Performance and design of dehumidifier wheels.
INTRODUCTION
Traditionally, dehumidification equipment was used in some spaces with special requirements, such as in some electronic component manufacturing shops and some ICUs in hospitals. Dehumidification can be accomplished using either active desiccants (solid or liquid) or cooling coils (Mumma 2001). As a general HVAC design rule for commercial and institutional buildings, cooling coils have been a better choice when the required dewpoint temperature is above 40[degrees]F (4[degrees]C). On the other hand, active desiccants are a better choice when the dewpoint temperature is below 40[degrees]F (4[degrees]C). Sensitized by litigation regarding indoor air quality problems that are often related to mold and moisture problems, building owners have been more willing to invest in better HVAC designs with improved dehumidification capabilities. Their interest is also prompted by comfort problems caused by the high internal relative humidity and moisture load caused by higher ventilation rates and building envelope leakage rates (Harriman et al. 2001 and Harriman and Judge 2002). Dehumidifier wheels, or desiccant dryer wheels, must have their desiccantcoated surfaces periodically regenerated to a dry condition once every rotation cycle, using a hot regenerative air. Although the literature for regenerative heat wheels has been developed over 85 years, the literature for regenerative dehumidifier and energy wheels goes back only 35 years. During this time, rigorous test standards have evolved for heat and energy wheels using effectiveness as the most important wheel performance factor (ASHRAE 2008). ANSI/ASHRAE Standard 1391998, Method of Testing for Rating Desiccant Dehumidification Utilizing Heat for the Regeneration Process (ASHRAE 1998), for testing dehumidification wheels appears to be less well known because nonstandard methods of testing and definitions of performance factors are still common in the literature. Zhang and Scott (1993) presented a unified approach for discretization of the analysis of thermal regenerator heat exchangers. Laplace transform and numerical techniques were compared for two models of regeneration operation. Changes of time scale and parameters were shown and compared. Overall wheel performance factors or comparisons with measured data were not presented; however, typical graphical timedependent outlet gas temperatures were shown for several cycles after a cold start and at steady state for countercurrent flow. Collier et al. (1990), Worek et al. (1991), Belding et al. (1991), Zheng and Worek (1993), and Zheng et al. (1993) used mostly numerical methods to investigate the performance of dehumidifier wheels using various desiccant types, a defined nondimensional time or wheel speed factor, number of transfer units, etc. as independent variables where the COP of the system and cooling capacity were investigated as dependent performance factors. They concluded that type 1A desiccants were best for dehumidifier wheels. Other findings from these papers are difficult to use by designers because the functional relationships between wheel design variables and performance factors remains hidden in the numerical code. There are no comparisons between measured data and numerical predictions; rather, their predictions are compared to other simulations and show good agreement. Example designs, defining all of the dimensional variables, are not presented. Using a small portion of a dehumidifier wheel, Czachorski et al. (1997) used a transient test method similar to the singleblow test method in a stationary dehumidifier wheel of Collier et al. (1992) where the airflow into the twowheel test sections is reversed at a selected time and all of the inlet and outlet temperatures, humidities, and flow rates are measured. Using the numerical model of Zheng and Worek (1993) and the results of Zheng et al. (1993), they predicted an optimum wheel speed of 16 rph for their particular wheel. Zhang and Niu (2002) developed a twodimensional, transient numerical model to study the effects of rotary speed, NTU, and exchanger surface area on rotary wheel performance. They found that the heat and mass transfer response for rotary desiccantcoated wheels depends on the speed of the wheel, as well as the inlet air conditions. Typical cyclic internal air temperature and humidity simulations were presented for energy and dehumidifier wheels that are similar to those of Zhang and Scott (1993) for heat transfer. For energy wheels, they presented simulated sensible and latent effectiveness versus NTU, specific area, and wheel speed. Their effectiveness results for the effect of wheel speed are not consistent with the data and simulations of Simonson et al. (2000) and the theoretical model of Shang and Besant (2008, 2009a, 2009b). Gao et al. (2005) used a numerical control volume method for the onedimensional NavierStokes equations to predict the transient and steady state of the outlet air temperature and humidity of a dehumidifier wheel moisture transport in each half of this wheel. Assuming fully developed turbulent flow in each flow channel of a dehumidifier wheel, they compared their simulations with measured data, but the agreement was not good. This disagreement may be due to their assumption of turbulent flow for flows that were laminar. Also, their sensitivity investigation and conclusion on the effect of the shape of the flow channels in the wheel matrix does not appear to be consistent with the simulation models of Simonson and Besant (1998) and other researchers or the theoretical models of Shang and Besant (2008, 2009a, 2009b) where flow channel total surface area per unit face area of the wheel was shown to be the dominant flow channel geometry factor. For dehumidifier wheels, they made similar comparisons using a nonstandard definition of dehumidification effectiveness and dehumidification power, so direct comparisons are not possible. Nonetheless, their prediction of the effect of wheel speed does not appear to be consistent with the predictions of Shang and Besant (2009a, 2009b). Jia et al. (2006) presented property data and parametric data for two nonstandard performance indices for two similar dehumidifier wheels: one coated with silica gel (approximate particle size range 30 < dp < 60 nm) and another coated with composite silica gel and LiCl particles (approximate size range 10 < dp < 30 nm). They concluded that there was a 50% improvement in the moisture removal capacity for the wheel coated with their new composite coating compared to the silica gelcoated wheel. It is not clear from this paper just what mass fractions of each particle species were used in their composite coating, nor what was the average mass density and thickness of the two coatings. Shang and Besant (2009a and 2009b) discuss how particle size differences may be very important for water vapor sorption, because the specific surface area in a particle bed will vary inversely with the particle diameter. So, the performance difference in Jia's measured test data may have been strongly influenced by this particle size difference and the composition. In addition, since LiCl has a very low deliquenscence humidity, the performance of any dehumidifier wheel that uses LiCl in its coating can be expected to deteriorate with the number of cycles of exposure at high humidities (Belding et