Relationship between energy wheel speed and effectiveness and its transient response, Part I: mathematical development of the characteristic time constants and their relationship with effectiveness.

ABSTRACT

Testing energy wheels to determine their effectiveness under steady-state operating conditions has proved to be difficult, time consuming, and costly if accurate results are to be obtained. This paper investigates the theoretical feasibility of using transient response tests on a stationary energy wheel to determine the effectiveness of the same energy wheel rotating at a specified rate.

In this paper mathematical models for predicting first the characteristic time constants and, second, the effectiveness of an energy wheel are developed using only the expected transient response functions for the wheel. These time constants for sensible and latent (phase change) energy transient response are directly related to the corresponding sensible and latent effectiveness of an energy wheel operating at steady state at a specified rotor speed. It is concluded that it is feasible to use a simple transient parallel flow test on a stationary energy wheel and use these data to predict the sensible and latent effectiveness of the same energy wheel operating at a known speed. Furthermore, the energy wheel and HVAC designer can use the equations developed for the time constants and the effectiveness to provide guidance toward better performance and control.

INTRODUCTION

Testing energy wheels to determine their effectiveness under specified steady-state operating conditions requires large test facilities, expensive instrumentation and data acquisition capabilities, and extensive online analysis of large quantities of data (Ciepliski et al. 1998; Simonson et al. 1999a, 1999b).

Several test standards have been developed by international agencies for testing heat exchangers under steady-state conditions. These have been reviewed by Ciepliski et al. (1998). Three of these standards are explicitly for air-to-air heat exchangers and they include ANSI/ASHRAE Standard 84-1991 (ASHRAE 1991). Two other standards for liquid-to-air heat exchangers have similar test requirements.

The proposed revisions to ANSI/ASHRAE Standard 84-1991 focus mostly on reducing the uncertainties of performance or effectiveness testing of air-to-air energy recovery systems at steady-state test conditions while maintaining balances of inlet and outlet dry air, water vapor, and energy flows within acceptable uncertainties. Ciepliski et al. (1998) used a test facility as shown in Figure 1 to continuously monitor all of the necessary air properties while determining the effectiveness.

[FIGURE 1 OMITTED]

For large energy wheels, large expensive test facilities are required because the inlet supply air must be chosen to simulate specified ambient air steady-state summer or winter test conditions (ARI Standard 1060-2001). Ciepliski et al. (1998) found that several hours of preconditioning are required at one test condition before steady-state conditions are achieved and all the air, water vapor, and energy balances are maintained within acceptable uncertainty limits.

Shah (1981), Klein et al. (1990), Simonson et al. (1999b), and others noted that although the airflows operate at steady state, the thermal and water vapor transfer processes on the rotary wheel matrix are transient for regenerative heat or energy wheels. It is hypothesized that if we could accurately measure the transient characteristics of an energy wheel, then we should be able to predict the effectiveness of an energy wheel. Furthermore, such a transient test need not be done on a rotating wheel and, because the flow channel matrices of energy wheel cores are nearly uniform, only a small fraction of the wheel would need to be tested. That is, a transient test could be devised to measure the outlet air properties for a stationary energy wheel subject to a transient step change in inlet air properties using a device that occupies a small fraction of the wheel surface. Using these transient response data for an energy wheel we should be able to predict the effectiveness of the same wheel for any HVAC application with similar, but not identical, inlet operating conditions.

Figure 2 shows a possible parallel airflow configuration for a transient step change test of a stationary energy wheel. In this configuration the two inlet tubes--each with an inlet steady-state flow, temperature, and humidity--are assumed to be interchanged at time t=0 while the outlet flow tube air properties are continuously measured so that these outlet properties change from their initial condition toward a final steady-state condition. During this process, the energy wheel matrix, first exposed to an initial steady-state measured airflow, temperature, and humidity before the switch or interchange of inlet flow tubes, experiences a step change in either the inlet air temperature or humidity. The corresponding outlet tube air temperature or humidity sensors can be measured to determine the response of the heat or water vapor interactions with the wheel matrix as the air flows from the inlet to the outlet tubes so that hypothetically these responses will be directly related to the sensible energy or moisture transfer effectiveness of the energy wheel. The development of these relationships for the outlet air properties is the object of this paper. This paper is an extension of the paper by Abe et al. (2006b), which presents equations for the development of equations for effectiveness of energy wheels with empirically measured time constants. This paper develops these time constants from basic principles.

Figure 2 shows the airflow configuration schematic, which is designed with a rotary inlet switch plate on a stationary air-to-air energy wheel to rapidly interchange the inlet tubes. It is described in detail in Part II of this paper.

TRANSIENT PROBLEM FORMATION

The diameter of the flow tubes in Figure 2 need only be large enough to avoid perimeter edge effects (Shang and Besant 2005a) but include the typical flow channel hydraulic diameter variations that characterize the particular energy wheel under test (Shang and Besant 2004, 2005b, 2005c). Figure 3 shows one typical flow channel between one of the inlet and outlet flow tubes shown in Figure 2. The characteristic geometry of this typical flow channel is the tube length, L, as shown in Figure 3a, wavelength, 2a, and wave amplitude, 2b, of a corrugated shape as shown in Figure 3b. The hydraulic diameter and area of the flow channel are [d.sub.h] and A, respectively. The perimeter of the corrugated shape is P. For some other flow channel geometric shapes, such as triangular, rectangular, hexagonal, etc., the analysis is similar to that for corrugated tube shape, which is one of the widely used flow channel geometries.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The basic equations for the airflow inside this flow channel are:

1. Dry air continuity, which, for steady mass flow rate, [dot.m], is

[dot.m.sub.a]=[[rho].sub.a][bar.U]A, (1)

where the bulk mean dry air density, [[rho].sub.a], and air speed, [bar.U], are invariant at any cylindrical type cross section of area of the flow channel (Shang and Besant 2005b),

A = [1/4]P[d.sub.h], (2)

where P is the perimeter and [d.sub.h] is the hydraulic diameter.

2. Water vapor continuity, which 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],

A[[[partial derivative][[rho].sub.w]]/[partial derivative]t] + [bar.U]A[[[partial derivative][[rho].sub.w]]/[partial derivative]z] = P[h.sub.w]([[rho].sub.w,m] - [[rho].sub.w]), (3)

where [[rho].sub.w,m] is the wheel matrix surface water vapor density.

