Effectiveness of desiccant coated regenerative wheels from transient response characteristics and flow channel propertiespart I: development of effectiveness equations from transient response characteristics.
INTRODUCTION
Twentyfive years ago airtoair exchangers were rarely used to recover heat from exhaust air in buildings, but now that they can transfer both heat and water vapor, they are included in most HVAC designs for commercial buildings. When energy wheels are employed in HVAC designs, the capacity of auxiliary heating and cooling equipment is reduced (Asiedu et al. 2004, 2005). ANSI/ASHRAE Standard 842008, Method of Testing AirtoAir Heat Exchangers (ASHRAE 2008), sets out a method of test to determine the performance of airtoair exchangers for transferring heat and water vapor between supply and exhaust air streams, as shown in Figure 1. At any operating condition, this performance is characterized by the determination of the dimensional [delta][p.sub.s] and [delta][p.sub.e] for pressure drop in the supply and exhaust air streams and six dimensionless factors: [[epsilon].sub.s], [[epsilon].sub.l], [[epsilon].sub.t], EATR, OACF, and RER for the sensible, latent or moisture, and total energy effectiveness, exhaust air transfer ratio, outdoor air correction factor, and recovery efficiency ratio. These performance factors will, in general, vary for each operating condition, so ARI Standard 1060, Rating AirtoAir Energy Recovery Equipment (ARI 2005) restricts its certification effectiveness tests to only one summer and winter test condition. To test large commercial airtoair exchangers, these performance factors require a large test facility with expensive instrumentation and, most importantly, a sophisticated online data acquisition and analysis methodology. As a consequence, only research or special test laboratories can be expected to acquire and analyze accurate test data for a few operating conditions, and then only when testing small or mediumsized commercial exchangers. [FIGURE 1 OMITTED] The desiccant coated airdrying or dehumidifier wheel is closely related to the energy wheel because it too transfers heat and water vapor. The energy wheel requires no external power other than for wheel rotation while the dehumidification wheel requires a high regeneration air temperature and flows that will differ from the supply air. The construction of these drying wheels, with desiccant coatings on the wheel matrix surfaces, is nearly identical to that of energy wheels; however, the operating conditions differ with respect to the regeneration temperature, wheel speeds, airflow speeds, and the fraction of the wheel used for regeneration. For example, desiccant drying wheel speeds of less than 0.5 rad/s are typically 10 to 100 times lower than energywheel speeds, and inlet air temperature differences for drying wheels are typically 10 to 20 times larger than those for energy wheels at ARI certification operating conditions. The performance of desiccant drying wheels may be characterized by a similar set of factors as energy wheels; however, some factors, such as sensible and total energy effectiveness, EATR and OACF are much less significant and are usually not considered while the regenerator input energy needs to be characterized by a dimensionless coefficient of performance (COP) (Charoensupaya and Worek 1988; Van den Bulck et al. 1988). A question that may be asked here is What is physically occurring in these two different regenerative wheel applicationsair drying and energy recoverythat accounts for the two different types of wheel performance? Decades of research on desiccant coated energy wheels, which are now common in HVAC designs in North America and elsewhere, has revealed that great care must be taken to minimize laboratory testing errors and uncertainty in obtaining the new ASHRAE performance factors (Simonson and Besant 1998; Shang et al. 2001). By and large, field testing is impractical. Alternatively, validated simulation methods and correlations can be used to predict the difficulttomeasure effectiveness, but it still requires many flow channel property data (Simonson et al. 1999; Simonson and Besant 1999), so different tests and measurements are required to get these important flow channel properties. Recently, Abe et al. (2006a, 2006b) devised a simple transient test method in an attempt to determine the characteristic temperature and humidity step response of a stationary energy wheel and, using an analytic model, predict the sensible and latent energy effectiveness. Considering the uncertainty in their data, the agreement between these predictions using only transient test data and steadystate test data appeared to be satisfactory, but small bias differences with steadystate test data suggested that corrections should be applied to these transient test results, because there were small differences between the transient test predictions and the steadystate test data. The literature on desiccant drying wheels or rotary dehumidifiers also includes extensive research investigations over the past few decades. Jurinak and Mitchell (1984) used a finite difference model to investigate the effect of