al. 1991). Wang et al. (2005) and Abe et al. (2006a, 2006b) used a different transient test to determine the time response of small sections of stationary energy wheels in order to predict their effectiveness using a theoretical model that employed well known parallel and counterflow heat exchanger effectiveness equations where NTU and airflow capacitance ratio are the two independent parameters. Using basic equations and wheel flow channel properties, Shang and Besant (2008) presented a mathematical model to predict the sensible effectiveness of a rotary regenerative wheel for equal flow areas and mass flow rates in the supply and exhaust streams. They presented an analytical equation for predicting the fully developed flow sensible effectiveness of an energy wheel that only depends on the wheel speed and time constant and includes corrections for entrance, axial conduction, carryover, sorption phase change, and manufacturing effects. Comparisons of this model with data show agreement within the uncertainty bounds for energy wheels. Shang and Besant (2009a and 2009b) also presented a mathematical model based on fundamental heat and mass transfer for predicting the latent effectiveness of energy wheels that have equal airflow areas and balanced flows. The equation for the transient humidity step response of an energy wheel is developed from physical principles using a similar model to that used for the sensible energy response. This fully developed flow model is used to derive a simple characteristic moisture transfer time constant in terms of the flow channel and its desiccantcoating properties and the airflow velocity. Scanning electron microscope (SEM) images and photos for one typical desiccant coating showed particles bonded on to one aluminum matrix. Isotherm data presented for the moisture content of these coatings and similar desiccant particles at several temperatures show significantly lower moisture content isotherms for the coatings than the for the particles. Calculated equilibrium isotherm moisture content changes, when compared to transient moisture content change correlations for a step change in humidity, showed agreement within the uncertainty limits for time periods of 1000 s or more. These longduration tests are akin to the behavior or characteristics of some dehumidifier or desiccant dryer wheels. For very short time exposures (e.g., 1 s or 2 s), which is typical for energy wheels, the sorption characteristics or time constants appear to change and become more uncertain, so a new, more convenient to use and accurate method was used to deduce the sorption time constants from measured data. This time constant and the wheel speed were then used to determine the predicted effectiveness, which compared well with measured data for two typical energy wheels. PERFORMANCE FACTORS FOR DEHUMIDIFIER WHEELS A typical dehumidifier wheel system is shown in Figure 1. In this system, there are two flow streamssupply air and regeneration air. As the dehumidifier wheel rotates, the water molecules are removed from the supply air and transferred to the regeneration air. Adjacent to this dehumidifying device is a heating device to heat the regenerator inlet air to a temperature much higher than the supply air. This elevated temperature is necessary to regenerate the desiccant coating on the wheel matrix for dehumidification of the supply air. The heating device can be an electric or gas heater, and it may be preheated upstream by a supply of waste or solarheated air. [FIGURE 1 OMITTED] Dehumidifier wheels have very different operating conditions than energy wheels, although their construction is usually similar and in some cases they are identical. Just how each wheel matrix design and desiccant coating design impact dehumidifier wheel performance is not well explained in the literature. A comprehensive theory of how dehumidifier wheels can be expected to perform at various wheel speeds, face velocities, and fractions of the regenerator face area relative to the supply air face area would not only give the manufacturer of the wheels better design tools, it would give the HVAC system designer improved wheel selections and specification of operating conditions. The purpose of this research is to sort through this dearth of information. In applications of desiccant dehumidifiers in HVAC systems, the coefficient of performance (COP) is often used for evaluating their performance. The COP is a dimensionless number and is defined as COP = [latent energy exchange rate/net auxiliary energy input rate], (1) where the net energy input rate is taken to be the auxiliary heat energy necessary for regeneration (Harriman and Kosar 2004). Equation 1 uses the energy balance of the first law to analyze and design the dehumidifying system, as shown in Figure 1. It is clear that the waste heat utilization in the system design can reduce, and possibly eliminate, the purchase of thermal energy for regeneration. Due to external heat added, the performance of a dehumidifier wheel cannot be uniquely characterized by the COP unless, as is specified in ASHRAE Standard 1391998 (ASHRAE 1998), the inlet air properties to the air heater are specified. Prior to considering the capital cost and the cost effectiveness of this system, it is convenient to only consider this simple air dryer system without other HVAC equipment when evaluating the performance of dehumidifier wheels. The main objective of this system is to remove moisture from the supply air so that the bulk mean outlet air humidity, [[bar.W].sub.so], is significantly lower than the inlet humidity, [W.sub.si]. At the same time, the energy input rate to the regenerator auxiliary air heating device should be low, and the temperature rise in the supply air, [[bar.T].sub.so]  [T.sub.si], should be low. In general, both the operation of dehumidifier wheels and the heating device must be considered as a system when auxiliary heating is required. Unlike energy wheels with normally equal ventilation supply and exhaust mass flow rates and equal flow area regions of the wheel sector surface (i.e., 180[degrees]), the regenerator air mass flow rate, [m.sub.r], need not equal the supply air rate, [m.sub.s]. Also, the dehumidifier wheel angular segment for regeneration, [[theta].sub.r], does not have to equal the supply air segment, [[theta].sub.s]. These dehumidifier wheel design objectives and constraints can be presented in a set of dimensionless performance factors for the dehumidifier wheels and in another set for the system that includes the dehumidifier wheels and the auxiliary heater. These are summarized momentarily. Water vapor sorption flux at any temperature on a surface is proportional to the difference between the adjacent air relative humidity and the surface equilibrium relative humidity. For any inlet air humidity ratio, W, the relative humidity is inversely proportional to its saturation vapor pressure, [p.sub.ws](T). For each selected operating condition, the dehumidifying performance of a dehumidifier wheel can be characterized by its water vapor removal effectiveness for the supply air, which, for dehumidifier wheels, should be defined as a ratio of actual water vapor mass rate in the supply divided by the maximum possible mass rate when it is adjusted for the same inlet temperatures for the supply and regeneration, as seen in Equation 2: [[epsilon].sub.w] = [[[m.sub.s]([W.sub.so]  [W.sub.si])]/[[m.sub.m]([W*.sub.ri]  [W.sub.si])]] = [[m.sub.s]/[m.sub.m]][[[DELTA]W]/[[DELTA]W*]] (2) where [m.sub.m] = min{[m.sub.s], [m.sub.r]}, and the regenerator humidity ratio is modified for the same temperature as the supply inlet using Equation 3: [W*.sub.ri] = [[W.sub.ri][p.sub.ws]([T.sub.si])/[p.sub.ws]([T.sub.ri])] (3) For a given operating condition, the value of [[epsilon].sub.w] should be high, as it is the primary factor used to characterize each dehumidifier wheel. This type of dimensionless factor is widely used for heat exchanger design, because it can be taken to be nearly constant over a significant range of operating conditions. Often, dehumidifier wheels use a range of operating conditions where this assumption would be misleading. It is suggested by Shang and Besant (2009a and 2009b) that because [[epsilon].sub.w] is so dependent on operating conditions (i.e., wheel speed, air mass rate, and the inlet air temperature and humidity) that perhaps more needs to be done to define these conditions, or perhaps, slightly different characteristics should be used. For example, flow channel time constants, which themselves will depend on air mass rate and the direction of moisture transfer (i.e., adsorption or desorption), may be a better factor to characterize a dehumidifier wheel, because it eliminates wheel speed for the measurement of the time constant (Shang and Besant 2009b). The wheel speed can be reintroduced explicitly into the theoretical equation for the fully developed flow effectiveness. The cost of testing regenerative wheels, using the transient test method of Shang and Besant, is also expected to be very much lower than a steadystate test. Equation 2 makes it clear that [[epsilon].sub.w] can be readily increased by letting the mass flow ratio, [m.sub.s]/[m.sub.m], become large. This would imply that the regenerator air mass rate, [m.sub.r], should be very small compared to the supply flow. However, this will defeat the main objective of this study, which is to remove as much moisture from the supply air as practical. Since the actual moisture removal rate is given by or [m.sub.s][DELTA]W, or [[epsilon].sub.w][m.sub.m][DELTA]W*, a low [m.sub.m] will lower the water vapor transfer rate in the supply air. We need to compare [m.sub.s][DELTA]W against the product of the supply air mass rate, [m.sub.s], and [DELTA]W*. This gives a water vapor difference mass ratio of [[[DELTA]W]/[[DELTA]W*]] = [r.sub.w][[epsilon].sub.w], (4) where [r.sub.w] = [m.sub.m]/[m.sub.s]. Some authors have called this type of ratio the water vapor capacity ratio or dehumidification efficiency (Zheng and Worek 1993). This ratio should be regarded as a more important performance factor for dehumidifier wheels than effectiveness. So, it should be as large as practical. For energy wheels that operate with equal air mass rates for the supply and exhaust, [r.sub.w] would be 1.0, so the wheel design objectives characterized by Equations 2 and 4 would be identical. Dehumidifier wheel designers don't necessarily select [r.sub.w] = 1.0, so Equation 4 is very important. For the wheelheater system, the main design objective is to have the phase change or latent energy rate saved. The moisture removal from the supply air should be large compared to the auxiliary heat input rate in the auxiliary air heater in the regenerator. Assuming constant properties [h.sub.ad] and [c.sub.pa], the COP for water vapor phase change or latent energy for the heaterdehumidifier wheel system is [COP.sub.l] = [[[m.sub.s][h.sub.ad]([W.sub.si]  [W.sub.so])]/[[m.sub.r][c.sub.pa]([T.sub.ho]  [T.sub.hi])]]*[[eta].sub.h] (5) or [COP.sub.l] = [[epsilon].sub.w]*[H*.sub.r]*[[m.sub.m]/[m.sub.r]]*[[eta].sub.h]. (6) And where [H*.sub.r] = [[[h.sub.ad][DELTA]W*]/[([c.sub.p][DELTA][T.sub.h]).sub.a]] (7) is a dimensionless moisture energy to auxiliary sensible energy input ratio, which is represented as the process slope on a psychrometric chart for the regenerator system, and [[eta].sub.h] is the auxiliary heater efficiency. [H*.sub.r] can be made large by operating the auxiliary heater with [DELTA][T.sub.h] as small as practical (i.e., it can be reduced by avoiding overheating of the desiccant coating and using waste heat and/or solar collectors to preheat the air delivered to the regenerator heater). A design constraint for dehumidifier wheelheater systems is that the supply air heat rate needs to be small relative to the auxiliary energy heat input rate. That is, the sensible energy rate ratio or COP for the dehumidifier wheel sensible energy transfer should be small. The COP for sensible energy transfer for the heaterdehumidifier wheel system is [COP.sub.s] = [[[m.sub.s][c.sub.pa]([T.sub.so]  [T.sub.si])]/[[m.sub.r][c.sub.pa]([T.sub.ho]  [T.sub.hi])/[[eta].sub.h]]] (8) or [COP.sub.s] = [[epsilon].sub.s][[eta].sub.h][[[m.sub.m][DELTA][T.sub.M]]/[[m.sub.r][DELTA][T.sub.h]]], (9) where [DELTA][T.sub.M] = [T.sub.ri]  [T.sub.si], and the sensible effectiveness of the dehumidifier wheel is [[epsilon].sub.s] = [[[m.sub.s][c.sub.pa]([T.sub.so]  [T.sub.si])]/[[m.sub.m][c.sub.pa]([T.sub.ri][T.sub.si])]], (10) where [m.sub.s]/[m.sub.r] and [DELTA][T.sub.M]/[DELTA][T.sub.h] are selected design ratios for each dehumidifier wheel application. It should be noted that this definition of dehumidifier wheel sensible energy rate ratio differs from the sensible effectiveness for an energy wheel, because the difference between the air heater outlet and inlet temperature, [DELTA][T.sub.h], can have any value when the regenerator air is preheated by waste heat before it enters the auxiliary heater. Equation 9 shows that when the conventional definition of sensible effectiveness is used (i.e., [[epsilon].sub.s] in Equation 10), it must be multiplied by [[eta].sub.h], a temperature difference ratio, [DELTA][T.sub.M]/[DELTA][T.sub.h], and a mass flow rate ratio [m.sub.s]/[m.sub.r]. In a given HVAC design, there may not be any good opportunity to significantly change the temperature ratio in Equation 9, because [DELTA][T.sub.M] is somewhat fixed by the desiccantcoating isotherm and the need to achieve a lowhumidity ratio in the supply air. [DELTA][T.sub.h] may be fixed by the HVAC design and the lowcost availability of waste heat, so if the supply and regenerator airflow rates are equal, the only dehumidifier wheel design option for Equation 9 is to design the matrix and select an operating speed so that the sensible effectiveness, [[epsilon].sub.s] is small. To investigate this complex problem analytically, first a simple decoupled base case model for fully developed flow is developed. Then, this theoretical result is corrected for other effects, especially the coupling of heat and water vapor transfer. In this base case model, it is assumed that the desiccant coating in each flow channel is uniform and each flow channel has the same properties. The two main assumptions in the development of this base case model are that the airflow in each channel is fully developed and laminar and the heat transfer is decoupled from the water vapor transfer. Later, corrections will be made to the predicted effectiveness to account for the effects of entrance, heat conduction, carry over, phase change, flow channel size variations, and sorption isotherms. When each of these corrections is small, they are independent, and the accuracy of the final prediction will be good (Shang and Besant 2008 and 2009a). However, phase change and sorption isotherm temperature effects are coupled and are much larger for dehumidifier wheels, so these corrections are added in an iterative manner that couples the heat and water vapor transfer. Figure 2 shows the schematic of a typical dehumidifier wheel with unequal flow areas for the regeneration and supply airflows. The shaded area is the regeneration flow area with the angle [[theta].sub.r], and the rest area of the wheel is for the supply airflow, which is in the same direction (i.e., parallel flow). Often, the angle is 0 < [[theta].sub.r] [less than or equal to] [pi]/2 for dehumidifier wheels so that [pi] < [[theta].sub.s] [less than or equal to](2[pi]  [[theta].sub.r]). One typical corrugated airflow channel coated with a desiccant bonded to all surfaces, which in total comprises the wheel matrix, is shown in Figure 3. Each flow channel rotates about the wheel axis and is subjected to a step change in inlet air properties as it cyclically passes into the regeneration airflow and then into the supply airflow. [FIGURE 2 OMITTED] [FIGURE 3 OMITTED] The basic equations for thermal energy balance and water vapor mass conservation for the airflow inside any typical flow channels are the following. Thermal Energy Assuming that phase change energy corrections can be treated as a correction that can be applied later (Shang and Besant 2008), for the air the thermal energy can be characterized by its bulk mean air temperature, T', which varies with time, t, and axial position, z. The thermal energy equation for the air can be written as [rho][c.sub.[upsilon]]A[[[partial derivative]T']/[[partial derivative]T]] + [rho][c.sub.p]VA[[[partial derivative]T']/[[partial derivative]z]] = Ph([T.sub.m]'  T'), (11) where [T'.sub.m] is the local matrix surface temperature, h is the convective heat transfer coefficient, [rho] is the bulk mean density of the dry air and water vapor mixture, [c.sub.[upsilon]] and [c.sub.p] are the bulk mean air mixture specific heat at constant volume and constant pressure, respectively. For the matrix, the thermal energy differs from Equation 11, due to axial heat conduction and phase change heat source and no convection, as seen in Equation 12: [A.sub.m][[rho].sub.m][c.sub.pm][[[partial derivative][T'.sub.m]]/[[partial derivative]t]] = [A.sub.m][k.sub.m][[[[partial derivative].sup.2][T'.sub.m]]/[[partial derivative][z.sup.2]]] + hP(T'  [T'.sub.m]) + [A*.sub.m][h.sub.a][[[partial derivative][[rho].sub.w,m]]/[[partial derivative]t]] (12) where [A*.sub.m] is the area of the matrix flow channel adjacent to the airflow channel, which can adsorb and desorb water vapor, [[rho].sub.m][c.sub.pm] is the density and specific heat product for the matrix, [k.sub.m] is the mean thermal conductivity of the matrix, and [h.sub.a] is the heat of water vapor sorption, which is taken to be nearly equal to the heat of evaporation, [h.sub.fg], in most cases. Not included in Equation 12 is heat conduction normal to the flow direction (i.e., x  y plane of the matrix), because it is assumed to be negligible. Water Vapor Continuity The water vapor continuity for the airflow channel can be expressed as A[[[partial derivative][[rho]'.sub.w]]/[[partial derivative]t]] + VA[[[partial derivative][[rho]'.sub.w]]/[[partial derivative]z]] = P[h.sub.w]([[rho]'.sub.w,m]  [[rho]'.sub.w]), (13) where [[rho]'.sub.w,m] is the wheel matrix surface water vapor density, and it includes time and spatial variations for the bulk mean water vapor density, [[rho]'.sub.w], and a flow channel perimeter surface water vapor convection coefficient, [h.sub.w]. In the airflow channel matrix, water vapor continuity can be written as [A*.sub.m][[[partial derivative][[rho]'.sub.w,m]]/[[partial derivative]t]] = P[h.sub.w]([[rho]'.sub.w]  [[rho]'.sub.w,m]). (14) An assumption implied by Equation 14 is that the matrix mean water vapor density, [[rho].sub.w,m], and surface mass density of water, [[rho]'.sub.w,m], are equal (i.e., internal matrix water concentration differences are negligible). BASE CASE EFFECTIVENESS CORRECTIONS A performance study for fully developed flow in each airflow channel in the matrix will not directly include the effects of the entrance enhancement, heat conduction in the matrix, carryover of regeneration gas to the supply, phase change caused by water vapor sorption in the desiccant coating, and flow channel size variations. These are treated as corrections to the sensible effectiveness by Shang and Besant (2008). Corrections for water vapor mass effectiveness include the entrance, sorption temperature, carryover, and flow channel variation effects (Shang and Besant 2009a and 2009b). When each correction is small when compared with the fully developed flow theoretical estimate of the sensible energy effectiveness, [[epsilon].sub.FD], and the mass flow ratio of air is one, each correction can be calculated separately and independently added to get the net total correction using the first term of a Taylor series expansion for sensible effectiveness, i=1, that is [[[DELTA][[epsilon].sub.tot,i]]/[[epsilon].sub.FD,i]] = [[[DELTA][[epsilon].sub.ent,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.cond,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.co,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.pc,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.var?,i]]/[[epsilon].sub.FD,i]], (15) where [DELTA][[epsilon].sub.ent,i] is the flow channel entrance flow correction to [[epsilon].sub.FD,i], [DELTA] [[epsilon].sub.cond,i] is the energy wheel matrix heat conduction correction to [[epsilon].sub.FD,i], [DELTA] [[epsilon].sub.co,i] is the correction due to the carryover effect, [DELTA][[epsilon].sub.pc,i] is the correction due to phase change of water as it is transferred either to or from the matrix, and [DELTA][[epsilon].sub.var,i] is the correction due to the energy wheel flow channel hydraulic diameter variations. It is shown by Shang and Besant (2008) that the phase change correction will become very large as the wheel speed goes toward zero. For example, this correction will exceed 20% for wheel speeds less than 5 rpm. Typically, this correction will cause the sensible effectiveness to be much lower than the water vapor effectiveness for dehumidifier wheels. In the discussion on performance factors for dehumidifier wheels, this is a desirable feature. However, at 1 rpm, we might expect this correction to exceed 50% making this extrapolation unreliable using Equation 15. Later we show that this correction can be included using an iterative method. Each of these corrections is considered separately and the net effect on the effectiveness is calculated using the equation [epsilon] = [[epsilon].sub.FD](1 + [[DELTA][[epsilon].sub.tot]]/[[epsilon].sub.FD]). (16) For water vapor mass transfer, the effectiveness corrections are expressed in a similar equation to Equation 15, i.e.: [[[DELTA][[epsilon].sub.tot,i]]/[[epsilon].sub.FD,i]] = [[[DELTA][[epsilon].sub.ent,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.sorp,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.co,i]]/[[epsilon].sub.FD,i]] + [[[DELTA][[epsilon].sub.var,i]]/[[epsilon].sub.FD,i]] (17) where i= 2 and [DELTA][[epsilon].sub.sorp,i] is the sorption temperature flow channel correction to [[epsilon].sub.FD,i] and [[epsilon].sub.w] is calculated using Equation 16. Since the