FLUID FLOW Introduction Fluid flow is an important part of many processes, including transporting materials from one point to another, mixing of materials, and chemical reactions. In this experiment, you will investigate fluid flow in a pipe network and will explore several methods (rotameter, orifice and venturi meters) for measurement of the fluid flow rate. You will also explore effects of the skin friction, pipe network configuration, and pipe fittings (tees, elbows, etc.) on the pressure drop across a pipe. Additionally, you will characterize the behavior of the pump which drives fluid flow throughout the pipe network, make predictions about operating points of the system, and validate those predictions experimentally. The main independent variables in this experiment are: Feed flow rate Fluid flow network configuration General Description The Fluid Flow system uses tap water from the building water supply, held in holding tanks on the first and third floors of the Unit Ops lab. The water is pressurized using a centrifugal pump and flows through the pipe network on the first floor in a closed loop, which recirculates back to one of the holding tanks. The water flow rate at the inlet of the pipe network can be adjusted using a globe valve and is measured using a rotameter. The pipe network contains two Schedule 80 PVC pipes of nominal diameter 0.75 and 1.0. Note that nominal pipe diameters usually differ from their actual diameters. In particular, the actual inner diameters of the 0.75 and 1.0 Schedule 80 pipes are 0.742 and 0.957, respectively 1. Each of the pipes contains a combination of tees, elbows, orifice plates, venturi meters, and valves. A tee is a pipe fitting that joins one pipe run to another that runs in a perpendicular direction to the main run. An elbow is used to change the direction of fluid flow. Venturi meters and orifice plates are flow measurement instruments which use the Bernoulli principle to measure flow rates, as described further below. All throughout the pipe network, there are small metal manometer ports, which are used to measure the pressure differences between any two points in the pipe network using a differential pressure gauge. In addition, there are several pressure gauges built into the network. 1 Appendix A5 of Ref. [1] 1-1
Theory Mechanical Energy Balance For each pipe segment, we can write the mechanical energy balance equation: 1 2 v in 2 + p in ρ + z ing = 1 2 v 2 out + p out ρ + z outg + F loss (1) Here, vin = inlet fluid velocity vout = outlet velocity of fluid zin = elevation height of the inlet zout = elevation height of the outlet pin = inlet pressure pout = outlet pressure ρ = fluid density Floss = energy loss due to friction (per unit mass). Friction Losses The friction losses depend on the type of the flow (laminar or turbulent) and pipe elements (valves, elbows, tees, etc.). A common approach to characterization of friction losses is to use the Fanning friction factor f defined as the friction force per unit surface area divided by the kinetic energy per unit volume (ρv 2 /2). Friction losses in a straight circular pipe of constant diameter can be expressed as F loss = 4f ΔL v 2 (2) D 2 Here, ΔL is the pipe length, D is the pipe inner diameter, v is the fluid velocity averaged over the pipe cross-section, and f is the Fanning friction factor. The Fanning friction factor for laminar flows is f = 16 (3) Re where Re = Dvρ/μ is the Reynolds number (ρ and μ are the fluid density and viscosity, respectively). The Fanning friction factor for turbulent flows is given by empirical relationships, such as the Colebrook equation or the Moody diagram. These relationships usually involve a new parameter ε corresponding to the pipe roughness. It depends on multiple factors, including the material from which the pipe is made and degree of corrosion. The flow network in our lab consists of pipes made from PVC. In addition to the pipes, the fluid flow network contains various fittings, including valves, tees, and elbows. Friction losses in pipe fittings are described using the loss factor Kf, v 2 F loss = K (4) f 2 Empirical values of Kf for various fittings are available in the literature [1]. 1-2
Valves In addition to losses due to skin friction and friction in pipe fittings, the pressure drop across a pipe can be influenced by valves. Common valve types include ball valves, globe valve, needle valves, gate valves, plug valves, and many more. The Fluid Flow system in the Unit Ops lab incorporates one globe valve and multiple ball valves shown in Figure 1. Figure 1. Cutaway view of (a) globe valve and (b) ball vale (www.wikipedia.org). Globe valves operate using a movable plug which can descend into its seat in order to close the valve. The height of the plug is manipulated manually using a handwheel. The valve is open when the plug is fully raised, allowing fluid to flow beneath it, and closed when the plug is fully descended into its seat. The design of the globe valve allows for gradual adjustment of flow, therefore making it