3. Thermal energy for the air, which is characterized by its bulk mean air temperature, T,

[rho][c.sub.v]A[[partial derivative]T/[partial derivative]t] + [rho][c.sub.p][bar.U]A[[partial derivative]T/[partial derivative]z] = Ph([T.sub.m] - T), (4)

where the matrix surface temperature is [T.sub.m], the convective heat transfer coefficient is h, [c.sub.v] and [c.sub.p] are the bulk mean air mixture specific heat at constant volume and constant pressure, respectively, and the bulk mean density of the dry air and water vapor mixture is

[rho] = [[rho].sub.a] + [[rho].sub.w]. (5)

In the above equations, the water vapor mass balance (Equation 3) and energy balance (Equation 4) are coupled to the matrix properties, so independent balance equations are needed for one-half of the flow channel matrix adjacent to the airflow channel. These are:

4. Matrix water continuity

[A*.sub.m][[[partial derivative][[rho].sub.w,m]]/[partial derivative]t] = P[h.sub.w]([[rho].sub.w] - [[rho].sub.w,m]) (6)

where [A*.sub.m] is the area of the matrix flow channel adjacent to the airflow channel shown in Figure 3, which can adsorb and desorb water vapor. An assumption implied by Equation 6 is that the matrix mean and surface mass density of water, [[rho].sub.w,m], are equal (i.e., internal matrix water concentration differences are negligible).

5. Matrix thermal energy

[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] (7)

where [[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 that is taken to be the heat of evaporation, [h.sub.fg], in most cases. An important term not included in Equation 7 is heat conduction in the x-y plane of the matrix (i.e., inter-tube heat conduction due to a temperature gradient in the flow tubes in the plane of the inlet [or outlet] surface of the wheel). Abe et al. (2006d) considered this heat conduction correction in a typical energy wheel matrix constructed using desiccant-coated aluminum matrix of thickness 0.084[+ or -]0.005 mm (composed of 0.052 mm aluminium and 0.016 mm desiccant layer on each side) and showed how the measured temperature results could be adjusted to correct for this effect. Hence, this effect can be neglected in the problem formulation, provided it is corrected for in the measured results.

Heat conduction in the matrix in the axial z direction will exist in regenerative wheels, but Holmberg (1979), Maclaine-Cross and Banks (1972), and Simonson et al. (1999b) show that this effect is negligible under typical operating conditions of heat or energy wheels.

It is seen that Equations 3, 4, and 7 each have terms with property variations in the flow channel axial direction, z. These equations can be integrated in the z direction from flow tube inlet (z = 0) to outlet (z = L) to give

AL[[d[bar.p.sub.w]]/dt] + A[bar.U]([[rho].sub,w,o] - [[rho].sub.w,i]) = PL[h.sub.w]([bar.[rho].sub.w,m] - [bar.[rho].sub.w] (3a)

AL[bar.[rho]][c.sub.v][d[bar.T]/dt] + [bar.U]A[rho][c.sub.p]([T.sub.o] - [T.sub.i]) = Ph([bar.T.sub.m] - [bar.T]) (4a)

and

[A.sub.m]L[[rho].sub.m][c.sub.pm][[d[bar.T.sub.m]]/dt]

= [A.sub.m][k.sub.m][([[partial derivative][T.sub.m]]/[[partial derivative].sub.z])[.sub.o] - ([[partial derivative][T.sub.m]]/[[partial derivative].sub.z])[.sub.i]] + PLh([bar.T] - [bar.T.sub.m] + [A*.sub.m]L[h.sub.a][[d[bar.[rho].sub.w,m]]/dt] (7a)

where the superscript bar denotes the average property value over the tube length L.

It will be shown that some of these terms are much smaller than others in these equations, so they can be deleted.

Dividing Equation 3a by A[bar.U]([[rho].sub.w,o] - [[rho].sub.w,i] and Equation 4a by [bar.U]A[bar.[rho]][c.sub.p]([T.sub.o] - [T.sub.i]) gives the dimensionless forms of these equations:

[L/[[bar.U]([[rho].sub.w,o] - [[rho].sub.w,i])]][[d[bar.[rho].sub.w]]/dt] + 1 = [N.sub.m]([[bar.[rho].sub.w,m] - [bar.[rho].sub.w]]/[[[rho].sub.w,o] - [[rho].sub.w,i]]) (3b)

([c.sub.v]/[c.sub.p])[L/[[bar.U]([T.sub.o] - [T.sub.i])]][d[bar.T]/dt] + 1 = N([[bar.T.sub.m] - [bar.T]]/[[T.sub.o] - [T.sub.i]]) (4b)

where

[N.sub.m] = [4[h.sub.w]L]/[[d.sub.h][bar.U]] (8)

and

N = 4hL/[[[rho].sub.a][bar.U][d.sub.h][c.sub.p,a]] (9)

are the number of transfer units for water vapor and heat transfer, respectively.

For typical energy wheels with airflow transport times, L/[bar.U], ranging from 0.02 to 0.2 seconds, the dimensionless transient terms in Equations 3b and 4b can be shown to range from 0.001 to 0.01 during the first part of the transient for a step change using the data reported by Wang et al. (2005) and Abe et al. (2006a). As time progresses after the early part of the step change, these dimensionless transient terms become even smaller. That is, they are always less than 1% for typical operating conditions of energy wheels, so they can be neglected.

This finding, with demonstration by transient data, has always been assumed in the analysis of energy and heat wheels (see, for example, Shah [1981], Klein et al. [1990], and Simonson and Besant [1999a, 199b]); now it is quantified experimentally.

Neglecting the transient terms in Equations 3b and 4b means that the matrix properties [bar.[rho].sub.w,m] and [bar.T.sub.m] are related to the air properties through algebraic equations. Furthermore, because the mean air properties [bar.[rho].sub.w] and [bar.T] can be expressed as the average of the inlet and outlet properties, the wheel time variable properties [[rho].sub.w,o] and [T.sub.o] tend to differ from the constant inlet properties as given by the equations,

[[rho].sub.w,o] - [[rho].sub.w,i] = [[2[N.sub.m]]/[2 + [N.sub.m]]][bar.[rho].sub.w,m] (10)

and

[T.sub.o] - [T.sub.i] = [2N/[2 + N]][bar.[rho].sub.m]. (11)

It is noted that the inlet and outlet air properties could be readily measured, but [bar.T.sub.m] and [bar.[rho].sub.w,m] are not convenient to measure. These simple results do not imply that the transient terms in Equation 7a for the wheel matrix are small; in fact, these are usually dominant terms in the equation for a transient process. The term that is negligible is the axial conduction at z = 0 (or the inlet) and at z = L (or the outlet), where the matrix is only in contact with air that has a thermal conductivity that is [10.sup.4] times smaller than that of an aluminum matrix, so this term is neglected. To nondimensionalize Equation 7a we can divide by the convective term, as we did in Equations 3b and 4b, and note that this convective term in the matrix is equal to the convective term in the thermal energy equation for the air, so the magnitude of terms in Equation 7b will be comparable to those in Equation 4b.

