# Generalized performance maps for single- and dual-speed residential heat pumps.

INTRODUCTION

Computationally efficient models for building simulations are usually provided as empirical models constructed from experimental data. A recent ASHRAE research project (Brandemuehl and Wassmer 2009) compared three methods of modeling residential HVAC equipment. The project compared the empirical correlation-based DOE-2 building simulation program's residential systems (RESYS) model (Winkelmann et al. 1993) against the ASHRAE secondary toolkit bypass model (Brandemuehl et al. 1993) and a recently developed component model approach (Brandemuehl and Wassmer 2009) that requires information from only a single data point. The project found the DOE-2 model was the most accurate at predicting performance in cases where there was more performance data available while the ASHRAE toolkit model predicted performance well in cases where there was a significant amount of wet coil operation data available. The component-based model performed better than the DOE-2 and ASHRAE toolkit models with default coefficients but worse than the DOE-2 and ASHRAE toolkit models with custom coefficients. All three of these approaches predict steady-state performance and make use of degradation coefficients to account for reductions in efficiency due to unit cycling.

A generic rating-data-based (GRDB) modeling method developed by Yang and Li (2010) requires less computational effort than the ASHRAE toolkit model (Brandemuehl et al. 1993). This approach requires only the typically available manufacturer published data. It was not appropriate for this study since more performance data was available that could produce a more accurate model using a different modeling approach.

National Renewable Energy Laboratory (NREL) conducted a study of 260 cooling-only and 200 reversible ducted heat pumps rated between SEER 13 and SEER 21 based on manufacturer performance tables and found that it was possible to use a single model to describe the performance of heat pumps from various manufacturers with the same Seasonal Energy Efficiency Ratio (SEER) rating (Cutler et al. 2012). The NREL study's models were based on a data set containing single- and dual-compressor-speed units. The NREL approach of using a generalized map for similar heat pumps is similar to that used for ducted equipment in this paper with the exception that NREL worked with a larger but less detailed data set. The results from the NREL study show that using a generalized approach of one family of heat pumps can be representative of heat pumps on the market with the same SEER rating while the more detailed data set available for this study adds confidence in the accuracy of the modeling approach.

The ASHRAE secondary toolkit bypass model (Brandemuehl et al. 1993) uses polynomials and a bypass factor approach derived from experimental data for direct-expansion coils. The DOE-2 RESYS routine (Winkelmann et al. 1993) is widely used for modeling single-speed systems. The toolkit model is the basis for the generalized single- and dual-speed ducted heat pump performance maps developed in this study; the results are compared with the ASHRAE toolkit model's default correction coefficients in cooling. The generalized heating operation maps are compared against the DOE-2 RESYS routine with default coefficients. Both the toolkit and RESYS models' default coefficients are described in the 1981 DOE-2 manual (LBNL 1981), hence the need to update them with coefficients that are more representative of the equipment currently on the market. The models described in this paper were developed for use as part of a larger study by Holloway (2013) that sought to quantify the performance differences between ducted unitary equipment and ductless split systems. Holloway implemented four heat pump types and compared their performance across prototypical United States home types and climates using a building energy simulation program.

HEAT PUMP PERFORMANCE DATA

The data used to model the heat pumps in this study were generated using a validated detailed component-based simulation program provided by the manufacturer of the heat pumps. The data-generation simulation program is used by the manufacturer to generate published energy-efficiency ratings for their products. Its outputs are accepted by the U.S. Department of Energy (DOE) as an alternative rating method (DOE 2011). The data set includes seven single-compressor-speed units rated SEER 13 with capacities from 1.5 to 5 tons (3.5 to 17.6 kW) in increments of 0.5 tons (1.76 kW) and four dual-compressor-speed units rated SEER 16 with capacities from 2 to 5 tons (7.0 to 17.6 kW) in 1 ton (3.5 kW) increments. In generating the data, the heat pumps were assumed to be at steady-state operation under standard atmospheric pressure, with 10[degrees]F (5.6[degrees]C) superheat and 25 ft (7.62 m) of refrigerant piping, half exposed to the outdoor and half exposed to the indoor conditions. The refrigerant charge level was determined at Air-Conditioning, Heating, and Refrigeration Institute (AHRI) cooling rating conditions of 95[degrees]F (35[degrees]C) ambient dry-bulb temperature and 67[degrees]F (19.4[degrees]C) indoor wet-bulb temperature with the cycle simulation subcooling set to 10[degrees]F (5.6[degrees]C) (AHRI 2008). The same charge level determined in cooling mode was used for heating mode.

