ME332 FLUID MECHANICS LABORATORY

Transcription

1 ME332 FLUID MECHANICS LABORATORY Mihir Sen Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN Contents January 8, 2003 Spring 2003 Before break: Experiments 1 6 After break: Experiments Force on vertical wall 3 2 Free surface under rotation 4 3 Force of jet 5 4 Capillary viscometer 6 5 Valves 6 6 Fittings 7 7 Water table 9 8 Reynolds experiment 9 9 Mean velocity Entrance effects and losses LFE and velocity profile 15 1

2 12 Heat exchanger Thrust of propeller Brookfield viscometer Total-head tube Centrifugal pump Piston in cylinder Falling-sphere viscometer Fan Heat transfer coefficient Hydronics network Optical techniques 28 Appendices 28 Calibration Local and mean velocities Head change Major losses Minor losses Total pressure Mean velocity from local velocity measurements Pump characteristics Load cell Rotameter Scanivalves Type T thermocouple Differential pressure transducer

3 1 Force on vertical wall The objective is to determine the hydrostatic force on a plane vertical wall and to compare the result with analysis. The experimental apparatus is shown in Fig. 1. The wall is free to articulate about its lower edge. The force on the load cell can be measured as a function of the two water depths h 1 and h 2. Load cell String Wall h 1 L L w W h 2 Articulation Figure 1: Hydrostatic force on vertical wall. The theoretical force on the load cell is given by F t = 1 6L ρgw(h3 1 h3 2 ) (1) where L is the distance fom the articulation to the location of the load cell, ρ is the fluid density, g is the acceleration due to gravity, and w is the width of wall. The friction at the articulation has been neglected. The output of the load cell can be calibrated (without water in the chambers) by using known weights W. The distance to the point of application of the force due to the weights is L w, which is different from L. IfF L is the force on the load cell during calibration, then F L L = WL w. (2) 3

4 ω r z Figure 2: Set-up for free surface in rotating cylinder. 2 Free surface under rotation The objective is to measure the geometry of the free surface of water in solid-body rotation as shown in Fig. 2, and to compare the results with theory. The equation of a free surface under rotation can be shown to be z = ω2 r 2 2g (3) where the origin is at the bottom of the paraboloid, z is the vertical coordinate measured upwards, r is the radial coordinate, ω is the radian speed of rotation, and g is the acceleration due to gravity. The geometry of the surface is measured by a traverse that is free to move both horizontally and vertically. For comparison with theory the measured data has to be reduced to (z, r) coordinates. 4

5 Load cell Control volume Water jet 3 Force of jet Figure 3: Schematic of force due to a water jet. The objective is to measure the force of a water jet on a flat surface and to compare the results to theory. The jet is schematically shown in Fig. 3. Consider the control volume indicated in the diagram where we will apply the steady-state momentum equation (13). Momentum flux in the vertical direction exists only at the entrance. If the flow rate is high enough, the fluid leaves the plate horizontally so that at the outlet of the control volume there is no component of the flow velocity in the vertical direction. The pressure at the boundary of the control volume is everywhere atmospheric, so the force due to that is zero. There is a force that the plate exerts on the fluid in the control volume, T, and this is downwards. The gravity force on the fluid, W, is also downwards. Thus, taking the positive direction to be upwards, we have T W = ρv 2 A (4) where ρ and V are the density and the mean velocity of the fluid, respectively, and A is the cross-sectional area of the jet. W is negligible since 1 cm 3 of water weighs less than 0.01 N, and T is much larger than that (if not, then W has to be included). In terms of the flow rate equation (4) is T = ρq2 (5) A where Q = VAis the volume flow rate. As a reminder, this formula is applicable only when the flow out of the control volume is in the horizontal direction and the weight of the fluid is small. The volume flow rate is measured by a rotameter and the force on the flat plate by a load cell. The flow rate can be varied to obtain a force vs. flow rate curve. The theoretical and experimental values can be compared. 5

