# An improved method for determining heat transfer fin efficiencies for dehumidifying cooling coils (RP-1194).

INTRODUCTION

Finned-tube heat exchangers are widely applied for cooling and dehumidifying moist air, and there is much literature related to modeling their steady-state performance. For instance, the 2000 ASHRAE Handbook-HVAC Systems and Equipment (ASHRAE 2000) and ARI Standard 410-2001: Forced-Circulation Air-Cooling and Air-Heating Coils (ARI 2001) present the same model for predicting the total and sensible energy transfer rates for wetted surface cooling coils. For the air side, the total (sensible and latent) energy transfer rate for wetted surfaces is proportional to the enthalpy difference between the airstream and saturated air at the temperature of the coil surface, while the sensible (convective) heat transfer is proportional to the temperature difference between the airstream and coil surface. A log mean temperature difference approach is utilized for estimating heat transfer between the coolant in the tubes (e.g., water) and the tube surface. Elmahdy and Mitalas (1977) and Braun et al. (1989) developed nearly equivalent and simpler models that transform the water stream to an equivalent airstream saturated with water vapor. With this transformation, the driving potential for the total energy transfer across the entire wetted section of the heat exchanger is written in terms of the enthalpy difference between the airstream and saturated air at the water temperature. The sensible heat transfer is determined using the model presented in the 2000 ASHRAE Handbook--HVAC Systems and Equipment (ASHRAE 2000) and ARI Standard 410 (2001).

Each of these models uses fin efficiency to simplify calculation of steady-state heat and mass transfer for the air side. They typically employ separate fin efficiencies for convective (sensible) heat transfer and combined heat and mass transfer. For convective heat transfer, fin efficiency accounts for the effect of the fin temperature distribution on total convective heat transfer to the fins. For combined heat and mass transfer, fin efficiency characterizes the impact of the distribution of wetted surface conditions on total energy transfer to the fins. Analytical expressions and correlations for heat transfer fin efficiencies have been developed for a wide variety of fin geometries based on a dry analysis, and these same expressions are typically applied for combined heat and mass transfer using modified properties (Kuehn et al. 1998; Xia and Jacobi 2005).

All of the existing cooling coil models that employ the fin efficiency concept utilize relationships developed for dry fins (i.e., no condensation) in determining sensible (convective) heat transfer. However, the fin temperature distribution is different for wet and dry fins and, therefore, the use of a dry fin efficiency relationship for convective heat transfer is not strictly correct under wet conditions. This paper develops a correction factor for existing fin efficiency relationships that allows a better estimate of convective heat transfer fin efficiencies under wet conditions. The improved method is validated using results obtained from a two-dimensional numerical analysis and experiments performed on an eight-row cooling coil.

DEVELOPMENT

This section develops a method for calculating sensible (convective) heat transfer fin efficiency under wet conditions for a classical straight fin geometry with uniform cross section. However, the method can be applied to any fin geometry where fin efficiency equations exist. Figure 1 illustrates the fin geometry and boundary conditions. In order to provide the proper background, classical fin efficiency relations for heat transfer under dry conditions and heat and mass transfer under wet conditions are first presented.

[FIGURE 1 OMITTED]

Fin efficiency for a dry fin is developed from the solution to a one-dimensional heat conduction problem along the fin height direction. Assuming the fin tip is adiabatic, uniform heat conductivity for the fin, [k.sub.f], and uniform convection coefficient across the fin, [h.sub.a-f], the fin efficiency for heat transfer only, [[eta].sub.f], is expressed as

[[eta].sub.f] = [[tan h(m*[H.sub.f])]/[m*[H.sub.f]]], (1)

where

m = [square root of [[h.sub.[a - f]]/[[k.sub.f]*t]]] (2)

and where [H.sub.f] is the fin height and t is half of the fin thickness (Kuehn et al. 1998).

Fin efficiency for a wet fin is developed using an analogous approach by adding the additional assumptions that the air specific heat, [C.sub.p,a], is constant and Lewis number is unity (Kuehn et al. 1998; Xia and Jacobi 2005). The fin efficiency for combined heat and mass transfer, [[eta]*.sub.f]<io> is

[[eta].sub.f.sup.*] = [[tan h([m.sup.*]*[H.sub.f])]/[[m.sup.*]*[H.sub.f]]], (3)

where

[m.sup.*] = [square root of [1/[[[[C.sub.p,a]/[[h.sub.[a - f].sup.*]*[C.sub.s]]] + [[t.sub.w]/[k.sub.w]]]/[[k.sub.f]*t]]]] (4)

and where [k.sub.w] and [t.sub.w] are the thermal conductivity and thickness of the condensate water film; [h*.sub.a - f] is the convection coefficient for a wet fin, which is somewhat higher than for a dry fin; and [C.sub.s] is the air saturation specific heat (Braun et al. 1989) defined as the derivative of the saturation air enthalpy with respect to temperature. In practice, is evaluated at the base temperature of the fin so that

[C.sub.s] = [[[d[h.sub.s]]/dT]|.sub.T = [T.sub.b]], (5)

where is [h.sub.s] saturated air enthalpy and [T.sub.b] is the fin base temperature.

Equation 4 was developed assuming film condensate and uniform film thickness along the fin. However, the film thickness is not uniform in practice, and drop-wise condensate occurs as well. It is common to exclude the condensate conduction term and utilize an air-to-fin convection heat transfer coefficient that accounts for the effects of condensate. Thus, Equation 4 becomes

[m.sup.*] = [square root of [[[h.sub.[a - f].sup.*]*[[C.sub.s]/[C.sub.p,a]]]/[[k.sub.f]*t]]]. (6)

In modeling a wetted fin surface, it is also necessary to determine fin efficiency for heat transfer only in order to determine sensible heat transfer. In previous modeling approaches, dry fin equations, such as Equation 1, were applied for this purpose. However, a temperature profile for a wet fin is not identical to that for a dry fin. An improved method for determining sensible heat transfer fin efficiency under wet conditions is developed by utilizing the fin temperature profile determined from the solution to the combined heat and mass transfer problem. This analysis provides a distribution of saturation air enthalpy along the fin ([T.sub.f]), which also uniquely determines the fin temperature distribution ([h.sub.s,f]). The relationship between the saturated air enthalpy and fin temperature profiles is approximated using the saturation specific heat determined with Equation 5. Assuming [C.sub.s] is constant over the relatively small temperature range within a fin,

[C.sub.s] = [[[h.sub.s,f](y) - [h.sub.s,b]]/[[T.sub.f](y) - [T.sub.b]]], (7)

where [T.sub.f] is the local fin temperature, [h.sub.s,f] is the local saturated air enthalpy evaluated at [T.sub.f], and [h.sub.s,b] is hs at the fin base temperature, [T.sub.b]. The temperature distribution at any location can then be determined from knowledge of the saturation air enthalpy profile using

[T.sub.f](y) - [T.sub.b] = [[[h.sub.s,f](y) - [h.sub.s,b]]/[C.sub.s]]. (8)

Rearranging Equation 8 gives

[[[T.sub.f](y) - [T.sub.a] + [T.sub.a] - [T.sub.b]]/[[T.sub.b] - [T.sub.a]]] = [[[h.sub.s,f](y) - [h.sub.a] + [h.sub.a] - [h.sub.s,b]]/[[C.sub.s]*([T.sub.b] - [T.sub.a])]], (9)

leading to the dimensionless fin temperature profile relation

[[[T.sub.f](y) - [T.sub.a]]/[[T.sub.b] - [T.sub.a]]] = 1 + [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]]*[[[h.sub.s,f](y) - [h.sub.a]]/[[h.sub.s,b] - [h.sub.a]]] - [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]]. (10)

The convective (sensible) heat transfer fin efficiency is defined as the ratio of the actual fin heat transfer to the heat transfer that would occur if the entire fin were at the base temperature. Under wetted surface conditions, the heat transfer fin efficiency is determined from the dimensionless temperature distribution as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII (11)

Defining a heat transfer fin efficiency correction factor as

[C.sub.F] = [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]], (12)

[[eta].sub.f] can be related to the heat and mass transfer fin efficiency according to

[[^.[eta]].sub.f] = 1 - [C.sub.F]*(1 - [[^.[eta]].sub.f.sup.*]). (13)

Equation 13 is a general expression for estimating heat transfer fin efficiency from combined heat and mass transfer fin efficiency and operating conditions when condensation occurs.