[[[A.sub.m][[rho].sub.m][c.sub.p,m]]/[Ph([bar.T] - [bar.T.sub.m])]][[d[bar.T.sub.m]]/dt] = 0 + 1 + [[[A.sub.m][h.sub.a]]/[Ph([bar.T] - [bar.T.sub.m])]][[d[bar.[rho].sub.w,m]]/dt] (7b)

This equation implies that we could do two types of experiments: case 1, where d[bar.[rho].sub.w,m]/dt = 0 when the inlet air is dry before and after the step change in temperature, and case 2, where both d[bar.[rho].sub.w,m]/dt and d[bar.T.sub.m]/dt are finite during an adiabatic process experiment with mass transfer.

In the adiabatic process case 2, the inlet air temperature is the same in both inlet tubes before and after the step change in humidity; this will lead to an adiabatic mass transfer process in which the outlet temperature differs from the inlet by only a small amount (e.g., less than 2[degrees]C). This small change in the outlet temperature is caused by the phase change on the surface of the wheel matrix. By keeping the inlet air temperature difference to zero in case (2) (i.e., an adiabatic mass transfer process for a step change in the inlet air water density), the heat transfer effects are restricted to only those induced by the phase change of water as it is adsorbed or desorbed by the flow channel desiccant layer. For example, with a large change in the inlet relative humidity (e.g., 6% RH to 60% RH) the data presented in Part II of this paper show that the temperature changes in the airflow through the channel are less than 2[degrees]C.

For the first case (e.g., no mass transfer) Equation 7b d[bar.[rho].sub.w,m]/dt = 0 but d[bar.T.sub.m]/dt is finite due to a step change in the inlet air temperature, i.e., there is heat transfer but no phase change due to water vapor adsorbed or desorbed in the matrix. Two types of boundary conditions are thought to characterize the practical limits of most heat transfer processes. These are: constant heat flux at the tube wall surface along the length of each flow channel at any instant in time, and constant tube wall temperature along the tube wall surface at any instant in time. Again it is noted that the airflow has an average time duration inside each tube of length L of L/[bar.U], a very short time compared to the measured time constants in Part II. For fully developed flow and heat transfer inside each tube, the constant heat flux boundary condition, denoted by subscript H, results in a linear relationship between the mean air temperature T and distance along the flow channel z (Arpaci and Larsen 1984) so that

[bar.T.sub.H] = [1/2]([T.sub.o] + [T.sub.i]), (12)

so that substituting Equations 11 and 12 into Equation 7b gives

[d[T.sub.o]/dt] + [1/[[tau].sub.H]][T.sub.o] = ([1 - N]/[[tau].sub.H])[T.sub.i], (13)

where [[tau].sub.H] is the time constant for the transient heat transfer process. Since the temperature scale is arbitrary, we can choose [T.sub.i] = 0 giving

[d[T.sub.o]/dt] + [1/[[tau].sub.H]][T.sub.o] = 0. (14)

Assuming that Equation 7b accurately models the physics of this process, the time constant [[tau].sub.H] is

[[tau].sub.H] = [1/h]([2.summation over (i=1)] [[delta].sub.i][[rho].sub.i][c.sub.p,i])[.sub.m]([1/2]N + 1), (15)

where

[[A.sub.m][[rho].sub.m][c.sub.p,m]]/P = ([[delta].sub.sm][[rho].sub.sm][c.sub.p,sm] + [[delta].sub.d][[rho].sub.d][c.sub.p,d]) = ([2.summation over (i=1)][[delta].sub.i][[rho].sub.i][c.sub.p,i])[.sub.m]. (16)

For the case of a constant temperature, [T.sub.m], of the matrix tube wall in a fully developed flow, the temperature distribution of the bulk mean airflow follows an exponential decay with distance along the tube, z, at any time so that relative to the tube matrix wall temperature, [T.sub.m], the bulk temperature is

T - [T.sub.m] = ([T.sub.i] - [T.sub.m])[e.sup.-z/z*] (17)

and an average bulk temperature of air is

[bar.T.sub.T] - [T.sub.m] = [z*/L]([T.sub.i] - [T.sub.m]), (18)

where

z* = [[d.sub.h.sup.2][bar.U]]/[4[[alpha].sub.T]N[u.sub.T]], (19)

which has a practical range for typical operating energy wheels of 0.01 < z* < 0.001 m when the Nusselt number N[u.sub.T] = 3.66 for tube flow, [[alpha].sub.T] = thermal diffusivity, and the hydraulic diameter, [d.sub.h], and mean air speed, [bar.U], take on typical values.

Again substituting into Equation 7b, the equation for the outlet temperature [T.sub.o] will follow the differential equation 

[d[T.sub.o]/dt] + [1/[[tau].sub.T]][T.sub.o] = [z*/L][T.sub.i], (20)

where

[[tau].sub.T] = [[tau].sub.H]/(1 - z*/L) (21)

differs only slightly from Equation 15. That is, the time constants of both boundary conditions vary by less than 10% for typical wheels so we will consider only one of these time constants (Equation 15) in our discussion of the transient heat transfer characteristics.

Both Equations 13 and 20 imply that change in the outlet air temperature, [DELTA][T.sub.o], will behave as a simple first order system, as in Equation 14, with a characteristic time constant [tau] so that for a step change in the inlet air temperature, [DELTA][T.sub.i] (where [T.sub.i] is taken to be zero for t[greater than or equal to]0), the outlet temperature change will be

[[DELTA][T.sub.o]]/[[DELTA][T.sub.i]] = [e.sup.-t/[tau]] (22)

for a step decrease, and, for a step increase, we get

[[DELTA][T.sub.o]]/[[DELTA][T.sub.i]] = 1 - [e.sup.-t/[tau]]. (23)

Using typical values for the terms for the time constant in Equation 15, we find that 8 < [tau] < 30 seconds. For example, a typical energy wheel with the geometric, thermal, and flow conditions in Table 1 results in N = 10.6 and a time constant of 7.1 seconds. It is noted that this calculation of fully developed heat transfer in a corrugated flow channel of an energy wheel as shown in Figure 3 includes modifications for an aspect ratio 2a/2b = 0.3 so that the heat transfer coefficient, h, is two-thirds of the value of h for the same mean velocity through a circular cylinder of the same hydraulic diameter, 1.0 mm (Shang and Besant 2005b). Equations 9 and 15 imply that the designer of an energy wheel can readily alter the heat transfer transient response of an energy wheel by simply changing the flow channel geometry or air velocity. For example, increasing the flow channel length, L, or matrix thickness, [[delta].sub.sm] and [[delta].sub.d], but decreasing the hydraulic diameter, [d.sub.h], will increase this time constant. Increasing the air speed, [bar.U], will, however, decrease [tau]. For fully developed laminar flow, the heat transfer coefficient, h, will have a small effect because it appears in the numerator in Equation 9 and explicitly in the denominator in Equation 15. It is clear that these equations imply that there could only be a small difference between summer and winter test conditions caused primarily by the change in the air specific heat, which will be less than 0.1% for the ARI summer and winter test conditions when the mass flow rates of dry air are the same for each.

