F L U I D S Y S T E M D Y N A M I C S

Transcription

1 F L U I D S Y S T E M D Y N A M I C S T he proper design, construction, operation, and maintenance of fluid systems requires understanding of the principles which govern them. These principles include thermodynamics, fluid statics and dynamics, heat and mass transfer, chemistry, and the like. Review of Fluid Flow Fundamentals The following sections discuss some of the general principles governing fluid flow in conduits. Our goal is to develop a mathematical model to assist in quantitatively analyzing a fluid system. The model should represent any variables (such as position, velocity, acceleration, pressure, temperature, density, enthalpy and entropy of the fluid) that change during the flow. The changes predicted by the model must be consistent with the laws that govern fluid flow. Although general flow conditions can be very complex, some simplifying assumptions are made for systems involving closed conduits (ducts or pipes). The most typical and useful assumptions are as follows: The flow is one-dimensional; changes in variables perpendicular to the flow are accounted for by using average values. The flow is steady; time variations in properties are small and centered around a constant value for each property. The fluid is essentially incompressible; (a reasonable assumption for liquids and considered true for gases if the fluid density does not change throughout the system by more than 7%). Effects due to surface tension and Coriolis forces are ignored. Bernoulli Equation The Bernoulli equation defines and relates a number of parameters within a stream line of a fluid system. One simplified form of the equation states that at any point in the system, the amount of energy in one unit of an incompressible fluid consists of the stored energy plus the kinetic energy, if we disregard any work or heat added to or removed from the system. The stored (or "potential") energy includes the flow work and an elevation term above some arbitrary datum. Thus, at a given point in the system: Compression term + kinetic energy term + elevation term = total energy (.1) In a fluid system with no losses, the sum of these three terms is a constant throughout the system. This simplified form of the equation can be written in any one of three versions, as follows: Energy per unit mass of fluid (Specific energy version) " " " weight ( "Head" version) " " " volume ("Pressure" version)

2 Most practicing engineers in this country use the so-called two-pound system, which incorporates both the pound mass and the pound force. In this system, g c has the numerical value of 3.17 and the units value shown below, so F = ma/g c Table.1 List of Variables, Symbols, and Units VARIABLE SYMBOL UNITS Density ρ (rho) lbm/ft 3 Elevation Z ft Force f lbf Gravitational constant g c lbm-ft/lbf-sec Local grav. acceleration g ft/sec Mass m lbm Pressure P lbf/ft Specific weight γ (gamma) lbf/ft 3 Velocity V ft/sec Weight w lbw (lbf) This table includes specific weight. The relationship between specific weight (γ) and density (ρ), for this set of units, (in an easy-to-remember form), is: γ ρ = g gc The net units of each side are lbf/lbm (.) Using the symbols from Table.1, the specific energy version of the Bernoulli equation (ignoring work and heat in or out) becomes: P ρ V g + + Z g g c c = constant Each term is in ft-lbf per lbm (.3) The form of the equation depends upon the set of units chosen for the analysis. Most texts on fluid dynamics use the pound as the force unit and the slug as the mass unit. 1 Similar equations can be developed for the head version (ft-lbf per lbw, or ft), and the pressure version (ft-lbf per ft 3, or lbf per ft ). The head version is commonly used in liquid (sometimes called hydraulic) systems. It requires that each term be in feet of the fluid that is flowing. Air systems involving fans (as distinguished from compressed air systems) usually use the pressure version of the equation, because feet of air are difficult to measure and involve large numbers. In the pressure version, each term must be in lbf/ft but the convenience of the water manometer as a measuring device has led to the convention of expressing the pressure in equivalent inches of water column (sometimes called inches water gage, abbreviated WG, or H O). This is discussed in detail in Handout #4. 1 In the lbf-slug system, the use of g c is avoided, because density is in slugs/ft 3 and one slug = 1 lbf-sec /ft.

