input pressure range Output pressure precision Min output pressure step Max output pressure Operation temperature range Input voltage range Weight Power consumption Min 10 23 1 9 Min 10 23 1 9 MFCS 25 mBar Typ Max Unit 500 600 mbar 25 full scale 23 ubar 25 mbar 25 50 C 24 25 V 2 25 kg lt 16 W MFCS 70 mBar Typ Max Unit 500 600 mbar lt 2 5 full scale 63 5 ubar 69 mbar 25 50 C 24 25 Vv 2 25 kg lt 16 W Comment Better if calibrated see Quick start Depends on model Comment Better if calibrated see Quick start Depends on model MFCS User Manual version 1 2 www fluigent com 15 Characteristics Input pressure range Output pressure precision Min output pressure step Max output pressure Operation temperature range Input voltage range Weight Power consumption Characteristics Input pressure range Output pressure precision Min output pressure step Max output pressure Operation tempe
18. ne U I a e E l I I a l i i ii A I Fi I wer CC Toon NE en Maximum presswg 256 25724 25718 56 E Dpetaiing Run Requested messurq som eso 3524 12008 I L eap Do Measured pressure mbar ED Brae aan 11994 I Feed back coeficient DS aal Hl 504 Manual tauy Te Auto TW Rud aio Te Ab Active contig 5 Status Numeric Configuration Block Selector Feed Back Figure 4 The MFCS Control software consists in 6 main fields fields limited by red dashed line and two additional control buttons The pull down menus are used for file and configuration operations calibration and provide information about the system The display field is dedicated to the trace of the output pressure MECS User Manual version 1 2 www fluigent com 9 3 1 Pull down menu File menu operations SAVE SAVE AS and OPEN The current configuration parameters that includes all the configurations definition can be stored by selecting the Save or Save as item and read with the Open item The datas stored are in the ASCII format and can be edited with notepad or automatically created The format is self explaining SAVE LOG The Save log option allows to trace back all executed operations and to analyze them in retrospect SAVE TRACE The Save trace option allows to storage the measured data in a text file It is further possible to select different acquisition rates 0 1s 1s and
19. of equal flow resistance merging into a third channel in which the flow should not be affected e g changing the mixing ratio of two reagents without changing the flow downstream Only two channels can be coupled in anti parallel sense Parallel and anti parallel couplings can be combined freely BMCS Control Be Setup Han M FLUIGENT EiDecumenks and Settinge cBureadlntitied mics Figure 5 Anti parallel coupling allows changes in the pressure difference of two channels without changing the overall pressure Parallel coupling is needed if e g the injection rate of two different liquids needs to be changed simultaneously Both function can be combined Numeric bloc This block provides useful information on the settings and measured values for each channel Maximum pressure The maximum pressure that each pressure sensor can measure is displayed in the first line Requested pressure This field displays the requested pressure corresponding to the slider button position It is also possible to enter numerical values directly by clicking in this box and typing the desired value on keyboard Measured pressure The value displays the actual measured pressures This allows a real time monitoring of the regulation and diagnosis of malfunction see below The time evolution of this value is shown for all channels in the graphic display each channel in a different co
20. rature range Input voltage range Weight Power consumption Min 10 23 1 9 MFCS 350 mBar Typ Max Unit 800 900 mbar 25 full scale 0 32 mbar 343 mbar 25 50 C 24 25 V 2 25 kg lt 16 W MECS 1000 mBar Min 10 23 1 9 Typ Max Unit 1 3 1 4 bar lt 25 full scale 0 95 mbar 1033 mbar 25 50 C 24 25 V 2 25 kg lt 16 W Comment Better if calibrated see Quick start Depends on model Comment Better if calibrated see Quick start Depends on model MFCS User Manual version 1 2 www fluigent com 16 Appendix B Unit Conversion Table value J factor xkPa bar unit gt 1 kPa 1 0 01 1 bar 100 1 1 psi 6 895 68 95x10 1 inch H2O 0 249 2 49x10 Example 10 kPa 10 kPa x 0 07 bar kPa 0 1 bar psi 0 145 14 5 1 3 612x10 inch H2O 4 016 401 6 27 68 1 MFCS User Manual version 1 2 www fluigent com 17 Appendix C Hydrodynamic Resistance of Rectangular Channels and Networks Pressure difference Ap the volume flow rate Q or the averaged velocity u and the friction coefficient C are related through ab Ces Ap abu Me H L a with channel dimensions a and b with a b gt gt 1 channel length and viscosity u 9 5 x 10 Pas for H20 at 25 C The relation between the friction C and the aspect ratio amp a b can be calculated Stone Stroock Ajdari Ann Rev Fluid Mech 2004 SR ey ii SEH 9 n Le 3
21. vestigation Configuration folders CONFIG Microfluidic experiments often involve switching between different sets of predefined flow configurations MFCScontrol allows you to define and store up to 8 different configurations Switching between configurations is performed by clicking on the corresponding folder button at the bottom of the numeric block In the direct control mode Direct control button white switching between different configurations is immediately effective on the output pressures In the setting mode Direct control button inactive the pressures corresponding to the selected configuration are effective only after the Run button has been pressed Active config This display shows the current number of the currently active configuration folder MFCS User Manual version 1 2 www fluigent com 13 Stand alone operation For long operation times under fixed pressure settings the computer can be disconnected from the MFCS unit after switching to the setting mode The system continues maintaining the previously set pressure values Any modification of these values needs to connect the computer again To avoid any risk of uncontrolled pressure variations in the microfluidic system please re connect and disconnect only in the setting mode Direct control switched off 3 6 Automation of Protocols For long period or complex operation the MFCSmat software is recommended This permits to progra
22. with 2n 1 a Ba gt TER p 1 12 0 08 0 06 LA 0 04 0 02 Jl 1 10 100 x Figure 6 This graph shows the friction coefficient as a function of the aspect ratio of the channel and can be used to estimate the flowrate due to a given pressure difference Typical values are amp 1 gt C 0 03512 amp 2 gt C 0 05708 amp 10 gt C 0 07688 Example A channel of 10x100um with a length of 10mm contains 0 01 ul Applying a pressure difference of 10 Pa 100 ubar gives a volume flow rate of about Q 0 0005 ml min MFCS User Manual version 1 2 www fluigent com 18 and an averaged flow velocity of u 5 m min A pressure of 100 000 Pa 1 bar is 10 000 times higher thus the flow rate and velocity are also 10 000 times increased Since Ohm s law also applies to flow of other particles than electrons e g water we can transfer the formulas For one single channel we have for the volume flow Q as a function of the externally applied pressure difference p and the resistance R Q p R Q abu The pressure pc at an arbitrary position in a single channel with partial resistances R and Re of the two halves of the channel is Q Q gt e e 5 _ PR PpR R R The total resistance is R R UR2 the flow yields Q ip1 pc1 Ri1 Re is equivalent If the two parts are of equal resistance equal length and diameter we find the well known average value pe 0 p10p20 2
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