1、 2016 ASHRAE 225ABSTRACTThispaperdetailsexperimentalmeasurementsandmath-ematicalmodelingofairflowthroughaperforatedductwithanopenareaof22%thatiscappedattheend.Measurementswereconducted on ducts with uniform diameters of 12, 10, and 8 in.(0.30,0.25,and0.20m).Allductswere20ft(6.10m)long,andinlet flow
2、rates ranged from approximately 350 to 700 cfm(165 to 330 L/s). Flow rates were measured along the lengthof the duct using the pitot traverse method. The static pressurewas also measured. The flow through the duct was modeledassuming one-dimensional flow, and a differential equationwasderivedusingth
3、emass,momentum,andenergyequations.Theresultingdifferentialequationwassolvednumericallyandthe results were compared to the experimental measurements.Good agreement was achieved when comparing the experi-mental and model flow rates for all test runs with a maximumdifferenceof14.0%andanaveragedifferenc
4、eof2.0%.Resultsfor the static pressure showed the same trends between theexperiments and the model. The pressure was largest at thecapped end of the duct, where the experimental measurementsexceeded the model results by a maximum of 21.8%.INTRODUCTIONA review of the literature shows that many of the
5、 appli-cations for perforated air ducts are for agricultural applica-tions, such as grain aeration and grain drying. Perforated airsupply ducts are also of interest in comfort air-conditioningapplications because they can deliver a large volume of air atrelatively low velocity and therefore low nois
6、e and turbulencelevels.Unfortunately,thereisalackofresourcesforpracticingengineers to design perforated duct systems. This paper pres-ents a simplified mathematical model that can be used tocalculate airflow rates through perforated ducts. Comparisonto experimental results is included for validation
7、 purposes.The word duct is the preferred term used in this paper todescribe the conduit through which fluid flows, but manifoldand tube are other terms that are found in the literature.Shove(1959)studiedairflowthroughperforatedductsforagricultural applications. He provided a comprehensivereviewofpre
8、viousworkonthesubjectanddevelopedadiffer-ential equation based on the momentum equation in the axialdirection, which could be solved to determine the static pres-sureandairflowratealongtheduct.Hesolvedtheequationfora finite length along the duct and compared the solution toexperimentalmeasurementsof
9、airflowthrougha5in.(0.13m)diameter duct with 36% open area. Shove (1959) investigatedbothdivergingandconvergingflows.Attemptstomeasurethedischarge angle for flow to and from the duct through theperforations were unsuccessful.SteeleandShove(1969)presentedchartsforthedesignofperforated duct systems fo
10、r both diverging and convergingflows. The charts looked at two separate cases: 1) uniformintake or discharge and 2) uniform openings. For case 2, thegoverning differential equation was solved numerically anddesignchartswerepresentedtocalculateparametersalongthelength of the duct: static pressure, ou
11、tflow and inflow, and thetotal volumetric flow rate through the duct. The differentialequations were based on the work presented by Shove (1959).El Moueddeb et al. (1997a) developed a model forairflowthroughperforatedducts.Theysolvedthefundamen-talequationsofmass,momentum,andenergyassumingone-dimens
12、ional flow. They followed this paper with a secondpublication(ElMoueddebetal.1997b),whichwasanexper-imental study to validate the model. The experimental appa-Modeling Airflow through a Perforated DuctJesse Maddren, PhD, PE John FarrellMember ASHRAEAlan Fields Cesar JarquinJesse Maddren is a profess
13、or in the Mechanical Engineering Department at California Polytechnic State University, San Luis Obispo, CA.John Farrell is an associate engineer at MHC Engineers, Inc., San Francisco, CA. Alan Fields is a design and engineering AHJ and utilityspecialist at Sungevity, Oakland, CA. Cesar Jarquin is a
14、 manufacturing engineer at Glenair, Inc., Glendale, CA.ST-16-023Published in ASHRAE Transactions, Volume 122, Part 2 226 ASHRAE Transactionsratus was a rectangular duct constructed from plywoodmeasuring 23.5 11.5 in. (0.60 0.29 m). The duct hadreplaceable sides so the open area could be varied from
15、0.5%to 2.0%. Due to the low open area, the governing equationswere solved for discrete locations upstream and downstreamofindividualopenings.Theyfoundgoodagreementbetweentheir model and experimental results for the static pressureand discharge angle along the length of the duct.More recently, Chen a
16、nd Sparrow (2009) calculated fluidflow through a perforated manifold using three-dimensionalcomputational fluid dynamics (CFD). The paper focused onthe geometry of the manifold openings and its effect on flowuniformity and exit angle. Lee et al. (2012) also used a CFDmodeltocalculatetheflowofwaterth
