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Chapter10.qxd 6/15/04 2:32 PM Page 299 Chapter 10 Flow, Pumps, and Piping Design The distribution of fluids by pipes, ducts, and conduits is essential to all heating and cooling systems. The fluids encountered are gases, vapors, liquids, and mixtures of liquid and vapor (two-phase flow). From the standpoint of overall design of the building system, water, vapor, and air are of greatest importance. This chapter deals with the fundamentals of incompressible flow of fluids such as air and water in conduits, considers the basics of centrifugal pumps, and develops simple design procedures for water and steam piping systems. Basic principles of the control of fluid-circulating systems—including variable flow, secondary pumping, and the relationship between thermal and hydraulic performance of the system—are covered. 10-1 FLUID FLOW BASICS The adiabatic, steady flow of a fluid in a pipe or conduit is governed by the first law of thermodynamics, which leads to the equation P1 V2 gz P V2 gz g l + 1 + 1 = 2 + 2 + 2 +w+ gc gc gc f ρ1 2 gc ρ2 2 gc (10-1a) where: P = static pressure, lbf/ft2 or N/m2 ρ = mass density at a cross section, lbm/ft3 or kg/m3 V = average velocity at a cross section, ft/sec or m/s g = local acceleration of gravity, ft/sec2 or m/s2 gc = constant = 32.17 (lbm-ft)/(lbf-sec2) = 1.0 (kg-m)/(N-s2) z = elevation, ft or m w = work, (ft-lbf)/lbm or J/kg lf = lost head, ft or m Each term of Eq. 10-1a has the units of energy per unit mass, or specific energy. The last term on the right in Eq. 10-1a is the internal conversion of energy due to friction. The first three terms on each side of the equality are the pressure energy, kinetic energy, and potential energy, respectively. A sign convention has been selected such that work done on the fluid is negative. Another governing relation for steady flow in a conduit is the conservation of mass. For one-dimensional flow along a single conduit the mass rate of flow at any two cross sections 1 and 2 is given by ṁ = ρ1V1 A1 = ρ2 V2 A2 (10-2) where: m = mass flow rate, lbm/sec or kg/s A = cross-sectional area normal to the flow, ft2 or m2 299 Chapter10.qxd 6/15/04 2:32 PM Page 300 300 Chapter 10 Flow, Pumps, and Piping Design When the fluid is incompressible, Eq. 10-2 becomes Q̇ = V1 A1 = V2 A2 (10-3) where: Q̇ = volume flow rate, ft 3 /sec or m 3 /s Equation 10-1a has other useful forms. If it is multiplied by the mass density, assumed constant, an equation is obtained where each term has the units of pressure: P1 + ρgl f ρ1V12 ρ1gz1 ρ V 2 ρ gz + = P2 + 2 2 + 2 2 + ρw + 2 gc 2 gc gc gc gc (10-1b) In this form the first three terms on each side of the equality are the static pressure, the velocity pressure, and the elevation pressure, respectively. The work term now has units of pressure, and the last term on the right is the pressure lost due to friction. Finally, if Eq. 10-1a is multiplied by gc/g, an equation results where each term has the units of length, commonly referred to as head: gc P1 V12 g P gw V2 + + z1 = c 2 + 2 + z2 + c + l f 2g 2g g ρ1 g ρ2 g (10-1c) The first three terms on each side of the equality are the static head, velocity head, and elevation head, respectively. The work term is now in terms of head, and the last term is the lost head due to friction. Equations 10-1a and 10-2 are complementary because they have the common variables of velocity and density. When Eq. 10-1a is multiplied by the mass flow rate m and solved for mw = W, another useful form of the energy equation results, assuming ρ = constant:  P − P2 V12 − V22 g( z1 − z2 ) g  W˙ = m˙  1 l  + + − 2 gc gc gc f   ρ (10-4) where: Ẇ = power (work per unit time), ft-lbf or W sec All terms on the right-hand side of the equality may be positive or negative except the lost energy, which must always be positive. Some of the terms in Eqs. 10-1a and 10-4 may be zero or negligibly small. When the fluid flowing is a liquid, such as water, the velocity terms are usually rather small and can be neglected. In the case of flowing gases, such as air, the potential energy terms are usually very small and can be neglected; however, the kinetic energy terms may be quite important. Obviously the work