Figure 2 shows saturated air enthalpy as a function of temperature, along with example air states at the fin base (point b) and in the local bulk airstream away from the fin (point a). The correction factor in Equation 12 is the ratio of the slope of the line connecting points b and a and the slope at point b of the saturation line ([C.sub.s]). It should be clear from this depiction that the correction factor is always less than or equal to one, approaching one as the air nears thermal equilibrium with the fin base (i.e., saturated air at the fin base temperature). For this limiting case, the heat transfer fin efficiency is equal to the heat and mass transfer fin efficiency. At other conditions, the heat transfer fin efficiency is greater than the heat and mass transfer fin efficiency.

[FIGURE 2 OMITTED]

In order to apply the modified heat transfer fin efficiency of Equation 13 within a cooling coil model, such as those presented by ARI (2001), Elmahdy and Mitalas (1977), Braun et al. (1989), and Kuehn et al. (1998), it is necessary to specify representative values for the tube surface and air conditions for use in Equation 12. For any individual control volume within an overall coil model, the local temperature of tube material, which is considered to be uniform, is used to evaluate [T.sub.b] and [h.sub.s,b]. The air condition varies within the control volume, and it was found by numerical study that simply using the local air inlet conditions for [T.sub.a] and [h.sub.a] works well.

NUMERICAL VALIDATION

The simple geometry depicted in Figure 3 was studied numerically in order to evaluate improvements associated with the new approach for determining heat transfer fin efficiency. It is a single finned tube with a counter-flow arrangement. Table 1 gives geometrical and heat transfer parameters associated with the numerical study.

[FIGURE 3 OMITTED]

The parameters of Table 1 were chosen to be consistent with an existing eight-row cooling coil that has been studied extensively in a laboratory environment. A detailed description of the coil tested in the laboratory is provided in the section "Experimental Validation." Two different models were developed for this geometry in order to evaluate the improvement associated with the modified heat transfer fin efficiency for wet surfaces: (1) a two-dimensional, finite-volume reference model and (2) a model that incorporates fin efficiencies for both heat transfer and heat and mass transfer. The models employ the same basic assumptions utilized in developing the fin efficiencies in order to isolate the impact of the improved method for determining heat transfer fin efficiency.

Reference Model

The two-dimensional reference model for the fin neglects conduction across the fin thickness but considers conduction along the fin height (y) and length (x). Because of symmetry, only the fin above the tube is modeled. For any point within the fin, the steady-state energy equation is

[k.sub.f]*2t*([[[partial derivative].sup.2][T.sub.f]]/[[partial derivative][x.sup.2]] + [[[partial derivative].sup.2][T.sub.f]]/[[partial derivative][y.sup.2]]) - [[q.sup.?].sub.[a - f]] = 0, (14)

where [q".sub.a-f] is the local energy transfer flux from the airflow stream to the fin at any point with coordinates x and y.

The edges of the fins are assumed to be adiabatic, and the thermal resistance of the tube attached to the fin base is neglected. With these assumptions, the boundary conditions for the fin model are

x = 0,[[[partial derivative][T.sub.f]]/[[partial derivative]x]] = 0 and x = [L.sub.f],[[[partial derivative][T.sub.f]]/[[partial derivative]x]] = 0 (15)

and y = 0,[k.sub.f]*[[[partial derivative][T.sub.f]]/[[partial derivative]y]]*2t - [[q.sup.'].sub.[t - w]]/2 = 0 and y = [H.sub.f],[[[partial derivative][T.sub.f]]/[[partial derivative]y]] = 0, (16)

where [q'.sub.t-w] is the heat transfer rate from the tube to the water flow stream per unit length in the x direction. One half of [q'.sub.t-w] is from the fin base above the tube, and the other half is from the fin base below the tube.

The airflow is assumed to have a uniform velocity throughout the entire fin heat exchanger, and the water flow is assumed to be incompressible with a constant specific heat. The energy equations for the air and water are written as

[m.sup.'].sub.a]*[[d[h.sub.a]]/dx] = - [[q.sup.?].sub.[a - f]] (17)

and

[C.sub.w]*[[dT.sub.w]/dx] = [[q.sup.'].sub.[t - w]], (18)

where [m'.sub.a] is the air mass flow rate per unit height of fin, [h.sub.a] is the local air enthalpy, [C.sub.w] is the water heat capacitance rate, and [T.sub.w] is the local bulk water temperature. The water temperature has a one-dimensional variation in the direction of water flow, whereas the air temperature varies in two dimensions because of a dependence of the heat flux on fin temperature.

Conservation of mass for the water vapor within the airstream leads to

[[m.sup.'].sub.a]*[[d[W.sub.a]]/dx] = - [[m.sup.'].sub.c], (19)

where [W.sub.a] is the local air humidity ratio away from the surface and .m".sub.c] is the mass flow rate of condensate per unit surface area of the fin.

The boundary conditions for the air and water flows are

x = [L.sub.f],[h.sub.a] = [h.sub.a,in,HX],[W.sub.a] = [W.sub.a,in,HX] (20)

and

x = 0,[T.sub.w] = [T.sub.w,in,HX], (21)

where [h.sub.a,in,Hx] and [W.sub.a,in,Hx] are the heat exchanger air inlet enthalpy and humidity ratios, and [T.sub.w,in,Hx] is the inlet water temperature to the heat exchanger.

The heat transfer rate between tube and water is

[[q.sup.'].sub.[t - w]] = [([T.sub.t] - [T.sub.w])/[[R.sup.'].sub.[t - w]]], (22)

where [T.sub.t] is the local tube temperature, which is considered to be equal to the fin base temperature, and [R'.sub.t-w] is the convection heat transfer resistance between tube and water per unit length.

The energy flux from the air to a particular point on the fin depends upon whether moisture condenses or not. If the local fin surface temperature is above the local dewpoint of the air, then the surface is dry and the condensate term in Equation 19 is zero. In this case, the local heat flux to the fin from the air for Equations 14 and 17 is due to convective heat transfer only and is determined as

[[q.sup.?].sub.[a - f]] = [h.sub.[a - f]]*([T.sub.a] - [T.sub.f]), (23)

where [T.sub.a] is the local air inlet temperature, which is a function of [h.sub.a] and [W.sub.a], and [h.sub.a-f] is a global heat transfer coefficient for the heat exchanger under dry conditions.

When the fin temperature is below the dew point temperature of the inlet air locally, dehumidification occurs and the fin is wet. In this case, the condensate flux and energy flux due to heat and mass transfer are generally given as

[[m.sup.?].sub.c] = [h.sub.d]*([W.sub.a] - [W.sub.s,f]) (24)

and

[[q.sup.?].sub.[a - f]] = [h.sub.[a - f].sup.*]*([T.sub.a] - [T.sub.f]) + [h.sub.d]*([W.sub.a] - [W.sub.s,f])*[h.sub.fg,f], (25)

where [h.sub.d] is the mass transfer coefficient, [W.sub.s,f] is the humidity ratio for saturated air at the surface with a temperature [T.sub.f], [h.sub.fg,f] is the heat of water vaporization at [T.sub.f], and [h*.sub.a-f] is a global heat transfer coefficient for the heat exchanger under wet conditions (Kuehn et al. 1998).

Equation 24 can be rewritten using the Lewis relation [Le = [h*.sub.a-f]/([h.sub.d] [C.sub.p,a])], the relation for mixture enthalpy of ideal gas mixtures with constant specific heats [h = [C.sub.p,a] (T -[T.sub.ref]) + W [h.sub.fg]], and the assumption that the heat of vaporization has negligible dependence on temperature, so that

[[q.sup.?].sub.[a - f]] = [[h.sub.[a - f].sup.*]/[C.sub.p,a]]*([h.sub.a] - [h.sub.s,f])*[1/Le] + [h.sub.[a - f].sup.*]*([T.sub.a] - [T.sub.f])*(1 - 1/Le), (26)

where [C.sub.p,a] is the specific heat of the air-water vapor mixture.

