3 Pressure Losses During Fluid Flow Through Straight Pipes and Local Resistances

Transcription

1 3 Pressure Losses During Fluid Flow Through Straight Pipes and Local Resistances Oldřich Holeček, Lenka Schreiberová I Basic relationships and definitions Description of fluid flow is based on the continuity equation, which expresses the law of conservation of mass and Bernoulli's equation, expressing the law of conservation of mechanical energy. As the volumetric flow rate V is given by the product of mean velocity and flow cross-section S V S, (3-) the mass balance for the system defined by point and assuming steady state can be expressed as S S. (3-) When the density is constant, the equation can be simplified to S S. (3-3) Motion of the real fluids brings friction. When the real fluid is flowing through the device, the loss of mechanical energy by friction and turbulence occurs. The magnitude of these losses is often expressed by an equivalent pressure difference, called pressure loss. Loss of pressure occurs when the fluid flows through the straight pipe and so called local resistances, which is a term for shaped pieces, all type of fittings and locations in the pipe line, in which there is a sudden change in the pipe cross-section. In this work we deal with the pressure losses in a straight pipe and fittings. To evaluate the loss of energy, Bernoulli equation (in the specific energy form) will be used p p h g h g edis,. (3-4) Index and marks the beginning and the end of the measured pipe section, p is pressure, h is geometric height in the section or, is velocity of the flowing fluid, g is the gravitational acceleration, is the density and e dis, is the loss of specific energy between sections and respectively. Specific energy loss is related to the pressure drop by relation dis, dis, p e. (3-5) Energy, that is lost in local resistance is characterized mostly by the coefficient of local resistance. It is defined by the relation e. (3-6) dis, / Specific energy that is lost during the flow in the straight pipe can be calculated using friction coefficient 3-

2 l/ d e dis, /, (3-7) where l is the length of the pipe with internal diameter d. The pressure change p = p - p can be calculated from Bernoulli's equation (3-4) p h h g edis,. (3-8) If the height of selected pipe sections and is the same, i.e., h = h than equation (3-8) simplifies to the form p edis,. (3-9) If the sections and have the same cross-section area, then derived from the equation of continuity, the fluid velocity is the same and equation (3-9) simplifies to the relationship p e p. (3-0) dis, dis, Relationship between the difference of the heights of the manometric fluid h in manometer and pressure drop p can be calculated from p = h ( m - ) g, (3-) where m is density of the manometric liquid. Combining relations (3-7) and (3-) we get the formula for calculating the coefficient. p ( ). (3-) Friction coefficient is calculated by the combination of the relations (3-8) and (3-) p d (l ). (3-3) Velocity is determined by both the volumetric flow rate and pipe diameter - equation (3-). The friction coefficient and coefficient of local resistance as well, generally depend on the fluid density and viscosity, flow velocity, characteristic dimension of the system (usually the internal diameter of the pipe), and the internal roughness of the pipe. The theory of similarity shows, that this dependence can be expressed as a single dimensionless variable - the Reynolds number Re. The Reynolds number for a tube with a circular cross-section is defined as Re = d /. (3-4) The dependence of on Re has been experimentally determined in wide range of conditions (by similar procedure as described above) and is shown in Fig

3 Fig. 3.0 as a function of Reynolds number and d For the flow inside the pipe, the flow characterised by the Reynolds number lower than 300 is called laminar flow. Laminar flow occurs when a fluid flows in parallel layers and momentum transfer in the direction perpendicular to the direction of flow occurs only at the molecular level (viscosity). The friction coefficient within this area depends only on Reynolds number. Contrary to laminar flow, turbulent flow (Re > fully developed turbulent flow) is characterised by a strong transfer of momentum in the direction perpendicular to the direction of flow due to macroscopic turbulent vortices. The formation of vortices supports the increase of the flow velocity and sudden changes in the direction of the fluid flow leading to strong increase in energy loss. The sudden changes in the flow direction occurs when the fluid is flowing through the local resistances and also when flowing around the small inequalities (protrusions) on the pipe wall. The mean height of the inequalities is called the absolute pipe roughness and its values are tabulated. Even if the flow in the axis of the pipe is turbulent, in the vicinity of the pipe wall is the liquid decelerated and the speed is lower. A laminar layer is created on the pipe wall and its thickness decreases with increasing Re number. At higher Re number, the protrusions on the pipe wall start to protrude from a laminar sublayer to the turbulent core and help to increase the intensity of the turbulence. The coefficient of the friction is then only the function of relative roughness (/d) and is independent on the Re number (fluid velocity). This qualitative interpretation explains why the values of the coefficient of the local resistance given in Chemical-Engineering tables are constants independent of Re number. It is assumed that in the shaped pieces and fittings high turbulence intensity is always obtained. 3-3

