Fluid Flow Equipment: Water reservoir, output tubes of various dimensions (length, diameter), beaker, electronic scale for each table. Computer and Logger Pro software. Lots of ice.temperature probe on computer, stopwatch on computer. Large wrench, calipers, meterstick. Purpose: Understand and be able to calculate the fluid flow of a viscous fluid through a pipe due to a pressure difference (Poiseuille s law). Flow rate dependence on pressure difference, length, radius or cross section area. Observe temperature dependence of viscosity. Background. When a fluid experiences a pressure difference between two linked locations, it will flow. The flow rate depends on the link (in this experiment a tube) between the two areas of different pressure. The rate at which the fluid flows, V/ t, (V is volume and t is time), is determined by the resistance R, of the link. Schematically, Fig. 2 shows the link between reservoir (1) and reservoir (2), P 1 > P 2 Link Flow» and mathematically, P 1 P 2 Figure 1. Flow Model P 1 P 2 = P = R V/ t. (1) The resistance R is determined by three characteristics of the experiment: 1.) length L of the link, 2.) cross-sectional area A of the link and 3.) the type of fluid. In the simplest model, we expect the resistance R of the link to be proportional to L, and inversely proportional to A: L R a, (2) A where a is a constant characteristic of the fluid. [To see this, think of water flowing through a pipe. If a certain amount of water is flowing, there will be a pressure drop along the pipe. If a second identical pipe is connected in series with the first, then the total pressure drop along both pipes will have to be twice as big to keep the flow constant. So the resistance of the two pipes together will be twice as big. On the other hand, if you connect a second pipe of the same length in parallel with the first, then twice as much water will flow, but the pressure drop will be unchanged, so the total resistance of the two pipes must only be half as big as for a single pipe.] Modeling Flow- 1
The above is true if the fluid is non-viscous, but water, like most fluids, is viscous: Viscous Fluid Flow: For a viscous fluid (like water) the flow rate and pressure difference are still related as in Eq.(1). But for a viscous fluid the resistance R is not given anymore by Eq.(2). The resistance to the flow is caused by friction between the tube and the water and between the water molecules themselves. This friction is characterized by the coefficient of viscosity (eta) of the fluid. Your textbook has a table of the coefficients of viscosity for several fluids. The units of viscosity are N/m 2.s = Pa.s. A commonly used unit (onetenth as large) is the poise, P; (1 P = 0.1 Pa.s) or the centipoise cp; (1 cp = 0.001 Pa.s). [The viscosity of water at 20 degrees C is about 1 cp.] The difference between non-viscous flow and viscous flow is illustrated in Figure 2. Non-viscous flow Viscous flow Figure 2. Velocity Vectors for Fluid Flow in a Tube for a.) non-viscous and b.) viscous fluids For a viscous flow, the interaction with the wall slows the fluid and, in fact, the fluid velocity at the wall is zero. Figure 2 shows how the velocity profile changes across the tube for non-viscous flow and for viscous flow of the fluid. Poiseuille showed that because of this non-uniform laminar flow, the actual equation for the resistance R for viscous flow differs from Eq. (2) and instead is given by L 8 L R 8, (3) 2 4 A r where L is the length of the tube, r is the radius of the tube and A is the cross sectional area of the tube. [This is not an obvious result, but you can see that as the tube diameter increases less of the liquid is close to the wall, reducing the resistance. This gives the extra factor of A -1 in R above.]. Think about the effect of the A -2 or r -4 dependence of R on the flow rate of blood in a human artery. Suppose cholesterol deposits narrow the radius of an artery by one half. Then the resistance R will increase by a factor of 16 and the flow rate of blood will decrease by a factor of 16, which is rather dramatic. Blood is a viscous fluid, but also ordinary water is a viscous fluid as this experiment will show. Actually the viscosity of blood (blood) is approximately three times the viscosity of water (water). Modeling Flow- 2