The reader should only use Equation 15 as a guide for design because experimental tests must be used to determine secondary effects not considered in this model, and the uncertainty in the terms used in the model is significant. For example, flow channel variations in hydraulic diameter could become important especially as [d.sub.h] is decreased to increase [tau] (Shang and Besant 2005a, 2005b, 2005c). Secondary heat transfer effects--not directly included in this model--need to be dealt with using the method outline by Abe et al. (2006d).

For case 2, the adiabatic mass transfer process for a wheel subjected to a step change in the inlet relative humidity while the inlet air temperatures are equal, we start with Equation 7b written in the form of Equation 7a,

[A.sub.m][[rho].sub.m][c.sum.p,m][d[bar.T.sub.m]/dt] - [A*.sub.m][h.sub.a][d[bar.[rho].sub.w,m]/dt] = Ph([bar.T] - [bar.T.sub.m], (7c)

and note that both d[bar.T.sub.m] / dt and ([bar.T] - [bar.T.sub.m]) are small so they should be written in terms of water vapor density. Using Equation 10 with [[rho].sub.w,i]=constant gives an equation in terms of the outlet air water vapor density, [[rho].sub.w,o].

[d[bar.[rho].sub.w,m]]/dt = ([[N.sub.m] + 2]/2[N.sub.m])[d[[rho].sub.w,o]/dt] (24)

Assuming that for an adiabatic process in the airflow through the flow channel the enthalpy difference is negligible, e.g., if the moisture content of the air decreases, the air temperature will increase according to the relationship

([T.sub.i] - [T.sub.o])/([[rho].sub.w,i] - [[rho].sub.w,o]) = [h.sub.fg]/[[[rho].sub.a][c.sub.p,a]], (25)

implying that if there is no change from inlet to outlet in air temperature there will be no change in the water vapor density. Then Equation 11 for the case of [T.sub.i] = constant gives

[d[bar.T.sub.m]/dt] = -([2 + N]/2N)[[h.sub.fg]/[[rho].sub.a][c.sub.pa]][d[[rho].sub.w,o]/dt]. (26)

Assuming that [T.sub.i] can be set equal to zero for t [greater than or equal to] 0 and substituting Equations 25 and 26 into Equation 7c, we can write

[d[[rho].sub.w,o]/dt] + [1/[[tau].sub.m]][[rho].sub.w,o] = [1/[[tau].sub.m]][[rho].sub.w,i], (27)

where

[[tau].sub.m] = [lambda][[[[I.summation over (i-1)][[delta].sub.i][[rho].sub.i][c.sub.p,i]]/h]([1/2]N + 1) + [[[delta].sub.m]/h][[h.sub.a]/[h.sub.fg]]([1/2][N.sub.m] + 1)[N/[N.sub.m]][[rho].sub.a][c.sub.p,a]]. (28)

It is necessary to introduce the dimensionless water vapor sorption coefficient, [lambda], which accounts for the moisture sorption efficacy of the desiccant coating on the flow channel matrix during a transient process; otherwise, the first term in Equation 28 would imply that an energy wheel with no desiccant coating (i.e., a heat wheel) would have the identical time constant for moisture transfer as the heat transfer process time constant. If there were no desiccant coating, the time constant in Equations 27 and 28 must be zero for moisture transfer, implying an instantaneous outlet response for a change in the water vapor density at the inlet. In this case, [lambda] = 0. On the other hand, if the flow channel has a very thick and effective desiccant coating, the first term in Equation 28 is controlled by the adiabatic coupled mass (water vapor) and heat transfer process, so in this special limiting case, [lambda] [right arrow] 1.0. For typical energy wheels 0 < [lambda] < 1.0 and [lambda] will approach 1.0 when the desiccant layer is thick and the desiccant material used has a high moisture adsorption capacity at typical operating temperatures and relative humidities (e.g., silica gel, which has an internal pore structure in each particle that is much bigger than the water molecule [e.g., pore sizes greater than 10 [Angstrom], Pesaran and Mills (1987a, 1987b)] and the relatively large internal surface areas as well as the external particle surface area have a strong affinity for polar water molecules with a molecular diameter of 2.8 [Angstrom]. Desiccants with a lesser moisture adsorption capacity at these operating conditions (e.g., 4 [Angstrom] molecular sieve) will have a slightly lower [lambda] because water adsorption layers are mostly on the external particle surfaces, but a 3 [Angstrom] molecular sieve, which has an internal pore structure only very slightly larger than the water molecule diameter at 2.8 [Angstrom], may have a [lambda] value that depends entirely on the external surface area of the particles.

More recently, some new hydrophilic-coated surfaces have been developed that are composed of chemically doped porous structures with polar molecules, similar to the well-known desiccants above, imbedded into porous hydrocarbon films or plastic surfaces. The water vapor sorption properties of these porous materials are only now under investigation.

The thickness of the desiccant layer, the particle sizes, and the bonding of these particles to the metal surface are also very important in determining [lambda]. For a step change transient process and a given type of coating (e.g., 5 to 10 m silica gel particles) bonded on each flow channel surface, it is expected [lambda] will be an empirical function of the desiccant thickness [[delta].sub.d] as measured and modeled by Pesaran and Mills (1987) for thin layers of silica gel particles. Their model and the limit conditions for [lambda] imply that [lambda] will vary with coating thickness as shown in Figure 4 and as given by

[lambda] = 1 - [e.sup.-[[delta].sub.d]/[[delta]*.sub.d]], (29)

where [[delta]*.sub.d] is the characteristic thickness for a desiccant coating on a flow channel matrix surface and can be determined by measurement. Equation 29 states that water vapor will not penetrate the entire desiccant thickness with equal efficacy during a transient adsorption process and that [lambda] for adsorption will be slightly smaller than the value for desorption, so the wheel designer will need to trade off desiccant thickness, [[delta].sub.d], with other practical factors. Thus, the effective moisture transfer thickness of the desiccant layer of thickness [[delta].sub.d] is [lambda][[delta].sub.M] where [[delta].sub.M] = [[delta]*.sub.d]/(1 - [e.sup.-1]) is assumed to be constant for a transient process. This implies that the energy wheel designer will want to select the actual desiccant thickness to satisfy

[[delta].sub.d] [much greater than] [[delta]*.sub.d], (30)

but selecting a very large value will be wasteful and an extra thick layer may cause additional time delays in the heat transfer process. Also, both Pesaran and Mills (1987a, 1987b) and Sun and Besant (2006) showed that the size of the silica gel particle is important for transient processes with moisture adsorption (i.e., the smaller the particles the greater the moisture adsorption per unit mass at any time); on the other hand, too small a particle thickly covered in bonding material on the surface of a wheel matrix will lose some of this adsorption capacity due to the masking effect of the bonding material. Thus, the desiccant particle-bonding material and its mass per unit area of matrix surface is also important. Also, glues used to join the surfaces of the matrix sheet into a flow channel shape (e.g., corrugated) need to be kept small in area and only sufficient to meet the mechanical strength requirements of the wheel matrix; otherwise, they too will mask the mass transfer area for water vapor and increase the thermal time constant (see Part II of this paper [Abe et al. 2006a]).