It is most common to assume that the Lewis number is unity for air-water vapor mixtures at atmospheric pressure. For this assumption, Equation 26 reduces to the following commonly used form (Kuehn et al. 1998):

[[q.sup.?].sub.[a - f]] = [[h.sub.[a - f].sup.*]/[C.sub.p,a]]*([h.sub.a] - [h.sub.s,f]) (27)

The differential equation describing the fin (Equation 14) was discretized using a second-order centered finite-difference method (Incropera and DeWitt 1996). The corresponding boundary conditions (Equations 15 and 16) were discretized using a first-order finite-difference method (Incropera and DeWitt 1996), and the equations for [q'.sub.t-w] (Equation 22) and [q".sub.a-f] (Equations 23, 26, or 27, depending on the fin condition and the assumption of ) were discretized using a first-order upwind scheme, which means that the local fluid conditions inherit their inlet conditions (Murthy and Mathur 1998). The discretized equations were solved using a line-by-line tri-diagonal matrix algorithm (Murthy and Mathur 1998). After solving these equations,[q".sub.a-f], [m".sub.c], [w'.sub. t-w] and were calculated for each control volume. The air and water energy equations (Equations 17 and 19 for the air and Equation 18 for the water) were discretized again using the first-order finite difference approach and were solved directly for the local outlet conditions. An iterative solution was necessary to couple the solution of the fin conduction problem to the solution of the air and water energy equations. At each iteration, it was necessary to compare all of the local surface temperatures to the local air dew-point temperatures to determine the condensate and energy fluxes. The final solution gives two-dimensional distributions for fin temperature, air temperature and humidity, and surface fluxes along with one-dimensional water temperature distributions.

Model Employing Fin Efficiencies

Typical heat exchanger and cooling coil models utilize fin efficiencies to account for temperature and enthalpy distributions along the fin height (y) direction and neglect fin conduction in the longitudinal (x) direction. The fin efficiencies are derived assuming that the local air states are uniform in the y direction but vary in the x direction. In addition, the specific heat of the air is assumed to be constant. With these assumptions, the energy equation for the air at a dry location in the heat exchanger of Figure 3 can be expressed as

[m.sub.a]*[C.sub.p,a]*[[dT.sub.a]/dx] = [[eta].sub.f]*[h.sub.[a - f]]*[H.sub.f]*([T.sub.t] - [T.sub.a]), (28)

where [m.sub.a] is the air mass flow rate and [T.sub.a] represents the bulk air temperature across the entire fin in the y direction at any x. The fin efficiency accounts for the temperature distribution in the fin and allows the heat transfer rate to be expressed as a function of the base temperature of the fin, which is also the tube surface temperature.

For wet sections within the coil, the assumption of a Lewis number of unity and an overall fin efficiency for heat and mass transfer are employed. In this case, the air energy equation is

[m.sub.a]*[[d[h.sub.a]]/dx] = [[eta].sub.f.sup.*]*[h.sub.[a - f].sup.*]*[H.sub.f]*([h.sub.s,t] - [h.sub.a]), (29)

where [[eta]*.sub.f] is the heat and mass transfer fin efficiency and [h.sub.s,t] is the saturated air enthalpy at [T.sub.t]. The fin efficiency accounts for the distribution of saturated air enthalpies along the fin height.

Equation 29 is applied when the local tube surface temperature is below the local air dew-point temperature; otherwise, only Equation 28 is applied. Unlike the reference model, the models employing fin efficiencies do not have the ability to track the transition of wet-to-dry surfaces along the fin height direction. They can only track the transition along the direction of airflow.

For wet sections, it is also necessary to consider a mass balance on the water within the airstream in order to fully characterize the air state. In order to determine the air temperature variation, a water vapor mass balance ([m.sub.a] [dw.sub.a]/dx = [-m'.sub.c]).can be combined with a basic energy balance ([m.sub.a] [dh.sub.a]/dx = [[eta].sub. f] [h*.sub.a-f] [H.sub. f] ([T.sub.a] - [T.sub.t]) - [m.sub. c] [h.sub. fg]) and the mixture enthalpy relation (h = [C.sub.p,a] (T -[T.sub.ref]) + W [h.sub.fg]) to give

[m.sub.a]*[C.sub.p,a][[dT.sub.a]/dx] = [[^.[eta]].sub.f]*[h.sub.[a - f].sup.*]*[H.sub.f]*([T.sub.t] - [T.sub.a]), (30)

where [[eta].sub.f] is the heat transfer fin efficiency for a wet surface. This relationship is nearly the same as the energy balance for a dry section (Equation 28), except that the heat transfer coefficient and fin efficiency are different because of the influence of the wetted surface. The traditional models for cooling coils typically employ a heat transfer efficiency derived for dry surfaces in determining the air temperatures for wet sections of the coil.

The energy equation for the water stream and the heat exchanger boundary conditions are identical to those presented for the reference model. These relations, along with Equations 28-30 and property functions, are sufficient to determine the one-dimensional variation in water temperature, tube temperature, air enthalpy, and air temperature in the flow direction within the heat exchanger.

Existing models for cooling coils, such as those presented by ARI (2001), Elmahdy and Mitalas (1977), Braun et al. (1989), and Kuehn et al. (1998), utilize a combination of analytical and numerical techniques to solve the equations. They involve determination of the point for transition between dry and wet sections of the coil and use of separate analytical solutions to the basic equations for each of these sections that arise from the use of some simplifying assumptions. However, in order to be consistent with the reference model and isolate the effects of utilizing the improved heat transfer fin efficiency, a numerical solution to the modeling equations was implemented. In order to evaluate the improvements associated with the use of heat transfer fin efficiency correction presented in this paper, the model described in this section was solved for two different cases for heat transfer fin efficiency in Equation 30: (1) a dry heat transfer fin efficiency (Equation 1) and (2) a corrected heat transfer fin efficiency based on a wet analysis (Equation 13).

The differential equations were discretized and solved numerically using the first-order finite-difference method and upwind scheme (Murthy and Mathur 1998). The Gauss-Seidel iteration (Incropera and DeWitt 1996) was used to update the boundary conditions that couple the different finite-difference equations in the solution of the counter-flow arrangement. For each air control volume, it is necessary to determine whether moisture is condensing or not. This is evaluated by comparing the surface temperature from a dry analysis with the local air inlet dew point.

Model Comparisons

Table 2 gives sample inlet conditions considered for comparisons of numerical predictions for the different models compared in this study. The inlet relative humidity was varied between 40% and 80% with all other inlet conditions held constant. These conditions were chosen for presentation because the heat exchanger transitions from partially wet at the lowest humidity to fully wet conditions at the highest relative humidity.

Model results are compared in terms of their ability to predict total and sensible energy transfer rates for the heat exchanger as determined by

[q.sub.tot] = [C.sub.w]*([T.sub.w,out,HX] - [T.sub.w,in,HX]) (31)

and

[q.sub.sen] = [C.sub.a]*([T.sub.a,in,HX] - [T.sub.a,out,HX]), (32)

where the subscript "HX" denotes that values are for the heat exchanger instead of local control volumes.

Figure 4 gives comparisons between predictions of total energy transfer rates for the reference and fin efficiency models as a function of inlet air relative humidity. The total energy transfer rate increases with relative humidity because of increased latent energy transfer. Predictions of total energy transfer rates for the fin efficiency model are independent of the heat transfer fin efficiency used for wet sections (only the combined heat and mass transfer fin efficiency influences the total energy transfer rate in the wet section). Therefore, only a single result is presented for this approach. For the reference model, results are presented for both a Lewis number of unity and 0.9. According to Kuehn et al. (1998), air-water vapor mixtures have a Lewis number of around 0.9 at atmospheric conditions for a wide range of temperatures and humidities. A Lewis number of unity is assumed for the model that employs fin efficiencies and for most traditional cooling models.

[FIGURE 4 OMITTED]

Figure 4 shows that errors associated with the assumption of a Lewis number of unity are very small for representative operating conditions. The model employing fin efficiency also gives total energy transfer rate results that are close to the reference model. They differ slightly because the reference model accounts for fin conduction in the longitudinal (x) direction and allows for partially wet and dry conditions in the fin height direction. Similar results were obtained for a wide variety of operating conditions. The heat and mass transfer fin efficiency provides a very good characterization of the saturated air enthalpy distribution on the fin.

Figure 5 gives comparisons between predictions of sensible energy transfer rates for the reference and fin efficiency models. The sensible heat transfer rate decreases with entering air relative humidity because increased latent energy transfer causes an increase in the heat exchanger surface temperature. The reference model results were obtained for a Lewis number of unity. For the fin efficiency model, results were determined using heat transfer fin efficiencies in the wet section that were derived for both dry surfaces (Equation 1) and wet conditions (Equation 13). The use of the modified heat transfer fin efficiency gives predictions that are in much closer agreement with the two-dimensional results as compared with the use of dry heat transfer fin efficiency relations. The errors associated with using a heat transfer fin efficiency derived for dry conditions in the wet section increase as more of the heat exchanger is wetted (i.e., at higher inlet relative air humidities). For the results of Figure 5, the errors in sensible heat transfer were reduced from a maximum of about 12% to less than 1% by applying the improved heat transfer fin efficiency. Similar results were obtained for a wide variety of operating conditions where moisture was condensing on the fin.