matrix properties on the performance of a counterflow rotary dehumidifier. First assuming infinite transfer coefficients, Van den Bulck et al. (1985) developed correlations for the humidity and enthalpy effectiveness of rotary heat and water vapor transfer wheels. Zheng and Worek (1993) presented a numerical model similar to that of Jurinak and Mitchell to simulate the combined heat and mass transfer processes that occur in a rotary dehumidifier and investigate the effect of the rotational speed on the performance of the dehumidifier. Van den Bulck and Klein (1990) used a singleblow transient test procedure to determine the overall heat and mass transfer coefficients of dehumidifier matrices. Their analysis techniquebased upon the transformation of the model partial differential equations into a set of ordinary differential equationsand the temperature and massfraction distributions are modeled by a system of nonstiff ordinary differential equations, which can be integrated numerically. More recently, Golubovic et al. (2006) presented sorption property data for different types of molecular sieves in equation form and investigated the influence of different assumptions for heat of sorption and equilibrium equation of molecular sieve on predicted optimum performance of a rotary dehumidifier. Some authors have considered both energy wheels and rotary dehumidifiers. For example, Zhang and Niu (2002) assumed desiccant film equilibrium in their numerical models for both recovery energy wheels and rotary dehumidifying wheels and predicted moisture effectiveness for each type of wheel. Most recently, Shang and Besant (2008) presented a theoretical analysis that can be used to correct transient test data for the sensible effectiveness of energy and heat wheels and showed that accurate sensible effectiveness values can be predicted using only the property data of the wheel matrix flow channels, wheel speed, and the inlet airflow properties. The questions that now need to be considered are: Can a similar theoretical transient model, combined with appropriate corrections, be developed for the latent or moisture transfer effectiveness in energy wheels? Can this model be applied to desiccant drying wheels, albeit with slightly modified flow channel properties? In this paper, the theoretical model for sensible effectiveness (Shang and Besant 2008) is modified to predict the latent energy response of the flow channels, outlet air humidity, and the energy wheel effectiveness. Using the properties of the desiccant coating on the flow channels of energy wheels, this model is modified to show the latent energy or moisture transfer effectiveness. Using these models, the sensible, latent, and total effectiveness can be predicted for energy wheels knowing only the operating conditions and the flow channel property data. The potential savings, resulting from this prediction method, will accrue from the avoidance of the high cost of ARI effectiveness certification testing for manufacturers (ARI 2005). More importantly, this theory will assist the design of highperformance energy wheels, and the design of HVAC systems that employ energy wheels, that must operate under a wide range of operating conditions. Finally, manufacturers and HVAC designers can be assured of targeted effectiveness values of energy wheels for specified operating conditions, provided they maintain good quality control for the manufacture of the airflow channels in each wheel matrix. Rotary dehumidifiers, which also require a high moisture transfer effectiveness, will need a modified analysis before they can be designed and operated to maximize their performance. Effectiveness of heat recuperators follow simple correlations with two independent dimensionless parameters for airtoair exchangers (Kays and London 1984). This effectiveness is used by designers to establish the heat rate and energy savings provided by a heat exchanger. Since effectiveness tends to be nearly constant over a range of operating conditions for heat exchangers, it is the preferred method of design. In addition, regenerative heat wheels, operating over a narrow range of wheel speeds, tend to follow similar correlations with three independent dimensionless parameters (Kays and London 1984). The sensible and latent effectiveness of energy wheels is a more complex function of several dimensionless variables (Simonson and Besant 1999) and is much more sensitive to inlet operating conditions than heat recuperators or heat wheels. This new theoretical method to predict effectiveness raises fundamental questions about whether effectiveness should be specified as the primary performance factor for the design of energy wheels since it is not exactly constant for an energy wheel and it can be directly determined for any operating condition knowing the properties of the wheel flow channels. For wheel design purposes, it may be more important to simply provide the data for the wheel matrix flow channels and let the energy wheel designer or HVAC system designer use the appropriate algorithms to determine the effectiveness for each operating condition. Of course, the HVAC system designer will want the effectiveness