Chapter 8, Problem 8.117P

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 significant amount of liquid so that its mass and temperature canbe accurately measured),arrays of microchannels aretypically used. Consider an array of microchannels ofcircular cross section, each with a nominal diameter of $50\mu \text{m}$ , fabricated into a copper block. The channels are 20 mm long, and the block is held at 310 K. Water at an inlet temperature of 300 K is forced into the channels from a pressurized plenum, so that a pressure difference of $2.5\times {10}^6$ Pa exists from the entrance to the exit of each channel. In many microscale systems, the characteristic dimensions are similar to the tolerances that can be controlled during the manufacture of the experimental apparatus. Hence, careful consideration of the effect of machining tolerances must be made when interpreting the experimental results.
(a) Consider the case in which three microchannels are machined in the copper block. The channel diameters exhibit some deviation due to manufacturing constraints and at of actual diameter $45\mu \text{m}$ , $50\mu \text{m}$ , and $55\mu \text{m}$ respectively. Calculate the mass how rate through each of the three channels, along with the mean outlet temperature of each channel.
(b) If the water exiting each of the three channels is collected and mixed in a single container, calculate the average how rate through each of the three channels and the average mixed temperature of the water that is collected from all three channels.
(c) The enthusiastic experimentalist uses the averageflow rate and the average mixed outlet temperature to analyze the performance of the average $50\mu \text{m}$ diameter channel and concludes thatflow rates and heat transfer coefficients areincreased and decreased. respectively, by about 5% when forced convection occurs in microchannels. Comment on the validity of the experimentalist’s conclusion.