6 Manometer tube Flow 1 2 Settling chamber 4 Capillary viscometer Figure 4: Flow in capillary tube The objective is to determine the viscosity of water from the volume flow rate in a tube. The critical Reynolds number for transition can also be found. A schematic of the apparatus is shown in Fig. 4. The water is gravity driven from an overhead tank. The pressure drop and the volume flow rate can be measured. The manometer measures the gage pressure in the setting chamber. The volume flow rate is measured by timing a certain volume of liquid in a graduated cylinder. Laminar flow For laminar flow, the flow rate through a horizontal pipe of circular cross section is given by Q = πd4 p (6) 128µL where Q is the volumetric flow rate, p = p 1 p 2 is the pressure drop, L is the distance between the points where the pressures are measured, D is the pipe inner diameter, and µ is the viscosity of the fluid. Entrance effects have been neglected. Transition to turbulence The pressure drop versus flow rate relation for laminar flow in a pipe is linear, while for turbulent flow it is not. Thus the p-q curve will indicate transition. 5 Valves The objective is to determine the loss coefficients for several valves as a function of valve openings. The loss coefficient, K, is defined in equation (37). The flow rate is measured by a rotameter, and a differential pressure transducer measures the pressure drop. The valve openings are measured in terms of angle, or better as fractions of the fully open angle. The valves available are: ball, gate, globe, needle and stop, 6

7 Flow Direction Ball Valve Flow Direction Flow Direction Gate Valve Globe Valve Flow Direction Flow Direction Needle Valve Stop Valve Figure 5: Sectional views of various valves. schematic drawings of which are shown in Fig. 5. A layout of the valves is shown in Fig Fittings The objective is to determine the loss coefficients and equivalent pipe lengths for several fittings. The fittings available are (i) a tee-junction (that can be tested in three different ways), (ii) a 90 elbow, (iii) a reduction and (iv) an expansion. A straight pipe is also available to determine the friction factor from pressure drop measurements using equation (36). The positions of the different fittings are indicated in Fig. 7. Equation (37) is valid for the tee-junction and the elbow for which the mean velocity does not change. For the reduction and expansion, however, there are area changes and consequent changes in mean velocity. In this case the loss coefficient, K, may be defined in terms of the mean velocity, V s, at the smaller of the two sections. 7

8 Flow Direction Pressure Tap Stop Valve Gate Valve Globe Valve Needle Valve Ball Valve Figure 6: Location of valves. Flow Direction Reduction Fitting Expansion Fitting Tee Fitting Straight Pipe Ball Valve Elbow Fitting Pressure Tap (typical) Figure 7: Location of fittings. 8

9 Combining this with equation (35) where z 1 = z 2,weget p 1 p 2 ρg + V 2 1 V 2 2 2g = K V 2 s 2g (7) instead of equation (37), where 1 is upstream and 2 downstream of the fitting. The loss coefficient can be calculated from this equation. For an expansion V s = V 1 and for a contraction V s = V 2. 7 Water table The objective is to observe the flow around objects using dye injection, especially the phenomena of separation and vortex shedding. A schematic of the water table is shown in Fig. 8. The velocity of the flow can be measured by inducing a disturbance in the dye stream and clocking its motion downstream. A camera is placed above the test zone where the object is to be placed. The flows can be photographed, downloaded to the laboratory computer, and transferred to an AFS account where Photoshop can be used to edit the picture for inclusion in the report. The two objects to be tested are: (i) Cylinder: Describe the flow patterns at increasing Reynolds numbers; the Reynolds number is usually based on the diameter of the cylinder. In the vortex shedding mode (Karman vortex street) determine the Strouhal number St = fd/v,wheref is the frequency of vortex shedding, D is the cylinder diameter, and V is the free stream velocity. Remember that vortices are shed from both sides of the cylinder, even though only one side may be visible with the dye. (ii) Flat plate: The angle of attack of the plate with respect to the flow can be varied. Describe the flow patterns at increasing angles of attack; determine the approximate angle at which the flow separates from the plate. For a plate at a certain angle of attack with respect to the flow, the separation, when it happens, will be on the surface facing away from the flow. 8 Reynolds experiment The objective of this experiment, originally carried out by Reynolds, is to observe the nature of the difference between laminar, transitional and turbulent flows, and to determine the critical Reynolds number for transition. The apparatus is schematically shown in Fig. 9. The flow is visualized by dye injection. The flow rate can be measured by using a graduated flask and stopwatch; the manometer will give the pressure drop between that point and the end of the pipe. (a) Describe your observations in words. Take several readings to estimate the critical Reynolds number. (b) Plot the pressure drop as a function of flow rate; it should be linear for laminar and nonlinear for turbulent flow. 9

10 Flow straigtener Object Test section Dye injector Figure 8: Water table. Dye injector Manometer Settling chamber Figure 9: Reynolds experiment. 10