The cooling simulations were performed for each unit with ambient temperatures from 75[degrees]F to 115[degrees]F (23.9[degrees]C to 46.1[degrees]C) in 5[degrees]F (2.8[degrees]C) increments and indoor wet-bulb temperatures from 49.4[degrees]F to 77[degrees]F (9.8[degrees]C to 25[degrees]C) with a constant indoor dry-bulb temperature of 80[degrees]F (26.7[degrees]C). The heating simulations had ambient temperatures from 20[degrees]F to 55[degrees]F (-6.7[degrees]C to 12.8[degrees]C) in 10[degrees]F (5.6[degrees]C) increments and indoor dry-bulb temperatures from 55[degrees]F to 75[degrees]F (12.8[degrees]C to 23.9[degrees]C). The low-stage operation data for dual-stage equipment was generated by running the detailed simulation at 70% compressor speed turndown ratio. Each unit was simulated under three indoor fan speeds. Defrost operation was not modeled and the mapping results in this paper may not be appropriate when there is a significant buildup of ice that may occur at low ambient conditions.

MODELING APPROACH

Model Outline

The single- and dual-speed units were modeled using the ASHRAE HVAC2 Toolkit model (Brandemuehl et al. 1993), which captures the effect of the indoor airflow rate and the indoor and outdoor air temperatures and humidity through Equations 1-6:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

COP = [[COP.sub.rat]][[f.sub.cop,t]]([T.sub.in], [T.sub.amb])[f.sub.cop,m]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) (2)

[f.sub.cap,t] = [a.sub.1] + [b.sub.1][[T.sub.amb]] + [c.sub.1][[T.sup.2.sub.amb]] + [d.sub.1][[T.sub.in]] + [e.sub.1][[T.sup.2.sub.in]] + [f.sub.1][[T.sub.in]][[T.sub.amb]] (3)

[f.sub.cap,m] = [X.sub.1] + [Y.sub.1]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) (4)

[f.sub.cop,t] = [a.sub.2] + [b.sub.2][[T.sub.amb]] + [c.sub.2][[T.sup.2.sub.amb]] + [d.sub.2][[T.sub.in]] + [e.sub.2][[T.sup.2.sub.in]] + [f.sub.2][[T.sub.in]][[T.sub.amb]] (5)

[f.sub.cop,m] = [X.sub.2] + [Y.sub.2]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) (6)

where

[??] = cooling or heating capacity [[??].sub.rat] = capacity at a rating condition [[??].sub.a, in] = indoor volumetric airflow rate [[??].sub.a, in, max] = the unit's maximum indoor air volumetric flow rate

[T.sub.in] = indoor air entering wet-bulb temperature for cooling operation and indoor air entering dry-bulb temperature for heating operation, [degrees]F

[T.sub.amb] = outdoor air entering dry-bulb temperature, [degrees]F

The variables [a.sub.1] through [f.sub.2] and [X.sub.1] through [Y.sub.2] are coefficients found through regression, and [f.sub.cap, m] and [f.sub.cap, t] are correction factors for airflow rate and temperature, respectively.

The rating condition for cooling mode is the AHRI standard 95[degrees]F (35[degrees]C) ambient dry-bulb temperature and 67[degrees]F (19.4[degrees]C) indoor wet-bulb temperature (AHRI 2008). The heating mode rating condition is 47[degrees]F (8.3[degrees]C) ambient dry-bulb temperature, 70[degrees]F (21.1[degrees]C) indoor dry-bulb temperature, and 60[degrees]F (15.6[degrees]C) wet-bulb entering indoor air temperature.

Sensible Heat Ratio and Dry Coil Operation

The relations in Equations 1-6 make use of indoor inlet wet-bulb temperature and only provide total cooling capacity for wet coils. However, it is also necessary to determine the sensible heat ratio (SHR) for cooling mode and to determine the performance for dry coils (SHR = 1) where the wet-bulb temperature has little influence on capacity. The SHR model is based on the ASHRAE toolkit model's (Brandemuehl et al. 1993) bypass factor approach, which involves solution of Equations 7-10 for apparatus dew-point condition given inlet conditions and outlet enthalpy determined from total capacity mode along with the heat exchanger's number of transfer units (NTU), [NTU.sub.rat], and [[??].sub.a, in, max]:

SHR = [[h([T.sub.evap,in], [[omega].sub.adp]) - [h.sub.adp]]/[h.sub.evap, in] - [h.sub.adp]] (7)

[h.sub.adp] = [h.sub.evap,in] - [[[h.sub.evap,in - [h.sub.evap,out]]/1 - BF] (8)

BF = [e.sup.-NTU] (9)

NTU = [[[NTU.sub.rat]]/([[??].sub.a,in]/[[??].sub.a,in,max])] (10)

For a dry coil, the toolkit model will predict a sensible cooling capacity greater than the total capacity and iteratively solves Equations 7-10 for both the inlet wet-bulb temperature and apparatus dew-point humidity ratio that give a unity SHR.

Heating Mode Operation

In heating mode, the relations in Equations 1-6 are applied with [T.sub.amb] as a dry-bulb temperature because the winter air temperature is assumed to be very dry and the impact of evaporator moisture removal on energy consumption in heating mode is neglected.