All of the above discussion implies that [lambda] is a complex function of this type of desiccant and bonding material and glue, particle size, and total thickness. It is best determined by the experimental tests presented in Part II of this paper (Abe et al. 2006a).

Again, using typical values of the terms in Equation 28 and [lambda] equal to 0.5 we find that 4 < [[tau].sub.m] < 15 seconds, which is similar to that for Equation 15. It is interesting to note that only the first term in Equation 28 is significant, indicating that the phase-change energy is temporarily stored in the matrix solid before it is convected out, so Equation 28 can be written as

[[tau].sub.m] = [lambda] x [tau], (31)

where [tau] is given by the time constant for heat transfer by Equation 15.

Similar to Equations 22 and 23, the solutions of interest for Equation 27 are

[DELTA][[rho].sub.w,o]/[DELTA][[rho].sub.w,i] = [e.sup.-t/[[tau].sub.m]] (32)

for a step decrease in [DELTA][[rho].sub.w,i] and by

[DELTA][[rho].sub.w,o]/[DELTA][[rho].sub.w,i] = 1 - [e.sup.-t/[[tau].sub.m]] (33)

for a step increase.

It is clear that Equations 14 and 27 are both of the form

[dy/dt] + ay = af(t), (34)

where

a = 1/[tau]. (35)

This type of equation, which is well known in the mathematical literature (Kreyszig 1999), is a linear first ordinary differential equation and it applies if this is either a transient test on a stationary wheel or the regenerative wheel is rotating at constant speed with steady-state inlet air properties.

Although we could consider more complex processes than the two cases considered for only heat transfer and adiabatic mass transfer with no inlet temperature difference, all the important information required to finally obtain the sensible, latent, and total effectiveness of an energy wheel can be deduced from the two processes above. The remainder of this paper outlines the equations required to get these effectivenesses.

Again it is noted that although we can estimate the time constants for heat transfer and adiabatic mass transfer and use the equations for guiding design operating modifications, we will need to get measured results if they are going to be used to predict the effectiveness because there will be some further important time delays in the heat transfer and moisture diffusion in the desiccant particle coatings that are not accounted for in the simple physical model presented above. This is done in Part II of this paper (Abe et al. 2006a).

STEADY-STATE ENERGY WHEEL RESPONSE

In HVAC applications, energy wheels operate in a steady-state counterflow mode for the inlet and outlet airflows for both the supply and exhaust side of the exchanger. As well, the wheel matrix is subjected a series of step changes of flow direction and inlet properties as each flow channel in the matrix is exposed to the inlet supply or exhaust flows of equal duration for a wheel rotating at a steady speed. This sequence of step changes in inlet air property f(t) (e.g., temperature or humidity) that the wheel matrix experiences for the inlet conditions is shown in Figure 5.

First we will analyze the case of steady-state parallel flow through the wheel matrix (i.e., both the exhaust and supply flow directions are the same) and then we will consider the more practical case of counterflow. The reason for doing the parallel flow case first is that this steady-state flow configuration is the same as that proposed for the transient test on a stationary wheel except that the step changes are reversed in inlet air properties every 180[degrees] of wheel rotation, but not flow direction. The matrix of an energy wheel rotating between hot and cold airstreams will be subject to a step change in the inlet conditions that are periodic, steady, and rectangular. The rectangular input function is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

where [theta] is the angle of wheel rotation for each flow channel in the matrix.

The infinite steady-state sequence of periodic inlet conditions shown in Figure 5 is best represented mathematically as a Fourier series. In the mathematical literature these inlet property changes are referred to as a forcing function,

f(t) = [a.sub.0] + [[infinity].summation over (n = 1)]([a.sub.n]cosn[omega]t + [b.sub.n]sinn[omega]t), (37)

where n is a positive integer, [omega] is the angular speed (rad/s), and t is time. Kreyszig (1999) and Abe et al. (2006b) show that when the Fourier coefficients [a.sub.n] and [b.sub.n] are evaluated for the sequence of square waves shown in Figure 5, Equation 34 becomes

f(t) = [4/[pi]][[infinity].summation over (n = 1)][sinn[omega]t/n] for n = 1,3,5,.... (38)

When this forcing function is used in the differential equation for the outlet air temperature (Equation 13) or humidity (Equation 27), y in Equation 34 represents either the outlet dimensionless temperature or humidity for one airflow channel in the matrix.

[FIGURE 5 OMITTED]

Substituting Equation 38 as the input forcing function into Equation 34, the periodic output (or steady-state solution) is:

y(t) = [[infinity].summation over (n = 1)][4a/[n[pi][square root of ([a.sup.2] + (n[omega])[.sup.2])]]][sin(n[omega]t - [alpha])] for n = 1,3,5,..., (39)

where [alpha] is a phase shift angle,

[alpha] = tan[.sup.-1.([[infinity].summation over (n = 1)][n[omega]/a])] (40)

for n = 1, 3, 5, ....

Since the magnitude of the terms in the infinite series in Equation 39 decreases rapidly with increasing n, it is found that including n [greater than or equal to] 21 is unnecessary in the calculations for the time constants used. Therefore, terms n [greater than or equal to] 21 are neglected in this paper, giving an error of approximately 0.0001%.