11 Flow Meter Valve Manifolds Flow Control Valve Test Unit Pump 9 Mean velocity Figure 10: Pumping station. The pumping station on which the flow measurement devices are mounted is shown in Fig. 10. The arrangement of the devices on the mounting frame is shown schematically in Fig. 11; the devices themselves are inside and cannot be seen. For calibration purposes the flow rate is measured using a rotameter. Venturimeter The objective is to (i) calibrate the meter (i.e. find the pressure difference versus flow rate curve), (ii) determine the head loss, and (iii) determine the discharge coefficient. The volume flow rate in the Venturi shown in Fig. 12 is given by Q = C 2(p 1 p 2 ) ρ ( ) (8) 1 1 A 2 2 A 2 1 where p 1 and A 1 are the pressure and tube cross-sectional area at location 1, and p 2 and A 2 the values at the throat. C is a discharge coefficient which takes into account frictional losses; C = 1 implies no losses. The head loss due to the Venturi should be determined between the two pressure taps 1 and 3; they have the same cross-sectional areas with centers at the same level. Orifice meter The objective is to (i) calibrate the meter (i.e. find the pressure difference between sections 1 and 2 in Fig. 13 versus flow rate curve), (ii) determine the loss coefficient as a function of flow rate, and (iii) determine the head loss as a function of flow rate. 11

12 Static Pressure Tap Total Head Probe Flow Directon Tap #2 Tap #1 Total Head Tube Tap #3 Tap #2 Tap #1 Orifice Meter Tap #3 Venturi Meter Figure 11: Location of flow measuring devices Flow Figure 12: Venturimeter 12

13 1 2 3 Flow Figure 13: Orifice meter Pressure taps Flow direction Orifice meter Vacuum pump Figure 14: Air flow for entrance effects and friction factor. The loss coefficient and head loss are measured between the most separated pressure taps 1 and Entrance effects and losses The objective is to determine (i) the pressure distribution in the entrance region with and without the bell-mouth, and (ii) the friction factor in the fully-developed region. The flow of air is induced by a vacuum pump. The flow rate is measured by an orifice meter using a calibration constant and equation (37). There are several pressure taps along the length of the pipe, as shown in Fig. 14. The computer records the pressures and the positions of the pressure tap. The pressures along the pipe (not the orifice plate pressures) are displayed on the screen. The experiment can be done with and without the special bell-mouth fitted to the entrance of the pipe. The entrance without the bell-mouth exhibits a separation, as shown schematically in Fig. 15, which affects the pressure pattern. On the other hand the pipe with the bell-mouth, as shown in Fig. 16, has a much smoother entrance and does not have the separation region. The ports selected by the motorized Scanivalve are in sequence as one moves along the tube from the entrance towards the fan. Note that the first reading comes from a port not connected to the tube since it is used to read the pressure at the entrance, i.e. the atmospheric pressure. The orifice meter has the last two connections. This pressure reading is then used to compute the gage pressures recorded in the data file. 13

14 Figure 15: Streamlines at entrance to pipe without bell mouth. Figure 16: Streamlines at entrance to pipe with bell mouth. 14

15 Flow LFE Vacuum pump Total-head rake Pressure taps Figure 17: Set-up for calibration of LFE and velocity profile measurement. 11 LFE and velocity profile The objective is to (i) determine the velocity profile in a pipe and (ii) to calibrate the LFE (i.e. find the pressure difference versus flow rate curve). An LFE (laminar flow element) is a device that is often used to measure the flow rate in a pipe. An LFE and total-head rake are mounted on the same circuit as shown in Fig. 17. The rake provides the velocity profile from which the flow rate can be calculated. A schematic of the LFE is shown in Fig. 18 and a photograph in Fig. 19. The pressure difference across it is a measure of the flow rate through it. The rake of total head tubes consists of 20 tubes that are 0.1 in. apart. The rake is symmetrically placed in the duct. The density of the air can be calculated from the ideal gas law at the measured temperature and static pressure. The inputs to the manual Scanivalve are from the pressure transducer. Therefore, when switch position 1 is chosen, the connected pressure source is output to the transducer. The twenty tubes of the total head rake have been connected in sequence. For each position on the switch the port directly behind is connected. Each total head tube thus measures the local velocity at a different location. After calculating the local velocities, the velocity profile in the duct can be graphed. Remember that the velocity at the wall is zero. Turbulent flows have a flattened velocity profile compared to laminar flows which have parabolic profiles. The volume flow rate, Q, may be numerically determined from the local velocity measurements made by the total head tubes, V (r), using equation (43). The volume flow rate measured by the LFE and the total head rake should be the same. This enables the LFE to be calibrated. 12 Heat exchanger The objective is to check for energy balance in a concentric tube heat exchanger in parallel and counterflow. Hot and cold water from the tap are used for the two sides of the heat exchanger. By closing and opening some valves, either parallel flow or counterflow can be obtained. The temperatures are measured using type T thermocouples (i.e. copper-constantan) with an ice-point cell as reference. Remember that it takes some time for all parts of the apparatus to come to thermal equilibrium; so give it time to settle down. 15