to estimate the energy saved in particular applications. It is anticipated that knowledge of the flow channel properties would level the playing field among various energy wheel manufacturers and prevent unsubstantiated effectiveness claims, because effectiveness could be readily computed for a given wheel matrix operating under specified conditions. FULLY DEVELOPED FLOW AND CORRECTIONS FOR SENSIBLE EFFECTIVENESS Regenerative energy wheels achieve high values of sensible and latent effectiveness by transferring heat and water vapor on large surface areas of the rotor matrix. Exchanger surface areas per unit volume of 3000 to 4500 [m.sup.2]/[m.sup.3] are about ten times larger in energy wheels than those used in liquidtoair coil tube exchangers for HVAC systems. The airflow in energy wheels is laminar, with flow tube Reynolds number ranging from 150 to 800. The essential heat and mass transfer characteristics are, with the exception of a flow tube leading edge region, constant along the flow tube length. The equations for these characteristics are well known in the literature, making it possible to develop a simulation model for energy wheels based on physical principles (Simonson and Besant 1998), which is then validated against measured effectiveness values (Simonson et al. 1999), and used to develop correlations for effectiveness (Simonson and Besant 1999). Such a model must not only include adjustments or corrections for the flow tube entrance but, in addition, axial heat conduction in the flow tube matrix in a counterflow arrangement, carryover between the supply and exhaust due to wheel rotation, phase change temperature effects as water vapor is adsorbed or desorbed by the flow tube desiccant coating with their temperature dependent properties. It is interesting to observe that each of these effects are usually small but significant for typical operating conditions, so the corrections are essential if one is to accurately predict values of effectiveness. Finally, Shang and Besant (2005) showed that flow channel hydraulic diameter variations due to manufacturing tolerances could lower the effectiveness, even though the effect is usually small. For a welldesigned energy wheel, we assume that each of these corrections cause changes to the effectiveness compared to the base case or fully developed flow case for each flow channel in the wheel matrix. First, the sensible effectiveness equations for fully developed flow are presented ([[epsilon].sub.S,FD]), and then corrections are added for the effects due to entrance ([delta][[epsilon].sub.S,ent]), axial heat conduction ([delta][[epsilon].sub.S,cond]), carryover ([delta][[epsilon].sub.S,co]), water vapor phase change ([delta][[epsilon].sub.S,pc]), and flow channel size variations ([delta][[epsilon].sub.S,var]): [[epsilon].sub.S] = [[epsilon].sub.S,FD] + [delta][[epsilon].sub.S,ent] + [delta][[epsilon].sub.S,cond] + [delta][[epsilon].sub.S,co] + [delta][[epsilon].sub.S,pc] + [delta][[epsilon].sub.S,var] (1) where it is implied that each of these corrections are smaller than the fully developed flow base case and independent. For correction, i, is defined as follows: [delta][[epsilon].sub.S,i] = [delta][[epsilon].sub.S,i][delta][[epsilon].sub.S,FD] (2) and the relative size of each sensible energy effectiveness correction is [[[delta][[epsilon].sub.S,i]]/[[delta][[epsilon].sub.S,FD]]]. (3) Figure 1 shows the flow and inlet and outlet stations for a regenerative wheel with counter flow. The development of the equation for the fully developed flow sensible energy effectiveness, [delta][[epsilon].sub.S,FD], starts with the balance of thermal energy for one flow channel of area, A, perimeter, P, and hydraulic diameter, [d.sub.h], in the wheel matrix, as shown in Figure 2. Air flowing through this channel at a constant bulk mean properties, [V.sub.a], [[rho].sub.a], and [c.sub.pa] for speed, density, and specific heat. Since the Biot number for heat convection is small (i.e., Bi<0.1) at any position within each flow channel, only wall temperature changes along the length of a flow channel are significant. The mean or average wall temperature of the matrix surface along the channel length, L, at any one time is [[bar.T].sub.m], and the corresponding bulk mean air temperature averaged over the length at the same time is [bar.T]. Both the matrix surface temperature, [T.sub.m], and the air temperature, T, have a temperature distribution at any time which varies linearly with z (Romie 1979; Bahnke and Howard 1964); however, these linear distributions will change with time as the outlet air temperature changes with wheel rotation. Since phase change effects are considered to be one of the corrections, they are not included in the development of [delta][[epsilon].sub.S,FD]. [FIGURE 2 OMITTED] The thermal energy equation for one flow channel is [([Mc.sub.p]).sub.m][[d[[bar.T].sub.m]]/[dt]] =  hPL([[bar.T].sub.m]  [bar.T]), (4) which is a balance between energy storage rate and the convective heat rate from the air flowing through the channel and where [(M[c.sub.p]).sub.m] is the mass specific heat product for the flow channel, and hPL is the heat convection coefficient