1、VOLUME 10, NUMBER 1 HVAC two years from now, the minimum SEER will be 12. From 1978to 1998, the average efficiency of centrifugal chillers increased by 36% and the efficiency of thebest chillers increased by 50% (American Standard 1999). With improvements in the efficien-cies of boilers, furnaces, h
2、eat exchangers, compressors, motors, fans, and pumps, there areexceptional opportunities for reducing energy use in buildings. Similar improvements have beenachieved in refrigerant technologies, contaminant removal, sensors, and controllers with corre-sponding impact on other environmental factors.
3、2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC Marques and Melo 1993). Perhaps themost challenging of these is void fraction determination.At a given cross section of a tube, the void fraction is defined as the fraction of ar
4、ea occupiedby vapor. While mass quality can be determined using conservation equations, in general, voidfraction cannot be directly calculated and must be modeled in some manner. Rice (1987) pre-sented a comprehensive review of the available void fraction models. The void fraction correla-tions of H
5、ughmark (1962), Premoli et al. (1971), Tandon et al. (1985), and Baroczy (1965) wererecommended, since they yield the highest charge predictions for condensers and the best over-all agreement with experimental data. Rice stated that there are insufficient data to recommendone over the others. He not
6、ed that the Hughmark method may overpredict charge in the con-denser yet still yield good agreement with the total charge by way of error cancellation withrespect to unaccounted charge elsewhere in the system.Todd M. Harms is an advanced engineer at Owens Corning, Granville, Ohio. James E. Braun is
7、a professor and Eck-hard A. Groll is an associate professor in the Department of Mechanical Engineering, Purdue University, West Lafayette,Ind. 2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC Charge, 5.1 kg)(kW) (kW)Pdis(kPa)
8、(g/s)Tsub(C) (kW)COPMeasured Results 17.62 23.43 1882 113.0 5.98 4.54 2.94Baseline Model 17.45 23.29 1883 109.9 10.57 4.44 2.96Condenser, hi20% 17.44 23.30 1892 109.9 10.72 4.46 2.95Condenser, hi+20% 17.46 23.28 1877 109.9 10.47 4.43 2.97Evaporator, hi20% 17.41 23.20 1882 109.6 10.13 4.44 2.95Evapor
9、ator, hi+20% 17.48 23.32 1883 110.0 10.57 4.44 2.97Condenser, ho20% 17.40 23.32 1918 109.9 11.03 4.52 2.91Condenser, ho+20% 17.49 23.22 1860 109.8 9.82 4.39 3.00Evaporator, ho9%a17.33 23.16 1879 109.2 10.55 4.44 2.95Evaporator, ho+20% 17.71 23.56 1888 111.3 10.58 4.45 3.00Condenser, ftp20% 17.45 23.
10、29 1882 109.9 10.62 4.44 2.96Condenser, ftp+20% 17.45 23.25 1883 109.9 10.08 4.44 2.96Evaporator, ftp20% 17.46 23.26 1883 109.9 10.15 4.44 2.96Evaporator, ftp+20% 17.45 23.28 1882 109.8 10.56 4.44 2.96a. An unstable operating condition exists at ho20%QevapQcondmrefWcomp 2004. American Society of Hea
11、ting, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC Charge, 5.1 kg)(kW) (kW)Pdis(kPa) (g/s)Tsub(C) (kW)COPMeasured Results 17.62 23.43 1882 113.0 5.98 4.54 2.94Homogeneous 17.53 23.70 2028 109.2 15.66 4.78 2.82Yashar et al. (2001) 17.52 23.38 1945 109.5 12.11
12、 4.59 2.90Tandon et al (1985) 17.51 23.43 1922 109.6 12.31 4.53 2.93Zivi (1964) 17.49 23.31 1909 109.7 11.06 4.50 2.94Baroczy (1965) 17.45 23.29 1883 109.9 10.57 4.44 2.96Yashar et al. + slug flowa17.50 23.41 1914 109.6 11.92 4.52 2.94Tandon et al. + slug flowa17.47 23.27 1893 109.8 10.52 4.47 2.95Z
13、ivi + slug flowa17.49 23.39 1909 109.7 11.80 4.50 2.94Baroczy + slug flowa17.45 23.29 1881 109.9 10.51 4.44 2.96a. Taitel and Barnea (1990)QevapQcondmrefWcomp 2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC Romie 1984; Roetzal
14、 andXaun 1992). Other models solved the PDE of one finned element of the cross-flow heatexchanger and used the solution to these elements to determine the solution of the heatexchanger (Reichert et al. 1988; Kabelac 1989). The computation time of these finite elementsolutions was fairly lengthy. Sti
15、ll other models considered a discrete time solution of the heatexchanger dynamics (Underwood 1990). Tamm was the first to develop a dynamic multi-row counterflow coil model (Tamm 1969).His model, like that of Gartner and Harrison, was interpreted in the frequency domain. Theterms needed in the solut
16、ion grew exceedingly numerous as the number of coil passes increased.The finite element models of Reichert et al. (1988), as well as Kabelac (1989), investigatemulti-pass heat exchangers as well. Both publications address the lengthy amount of time thatcounterflow arrangement models need to converge
