1 FLOW IN PIPES Mark J McCready University of Notre Dame July 24, 2017
2 OVERVIEW This lecture will provide the simplest framework to explain The three forces at that are important to fluid flow in pipes The three equations that needed The three basic flow situations that comprise process pipe flows
3 IMPERIAL FLOWSHEET H100
4 BASIC FORCES THAT AFFECT A FLOWING FLUID Gravity: We observe that liquids flow down-hill and hence know that we will have to overcome the action of gravity to pump the liquid to top of the column F g = m g Friction caused by the viscosity of the liquid or gas: From the observations that pipes and tubes of different diameters are used in the same systems, presumably for different flow rates of the same liquid, (or when we try to drink using straws of different diameters, we expect that there is a resistance to fluid flow that must be overcome) 2 F v = f(μ) ==> 1/2 k v With reference to the straw, but when we are expelling air, (or simply playing the tuba), to make the fluid move, we need to make the pressure higher on one side of the tube. In a process, we use a pump to create the pressure increase F p = p A ==> Δp A
8 PIPE SIZES We have to presume that engineers have picked the sizes of pipes for different applications in some sensible (optimal) fashion. Likewise, we expect that over millions of years, evolution has determined what size blood conduits should be. Thus one of the questions we would like to answer is how do we determine how large a pipe should be for a given flow rate The essence of this calculation is to determine the relationship between flow rate and pressure drop for liquids and gases in pipes.
9 ENGINEERING TRADEOFF Capital costs Operating Costs Optimal pipe diameter is a classic problem For engineers and It is evident in your circulatory system! Murray s Law
11 DIMENSIONAL REASONING AND DIMENSIONAL ANALYSIS Before we address the specific questions related to fluid flow, it is worthwhile to introduce a general concept, that provides useful insights and quantitative results, across all transport process and many other topics in chemical engineering and other technical disciplines That we can gain understanding of fundamental behavior by first identifying quantities such as a time scale or a length scale associated with a process or device. Then by comparing these scales to an alternative scale of the same fundamental dimension.
12 DIMENSIONAL REASONING Simplest example: compare the same variable for a process at two different times Suppose that in one device, 1 mole of gas at STP is trapped in a piston and the initial volume is increased from 22.4 l to 44.8 l. In a second device, 10 moles of gas at STP are trapped in a piston and the initial volume is increased to from 224 l to 448 l. We realize the final temperature and pressure in both pistons is the same and only the ratio of volume 2 to volume 1 is needed to characterize the process. This comparison of like quantities tells us exactly how compare the two processes However, we would find it nonsensical to compare the initial volume to the final temperature.
13 ISOTHERMAL COMPRESSION If we want the temperature to be constant within the piston as we compress from V 1 to V 2, how slowly do we need to conduct the process? Intuition tells us that the one on the right will have to be slower. This is a statement saying we need to compare the time, τ, of compression (process time) to the time scale of heat conduction. To have identical outcomes, we need τ conduction /τ compress <<1 for both cases
14 LET S APPLY DIMENSIONAL ANALYSIS TO A FLUID FLOW PROBLEM A field of 40 wind turbines covers about 1400 acre This field is producing 56 MW of power for a wind speed of 10 m/s (22 mph) which is about the optimal/maximal value This is.04 MW/acre A coal fired power plant would produce about 10 MW/acre! If the wind speed drops to 5m/s, how much power will the field produce? With no reference to formulas is it possible to get an answer? Does engineering analysis have to be really complex? 14
15 ENGINEERS LIKE TO COMPARE THINGS If I asked:.. how far is it to Chicago? would you answer? a couple of hours or... about 90 miles If I asked:.. is a meter a long distance? what would you say No, compared to the distance to Chicago Yes, compared to a micron For our conclusion to be valid we need to compare like (same dimensions) quantities.
16 POWER AND WIND SPEED? How does the power generated by the windmill change with wind speed? How is power being generated? Wind flows through area swept by blades Windmill converts this kinetic energy to electric power Air flow in Air flow out 16 Power out
17 POWER AND WIND SPEED? How does the power generated by the windmill change with wind speed? Let s see if we can figure this out based on dimensional reasoning Power is work/time which is force * distance/time which is mass* acceleration *distance/time Thus we could write power = m l / t 2 l / t = ml 2 What variables could be used? t 3 17
18 EQUATION FOR POWER FROM WIND Windspeed, blade diameter, air density v [=] l/t d, r [=] l Density of air ρ [=] m/l 3 Arrange these variables to get dimensions of power: power ~ ρv 3 d 2 [=] ml 2 If the wind speed is cut in half, the power reduced to 1/8! So our 40 wind turbines will produce about. 7 MW! t 3 18
19 WIND TURBINE POWER About a factor of 7! 19
20 FUNDAMENTAL FLUID MECHANICS For steady flow in a pipe, the forcing from the pressure difference is exactly balanced by the friction from the fluid viscosity An equivalent way to think of this is that the work being done to move the fluid is all being converted to heat.