[FIGURE 5 OMITTED]

Figure 6 shows variation in local heat transfer and heat and mass transfer fin efficiencies as a function of position along the airflow stream for an air inlet relative humidity of 60%. Heat transfer fin efficiency results are presented for both the dry analysis (original model) and the new approach that corrects for the wetted fin. The initial 20% of the fin surface is dry, so only the heat transfer fin efficiency for dry surfaces applies. After condensation starts, the heat and mass transfer fin efficiency for wet surfaces applies and increases as the surface becomes wetter when approaching the chilled-water inlet. In contrast, the heat transfer fin efficiency for the wet surface decreases in the direction of the airflow and approaches the heat and mass transfer fin efficiency as the air approaches saturation. The error associated with using a dry local fin efficiency is largest for saturated air conditions and is greater than 15% at the air outlet condition for the results of Figure 6.

[FIGURE 6 OMITTED]

EXPERIMENTAL VALIDATION

The improved method of determining heat transfer fin efficiency under wet surface conditions was implemented within an overall cooling coil model for a coil that has been extensively tested in a laboratory as described by Zhou and Braun (2005). This is an eight-row coil with wavy fins and a fin spacing of eight fins per inch, which is representative of large commercial applications. Table 3 provides the parameters that characterize the coil, and Figure 7 depicts its flow circuiting.

[FIGURE 7 OMITTED]

The coil was tested using a psychrometric wind tunnel that allows control of the coil inlet air temperature, humidity and flow rate, and water temperature and flow rate. Sufficient instrumentation was installed to determine air-side and water-side heat transfer rates under steady-state conditions (Zhou and Braun 2005).

The cooling coil model used for implementing and evaluating the improved fin efficiency approach was presented in ARI Standard 410 (2001). It separates the coil into dry and wet sections and applies the log mean temperature difference method to the dry section and the log mean enthalpy difference method to the wet section. Both the original and improved methods for heat transfer fin efficiency were implemented for comparison with experimental results.

Table 4 gives sample comparisons between model predictions and experimental results for a range of conditions where the coil was partially and fully wetted. Predictions for total energy transfer rate are within 3.2% of the values determined from experimental measurements, which is less than the measurement uncertainty and energy balance error between air-side and water-side measurements (6%) for the test setup. The model predictions for total energy transfer rate are independent of the approach used to characterize the heat transfer fin efficiency in the wet section. Predictions of sensible heat transfer are within 1.2% of the measurements for the modified heat transfer fin efficiency approach, which is an improvement compared to the 8.4% difference associated with the original method.

Figure 8 shows comparisons for sensible heat ratio (SHR), defined as the ratio of the sensible to total energy transfer rates. The SHR predictions by the original model are beyond the uncertainty bounds for the experimental measurements. The modified heat transfer fin efficiency model improves these predictions into or on the uncertainty bounds and provides greater relative improvements in SHR at lower values where the coil is wetter.

[FIGURE 8 OMITTED]

CONCLUSIONS

This paper develops and presents an improved method for determining heat transfer fin efficiency for wetted fins. The heat transfer fin efficiency is determined using a correction factor applied to the heat and mass transfer fin efficiency (Equation 13). Detailed numerical and experimental results show that the method provides improved predictions of sensible heat transfer rates compared to the use of dry surface fin efficiency relations, especially for conditions where cooling coils are fully wetted. The improved method is readily implemented in existing models and does not result in additional complexity.

ACKNOWLEDGMENTS

This study was funded by ASHRAE (ASHRAE RP-1194). Their support is gratefully appreciated.

NOMENCLATURE

C = specific heat, kJ/kg K

C = heat capacitance rate, kW/K

t = half of fin thickness, m

h = convection coefficient, W/[m.sup.2] K or specific enthalpy, kJ/kg

H = height, m

[h.sub.d] = mass transfer coefficient, kga/s [m.sup.2]

[h.sub.fg] = heat of water vaporization, kJ/kgw

k = heat conductivity, W/m K

L = length, m

Le = Lewis number

m = mass flow rate, kg/s

q = heat transfer rate, W

R = heat resistance, K/W

RH = relative humidity, %

T = temperature, [degrees]C

W = humidity ratio, [kg.sub.w]/[kg.sub.a]

Greek symbols

[[eta].sub.f] = heat transfer fin efficiency for dry surface

[[eta].sub.f] = heat transfer fin efficiency for wet surface

[[eta]*.sub.f] = heat and mass transfer fin efficiency for wet surface

Subscripts

a = air

b = fin base

c = condensate

f = fin

in = inlet

out = outlet

s = saturation

sen = sensible

t = tube

tot = total

w = water

Superscripts

* = wet fin

' = per unit length

" = per unit surface area

REFERENCES

ARI. 2001. ARI 410-2001: Forced-Circulation Air-Cooling and Air-Heating Coils. Arlington, VA: Air-Conditioning and Refrigeration Institute.

ASHRAE. 2000. 2000 ASHRAE Handbook-HVAC Systems and Equipment. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Braun, J.E., S.A. Klein, and J.W. Mitchell. 1989. Effectiveness models for cooling towers and cooling coils. ASHRAE Transactions 95(2):164-74.

Elmahdy, A.H., and G.P. Mitalas. 1977. A simple model for cooling and dehumidifying coils for use in calculating energy requirements for buildings. ASHRAE Transactions 83(2):103-17.

Incropera, F.P., and D.P. DeWitt. 1996. Fundamentals of Heat and Mass Transfer. New York: John Wiley & Sons.

Kuehn, T.H., J.W. Ramsey, and J.L. Threlkeld. 1998. Thermal Environmental Engineering. Upper Saddle River, NJ: Prentice-Hall.

Murthy, J.Y., and S.R. Mathur. 1998. Numerical methods in heat, mass, and momentum transfer. Lecture notes.

Xia, Y., and A.M. Jacobi. 2005. Air-side data interpretation and performance analysis for heat exchangers with simultaneous heat and mass transfer: Wet and frosted surfaces. International Journal of Heat and Mass Transfer 48:5089-102.

Zhou, X., and J.E. Braun. 2005. Dynamic modeling of chilled water cooling coils. ASHRAE 1194-RP Final Report, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., Atlanta.

Xiaotang Zhou, PhD Associate Member ASHRAE James E. Braun, PhD Fellow ASHRAE Qingfan Zeng, PhD

Xiaotang Zhou is an Engineering Leadership Program associate for the Carrier Corporation, Syracuse, NY. James E. Braun is a professor of mechanical engineering for Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN. Qingfan Zeng is a senior engineer for Carrier China, Shanghai, China.

Received May 26, 2006; accepted June 11, 2007

This paper is based on findings resulting from ASHRAE Research Project RP-1194.

Finned-tube heat exchangers are widely applied for cooling and dehumidifying moist air, and there is much literature related to modeling their steady-state performance. For instance, the 2000 ASHRAE Handbook-HVAC Systems and Equipment (ASHRAE 2000) and ARI Standard 410-2001: Forced-Circulation Air-Cooling and Air-Heating Coils (ARI 2001) present the same model for predicting the total and sensible energy transfer rates for wetted surface cooling coils. For the air side, the total (sensible and latent) energy transfer rate for wetted surfaces is proportional to the enthalpy difference between the airstream and saturated air at the temperature of the coil surface, while the sensible (convective) heat transfer is proportional to the temperature difference between the airstream and coil surface. A log mean temperature difference approach is utilized for estimating heat transfer between the coolant in the tubes (e.g., water) and the tube surface. Elmahdy and Mitalas (1977) and Braun et al. (1989) developed nearly equivalent and simpler models that transform the water stream to an equivalent airstream saturated with water vapor. With this transformation, the driving potential for the total energy transfer across the entire wetted section of the heat exchanger is written in terms of the enthalpy difference between the airstream and saturated air at the water temperature. The sensible heat transfer is determined using the model presented in the 2000 ASHRAE Handbook--HVAC Systems and Equipment (ASHRAE 2000) and ARI Standard 410 (2001).