4 The arrangement of the apparatus in the laboratory allows us to verify this assumption. II Objective of the work. Determine the friction coefficient for a given straight pipe and the coefficient of the local resistance for the given valves.. Plot the dependence of the friction coefficient on the Reynolds criterion. 3. For the coefficients of local resistance determine the average value from the values measured at different volumetric flow rates. III Description of the apparatus A schematic diagram of the apparatus is given in Figure 3.. Water is pumped from the storage tank by a centrifugal pump through one of a pair of valves (4) into one of two rotameters (5). After the rotameter the water flow is divided into three branches A, B and C, which differ in diameter. In the direction of the flow, each branch has at first straight pipe section and then one of the measured armatures (7, 8 or 9). All measured sections are permanently connected to a manometer () through the connection module () using the hoses. The dashed line in Fig. 3. schematically shows the connection of the armature 7 through the connection module () to manometer. Other parts of the piping system are connected analogically. To measure the pressure loss open only the two valves or taps (5-0) leading to the module, that corresponds to the section you want to measure (straight pipe A, B, C or one of the three armatures 7, 8 or 9). All other valves are closed. Differential pressure gauge (manometer) for measuring the pressure loss in each section of the apparatus consists of a glass U-tube filled with a manometric liquid immiscible with water. The manometer is equipped with two vent valves () and a shortcut tap (3). Valves 0 and are used for venting the apparatus (removing the gas from the apparatus). The gate 6 is used to regulate the flow rate of water through the apparatus. 3-4

5 a 0a 8a 9a 0a b 9b 0b 5a 6a 7a 4 8 A B C 5b 6b 7b a b 3 5a 4a 0b b 5b 4b Fig. 3. Scheme of the apparatus 3 storage tank 8 slanted-seat valve 5a,b - valves for pressure gauge connection for direct pipe A switch the pump 9 valve 6a,b - valves for pressure gauge connection for direct pipe B 3 pump 0a, b valves for venting pipes 7a,b - valves for pressure gauge connection for direct pipe C 4 valves - differential pressure gauge (manometer) 8a,b - valves for connecting the pressure gauge on the armature 7 5 rotameters valves for venting pressure gauge 9a,b - valves for connecting the pressure gauge on the armature 8 6 gate 3 - manometer shortcut tap 0a,b - valves for connecting the pressure gauge on the armature 9 7 gate 4 thermometer a,b interface modules A, B, C - straight pipe sections (Note: Taps will be placed instead of valves 5 0 in the future.) IV Work description IV. Preparation of the apparatus for measurement ) Make sure that the storage tank is filled with water, if not, fill it with distilled water. ) Remove the air from the apparatus (venting): Open the valves 7, 8 and 9 and the vent valves 0. Open both valves 4 (two turns) and close the gate 6. Make sure that all valves or taps (5-0) on the connecting hoses leading to 3-5