The DOE-2.1 RESYS routine (Winkelmann et al. 1993) that was used as a benchmark in heating operation arrives at the correction factors in Equations 1 and 2 using the cubic relations in Equations 11 and 12 instead of Equations 3 and 5. The form of the air mass flow rate corrections for capacity is the same as Equation 4, but the air mass flow correction for coefficient of performance (COP) is different from Equation 6 and is shown in Equation 13:

[f.sub.cap, t] = ([a.sub.3]) + [b.sub.3][T.sub.amb] + [c.sub.3][T.sup.2.sub.amb] + [d.sub.3][T.sup.3.sub.amb] (11)

[f.sub.cop, t] = [([a.sub.4] + [b.sub.4][T.sub.amb] + [c.sub.4][T.sup.2.sub.amb] + [d.sub.4][T.sup.3.sub.amb]).sup.-1] (12)

[f.sub.cop,m] = [[[X.sub.3] + [Y.sub.3]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) + [Z.sub.3][([[[??].sub.a,in]/[[??].sub.a,in,max]]).sup.2]].sup.-1] (13)

The RESYS routine was developed from ratings data and manufacturers' catalog data for units delivering 3 tons (10.6 kW) at the heating rating condition with 1200 cfm (566 L/s) rated airflow.

Estimation of Coefficients using Linear Regression

All the coefficients of [T.sub.in], [T.sub.amb], and [[??].sub.a, in]/[[??].sub.a, in, max] in Equations 3-6 were found through linear regression. In order for linear regression to be performed on Equation 1, which is nonlinear in the unknown coefficients, [f.sub.cap, m] was set equal to 1 and then the temperature correction factor coefficients for capacity in Equation 3 were found through linear regression using only data points where [[??].sub.a, in] was equal to [[??].sub.a, in, max]. The coefficients [a.sub.1] to [f.sub.1] were fit to the data by minimizing the sum of the squared fractional residuals as shown in Equation 14 with the true temperature correction factor from the data determined using Equation 15:

Error = [summation][([[[f.sub.cap,t,predicted] - [f.sub.cap,t,data]]/[f.sub.cap,t,data]]).sup.2] (14)

[f.sub. cap, t, data] = [[??]/[[??].sub.rat]] (15)

This approach assumes that temperature and airflow rate are decoupled and it is possible to capture the effect of temperature changes completely with data containing fixed airflow rate. These assumptions are reasonable as the airflow rate can be set by the heat pump control system independently of the temperature and the data show a similar effect of temperature changes across different airflow rates. To maintain consistency with the previous step of fixing [f.sub.cap,m] equal to 1 when [[??].sub.a, in] is equal to [[??].sub.a, in, max], the airflow correction factor coefficients [X.sub.1] and [Y.sub.1] must satisfy Equation 16, which is an equality constraint. This constraint was applied by substitution, as shown in Equations 17-20. The resulting Equation 20 is a modification of Equation 1, effectively constraining the value of [f.sub.cap,m] to be equal to 1 when [[??].sub.a, in] is equal to [[??].sub.a, in, max]:

[Y.sub.1] + [X.sub.1] = 1 (16)

[f.sub.cap, m] = [X.sub.1] + [Y.sub.1]([[[??].sub.a,in]]/[[??].sub.a,in,max]) (17)

[f.sub.cap,m] = (1 - [Y.sub.1]) + [Y.sub.1]([[[[??].sub.a,in]]/[[??].sub.a, in max]]) (18)

[f.sub.cap,m] - 1 = [Y.sub.1]([[[[??].sub.a,in]]/[[??].sub.a,in,max]] - 1) (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[Y.sub.1] was found by linear least-squares regression, minimizing the sum of the squared residuals as shown in Equation 21. [X.sub.1] was then found by substituting [Y.sub.1] into the constraint, Equation 16. The COP correction factor coefficients were found using the same procedure as the capacity correction factor coefficients.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

The rated NTU that produced the best SHR estimation was found by minimizing Equation 22, the sum of the square of the residuals between the actual SHRs and those found from evaluating the expression in Equation 7. A Newton's method minimization routine was used to find the rated NTU since SHR is a nonlinear function of rated NTU.

Error = [summation][([SHR.sub.predicted] - [SHR.sub.data]/[SHR.sub.data]).sup.2] (22)

Dual-Speed Heat Pump Models

For dual-speed units, the same approach outlined for single speed units was used. The high compressor speed was treated as a separate unit from the low compressor speed operation, resulting in two sets of cooling coefficients and two sets of heating coefficients for each unit.

GENERALIZED HEAT PUMP MODEL RESULTS

It was observed that the correction factors for single-speed units were similar for given indoor and outdoor conditions. The same was true for the dual-speed units. To simplify the models for use in building simulations, the data for each heat pump family (single- and dual-speed compressors) and operating mode (heating or cooling) was aggregated and aggregate models for each heat pump family were fit using Equations 1-10. The unit's rating data was also applied to the ASHRAE secondary toolkit's (Brandemuehl et al. 1993) direct-expansion cooling model and the DOE-2 model (Winkelmann et al. 1993) for heating with default coefficients. A comparison of the correction coefficients developed in this study with the established models is helpful for deciding if heat pump efficiency changes over time necessitate an update to the established models.