Each of these models uses fin efficiency to simplify calculation of steady-state heat and mass transfer for the air side. They typically employ separate fin efficiencies for convective (sensible) heat transfer and combined heat and mass transfer. For convective heat transfer, fin efficiency accounts for the effect of the fin temperature distribution on total convective heat transfer to the fins. For combined heat and mass transfer, fin efficiency characterizes the impact of the distribution of wetted surface conditions on total energy transfer to the fins. Analytical expressions and correlations for heat transfer fin efficiencies have been developed for a wide variety of fin geometries based on a dry analysis, and these same expressions are typically applied for combined heat and mass transfer using modified properties (Kuehn et al. 1998; Xia and Jacobi 2005).

All of the existing cooling coil models that employ the fin efficiency concept utilize relationships developed for dry fins (i.e., no condensation) in determining sensible (convective) heat transfer. However, the fin temperature distribution is different for wet and dry fins and, therefore, the use of a dry fin efficiency relationship for convective heat transfer is not strictly correct under wet conditions. This paper develops a correction factor for existing fin efficiency relationships that allows a better estimate of convective heat transfer fin efficiencies under wet conditions. The improved method is validated using results obtained from a two-dimensional numerical analysis and experiments performed on an eight-row cooling coil.

DEVELOPMENT

This section develops a method for calculating sensible (convective) heat transfer fin efficiency under wet conditions for a classical straight fin geometry with uniform cross section. However, the method can be applied to any fin geometry where fin efficiency equations exist. Figure 1 illustrates the fin geometry and boundary conditions. In order to provide the proper background, classical fin efficiency relations for heat transfer under dry conditions and heat and mass transfer under wet conditions are first presented.

[FIGURE 1 OMITTED]

Fin efficiency for a dry fin is developed from the solution to a one-dimensional heat conduction problem along the fin height direction. Assuming the fin tip is adiabatic, uniform heat conductivity for the fin, [k.sub.f], and uniform convection coefficient across the fin, [h.sub.a-f], the fin efficiency for heat transfer only, [[eta].sub.f], is expressed as

[[eta].sub.f] = [[tan h(m*[H.sub.f])]/[m*[H.sub.f]]], (1)

where

m = [square root of [[h.sub.[a - f]]/[[k.sub.f]*t]]] (2)

and where [H.sub.f] is the fin height and t is half of the fin thickness (Kuehn et al. 1998).

Fin efficiency for a wet fin is developed using an analogous approach by adding the additional assumptions that the air specific heat, [C.sub.p,a], is constant and Lewis number is unity (Kuehn et al. 1998; Xia and Jacobi 2005). The fin efficiency for combined heat and mass transfer, [[eta]*.sub.f]<io> is

[[eta].sub.f.sup.*] = [[tan h([m.sup.*]*[H.sub.f])]/[[m.sup.*]*[H.sub.f]]], (3)

where

[m.sup.*] = [square root of [1/[[[[C.sub.p,a]/[[h.sub.[a - f].sup.*]*[C.sub.s]]] + [[t.sub.w]/[k.sub.w]]]/[[k.sub.f]*t]]]] (4)

and where [k.sub.w] and [t.sub.w] are the thermal conductivity and thickness of the condensate water film; [h*.sub.a - f] is the convection coefficient for a wet fin, which is somewhat higher than for a dry fin; and [C.sub.s] is the air saturation specific heat (Braun et al. 1989) defined as the derivative of the saturation air enthalpy with respect to temperature. In practice, is evaluated at the base temperature of the fin so that

[C.sub.s] = [[[d[h.sub.s]]/dT]|.sub.T = [T.sub.b]], (5)

where is [h.sub.s] saturated air enthalpy and [T.sub.b] is the fin base temperature.

Equation 4 was developed assuming film condensate and uniform film thickness along the fin. However, the film thickness is not uniform in practice, and drop-wise condensate occurs as well. It is common to exclude the condensate conduction term and utilize an air-to-fin convection heat transfer coefficient that accounts for the effects of condensate. Thus, Equation 4 becomes

[m.sup.*] = [square root of [[[h.sub.[a - f].sup.*]*[[C.sub.s]/[C.sub.p,a]]]/[[k.sub.f]*t]]]. (6)

In modeling a wetted fin surface, it is also necessary to determine fin efficiency for heat transfer only in order to determine sensible heat transfer. In previous modeling approaches, dry fin equations, such as Equation 1, were applied for this purpose. However, a temperature profile for a wet fin is not identical to that for a dry fin. An improved method for determining sensible heat transfer fin efficiency under wet conditions is developed by utilizing the fin temperature profile determined from the solution to the combined heat and mass transfer problem. This analysis provides a distribution of saturation air enthalpy along the fin ([T.sub.f]), which also uniquely determines the fin temperature distribution ([h.sub.s,f]). The relationship between the saturated air enthalpy and fin temperature profiles is approximated using the saturation specific heat determined with Equation 5. Assuming [C.sub.s] is constant over the relatively small temperature range within a fin,

[C.sub.s] = [[[h.sub.s,f](y) - [h.sub.s,b]]/[[T.sub.f](y) - [T.sub.b]]], (7)

where [T.sub.f] is the local fin temperature, [h.sub.s,f] is the local saturated air enthalpy evaluated at [T.sub.f], and [h.sub.s,b] is hs at the fin base temperature, [T.sub.b]. The temperature distribution at any location can then be determined from knowledge of the saturation air enthalpy profile using

[T.sub.f](y) - [T.sub.b] = [[[h.sub.s,f](y) - [h.sub.s,b]]/[C.sub.s]]. (8)

Rearranging Equation 8 gives

[[[T.sub.f](y) - [T.sub.a] + [T.sub.a] - [T.sub.b]]/[[T.sub.b] - [T.sub.a]]] = [[[h.sub.s,f](y) - [h.sub.a] + [h.sub.a] - [h.sub.s,b]]/[[C.sub.s]*([T.sub.b] - [T.sub.a])]], (9)

leading to the dimensionless fin temperature profile relation

[[[T.sub.f](y) - [T.sub.a]]/[[T.sub.b] - [T.sub.a]]] = 1 + [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]]*[[[h.sub.s,f](y) - [h.sub.a]]/[[h.sub.s,b] - [h.sub.a]]] - [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]]. (10)

The convective (sensible) heat transfer fin efficiency is defined as the ratio of the actual fin heat transfer to the heat transfer that would occur if the entire fin were at the base temperature. Under wetted surface conditions, the heat transfer fin efficiency is determined from the dimensionless temperature distribution as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII (11)

Defining a heat transfer fin efficiency correction factor as

[C.sub.F] = [1/[C.sub.s]]*[[[h.sub.s,b] - [h.sub.a]]/[[T.sub.b] - [T.sub.a]]], (12)

[[eta].sub.f] can be related to the heat and mass transfer fin efficiency according to

[[^.[eta]].sub.f] = 1 - [C.sub.F]*(1 - [[^.[eta]].sub.f.sup.*]). (13)

Equation 13 is a general expression for estimating heat transfer fin efficiency from combined heat and mass transfer fin efficiency and operating conditions when condensation occurs.

Figure 2 shows saturated air enthalpy as a function of temperature, along with example air states at the fin base (point b) and in the local bulk airstream away from the fin (point a). The correction factor in Equation 12 is the ratio of the slope of the line connecting points b and a and the slope at point b of the saturation line ([C.sub.s]). It should be clear from this depiction that the correction factor is always less than or equal to one, approaching one as the air nears thermal equilibrium with the fin base (i.e., saturated air at the fin base temperature). For this limiting case, the heat transfer fin efficiency is equal to the heat and mass transfer fin efficiency. At other conditions, the heat transfer fin efficiency is greater than the heat and mass transfer fin efficiency.

[FIGURE 2 OMITTED]

In order to apply the modified heat transfer fin efficiency of Equation 13 within a cooling coil model, such as those presented by ARI (2001), Elmahdy and Mitalas (1977), Braun et al. (1989), and Kuehn et al. (1998), it is necessary to specify representative values for the tube surface and air conditions for use in Equation 12. For any individual control volume within an overall coil model, the local temperature of tube material, which is considered to be uniform, is used to evaluate [T.sub.b] and [h.sub.s,b]. The air condition varies within the control volume, and it was found by numerical study that simply using the local air inlet conditions for [T.sub.a] and [h.sub.a] works well.

