1、308 2008 ASHRAE ABSTRACTDynamic cooling coil models are useful in the develop-ment and evaluation of feedback controllers and fault detectionand diagnostics algorithms. Existing models in the literatureare forward models, which require detailed specification ofphysical parameters such as geometry an
2、d material proper-ties. On the other hand, inverse models typically requiremeasurements and parameter estimation techniques to identifyunknown parameters and are particularly useful for onlineapplications where models are part of a control and/or diag-nostic algorithm. This paper develops a simplifi
3、ed distributedinverse model for transient performance of cooling coils thatis based on a forward model recently presented in the litera-ture. The inverse model employs lumped parameters forconductances and capacitances. Simple empirical forms areemployed for characterizing the impact of flow rates o
4、n air-side and water-side conductances. Unknown parameters aredetermined using a two-step approach that requires a smallamount of steady-state and transient training data. Dividingthe parameter estimation process into two separate stepsgreatly reduces the training data requirements. A case study isp
5、resented for an 8-row cooling coil that was tested in a labo-ratory environment. The case study highlights the ability of themodel to extrapolate performance when trained with a limitedamount of data. INTRODUCTIONZhou and Braun (2005, 2007a, 2007b) presented thedevelopment and validation of a relati
6、vely simple model fortransient behavior of cooling and dehumidifying coils. Themodel provides very accurate predictions at significantlyreduced computational requirements as compared with moredetailed finite-difference models that have been presented inthe literature, e.g., McCullagh et al. (1969) a
7、nd Chow (1997).The computational advantages of the model were realizedthrough incorporation of steady-state performance indicesthat characterize steady-state temperature profiles in the fins(fin efficiency), air (air-side effectiveness), and water (water-side effectiveness). The use of these perform
8、ance indicesreduces the number of state variables needed to characterizeperformance as compared with typical finite-differencingapproaches and ensures that the model approaches steady-state or quasi-steady behavior when subjected to static orslowly changing boundary conditions. The simplicity and sp
9、eed of this simplified model lendsitself to online applications. However, it is necessary to havea means of estimating overall coil parameters. The necessaryparameters could be determined from detailed information fortube and fin dimensions and materials. However, this infor-mation may not be readil
10、y available for application to an exist-ing coil in the field. An inverse model, such as the onedescribed in this paper, is useful for an on-line application andinvolves learning basic model parameters through a trainingprocess that attempts to minimize differences betweenmeasurements and model pred
11、ictions. The current paper pres-ents a methodology for characterizing coil characteristics andestimating parameters that utilizes both steady-state and tran-sient measurements. The methodology is demonstrated usingmeasurements for a coil tested under laboratory conditions(see Zhou and Braun, (2005,
12、2007b). Simplified Transient Cooling Coil ModelThe model of Zhou and Braun (2005, 2007a) applies tothe counter cross-flow cooling coil presented in Figure 1.An Inverse Model for Transient Cooling and Dehumidifying Coil PerformanceXiaotang Zhou, PhD James E. Braun, PhDAssociate Member ASHRAE Fellow A
13、SHRAEXiaotang Zhou is a Staff Engineer of Truck/Trailer Engineering for Carrier Transicold Division, Syracuse, NY. James E.Braun is a professorof mechanical engineering for Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN.NY-08-037 (RP-1194)2008, American Society of Heating, Refrig
14、erating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions, Volume 114, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.ASHRAE Transact
15、ions 309Chilled water enters at the back row of the coil and flows in aserpentine arrangement. Air flows over finned tubes and iscooled and possibly dehumidified due to contact with coldsurfaces. The simplified transient model was developed byapplying energy balances to the air, water, tube, and fin
16、 mate-rial within each row of the cooling coil. The modeling assump-tions, resulting equations, and numerical solution method arepresented here for completeness and to identify the lumpedparameters that need to be identified through training withmeasurements.The following assumptions are employed by
17、 the model:1. Water is incompressible.2. Ideal gas mixture for air and water vapor.3. Constant densities and specific heats for air, water, andfin and tube material.4. Negligible conduction in air and water flow directions forboth fluids.5. Negligible conduction for tube material in the water flowdi