surface area product for the interface between the air and the flow channel. The average temperature of the air at any time inside the flow channel is the average of the inlet and outlet air temperature: [bar.T] = [1/2]([T.sub.i] + [T.sub.o]) (5) At any time, the thermal energy loss or gain rate by the flow channel is balanced by a corresponding gain or loss rate by the air which is written as follows: hPL([[bar.T].sub.m]  [bar.T]) = [(m[c.sub.p]).sub.a]([T.sub.o]  [T.sub.i]) (6) Here, the heat transfer process in each flow channel is dominated by the convection resistance [(hPL).sup.1]. Noting that the inlet air temperature, [T.sub.i], is constant any time the flow channel is in either the supply or exhaust air stream, we can combine Equations 46 to get an equation for the outlet air temperature: {[[(M[c.sub.p]).sub.m]/[(m[c.sub.p]).sub.a]][[[[1/2] + ([rho][c.sub.p]V).sub.a][d.sub.h]]/[4hL]]}[[d[T.sub.o]]/[dt]] + [T.sub.o] = [T.sub.i] (7) where the bracketed term, rewritten as [[tau].sub.S] = [[4([rho][c.sub.p][delta]).sub.m]/([rho][c.sub.p]V).sub.a](L/[d.sub.h])[[1/2] + [1/4St]([d.sub.h]/L)], (8) is the time constant for the outlet air, and St is the Stanton number for the flow channel [h/[([rho][c.sub.p]V).sub.a])]. As implied by Equation 4, the time constant for the solid components of the flow channel is as follows: [[tau].sub.m] = [[(M[c.sub.p]).sub.m]/[hPL]] = [[([rho][c.sub.p][delta]).sub.m]/[([rho][c.sub.p]V).sub.a]][1/[St]] (9) In the subsequent analysis of the heat transfer effects during an adiabatic moisture transfer process, we use these equations again where the forcing function is time dependent. It is interesting to note that the convective heat transfer coefficient, h, is a constant for fully developed laminar flow in a flow channel of a specified profile, so the mass flow rate of air is the only parametric term that can be changed for a given wheel. As the energy wheel rotates at an angular speed, [omega], the inlet temperature, [T.sub.i], of each flow channel switches from supply inlet (1) to exhaust inlet (3) and back again (as shown in Figure 1) every instant the flow channel passes under one of the two contact seals that separates the supply from the exhaust. In this analysis, it is assumed that this occurs with each 180[degrees] of wheel rotation. It is also assumed that the supply and exhaust mass flow rates are equal. Then this flow channel inside a wheel matrix will have the inlet air go through a series of cyclic step changes and the outlet air temperature will be determined using only Equation 7. Shang and Besant (2008) show that the corresponding sensible effectiveness for fully developed parallel flow in an energy wheel, which operate at speeds greater than 0.5 rad/s (5 rpm), will be as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10a) [[epsilon].sub.S,FD  PF] = [1/2](1  [e.sup.[  2N]]) (10b) where the number of transfer units, N, is a function of the product of wheel speed, [omega], and time constant, [[tau].sub.s]. For equal flow areas for supply and exhaust we get the following: N =  [1/2]ln[1  [[2[omega][[tau].sub.S]]/[pi]]*[[(1  [e.sup.[ [pi]/[omega][[tau].sub.S]]])].sup.2]/[(1  [e.sup.[  2[pi]/[omega][[tau].sub.S]]])]] (11) For the same value of [omega][[tau].sub.s], the value of N for a counterflow energy wheel will be the same as for parallel flow, so the effectiveness for a counterflow energy wheel will be as follows: [[epsilon].sub.S,FD  CF] = [N/1 + N] (12) Equations 8, 10, 11, and 12 provide the sensible energy effectiveness model for fully developed airflow in regenerative wheels. For desiccant drying wheels that are used to transfer water vapor out of the supply air and that operate at speeds less than 0.5 rad/s (5 rpm), the product [omega][[tau].sub.s] may become a small characteristic angle with respect to one half revolution of [pi] radians, so Equation 11 will not converge as rapidly. More importantly, axial heat conduction effects in the wheel matrix flow channels, which are discussed below, will become very large for metallic matrix channels. Abe et al. (2006a and 2006b) showed that transverse heat conduction effects in the plane of wheel rotation alter the results for sensible effectiveness when transient tests are used for a small part of a wheel. As a consequence of these matrix heat transfer effects and the desire by desiccant drying wheel users to minimize the transfer of heat into the supply air from the hightemperature regeneration air, dryer wheel speeds are either reduced, or nonmetallic matrix materials are used in their construction, or a purge section is used to drop the temperature of the wheel matrix to a value close to that of the supply air. Consequently, only the moisture transfer effectiveness is important for desiccant drying wheels. For energy wheels, we need to apply each of the aforementioned corrections in Equation 12 to get the predicted sensible effectiveness, [[epsilon].sub.s]. These are presented and discussed by Shang and Besant (2008). FULLY DEVELOPED FLOW FOR WATER VAPOR TRANSFER The convection and diffusion of water vapor in airflow through the flow channel is similar to the convection and conduction of heat in air, so the