17、roughaperforatedtube;these results were compared to experimental measurements.Their investigation considered rectangular tubes with rectan-gular perforations and 2.1% to 10.7% open area. The flowuniformity and discharge angle were studied as a function ofthe open area, spacing of the openings, and t
18、ube wall thick-ness.MATHEMATICAL MODELAirflow within a perforated duct is modeled assumingone-dimensional flow in the axial direction. The duct isassumed to have a round cross section and the flow is axisym-metric. The differential control volume of width dx in the flowdirection is shown in Figure 1
19、. The outflow through the perfo-rationshasbothaxialandradialcomponents,anditisassumedthat the outflow can be characterized by a single velocityvectoratananglearelativetotheductwall.Duetotheoutflow,the velocity at the outlet Vx+dxis less than the velocity at theinlet Vx. Friction at the duct walls ex
20、erts a differential force,F, opposite the flow direction as shown in Figure 1.Conservation of MassThe continuity equation (Munson et al. 2013) is(1)and the flow is assumed to be steady and incompressible.There are three control surfaces at the left-hand side (x), theright-hand side (x+dx), and throu
21、gh the perforations (o), asshown in Figure 1. The duct is assumed to have a constantcross-sectional area, A, and the velocity is assumed to beuniform at the control surfaces. Therefore, the continuityequation yields(2)The velocity at the right-hand side of the control volumecan be expressed as(3)and
22、 the continuity equation reduces to(4)Linear Momentum EquationThe linear momentum equation for a non-acceleratingcontrol volume (Munson et al. 2013) is(5)Neglecting body forces, the only other forces acting onthe control volume are the forces due to the pressure p at x andx+dx and the force due to f
23、riction as shown in Figure 1. Thereare no pressure forces at the perforated duct surface since thesurroundings are assumed to be at atmospheric pressure.Applyingthemomentumequationinthex-directionatsteady-state and assuming the velocity is uniform at all controlsurfaces yields(6)Due to friction at t
24、he surfaces of the duct, the velocity isnot uniform at each control surface. Assuming turbulent flow,a correction factor for the momentum equation can be deter-minedsimilarlytothekineticenergycoefficientfortheenergyequation. This correction factor was evaluated for turbulentflow using a power-law ap
25、proximation to the velocity profilewith n = 7 (Re 70,000), and the correction was determinedto be less than 3%. This was deemed small enough to neglect.It is customary to express the friction force in terms of afriction factor f,or(7)Figure 1 Differential control volume for mathematicalmodel.t- dCVV
26、CSAd+0=VxA Vxdx+A Vosin dAo+ 0=Vxdx+VxdVxdx- dx+=AdVxdx- dx Vosin dAo+0=Ft- VdCVVVCSAd+=pxApxdx+A FVx2A Vxdx+2A Vo2 dAocossin+=F fdxD-Vx22-A=Published in ASHRAE Transactions, Volume 122, Part 2 ASHRAE Transactions 227Also, the changes in pressure and velocity in Equation 6can be expressed similarly
27、to that for the velocity inEquation 3. Substituting the result from the continuity equa-tion (Equation 4) and simplifying yields(8)where the x subscript has been dropped for convenience.Steele and Shove (1969) defined a constant(9)and experimentally measured K to equal 1.5 for divergingflows and 1.7
28、 for converging flows. El Moueddeb et al.(1997b) determined the constant to be 1.0 from the energyequation and reported good agreement with this result andexperiments. This is discussed later in this paper when weconsider the energy equation for the control volume.The average velocity can be defined
29、 in terms of the volu-metric flow rate, V = Q/A, and the differential outflow throughthe perforations can be expressed in terms of the equation forflow through a single orifice within a larger concentric tube(Munson et al. 2013):(10)where Kpis the orifice flow coefficient and pais the atmo-spheric p
30、ressure outside the duct. The differential outflow canbe expressed in terms of the variation of the flow through theduct using continuity, and Equation 10 can be solved for thepressure difference(11)where is the fraction of open area due to the perforations.Taking the derivative of Equation 11 yield
31、s(12)Substituting Equation 12 into Equation 8 gives(13)which is a second-order, nonlinear differential equation forQ(x). Steele and Shove (1969) continued by letting(14)and(15)which then yields the following for the momentum equation:(16)This equation can be solved numerically; the method isoutlined