Figure 6 shows the steady-state periodic response y(t) of a system for one cycle with various time constants when the rectangular periodic input function f(t) in Equation 34 has an angular frequency of (a) [omega] = [pi]/3 rad/s (10 rpm) and (b) [omega] = 2[pi]/3 rad/s (20 rpm) and the exchanger inlet flows are both in the same direction (i.e., parallel flow). Comparison of Figures 6a and 6b shows that as the wheel speed increases, the amplitude of the system response is reduced. This figure shows the effects of the angular frequency and time constant on the output for a first-order linear system. The output of a system with a small time constant follows the input function more closely than a system with a large time constant. In many applications, it is desirable for the output to follow the input closely, but the opposite is the case for a regenerative energy wheel application. The figure also shows that, as the wheel speed increases, the amplitude decreases and the phase shift angle increases.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

EFFECTIVENESS

The performance of air-to-air exchangers for transferring energy is characterized by their effectiveness. The effectiveness of a heat exchanger is defined as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate (for an infinite heat transfer surface area). The definition of performance effectiveness of an air-to-air heat/energy device is

[epsilon] = [[dot.m.sub.s]([X.sub.1] - [X.sub.2])]/[[dot.m.sub.min]([X.sub.1] - [X.sub.3])], (41)

where

[epsilon] = sensible, latent, or total energy effectiveness;

X = dry-bulb temperature ([degrees]C), humidity ratio ([kg.sub.water]/[kg.sub.dry air]), or enthalpy (kJ/[kg.sub.a]);

[dot.m.sub.s] = mass flow rate of dry air in the supply airstream (kg/s);

[dot.m.sub.e] = mass flow rate of dry air in the exhaust airstream (kg/s); and

[dot.m.sub.min] = minimum value of either [dot.m.sub.s] or [dot.m.sub.e] (kg/s).

Parallel Flow Exchangers

The thermal energy transfer process implied in Figure 6 can be best explained for one flow channel in a regenerative wheel operating in parallel flow by considering a parallel flow wheel with one time constant for heat transfer. The rate of energy transfer or heat rate to the supply air at any time for one flow channel with mass flow rate of air [dot.m.sub.a] is

q = ([dot.m.sub.a][c.sub.p,a])[.sub.s]([T.sub.1] - [T.sub.2]), (42)

while the maximum possible heat rate is

[q.sub.max] = ([dot.m.sub.a][c.sub.p,a])[.sub.min]([T.sub.1] - [T.sub.3]), (43)

where [T.sub.1] and [T.sub.3] are the inlet supply and exhaust air temperatures, which are assumed to be constant, and [T.sub.2] is the outlet temperature of the supply air, which changes with time as the wheel rotates the flow channel from 0[degrees] to 180[degrees] on the supply side.

If the mass flow rate through the airflow channel is the same on the exhaust side (i.e., from 180[degrees] to 360[degrees]), the time average effectiveness of each flow channel while it is in the supply side will be

[epsilon] = (q/[q.sub.max])[.sub.avg] = [1/[pi]][[pi].[integral].0]([[T.sub.1] - [T.sub.2]]/[[T.sub.1] - [T.sub.3]])d[theta]. (44)

Graphically, this effectiveness can be represented as an area ratio, so for the case of parallel flow shown in Figure 6, we can show the effectiveness as the ratio of output area divided by the input area.

It is clear from Figures 6 and 7 that the effectiveness of a parallel flow exchanger with equal supply and exhaust air mass flow rate increases as both the time constant, [tau], and wheel speed, [omega], increase and the maximum possible effectiveness will be 0.5 or 50% as either [tau] and [omega] tend toward infinity.

It will be seen later that the effectiveness of the same flow channel in a counterflow exchanger can easily exceed 50% but it cannot exceed 100%. It is also interesting to note that while the flow channel is in the exhaust side where it is regenerated to the condition at [theta] = 0, it has no direct impact on the effectiveness.

Graphical results similar to Figures 6 and 7 could be presented for the outlet air moisture content or water vapor density, so we return again to the general mathematical formulation where y(t) can represent either temperature or water vapor density.

Since the areas under the output and the input are both symmetrical about the axis y = 0 for a periodic input, as shown in Figure 6, the average is taken over one-half cycle, and Equation 44 is expressed as

[[epsilon].sub.PF] = 1 - [w/2[pi]][[integral].sup.[pi]/[omega].sub.0][1 + y(t)]dt, (45)

where y(t) is given by Equation 39. After integration, Equation 45 becomes

[[epsilon].sub.PF] = 0.5 - [1/[[pi].sup.2]][[infinity].summation over (n = 1)][4[a.sup.2]/[n.sup.2][[a.sup.2] + (n[omega])[.sup.2]]]for n = 1,3,5,.... (46)

[FIGURE 7 OMITTED]

It must be noted that though the same formula (Equation 44) is used for both sensible and latent effectiveness, they are determined independently. Sensible effectiveness is determined from the transient data obtained when the energy wheel is subjected to a step change in inlet air temperature, while the latent effectiveness is determined from the transient data obtained when the energy wheel is subjected to a step change in inlet air absolute humidity.

It follows from Figure 6 and the previous statements that an exchanger that operates with balanced supply and exhaust airflows or Cr = 1 will have maximum effectiveness as the amplitude ratio approaches 0 in Figure 6 and the phase lag approaches 90[degrees], as given by Equation 40. Therefore, for an energy exchanger, a lower output amplitude results in a lower amplitude ratio and the exchanger effectiveness will be higher. The effectiveness of a parallel flow exchanger ([[epsilon].sub.PF]) can also be expressed as (Abe et al. 2006b)

[[epsilon].sub.PF] = [1/2][1 - [8/[[pi].sup.2]][[infinity].summation over (n = 1)]([B.sub.n]/n)[.sup.2]]for n = 1,3,5,..., (47)

where

[B.sub.n] = [a/[square root of ([a.sup.2] + (n[omega])[.sub.2])]] for n = 1,3,5,.... (48)

Figure 8 shows the effectiveness [[epsilon].sub.PF] as the wheel speed changes for various time constants (1/a = [tau]). This figure shows that the effectiveness increases as both the time constant and wheel speed increase. It should be noted that increasing the energy wheel speed to a high value will have another less desirable consequence. It will increase the carryover of exhaust air to the supply side; so manufacturers limit the wheel speed to a practical range (e.g., 20 to 40 rpm) and the carryover is a small fraction of the total airflow rate (Simonson et al. 1999a, 1999b). This leaves the time constant as the main parameter that needs to be examined. As the time constant increases, the output amplitude decreases (as shown in Figure 6), thereby reducing the amplitude ratio. Equation 47 shows that a reduction in the amplitude ratio increases the effectiveness. In Figure 8, the effectiveness increases from 0 toward 50% as the wheel speed increases from 0 rpm to 50 rpm for time constants of 4 s and 10 s. At very large wheel speeds, the effectiveness approaches (50%) as expected for a parallel flow heat exchanger (recuperator) with Cr = 1. As an example, the effectiveness is 49.9% at 20 rpm when [tau] = 10s.