equations for balance of water mass in the desiccant coating of each flow channel is expected to be similar to those for sensible energy and heat transfer. However, the physics of water vapor transport and sorption in the porous desiccant wheel matrix film coatings is different from that for heat transfer. Similar to Equation 1, the latent effectiveness of an energy wheel can be expressed as the fully developed flow channel effectiveness plus corrections: [[epsilon].sub.l] = [[epsilon].sub.l, FD] + [delta][[epsilon].sub.l,ent] + [delta][[epsilon].sub.l,co] + [delta][[epsilon].sub.l,temp] + [delta][[epsilon].sub.l,var] (13) where it is implied that each of these corrections are small and independent (Van Dyke 1964), so for correction, k, it is defined as follows: [delta][[epsilon].sub.l,k] = [[epsilon].sub.l,k]  [[epsilon].sub.l,FD] (14) and the relative size of each correction is [[[delta][[epsilon].sub.l,k]]/[[epsilon].sub.l,FD]]. (15) The equation for the balance of water mass at any time, but averaged over the length, L, of the flow tube shown in Figure 2 is [M.sub.d][d[bar.u]/dt] =  [h.sub.w]PL([[bar.[rho]].sub.w,da]  [[bar.[rho]].sub.w,a]), (16) where [M.sub.d] is the mass of desiccant, [bar.u] is the average moisture content of the desiccant, [h.sub.w] is the convection coefficient for fully developed flow for water vapor transfer at the aircoating interface, [[bar.[rho]].sub.w,da] is the average density of water vapor on the same interface, and [[bar.[rho]].sub.w,a] is the average bulk mean density of water vapor in the airflow channel. This last term is simply the average of the inlet and outlet water vapor density for the airflow channel [[bar.[rho]].sub.w,a] = [1/2]([[rho].sub.w,ai] + [[rho].sub.w,ao]). (17) At any time, there is a balance between the water vapor lost by the desiccant coating at the interface of the channel airflow and the desiccant coating and gained by the flowing air for desorption and vice versa for adsorption: [h.sub.w]PL([[bar.[rho]].sub.w,da]  [[bar.[rho]].sub.w,a]) = [m.sub.a]([W.sub.o]  [W.sub.i]) (18) where [W.sub.o] and [W.sub.i] are the humidity ratios of air, [[rho].sub.w]/[[rho].sub.a], at the outlet and inlet, and [m.sub.a] is the mass flux of dry air in the flow channel. Combining Equations 17 and 18 shows that the interface water density at the desiccantair interface, [[bar.[rho]].sub.w,da], is directly proportional to [w.sub.o] when [W.sub.i] is constant. [[bar.[rho]].sub.w,da] = ([m.sub.a]/[[h.sub.w][P.sub.d]L] + [[rho].sub.a]/2)[W.sub.o]  ([m.sub.a]/[[h.sub.w][P.sub.d]L]  [[rho].sub.a]/2)[W.sub.i] (19) It is important to note that [[bar.[rho]].sub.w,da] for the desiccant surface is different from the water density average throughout the bed of desiccant at any time in the film coating, [[bar.[rho]].sub.w,d], because the moisture content at any time for a coating of homogeneous density, [[rho].sub.d], is given by the following: [bar.u] = [[bar.[rho]].sub.w,d]/[[rho].sub.d] (20) and this varies normal to the airdesiccant interface throughout the coating. Generally, the moisture content of the desiccant coating at thermodynamic equilibrium, u, will be a nonlinear function of the adjacent air relative humidity and temperature (Ruthven 1984); however, over the typical operating range (e.g., the selected ARI summer or winter test conditions), the sensitivity of coefficients for u, [([[partial derivative]u]/[[partial derivative][phi]]).sub.T] and [([[partial derivative]u]/[[partial derivative]T]).sub.[phi]], can be assumed constant for typical inlet operating conditions. Furthermore, when this moisture content is averaged over the length of the flow channel at any time, [bar.u] will have a small variation with temperature because the average temperature of the matrix will only change by a small amount during each wheel cycle. This temperature sensitivity can cause a small change in the latent energy effectiveness, [[epsilon].sub.1], and it will be shown that this temperature sensitivity is small for energy wheels but not for dehumidifier wheels. Now we need to combine Equations 16 and 18 to express [bar.u] as a function of the average relative humidity. It is convenient to introduce the normalized variables w* and W* for average moisture content of the matrix and the air so that Equation 16 can be written as follows: [[tau]*.sub.w][[dw*]/dt] =  W* (21) where the sorption time constant for the step change is: [[tau]*.sub.w] = [[[M.sub.d][delta][u.sub.M]]/[[m.sub.a][delta][W.sub.M]]]. (22) w* = [[[bar.u]  [[bar.u].sub.initial]]/[[[bar.u].sub.final](t[right arrow] [infinity])  [[bar.u].sub.initial]]] = [[[delta]u]/[[delta][u.sub.M]]] (23) is a dependent variable that is not convenient to measure during a step change or a wheel rotation. On the other hand, the air humidity ratio, W* = [[[W.sub.o]  [W.sub.i]]/[[W.sub.final](t[right arrow] [infinity])  [W.sub.i]]] = [[[delta]W]/[[delta][W.sub.M]]], (24) is a dependent variable that is more convenient to measure for a step change or at steadystate operating conditions. For both Equations 23 and 24, the full reference humidity step changes are used to normalize w* and W*. The mass of moisture interactive desiccant and mass flow rate of airflow are as follows: [M.sub.d] = [[bar.