Having a single heat pump model for a family of units allows approximate simulation of other similar equipment when the only data available is performance at rating conditions. It also enables precise and consistent sizing of equipment that is needed when performing parametric studies using building simulations.

Figures 1 through 4 use parity plots to compare the prediction accuracy for capacity and power consumption of using a generalized model for each family against the accuracy of using the default correction factor coefficients. The dimensionless values on the axes of Figures 1 through 3 are the absolute values of capacity and power normalized against the values at the rating condition. Figures 1 through 3 show the single-speed heat pump results while Figure 4 shows the results for dual-speed heat pumps. The SHR predictions for cooling conditions are shown in Figure 5. For clarity, the parity plots show every other data point.

For the single-speed heat pump family (Figures 1-3), the results for the updated generalized mapping show good agreement with the predicted capacity and power consumption, having a maximum power prediction error of 7.2% in cooling mode and maximum capacity prediction error of 10.7%. In contrast, use of the default parameters associated with the ASHRAE toolkit and DOE-2.1 RESYS models shows poorer performance for this family of units, particularly for power predictions. Figure 1 shows larger prediction errors for the default parameters for cooling capacity occur at lower outdoor ambient conditions of 75[degrees]F to 80[degrees]F (23.9[degrees]C to 26.7[degrees]C). However, capacity limitations are usually not encountered at these low ambient conditions and the model would perform well under normal cooling operation. Figure 2 shows that the default RESYS routine map does not take the indoor temperature variation into account, reducing accuracy when indoor temperature is well above or below the 70[degrees]F (21.1[degrees]C) rating condition. However, most homes have heating setpoints above 60[degrees]F (15.6[degrees]C), where the default map is more accurate.

Figures 3 and 4 show that there is a much greater range of errors in power predictions when using the default correlations, especially for the dual-speed heat pump. The capacity and power prediction accuracy of the generalized map for the dual-speed heat pumps is comparable to that for the single-speed units. However, the accuracy of using the default parameters for the power model is significantly worse than for the single-speed case, as shown in Figure 4. Figure 4, like Figure 2, shows the effect of the heating mode RESYS model not accounting for indoor temperature effect. Although not shown, high-compressor-speed results were similar to those for low speed. Figure 5 shows that predictions of SHR in cooling are similar for the updated and default parameter models.

Tables 1 and 2 compare statistics regarding accuracy of models obtained using individualized coefficients as well as the generalized map developed for this study and the default toolkit model. The last two rows of Table 1 and most of the high-compressor-speed cases in Table 2 show a negative coefficient of determination for the toolkit model power predictions with default parameters, making the prediction for the 4 and 5 ton (14.1 and 17.58 kW) dual-speed units in heating mode worse than simply using the mean value for all predictions. One reason for this is that the capacity and COP prediction errors combine, resulting in a large error in power prediction. The default RESYS routine map was developed from single-speed heat pumps, and applying it to dual-speed heat pumps reduces the accuracy.

Equipment Mapping Coefficients and Characteristics

Tables 3 and 4 contain the capacity and COP correction factor coefficients, respectively, for the generalized mapping described in this study. The coefficients require the temperatures in Equations 1-5 to be given in [degrees]F.

Figure 6 shows the variation of the temperature-dependent capacity and COP correction factors, [f.sub.cap,t] and [f.sub.cop,t], for single-speed heat pumps in cooling mode. The default coefficients underpredict the capacity at the lower ambient temperature conditions, reflecting the same lower accuracy at low ambient conditions as Figure 1. The influence of indoor wet-bulb temperature on capacity is similar for both models, with higher wet-bulb temperatures giving higher total cooling capacity.

Figure 7 shows the variation of [f.sub.cap,t] and [f.sub.cop,t] for single-speed heat pumps in heating mode. The main distinction from the cooling mode models is that the RESYS model (Winkelmann et al. 1993) does not account for variation of indoor temperature in predicting capacity and COP. This accounts for the horizontal spread of predictions in the parity plot for capacity in Figures 2 and 4. The default coefficients underpredict capacity at low ambient conditions, indicating that newer equipment on the market has better low-temperature performance than equipment in the past. The predictions based on default coefficients and the updated generalized map predictions are most different at ambient temperatures further away from the rating condition.

Figure 8 shows that the single-stage heat pump generalized map cooling mode airflow correction factors, [f.sub.cap, m] and [f.sub.cop, m], are similar to the original ASHRAE toolkit model (Brandemuehl et al. 1993) correction factors. Indoor unit airflow has a smaller influence on COP and capacity than the indoor and outdoor air temperatures, as shown by the smaller range of the airflow correction factor values. The generalized map airflow correction factors are consistently higher than those given by the default coefficients. The generalized map heating mode airflow correction factors are similarly close to those of the original ASHRAE toolkit model.