sorption correction is large for dehumidifier wheels it too will be added in an iterative methodso Equations 15, 16, and 17 will be coupled. Base Case Assuming fully developed flow in each flow channel and no coupling between heat and water vapor transfer and integrating Equations 11, 12, and 13 over the length of the flow channel gives the average air temperature, T(t), matrix temperature, [T.sub.m], and water vapor density, [[rho].sub.w](t), inside a flow channel. Shang and Besant (2008) show that for fully developed flow this average air temperature and water vapor density can be written as ([T.sub.i] + [T.sub.o])/2 and ([[rho].sub.i] + [[rho].sub.o])/2, respectively, where the subscripts i and o refer to the inlet and outlet conditions. These substitutions allow us to write an ordinary differential equation for the outlet air temperature with the inlet air temperature as the known forcing function (Shang and Besant 2008): {[[4[([rho][delta][c.sub.p]).sub.m]L]/[[([rho]V[c.sub.p]).sub.a][d.sub.h]]][[1/2][[([rho]V[c.sub.p]).sub.a][d.sub.h]]/4hL]}[[d[T.sub.o]]/dt] + [T.sub.o] = [T.sub.i] (18) Similarly for water vapor transfer we can write a single ordinary differential equation for the outlet air water vapor ratio with the inlet water vapour ratio as the know forcing function (Shang and Besant 2009a): [4[beta]([[bar.[rho]].sub.d]/[[bar.[rho]].sub.a])([[bar.[delta]].sub.d]/[d.sub.h])[([[partial derivative][bar.u]]/[[partial derivative][phi]]).sub.T][1/[[W.sub.s]([bar.T])]](L/[V.sub.a])][[d[W.sub.o]]/dt] + [W.sub.o] = [W.sub.i] (19) The terms in the brackets in Equations 18 and 19 are interpreted as characteristic time constants, [[tau].sub.1] and [[tau].sub.2], of the outlet temperature, [T.sub.o](i.e., [[tau].sub.s]), and water vapor humidity ratio, [W.sub.o] (i.e., [[tau].sub.m]) during a step change in the inlet temperature [DELTA][T.sub.M] and humidity [DELTA][W.sub.M]. These time constants are dependent only on the properties of the typical flow channel and its coating and the air velocity, [V.sub.a], and density, [[rho].sub.a]. The dynamic operation of a dehumidifier wheel matrix rotating between two air streams with constant but different inlet temperatures and humidities causes each flow channel to behave like a linear system with a steady state rectangular periodic input forcing function. Considering Equations 18 and 19 for one flow channel they can be rewritten as [[tau].sub.i][[d[[THETA].sub.i]]/dt] + [[THETA].sub.i] = f(t), (20) where i = 1, 2 for thermal response and mass transfer response, respectively. [[THETA].sub.1](t) = ([T.sub.o]  [T.sub.i])/([DELTA][T.sub.M]) is the dimensionless temperature, and [[THETA].sub.2](t) = ([W.sub.o]/[W.sub.i])/([DELTA]W*) is the dimensionless humidity. [DELTA][T.sub.M] and [DELTA]W* represent the maximum (or minimum) difference between the regenerator and supply inlet temperatures or humidities such that both [[THETA].sub.1](t) and [[THETA].sub.2](t) are positive. f(t) is the dimensionless periodic inputforcing function representing the inlet property changes that occur as step changes as each flow channel switches from the regenerator inlet air to the supply air or vice versa. It is written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21) where [[theta].sub.r] is the regeneration sector angle of the wheel (as shown in Figure 2), and [omega] is the angular frequency of the input (i.e., the speed of wheel rotation in rad/s). Using the assumption of constant thermal properties, flow channel geometry, mass flow rate (i.e., [[rho].sub.r][V.sub.r] = [[theta].sub.s][V.sub.s] is the same in each flow channel), and the Fourier transform for Equation 20, Kaplan (1962) and Kuhfittig (1978) show that the steadystate solution for the periodic step change in inlet properties (Equation 21) is the following: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22) where the time constants, [[tau].sub.i] (given by Equations 18 and 19), do not change from the regenerator to the supply sections. The periodic input (Equation 21) and steadystate output, [[THETA].sub.i](t), are shown in Figure 4 for one time cycle from 0 to 2[pi]/[omega] s. It can be seen by integrating Equation 22, with respect to time over one cycle, that the shaded area with the solid lines for each flow channel airflow sensible energy (or mass of water vapor) change while it is in the regeneration sector equals the shaded area in the supply air section of the wheel. That is, sensible energy (or water vapor mass) transfer rates must be equal but opposite in sign for the regenerator and supply air at steady state. [FIGURE 4 OMITTED] The assumption of equal mass flow rates in each flow channel, while convenient to illustrate the behavior of a regenerator wheel, implies equal time constants in the regenerator and supply air. This is an unnecessary assumption. That is, dehumidifier wheels often operate with unequal mass flow rates in unequal flow area wheel regions for the regenerator sector and the supply air sector. The mass flow of dry air through each wheel sector (regenerative or supply air) and inlet air properties are cyclic. Then [[tau].sub.ri] [not equal to] [[tau].sub.si] and [[theta].sub.ri] [not equal to] [[theta].sub.si] where [[theta].sub.ri] + [[theta].sub.si] = 2[pi]. For this general case, it is necessary to transform the time scale in seconds to a dimensionless time scale, t*, for Equation 20, which is now written as [[d[[THETA].sub.i]]/dt*] + [[THETA].sub.i] = f*(t*), (23) where t* = t/[[tau].sub.si] for the supply and t* = t*/[[tau].sub.ri] for the regenerator sector of the wheel. This change in the independent variable alters the dimensionless duration for each step in the forcing function of each step change, as seen in Equation 24: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24) where [[PSI].sub.i] = [[psi].sub.ri][[psi].sub.si]/([[psi].sub.ri][[psi].sub.si]), [[psi].sub.ri] = [omega][[tau].sub.ri]/[[theta].sub.ri], and [[psi].sub.si] = [omega][[tau].sub.si]/[[theta].sub.si]*[[psi].sub.ri] = ([[omega][[tau].sub.ri])/[[theta].sub.ri], and [[psi].sub.si] ([omega][[tau].sub.si])/[[theta].sub.si] can be physically interpreted as either angle ratios or time constant divided by the period of exposure for the regenerator side and supply side, respectively. For typical dehumidifier wheels [[psi].sub.ri] [greater than or equal to] [[psi].sub.si]. By altering the time scale in this manner, the new one is nondimensionalized and normalized with a period of time constant. Through their definitions in Equations 18 and 19, the mass flow rates change as the flow channel switches from the regenerator to the supply side and back again. Figure 4, when revised with these definitions, appears as Figure 5. [FIGURE 5 OMITTED] The solution of the differential Equation 23 and bounding conditions (Equation 24) is similar to Equation 22: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25) Since each flow channel is assumed to behave exactly like others in the wheel, they all will have the same output area divided by input area as in Figure 5. The effectiveness of a heat exchanger is defined as the ratio of the actual heat transfer rate to the possible maximum heat transfer rate. For this input and output transient response system at steady state, the following definition for a parallel flow heat exchanger can be used: [[epsilon].sub.PF] = [actual heat (or mass) transfer rate/maximum possible