Activity 1: Poiseuille s Law. The lab setup consists of a tank for water with an outlet at the bottom to which you can attach tubes of various lengths and diameters, Fig. 3. The pressure at the bottom of the tank is determined by the combined weight of the atmosphere and the column of water. [What is the pressure at the other end of the horizontal outlet tube?] The flow is determined by weighing the volume of water that flows into the beaker in a given time. Beaker or cup Electronic scale Figure 3. Poiseuille s Flow Apparatus [The time interval should be short enough that the water level in the reservoir does not appreciably change. Why?] Your experiment will be to determine the flow rate for several different dimension tubes and/or for several different water heights. From your measurements you will verify Poiseuille s Law and determine the viscosity of water. Activity 2: Temperature dependence of viscosity. Since there is a substantial temperature dependence of viscosity, we will now repeat part of the experiment of Activity 1 with water of different temperatures and determine the temperature dependence of the viscosity of water. Modeling Flow- 3
Fluid Flow (preliminary questions) Name: Section/TA: Date: Show your work. 8 L L 1. From the two equations, P 1 P 2 = P = R V/ t, and R 8 4 2, r A figure out the units of the coefficient of viscosity. Show work. 2. A team of fire fighters works with a water hose of 50 m length and internal diameter of 8 cm. The available water has temperature of 4 degrees C and has coeficient of viscosity of 1.55 cp (centipoise). The pumper maintains a pressure of 4 atmosphere. What is the water flow rate out of the (horizontal) hose? 3. If the fire fighters were to cut the hose in two halves and then connect each of the two hoses directly to the pumper, assuming that the pumper has two outlets. In this new setup, what is the total water flow rate out of the pumper (sum of two hoses)? 4. Consider again the situation as in question 2 above. What is the flow rate if the water has a temperature of 20 degrees C, where the coefficient of viscosity is 1.00 cp (centipoise). Modeling Flow- 4
Report -- Fluid Flow Name: Section/TA: Date: Partners (full names): (Fluid Flow, Poiseuille s Law) There are 3 types of inner diameter of tubes (d = 3mm, 4mm, 5mm). The flow rate is best determined by measuring the time needed for a total outflow of respectively 200 gram, 300 gram, 400 gram. After each measurement return all water to the big container, so the original water level is restored. IMPORTANT : Give all numbers in 3 or 4 significant figures. Activity 1: Poiseuille s Law Flow Rate. Fill the water reservoir to a level about 11 cm above the outlet spigot and attach one of the longer tubes to the spigot. Now measure the flow rate in units of gram/second. Verify that the flow rate varies if 1.) you hold the tube horizontal in one case, and if 2.) you let the open end be near the bottom of the container. Height difference between beginning and end of the tube = Height difference of water level and spigot level = Pressure difference over the tube = Tube position Time interval (sec) Water output (gram) Flow rate (gram/sec) Horizontal Towards bottom How much is the difference in flow rate and how much is the difference in height of the open tube end in both cases? Explain the connection between height difference and pressure difference. Modeling Flow- 5
From now on you should make sure that the tube is always horizontal. Pressure dependence. Measure for the same tube the flow rate at 3 different heights of the water level in the big reservoir (for example approximately 11, 16, and 21 cm height above the spigot). Height water level Time interval (sec) Water output (gram) Flow rate (gram/sec) Make a plot of flow rate as function of the height. Include in your graph the point of zero flow rate at zero height (but do not include it in your fit). Is there a linear relation between rate and height? Correlation = Why would this indicate a linear relation between flow rate and pressure difference over the tube? Note that pressure P 1 is at the beginning of the tube at the spigot and P 2 is the pressure at the end of the tube in open air. From now on leave the water level at the highest of the 3 levels you considered above. Radius dependence. Choose a set of 3 tubes of approximately the same length but different diameter. You should prefer the set of 54-55 cm, and you will need to coordinate with other users. Modeling Flow- 6
