which depend on the tracer injection and measurement positions, 0 and . It was then noticed that the cos ( 0 / ) expressions in eq 7 would equal zero in all odd terms ( = 1, 3, 5, . . .) if the feed point was brought to the middle ( 0 / = 0.5), and as a consequence, all odd terms of eq 3 would disappear and the limiting rate constant would be found in the second term ( = 2) associated with a 4-fold rate (eq 6). In other words, positioning the feed in the middle instead of the top or the bottom reduces the mixing time-scale to a quarter in theory.