16 P 1 P 2 Flow Metering element Flow straighteners Figure 18: Schematic of a LFE. Figure 19: Photograph of a LFE. 16

17 T in 2 T w i T w o T 2 out T in 1 T 1 out Figure 20: Schematic of heat exchanger in parallel flow. Parallel flow In a parallel flow arrangement the two streams are in the same direction as shown in Fig. 20. The flow of hot water in the inner tube is designated by the subscript 1 and the cold water in the outer by 2. The heat and mass flow rates are q and ṁ; c is the specific heat at constant pressure of the fluid. The heat given up by the hot water is q 1 = ṁ 1 c(tin 1 T out 1 ) (9) since the velocity at the inlet and outlet do not change appreciably. The heat received by the cold water is similarly q 2 = ṁ 2 c(tout 2 T in 2 ). (10) Some heat is lost to the room. We can estimate the conduction through the cylindrical insulation as q cond = 2πkL(T i w To w ) (11) ln(r o /r i ) where k is the thermal conductivity of the insulation which must be given to you, and Ti w and To w are the inner and outer wall temperatures. Remember that q cond will be negative if To w >Ti w. The geometry of the insulation is shown in Fig. 21, the dimensions of which can be measured. For perfect heat balance we must have q 1 = q 2 + q cond. (12) The left and right sides of the equation can be determined and compared. Counterflow The counterflow arrangement is similar to the above, except that one of the flow directions is reversed. 13 Thrust of propeller The objective is to measure the static thrust developed by a ducted propeller at different rotational rates and compare to theory. The actuator disk theory of Rankine 17

18 r o L r i Figure 21: Geometry of the insulation. 1 2 Control volume LED pair Figure 22: Schematic of propeller thrust experiment. for a free, unducted propeller is not applicable here. A schematic of the experiment is shown in Fig. 22. For steady flow, the force on the fluid in a control volume CV is given by F = VρV da (13) CS where CS is the surface of CV. The suggested control volume is indicated by dashed lines in the figure. The component of the equation in the flow direction is F = ρv 2 da. (14) CV Assuming a uniform velocity 1 across sections 1 and 2 (indicated by an overbar), this reduces to F = ( ρ 1 V ρ 2V 2 2) A. (15) Since ρ 1 V 1 A = ρ 2 V 2 A = ṁ by mass conservation, we get F = ṁ( V 1 + V 2 ). (16) 1 If the velocity is not uniform, as may be the case especially in the downstream location, local velocities have to be measured and the integral obtained numerically. Here we will assume the velocity to be uniform. 18

19 Because the density change of the air is small we can assume ρ 1 = ρ 2,sothatV 1 = V 2. Thus, the sum of forces on the control volume due to the propeller and the pressures must be zero. From this we get T =(p 2 p 1 ) A (17) where T is the thrust, p i is the pressure at i, anda is the cross-sectional area. The force of the propeller on the fluid is towards the right in the figure, and that of the fluid on the propeller towards the left. The pressure difference between inlet and outlet is indicated by the differential pressure transducer; the force on the propeller is measured by a load cell. Compare the measured values of the thrust with the theoretical values. A blade-passing signal is provided by a LED-photocell pair 2 ; its frequency can be measured from which the rotational rate of the propeller can be determined. Notice that the blades cut the light beam twice in a revolution. Plot the propeller thrust as a function of its rotational rate. On dimensional grounds, it is predicted that the thrust should be proportional to the square of the rpm; check if this is so. 14 Brookfield viscometer The objective of this experiment is to determine the viscosity of a fluid for different temperatures. In this instrument, also called Synchro-Lectric viscometer, the torque needed to rotate a cylinder or spindle is measured. Within a limited range of speeds, the torque, T, is proportional to the fluid viscosity, µ, and the rotation rate, ω, i.e. T = Kµω. (18) Since the geometry of the container in which the fluid is kept is not simple, K cannot be determined analytically. The cylinder is driven by a synchronous electric motor through an 8-speed gearbox and a spiral spring, schematically shown in Fig. 23. The spring is a metallic strip in the form of a spiral; the motor shaft is attached to the outer edge of the spiral and the cylinder shaft to its center. The rotation rate of the cylinder is precisely known for the 8 settings. The angular deflection of the spring, measured by the pointer against the graduated disk, is proportional to the torque being transmitted. Additional information about this viscometer can be found in: It is a Brookfield dial viscometer, also known as an LVT viscometer. The instrument calibration constant K is found by using a fluid of known viscosity, after which the viscosity of any other fluid can be experimentally determined. The fluid is placed on an electric heater and its viscosity and temperature are measured at the same time while it is being heated. A thermometer immersed in the fluid gives its temperature. 2 LED = Light Emitting Diode. 19

