An experiment is devised to measure liquid flow and convective heat transfer rates in micro-scale channels. The mass flow rate through a channel is determined by measuring the amount of liquid that has flowed through the channel and dividing by the duration of the experiment. The mean temperature of the outlet fluid is also measured. To minimize the time needed to perform the experiment (that is, to collect a significant amount of liquid so that its mass and temperature can be accurately measured), arrays of micro-channels are typically used. Consider an array of micro-channels of circular cross section, each with a nominal diameter of 50 pm, 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 x 106 Pa exists from the entrance to the exit of each channel. In many micro scale 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 micro-channels are machined in the copper block. The channel diameters exhibit some deviation due to manufacturing constraints and are of actual diameter 45μm, 50μm, and 55μm, respectively. Calculate the mass flow 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 flow 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 average flow rate and the average mixed outlet temperature to analyze the performance of the average (50μm) diameter channel and concludes that flow rates and heat transfer coefficients are increased and decreased, respectively, by about 5% when forced convection occurs in micro-channels. Comment on the validity of the experimentalist's conclusion.