term will be zero when no pump, turbine, or fan is present. The total pressure, a very important concept, is the sum of the static pressure and the velocity pressure: ρV 2 P0 = P + (10-5a) 2 gc In terms of head, Eq. 10-5a is written gc P0 g P V2 = c + g g 2g (10-5b) Chapter10.qxd 6/15/04 2:32 PM Page 301 10-1 Fluid Flow Basics 301 Equations 10-1c and 10-4 may be written in terms of total head and with rearrangement of terms become gc P01 − P02 gw + ( z1 − z2 ) = c + l f ρ g g (10-1d) This form of the equation is much simpler to use with gases because the term z1 – z2 is negligible, and when no fan is in the system, the lost head equals the loss in total pressure head. Lost Head For incompressible flow in pipes and ducts the lost head is expressed as lf = f L V2 D 2g (10-6) where: f = Moody friction factor L = length of the pipe or duct, ft or m D = diameter of the pipe or duct, ft or m V = average velocity in the conduit, ft/sec or m/s g = acceleration due to gravity, ft/sec2 or m/s2 The lost head has the units of feet or meters of the fluid flowing. For conduits of noncircular cross section, the hydraulic diameter Dh is a useful concept: Dh = 4(cross-sectional area) wetted perimeter (10-7) Usefulness of the hydraulic diameter concept is restricted to turbulent flow and crosssectional geometries without extremely sharp corners. Figure 10-1 shows friction data correlated by Moody (1), which is commonly referred to as the Moody diagram. Table 10-1 gives some values of absolute roughness for common pipes and conduits. The relative roughness may be computed using diameter data such as that in Tables C-1 and C-2. The friction factor is a function of the Reynolds number (Re) and the relative roughness e/D of the conduit in the transition zone; is a function of only the Reynolds number for laminar flow; and is a function of only relative roughness in the complete turbulence zone. Note that for high Reynolds numbers and relative roughness the Table 10-1 Absolute Roughness Values for Some Pipe Materials Absolute Roughness e Type Commercial Steel Drawn Tubing or Plastic Cast Iron Galvanized Iron Concrete Feet mm 0.000150 0.000005 0.000850 0.000500 0.001000 0.4570 0.0015 0.2591 0.1524 0.3048 Chapter10.qxd 6/15/04 2:32 PM Page 302 302 Chapter 10 Flow, Pumps, and Piping Design 0.09 0.08 Critical zone Laminar Transition zone zone Complete turbulence, rough pipes 0.05 0.04 0.07 0.06 0.03 0.05 0.03 0.01 0.008 0.006 0.004 0.025 0.002 0.02 0.015 Sm 0.001 00008 0.0006 0.004 oo th pip Relative roughness e/D 2 If L D Friction factor f = 0.015 0.04 r 4/R =6 ( ) V2g 0.02 0.0002 es 0.0001 0.01 0.00005 0.009 0.008 0.00001 103 2 3 4 5 6 8 104 2 3 4 5 6 8 105 2 3 4 5 6 8 106 2 3 4 5 6 8107 DVp ρ e e Reynolds number Re = = 0.000001 = 0.000005 µ D D 2 3 4 5 6 8 108 Figure 10-1 Friction factors for pipe flow. friction factor becomes independent of the Reynolds number and can be read directly from Fig. 10-1. Also, in this regime the friction factor can be expressed by 1 = 1.14 + 2 log( D/e) f (10-8) Values of the friction factor in the region between smooth pipes and complete turbulence, rough pipes can be expressed by Colebrook’s natural roughness function   1 9.3 = 1.14 + 2 log( D/e) − 2 log 1 +  Re(e/ D) f  f  The Reynolds number is defined as Re = ρVD VD = µ v (10-9) (10-10) where: ρ = mass density of the flowing fluid, lbm/ft3 or kg/m3 µ = dynamic viscosity, lbm/(ft-sec) or (N-s)/m2
= kinematic viscosity, ft2/sec or m2/s The hydraulic diameter is used to calculate Re when the conduit is noncircular. Appendix A contains viscosity data for water, air, and refrigerants. The ASHRAE Handbook, Fundamentals Volume (2) has data on a wide variety of fluids. To prevent freezing it is often necessary to use a secondary coolant (brine solution), possibly a mixture of ethylene glycol and water. Figure 10-2 gives specific gravity and Chapter10.qxd 6/15/04 2:32 PM Page 303 10-1 Fluid Flow Basics 303 Figure 10-2a Specific gravity of aqueous ethylene glycol solutions. (Adapted by permission from ASHRAE Handbook, Fundamentals Volume, 1989.) Figure 10-2b Viscosity of aqueous ethylene glycol solutions. (Adapted by permission from ASHRAE Handbook, Fundamentals Volume, 1989.) Chapter10.qxd 6/15/04 2:32 