[rho]].sub.d][[bar.[delta]].sub.d][[bar.P].sub.d]L[beta] (25) [m.sub.a] = [[bar.[rho]].sub.a][[bar.V].sub.a][1/4]P[d.sub.h] (26) where [[bar.[rho]].sub.d] = [1/L][L.[integral].0][[rho].sub.d]dx is the average desiccant coating density; [[bar.[delta]].sub.d] = [1/L][L.[integral].0][[delta].sub.d]dx is the average desiccant thickness; [[bar.P].sub.d] = [1/L][L.[integral].0][P.sub.d]dx is the average perimeter of desiccant in the flow channel; and [beta] is the mass fraction of desiccant film that is exposed and at equilibrium for water sorption during the step change. There have been many investigations of moisture transfer and sorption in porous beds of materials, such as insulation and fertilizer particles. When an effective diffusion coefficient can be used to characterize the diffusion and sorption in the bed, it will transfer water as a diffusion process with a much lower effective diffusion coefficient (Olutimayin and Simonson 2005). This coefficient, [beta], which depends on the properties of the desiccant coating, will only go toward 1.0 as the time after the start of a step change becomes very large (i.e., the case of very slowly rotating dehumidifiers); but, for very short time durations (i.e., energy wheels), it will be smaller because the change in moisture content of the matrix desiccant may not saturate the total desiccant depth, [[delta].sub.d]. When [bar.P] = P, [[tau]*.sub.w], can be written as a product or ratio, so Equations 25 and 26 give an explicit relation among four dimensionless numbers and (L/[V.sub.a]): [[tau]*.sub.w] = 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]) (27) where [W.sub.s]([bar.T]) is the saturation humidity ratio of air at the mean temperature of the inlet air temperatures and [[([partial derivative][bar.u]/[partial derivative][phi])].sub.T] is the average slope of the moisture adsorption isotherm for the desiccant for the two inlet temperature conditions. Although Equations 16 and 18 imply that we have two dependent variables, u and [W.sub.o], Equation 21 indicates that the dimensionless normalized moisture content of both the desiccant coating, w*, and the air, W*, are equal at any time; so we can rewrite Equation 21 as an ordinary firstorder differential equation similar to Equation 7: [[tau]*.sub.w][[dW*]/[dt]] + W* = 0 (28) The problem with this equation is that the physics of the water vapor diffusion process into the desiccant coatings needs to be explained in more detail before [[tau]*.sub.w]can be evaluated. TIME CONSTANTS FOR LATENT STEP RESPONSE Figure 3 shows a schematic of the cross section of a metal sheet coated with desiccant particles on both sides of the metal sheet. The particles are bonded on the surfaces of the metal sheet. [FIGURE 3 OMITTED] A scanning electron microscope image of a typical energy wheel matrix flow channel element is shown in Figure 4. This desiccant coated surface shows an image of the plan view (a) and one crosssection view (b) with desiccant particles on both sides of an aluminum sheet. The bonding material, which is not clearly seen in plan view, is very clearly seen in crosssection. This implies that the particles near the airparticle interface should be able to respond more quickly to a change in vapor pressure than those several particle layers away from this interface, and those near the aluminum will be slowest. The images for the molecular sieve coated wheel are similar to Figure 4. The desiccant coating properties are summarized in Table 1. [FIGURE 4 OMITTED] Table 1. Typical Desiccant Coating Properties of Flow Channels with Estimated Thermal Properties for Two Different Desiccant Coatings Particle Fraction of Particle Interparticle Coating Particle Energy Size Void Depth Obstructed Wheel Range, Size Range, Range, by Bonding Coating [micro]m [micro]m [micro]m Material Molecular 0.5~2.5 1~20 34~38 0.1~0.6 sieve Silica gel 0.4~2.0 0.5~5 19~29 0.2~0.7 Particle Specific Coating Heat for Aluminum Aluminum Energy Density Range Coating Foil Foil Wheel [[rho].sub.d], [c.sub.p], Thickness, Density, Coating kg/[m.sup.3] J/kg*K [micro]m kg/[m.sup.3] Molecular 1600~1900 700 30 2700 sieve Silica gel 1700~2000 780 20 2790 Specific Heat for Energy Aluminum Wheel [c.sub.p], Coating J/kg*K Molecular 902 sieve Silica gel 880 The equilibrium sorption properties of the particles and the desiccant coatings are both of direct interest for energy wheels because they imply the maximum change in moisture content that can occur between any two inlet temperature and humidity conditions. Figure 5 shows these isotherms for the molecular sieve and silica gel particles and coatings. These graphs show that the particles alone behave differently than the coatingsi.e., they have significantly lower moisture contents and lower gradients of moisture content for the typical range of operating conditions for energy wheels. In addition, the changes in these isotherms are likely to be very small for energy wheels but more significant for dehumidifier wheels because they are subjected to large temperature changes. [FIGURE 5 OMITTED] Experimental studies of the transient humidity step change response of the flow channels in typical energy wheels showed that, using 1000 data (the time interval between two measurements is one second), the best correlation equation fit (i.e., [r.sup.2]>0.99) for both molecular sieve and silica gel between outlet humidity response [delta][W.sub.o] after a step input increase, [delta][W.sub.i] (adsorption) was always of the following form (Wang et al. 2005): W* = [[[delta][W.sub.o]]/[[delta][W.sub.i]]] = 1  [X.sub.1][e.sup.