ideal for applications involving throttling. Ball valves operate using a hollow, perforated ball which rotates to control the flow through it. The orientation of the ball is manipulated using the quarter-turn handle. The valve is open when the perforation in the ball aligns with the direction of the pipe, and the valve is closed when the perforation is perpendicular to the pipe. The ball valves are designed primarily for on/off manipulation and are not recommended for applications requiring gradual throttling of the fluid flow. Valve resistance to the fluid flow is usually characterized by the valve flow coefficient defined as [2] C v = Q ( SG ΔP ) 1/2 (5) Here, Q is the flow rate (in gallons per minute), ΔP is the pressure drop across the valve (in psi), and SG is the specific gravity of the process fluid (i.e., fluid flowing through the valve). Note that Cv is closely related to the loss factor Kf defined in Eq. (4). Cv and Kf contain essentially the same information, but Cv is more commonly used in industry to characterize valve performance. Another interpretation of Eq. (5) is that Cv is the flow rate of water through a valve when the pressure drop across the valve is 1 psi. A valve characteristic curve shows dependence of Cv on the degree of valve opening (e.g., 50% open, 100% open, etc.). Examples of valve characteristic curves are shown in Figure 2. The flow rate shown in this plot corresponds to flow with the pressure drop of 1 psi across the valve, i.e. the y-axis of this plot shows the ratio of Cv of a partially open valve to Cv of a fully open valve. 1-3
Figure 2. Examples of valve characteristic curves (www.flowserveperformance.com). Figure 2 displays characteristic curves for three common categories of valves. Quick open valves are designed, as the name implies, to achieve a high flow rate when the valve has only been opened a small amount. Linear valves operate so that Cv is proportional to percentage of the valve opening. Equal percent valves are designed such that the slope of their characteristic curve increases as the valve is opened. In the Fluid Flow lab you will measure characteristic curves for the globe and ball valves. Analysis of Pipe Networks The mechanical energy balance equation (1) is valid only for individual pipe segments. To apply this equation to a pipe network, it is necessary to split the network into segments without branches and apply Eq. (1) separately to each segment. The mechanical energy balances for individual pipe segments should be connected using the mass balances and additional conditions on pressure changes, which should be deduced from the network structure. To illustrate this approach, consider a simple network shown in Figure 3. A pipe branches into two segments (referred to as Pipe 1 and Pipe 2 in the figure) at some point A. These segments then reconnect downstream at some point B. Figure 3. Example of a simple pipe network. Different segments of the pipe network are shown by different colors. 1-4
Let us discuss how one can determine flow rates Q 1 and Q 2 through the pipes 1 and 2 if the total flow rate Q is known. The mass balance for this system can be written as Furthermore, Q = Q 1 + Q 2 (6) p A,1 = p A,2, (7) where pa,1 and pa,2 are pressures in pipes 1 and 2 at the branch point A. A similar relationship holds for the point B. Therefore, the pressure drops across pipes 1 and 2 are the same, p 1 = p 2. (8) Pressure drops across individual pipe segments can be determined from the mechanical energy balance (1). Let us assume for simplicity that vin = vout and zin = zout for each of the pipe segments. Then Eq. (1) simplifies to p i ρ = F i(q i ), i = 1, 2. (9) Here, Fi is the friction losses in the i-th pipe segment (i = 1, 2) that can be expressed in terms of the flow rates and the friction factors, as discussed above. Combining Eqs. (8) and (9), we obtain F 1 (Q 1 ) = F 2 (Q 2 ). (10) Assuming that the friction factors are known, Eq. (10) contains two unknowns, Q 1 and Q 2. Combining this equation with the mass balance (6), we obtain a system of two equations with two unknowns. As evident from the above discussion, analysis of pipe networks requires knowledge of friction coefficients in individual pipe segments. In this lab, you will obtain these coefficients experimentally by performing experiments with fluid flowing through only one branch of a network at a time. You will then use these data to make predictions for flow rates through a pipe network and verify these predictions experimentally. Flowmeters The pipe network in the Unit Ops Lab contains three flowmeters: a rotameter, a venturi meter and an orifice meter. Principles of operation of these devices are briefly reviewed below. A rotameter is a vertical conical tube with a float in it, as shown schematically in Figure 4. The float is free to move in the vertical direction and its steady-state location is directly related to the fluid flow rate. This follows from the force balance between the drag, gravity, and buoyance forces (Fd, Fg, and Fb) acting on the float. Out of these three forces, only the drag force depends on the flow rate of the fluid, F d = C D ρa p v 2 2, (11) 1-5