The aim is to measure the flow rate for 3 values of radius r but with the same length and same pressure difference. L = Height difference = Pressure difference = Radius r of tube Time interval (sec) Water output (gram) Flow rate (gram/sec) 1.5 mm 2.0 mm 2.5 mm Make a plot of flow rate versus r 2 and another plot of flow rate versus r 4. Include the datapoint (zero flow at zero radius) but do not include it in your fits. Which plot allows for the better linear fit? Correlation ( r 2 ) = Correlation ( r 4 ) = Length dependence. Choose a set of 3 tubes of the same radius but different length. The aim is to measure the flow rate for 3 different values of length L but with the same radius and same pressure difference. r = Height difference = Pressure difference = Modeling Flow- 7
Length L of tube Time interval (sec) Water output (gram) Flow rate (gram/sec) Make a plot of flow rate versus L -1. Include the datapoint (zero flow at infinite L) but do not include it in your linear fit. Does this plot allow for a linear fit? Correlation = NOTE: In most cases the L dependence of the flow rate needs important correction terms which are due to turbulence effects at begin and end points of the tube. These correction terms are largest for the short tubes. Therefore in the next section where we determine the value of the viscosity parameter η, we prefer to use one of the longer tubes. Viscosity of water. At this stage it is important to use SI units throughout (L, A 2, flow rate 3 /sec, pressure in Pascal) Assume that Poiseuille s Law of Eq. 1 and Eq. 3 holds. The aim is to determine the viscosity parameter η. Choose one of your earlier longer tubes. For this tube determine the ratio L / A 2. Water is viscous. Therefore the resistance of the tube depends on L / A 2. Fill the tank every time to the same level and determine each time the fluid height and calculate P 1 P 2. (Pressure difference between water surface and outlet valve). Why is this also the pressure difference over the horizontal tube? Find for this tube the flow rate by weighing the water that flows into the beaker in a certain amount of time. For the same tube (and same height of water level) repeat this procedure three times and average to improve your accuracy. Modeling Flow- 8
Give your results for this tube in a systematic form: L = r = A = 2 L/ A 2 = -3. Height difference of water level and spigot = Pressure difference along the tube = In the Table below report the three measurements of: time, water volume, flow rate in the required units (3 significant figures). Time interval (s) Water volume Flow rate (m^3/s) Average flow rate for (m^3) the three trials (1 st trial)= *** (2 nd trial)= *** (3 rd trial)= From pressure difference and average flow rate calculate the value of the coefficient of viscosity. IMPORTANT: At what water temperature was this measurement done? Modeling Flow- 9
Activity 2: Temperature Dependence of Viscosity of water. Lower the temperature of the water by dumping an amount of ice in the fluid, stir until the ice is melted. Measure the temperature and record it.. Repeat the procedure measurement of viscosity of activity 1 using your last tube. Do this at a low temperature and at a high temperature (you already have results at room temperature). You can change temperature of the water by dumping ice in the water and stirring until it is all melted. (Aim for a low temperature in the 0 C - 5 C range. For the high temperature use warm tap water.) Each time, measure the temperature of the water with the computer temperature probe and record it. Also measure the height of the water after the ice has melted. Record the values of the measured quantities. L = A = 2 L/ A 2 = -3. After stirring and ice is melted, Water Temperature = Height difference = Pressure difference = Time interval (s) Water volume Flow rate (m^3/s) Average flow rate for (m^3) the three trials (1 st trial)= *** (2 nd trial)= *** (3 rd trial)= Calculate average flow-rate, then calculate the value of the coefficient of viscosity. Modeling Flow- 10
After adding warm water (well above room temperature), Water Temperature = Height difference = Pressure difference = Time interval (s) Water volume Flow rate (m^3/s) Average flow rate for (m^3) the three trials (1 st trial)= *** (2 nd trial)= *** (3 rd trial)= Calculate average flow-rate, then calculate the value of the coefficient of viscosity. Summary Activity 2: Give a summary of the values of viscosity at different temperatures. Modeling Flow- 11