Figure 9 shows the dual-stage heat pumps' airflow correction factors, [f.sub.cap,m] and [f.sub.cop,m], in heating mode. Unlike the RESYS default coefficients, the updated generalized map predicts a decrease in capacity with increasing airflow. This counterintuitive trend was seen in individual data for each dual-speed heat pump operating in heating mode. Figure 10 illustrates the issue for the high-compressor-speed cases.

For many of the individual cases, the heating capacity increases with decreasing indoor airflow, particularly at the higher flows. One of the reasons is that the condensing temperature and pressure increase to compensate for a lower heat transfer conductance with decreasing airflow. In some operating ranges, the larger pressure rise across the compressor leads to increased compressor power that causes a higher heat rejection requirement for the condenser. However, results for the 5 ton (17.6 kW) unit given in Table 5 show that the compressor power increase accounted for only a portion (87 Btu/h[26W] outofthe456Btu/h[134W]) of the increase in net heating capacity when airflow decreased from 2080 to 2010 scfm (1.179 to 1.139 kg/s).

The remaining increase in capacity is due to the indoor fan. For the 5 ton unit, the indoor fan power increased by 424 Btu/h (124 W) when airflow changed from 2080 to 2010 scfm, as shown in Figure 11a. The increased fan power is added to the refrigerant-side capacity to give a higher net capacity. Figure 11a indicates that there is a peak in the fan power with respect to changes in airflow. For each compressor speed of the dual-stage units, the airflow was varied by changing the external static pressure (ESP) while holding the fan speed constant. A typical fan power curve (Dayton 2013) for a fixed-speed fan is shown in Figure 11b. To decrease the airflow, the system resistance curve is shifted to the left, leading at first to an operating point with higher fan power. However, as the system resistance curve is shifted further to the left to give lower airflow, the fan power decreases after going past the peak in the fan power curve. The linear fit used for [f.sub.cap,m] does not capture the peak in the fan power curve but instead gives a monotonic trend. However, the correction factor is small (less than 3%) and the model is sufficiently accurate.

CONCLUSIONS

The updated generalized single- and dual-compressor-speed heat pump maps provide an update to the ASHRAE toolkit (Brandemuehl et al. 1993) and RESYS (Winkelmann et al. 1993) models and correlate the data within approximately 10% for both capacity and power consumption. These maps were more accurate than the widely used ASHRAE toolkit model with default coefficients and DOE 2.1's default model at predicting the performance of the families of heat pumps under study. In particular, the default maps performed poorly at predicting the power consumption for dual-speed heat pumps in heating operation. The accuracy of the default correlations decreases for operating conditions further away from the rating conditions, and custom coefficients should be considered for these off-design conditions when appropriate performance data is available. Defrost operation was not modeled, and the mapping results in this paper may not be appropriate when there is a significant buildup of ice that may occur at low ambient conditions.

Updating the model to reflect modern equipment on the market after a period of 30 years since the original RESYS routine is useful to building designers, government regulators, electrical utility companies, and HVAC equipment manufacturing firms. These stakeholders can make more informed choices when selecting space-conditioning equipment than they would have been able to using the published SEER and (Heating Seasonal Performance Factor) HSPF performance ratings or the originally published models' default coefficients. The updated generalized maps can also be used in building energy simulations.

ACKNOWLEDGMENTS

The authors thank Ingersoll Rand Inc. for supporting this research project.

REFERENCES

AHRI. 2008. AHRI Standard 210/240-2008, Performance Rating of Unitary Air-Conditioning and Air-Source Heat Pump Equipment. Arlington, VA: Air-Conditioning, Heating, and Refrigeration Institute.

Brandemuehl, M. J., S. Gabel, and I. Andresen. 1993. HVAC 2 Toolkit: A Toolkit for Secondary HVAC System Energy Calculations [CD]. Atlanta: ASHRAE.

Brandemuehl, M.J., and M. Wassmer. 2009. Updated energy calculation models for residential HVAC equipment. Report ASHRAE 1197 RP. Atlanta: ASHRAE.

Cutler, D., J. Winkler, N. Kruis, and C. Christensen. 2012. Improved modeling of residential air conditioners and heat pumps for energy calculations. Report NREL/TP5500-56354. Golden, CO: National Renewable Energy Laboratory.

Dayton. 2013. Fan Curves. Dayton Fans. www.solutionsforair.com/dayton_content/pdfs/FanCurves.pdf

DOE. 2011. Units to be tested. Code of Federal Regulations, 10 CFR 430.24(m). Washington, DC: Government Printing Office.

Holloway, S.O. 2013. An annual performance comparison of various heat pumps in residential applications. Master's Thesis, Purdue University, West Lafayette, IN.

LBNL. 1981. DOE-2, Version 2.1A. Berkeley, CA: Lawrence Berkeley National Laboratory.