heat (or mass) transfer rate] (26) The net sensible energy (or water vapor mass) rate in the supply side (i.e., area under the supply side in Figure 4 or 5) divided by the maximum possible sensible energy transfer (or water vapor mass) rate (i.e., the area with dashed shading lines) is defined as the sensible (or water vapor) effectiveness for parallel flow. The resulting sensible energy (or water vapor mass) effectiveness is a function only of two dimensionless parameters, [[psi].sub.ri] and [[psi].sub.si], as seen in Equation 27: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27) where i = 1, 2 for thermal response and mass transfer response, respectively. Figure 6 shows [[epsilon].sub.i] for parallel flow rotary regenerative heat exchangers using [[psi].sub.ri] as the independent variable and [[psi].sub.si]/[[psi].sub.ri] as a parameter. The ratio, [r.sub.i], is defined as [FIGURE 6 OMITTED] [r.sub.i] = [[[psi].sub.si]/[[psi].sub.ri]], (28) which equals [[theta].sub.r]/[[theta].sub.s] for the special case of [[theta].sub.min] = [[theta].sub.r] and [[tau].sub.ri] = [[tau].sub.si]. Equation 27 can be rewritten as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29) which is the same as Equation 27, except that [[epsilon].sub.PF, i] is written as a function of [[psi].sub.ri] and [r.sub.i], which equals the corresponding heat capacity rate ratio, [C.sub.min]/[C.sub.max], when the specific heats are equal for the regenerator and supply sides, as presented in most textbooks for heat transfer (e.g., Incropera and DeWitt 2005). A comparison of [[epsilon].sub.PF, i] in Figure 6, with similar curves for heat exchangers, shows very similar curves, even though [[psi].sub.ri] is used on the abscissa in Figure 6, while heat transfer textbooks use number of transfer units, N. This result implies that a relationship exists between [[epsilon].sub.PF](N, R) and [[epsilon].sub.PF]([[psi].sub.ri], [[psi].sub.si]). For parallelflow heat exchangers, Kays and London (1964) show that the effectiveness is [[epsilon].sub.PF] = [[1  exp[(1 + R)N]]/[1 + R]], (30) where R = [C.sub.min]/[C.sub.max]. Therefore, the number of transfer units is also a function of [[psi].sub.ri], and [[psi].sub.si] is given by N =  [1/[1 + R]]ln[1  (1 + R) * [[epsilon].sub.PF]([[psi].sub.ri], [[psi].sub.si])]. (31) For fully developed flow with the same inlet conditions and small property changes in the air, the value of N is the same for parallel and counterflow exchangers, so the effectiveness for counterflow is given by [[epsilon].sub.CF] = [[1  exp[  (1  R)N]]/[1  R * exp[  (1  R)N]]]. (32) Figure 7 shows [[epsilon].sub.i] for counterflow rotary regenerative heat exchangers using [[psi].sub.ri] as the independent variables in place of N * [r.sub.i] is assumed to be equal to R. Again, a comparison with the heat transfer [[epsilon].sub.CF] with N in place of [[psi].sub.ri] shows a strong similarity. [FIGURE 7 OMITTED] Both Figures 6 and 7 show that for [[psi].sub.ri] < 0.35, [[epsilon].sub.PF, i] = [[epsilon].sub.CF, i] = [[epsilon].sub.i] = [[psi].sub.ri]. (33) For this case [[epsilon].sub.i] is the same for parallel and counterflow and is equal to just [[epsilon].sub.ri], independent of [r.sub.i]. Based on only this base case effectiveness, dehumidifier wheels, which operate at low speeds such that [[epsilon].sub.ri] < 0.35, can operate with any face velocity or wheel sector angle devoted to regeneration, and the effectiveness will equal [[epsilon].sub.ri]. As noted in the discussion on performance factors for dehumidifier wheels, the effectiveness, [[epsilon].sub.PF, i] or [[epsilon].sub.CF, i], are not the best dimensionless factors to characterize dehumidifier wheels; rather, we should use Equation 3 to develop graphs showing [r.sub.i][[epsilon].sub.PF, i] or [r.sub.i][[epsilon].sub.CF, i] versus [[psi].sub.ri], with [r.sub.i] as a parameter. Assuming [r.sub.w] = [r.sub.i] = [r.sub.2], these graphs, presented in Figures 8 and 9, show that the best selection of [r.sub.i] will be 1.0 for all values of [[psi].sub.ri]. That is, the air mass flow rate through the regenerative sector of the wheel should equal that of the supply air sector when the water vapor desorption characteristics do not differ significantly from the adsorption characteristics of the dehumidifier wheel (Shang and Besant 2009a, 2009b). For equal air mass flux on the faces of each of these wheel sectors, the regenerative sector angle, [[theta].sub.r], should equal the supply sector angle, [[theta].sub.s]. [FIGURE 8 OMITTED] [FIGURE 9 OMITTED] Figures 8 and 9, similar to Figures 6 and 7, imply that the performance factor, [r.sub.i][[epsilon].sub.i], or water vapor mass ratio, [DELTA]W/[DELTA]W*, dramatically declines as the wheel speed decreases to zero, because [[psi].sub.ri] = [omega][[tau].sub.ri]/[[theta].sub.ri] goes to zero with wheel speed. This decline in [r.sub.i][[epsilon].sub.i] with wheel speed means that for water vapor the actual reduction in the supply air humidity ratio, [DELTA]W, will decline with respect to the maximum change, [DELTA]W*, in Equation 4. That is, the speed of the dehumidifier wheels should not be so low that they transfer an insignificant fraction of the supply air inlet humidity. Unlike Figures 6 and 7, which showed that the effectiveness [[epsilon].sub.i] was equal to [[psi].sub.ri] for [[psi].sub.ri] < 0.35, Figures 8 and 9 show that [r.sub.i] should be equal to 1.0, because for [[psi].sub.ri] or [[psi].sub.2] < 0.35, [[[DELTA]W]/[[DELTA][W.sub.M]]] = [r.sub.2][[epsilon].sub.2] = [r.sub.2][[psi].sub.i], (34) and this will result in the maximum drying rate for the supply air. In this case, [[psi].sub.2] = [[psi].sub.r2] = [[psi].sub.s2]. Looking at the definitions of [r.sub.i] in Equation 28 and [[psi].sub.ri] in Equations 19 and 24, this means that, regardless of the size of each sector angle [[theta].sub.r] and [[theta].sub.s], the mass flow rate of air through the regenerator sector of the wheel should equal that in the supply sector. The problem that will occur if the wheel speed is high is not that more moisture will be transferred. The problem is that the heat transfer into the supply air will increase with wheel speed, which is why it is important to select the operating conditions so that the sensible energy effectiveness, [[epsilon].sub.1], is low while the water vapor transfer effectiveness, [[epsilon].sub.2], is high. This is a difficult design compromise or tradeoff because, as implied in Figures 6, 7, 8, and 9, for fully developed flow both the water vapor transfer effectiveness and the sensible energy effectiveness have similar functional relationships with and [[psi].sub.ri] and [r.sub.i]. Wheel Matrix and Flow Channel Design Considerations for the Base Case Using the theoretical model developed by Shang and Besant (2009a, 2009b), guidance for the design desiccant coating for a dehumidifier wheel can be stated so that the time constant for heat transfer will be significantly less than the time constant for moisture transfer for the same flow conditions. This may be achieved using the following design guide: 1. Select the best type of desiccant. 2. Use a bonding material and coating method that does not excessively obstruct the water vapor transfer. 3. Choose small desiccant particle sizes so that the specific surface area of the coating is large. 