3 In reality, there are always losses in a system, so the Bernoulli equation is usually written 1, (in this example, in the "head" version) P 1 γ + V 1 + Z 1 = g P γ V + + Z g + h L (.4) where subscripts 1 and are two points in the system, 1 being upstream of, and h l represents the loss of head. This can be rewritten to solve for h L as follows: Flow Regimes P P V V γ g ( ) + Z Z = h L (.4.1) The loss of energy in the direction of flow depends on the flow regime in the fluid. The flow regime determines the velocity gradient at any given position x,y along the flow path, but the shape of the conduit (i.e., flat plate, parallel plates, circular pipe, rectangular duct, etc.) determines the shape of the velocity profile. The two regimes are LAMINAR and TURBULENT with a somewhat indeterminate transitional zone between them, called the CRITICAL ZONE. The value of the local Reynolds Number (Re) and the nature of the conduit determine the flow regime at any point in the system. For circular conduit, if Re < 300, laminar flow exists. If Re > 6000, turbulent flow exists. In between, the flow is critical and exact analysis is difficult. Reynolds Number is the ratio of the inertial forces in the fluid to the viscous damping forces, so a low value of Re yields smooth orderly (laminar) flow, wherein definite stream lines are maintained. Large Re causes chaotic (turbulent) flow with appreciable velocity variations within the average velocity in the direction of flow. Since the local Re across a section varies from point to point as velocity varies, it is convenient to use a Re based on the mean velocity at the section. A velocity gradient occurs in the fluid in contact with the boundary surface, the BOUNDARY LAYER. Its thickness increases in the direction of flow. If the boundaries form a closed uniform conduit of sufficient length, flow becomes "established," or "fully-developed," meaning the velocity profile will be the same in all succeeding sections, provided that the Reynolds number remains constant. This is usually the case for piping systems and ducted systems. Algebraically, for closed conduit of circular cross-section: or Re = VDρ µ Re = DV ν µ Is absolute viscosity [.3] (.5) ν (Kinematic viscosity) is defined as µ/ρ (.6) One must select units of viscosity such that the Re number is dimensionless. Surface Roughness 1 The kinetic energy factor, α, which belongs with the velocity head term, is nearly 1.0, and is frequently omitted, as shown in this section. Equation numbers in brackets appear in the course text, Ventilation for Control of the Work Environment.

4 In turbulent flow, the conduit affects friction losses because of the relationship between the fluid boundary and the surface of the conduit. Every conduit, no matter how smooth, has "peaks and valleys' in the surface. Thus there is an ABSOLUTE ROUGHNESS, ε, defined as the average distance between the peaks and the valleys. Table. shows typical values for ε for some commonly used conduits in fluid systems. Table. Values for Absolute Roughness, ε Absolute Roughness Category Examples Value of ε, in feet Average Range 1. Very smooth Drawn tubing; glass pipe 5.0 E Smooth Cooper or plastic tubing.5 E Fairly smooth New steel pipe; aluminum duct 1.5 E Average Galvanized iron duct(*) 5.0 E Fairly rough New cast iron pipe; spiral galv. duct 6.0 E Rough Old cast iron pipe; 1.5 E lined duct (glass fiber insulated) 7. Very rough Concrete pipe or duct; 7.0 E flexible duct improperly supported (*)based on forty transverse joints per 100 feet for ducts, and screwed or welded fittings for pipe. The effect caused by the conduit roughness depends on whether the fluid boundary layer is "deeper" than the peaks; i.e., whether or not the peaks protrude up through the boundary layer. In laminar flow, the boundary layer extends completely across the diameter of the conduit so surface roughness is not a factor. In turbulent flow, the boundary layer is relatively thin, and may or may not submerge the peaks. If the boundary layer is about four times or more as thick as the height of the peaks, the system is called "hydraulically smooth." If the boundary layer is less thick than that, the system is "hydraulically rough." Most piping and duct systems fall in the latter category. Relative Roughness Obviously, because a given value of absolute roughness, ε, will have a greater effect on friction in a small diameter conduit than in a large one, (in turbulent flow), a more meaningful parameter called relative roughness, ε/d, is used. Relative roughness is defined as the absolute roughness divided by the conduit diameter, and is a unitless number. ENERGY LOSSES In general, two types of energy loss occur in fluid systems. One type of loss is caused by the shear stresses set up within the boundary layer of the fluid and is commonly called friction loss. The other type of loss, called dynamic loss, occurs because of separation of the fluid from the boundary layer, or by acceleration or deceleration, all of which occur because of changes in direction, expansion or contraction, and converging or diverging fluid streams. We will consider first the frictional losses.