where CD is the drag coefficient, Ap is the area of the projection of the float onto the horizontal plane, and v is the fluid velocity directly below the float [1]. This velocity depends on both the fluid flow rate Q and the location of the float: due to the conical shape of the rotameter tube, its cross-section area As depends on the height. Figure 4. Schematics of a rotameter at (a) low and (b) high fluid flow rates. F d, F g, and F b, and are the drag, gravity, and buoyancy forces, respectively. The force balance can be written as (Q/A s ) 2 C D ρa p = V 2 f (ρ f ρ)g, (12) where Vf and ρf are the volume and the density of the float and g is the acceleration due to gravity. Therefore, for each fluid flow rate Q, the float reaches a steady-state at a height corresponding to the rotameter cross-section As satisfying Eq. (12). This is illustrated in Figure 4. A venturi meter is a tube of non-constant diameter (see Figure 5). Variation of the tube diameter leads to variation of the fluid velocity and, hence, pressure inside the meter. There are two pressure taps located at the widest and the narrowest locations of the tube. Therefore, we can determine the flow rate by measuring pressures p 1 and p 2 at these locations and substituting them into the Bernoulli equation. For an incompressible fluid, the pressure drop is related to the flow rate by the following formula [1]: Q = πd 2 2 C d 4 (1 (D 2 /D 1 ) 2(p 1 p 2 ) 4 ρ (13) Here, D 1 and D 2 are the pipe diameters at the pressure tap locations and C d is the discharge coefficient. In the absence of friction losses, C =1. In most venturi meters, C d d is very close to 1. An orifice meter is a plate with a machined hole in the center (see Figure 6). The flow rate is determined by measuring the pressure drop as the flow passes through the plate. Eq. (13) still holds for orifice meter with D 1 and D 2 being diameters of the pipe and the orifice hole, respectively. The friction losses in the orifice meter are much larger than in the venturi meter and a typical value of the orifice discharge coefficient C is 0.6. Precise value of C d d should be determined experimentally. 1-6
Figure 5. Venturi meter. This image is taken from documentation by Lambda Square Inc. for the venturi meter installed in the Fluid Flow system of the Unit Ops Lab. Complete documentation (including information on dimensions) of this venturi meter is available on the lab website. Figure 6. Orifice flow meter [3]. Both venturi and orifice flow meters lead to a permanent pressure loss, i.e. pressure downstream from these meters does not fully recover to the pressure p 1 at their inlets. The permanent pressure loss in a venturi meter is typically about 10%. The permanent pressure loss in an orifice is [1] p 1 p 3 = (1 β 2 )(p 1 p 2 ) (14) Here, p 3 is the pressure 4-8 pipe diameters downstream from the meter and β = D 2 /D 1. Centrifugal Pumps The Fluid Flow system uses a centrifugal pump to make water flow through the pipe network. A typical design of a centrifugal pump is shown in Figure 7. Fluid enters the pump through a suction line towards the eye of a rotating impeller. The impeller accelerates the fluid and spreads it in the radial direction. The fluid is then collected in the volute chamber at the periphery of the impeller and is pushed out through the discharge line. 1-7
Figure 7. Cutaway view of a centrifugal pump (adapted from [4]). The pump is connected to an electric motor through the driveshaft flange. Cavitation Cavitation is formation and implosion of small gas bubbles in a liquid. Cavitation may arise in centrifugal pumps due to the pressure decrease in the eye of the impeller. The decrease in pressure leads to a decrease in the boiling temperature, which leads to the bubble formation. The vapor bubbles collapse as they pass from low to high pressure zones in the pump. When this happens the liquid strikes metal parts at a very high speed. Cavitation sounds like pumping rocks as the implosion of the bubbles results in popping noise. Cavitation decreases the pump efficiency as the energy is lost in expanding the bubbles in the low pressure region and compressing them in the high pressure region inside the pump. Furthermore, cavitation may result in the pump failure. Therefore, pipe networks containing centrifugal pumps must be designed so that the pumps operate away from conditions that likely to cause cavitation. These conditions are described below. Shut-off Conditions: A pump can elevate liquid in a vertical tube up to a point where the weight of the liquid and gravity permit no more elevation. At this point the flow rate of the liquid is zero. The maximum liquid elevation achievable by the pump is referred to as the shut-off head and the corresponding pressure is the shut-off pressure. If the pump is operated near the shut-off point, the pump will vibrate and the liquid will heat up resulting in cavitation, which is likely