6 manometer are closed. Run the pump (switch ) and wait until water coming back into the storage tank from the hoses connected to venting valves 0 is without bubbles. Turn off the pump and close both venting valves 0. 3) Close two valves from the valve 7, 8 or 9 in the two branches, were will be no measurement. Check, that the valve on the measured branch is fully opened. Open (two or three turns) the two valves 5-0 connecting the measured branch with the manometer. If tap(s) was placed instead of valve(s) then open taps (connecting the measured branch with the manometer) fully. 4) Vent the manometer before measurement. The presence of bubbles in the hose connecting the manometer with the measured section causes error of the measured pressure difference. Open the valve 4b (about three turns) leading to the larger rotameter and close the valve 4a leading to the smaller rotameter. Open the gate 6 (two turns). Then fully open the shortcut tap 3. Turn on the pump and carefully open one of the valves for venting the manometer. During this step, there is a danger of ejecting the manometric liquid from the tube of the manometer, which is undesirable because it is toxic and expensive. If the water coming from the hose attached to the venting valve is without bubbles, the arm of the manometer is vented. Close the venting valve and vent in the same way the other arm of the manometer. After venting the manometer connecting hoses, close the shorting tap 3 and manometer is ready for use. To verify, that the manometer was vented properly, close both valves 4 leading to rotameters. With no volumetric flow the manometer must show zero pressure difference (no difference between the levels of the manometric liquids in U-tube). 5) Find the widest range of the volumetric flows for measuring the pressure loss of the particular section attached to manometer: The minimum measurable flow rate is written at a smaller rotameter calibration equation, but it can happen that at this flow rate level you cannot read a loss of pressure. Then minimal flow rate is the flow rate at which the pressure drop is still measurable. Maximum flow rate is determined by turning on the pump and slowly opening the inlet valve 4a to a larger rotameter, (close the smaller rotameter) and at the same time opening the gate 6. Observe simultaneously the rotameter and manometer. The maximum flow rate is limited by either the difference of the levels of the manometric liquid reaches the full scale or the measuring range of the rotameter is exceeded or the top performance of the pump is reached. The step 4 and 5 has to be repeated each time you link to a new measured section. IV. Measurement Before starting the measurement, read the water temperature on the thermometer 4 and write it down to the form. Divide the difference between the maximum and the minimum flow rate found in point 5 of the previous section, into equal number of points according to the number of lines in the form. Set the desired flow rate by gate 6 and read and write down the pressure drop. At low 3-6

7 flow rates (less than 50 divisions) close a little bit the valve 4 under the rotameter you are using. IV.3 Finishing the work After all required straight pipes or valves are measured, read and write down the water temperature. Close both valves 4, switch off the pump and close the gate valve 6. V Safety precautions. Do not climb on the apparatus.. Avoid contact with the pump when running. 3. Change the flow rate of water slowly to avoid pressure surge in the apparatus. VI Processing of the measured values a) Calculate the flow rate from the calibration equation of rotameter, written on the board by the apparatus. b) Calculate the flow velocity from the measured volumetric flow rate using equation (3-). Internal diameter and length of the pipes are also on the board. When calculating the velocity of water flowing through the local resistance (armature) use, as the cross-sectional area, the cross section of the attached pipe. c) Calculate pressure drop from the differential manometer readings using equation (3-). Consider all quantities in SI units! The density of the manometric liquid is given on the board. d) Calculate Reynolds criterion from formula (3-4). Density and viscosity of water, at the average water temperature during the measurement, can be obtained from the tables. e) Calculate the resistance coefficient for local resistances from equation (3-). f) Calculate the coefficient of friction for straight pipes from equation (3-3). g) Plot the dependence of the friction coefficient on the Reynolds criterion, use semilogarithmic coordinates. (Careful! Do not plot the logarithm value of Reynolds number. Use only the logarithmic scale on the x axis). Do not forget to write down the rotameter calibration equation and all the information given at the apparatus board before you leave the lab. VII Symbols d pipe internal diameter m e dis specific energy lost m s - g acceleration of gravity m s - h geometric height m l pipe length m 3-7

8 p pressure in the pipe Pa Re Reynolds criterion (Reynolds number) S flow area m v fluid velocity m s - V volumetric fluid flow rate m 3 s - h height difference in the manometer levels m p differential pressure (here pressure drop) Pa absolute roughness m dynamic viscosity Pa s coefficient of friction in the straight pipe (friction factor) fluid flowing density kg m -3 m manometric fluid density kg m -3 coefficient of local resistance (zeta factor) VIII Review Questions. What is the aim of the work, what variables do you set and what do you measure?. What do you do before measuring? 3. When and how do you vent the apparatus? 4. When and how do you vent the manometer? 5. How do you make sure that the manometer liquid won t run away from the manometer when venting? 6. Can you touch the valves when the liquid is flowing through them? May be the centrifugal pump running, if the outlet valve is closed? 7. Can you measure the fluid flow with smaller and larger rotameters at the same time? 8. How do you read from the U manometer? How do you verify that the manometer is properly vented? 9. How much do you close the valve on the section that won t be used for measurement? 3-8