20 8-speed motor and gearbox Graduated disk Motor shaft Spiral spring Pointer Cylinder shaft Rotating cylinder 15 Total-head tube Figure 23: Brookfield viscometer. The objective is to find the local velocity at a point as a function of the mean velocity. The local velocity is measured by a total-head tube, and the mean velocity is determined from the flow rate measured by the rotameter. A total head probe, shown in Fig. 24, is similar to the Pitot tube. The facility is set up to measure the difference between the total and static pressures (i.e. it measures the dynamic pressure). From equation (39), we get 2(p0 p) V =. (19) ρ The total-head tube measures the local velocity, i.e. the velocity at a point not the mean velocity. 16 Centrifugal pump The objective is to determine the characteristic curves (i.e. the head and the theoretical power versus flow rate) for different pump speeds,. The efficiency is not measured here. Determine also the nondimensional and dimensional specific speeds. The pump is mounted in a loop shown in Fig. 25. There is an air vent to remove any air in the loop. The flow rate is varied by a butterfly valve, and is measured by a paddle-wheel flow meter the output of which is read by a panel meter. The difference in pressures between the inlet and outlet of the pump is measured by a differential transducer and measured by another panel meter. The pump speed can be varied and the frequency of the voltage is proportional to its rpm. The water for the loop 20

21 Static pressure tap Totalhead tube Figure 24: Total head tube. Panel meters Air vent Motor drive Butterfly valve To building water line Butterfly valve controller Check valve Pressure gage Pressure regulator Expansion tank Paddle-wheel flow meter Pump Figure 25: Schematic of test loop. comes from the building water line and there is a double-check reduced pressure vent system to prevent water from going back into the line (this is always necessary when installing a pump) and an expansion tank for changes in volume due to temperatures. The pump is schematically shown in Fig. 26. For an incompressible fluid such as water, the change in enthalpy due to change in its internal energy is negligible. Since the enthalpy is defined as h = u + p (20) ρ where u is the internal energy, p is the pressure, and ρ is the fluid density, the expression for the head, equation (44), simplifies to H = p ρg + V 2 2g + z p ρg + V 2 2g + z. (21)

22 Piston in cylinder Figure 26: Schematic of centrifugal pump. The objective is to confirm the linear relationship between the force due to pressure and the pressure itself. The experimental apparatus consists of three piston and cylinder arrangements, one of which is shown in Fig. 27. The air pressure can be varied and additional weights can be placed on top of the piston to compensate for it. Before each measurement the piston is brought back to the same level. The downward force on the piston is the sum of the weight of the piston itself, W p, the additional weights placed on top, W a, and the spring force, F s. The force upward is due to the air pressure acting on the piston face. At equilibrium, the sum of the forces should be zero. Thus pa = F s + W p + W a (22) where p is the gage pressure of the air below the piston, and A is the face area of the piston. The effect of the atmospheric pressure on both sides of the piston cancels out. F s is a constant since the piston is brought back to the same position every time as additional weights are added. Plotting W a vs. p should, from equation (22), give a straight line for each one of the three cylinders. From the slope and the intercept of the graph, the area A and the combined force (F s + W p ) can be found for each cylinder. 18 Falling-sphere viscometer The objective is to determine the viscosity of a fluid using the Stokes drag law and the drag coefficient vs. Reynolds number relation. Several spheres are let fall through the fluid as shown in Fig. 28, and for each the terminal velocity is measured. 22