17、 on a solution. Chris C. Delnero is with Lockheed Martin, Moorestown, N.J. Douglas C. Hittle is director of the Solar Energy Appli-cations Laboratory, Dave Dreisigmeyer is in the Mathematics Department, Peter M. Young and Michael L. Andersonare in the Department of Electrical and Computer Engineerin
18、g, and Charles W. Anderson is in the Department of Com-puter Science, Colorado State University, Fort Collins, Colo. 2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC&R Research, Vol. 10, No. 1, January 2004. For personal use on
19、ly. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.22 HVAC&R RESEARCHMcCutchan (1973) investigated the time solution of a cross-flow, water-to-air heat exchanger.His thesis extended the work of Gartner
20、and Harrison by developing a first principles model of afinned serpentine cross-flow heat exchanger. The mixed partial differential equation thatresulted was considered too difficult to solve when McCutchans research was published.Instead, McCutchan divided the dynamics of the coil into two separate
21、 actions and used super-position to determine model predictions. PDE MODEL In order to simulate complex HVAC control schemes such as MIMO (multi-input multi-out-put) controllers that utilize several changing heating coil inlet conditions at the same time, morecomplex coil dynamic models must be deve
22、loped. The dynamic model presented here is the firststep in developing a more complex model, and it is an extension of the model presented in thepaper by Pearson et al. (1974). Their model is developed for a single-pass, cross-flow, hotwater-to-air, finned tube heat exchanger, but it can also be ext
23、ended to a multi-pass heatexchanger such as the one used in this study. The partial differential equation model discussed intheir paper was developed from first principle energy balances. However, at that time, no solu-tion to this PDE was available and, hence, approximate solutions were developed (
24、we willdevelop an exact solution to this PDE). This model looks at the coil dynamics for the case of astep change in hot water flow rate initially having no flow and no temperature gradient from thecoil water to the air flowing across the coil. This model is not valid for the case where the air isco
25、oled by the heat exchanger. Condensation on the fins during air cooling would add complexityto the energy balance presented in Equation 2 while causing the outlet air temperature to bedependent on the moisture content of the airflow.Assumptions: 1. The densities and specific heats of the tube materi
26、al, fin material, water, and air are consid-ered to be constant and are evaluated at their mean value. 2. The heat capacity effects of the water and metal contained in the U-tube bends are accountedfor by distributing the U-bend metal and water throughout the finned portion of the coil. 3. Convectiv
27、e heat transfer coefficients on the air and water sides are independent of tempera-ture, time, and location and are evaluated at their mean temperature. 4. Conductive resistance through the tube wall is negligible. 5. Thermal resistance between the tube and fins is negligible. 6. Heat conduction in
28、the water and tube in the axial direction is negligible. 7. Conduction through the fins from row to row is negligible.18. Air temperature and velocity are constant throughout the entrance cross section to the heatexchanger. 9. The effective temperature difference between the metal and air for the he
29、at transfer purposeis based upon the log mean temperature difference between the metal and air. 10. Fin effectiveness is constant. Each run of the heat exchanger can now be modeled as a long finned tube heat exchanger asshown in Figure 1, where Lfis the length of an individual finned tube from suppl
30、y manifold toreturn manifold. The energy balance across an element of the heat exchanger shown in Figure 2 leads to the setof three equations with three unknowns (T, Tt, Tw), describing the transient heat transfer pro-cess. 1Work presented by Lage (2001) shows that for some heat exchanger configurat
31、ions and operating conditions, significantfin-to-fin heat transfer may occur. For low hot water temperature difference between adjacent tube passes, such as pre-sented in this paper, assumption 7 is considered valid. 2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, In