NUMERICAL VALIDATION

The simple geometry depicted in Figure 3 was studied numerically in order to evaluate improvements associated with the new approach for determining heat transfer fin efficiency. It is a single finned tube with a counter-flow arrangement. Table 1 gives geometrical and heat transfer parameters associated with the numerical study.

[FIGURE 3 OMITTED]

The parameters of Table 1 were chosen to be consistent with an existing eight-row cooling coil that has been studied extensively in a laboratory environment. A detailed description of the coil tested in the laboratory is provided in the section "Experimental Validation." Two different models were developed for this geometry in order to evaluate the improvement associated with the modified heat transfer fin efficiency for wet surfaces: (1) a two-dimensional, finite-volume reference model and (2) a model that incorporates fin efficiencies for both heat transfer and heat and mass transfer. The models employ the same basic assumptions utilized in developing the fin efficiencies in order to isolate the impact of the improved method for determining heat transfer fin efficiency.

Table 1. Characteristics of the Straight Finned-Tube Heat Exchanger Parameter Value Fin material Aluminum Fin material 237 W/m K thermal conductivity, [k.sub.f] Fin length, 0.3 m [L.sub.f] Fin height, 0.02 m [H.sub.f] Fin thickness, 2t 0.0002 m Dry fin convection 45.9 coefficient, W/[m.sup.2] K [h.sub.a-f] Wet fin convection 49.8 coefficient, W/[m.sup.2] K [h*.sub.a-f] Water-to-tube heat 0.31 m K/W transfer resistance per unit length, [R.sub.w-t.sup.']

Reference Model

The two-dimensional reference model for the fin neglects conduction across the fin thickness but considers conduction along the fin height (y) and length (x). Because of symmetry, only the fin above the tube is modeled. For any point within the fin, the steady-state energy equation is

[k.sub.f]*2t*([[[partial derivative].sup.2][T.sub.f]]/[[partial derivative][x.sup.2]] + [[[partial derivative].sup.2][T.sub.f]]/[[partial derivative][y.sup.2]]) - [[q.sup.?].sub.[a - f]] = 0, (14)

where [q".sub.a-f] is the local energy transfer flux from the airflow stream to the fin at any point with coordinates x and y.

The edges of the fins are assumed to be adiabatic, and the thermal resistance of the tube attached to the fin base is neglected. With these assumptions, the boundary conditions for the fin model are

x = 0,[[[partial derivative][T.sub.f]]/[[partial derivative]x]] = 0 and x = [L.sub.f],[[[partial derivative][T.sub.f]]/[[partial derivative]x]] = 0 (15)

and y = 0,[k.sub.f]*[[[partial derivative][T.sub.f]]/[[partial derivative]y]]*2t - [[q.sup.'].sub.[t - w]]/2 = 0 and y = [H.sub.f],[[[partial derivative][T.sub.f]]/[[partial derivative]y]] = 0, (16)

where [q'.sub.t-w] is the heat transfer rate from the tube to the water flow stream per unit length in the x direction. One half of [q'.sub.t-w] is from the fin base above the tube, and the other half is from the fin base below the tube.

The airflow is assumed to have a uniform velocity throughout the entire fin heat exchanger, and the water flow is assumed to be incompressible with a constant specific heat. The energy equations for the air and water are written as

[m.sup.'].sub.a]*[[d[h.sub.a]]/dx] = - [[q.sup.?].sub.[a - f]] (17)

and

[C.sub.w]*[[dT.sub.w]/dx] = [[q.sup.'].sub.[t - w]], (18)

where [m'.sub.a] is the air mass flow rate per unit height of fin, [h.sub.a] is the local air enthalpy, [C.sub.w] is the water heat capacitance rate, and [T.sub.w] is the local bulk water temperature. The water temperature has a one-dimensional variation in the direction of water flow, whereas the air temperature varies in two dimensions because of a dependence of the heat flux on fin temperature.

Conservation of mass for the water vapor within the airstream leads to

[[m.sup.'].sub.a]*[[d[W.sub.a]]/dx] = - [[m.sup.'].sub.c], (19)

where [W.sub.a] is the local air humidity ratio away from the surface and .m".sub.c] is the mass flow rate of condensate per unit surface area of the fin.

The boundary conditions for the air and water flows are

x = [L.sub.f],[h.sub.a] = [h.sub.a,in,HX],[W.sub.a] = [W.sub.a,in,HX] (20)

and

x = 0,[T.sub.w] = [T.sub.w,in,HX], (21)

where [h.sub.a,in,Hx] and [W.sub.a,in,Hx] are the heat exchanger air inlet enthalpy and humidity ratios, and [T.sub.w,in,Hx] is the inlet water temperature to the heat exchanger.

The heat transfer rate between tube and water is

[[q.sup.'].sub.[t - w]] = [([T.sub.t] - [T.sub.w])/[[R.sup.'].sub.[t - w]]], (22)

where [T.sub.t] is the local tube temperature, which is considered to be equal to the fin base temperature, and [R'.sub.t-w] is the convection heat transfer resistance between tube and water per unit length.

The energy flux from the air to a particular point on the fin depends upon whether moisture condenses or not. If the local fin surface temperature is above the local dewpoint of the air, then the surface is dry and the condensate term in Equation 19 is zero. In this case, the local heat flux to the fin from the air for Equations 14 and 17 is due to convective heat transfer only and is determined as

[[q.sup.?].sub.[a - f]] = [h.sub.[a - f]]*([T.sub.a] - [T.sub.f]), (23)

where [T.sub.a] is the local air inlet temperature, which is a function of [h.sub.a] and [W.sub.a], and [h.sub.a-f] is a global heat transfer coefficient for the heat exchanger under dry conditions.

When the fin temperature is below the dew point temperature of the inlet air locally, dehumidification occurs and the fin is wet. In this case, the condensate flux and energy flux due to heat and mass transfer are generally given as

[[m.sup.?].sub.c] = [h.sub.d]*([W.sub.a] - [W.sub.s,f]) (24)

and

[[q.sup.?].sub.[a - f]] = [h.sub.[a - f].sup.*]*([T.sub.a] - [T.sub.f]) + [h.sub.d]*([W.sub.a] - [W.sub.s,f])*[h.sub.fg,f], (25)

where [h.sub.d] is the mass transfer coefficient, [W.sub.s,f] is the humidity ratio for saturated air at the surface with a temperature [T.sub.f], [h.sub.fg,f] is the heat of water vaporization at [T.sub.f], and [h*.sub.a-f] is a global heat transfer coefficient for the heat exchanger under wet conditions (Kuehn et al. 1998).

Equation 24 can be rewritten using the Lewis relation [Le = [h*.sub.a-f]/([h.sub.d] [C.sub.p,a])], the relation for mixture enthalpy of ideal gas mixtures with constant specific heats [h = [C.sub.p,a] (T -[T.sub.ref]) + W [h.sub.fg]], and the assumption that the heat of vaporization has negligible dependence on temperature, so that

[[q.sup.?].sub.[a - f]] = [[h.sub.[a - f].sup.*]/[C.sub.p,a]]*([h.sub.a] - [h.sub.s,f])*[1/Le] + [h.sub.[a - f].sup.*]*([T.sub.a] - [T.sub.f])*(1 - 1/Le), (26)

where [C.sub.p,a] is the specific heat of the air-water vapor mixture.

It is most common to assume that the Lewis number is unity for air-water vapor mixtures at atmospheric pressure. For this assumption, Equation 26 reduces to the following commonly used form (Kuehn et al. 1998):

[[q.sup.?].sub.[a - f]] = [[h.sub.[a - f].sup.*]/[C.sub.p,a]]*([h.sub.a] - [h.sub.s,f]) (27)

The differential equation describing the fin (Equation 14) was discretized using a second-order centered finite-difference method (Incropera and DeWitt 1996). The corresponding boundary conditions (Equations 15 and 16) were discretized using a first-order finite-difference method (Incropera and DeWitt 1996), and the equations for [q'.sub.t-w] (Equation 22) and [q".sub.a-f] (Equations 23, 26, or 27, depending on the fin condition and the assumption of ) were discretized using a first-order upwind scheme, which means that the local fluid conditions inherit their inlet conditions (Murthy and Mathur 1998). The discretized equations were solved using a line-by-line tri-diagonal matrix algorithm (Murthy and Mathur 1998). After solving these equations,[q".sub.a-f], [m".sub.c], [w'.sub. t-w] and were calculated for each control volume. The air and water energy equations (Equations 17 and 19 for the air and Equation 18 for the water) were discretized again using the first-order finite difference approach and were solved directly for the local outlet conditions. An iterative solution was necessary to couple the solution of the fin conduction problem to the solution of the air and water energy equations. At each iteration, it was necessary to compare all of the local surface temperatures to the local air dew-point temperatures to determine the condensate and energy fluxes. The final solution gives two-dimensional distributions for fin temperature, air temperature and humidity, and surface fluxes along with one-dimensional water temperature distributions.