23 Additional weights W a W p Precompressed spring F s Piston pa Air-tight cylinder Pressurized air Figure 27: Piston in cylinder arrangement. Figure 28: Falling sphere 23

24 The drag coefficient, C D of a body dropping at constant velocity, V,isgivenby the equation ( ) 1 W F B = C D 2 ρv 2 A where W is the weight, F B is the buoyancy force, and ρ is the fluid density. A = πd 2 /4 is the frontal area, where D is the diameter of the sphere, and F B = ρgπd 3 /6, where g is the acceleration due to gravity. The Reynolds number is defined as Re = ρv D µ where µ is the viscosity of the fluid. If the Reynolds number is of order unity or smaller, we know that for a sphere moving in an infinite expanse of fluid which gives C D = 24 Re (23) W F B =3πµDV. (24) A hydrometer is available to measure the specific gravity of the fluid. From the lowest Reynolds number measurements the viscosity of the fluid can be calculated. Using this value of the viscosity and all the measurements, the C D vs. Re curve can be plotted. The presence of the wall will provide additional drag and hence tend to increase the drag coefficient, especially for the larger spheres. The Stokes viscosity law is applicable only to low Reynolds numbers in an infinite fluid (i.e. without walls). So if you use it to determine the viscosity from the terminal fall velocity for all spheres you will get apparent viscosities which are all different. However, only the viscosities for the low Reynolds number cases are the true viscosity. You can take the viscosity given by the smallest sphere which has the lowest Reynolds number to give the true viscosity. Then, using this value of true viscosity for all the other cases, you can compute the drag coefficient and Reynolds numbers for all spheres. From this you will obtain a C D vs. Re curve. 19 Fan The objective is to find the characteristics of the fan and its specific speed. The efficiency cannot be measured, but the rpm of the fan can be varied. The fan arrangement is schematically shown in Fig. 29. The temperatures at points 1 and 2 are measured by RTDs 3. The differential pressure transducer measures the gage pressure in the plenum at 1, which is really p 1 p 2 since the atmospheric pressure p 3 is the same as p 2. The flow rate is measured by the orifice meter upstream. There is a tachometer connected to the fan shaft that sends a periodic signal to the oscilloscope; the frequency of this signal gives the rate of rotation of the shaft. 3 The resistance of an RTD (Resistance Temperature Device) changes linearly with temperature. 24

25 p 2 Fan Orifice meter Plenum chamber 1 3 Figure 29: Schematic of set-up for fan measurements. Approximating air to be an ideal gas, the change in enthalpy is proportional to the change in temperature alone. That is h 2 h 1 = c p (T 2 T 2 ) (25) where T is the temperature and c p is the specific heat at constant pressure. Equation (44) becomes H = c pt g + V 2 2g + z c pt g + V 2 2g + z (26) A pressure head can also be defined as ( ) ( ) p p H p = (27) ρg ρg 2 1 Though this is not the true head as defined by work input per unit weight, it is often used by manufacturers as a measure of the pressure difference generated by the fan. 20 Heat transfer coefficient The objective is to find the heat transfer coefficient for convection from a hot fluid to the wall of a circular pipe. The set-up is schematically shown in Fig. 30. A blower moves the air along the pipe and an electric heater is used to warm it. There are a number of thermocouples along the length of the pipe to measure the air temperature as well as the temperature of the wall. Consider a section of the pipe shown in Fig. 31. The heat balance for the section is given by ṁc p ( T 1 a T 2 a 2 ) = hp L (T av a 1 T av w ) (28) where ṁ is the mass flow rate, c p is the specific heat of air at constant pressure, h is the heat transfer coefficient, P is the perimeter of the inner cross section of the pipe, and L is the length of the section. The superscript on the temperatures indicates the 25

26 Thermocouple (wall) Thermocouple (air) Heater Blower Flow Figure 30: Schematic of set-up to determine the heat transfer coefficient. 1 2 T 1 T 1 wall wall T 1 air Air T 2 air Figure 31: Section of pipe. location where the temperature is measured. The average temperature of the air in the section can be approximated by Ta av = 1 ( ) T 1 2 a + Ta 2 (29) and that of the wall by Tw av = 1 ( T 1 2 w + Tw) 2 (30) Measuring the other quantities, the heat transfer coefficient h for the section can be determined. There are a total of six similar sections on the pipe and the values of the heat transfer coefficient for each section can be obtained, and then averaged. The same procedure can be carried out at different heating rates. 21 Hydronics network The setup in Fig. 32 has one primary loop receiving chilled water and three secondary loops with hot water. The objective of the experiment is to control the temperature using different strategies. 26