To verify the effectiveness calculated from the linear system model, the results predicted from Equation 46 are compared with analytical solutions of Romie (1992) and Hausen (1983) for a parallel flow heat regenerator. Relating the key parameter in the linear system model (i.e., [tau] and [omega]) with the corresponding parameters in the analytical solution of Romie (1992) (i.e., N and Cr*) allows a direct comparison between the linear model used in this paper and the analytical solution. Figure 8 shows the thermal effectiveness, calculated from the linear model (i.e., Equation 46), is in very close agreement with the results predicted by Romie (1992). The average and maximum differences between the effectiveness values determined with Equation 46 and the analytical solution are 0.001 and 0.002, respectively.

[FIGURE 8 OMITTED]

Counterflow Exchangers

Since energy exchangers are almost always used in a counterflow and not parallel flow configuration, the effectiveness expression obtained thus far for a parallel flow regenerator, e.g., Equation 46, will now be related to a counterflow configuration. It is known that the effectiveness of a regenerator ([epsilon]) can be calculated as a product of the effectiveness of a recuperator ([[epsilon].sup.rec]) and a constant, which is a function of Cr* (Simonson and Besant 1999a; Kays and London 1984).

[epsilon] = [[epsilon].sup.rec] x f(Cr*) (49)

where Cr* represents the matrix heat (or moisture) capacity ratio on the supply or exhaust side. Therefore, the effectiveness of a parallel flow (PF) sensible heat regenerator can be calculated as

[[epsilon].sub.PF] = [[epsilon].sub.PF.sup.rec] x [f.sub.PF](Cr*), (50),

where [f.sub.PF](Cr*) is the constant that is a function of Cr* for an exchanger operating in a parallel flow configuration and [[epsilon].sub.PF.sup.rec] is the effectiveness of a parallel flow recuperator. The effectiveness of a parallel flow heat exchanger (recuperator) with Cr = 1 is given (Incropera and Dewitt 2002) as

[[epsilon].sub.PF.sup.rec] = 0.5(1 - [e.sup.-2[N.sub.PF]]). (51)

Using Equations 46, 49, and 50, a relationship between the number of transfer units, N, and the inverse time constant, a, and wheel speed, [omega], for a parallel flow heat regenerator can be established as (Abe et al. 2006b)

[N.sub.PF] = -0.5 ln[1 - [1/[f.sub.PF](Cr*)] + [[infinity].summation over (n = 1)][8[a.sup.2]/[f(Cr*)(n[pi])[.sup.2][[a.sup.2] + ([pi][omega])[.sup.2]]]]]for n = 1,3,5,.... (52)

Note that this equation allows the calculation of N from a, [omega], and [f.sub.PF](Cr*). Also, as a decreases (i.e., [tau] increases), N increases. Abe et al. (2006a) show that for practical applications of energy wheels, [f.sub.PF](Cr*) has the range

0.999 < [f.sub.PF](Cr*) < 0.9998 (53)

when Cr* [greater than or equal to] 5 for typical energy wheel operating conditions. Thus, the term [f.sub.PF](Cr*) can be assumed equal to 1 for practical energy wheels and N can be determined knowing only the time constant and the wheel speed.

As previously stated, the aim is to relate the effectiveness of a parallel flow regenerator to a counterflow configuration. The effectiveness for a counterflow (CF) sensible heat regenerator is presented by Kays and London (1984) as

[[epsilon].sub.CF] = [[epsilon].sub.CF.sup.rec][1 - [1/9(Cr*)[.sup.1.93]]], (54)

where

[[epsilon].sub.CF.sup.rec] = f(N,Cr). (55)

For typical energy wheels, Cr* [greater than or equal to] 5 for wheel speed 15 rpm, giving a practical range for the second term in Equation 54 as,

0.995 < [f.sub.CF](Cr*) < 0.9998, (56)

where [f.sub.CF](Cr*) means a function of Cr* of an exchanger operating in a counterflow configuration. Thus, the term [f.sub.CF](Cr*) can be assumed equal to 1 for practical energy wheels. Equations 53 and 56, which are approximately equal to 1, permit relating the effectiveness of a sensible heat regenerator operating with a parallel flow configuration to one with a counterflow configuration. For the same mass flow rate of air for both cases, the N for the counterflow (CF) regenerator is equal to the number of transfer units, N, for a parallel flow regenerator (PF), i.e.,

[N.sub.CF] = [N.sub.PF]. (57)

The statement that [N.sub.CF] = [N.sub.PF] when the mass flow rate and inlet property conditions are the same in counterflow and parallel flow results from the fact that convective heat transfer coefficients between the air and the matrix are the same whether the airflow is arranged in counterflow or parallel flow. Therefore, the effectiveness of a counterflow heat regenerator with Cr = 1 and [f.sub.CF](Cr*) [congruent to] 1 can be approximated as

[FIGURE 9 OMITTED]

[[epsilon].sub.CF] = [[epsilon].sub.CF.sup.rec] = [N.sub.CF]/[1 + [N.sub.CF]]. (58)

Substituting Equations 52 and 57 into Equation 58, the effectiveness for a counterflow regenerative heat wheel exchanger with Cr = 1 and [f.sub.CF](Cr*) [congruent to] 1 can be expressed as

[[epsilon].sub.CF] = -0.5ln[[[infinity].summation over (n = 1)][8[a.sup.2]/[(n[pi])[.sup.2][[a.sup.2] + (n[omega])[.sup.2]]]]]/{1 - 0.5ln[[[infinity].summation over (n = 1)][8[a.sup.2]/[(n[pi])[.sup.2][[a.sup.2] + (n[omega])[.sup.2]]]]]} for n = 1,3,5,.... (59)

Equation 59, therefore, shows that the effectiveness of a regenerative exchanger (i.e., energy wheel) with a counterflow configuration can be predicted when the time constant of the wheel and the wheel speed are known. The predicted effectiveness of a counterflow energy wheel with various time constants using Equation 59 is shown in Figure 9.

Note that the effect of assuming [f.sub.CF](Cr*) [congruent to] 1 on the predicted effectiveness is trivial, as shown in Figure 9. At a practical range of 20 to 40 rpm for typical energy wheels in HVAC systems, [f.sub.CF](Cr*) has little effect on the effectiveness. The maximum difference obtained is 0.02. This figure shows that as the time constant and wheel speed increase, the effectiveness increases. This is as expected because Equation 52 showed that energy wheels that have a larger time constant will have larger values of N. The larger the N, the larger the heat transfer surface area and, thus, the higher the effectiveness. To verify the effectiveness calculated from Equation 59, the results shown in Figure 9 are compared with results obtained from the effectiveness correlation of Kays and London (1984), showing good agreement in Figure 10.