Model Employing Fin Efficiencies

Typical heat exchanger and cooling coil models utilize fin efficiencies to account for temperature and enthalpy distributions along the fin height (y) direction and neglect fin conduction in the longitudinal (x) direction. The fin efficiencies are derived assuming that the local air states are uniform in the y direction but vary in the x direction. In addition, the specific heat of the air is assumed to be constant. With these assumptions, the energy equation for the air at a dry location in the heat exchanger of Figure 3 can be expressed as

[m.sub.a]*[C.sub.p,a]*[[dT.sub.a]/dx] = [[eta].sub.f]*[h.sub.[a - f]]*[H.sub.f]*([T.sub.t] - [T.sub.a]), (28)

where [m.sub.a] is the air mass flow rate and [T.sub.a] represents the bulk air temperature across the entire fin in the y direction at any x. The fin efficiency accounts for the temperature distribution in the fin and allows the heat transfer rate to be expressed as a function of the base temperature of the fin, which is also the tube surface temperature.

For wet sections within the coil, the assumption of a Lewis number of unity and an overall fin efficiency for heat and mass transfer are employed. In this case, the air energy equation is

[m.sub.a]*[[d[h.sub.a]]/dx] = [[eta].sub.f.sup.*]*[h.sub.[a - f].sup.*]*[H.sub.f]*([h.sub.s,t] - [h.sub.a]), (29)

where [[eta]*.sub.f] is the heat and mass transfer fin efficiency and [h.sub.s,t] is the saturated air enthalpy at [T.sub.t]. The fin efficiency accounts for the distribution of saturated air enthalpies along the fin height.

Equation 29 is applied when the local tube surface temperature is below the local air dew-point temperature; otherwise, only Equation 28 is applied. Unlike the reference model, the models employing fin efficiencies do not have the ability to track the transition of wet-to-dry surfaces along the fin height direction. They can only track the transition along the direction of airflow.

For wet sections, it is also necessary to consider a mass balance on the water within the airstream in order to fully characterize the air state. In order to determine the air temperature variation, a water vapor mass balance ([m.sub.a] [dw.sub.a]/dx = [-m'.sub.c]).can be combined with a basic energy balance ([m.sub.a] [dh.sub.a]/dx = [[eta].sub. f] [h*.sub.a-f] [H.sub. f] ([T.sub.a] - [T.sub.t]) - [m.sub. c] [h.sub. fg]) and the mixture enthalpy relation (h = [C.sub.p,a] (T -[T.sub.ref]) + W [h.sub.fg]) to give

[m.sub.a]*[C.sub.p,a][[dT.sub.a]/dx] = [[^.[eta]].sub.f]*[h.sub.[a - f].sup.*]*[H.sub.f]*([T.sub.t] - [T.sub.a]), (30)

where [[eta].sub.f] is the heat transfer fin efficiency for a wet surface. This relationship is nearly the same as the energy balance for a dry section (Equation 28), except that the heat transfer coefficient and fin efficiency are different because of the influence of the wetted surface. The traditional models for cooling coils typically employ a heat transfer efficiency derived for dry surfaces in determining the air temperatures for wet sections of the coil.

The energy equation for the water stream and the heat exchanger boundary conditions are identical to those presented for the reference model. These relations, along with Equations 28-30 and property functions, are sufficient to determine the one-dimensional variation in water temperature, tube temperature, air enthalpy, and air temperature in the flow direction within the heat exchanger.

Existing models for cooling coils, such as those presented by ARI (2001), Elmahdy and Mitalas (1977), Braun et al. (1989), and Kuehn et al. (1998), utilize a combination of analytical and numerical techniques to solve the equations. They involve determination of the point for transition between dry and wet sections of the coil and use of separate analytical solutions to the basic equations for each of these sections that arise from the use of some simplifying assumptions. However, in order to be consistent with the reference model and isolate the effects of utilizing the improved heat transfer fin efficiency, a numerical solution to the modeling equations was implemented. In order to evaluate the improvements associated with the use of heat transfer fin efficiency correction presented in this paper, the model described in this section was solved for two different cases for heat transfer fin efficiency in Equation 30: (1) a dry heat transfer fin efficiency (Equation 1) and (2) a corrected heat transfer fin efficiency based on a wet analysis (Equation 13).

The differential equations were discretized and solved numerically using the first-order finite-difference method and upwind scheme (Murthy and Mathur 1998). The Gauss-Seidel iteration (Incropera and DeWitt 1996) was used to update the boundary conditions that couple the different finite-difference equations in the solution of the counter-flow arrangement. For each air control volume, it is necessary to determine whether moisture is condensing or not. This is evaluated by comparing the surface temperature from a dry analysis with the local air inlet dew point.

Model Comparisons

Table 2 gives sample inlet conditions considered for comparisons of numerical predictions for the different models compared in this study. The inlet relative humidity was varied between 40% and 80% with all other inlet conditions held constant. These conditions were chosen for presentation because the heat exchanger transitions from partially wet at the lowest humidity to fully wet conditions at the highest relative humidity.

Table 2. Heat Exchanger Inlet Conditions for Numerical Study [T.sub.a [RH.sub.a [M.sub.a], [T.sub.w [M.sub.w], in], in], % g/s in], g/s [degrees]C [degrees]C 26.67 40-80 0.36 4.44 0.16

Model results are compared in terms of their ability to predict total and sensible energy transfer rates for the heat exchanger as determined by

[q.sub.tot] = [C.sub.w]*([T.sub.w,out,HX] - [T.sub.w,in,HX]) (31)

and

[q.sub.sen] = [C.sub.a]*([T.sub.a,in,HX] - [T.sub.a,out,HX]), (32)

where the subscript "HX" denotes that values are for the heat exchanger instead of local control volumes.

Figure 4 gives comparisons between predictions of total energy transfer rates for the reference and fin efficiency models as a function of inlet air relative humidity. The total energy transfer rate increases with relative humidity because of increased latent energy transfer. Predictions of total energy transfer rates for the fin efficiency model are independent of the heat transfer fin efficiency used for wet sections (only the combined heat and mass transfer fin efficiency influences the total energy transfer rate in the wet section). Therefore, only a single result is presented for this approach. For the reference model, results are presented for both a Lewis number of unity and 0.9. According to Kuehn et al. (1998), air-water vapor mixtures have a Lewis number of around 0.9 at atmospheric conditions for a wide range of temperatures and humidities. A Lewis number of unity is assumed for the model that employs fin efficiencies and for most traditional cooling models.

[FIGURE 4 OMITTED]

Figure 4 shows that errors associated with the assumption of a Lewis number of unity are very small for representative operating conditions. The model employing fin efficiency also gives total energy transfer rate results that are close to the reference model. They differ slightly because the reference model accounts for fin conduction in the longitudinal (x) direction and allows for partially wet and dry conditions in the fin height direction. Similar results were obtained for a wide variety of operating conditions. The heat and mass transfer fin efficiency provides a very good characterization of the saturated air enthalpy distribution on the fin.

Figure 5 gives comparisons between predictions of sensible energy transfer rates for the reference and fin efficiency models. The sensible heat transfer rate decreases with entering air relative humidity because increased latent energy transfer causes an increase in the heat exchanger surface temperature. The reference model results were obtained for a Lewis number of unity. For the fin efficiency model, results were determined using heat transfer fin efficiencies in the wet section that were derived for both dry surfaces (Equation 1) and wet conditions (Equation 13). The use of the modified heat transfer fin efficiency gives predictions that are in much closer agreement with the two-dimensional results as compared with the use of dry heat transfer fin efficiency relations. The errors associated with using a heat transfer fin efficiency derived for dry conditions in the wet section increase as more of the heat exchanger is wetted (i.e., at higher inlet relative air humidities). For the results of Figure 5, the errors in sensible heat transfer were reduced from a maximum of about 12% to less than 1% by applying the improved heat transfer fin efficiency. Similar results were obtained for a wide variety of operating conditions where moisture was condensing on the fin.