Winkelmann, F.C., B.E. Birdsall, W.F. Buhl, K.L. Ellington, E. Erdem, J.J. Hirsch, and S. Gates. 1993. DOE-2 Supplement (Version 2.1E). Berkeley, CA: Lawrence Berkeley National Laboratory.

Yang, H., and H. Li. 2010. A Generic Rating-Data-Based (GRDB) DX Coils Modeling Method. HVAC&R Research 16(3):331-53.

DISCUSSION

Kelly Kissock, Professor, University of Dayton, Dayton, OH: How much do your results vary with other manufacturers? Simbarashe Nyika: We cannot say for sure how our results compare with heat pumps from other manufacturers in the industry as we did not have access to this level of detailed data from other suppliers. However, we anticipate the difference is small. We consulted the recent study by NREL (Cutler et al. 2012) of 260 cooling-only and 200 reversible ducted heat pumps rated between SEER 13 and SEER 21 based on manufacturer performance tables and found that it was possible to use a single model to describe the performance of heat pumps from various manufacturers with the same SEER rating.

Cutler, D., J. Winkler, N. Kruis, and C. Christensen. 2012. Improved modeling of residential air conditioners and heat pumps for energy calculations. Report NREL/TP5500-56354. Golden, CO: National Renewable Energy Laboratory. www.nrel.gov/docs/fy13osti/56354.pdf. Hugh Henderson, Principal, CDH Energy Corporation, Cazenovia, NY: Your comparison to the ASHRAE secondary toolkit using the 1982 coefficient was not the best baseline. I would suggest using the EnergyPlus DX coil coefficient, which has been updated.

Simbarashe Nyika: At the time of writing the paper, the EnergyPlus Engineering Reference guide (DOE 2014) referenced the 1993 ASHRAE toolkit, which in turn references the 1982 coefficients. As of July 21, 2014, the EnergyPlus reference guide still shows the same information. The engineering reference did not list any default coefficients and asks for these as input from the user.

DOE. 2014. EnergyPlus Engineering Reference. Washington, DC: U.S. Department of Energy. http:// apps1.eere.energy.gov/buildings/energyplus/pdfs /engineeringreference.pdf.

Simbarashe Nyika

W. Travis Horton, PhD

Member ASHRAE

Seth O. Holloway

Associate Member ASHRAE

James E. Braun, PhD, PE

Fellow ASHRAE

Simbarashe Nyika is a cooling development engineer at Whirlpool Corporation, Benton Harbor, MI. Seth O. Holloway is a systems engineer at Ingersoll Rand, Tyler, TX. W. Travis Horton is an assistant professor of civil engineering and James E. Braun is a professor of mechanical engineering in the Ray W. Herrick Laboratories School of Mechanical Engineering at Purdue University, West Lafayette, IN.

Computationally efficient models for building simulations are usually provided as empirical models constructed from experimental data. A recent ASHRAE research project (Brandemuehl and Wassmer 2009) compared three methods of modeling residential HVAC equipment. The project compared the empirical correlation-based DOE-2 building simulation program's residential systems (RESYS) model (Winkelmann et al. 1993) against the ASHRAE secondary toolkit bypass model (Brandemuehl et al. 1993) and a recently developed component model approach (Brandemuehl and Wassmer 2009) that requires information from only a single data point. The project found the DOE-2 model was the most accurate at predicting performance in cases where there was more performance data available while the ASHRAE toolkit model predicted performance well in cases where there was a significant amount of wet coil operation data available. The component-based model performed better than the DOE-2 and ASHRAE toolkit models with default coefficients but worse than the DOE-2 and ASHRAE toolkit models with custom coefficients. All three of these approaches predict steady-state performance and make use of degradation coefficients to account for reductions in efficiency due to unit cycling.

A generic rating-data-based (GRDB) modeling method developed by Yang and Li (2010) requires less computational effort than the ASHRAE toolkit model (Brandemuehl et al. 1993). This approach requires only the typically available manufacturer published data. It was not appropriate for this study since more performance data was available that could produce a more accurate model using a different modeling approach.

National Renewable Energy Laboratory (NREL) conducted a study of 260 cooling-only and 200 reversible ducted heat pumps rated between SEER 13 and SEER 21 based on manufacturer performance tables and found that it was possible to use a single model to describe the performance of heat pumps from various manufacturers with the same Seasonal Energy Efficiency Ratio (SEER) rating (Cutler et al. 2012). The NREL study's models were based on a data set containing single- and dual-compressor-speed units. The NREL approach of using a generalized map for similar heat pumps is similar to that used for ducted equipment in this paper with the exception that NREL worked with a larger but less detailed data set. The results from the NREL study show that using a generalized approach of one family of heat pumps can be representative of heat pumps on the market with the same SEER rating while the more detailed data set available for this study adds confidence in the accuracy of the modeling approach.

