An experiment is devised to measure liquid flow

Chapter 8, Problem 8.117

(choose chapter or problem)

An experiment is devised to measure liquid flow andconvective heat transfer rates in microscale channels.The mass flow rate through a channel is determined bymeasuring the amount of liquid that has flowedthrough the channel and dividing by the duration ofthe experiment. The mean temperature of the outletfluid is also measured. To minimize the time needed toperform the experiment (that is, to collect a significantamount of liquid so that its mass and temperature canbe accurately measured), arraysof microchannels aretypically used. Consider an array of microchannels of circular cross section, each with a nominal diameter of50?m, fabricated into a copper block. The channelsare 20 mm long, and the block is held at 310 K. Waterat an inlet temperature of 300 K is forced into thechannels from a pressurized plenum, so that a pressuredifference of 2.5?106Pa exists from the entrance tothe exit of each channel.In many microscale systems, the characteristicdimensions are similar to the tolerances that can becontrolled during the manufacture of the experimentalapparatus. Hence, careful consideration of the effect ofmachining tolerances must be made when interpretingthe experimental results.(a) Consider the case in which three microchannelsare machined in the copper block. The channeldiameters exhibit some deviation due to manufac-turing constraints and are of actual diameter45?m, 50?m, and 55?m, respectively. Calculatethe mass flow rate through each of the three chan-nels, along with the mean outlet temperature ofeach channel.(b) If the water exiting each of the three channels iscollected and mixed in a single container, calculatethe average flow rate through each of the threechannels and the average mixed temperature of thewater that is collected from all three channels.(c) The enthusiastic experimentalist uses the averageflow rate and the average mixed outlet tempera-ture to analyze the performance of the average(50?m) diameter channel and concludes that flow rates and heat transfer coefficients areincreased and decreased, respectively, by about5% when forced convection occurs in microchan-nels. Comment on the validity of the experimen-talists conclusion.