4. Select a coating thickness so that it enhances the water vapor transfer time constant without excessively lengthening the sensible energy time constant. Other factors that could be changed in the wheel matrix flow channel design for the time constants in Equations 18 and 19 will tend to have the same effect on [[tau].sub.1] and [[tau].sub.2]. So, the net benefits of such changes will be very small. For a given inlet face velocity, the sensible energy time constant can be estimated to a reasonable uncertainty using only flow channel geometric and thermal properties as described in Shang and Besant (2008). The water vapor time constant cannot, however, be determined without testing the flow channels. Shang and Besant (2009b) present a method to determine this time constant, so an existing desiccantcoated wheel matrix can be tested during product development and before it is widely used in HVAC applications. The design choice of regenerative, [[theta].sub.r], (or supply, [[theta].sub.s]) flow sector angle is not expected to be a very sensitive factor; but it will be impacted slightly by the correction factors and subject to the optimum design choice where the mass flow rates of regenerative and supply air are nearly equal. These corrections tend to imply that [[theta].sub.r] should be nearly equal to [[theta].sub.s]. The design choice of wheel speed, [omega], is best considered with the corrections which are discussed in the next section. Corrections and Coupling of Heat and Water Vapor Transfer for Dehumidifier Wheels In addition to designing the wheel coating as above, it is important to consider the correction factors for the fully developed flow sensible and water vapor effectiveness. Only the phase change and sorption coupling corrections will be considered here because they dominate. As explained above for corrections to effectiveness, these are different for sensible energy and water vapor. Corrections to the base case sensible and water vapor effectiveness are more important for dehumidifier wheels than energy wheels, because the differences in the inlet air properties are much larger, and the wheel speeds are much lower. Temperature differences in the inlet air impact the air densities, viscosities, heat conduction, and diffusion coefficients and the desiccant coating sorption properties. Low wheel speeds cause the correction to the sensible energy effectiveness to become large due to phase change or water vapor adsorption. The outlet air temperature of each flow channel at the angular position where the flow channel switches to the regenerator sector on the supply side is elevated above that of the base model due to the heating effect of water vapor adsorption on the wheel. At the same time, the outlet air temperature of the regenerator side is decreased due to water vapor desorption. These phase change temperature effects, schematically shown in Figure 10a, indicate that these phase change effects act in such a manner that the sensible energy effective time constant, [[tau].sub.1j], is increased for a dehumidifier wheel relative to the case when there is no phase change (i.e., [[tau].sub.1] for only heat transfer) and the sensible energy effectiveness, [[epsilon].sub.1j], is increased compared with the base case of no phase change (i.e., [[epsilon].sub.1]). The temperature of the desiccant coating is raised for the supply side but decreased for the regeneration side, which will reduce the mass of water vapor that can be adsorbed into coating on the supply side and desorbed on the regeneration side. This modification of the water vapor difference ratio [[THETA].sub.2] versus t* is shown in Figure 10b. This implies that the water vapor effectiveness, [[epsilon].sub.2j], is coupled to the sensible energy effectiveness, [[epsilon].sub.1j]. Low wheel speeds cause these effects to be amplified, because as goes to zero, both [[psi].sub.1] and [[psi].sub.2] do so also, and, as seen in Figures 8 and 9, [r.sub.1][[epsilon].sub.1] and [r.sub.2][[epsilon].sub.2] go to zero directly with [[psi].sub.1] and [[psi].sub.2], respectively. [FIGURE 10 OMITTED] The sensible and latent effectiveness of dehumidifier wheels are coupled due to water vapor phase change and sorption temperature effects, respectively. They can be expressed using a set of iterative or successive approximations (Van Dyke 1964; Pletcher et al. 1988) where the basic approximate solution [[epsilon].sub.1] or [[epsilon].sub.2] base case is used to modify subsequent iterations. For sensible effectiveness, the equation can be written as [[epsilon].sub.1j] = [[epsilon].sub.1] + [K.sub.1]H * [[[epsilon].sub.2,j  1]/[[psi].sub.1,j  1]], (35) where H * = [[[h.sub.ad][DELTA]W*]/[[c.sub.p][DELTA][T.sub.M]]], (36) which is the ratio of the maximum possible phase change energy to sensible energy. For latent effectiveness, the equation can be written as [[epsilon].sub.2j] = [[epsilon].sub.2] + [K.sub.2][1/u][([[partial derivative]u]/[[partial derivative]T]).sub.[phi]][DELTA][T.sub.M][[[epsilon].sub.1,j  1]/[[psi].sub.2,j  1]], (37) where j = 1, 2,... and [[epsilon].sub.10] = [[epsilon].sub.1], [[epsilon].sub.20] = [[epsilon].sub.2], [[psi].sub.10] = [[psi].sub.1] and [[psi].sub.20] = [[psi].sub.2] * [[psi].sub.1,j  1]([[epsilon].sub.1,j1) and [[psi].sub.2,j1]([[epsilon].sub.2,j1])*[1/u][([[partial derivative]u]/[[partial derivative]T]).sub.[phi]] is the temperature sensitivity coefficient for the desiccant coating, as discussed in Shang and Besant (2009a). For the coating of the energy wheels tested, these values of 0.007 to 0.006 [K.sup.1] were measured for this coefficient. [K.sub.1] is an empirical constant that is expected to have a value in the range of 0.05 to 0.1. [K.sub.2] is another empirical coefficient that is expected to be about one half of [K.sub.1]. Laboratory experiments will be necessary to directly measure [[epsilon].sub.s] and [[epsilon].sub.w] to reduce the uncertainty range of [K.sub.1] and [K.sub.2] suggested here. In Equations 35 and 37, each successive approximation improves the result by increasing the effects of coupling between the two. Van Dyke (1964) discusses this type of successive iteration convergence, which limits the choice of [K.sub.1] and [K.sub.2] to small values, while noting that mathematical convergence is not considered in physical problems. These iteration calculations are terminated when the changes are no longer significant relative to the uncertainties. Then, [[epsilon].sub.1j] = [[epsilon].sub.s] and [[epsilon].sub.2j] = [[epsilon].sub.w]. Wheel Speed and Design TradeOffs Several authors have suggested that designers use optimum properties or performance of factors for dehumidifier wheels without completely defining the optimization problem for dehumidifier wheels. From the model presented here, one can easily conclude that if the only criterion for optimization was maximum moisture transfer rate then a high rotation speed should be used (e.g., maybe 5 or 10 rpm or higher) as well as a reasonably high regeneration temperature (e.g., 100[degrees]C [212[degrees]F]). However, then the sensible energy transfer rate would be so high that the outlet temperature would be unacceptable for the supply air. By investigating how the water vapor effectiveness and sensible energy effectiveness change with wheel speed for equal air mass flow rates in the supply and regeneration streams, one can see how these factors tradeoff at low wheel speeds. For example, for a dehumidifier wheel with average