The ASHRAE secondary toolkit bypass model (Brandemuehl et al. 1993) uses polynomials and a bypass factor approach derived from experimental data for direct-expansion coils. The DOE-2 RESYS routine (Winkelmann et al. 1993) is widely used for modeling single-speed systems. The toolkit model is the basis for the generalized single- and dual-speed ducted heat pump performance maps developed in this study; the results are compared with the ASHRAE toolkit model's default correction coefficients in cooling. The generalized heating operation maps are compared against the DOE-2 RESYS routine with default coefficients. Both the toolkit and RESYS models' default coefficients are described in the 1981 DOE-2 manual (LBNL 1981), hence the need to update them with coefficients that are more representative of the equipment currently on the market. The models described in this paper were developed for use as part of a larger study by Holloway (2013) that sought to quantify the performance differences between ducted unitary equipment and ductless split systems. Holloway implemented four heat pump types and compared their performance across prototypical United States home types and climates using a building energy simulation program.

HEAT PUMP PERFORMANCE DATA

The data used to model the heat pumps in this study were generated using a validated detailed component-based simulation program provided by the manufacturer of the heat pumps. The data-generation simulation program is used by the manufacturer to generate published energy-efficiency ratings for their products. Its outputs are accepted by the U.S. Department of Energy (DOE) as an alternative rating method (DOE 2011). The data set includes seven single-compressor-speed units rated SEER 13 with capacities from 1.5 to 5 tons (3.5 to 17.6 kW) in increments of 0.5 tons (1.76 kW) and four dual-compressor-speed units rated SEER 16 with capacities from 2 to 5 tons (7.0 to 17.6 kW) in 1 ton (3.5 kW) increments. In generating the data, the heat pumps were assumed to be at steady-state operation under standard atmospheric pressure, with 10[degrees]F (5.6[degrees]C) superheat and 25 ft (7.62 m) of refrigerant piping, half exposed to the outdoor and half exposed to the indoor conditions. The refrigerant charge level was determined at Air-Conditioning, Heating, and Refrigeration Institute (AHRI) cooling rating conditions of 95[degrees]F (35[degrees]C) ambient dry-bulb temperature and 67[degrees]F (19.4[degrees]C) indoor wet-bulb temperature with the cycle simulation subcooling set to 10[degrees]F (5.6[degrees]C) (AHRI 2008). The same charge level determined in cooling mode was used for heating mode.

The cooling simulations were performed for each unit with ambient temperatures from 75[degrees]F to 115[degrees]F (23.9[degrees]C to 46.1[degrees]C) in 5[degrees]F (2.8[degrees]C) increments and indoor wet-bulb temperatures from 49.4[degrees]F to 77[degrees]F (9.8[degrees]C to 25[degrees]C) with a constant indoor dry-bulb temperature of 80[degrees]F (26.7[degrees]C). The heating simulations had ambient temperatures from 20[degrees]F to 55[degrees]F (-6.7[degrees]C to 12.8[degrees]C) in 10[degrees]F (5.6[degrees]C) increments and indoor dry-bulb temperatures from 55[degrees]F to 75[degrees]F (12.8[degrees]C to 23.9[degrees]C). The low-stage operation data for dual-stage equipment was generated by running the detailed simulation at 70% compressor speed turndown ratio. Each unit was simulated under three indoor fan speeds. Defrost operation was not modeled and the mapping results in this paper may not be appropriate when there is a significant buildup of ice that may occur at low ambient conditions.

MODELING APPROACH

Model Outline

The single- and dual-speed units were modeled using the ASHRAE HVAC2 Toolkit model (Brandemuehl et al. 1993), which captures the effect of the indoor airflow rate and the indoor and outdoor air temperatures and humidity through Equations 1-6:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

COP = [[COP.sub.rat]][[f.sub.cop,t]]([T.sub.in], [T.sub.amb])[f.sub.cop,m]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) (2)

[f.sub.cap,t] = [a.sub.1] + [b.sub.1][[T.sub.amb]] + [c.sub.1][[T.sup.2.sub.amb]] + [d.sub.1][[T.sub.in]] + [e.sub.1][[T.sup.2.sub.in]] + [f.sub.1][[T.sub.in]][[T.sub.amb]] (3)

[f.sub.cap,m] = [X.sub.1] + [Y.sub.1]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) (4)

[f.sub.cop,t] = [a.sub.2] + [b.sub.2][[T.sub.amb]] + [c.sub.2][[T.sup.2.sub.amb]] + [d.sub.2][[T.sub.in]] + [e.sub.2][[T.sup.2.sub.in]] + [f.sub.2][[T.sub.in]][[T.sub.amb]] (5)

[f.sub.cop,m] = [X.sub.2] + [Y.sub.2]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) (6)

where

[??] = cooling or heating capacity [[??].sub.rat] = capacity at a rating condition [[??].sub.a, in] = indoor volumetric airflow rate [[??].sub.a, in, max] = the unit's maximum indoor air volumetric flow rate

[T.sub.in] = indoor air entering wet-bulb temperature for cooling operation and indoor air entering dry-bulb temperature for heating operation, [degrees]F

[T.sub.amb] = outdoor air entering dry-bulb temperature, [degrees]F

The variables [a.sub.1] through [f.sub.2] and [X.sub.1] through [Y.sub.2] are coefficients found through regression, and [f.sub.cap, m] and [f.sub.cap, t] are correction factors for airflow rate and temperature, respectively.