[  t/[[tau].sub.w1]]]  [X.sub.2][e.sup.[  t/[[tau].sub.w2]]] (29) where [X.sub.1] and [X.sub.2] are weighting factors such that [X.sub.1]+[X.sub.2] = 1 and [[tau].sub.w1] and [[tau].sub.w2] are two different time constants with [[tau].sub.w2] [much greater than] [[tau].sub.w1] and [X.sub.2]<[X.sub.1]. The second, much larger time constant appears to be a consequence of the water vapor's very slow effective diffusion toward and saturation of the layers of desiccant coating particles closest to the aluminum film, while the first time constant is more representative of the effective diffusion in the particle layers in the middle and closer to the airparticle interface when a large number of data are used in the correlation. For very short time intervals, the adiabatic process time constants of the humidity sensors (e.g., about 3.0 s) implies that there will be errors in any correlation for data analysis durations of less than three seconds. Similarly, for a step decrease in humidity (desorption), the best correlation is as follows: W* = [[W.sub.o]/[W.sub.i]] = [X.sub.1][e.sup.  t/[[tau].sub.w1]] + [X.sub.2][e.sup.[  t/[[tau].sub.w2]]] (30) with the magnitude of each of the parameters [X.sub.1], [X.sub.2], [[tau].sub.w1], and [[tau].sub.w2] nearly equal to those for adsorption. Table 2 shows these average data and their uncertainties for these parameters for the molecular sieve and silica gel coated wheels tested. Table 3 presents the geometric properties of the flow channels for these energy wheels. Table 2. Measured Time Constants for Molecular Sieve and Silica Gel Coatings for Two Energy Wheels with Corrugated Flow Channel Geometry [N.sub.d], Energy Wheel Number of [[tau].sub.w1] Coating Data [X.sub.1] (second) [X.sub.2] 1000 0.80 7.1[+ or ]0.7 0.20 MS 30 ~1.0 11.96 ~0 L = 100 mm [V.sub.a]= 2.0 m/s 20 ~1.0 11.22 ~0 10 1.0 9.75 0 1000 0.93 6.3[+ or ]0.8 0.07 SG 30 ~1.0 6.91 ~0 L = 100 mm [V.sub.a]= 2.0 m/s 20 ~1.0 6.83 ~0 10 1.0 6.51 0 Energy Wheel [[tau].sub.w2] Coating (second) 135[+ or ]15 MS L = 100 mm [V.sub.a]= 2.0 m/s 215[+ or ]75 SG L = 100 mm [V.sub.a]= 2.0 m/s Table 3. Geometric Properties of the Corrugated Energy Wheel Flow Channels Flow Mean Flow Wave Wave Channel Hydraulic Channel Energy Height, Length, Length, Diameter, Aspect Wheel mm mm mm mm Ratio, [eta] Molecular 1.30 3.81 100 1.15 0.34 sieve Silica gel 1.83 4.05 100 1.53 0.45 Manufacturing Energy Quality Factor, Wheel [sigma]/[d.sub.h] Molecular 0.07 sieve Silica 0.05 gel The time constants presented in Table 2 depend on the time duration of the data used in the correlation or the number of data, [N.sub.d]. It was found by Wang (2005) that these time constants decreased significantly with increasing time duration or number of data points, [N.sub.d]. Conversely, as [N.sub.d] decreases, [[tau].sub.w1] increases. For example, there was a 15% increase in the first time constant, [[tau].sub.w1], for a molecular sieve coated wheel as the number of points decreased from [N.sub.d] = 30 to 10 and, in addition, the weighting factor for [X.sub.1] approached 1.0. For an energy wheel, operating at a typical wheel speed of [omega] = 2.1 rad/s (20 rpm), step changes occur in the inlet humidity every half revolution, or 1.5 s, so we need to establish the average time constant, [[tau].sub.w], for 0[less than or equal to]t[less than or equal to][pi]/[omega] s for [X.sub.1] = 1.0. Using the method by Wang et al. (2005) we cannot directly measure humidity time responses with good accuracy for such a short time duration, because the time constant of the calibrated humidity sensor is 3.0 s for this process, which is nearly isothermal. We could, however, extrapolate the time constant data to the time period 0[less than or equal to]t[less than or equal to][pi]/[omega] using the time constant data from the correlations for longer durations (i.e., large data sets), but we must keep in mind that the accuracy of the extrapolation will decrease as [pi]/[omega] goes toward 0 s (i.e., the uncertainty will increase as the number of data points decrease and the transient characteristics of the sensor become more important). The extrapolation of [[tau].sub.1] and [X.sub.1] for [pi]/[omega][right arrow]0 using the silica gel data measured in Table 2 gives [[tau].sub.w1] = 6.51 s and [X.sub.1] = 1.0. These experimental findings imply that the desiccant coatings have significant time delays for water vapor interactions, and for step changes of very long time durations ([N.sub.d][much greater than]1), Equation 28 should be divided into two equations of the same form: [[tau].sub.wj] = [[dW*.sub.j]]/[dt]] + [W*.sub.j] = [X.sub.j,] j = 1,2 (31) where the total response at any time is given by W* = [W.sub.1]* + [W.sub.2]* (32) These