PM Page 304 304 Chapter 10 Flow, Pumps, and Piping Design viscosity data for water and various solutions of ethylene glycol and water. Note that the viscosity is given in centipoise [1 lbm/(ft-sec) = 1490 centipoise and 103 centipoise = 1 (N-s)/m2]. The following example demonstrates calculation of lost head for pipe flow. EXAMPLE 10-1 Compare the lost head for water and a 30 percent ethylene glycol solution flowing at the rate of 110 gallons per minute (gpm) in a 3 in. standard (Schedule 40) commercial steel pipe 200 ft in length. The temperature of the water is 50 F. SOLUTION Equation 10-6 will be used. From Table C-1 the inside diameter of 3 in. nominal diameter Schedule 40 pipe is 3.068 in. and the inside cross-sectional area for flow is 0.0513 ft2. The Reynolds number is given by Eq. 10-10, and the average velocity in the pipe is 110 gal/ min Q˙ V = = = 287 ft / min = 4.78 ft /sec A ( 7.48 gal/ ft 3 )(0.0513 ft 2 ) The absolute viscosity of pure water at 50 F is 1.4 centipoise, or 9.4 × 10-4 lbm/ (ft-sec), from Fig. 10-2b. Then 62.4( 4.78) (3.068 /12) Re = = 8.1 × 10 4 9.4 × 10 −4 From Fig. 10-1 the absolute roughness e is 0.00015 for commercial steel pipe. The relative roughness is then e/D = 12(0.00015/3.068) = 0.00058 The flow is in the transition zone, and the friction factor f is 0.021 from Fig. 10-1. The lost head for pure water is then computed using Eq. 10-6: l fw = 0.021 × 200 ( 4.78)2 × = 5.83 ft of water 3.068 /12 2(32.2) The absolute viscosity of the 30 percent ethylene glycol solution is 3.1 centipoise from Fig. 10-2b, and its specific gravity is 1.042 from Fig. 10-2a. The Reynolds number for this case is 1.042(62.4) ( 4.78) (3.068 /12) Re = = 3.8 × 10 4 3.1/1490 and the friction factor is 0.024 from Fig. 10-1. Then l fe = 0.024 × 200 ( 4.78)2 × = 6.66 ft of E.G.S. 3.068 /12 2(32.2) = 6.94 ft of water The increase in lost head with the brine solution is Percent increase = 100(6.94 − 5.83) = 19 percent 5.83 Chapter10.qxd 6/15/04 2:32 PM Page 305 10-1 Fluid Flow Basics 305 System Characteristic The behavior of a piping system may be conveniently represented by plotting total head versus volume flow rate. Eq. 10-1d becomes Hp = gc ( P01 − P02 ) + ( z1 − z2 ) − l f gρ (10-1e) Total head Hp where Hp represents the total head required to produce the change in static, velocity, and elevation head and to offset the lost head. If a pump is present in the system, Hp is the total head it must produce for a given volume flow rate. Since the lost head and velocity head are proportional to the square of the velocity, the plot of total head versus flow rate is approximately parabolic, as shown in Fig. 10-3. Note that the elevation head is the same regardless of the flow rate. System characteristics are useful in analyzing complex circuits such as the parallel arrangement of Fig. 10-4. Circuits 1a2 and 1b2 each have a characteristic as shown in Fig. 10-5. The total flow rate is equal to the sum of Qa and Qb and the total head is the same for both circuits; therefore, the characteristics are summed for various values of Hp to obtain the curve for the complete system, shown as a + b. Series circuits have a common flow rate and the total heads are additive (Fig. 10-6). More discussion of system characteristics will follow the introduction of pumps in Section 10-2. z2 – z1 • Volume flow rate Q Figure 10-3 Typical system characteristic. a 1 2 b Figure 10-4 Arbitrary parallel flow circuit. a+b a Total head Hp b • • • Qb Qa Q ab • Volume flow rate Q Figure 10-5 System characteristic for parallel circuits. Chapter10.qxd 6/15/04 2:32 PM Page 306 306 Chapter 10 Flow, Pumps, and Piping Design Total head Hp a+b Hab b Hb a Ha • Volume flow rate Q Figure 10-6 System characteristic for series circuits. Flow Measurement Provisions for the measurement of flow rate in piping and duct systems are usually required or indications of flow rate or velocity may be needed for control purposes. Common devices for making these measurements are the pitot tube and the orifice, or venturi meter. The pitot tube and the orifice meter will be discussed here. Figure 10-7 shows a pitot tube installed in a duct. The pitot tube senses both total and static pressure. The difference, the