32、 in the Numerical Solution section of this paper.Energy EquationThe energy equation for the differential control volume(Munson et al. 2013) can be expressed as(17)Assuming steady flow with no work and negligiblechanges in potential energy, the energy equation becomes(18)Thechangesininternalenergy,pr
33、essure,andvelocitycanbe expressed similarly to that for the velocity in Equation 3.Additionally, substituting the result from continuity(Equation 4) gives(19)where the x subscript has been dropped for convenience. Thesecond term on the right-hand side of Equation 19 goes tozero. The term in brackets
34、 is the change in energy per unitmass of the flow from the duct through the perforations, andsincethereisnoheattransferorwork,theenergyperunitmassis constant. Also, the mass flow rate is(20)and therefore the energy equation can be expressed as(21)1-dpdx-fD-V22-dVdx- 2 VVocos=K2VVocosV-=dQoKp2 ppa- d
35、Ao=p pa222D2Kp2-dQdx-2=dpdx-22D2Kp2-dQdx-d2Qdx2-=122D2Kp2-dQdx-d2Qdx2-f2D-Q2A2- KQA2-dQdx-=x1f2KD- Lx=Cf2642Kp2K3-=CdQdx1-d2Qdx12-dQdx1- QQ2+=QWt- edCVup-V22- gz+ +VCSAd+=QVxAup-V22-+xVxdx+Aup-V22-+xdx+VodAoup-V22-+osin+=QVAdudx-dx1-dpdx-dx VdVdx- dx+AdVdx- dx u uo1- ppoV2Vo22-+=mVA=Qm- du1-dpdx-dx
36、VdVdx- dx+=Published in ASHRAE Transactions, Volume 122, Part 2 228 ASHRAE TransactionsTheleft-handsideofEquation21isthelossinmechanicalenergyduetofriction,expressedintermsoftheenergyperunitmass, or(22)Combining Equations 8, 21, and 22, yields(23)and therefore K = 1.0 (see Equation 9). This is the s
37、ame resultfrom El Moueddeb et al. (1997b) and K will be assumed equalto 1.0 for all future calculations.EXPERIMENTAL METHODSA schematic of the experimental apparatus is shown inFigure2.Thefanwasadirect-drivecentrifugalinlinefan.Theoutlet of the fan was connected toa5ft(1.52 m) long, 12 13 in. (0.30
38、0.33 m) rectangular duct that included a flowstraightenerandarectangular-to-roundtransitiontotheperfo-rated duct. The perforated duct was 20 ft (6.10 m) long andthree different duct diameters were tested: 12, 10, and 8 in.(0.30, 0.25, and 0.20 m). The perforated duct was constructedof 22 gage galvan
39、ized sheet metal with spiral seams. Theperforationswerestaggeredholes0.093in.(2.4mm)diameterwith a spacing of 0.188 in. (0.48 mm), resulting in 22% openarea. The end of the duct was capped with no flow exiting.The volumetric airflow rate was measured at locationsalong the length of the duct using a
40、pitot traverse. The diameterof the pitot tube was 0.125 in. (3.2 mm) and it was inserted into0.20in.(5.1mm)diameterholesdrilledintotheperforatedduct.The log-linear method (ASHRAE 2013b) was used with sixmeasuring points per diameter along two perpendicular diame-ters. The uncertainty of the velocity
41、 measurement was 3% ofreading 7 ft (2.1 m) per minute. The flow rate was measuredalong the length of the duct in 2 ft (0.61 m) increments, exceptnear the end of the duct where it was measured in 1 ft (0.30 m)increments due to the greater outflow through the perforations.Thestaticpressurewasmeasuredr
42、elativetothesurround-ing atmospheric pressure. The measurement range of thedevice was from 0.0001 to 60.00 in. H2O (0.025 to 15000 Pa)and the uncertainty was 2% of reading 0.001 in. H2O(0.25 Pa) from 0.05 to 50.0 in. H2O (12 to 12000 Pa). Most ofthe static pressure measurements were below 0.0001 in.
43、 H2O(0.025 Pa) and it was only near the capped end of the ductwhen the static pressure was high enough to yield an accuratereading.The temperature and barometric pressure were measuredprior to each test. The barometric pressure was constant for alltests at 29.7 in. Hg (101 kPa). The temperature rang
44、ed from69.5Fto73.0F(20.8Cto22.8C).Testswereconductedforthree different flow rates for each duct diameter for a total ofnine tests. The static pressure was first measured along thelength of the duct using the static pressure port of the pitotstatic tube. The flow rate was then measured along the leng
45、thof the duct using the pitot traverse method. A sample of thedata for the 12 in. (0.30 m) diameter duct is shown in Figure 3.Thefigureshowstheflowrateasafunctionofthelengthalongtheduct,x,wherex=0ft(0m)istheinletandx=20ft(6.10 m)is the capped end. The figure shows that the there is a nearuniform dec
46、rease in the flow rate (uniform outflow) forapproximately the first 16 ft (4.88 m). The flow then dropsrapidlytozeroatthecappedend.Thetestsarelabeledwiththeduct size, 12 in. (0.30 m), and test run A, B, or C for eachdifferent flow rate. Additional tests with different duct diam-eters are labeled sim
47、ilarly. The flow rate at the entrance of theduct ranged from approximately 350 to 700 cfm (165 to 330 L/s)for all tests (all duct diameters).The experimental uncertainty in the measured flow rateswas calculatedusing themethods ofKim etal. (1993)with thepreviouslyreporteduncertaintyinthevelocitymeasurements.The estimated uncertainty in the flow ra
