Chapter 11 Fluid Flow 11.1 Purpose The purpose of this experiment is to measure water flow through capillary tubes at different pressures, to study resistance to flow using tubes of different diameter, and to measure series resistance of capillaries. Refer to appropriate sections of the textbook as necessary. 11.2 Introduction This experiment will develop skills in graphing and graphical analysis. 11.3 Theory Fluid flows through a pipe because of the pressure difference between the ends of the pipe. Q is the fluid flow rate; if V is the volume of water flowing through a pipe in a time t, fluid flow rate Q = V/t(ml./sec.) (11.1) For a given capillary (like arterioles and veinlets in the human body), Q increases as the pressure difference 1 δp = P 1 P 2 between the ends of the capillary increases. 1 To avoid confusion with uncertainties, the symbol δ will be used to mean difference in for this experiment.
11-2 Fluid Flow This experiment is a study of the relationship between Q and δp for water flow through capillary tubes. For a sufficiently small flow rate, Q is proportional to δp so we write δp = RQ where R is a constant called the capillary s resistance. Note the analogy to the electrical case where a potential difference δv causes a flow of charge (current I) through a resistance R; Ohm s Law is δv = RI. δp = RQ is Poiseuille s Law, valid whenever Q in (ml./sec.) is less than the capillary radius in mm. If Q exceeds this limit, flow ceases to be orderly (laminar) and becomes turbulent, and the flow law does not hold because R increases considerably. Resistance depends upon the fluid viscosity η and the capillary length L and radius r and for laminar flow, R = 8ηL πr 4 (11.2) i. e.: R L; R 1/r 4. Thus, if L is doubled, R is doubled: if r is doubled, R is decreased by 1/16. Consequently, small radius changes will cause relatively large changes in resistance, and thus also in flow for a given pressure difference. Don t confuse R and r in this experiment! 11.4 Procedure 11.4.1 Apparatus Two constant-level baths, 100 ml. graduated cylinder, Glass capillary tubes 1.60 ± 0.05 mm. dia., L 8 cm. 1.60 ± 0.05 mm. dia., L 16 cm. 0.75 ± 0.05 mm. dia., L 8 cm. watch or timer, tubing, clamps, stand, meter stick, and a beaker.
11.4 Procedure 11-3 11.4.2 Method Figure 11.1: Fluid Reservoirs at Different Heights The capillary tube is connected by rubber tubes to the drain outlets of two constant level baths as in Figure 11.1. With baths at different heights there is a pressure difference related to the difference in height (or head ) δh = h 1 h 2 of the upper surface levels of the two baths by the relationship where for water, and δp = ρgδh ρ = 1000kg/m 3 g = 9.8m/sec 2 1 cc = 1 ml = 10 6 m 3 Constant level baths maintain their upper surface level constant by means of continuous input and an overflow pipe. Side 1 is the upper level bath, side 2 is the lower level bath.
11-4 Fluid Flow Capillary tubes must be mounted horizontally and water must flow from the tap continuously and drain to the sink continuously. 1. With clamp 2 on, fill bath 2 using a graduated cylinder up to the overflow drain: this is where the flow volume V will be collected. Place a graduated cylinder under this drain via the drain-tube, open clamp 2 and get a continuous flow of water into the collecting graduated cylinder by making δh at least several cm. Try to exclude all bubbles from the tubing by pinching tubes at appropriate positions. Try to ensure that this flow is continuous for a few minutes, draining the graduated cylinder as required. 11.4.3 Exercise 1: Capillary Resistance 1. Use capillary tube A first, and get the apparatus running smoothly as described previously. Set δh to 20 cm. and again get the system running smoothly by letting flow occur for a couple of minutes. 2. Drain the graduated cylinder without interrupting water flow (use a beaker to catch flow while draining cylinder), replace it under flow, and record the time to the nearest second as water level passes the 15 ml. mark on the cylinder. Record the time again as the collected water level passes the 25 ml. mark. These collected volumes and times should be entered in an appropriate table. If using a stopwatch, the lap or split feature is useful here. The reason for starting the volume at something other than zero is that it reduces errors due to getting started ;ie. you can anticipate the events you are timing, so your reaction time should not play as big a part. 3. Still using tube A, repeat measurements with δh = 15, 10, and 5 cm. successively; it is important to start with highest pressure head first and work downward, having got smooth flow and bubble-free tubing at higher rates first. Note: If water being collected splashes into graduated cylinder the level readings are hard to make, thus, let the drops flow down the side of the graduated cylinder by touching the flow tubing to the inside wall of the graduated cylinder near the top.
11.4 Procedure 11-5 4. Repeat these measurements for tubes B and C, and use clamps 1 and 2 while making the change of capillary tubing. For each tube one must be careful not to exceed the limiting case for laminar flow, hence the maximum δh is important. For tube B, use δh roughly 15, 12, 9, and 6 cm. For tube C, use δh roughly 18, 15, 12, and 9 cm. Please do not alter tube connections to constant level baths as they are quite delicate. Capillary diameters are given; measure lengths L with a meter stick. What is the realistic uncertainty in the length of the tubes, based on how cleanly the ends are cut? Be sure to use this value in your calculations. 5. Proceed through the exercise and create a data table with entries for fluid flow and pressure head causing the flow. When the table is full, plot all the results as pressure δp vs. flow rate Q, with all data for all 3 tubes on the same graph sheet. (Is the origin a point?) Draw a straight line through the data for each of the tubes. 6. As δp = RQ, then R = δp/q which is the slope of the graph. Find R by this graphical method for each of the capillary tubes. Does the y-intercept agree with what you expect? Explain. As R = 8ηL πr 4, then r4 R/L = 8η/π = constant (11.3) Hence, using R as obtained graphically, r 4 R/L is calculated for each of the tubes, and the average of these three values is obtained. From this average, the viscosity of water at the top temperature is obtained. 11.4.4 Exercise 2: Resistance of Capillaries in Series 1. Connect capillary tubes A and B in series with a 2 inch piece of tubing. Ensure somehow that they lie horizontally on the same line. Measure Q for several values of δh and record in a table like that used in Exercise 1.
11-6 Fluid Flow 2. Plot the pressure vs. flow results on the same graph as used in Exercise 1, and calculate the resistance from the graph. 3. The result R series should equal the sum of the separate resistances: R series = R A + R B (11.4) Compare the result with the sum obtained from Exercise 1. (Note that this is also analogous to what happens with resistances in series in an electrical circuit.)