Learning Objectives. Lesson 6: Mathematical Models of Fluid Flow Components. ET 438a Automatic Control Systems Technology 8/27/2015
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1 Lesson 6: Mathematical Models of Fluid Flow Components ET 438a Automatic Control Systems Technology lesson6et438a.pptx 1 Learning Objectives After this presentation you will be able to: Define the characteristics of a fluid flow system Identify if a fluid flow is Laminar or Turbulent based on fluid and system parameters. Write mathematical models for fluid characteristics. Develop an analogy between electrical characteristic and fluid system characteristics. Solve for steady-state fluid flow using given mathematical modeling equations. lesson6et438a.pptx 2 1
2 Liquid Flow Characteristics Flow Volume of liquid/unit time (e.g. gallons/minute) Current (Coulombs/s) Pressure Flow Resistance force/unit area (e.g. lb/in 2 ) Pressure drop/unit flow (e.g. lb/in 2 /gal) Potential Difference Electrical Resistance Note: Flow resistance is either linear or non-linear. It depends on type of fluid flow, which is based on the fluid and piping parameters. lesson6et438a.pptx 3 Fluid Flow Classifications Types of liquid flow Laminar Flow - low velocity flows. Stream lines are parallel. Liquid flows in layers. Linear flow resistance. piping lesson6et438a.pptx 4 2
3 Fluid Flow Classifications Types of liquid flow Turbulent - relatively high velocity flow. Liquid swirls and spins as it flows. Non-linear flow resistance. Flow type determined by the Reynold's Number piping lesson6et438a.pptx 5 Determining Flow Types Reynold's Number r vd R m Where r = density of the fluid (kg/m 3 ) v = average velocity of the fluid (m/s) d = diameter of pipe (m) m = absolute viscosity of fluid (Pa-s) Note: Reynold's number is dimensionless Laminar flow : R<2000 Turbulent flow: R>4000 Transition flow: 2000< R< 4000 lesson6et438a.pptx 6 3
4 Determining Flow Types Computing average velocity Velocity profile changes across the cross section of pipes and ducts V=max Q v A Since pipe diameter is usually given V=0 4Q v 2 d Where: A = area of pipe (m 2 ) Q = flow (m 3 /s) d = pipe diameter (m) lesson6et438a.pptx 7 Laminar Flow Equations Laminar Flow Equations for Round Pipes P R R L L Q 128ml 4 d (Pa) Where: p = pressure drop (Pascals) R L = laminar flow resistance Q = flow (m 3 /s) l = length of pipe (m) m = absolute viscosity (Pa-s) d = pipe diameter (m) lesson6et438a.pptx 8 4
5 Turbulent Flow Equations Turbulent Flow Equations for Round Pipes 2 P K Q 8rf l K t 2 5 d R 2K Q t t t (Pa) 3 (Pa -s/m ) Where: f= friction factor (see table 3.3 p. 81 text) l = length (m) d = pipe diameter (m) r = density of liquid (kg/m 3 ) R t = turbulent flow resistance (Pa-s/m 3 ) p = pressure (Pa) Q = flow (m 3 /s) lesson6et438a.pptx 9 Laminar Flow Example Example 6-1: Oil at a temperature of 15 C flows in a horizontal, 1 cm diameter tube with a flow rate of 9.42 L/min. Tube length is 10 m. Find: Reynold's number, flow resistance, pressure drop in tube. Convert all units into SI units All conversion factors are in the textbook appendix Average velocity From Appendix A in textbook r=880 kg/m 3 m=0.160 Pa-s Compute Reynold s number R<2000 so flow in Laminar. Linear relationship between flow and pressure drop. lesson6et438a.pptx 10 5
6 Laminar Flow Solution (2) Compute Laminar flow resistance Compute pressure drop Convert to psi lesson6et438a.pptx 11 Turbulent Flow Example Example 6-2: Water at 15 C flows through a commercial steel pipe with a diameter of 0.4 inch with a flow rate of 6 gal/min. The line is 50 ft long. Find: Reynold's number, flow resistance, pressure drop in pipe. From Appendix A Density r=1000 kg/m 3 Viscosity m=0.001 Pa-s Convert all English units to SI units Find the average velocity lesson6et438a.pptx 12 6
7 Turbulent Flow Solution (2) Compute the Reynold s number R> 2000 so the flow is turbulent. Use the turbulent equations. Turbulent equations require friction factors from Table 3-3 on page 81 of textbook. Find f values from Table 3-3. Get values that bracket the Computed value. For commercial steel pipe: diameter 1-2 cm R between R a >10,000 R b <100,000 f between f a =0.035 f b =0.028 lesson6et438a.pptx 13 Turbulent Flow Solution (3) Friction factor, f, must be between f a and f b Now find K t Find R t lesson6et438a.pptx 14 7
8 Turbulent Flow Solution (4) Convert R t to English units Compute pressure drop Convert P to English units lesson6et438a.pptx 15 Liquid Flow Capacitance Liquid Flow Capacitance - increase in volume of liquid required to produce unit increase in pressure C L DV DP Where: C L = capacitance (m 3 /Pa) DV = volume change (m 3 ) Dp = pressure change (Pa) Derive relationship for C L. Pressure relationship DP rgdh Where: r = density of fluid g = acceleration due to gravity H = height of liquid in tank lesson6et438a.pptx 16 8
9 Liquid Flow Capacitance Derivation Find liquid flow capacitance in terms of tank parameters. DP rg DH DV DH A DV DP rg A DV DV CL DP DV rg A A rg 3 (m / Pa) C L A rg 3 (m / Pa) Final Equation lesson6et438a.pptx 17 Fluid Capacitance Examples Example 6-3: A tank has a diameter of 1.83 meters and a height of 10 ft. Determine the capacitance of the tank when it holds: a.) water b.) oil c.) kerosene d.) gasoline lesson6et438a.pptx 18 9
10 Fluid Capacitance Examples Parts c. and d. Note: As density of fluid decreases, the volume of liquid required to produce a unit increase in pressure increases. lesson6et438a.pptx 19 Fluid Inertance Amount of pressure drop required to increase flow rate by one unit/second. Analogy-electrical inductance. I L p DQ Dt Where I L = inertance (Pa/(m 3 /s 2 ) p = pressure drop (Pa) DQ/Dt = change in flow Inertance defined using physical parameters of pipe. rl I L A Where: A = area of pipe (m 2 ) r = density of liquid (Kg/m 3 ) l = length of pipe (m) lesson6et438a.pptx 20 10
11 Dead-Time Delay Dead-time Delay of Liquid - time required to transport liquid from one point to another in piping system or ducts. D t d v v = average velocity of fluid (m/s) D = distance traveled (m) Example 6-4: Determine the inertance of water in a pipe with a diameter of 2.1 cm and a length of 65 meters. lesson6et438a.pptx 21 ET 438a Automatic Control Systems Technology END LESSON 6: MATHEMATICAL MODELS OF FLUID FLOW COMPONENTS lesson6et438a.pptx 22 11
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