water vapor sorption [[tau].sub.2] = 30 s and average sensible energy time constant [[tau].sub.1] = 10 s with [[theta].sub.r] = [[theta].sub.s] = [pi], [[epsilon].sub.1], and [[epsilon].sub.2] versus wheel speed can be plotted, as in Figure 11. For low wheel speeds with [[psi].sub.1] and [[psi].sub.2] < 0.35, the ratio [[epsilon].sub.2]/[[epsilon].sub.1] = 3, which is a constant independent of wheel speed. For [[psi].sub.2] > 0.35, Figure 11 shows some of the nonlinear effects of wheel speed similar to Figure 9. Also plotted on this graph are the curves for [[epsilon].sub.s] and [[epsilon].sub.w] which include the corrections to [[epsilon].sub.1] and [[epsilon].sub.2] for phase change and sorption temperature effects. It can be seen from these low wheel speed curves that the ratio of expected effectiveness, [[epsilon].sub.w]/[[epsilon].sub.s], is not a constant independent of wheel speed. So, there are design choices or tradeoffs to be made. If the wheel speed is too low, the corrected sensible and water vapor effectiveness [[epsilon].sub.s] and [[epsilon].sub.w] will not only be very low; [[epsilon].sub.s] will remain positive while [[epsilon].sub.w] can even be negative as [omega] [right arrow] 0. When this happens, the total energy or enthalpy change will go to near zero for each airstream. Some manufacturers have employed this feature of desiccantcoated wheels to decrease the moisture content of the supply airstream after a cooling coil. [FIGURE 11 OMITTED] The design choice or tradeoff for dehumidifier wheels could be the selection of [[epsilon].sub.s] or the ratio [[epsilon].sub.w]/[[epsilon].sub.s], or the maximum value of [[epsilon].sub.s] or the minimum value of [[epsilon].sub.w] that is acceptable for each particular HVAC design. Each of these choices leads to a different wheel speed. For example, for a minimum value of [[epsilon].sub.w]/[[epsilon].sub.s] = 1.0 or 1.5, the wheel speed determined for [K.sub.1] = 0.1 and [K.sub.2] = 0.05 will be [omega] = 0.026 or 0.045 rad/s and for [K.sub.1] = 0.05 and [K.sub.2] = 0.025, [omega] = 0.014 or 0.025 rad/s, respectively. If, on the other hand, [[epsilon].sub.w]/[[epsilon].sub.s] = 2.0 was the minimum value, then this graph shows that, for the time constants used, this result cannot be achieved. This simple example suggests that dehumidifier wheel speeds might be better chosen by accurately knowing the values of [K.sub.1] and [K.sub.2], as well as the time constants and [[tau].sub.1], and [[tau].sub.2], plotting a graph similar to Figure 11. SUMMARY AND CONCLUSION In this paper, new performance factors are defined for dehumidifier wheels (i.e., the sensible and modified water vapor effectiveness and water vapor change mass ratio) and other heaterwheel system performance factors, such as COP, are discussed. A new theoretical model is developed for dehumidifier wheels, which is comprised of a base case, assuming fully developed laminar flow in each flow channel, and a decoupling of the heat transfer and water vapor transfer. This base case results in simple algebraic relationships between sensible energy and water vapor effectiveness and a dimensionless ratio of wheel speed times the time constant for the flow channel divided by the wheel sector angle for regeneration (or supply) airflow. New dimensionless graphs show these base case effectiveness relationships for each performance factor for both parallel and counterflow. These graphs are very similar to those found in heat transfer textbooks, but they differ in that the independent variables are explicit functions of wheel speed, time constants, and wheel flow sector angle. It is shown from this base case model that the mass flow rate of air through the supply and regenerator wheel sectors should be equal for maximum dehumidifying performance. Guidance is provided to manufacture wheel matrix designs that will have a sensible energy transfer time constant that is smaller than the water vapor transfer time constant. Methods are stated to determine or measure these resulting heat and water vapor time constants. The base case models for heat and water vapor effectiveness and water vapor mass ratio are corrected for the effects of flow channel entrance and heat conduction effects, as well as carryover and manufacturing variations, even though these corrections may be relatively small. It is shown that heat and water vapor transfer coupling effects cause the wheel performance factors to change significantly, especially as the wheel speed decreases to low values. These large corrections to sensible and water vapor effectiveness are added and coupled in an iterative matter that is well established in the literature. For example, the sensible effectiveness increases while the water vapor effectiveness decreases relative to the base case. For the water vapor effectiveness, it varies in such a manner that the water vapor effectiveness can become negative while the sensible effectiveness is positive as the wheel speed goes toward zero. It is recommended that more experimental research be done to more accurately determine the coupling coefficients between heat and water vapor effectiveness. ACKNOWLEDGMENT This research was financially supported by the National Science and Engineering Research Council of Canada (NSERC). NOMENCLATURE A = airflow crosssectional area, [m.sup.2] [c.sub.p] = specific heat capacity at constant pressure, J/(kg*K) [c.sub.v] = specific heat capacity at constant volume, J/(kg*K) [d.sub.h] = flow channel hydraulic diameter, m H* = operating condition factor, defined in Equation 36, dimensionless [H*.sub.r] = operating condition factor, defined in Equation 7, dimensionless h = heat transfer coefficient, W/([m.sup.2]*K) [h.sub.ad] = specific heat of phase change, J/kg [K.sub.1] = empirical constant, dimensionless [K.sub.2] = empirical constant, dimensionless L = flow channel length, m m = mass flow rate, kg/s N = number of transfer units, dimensionless P = flow channel perimeter, m p = pressure, Pa [r.sub.w] = mass flow ratio, dimensionless T = temperature, K T' = bulk mean temperature, K t = time, s t* = dimensionless time u = moisture content, kg/kg [bar.u] = average moisture content, kg/kg V = air velocity, m/s W = humidity ratio, [kg.sub.water]/[kg.sub.dry air] Greek Symbols [alpha] = coefficient, dimensionless [beta] = coefficient, dimensionless [DELTA] = difference [delta] = thickness, m [epsilon] = effectiveness, dimensionless [phi] = relative humidity, dimensionless [eta] = efficiency of heating device, dimensionless [THETA] = normalized temperature or humidity, dimensionless [theta] = angle, radian [rho] = density, kg/[m.sup.3] [tau] = time constant, s [omega] = angular speed, rad/s [PSI] = constant, dimensionless [psi] = constant, dimensionless Subscripts a = air CF = counterflow heat exchanger co = carryover cond = heat conduction d = desiccant e = exhaust ent = entrance effect FD = fully developed flow h = heat device i = inlet or i = 1, 2. l = latent energy M = maximum or minimum m = wheel matrix or mass transfer max = maximum min = minimum o = outlet PF = parallel flow heat exchanger pc = phase change r = regenerator air or sector s = sensible energy or supply air or sector t = total energy or enthalpy tot = total effect temp = temperature effect var = flow channel size varation w = water vapor REFERENCES Abe, O.O., R.W. 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