27 PID P1 HT 1 HT 2 P 10 T F F F HX BT HX CF HX VF 3 T 7 4 T 1 T Pump Control Valve Valve 8 P P P T P4 P HX cw 11 3 P Figure 32: Hydronics network. 27

28 22 Optical techniques The objective is to understand the process of taking pictures of flows with density variations using the techniques of (i) shadowgraph (also called the shadow method), and (ii) Schlieren, and to use them in two different flows. The Schlieren technique is more sensitive than the shadowgraph. The Schlieren image is characterized by an asymmetry in dark and light shades in a direction normal to the knife edge. Recommended reading: Sections of Elements of Gasdynamics, H.W. Liepmann and A. Roshko, John Wiley, A schematic of the optical set-up is shown in Fig. 33. For the Schlieren the knife-edge must be at the focal point, for the shadowgraph it must be removed. The images can be recorded by a videocamera and displayed on a computer screen; the still pictures can be sent to an AFS account for later editing. The image seen by the eye will be better than that on the computer screen since the eye is much sharper in detecting differences in shade. The report should include some examples of shadowgraphs and Schlieren pictures and explanations of the principle of operation of each method. Natural convection The first test flow is natural convection from an electrical heater. Air that comes into contact with the heater increases in temperature, and its density decreases. As a consequence it rises, and a plume is quickly formed above the heater when it is turned on. Describe the characteristics of the plume. You may want to align the heater two different ways with respect to the light beam. It takes a while for the plume to disappear when the heater is turned off. Air jet The second test flow is a jet of compressed air coming vertically downwards out of a nozzle. There is a density difference between the compressed air and the surroundings for a short distance from the nozzle which makes the jet visible over this length. Appendices Calibration Calibration is a concept that is frequently used in relation to measuring instruments, and it is worthwhile discussing it in further detail. Consider an instrument which has an input x and output y. Taking the example of a pressure transducer, x may be a pressure difference, and y may be the voltage. In general, the output will obey some law of the form y = f(x) (31) The calibration then is this curve. For every instrument, one needs the calibration curve to convert the output y into the input x. 28

29 Mirror Light source Screen Test zone Mirror Knife edge Figure 33: Schematic of set-up for shadowgraph and Schlieren. Many instruments have strongly nonlinear calibration curves which must be reported as a graph or table. Sometimes, however, the curve is simple like, for example, a straight line. Thus, we may have y = ax + b (32) Here a is a calibration (or sensitivity) constant being the slope of the y vs. x curve, and b is an offset. If this is the case, it is sufficient to provide the value of the two constants. Sometimes the same information is given in another form: the user is told that x psi gives a reading of y volts. Also since the offset is the reading of y for zero x, it is easily measured. Local and mean velocities The velocity at each point on the cross-section of a duct, V, is a local velocity. The mean or average velocity at that cross section, V,isgivenby V = Q A (33) Thevolumeflowrateis Q = where A is the cross section of the duct. Head change A V da. (34) The general definition of change in head between two sections 1 and 2 in a duct with different cross-sectional area is 4 h f = p ρg + V 2 2g + z p ρg + V 2 2g + z (35) Some authors multiply this expression by g. It is then no longer in units of length. 29

30 where 1 is upstream of 2. Here h f is the head change, p is the pressure, ρ is the fluid density, V is the mean velocity, z is the height above a reference, and g is the acceleration due to gravity. Major losses The friction factor f in fully developed flow of an incompressible fluid in a horizontal pipe of constant cross-sectional area is defined by h f = f L D V 2 (36) 2g where p 1 p 2 is the pressure drop along a length L of the pipe, and D is the inner diameter of the pipe. Section 1 is upstream of 2. Minor losses The loss coefficient, K, in a fitting is defined by h f = K V 2 2g (37) where K is the loss coefficient and V is the mean velocity. h f is defined in equation (35) and usually we assume z 1 = z 2. There is no confusion if V isthesameoneither side of the fitting. If, however, it is different, as in a contraction, it should be modified as in equation (7). The length of straight pipe that would have the same pressure drop is given by L eq = K f D (38) where L eq is the equivalent length, f is the friction factor and D is the diameter of the straight pipe. Total pressure The total (or stagnation) pressure in an incompressible fluid is p 0 = p ρv 2 (39) where p 0, p and ρv 2 /2 are the total, static and dynamic pressures, respectively. 30