[FIGURE 10 OMITTED]

For a given energy wheel and airflow rate, Figures 9 and 10 imply that if the wheel speed is altered, the effectiveness will change. This wheel speed can be used for part-load control of energy wheels. Asiedu et al. (2004, 2005) discussed the optimal design and part-load requirements of energy wheels used in stand-alone units or in duel heat and energy wheel cabinet units of HVAC ventilation applications. Since part-load operating conditions occur for more than half of the operating time for typical good HVAC system designs, wheel speed control could be used as the preferred method of part-load control for a wide range of weather conditions.

CONCLUSIONS

The hypothesis that transient tests on a stationary wheel may be used to predict energy wheel effectiveness is examined in this paper. Equations are developed from first principles for the time constant for a transient heat transfer test where both inlet air temperatures before and after a step change when the air is dry and for an adiabatic moisture transfer test with identical inlet air temperatures and a step change in the humidity. The time constant for heat transfer is found to be a simple function of the wheel geometry and its thermal properties as well as the inlet air speed. The time constant for adiabatic moisture transfer is found to be a product of the heat transfer time constant and a new empirical desiccant coating efficacy coefficient, and this moisture transfer time constant will be less than the corresponding heat transfer time constant. The equations for these time constants can be used to provide guidance for future energy wheel design and operation.

An analytical model is developed to predict the sensible and latent or moisture transfer effectiveness of a counterflow energy wheel knowing the time constants of the wheel for sensible energy transfer for dry air and the moisture transfer with an adiabatic process with no inlet temperature difference. This model shows that the effectiveness increases with both time constant and wheel speed increases. The time constants for heat transfer and moisture transfer are shown to be functions of the energy wheel matrix properties as well as the inlet airflow rate. Both sensible effectiveness and latent effectiveness of the energy wheel can be determined by using transient temperature and humidity step responses. During moderate outdoor temperature and humidity conditions, part-load control of the energy transfer rate in energy wheels may be best done through wheel speed control using the relationships presented in this paper. A novel setup for testing energy wheels is presented in Part II (Abe et al. 2006a), where only the inlet and outlet air properties are measured for transient processes for heat transfer and moisture transfer.

ACKNOWLEDGMENTS

Financial assistance from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Venmar CES, Saskatoon, is appreciated.

NOMENCLATURE 

a = characteristic of the system, inverse of the time constant [tau] (1/s)

[a.sub.n] = constant in Fourier series

A = flow channel area, [m.sup.2]

A' = amplitude ratio

[A.sub.m] = matrix cross-sectional area for one channel, [m.sup.2]

[A*.sub.m] = matrix desiccant coating cross-sectional area for one channel, [m.sup.2]

[A.sub.s] = heat and mass transfer surface area on the supply or exhaust side, [m.sup.2]

b = constant forcing or input function (K or kg/[m.sup.3]) or constant in Fourier series

B = constant defined in Equation 31

c = constant of integration

Cp = specific heat capacity, J/(kg x K)

Cr = ratio of the minimum to maximum heat capacity rate of the airstreams

Cr* = matrix heat (or moisture) capacity ratio on the supply or exhaust side

d = constant defined in Equation 42

[d.sub.h] = hydraulic diameter, m

e = constant defined in Equation 42

f = function of

f(t) = forcing function or external input, K or kg/[m.sup.3]

af(t) = experimental forcing function or external input, K/s or kg/([m.sup.3] x s)

h = convective heat transfer coefficient, W/(m.sup.2) x K)

[h.sub.m] = convective mass transfer coefficient, m/s

L = thickness of wheel, m

M = total mass of wheel, kg

[dot.m] = mass flow rate of dry air on the supply or exhaust side, kg/s

n = integer constant in Fourier series

N = number of transfer units

N[u.sub.T] = Nusselt number for fully developed flow and constant wall temperature in a tube

[N.sub.[omega]] = angular speed of the wheel, cycle/s

P = flow channel perimeter, m

Q = volume flow rate of air on the supply or exhaust side, [m.sup.3]/s

RH = relative humidity

rpm = revolutions per minute

t = time, s

T = bulk temperature, [degrees]C or K

u = mass fraction of water in the desiccant, k[g.sub.w]/k[g.sub.d]

[bar.U] = mean airflow velocity in an exchanger flow channel, m/s

U([eta]) = uncertainty in parameter [eta]

y = output or response of the system, the same as y(t)

z = axial coordinate, m

z* = dimensionless axial coordinate defined by Equation 19

Greek

[alpha] = phase shift angle, rad

[epsilon] = effectiveness

[delta] = flow channel matrix thickness, m

[[delta].sub.d] = desiccant coating thickness on flow channel matrix surface, m

[[delta]*.sub.d] = characteristic thickness for a desiccant coating on flow channel matrix surface, m

[[delta].sub.M] = effective thickness of desiccant coating on flow channel matrix surface, m

[[delta].sub.sm] = thickness of metal film for one flow channel, m

[theta] = angle of wheel rotation, rad

[lambda] = water vapor desiccant coating efficacy coefficient, dimensionless

[rho] = density, kg/[m.sup.3]

[tau] = time constant for heat transfer, s

[[tau].sub.m] = time constant for moisture transfer, s

[chi] = weighting factor of a time constant

[psi] = dependent variable (temperature, humidity ratio, or enthalpy) used in Equation 1

[omega] = angular frequency of the forcing function, rad/s

Subscripts

a = air

CF = counterflow

d = desiccant

e = exhaust

i = inlet

g = total gas phase (air and water vapor)

m = matrix (including support material, desiccant, and moisture)

min = minimum

mt = dimensionless moisture transfer group for energy wheels

o = outlet

PF = parallel flow

s = supply

sm = matrix metal substrate 

v = water vapor

Superscript

rec = recuperator

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Oyetope O. Abe

Robert W. Besant, PEng

Fellow ASHRAE

Carey J. Simonson, PhD, PEng

Member ASHRAE

Wei Shang, PhD

Oyetope O. Abe is an associate technical professional/field engineer at Halliburton Energy Services, Grand Prairie, AB, Canada. Robert W. Besant is professor emeritus, Carey J. Simonson is an associate professor, and Wei Shang is a research associate in the Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK, Canada.

Table 1. Geometrical, Thermal, and Flow Conditions for a Typical
Corrugated Matrix Energy Wheel (2a/2b = 0.3)

Flow Condition
[bar.U]  h
m/s      W/([m.sup.2] x K)  N

1.8      60                 11

Metal Substrate
[[delta].sub.sm]    [[rho].sub.sm]  [c.sub.p,sm]
m                   kg/[m.sup.3]    J/(kg x K)

0.07 x [10.sup.-3]  2702            903

Desiccant
[[delta].sub.d]      [[rho].sub.d]  [c.sub.p,d]
m                    kg/[m.sup.3]   J/(kg x K)

0.002 x [10.sup.-3]  350            700