[FIGURE 5 OMITTED]

Figure 6 shows variation in local heat transfer and heat and mass transfer fin efficiencies as a function of position along the airflow stream for an air inlet relative humidity of 60%. Heat transfer fin efficiency results are presented for both the dry analysis (original model) and the new approach that corrects for the wetted fin. The initial 20% of the fin surface is dry, so only the heat transfer fin efficiency for dry surfaces applies. After condensation starts, the heat and mass transfer fin efficiency for wet surfaces applies and increases as the surface becomes wetter when approaching the chilled-water inlet. In contrast, the heat transfer fin efficiency for the wet surface decreases in the direction of the airflow and approaches the heat and mass transfer fin efficiency as the air approaches saturation. The error associated with using a dry local fin efficiency is largest for saturated air conditions and is greater than 15% at the air outlet condition for the results of Figure 6.

[FIGURE 6 OMITTED]

EXPERIMENTAL VALIDATION

The improved method of determining heat transfer fin efficiency under wet surface conditions was implemented within an overall cooling coil model for a coil that has been extensively tested in a laboratory as described by Zhou and Braun (2005). This is an eight-row coil with wavy fins and a fin spacing of eight fins per inch, which is representative of large commercial applications. Table 3 provides the parameters that characterize the coil, and Figure 7 depicts its flow circuiting.

[FIGURE 7 OMITTED]

Table 3. Characteristics of Eight-Row Test Coil Coil Physical Parameter Eight-Row Coil Fin geometry Wavy Coil depth, m 0.264 Number of fins per inch (/0.0254 m) 8 Coil face width, m 0.6096 Coil face height, m 0.6096 Tube material Copper Tube outer diameter, m 0.0127 Tube thickness, m 0.0004 Tube longitudinal pitch, m 0.033 Tube transverse pitch, m 0.0381 Fin material Aluminum Fin thickness, m 0.0002

The coil was tested using a psychrometric wind tunnel that allows control of the coil inlet air temperature, humidity and flow rate, and water temperature and flow rate. Sufficient instrumentation was installed to determine air-side and water-side heat transfer rates under steady-state conditions (Zhou and Braun 2005).

The cooling coil model used for implementing and evaluating the improved fin efficiency approach was presented in ARI Standard 410 (2001). It separates the coil into dry and wet sections and applies the log mean temperature difference method to the dry section and the log mean enthalpy difference method to the wet section. Both the original and improved methods for heat transfer fin efficiency were implemented for comparison with experimental results.

Table 4 gives sample comparisons between model predictions and experimental results for a range of conditions where the coil was partially and fully wetted. Predictions for total energy transfer rate are within 3.2% of the values determined from experimental measurements, which is less than the measurement uncertainty and energy balance error between air-side and water-side measurements (6%) for the test setup. The model predictions for total energy transfer rate are independent of the approach used to characterize the heat transfer fin efficiency in the wet section. Predictions of sensible heat transfer are within 1.2% of the measurements for the modified heat transfer fin efficiency approach, which is an improvement compared to the 8.4% difference associated with the original method.

Table 4. Comparisons with Experimental Results Case [T.sub.a, [RH.sub.a, [M.sub.a] [T.sub.w, [M.sub.w] in], in], % in], [degrees]C kg/s [degrees]C kg/s 1 27.57 45.61 1.02 2.76 0.45 2 28.38 52.68 1.02 3.18 0.46 3 28.31 60.8 1.01 3.47 0.47 4 28.32 68.63 1 3.24 0.46 5 28.23 76.03 1 2.41 0.45 [q.sub.tot,]kW [q.sub.sen,] kW Case Exp. Original Improved Exp. Original Model Model Model 1 21.62 0.93 20.93 15.46 15.9 2 23.79 23.23 23.23 14.66 15.27 3 25.28 25.2 25.2 13.34 14.03 4 27.2 27.06 27.06 12.26 13.07 5 29.27 29.61 29.61 11.38 12.33 [q.sub.sen,] kW Case Improved Model 1 15.45 2 14.74 3 13.46 4 12.39 5 11.51

Figure 8 shows comparisons for sensible heat ratio (SHR), defined as the ratio of the sensible to total energy transfer rates. The SHR predictions by the original model are beyond the uncertainty bounds for the experimental measurements. The modified heat transfer fin efficiency model improves these predictions into or on the uncertainty bounds and provides greater relative improvements in SHR at lower values where the coil is wetter.

[FIGURE 8 OMITTED]

CONCLUSIONS

This paper develops and presents an improved method for determining heat transfer fin efficiency for wetted fins. The heat transfer fin efficiency is determined using a correction factor applied to the heat and mass transfer fin efficiency (Equation 13). Detailed numerical and experimental results show that the method provides improved predictions of sensible heat transfer rates compared to the use of dry surface fin efficiency relations, especially for conditions where cooling coils are fully wetted. The improved method is readily implemented in existing models and does not result in additional complexity.

ACKNOWLEDGMENTS

This study was funded by ASHRAE (ASHRAE RP-1194). Their support is gratefully appreciated.

NOMENCLATURE

C = specific heat, kJ/kg K

C = heat capacitance rate, kW/K

t = half of fin thickness, m

h = convection coefficient, W/[m.sup.2] K or specific enthalpy, kJ/kg

H = height, m

[h.sub.d] = mass transfer coefficient, kga/s [m.sup.2]

[h.sub.fg] = heat of water vaporization, kJ/kgw

k = heat conductivity, W/m K

L = length, m

Le = Lewis number

m = mass flow rate, kg/s

q = heat transfer rate, W

R = heat resistance, K/W

RH = relative humidity, %

T = temperature, [degrees]C

W = humidity ratio, [kg.sub.w]/[kg.sub.a]

Greek symbols

[[eta].sub.f] = heat transfer fin efficiency for dry surface

[[eta].sub.f] = heat transfer fin efficiency for wet surface

[[eta]*.sub.f] = heat and mass transfer fin efficiency for wet surface

Subscripts

a = air

b = fin base

c = condensate

f = fin

in = inlet

out = outlet

s = saturation

sen = sensible

t = tube

tot = total

w = water

Superscripts

* = wet fin

' = per unit length

" = per unit surface area

REFERENCES

ARI. 2001. ARI 410-2001: Forced-Circulation Air-Cooling and Air-Heating Coils. Arlington, VA: Air-Conditioning and Refrigeration Institute.

ASHRAE. 2000. 2000 ASHRAE Handbook-HVAC Systems and Equipment. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Braun, J.E., S.A. Klein, and J.W. Mitchell. 1989. Effectiveness models for cooling towers and cooling coils. ASHRAE Transactions 95(2):164-74.

Elmahdy, A.H., and G.P. Mitalas. 1977. A simple model for cooling and dehumidifying coils for use in calculating energy requirements for buildings. ASHRAE Transactions 83(2):103-17.

Incropera, F.P., and D.P. DeWitt. 1996. Fundamentals of Heat and Mass Transfer. New York: John Wiley & Sons.

Kuehn, T.H., J.W. Ramsey, and J.L. Threlkeld. 1998. Thermal Environmental Engineering. Upper Saddle River, NJ: Prentice-Hall.

Murthy, J.Y., and S.R. Mathur. 1998. Numerical methods in heat, mass, and momentum transfer. Lecture notes.

Xia, Y., and A.M. Jacobi. 2005. Air-side data interpretation and performance analysis for heat exchangers with simultaneous heat and mass transfer: Wet and frosted surfaces. International Journal of Heat and Mass Transfer 48:5089-102.

Zhou, X., and J.E. Braun. 2005. Dynamic modeling of chilled water cooling coils. ASHRAE 1194-RP Final Report, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., Atlanta.

Xiaotang Zhou, PhD Associate Member ASHRAE James E. Braun, PhD Fellow ASHRAE Qingfan Zeng, PhD

Xiaotang Zhou is an Engineering Leadership Program associate for the Carrier Corporation, Syracuse, NY. James E. Braun is a professor of mechanical engineering for Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN. Qingfan Zeng is a senior engineer for Carrier China, Shanghai, China.

Received May 26, 2006; accepted June 11, 2007

This paper is based on findings resulting from ASHRAE Research Project RP-1194.

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Author: | Zhou, Xiaotang; Braun, James E.; Zeng, Qingfan |
---|---|

Publication: | HVAC & R Research |

Article Type: | Technical report |

Geographic Code: | 1USA |

Date: | Sep 1, 2007 |

Words: | 6202 |

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