5 Friction Losses "The flow of any real (Newtonian) fluid is resisted by friction forces which arise because of velocity gradients." (FAN ENGINEERING, 8th edn., Buffalo Forge Co, R. Jorgensen, editor.) Many equations have been proposed to express the relationship between velocity, diameter, surface conditions, length, fluid properties, and the loss of energy due to friction. One, which is commonly used for flow of water in pipes, is the Darcy-Weisbach equation: L Lost head = f D V g where L is length of conduit, D is diameter, and f is a unitless friction factor (.7) In laminar flow, the surface roughness has no effect on frictional losses. It has been experimentally shown that for laminar flow, f = 64 Re [.8] (.8) This equation does not include any roughness factor terms. Also notice that the friction factor is greatest when the Re number is the smallest. For hydraulically smooth pipe, turbulent flow, f = Re. 5 (.9) This equation resembles the laminar flow version, in that surface roughness is not a factor. For hydraulically rough conduit, when the Re number exceeds about 4000, values of f vary with roughness as well as density and viscosity. The Colebrook equation (see any standard fluid mechanics text) indicates that both the relative roughness and Re number contribute to the value of f. In the fully-developed turbulent region, where Re exceeds 6000, the value of f depends strictly on the relative roughness, and is not influenced by the fluid's properties, as follows: f = 1 37 log. ε D (.10) Values of f have been presented graphically as the Moody Chart, (next page) which plots f vs. Re. Equations.8 and.9 plot as straight lines sloping upper left to lower right. The Colebrook equation covers the curved portion of the rest of the lines on the chart. Equation.10 plots the straight horizontal portion of the lines for various relative roughnesses. More recently, an equation by Churchill describes the entire chart, but it is an equation best suited for computer solution.

6 Duct Material Uncoated carbon steel, clean (Moody 1944) ( ft) PVC placstic pipe (Swim 198) ( ft) Aluminum (Hutchinson 1953) ( ft) Galvanized steel, longitudinal seams, 4 ft joints (Griggs 1987) ( ft) Galvanized steel, spiral seam with 1,, and 3 ribs, 1 joints (Jones 1979, Griggs 1987) ( ft) Galvanized steel, longitudinal seams,.5 ft joints (Wright 1945) ( ft) Fibrous glass duct, rigid Fibrous glass duct liner, air side with facing material (Swim 1978) (0.005 ft) Fibrous glass duct lines, air side spray coated (Swim 1978) (0.015 ft) Flexible duct, metalic, ( ft when fully extended) Flexible duct, all types of fabric and wire ( ft when fully extended) Concrete (Moody 1944) ( ft) Roughness Catagory Absolute Roughness Smooth Medium Smooth Average Medium Rough Rough 0.01

7

8 The Darcy-Weisbach equation somewhat simplifies a very complex problem, but is reasonably accurate for fully-developed wholly-rough turbulent flow. For air systems, a more exact (but still a compromise) version is as follows: P L = L C V D n 1.4 n 1 n ρ µ g c n (.11) The left side of the equation is in terms of pressure rather than head, but C can take care of that. The values for C and the exponent, n, are dependent on absolute roughness, and are available from table similar th one below. Condition Typical Surface Average Range n C Very Smooth Drawn Tubing ft Medium Smooth Aluminum Duct ft Average Galvanized iron duct.0005 ft Medium Rough Concrete pipe.003 ft Very Rough Riveted steel pipe.01 ft The rougher the conduit, the more nearly the pressure drop varies as the square of the velocity, the first power of the density, the zero power of viscosity, and inversely as the 1.4 power of the diameter. In common practice charts and tables for standard conditions are used, and corrections for roughness and density are made as needed. Viscosity effects are usually ignored. Whereas frictional losses in turbulent flow are dependent on several variables, as previously shown, friction losses in laminar flow are: V = 64 g L PL µ c D (.1) Notice that roughness and density are not factors in loss, but velocity, viscosity, and diameter have different influences than in turbulent flow. Virtually all-fluid systems operate in the turbulent range, for economic reasons, and many are hydraulically fully rough. There can be components (some filters, for example) through which the flow is laminar, but turbulent flow predominates in almost all systems. Equation (.7) for liquid systems, and its counterpart for air systems, equation (.11), can be rewritten in terms of the friction factor, f, as follows: f = hl h ( L ) D v, Where h L is head loss, and h v is v /g, (velocity head), (.13) This version says that "the Darcy friction factor is a dimensionless group which relates two other dimensionless groups: the ratio of the loss of total head between two points to the velocity head, and the ratio of the distance between those two points and some characteristic dimension which determines velocity." (From FAN ENGINEERING, 8th edition.) Rewriting (.13) suggests two formats for expressing losses, as follows: f L D hl = = KL, Where K L is a dimensionless loss coefficient (.14) h V