equations and the empirical data in Table 3 for the corrugated flow channels imply that the desiccant coating will sorb water quickly, as implied by [[tau].sub.w1], but the weighting factor, [X.sub.1], indicates the amplitude of this component of the response is restricted. The second, much larger time constant, [[tau].sub.w2], implies that, for typical wheel speeds, this component of the response will be negligible. This second time constant, [[tau].sub.w2], is much larger than the first one, because water vapor penetrates through many particle layers and into very small interstitial void spaces between particles or, for the case of silica gel, internal particle void spaces. The first time constant, [[tau].sub.w1], is thought to be mostly due to water vapor interactions on the wellexposed particle surfaces in these multilayer desiccant particle beds or coatings. If we compare the theoretical estimate of [[tau]*.sub.w] as given by Equation 27 using the data in Tables 2 and 4 with the measured data the first and most important time constant, [[tau].sub.w1], in Table 3 ([N.sub.d] = 1000), we can calculate the empirical coefficient, [beta]. For the molecular sieve coated flow channels, [[beta].sub.MS] = 1.30[+ or ]0.22, and for silica gel coated flow channels, [[beta].sub.SG] = 1.82[+ or ]0.30. For desiccant wheels of similar properties to those tested here, these results imply that Equation 27 should provide good flow channel design guidance for the effect of each parameter in Equation 27 for desiccant coated wheels; however, the estimated total uncertainty, which is mostly caused by uncertainties for the coating density, [[rho].sub.d], coating thickness, [[delta].sub.d], and isotherm slope, [[([partial derivative][bar.u]/[partial derivative][phi])].sub.T] in Equation 27, is very important. CONCLUSION Using the fact that the flow, heat transfer, and moisture transfer in the airflow channels are nearly fully developed so corrections are small, equations are presented for the sensible and latent energy effectiveness of energy wheels, which transfer both heat and water vapor, knowing that the cyclic changes in inlet airflow properties follow a series of step changes. For energy wheels that rotate at speeds greater than 0.5 rad/s (5 rpm), the fully developed sensible effectiveness of energy wheels can be calculated and corrected for entrance effects, axial heat conduction, carryover, and manufacturing tolerance variations using only the properties of the wheel matrix flow channels and the operating conditions. The correction for phase change in desiccant coated energy wheels requires an estimate of the latent effectiveness. The important flow channel desiccant coating properties that are needed for a complete latent effectiveness analysis are presented in this paper. The companion to this paper presents comparisons between the theoretical latent effectiveness predictions with measure data (Shang and Besant 2009). For desiccant drying wheels, which operate with high regeneration temperature (e.g., 100[degrees]C to 250[degrees]C) and at speeds less than 0.1 rad/s (1 rpm), axial and transverse heat conduction effects can be very important; so, to minimize heating of the supply air, it is important to either employ a heat purge section or slow the wheel down to less than 0.01 rad/s. NOMENCLATURE A = one flow channel area, [m.sup.2] [c.sub.p] = specific heat capacity, J/(kg*K) [d.sub.h] = flow channel hydraulic diameter, m h = convective heat transfer coefficient, W/([m.sup.2]*K) [h.sub.w] = mass transfer coefficient of water vapor, m/s L = flow channel length, m M =mass, kg m = mass flow rate, kg/s N = the number of transfer units, dimensionless P = flow channel perimeter, m p = pressure, Pa T = temperature, K t = time, s V = air velocity, m/s u = moisture content, kg/kg [bar.u] = average moisture content, kg/kg W = humidity ratio, [kg.sub.water]/[kg.sub.dry] air W* = normalized moisture content of air, dimensionless w* = normalized moisture content of matrix, dimensionless Greek Symbols [beta] = coefficient, dimensionless [delta] = thickness, m [delta] = difference [epsilon] = effectiveness, dimensionless [delta][[epsilon].sub.m] = effectiveness sensitivity coefficient, dimensionless [phi] = relative humidity, dimensionless [rho] = density, kg/[m.sup.3] [tau] = time constant, s [[tau].sub.w] = time constant of sorption for step change, s [omega] = angular speed, rad/s Subscripts a = air co = carryover cond = heat conduction d = desiccant da = interface of desiccant and air e = exhaust air side ent = entrance effect FD = fully developed flow FDCF = counterflow heat exchanger under fully developed flow conditions FDPF = parallel flow heat exchanger under fully developed flow conditions i = inlet or i = 1, 2, 3, 4, 5. k = 1, 2, 3, 4 l = latent energy M = maximum or minimum m = wheel matrix or mass transfer o = outlet pc = phase change s = sensible energy or supply air side or saturation T = constant temperature t = total energy or enthalpy temp = temperature effect var = flow channel size variation w = water vapor [phi] = constant relative humidity REFERENCES Abe, O.O., R.W. 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