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Example 11-12 How Many Capillaries? The inner diameter of the human aorta is about 2.50 cm, while that of a typical capillary is about 6.00 mm = 6.00 * 1026 m (see Figure 11-24). In a person at rest, the average flow speed of blood is about 20.0 cm>s in the aorta and about 1.00 mm>s in a capillary. Calculate (a) the volume flow rate of blood in the aorta, (b) the volume flow rate in a single capillary, and (c) the total number of open capillaries into which blood from the aorta is distributed at any one time. Set Up Figure 11-24 shows the situation. We are given the dimensions of the aorta and each capillary as well as the flow speed in each of these pipes. Our goal is to determine the volume flow rate (in volume per unit time, or m3 >s) in the aorta and in a capillary as well as the number of capillaries into which the aorta empties. The volume flow rate in a pipe is related to its cross-sectional area and the speed of the fluid moving in the pipe. Like water, blood acts like an incompressible fluid. (It will only compress appreciably under pressures much higher than those found in the body.) So we can use the ideas of the equation of continuity, including the idea that the volume flow rate through the aorta must be equal to the flow rate through all of the open capillaries combined. Solve (a) The volume flow rate in the aorta is equal to the product of its cross-sectional area and the flow speed of aortal blood 1vaorta = 20.0 cm>s = 0.200 m>s2. (b) Do the same calculations for a capillary, in which the flow speed is vcapillary = 1.00 mm>s = 1.00 * 10-3 m>s. Equation of continuity for steady flow of an incompressible fluid: (11-19) A1v1 = A2v2 1 = aorta 2 = all open capillaries combined v volume flow rate = Av Radius of aorta: raorta = 1>2 * 1diameter of aorta2 = 1>2 * 2.50 cm = 1.25 cm = 1.25 * 1022 m Cross-sectional area of aorta: Aaorta = pr 2aorta = p 11.25 * 10-2 m2 2 = 4.91 * 1024 m2 Volume flow rate in aorta: Aaorta vaorta = 14.91 * 10-4 m2 2 10.200 m>s2 = 9.82 * 10-5 m3 >s Radius of a capillary: rcapillary = 1>2 * 1diameter of capillary2 = 1>2 * 6.00 * 10-6 m = 3.00 * 1026 m Cross-sectional area of a capillary: Acapillary = pr 2capillary = p 13.00 * 10-6 m2 2 = 2.83 * 10211 m2 (c) Our results from (a) and (b) show that compared to the volume flow rate through a single capillary, the volume flow rate through the aorta is 3.47 * 109 times greater. The idea of continuity tells us that the combined volume flow rate through all the open capillaries must be equal to the volume flow rate through the aorta. We therefore learn the total number of open capillaries. A Volume flow rate in a capillary: Acapillary vcapillary = 12.83 * 10-11 m2 2 11.00 * 10-3 m>s2 = 2.83 * 10-14 m3 >s 9.82 * 10-5 m3 >s Volume flow rate in aorta = Volume flow rate in a capillary 2.83 * 10-14 m3 >s = 3.47 * 109 (Volume flow rate in aorta) = (total volume flow rate in all open capillaries combined) . . . so there must be 3.47 * 109 open capillaries. Reflect Our results show that the human circulatory system is truly extensive! As a check on our results, note that the combined cross-sectional areas of all capillaries is 9.82 * 1022 m2, which is 200 times greater Total cross-sectional area of all open capillaries combined: than the cross-sectional area of the aorta. By the equation of continuity, the flow speed in the capillaries should therefore be slower than in the aorta by a factor of 1> 12.00 * 102 2: That is, vcapillary = vaorta > 12.00 * 102 2 = 10.200 m>s2 > 12.00 * 102 2 = 1.00 * 10-3 m>s. This gives us back one of the numbers we started with, so our calculation is consistent. Area of all open capillaries combined 9.82 * 10-2 m2 = Area of aorta 4.91 * 10-4 m2 = 2.00 * 102 Aall open capillaries = (3.47 * 109)Acapillary = (3.47 * 109)(2.83 * 10211 m2) = 9.82 * 1022 m2