32、c. (www.ashrae.org). Published in HVAC&R Research, Vol. 10, No. 1, January 2004. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.VOLUME 10, NUMBER 1, JANUARY 2004 23Figure 1. Strai
33、ght line model of heat exchanger.Figure 2. Thermal energy balance on an element of the straight line heat exchangermodel. 2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC&R Research, Vol. 10, No. 1, January 2004. For personal u
34、se only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.24 HVAC&R RESEARCH(1)(2)(3)Define dimensionless B parameters asLet Equations 1, 2, and 3 be expressed in dimensionless form as (4)(5)(6)where solv
35、ing LMD for tgives an additional relationship,(7)Combining Equations 4 through 7 yields the mixed partial differential equation that describesthe outlet air temperature. (8)Equation 8 is the same as Equation 5 printed in Pearsons paper.In order to solve this PDE, the boundary and initial conditions
36、must be specified.In order to solve the boundary condition, the initial air outlet temperature distribution must beknown at the point x*= 0. The hot water inlet condition at x*= 0, wi, is constant and equal tounity by definition of . The solution at this point can be calculated by solving the energy
37、 bal-ance at the inlet of the coil for a steady-state condition. Using the energy balance shown in Fig-ure 2 for steady state, the energy balance yields(9)mwcpwTwx- wcpwAcwTwx-LtLf- hwAwTwTt()+ 0=macpaTaoTai()hwAwTwTt() cmTtt-+ 0=macpaTaoTai()ohaAaLMTD=B1mwcpwmacpaLf-B2hwAwmacpa-B3cmmacpaFT-B4ohaAam
38、acpa-B1wx*- B2wt()B1wt*-+0=xaoB2wt() B3tt*-+ 0=xaoB4LMD=xaoC4t.=B1B3B2C4-t*-xaot*-xaox*-+B3C4-B1C-+xaot*-B1C-xaox*- xao+0=xao0,0()Ci.= 2004. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC&R Research, Vol. 10, No. 1, January 2004. F
39、or personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.VOLUME 10, NUMBER 1, JANUARY 2004 25The boundary condition at x*= 0 is obtained by solving the energy balance of Equation 5 withthe h
40、elp of Equation 7.(10)Solving for the initial air outlet temperature distribution to get the first initial condition, thetime derivatives in Equation 8 are set to zero, and then xaois solved, yielding (11)The second initial condition, , is equal to 0 because the coil is at steady state at t = 0.(12)
41、The form of Equation 8 can be written in a way that makes it easier to work with. Let u = xaoand the derivatives be set as subscripts of u. Also, divide the constant on the second-order deriv-atives through and set the new constants to specified variables. By setting(13)Equation 8 can now be written
42、 as (14)with boundary value and initial conditions as (15)(16)(17)PDE SOLUTION The approach taken in this study is to separate the PDE given in Equation 15 into a boundaryvalue problem and an initial-boundary value problem, then combining the solutions of thesesubproblems by superposition to obtain
43、the general solution. Because the dynamic case that thisstudy investigates incorporates zero initial conditions, only the solution to the boundary valueproblem is implemented in the following sections. The solution to the initial-boundary valueproblem is still presented as a matter of completion for
44、 future dynamic coil models. xao0,t*()CfCiCf()expC4fB2f+B3f- t*+=xaoCiexp bi x*() .=xaox*,0()t*-xaox*,0()t*- 0=aB3C4-B1C-+B2C4B1B3-, = bB2C4CB3-= , cB2C4B1B3- ,=uttutxautbuxcu+ + 0=u 0,t*()CfCiCf()expC4fB2f+B3f- t*+ ft*()=ux*,0()Ciexp bix*()gx*()=utx*,0()0 hx*()= 2004. American Society of Heating, R
45、efrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC&R Research, Vol. 10, No. 1, January 2004. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.26 H
46、VAC&R RESEARCHFor a non-zero boundary value condition and zero initial conditions, subproblem 1 is defined as (18)From here on down, the nondimensional superscript “*” is dropped from all x and t variables. To simplify subproblem 1, we let (19)Equation 18 becomes (20)Taking the LaPlace transform and
47、 solving with the boundary condition gives (21)Using the LaPlace identities (22)(23)(24)(25)and gives (26)Using Equation 26 and 19, the solution to subproblem 1 becomes (27)For a boundary value condition forced to 0 and nonzero initial conditions, subproblem 2 isdefined as follows:uttutxautbuxcu+ +
48、0=u 0,t*()ft*()=ux*,0()0=utx*,0()0=uve2ba()xbt.=vttvtxAv+ 0 ,= x 0 , t 0 , Acab b2.+=vx*,s()Fs()es2A+s- x.=exsf s()0, for txft x() for t 0L1sFs()Ft()=L1 1s-eAxs-Jo2 Axt()=L1f s()gs() ft ()g ()d0t=vx,t()0for tx ,Ftx ()Jo2 Ax()d0txfor tx .u1x,t() vx,t()e2ba()xbt.= 2004. American Society of Heating, Refrigerating and Air-