18、rection.6. Quasi-steady water and air flow.7. Uniform air velocity across the coil cross section.8. Negligible energy storage within air.9. Negligible effect of water condensate retained on fin andtube outer surfaces when dehumidification occurs.10. Lewis number of unity for heat and mass transfer.1
19、1. The temperature profile within fins follows the steady-state profile allowing the use of heat transfer andcombined heat and mass transfer fin efficiencies. 12. The spatial variation in states for air flowing over a row offinned tube follows the steady-state profiles for transfer toa surface at a
20、constant temperature equal to the mean coilsurface temperature allowing the use of heat transfer andcombined heat and mass transfer effectiveness.13. The spatial variation in temperature for water flowingwithin a tube through a row follows the steady-stateprofile for heat transfer from a surface at
21、a constanttemperature equal to the mean coil surface temperatureallowing the use of heat transfer effectiveness.14. For each row, the time derivative for water temperature isthe same at each point within the water flow stream.The use of fin efficiencies and effectiveness modelsensures that the coil
22、model approaches steady-state or quasi-steady behavior when subjected to static or slowly changingboundary conditions and allows the use of relatively few statevariables. A “slowly changing boundary condition” meansthat the boundary condition changes relatively slowly in rela-tion to the dynamic res
23、ponse of the coil. This situation mightexist during periods where the building load requirements arechanging slowly and the control is stable. The coil dynamicsare important during abrupt load or setpoint changes.The quasi-steady assumption for water and air flowimplies that there are no time-deriva
24、tive terms involving theseflows and that the flow rates throughout the coil respondinstantaneously to changes in the inlet flow rates. The flowsare boundary conditions for the models that have transientchanges that may be imposed during the numerical solution ofthe model. Dry RowFor each discrete ti
25、me interval, each row is treated aseither all wet or all dry. For a dry row, the following differentialequation results from the set of assumptions listed above.(1)(2)where for each row, Tw,inis the water inlet temperature, Tw,outis the water outlet temperature, Tcis the mean temperature forcoil mat
26、erial, Ta,inis the air inlet temperature, Cwand Ccarethe heat capacitances of water and coil material (product ofmass and specific heat), is the total heat capacitance rateassociated with the water flow stream (product of mass flowrate and specific heat), and Rwand Raare the total thermalresistances
27、 for heat transfer between water and coil materialand coil material and air, respectively.The row heat capacitances are related to the total coil heatcapacitances according to(3)(4)where Nrowsis the number of tube rows.Figure 1 Schematic of a counter cross-flow cooling coil.CwdTw out,dt- CwTwout,Twi
28、n,()1Rw- Twin,Tc()+0=CcdTcdt-1Ra- TcTain,()1Rw- TcTwin,()+ 0=CwCwCwtot,Nrows-=CcCctot,Nrows-=310 ASHRAE TransactionsThe water-side heat resistance for each row is determinedwith(5)where wis the water-side heat transfer effectiveness for therow expressed as(6)and NTUw,totis the number of transfer uni
29、ts for the coil deter-mined with(7)where Aw,totis the total internal surface area of the tubes withinthe cooling coil.The air-side thermal resistance for each row is(8)where is the total heat capacitance rate associated with theair flow stream (product of mass flow rate and specific heat)and ais the
30、 row air-side heat transfer effectiveness deter-mined with(9)and where(10)where ais the overall fin efficiency for heat transfer, hais theconvection coefficient for air-side heat transfer, and Aa,totisthe air-side coil surface area.The row outlet air temperature is determined with(11)which is the in
31、let air temperature for the next row just down-stream in the direction of air flow. The outlet air enthalpy isdetermined using a psychrometric function with the outlet airtemperature and the inlet humidity ratio.Wet RowFor a wet row, the driving potential for combined heat andmass transfer is the di
32、fference between air enthalpy and satu-ration air enthalpy at the coil surface temperature seeElmahdy and Mitalas (1977) or Braun et al. (1989). In thiscase, Equation (2) is replaced with(12)where for each row, ha,inis the air inlet enthalpy, hs,cis thesaturation air enthalpy at the mean coil temper
33、ature Tc, andis the total resistance for heat and mass transfer betweencoil material and air determined with(13)and where is the total air mass flow rate and is the air-side heat and mass transfer effectiveness determined with(14)where(15)and where is the overall fin efficiency for combined heatand