velocity pressure, is measured with a manometer or sensed electronically. The pitot tube is very small relative to the duct size so traverses usually must be made when measuring flow rate. When Eq. 10-1a is applied to a streamline between the tip of the pitot tube and a point a short distance upstream, the following equation results (the head loss is assumed to be negligibly small, and the mass density constant): P P1 V12 P + = 2 = 02 ρ 2 gc ρ ρ (10-11a) P02 − P1 V2 = 1 = Pv ρ 2 gc (10-11b) or Static tube Pυu Static and velocity tube Manometer or draft gage Static pressure holes Figure 10-7 Pitot tube in a duct. Duct Chapter10.qxd 6/15/04 2:32 PM Page 307 10-1 Fluid Flow Basics 307 Solving for V1, P − P1   V1 =  2 gc 02  ρ   1/ 2 (10-12) Equation 10-12 yields the velocity upstream of the pitot tube. It is generally necessary to traverse the pipe or duct and to integrate either graphically or numerically to find the average velocity in the duct (2). Equations 10-2 and 10-3 are then used to find the mass or volume flow rate. When the pitot tube is used to measure velocity for control purposes, a centerline value is sufficient. EXAMPLE 10-2 A pitot tube is installed in an air duct on the center line. The velocity pressure as indicated by an inclined gage is 0.32 in. of water, the air temperature is 60 F, and barometric pressure is 29.92 in. of mercury. Assuming that fully developed turbulent flow exists where the average velocity is approximately 82 percent of the center-line value, compute the volume and mass flow rates for a 10 in. diameter duct. SOLUTION The mass and volume flow rates are obtained from the average velocity, using Eqs. 10-2 and 10-3. The average velocity is fixed by the center-line velocity in this case, which is computed by using Eq. 10-12. Since the fluid flowing is air, the density term in Eq. 10-12 is that for air, ρa. The pressure difference P02 – P1 is the measured pressure indicated by the inclined gage as 0.32 in. of water (y). The pressure equivalent of this column of water is given by g ρw gc 0.32  P02 − P1 =  ft  12  lbf = 1.664 2 ft P02 − P1 = y  32.2  lbf (62.4) lbmw  32.2  lbmw ft 3 To get the density of the air we assume an ideal gas: ρa = Pa (29.92) (0.491) (144) lbma = = 0.076 3 Ra Ta (53.35) (60 + 460) ft which neglects the slight pressurization of the air in the duct. The center-line velocity is given by Eq. 10-12, (2) (32.2) (1.644) 1/ 2 Vcl =  = 37.6 ft /sec  0.076  and the average velocity is V = 0.82Vcl = (0.82) (37.6) = 30.8 ft /sec The mass flow rate is given by Eq. 10-2 with the area given by Chapter10.qxd 6/15/04 2:32 PM Page 308 308 Chapter 10 Flow, Pumps, and Piping Design π  10  2 = 0.545 ft 2 4  12  m˙ = ρ a VA = 0.076 (30.8) 0.545 = 1.28 lbm/sec A= The volume flow rate is Q˙ = VA = 30.8 (0.545) 60 = 1007 ft 3 / min using Eq. 10-3. Flow-measuring devices of the restrictive type use the pressure drop across an orifice, nozzle, or venturi to predict flow rate. The square-edged orifice is widely used because of its simplicity. Figure 10-8 shows such a meter with the location of the pressure taps (3, 4). The flange-type pressure taps are widely used in HVAC piping systems and are standard fittings available commercially. The orifice plate may be fabricated locally or may be purchased. The American Society of Mechanical Engineers outlines the manufacturing procedure in detail (3). The orifice meter is far from being an ideal flow device and introduces an appreciable loss in total pressure. An empirical discharge coefficient is Q˙ C = ˙actual (10-13) Q ideal The ideal flow rate may be derived from Eq. 10-1a with the lost energy equal to zero. Applying Eq. 10-1a between the cross sections defined by the pressure taps gives P1 V12 P V2 + = 2 + 2 ρ 2 gc ρ 2 gc (10-14) To eliminate the velocity V1 from Eq. 10-14, Eq. 10-3 is recalled and V1 = V2 A2 A1 Vena contracta connections: P2 at vena contracta 1D, and 1 D, connections: P2 at D1/2 2 D1 Inlet pressure P1 connection (10-13a) Outlet pressure connection P2 t 5 or 6D1 D1 D2 Orifice 1 in. 1 in. Flange connections Figure 10-8 Recommended location of pressure taps for use with thin-plate and square-edged orifices according to the American Society of Mechanical Engineers (4).