to damage the pump. Run-out Conditions: When the flow rate of the liquid is too high, the liquid may leave the pump faster than it enters the pump. This creates a very low pressure in the pump resulting in a substantial decrease in the boiling point of the liquid and leading to vaporization cavitation. Pump run-out is the maximum flow-rate that can be developed by a pump without damaging the pump. The run-out conditions correspond to a very small pump pressure. Decreasing pressure below the run-out point may overload the pump motor. Low Net Positive Suction Head (NPSH): Since action of the impeller accelerates the liquid, the liquid pressure at the eye of the impeller is less than the pressure Ps at the pump suction. If the liquid pressure drops below its vapor pressure PVP, the liquid starts boiling, which leads to formation of vapor bubbles, i.e. cavitation. 1-8
The net difference between the head at the pump suction and the head corresponding to the vapor pressure of the liquid is referred to as available NPSH or (NPSH)A, i.e. (NPSH) A = P s P VP ρg (15) The available NPSH should be larger than the required NPSH (denoted as (NPSH)R) to avoid cavitation. (NPSH)R is determined experimentally and is usually provided by pump manufacturers. (NPSH)A may be affected by multiple factors, such as elevation of the feed tank above the pump suction, friction losses in the pipe connecting the feed tank and the pump suction, and the liquid temperature (which affects PVP). In this lab, you will experiment with different (NPSH)A corresponding to feed tanks on the 1 st and the 3 rd floors. Characteristic Curves and Operating Points Characteristic curves are a useful tool for analysis of behavior of a device or a system over a range of operating conditions. In the case of the Fluid Flow system, characteristic curves will be developed for the pump, valves, and the pipe network. These curves will then be utilized to determine operating points of the system at various conditions, as illustrated in Figure 8. Figure 8. Pump and system characteristic curves. A pump characteristic curve shows the relationship between the pressure head generated by the pump and the fluid flow rate, whereas a system curve shows the dependence of the pressure drop across a pipe network on the flow rate. Intersection of the pump and system characteristic curves corresponds to the operating point of the system. A pump characteristic curve shows the relationship between the fluid flow rate and the pressure head generated by the pump (i.e., pressure difference between the pump discharge and suction). In this experiment, you will obtain the pump curve by operating the pump at various flow rates (by adjusting a valve on the pump discharge line) and measuring the flow rates (using a rotameter) and the pressure head generated by the pump (using pressure gauges at the pump inlet 1-9
and outlet). Further details of the experimental procedure are described in the Operation Instructions for this lab. A system characteristic curve shows the relationship between the pressure drop across a pipe network and the flow rate of water travelling through the network. For a pipe segment with no branch points, the system curve is a graphical representation of the mechanical energy balance. For more complex networks, the system curve should be obtained using the approach outlined in the section on Analysis of Pipe Networks. Intersection of the pipe and system curves corresponds to an operating point of the system. This point satisfies the following requirements: (i) fluid flow rate through the pump is the same as that through the pipe network attached to the pump and (ii) pressure generated by the pump equals to the pressure necessary to overcome losses in the pipe network. The system curve depends on the pipe network configuration, which can be modified by opening/closing certain valves. For example, in the simple system shown in Figure 3, the system curve depends on whether the fluid flows through Pipe 1 only, Pipe 2 only, or both pipes in parallel. Therefore, the operating point of the system also depends on the network configuration. Moreover, even if the network configuration remains the same, one can change the system curve by partially closing one or more valves in the system, which changes the valve resistance to the fluid flow. Hence, changing the degree of a valve opening also changes the operating point of the system. In this lab, you will investigate effects of both the pipe network configuration and the degree of valve opening on the operating point. References [1] C. J. Geankoplis, Transport Processes and Separation Process Principles (Includes Unit Operations), 4th ed., Prentice Hall, 2003. [2] T. Blevins and M. Nixon, Control Loop Foundation -- Batch and Continuous Processes, 2011. [3] "Orifice plate," [Online]. Available: https://en.wikipedia.org/wiki/orifice_plate. [Accessed 12 January 2018]. [4] "Centrifugal Pump," [Online]. Available: https://en.wikipedia.org/wiki/centrifugal_pump. [Accessed 14 January 2018]. 1-10