31 Mean velocity from local velocity measurements Thevolumeflowrate, Q, can be determined from local velocity measurements by numerical integration. For a cylindrical pipe, equation (34) becomes Q =2π R 0 rv (r) dr (40) where V (r) is the local velocity at a given radius r, andr is the radius of the pipe. From Q we can calculate the mean velocity V = Q πr 2 using equation (33). To perform the integration in equation (40) from experimental data, we first divide the circular flow area along a diameter perpendicular to the measurement diameter to get two semi-circles. We find the flow rate in each semi-circle and then add them to get the total flow rate. Since we have V (r) measurements at specific radial points on each semi-circle, we will assume that the velocity profile between the two points is linear, so that V (r) =V 1 + V 2 V 1 (r r 1 ) for r 1 r r 2 (41) r 2 r 1 where V (r 1 )=V 1 and V (r 2 )=V 2 are the two measured velocities. A semi-circular strip between r 1 and r 2 is shown in Fig. 34. The volume flow rate through this strip is r2 Q = π rv (r) dr (42) r 1 Substituting for V (r) and integrating, we get [ { V 2 V 1 Q = π V 1 r 1 r 2 r 1 } r 2 2 r V 2 V 1 r2 3 r 3 ] 1 r 2 r 1 3 This gives the flow rate in one semi-circular strip. Similarly, find the flow rates for all the other such strips and add to get the flow rate in the semi-circle. Remember that the fluid velocity at the wall is zero. Determine the flow rate in both semi-circles and add them to get the flow rate across the entire cross section. Pump characteristics A working machine (such as a pump, compressor or fan) puts energy into a fluid stream while a power machine (like a turbine) takes energy out. The head of a fluid machine is defined as the rate of work put in or taken out per unit weight of fluid. It has units of length 5. For a working machine it is H = h g + V 2 2g + z h g + V 2 2g + z (44) outlet inlet 5 Remember that some books use a slightly different definition in which the right side is multiplied by g so that the head comes out in different units. 31 (43)

32 Velocity measurement points + + r 1 r 2 Figure 34: Strip for numerical integration of local velocity. where H is the head, h is the specific enthalpy, g is the acceleration due to gravity, V is the mean velocity, and z is the height above a reference. We have ignored the heat transfer from the machine. Since ρgq is the weight flow rate through the machine, the theoretical power to or from the machine is given by P = ρgqh (45) where P is the power, ρ is the fluid density, and Q is the volume flow rate. The fan or pump characteristic is a diagram which has the head, power and efficiency curves versus the volume flow rate. The curves are usually drawn for a series of constant rotational rates of the machine. The nondimensional specific speed for a working machine is given by N nd s = ωq1/2 (gh) 3/4 (46) where ω is the rotation rate in rad/s, Q the volume flow rate in m 3 /s, and H the head in m. There is also a commonly used dimensional specific speed Ns d = nq1/2 (47) H 3/4 where now n is the rotation rate is in rpm, Q is the volume flow rate in gpm (gallons per minute), and H istheheadinft. Load cell The load cells used in this laboratory are Model LCL made by Omega. More information can be found in their Webpage: 32

33 Flow Figure 35: Schematic of rotameter. asp/subsection.asp?subsection=f07&book=pressure Rotameter This is also called a float meter. It consists of a vertical, slightly conical tube with a ball or other freely movable object inside as shown in Fig. 35. The fluid flows upwards through the tube and the object rises to the point where its weight is equal to the drag of the fluid. Notice that as the object moves upwards the mean velocity of the fluid around it, and hence the drag force, decreases. Purchased rotameters are usually already calibrated; the position of the object gives the flow rate. Scanivalves A manual Scanivalve is analogous to a manual selector switch on an electronic device. It has several inputs that may be selected for connection to a single output. The motorized Scanivalve is a motorized version and operates on the same principle as the manual Scanivalve. Type T thermocouple The calibration of a type T thermocouple is ( T = C ) ( E mv ( ) C E 2 mv 2 ) ( C E mv 3 ) C E 4 (48) mv 4 where T is the temperature in Celsius, and E is the thermocouple emf in millivolts. Differential pressure transducer A differential pressure transducer gives a voltage corresponding to the difference between two pressures. There is usually an offset, i.e. the output is not zero even when 33

34 the two pressures are the same. 34