34、mass transfer (different from a, which is for heat transferonly). In this case, is the convection coefficient for air-sideheat transfer under dehumidification.The row outlet air enthalpy is(16)which is the inlet air enthalpy for the next row that is justdownstream in the direction of air flow.Numeri
35、cal SolutionZhou and Braun (2005, 2007a) investigated differentalgorithms for integrating Equations (1), (2), and (12) andrecommended an explicit method that uses the following finitedifference equations (see ziik (1994) for an overview offinite-difference methods).(17)(18)(19)where the superscript
36、0 represents the value at the previoustime.Rw1wCw-=w1 eNTUwtot, Nrows=NTUwtot,hwAwtot,Cw-=Ra1aCa-=Caa1 eNTUatot, Nrows=NTUatot,ahaAatot,Ca-=Taout,Tain,aTcTain,()+=CcdTcdt-1Ra*- hsc,hain,()1Rw- TcTwin,()+ 0=Ra*Ra*1a*Ma-=Maa*a*1 eNTUatot,* Nrows=NTUatot,*a*ha*Aatot,Ca-=a*ha*ha out,hain,a*+ hsc,hain,()
37、=CwTwout,Tw out,0()t- CwTwout,0Twin,0()+?1Rw- Twin,0Tc0()+0=CcTcTc0()t-1Ra- Tc0Tain,0()1Rw- Tc0Twin,()+ 0=CcTcTc0()t-1Ra*- hsc,0hain,0()1Rw- Tc0Twin,()+ 0=ASHRAE Transactions 311At any time, the analysis begins at the first row downstreamof the coil inlet air stream. For each row, a dry analysis is
38、per-formed first. A dry analysis involves the determination of Tw,out,Tc, and Ta,outfrom Equations (17), (18), and (11) for given ini-tial conditions, inlet conditions, and other required parameters.The equations are explicit in these three state variables. Theoutlet enthalpy is determined using a p
39、sychrometric functionwith the outlet air temperature and the inlet humidity ratio.To evaluate whether a wet analysis should be performedfor a particular row, the coil surface temperature is comparedwith the dewpoint of the inlet air which is determined from thecontrol volume inlet air enthalpy and t
40、emperature using apsychrometric function. If the coil surface temperature isgreater than the dewpoint temperature, then the dry analysis isappropriate. Otherwise, Equations (17), (19), and (16) areused to explicitly determine Tw,out, Tc, and ha,outwith the giveninitial conditions, inlet conditions,
41、and parameters. The outletair temperature is determined using Equation (11).At the start of the simulation, it is necessary to specifyinitial coil and air states. Zhou and Braun (2005, 2007a)recommended the use of a steady-state version of the simpli-fied model to provide quasi-static initial coil s
42、tates for thespecified initial boundary conditions. The transient terms areremoved from the energy balances of Equations (1), (2), and(12) and the model is solved for state variables associated withsteady-state operation at a given set of boundary conditions.This modeling approach was employed in th
43、e current paper fordetermining initial states for the transient model and for deter-mining heat transfer parameters in the first step of the param-eter estimation process described in the next section.Estimating Overall Coil Parameters from MeasurementsIn developing an inverse model, it is important
44、 to identifythe minimum number of parameters necessary to characterizeperformance. For the cooling coil model presented in theprevious section, transient performance can be predicted forspecified inlet flows and states of air and water if the followingparameters are known: NTUw,tot, NTUa,tot, Cw,tot
45、,Cc,tot, and Nrows. For a forward model, these parameters aredetermined from physical dimensions, properties, and heattransfer correlations/relations. For an inverse model, thenumber of tube rows is either known or can be assumed and theother parameters are determined using parameter estimation.The
46、heat capacitances of the water and material within thecoil are essentially constant, whereas the air and water-sidetransfer units depend strongly on flow rates. The air and waterflow rates and inlet states are boundary conditions for themodel and need to be measured for both training and applica-tio
47、n of the model. In order to simplify the analysis with rela-tively small impact on model accuracy, it is assumed that. This assumption implies that the heattransfer coefficients are the same for dry and wet coils and thefin efficiency for heat and mass transfer is equal to the fin effi-ciency for he
48、at transfer only.Furthermore, it is assumed that the air and water-sidetransfer units have the following dependence on flow rates(20)(21)where and are design (or maximum expected)values for air and water flow rates, NTUa,tot,desandNTUw,tot,desare air and water-side NTUs at the design flowrates, and naand nware empirical coefficients. Typically,NTUa,tot,desand NTUw,tot,desare in the range of 1 to 4, whereasnaand nware between 0.5 and 1. These ranges are useful inspecifying initial guesses for the parameter estimation.The p