9 This says that losses may be determined by either: or h L h = f L V D g L = K L or h L= V g ( TheDarcy Weisbach Equation) (.7) K L ( h ) ( Used for dynamic losses) (.15) V Frictional losses in straight pipe (generated in the boundary layer by shear forces) can be determined, as shown above, by calculation or the use of various charts. Charts that graph equation (.7) for various fluids and conduit materials are readily available and are commonly used, with corrections applied as necessary. These charts are "normalized" by plotting pressure (or head) loss per 100 feet of conduit vs. flow rate (rather than velocity) and are plotted on loglog coordinates to produce families of straight lines. One family of lines shows various conduit sizes, and another family shows lines of constant velocity. The next two pages show examples of typical charts for piping and duct systems. Note carefully the standardization and assumptions, which are implicit in each chart. For example, a standard density is specified, conduit materials and joining methods are mentioned, and on the chart for air, the exponents from equation (.11) are shown. An interesting situation arises when a given flow rate at a desired pressure drop per 100 is specified, and the intersection of those two coordinate lines falls between two available sizes of conduit. The issue of negotiability (see H0-1) arises. Do I change one of the parameters so as to intersect on an available size, or do I request the vendor to produce a size to meet my original conditions? In other words, what is negotiable in this situation? DYNAMIC LOSSES Dynamic losses occur in fittings because of fluid separation from the conduit wall, which causes resultant eddy currents. Whereas there is essentially one geometric possibility for straight pipe (with various L/D ratios), there are hundreds of shapes of fittings, such as: entrances (hoods, bellmouths, duct or pipe ends, inlet louvers) direction changes (elbows, sweeps) contractions, expansions (where conduit size changes) convergent or divergent fittings (tees and Y s, crosses) obstructions (filters, strainers, heat exchangers, valves, turbines) some types of exit (stacks with caps, screened exits, nozzles, outlet louvers Loss Coefficients Equation (.15) indicates that dynamic losses can be expressed as a function of the energy represented by the velocity of the fluid. The functional constant, K L, is a unique value for each type of fitting. Dynamic losses for the types of fitting listed above have been determined experimentally for a wide range of geometry's and sizes. They are published in many technical books and manufacturers catalogs.

10

11

12 Information relating to liquid (especially water) systems usually uses the symbol K L (or K) and requires that the terms be in feet of head of the fluid that is flowing. Data for air systems use the symbol c, and calculate loss of pressure. This will be discussed in more detail later. Important! Although frictional and dynamic losses are proportional to kinetic energy in the fluid, (V /g), velocity NEVER decreases because of losses. Velocity behaves in strict accordance with the continuity equation: Q = AV (.16) If flow rate, Q, and area, A, are fixed in a section of conduit, average velocity, V, is fixed, regardless of the losses in that section. Equivalent Length Another method for determining dynamic losses is to express the loss through a fitting in equivalent length. In other words, a fitting such as an elbow of a given size, material, and geometry will have the same resistance to flow as X feet of straight conduit of the same size. Adding the equivalent length of the fitting to the actual length of the conduit allows use of the Darcy-Weisbach, or similar, equation to calculate all of the losses in one step This method, commonly used for certain types of systems, such as plumbing systems, has some disadvantages: the available list of fitting types, sizes, and geometries is somewhat limited one must know the size of the fitting to determine its equivalent length (EL) Frequently the system losses must be determined before size is established. The EL must be estimated before the system is sized, then recalculated after sizing, then resized, recalculated, etc. Two or three iterations are usually sufficient, but the nuisance is obvious. Some sources recommend an initial estimate of 50% EL be added to the actual system length, but experienced designers can usually make an estimate which limits the iterations necessary to solve the problem. The wide availability of published loss coefficients favors the use of that method over the equivalent length method.