21 DIMENSIONAL REASONING APPLIED TO PIPE FLOW The variables that describe flow in a pipe are, μ, ρ, Δp, D, V, L We might worry that to formulate and solve a problem in fluid flow that the result would be different for each combination of values of these variables However, by applying dimensional reasoning to this problem using a more formal process of Dimensional Analysis we determine that these variables can be arranged into two dimensionless groups (the dimensional comparisons are built it) which should describe the behavior for this problem. The result is:
22 DIMENSIONLESS PARAMETERS FOR FLUID FLOW Reynolds number: ratio of inertia to viscous forces Friction factor: ratio of pressure forces to inertia
23 THIS FORMULATION WORKS! DATA FOR LAMINAR AND TURBULENT PIPE FLOW!" $%& ( ) + $, 2&%. From: M. M. Denn, Process Fluid Mechanics
25 BASIS OF DIMENSIONLESS PARAMETERS If you have water flowing in a 1 cm pipe or a 2 cm pipe, we might expect that the physics of these two situations is exactly the same even if the numbers are different. Similarly, suppose we have the same pipe and we have a flow of water compared to water with a sugar solution. We might again expect the same physics with just a quantitative difference. This idea is substantially strengthened when we take the differential equations for fluid motion (derived from F = m a), solve them and then do some analysis to show exactly the simplifications that be obtained mathematically. If this done you get an exact result that f = 16/Re
26 REGIMES OF PIPE FLOW Laminar Straight streamlines Viscous (molecular) transmission of momentum/stress in radial direction Turbulent Irregularity of the flow causes increased pressure drop Convective transport of momentum/stress You don t get to choose. The system and conditions determine which will occur!
27 DYE STREAK EXPERIMENT
28 TYPICAL CALCULATION What is the pressure change over 30 m of pipe, if the flow rate is m 3 /s, the diameter is 0.03 m and liquid is water, μ = kg/(m-s), ρ = 1000 kg/m 3? About kg/(m-s 2 ) =~ 0.2 ATM or 3 PSI. This is probably acceptable. If we made the pipe just 20% smaller, the pressure change would be 63,000 kg/(m-s 2 ) and the power necessary to pump would increase from 20W to 60W!
29 NUMBER CHECK
30 HOW BIG SHOULD THE PUMP BE? Perhaps we are really asking the power that will be necessary to pump the liquid
31 BERNOULLI EQUATION To include gravity, friction from all of the straight sections of pipe and the pressure change associated with all of the fittings, the typical equation that is most convenient is the Engineering Bernoulli Equation This can be derived from the first and second laws of thermodynamics and the equation can be applied between any 2 continuous flow paths in a process system.
32 TYPICAL CALCULATION Now what is the pumping power? Suppose that the flow rate is still 1 l/s of water, the total length of the pipe is still 30 m, the rise is 10 m and there are 6 elbows and a tank inlet Note we would do the calculation from 1 to 2 and if so, Δp is 0! The result is that it would take about 125 W. About 100 W is to pump against gravity, 20W for the straight pipe and only about 5W because of the fittings.
33 NUMBER CHECK
34 FRICTION FACTOR REYNOLDS NUMBER CHART ROUGHNESS, εε/d, ADDS AN ADDITIONAL PARAMETER
35 PUMPING FLUIDS Blowers and compressors are used for gases For low and moderate viscosity liquids, low to very high volumetric flow rates Centrifugal pumps are most common. The Imperial College pilot plant is a special case they use mostly positive displacement pumps with direct motor drive. The other key word is positive displacement This means that specific volumes of fluid are forced through the pump/ compressor/blower (with no back flow) so that the flow rate must be controlled by the speed of the motor/driving coupling.
36 CENTRIFUGAL PUMP Not positive displacement. A valve at the outlet can control the flow rate. These can be throttled down quite a bit without causing any damage. Pump Curves
37 DIRECT DRIVE POSITIVE DISPLACEMENT PUMPS
38 BLOWER Gases are pumped using blowers and compressors. Imperial has a lobe blower for the N2 recycle. Blowers usually provide some modest increase in pressure, but handle large volumes of air efficiently.
39 Gas In Roto r Main Body Rotor Trapped Volume Gas Out
40 COMPRESSOR Compressors usually provide some ratio increase in pressure. Hence the power requirements are significant. Come in all sizes Reciprocating compressors are generally used for smaller scale flows (<50 HP). Imperial Compressor is a single-stage ( piston ) reciprocating compressor.
41 Pistons on top
42 PIPE FITTINGS
43 WELDED JOINTS/BOLTED FLANGES
44 PIPE THREADS
45 CHEMICAL COMPATIBILITY
46 PIPES Standard sizes of fabrication schedule relates to wall thickness Chose material for appropriate compatibility 304 Stainless is fine for MEA
47 SUMMARY Pipe sizing and piping system design is done based on the pressure drop flow rate behavior that occurs This is well known and for simple fluids is captured in the single plot of friction factor and Reynolds number To design an entire pipe system and to determine the power necessary for pumping, the Engineering Bernoulli Equation is used. The material selected must be compatible with the fluids being pumped, but cost, weight and strength are also factors that must be considered.
PIPE FLOW General Characteristic of Pipe Flow Figure 1 Some of the basic components of a typical pipe system are shown in Figure 1. They include the pipes, the various fitting used to connect the individual
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