The rating condition for cooling mode is the AHRI standard 95[degrees]F (35[degrees]C) ambient dry-bulb temperature and 67[degrees]F (19.4[degrees]C) indoor wet-bulb temperature (AHRI 2008). The heating mode rating condition is 47[degrees]F (8.3[degrees]C) ambient dry-bulb temperature, 70[degrees]F (21.1[degrees]C) indoor dry-bulb temperature, and 60[degrees]F (15.6[degrees]C) wet-bulb entering indoor air temperature.

Sensible Heat Ratio and Dry Coil Operation

The relations in Equations 1-6 make use of indoor inlet wet-bulb temperature and only provide total cooling capacity for wet coils. However, it is also necessary to determine the sensible heat ratio (SHR) for cooling mode and to determine the performance for dry coils (SHR = 1) where the wet-bulb temperature has little influence on capacity. The SHR model is based on the ASHRAE toolkit model's (Brandemuehl et al. 1993) bypass factor approach, which involves solution of Equations 7-10 for apparatus dew-point condition given inlet conditions and outlet enthalpy determined from total capacity mode along with the heat exchanger's number of transfer units (NTU), [NTU.sub.rat], and [[??].sub.a, in, max]:

SHR = [[h([T.sub.evap,in], [[omega].sub.adp]) - [h.sub.adp]]/[h.sub.evap, in] - [h.sub.adp]] (7)

[h.sub.adp] = [h.sub.evap,in] - [[[h.sub.evap,in - [h.sub.evap,out]]/1 - BF] (8)

BF = [e.sup.-NTU] (9)

NTU = [[[NTU.sub.rat]]/([[??].sub.a,in]/[[??].sub.a,in,max])] (10)

For a dry coil, the toolkit model will predict a sensible cooling capacity greater than the total capacity and iteratively solves Equations 7-10 for both the inlet wet-bulb temperature and apparatus dew-point humidity ratio that give a unity SHR.

Heating Mode Operation

In heating mode, the relations in Equations 1-6 are applied with [T.sub.amb] as a dry-bulb temperature because the winter air temperature is assumed to be very dry and the impact of evaporator moisture removal on energy consumption in heating mode is neglected.

The DOE-2.1 RESYS routine (Winkelmann et al. 1993) that was used as a benchmark in heating operation arrives at the correction factors in Equations 1 and 2 using the cubic relations in Equations 11 and 12 instead of Equations 3 and 5. The form of the air mass flow rate corrections for capacity is the same as Equation 4, but the air mass flow correction for coefficient of performance (COP) is different from Equation 6 and is shown in Equation 13:

[f.sub.cap, t] = ([a.sub.3]) + [b.sub.3][T.sub.amb] + [c.sub.3][T.sup.2.sub.amb] + [d.sub.3][T.sup.3.sub.amb] (11)

[f.sub.cop, t] = [([a.sub.4] + [b.sub.4][T.sub.amb] + [c.sub.4][T.sup.2.sub.amb] + [d.sub.4][T.sup.3.sub.amb]).sup.-1] (12)

[f.sub.cop,m] = [[[X.sub.3] + [Y.sub.3]([[[[??].sub.a,in]]/[[??].sub.a,in,max]]) + [Z.sub.3][([[[??].sub.a,in]/[[??].sub.a,in,max]]).sup.2]].sup.-1] (13)

The RESYS routine was developed from ratings data and manufacturers' catalog data for units delivering 3 tons (10.6 kW) at the heating rating condition with 1200 cfm (566 L/s) rated airflow.

Estimation of Coefficients using Linear Regression

All the coefficients of [T.sub.in], [T.sub.amb], and [[??].sub.a, in]/[[??].sub.a, in, max] in Equations 3-6 were found through linear regression. In order for linear regression to be performed on Equation 1, which is nonlinear in the unknown coefficients, [f.sub.cap, m] was set equal to 1 and then the temperature correction factor coefficients for capacity in Equation 3 were found through linear regression using only data points where [[??].sub.a, in] was equal to [[??].sub.a, in, max]. The coefficients [a.sub.1] to [f.sub.1] were fit to the data by minimizing the sum of the squared fractional residuals as shown in Equation 14 with the true temperature correction factor from the data determined using Equation 15:

Error = [summation][([[[f.sub.cap,t,predicted] - [f.sub.cap,t,data]]/[f.sub.cap,t,data]]).sup.2] (14)

[f.sub. cap, t, data] = [[??]/[[??].sub.rat]] (15)

This approach assumes that temperature and airflow rate are decoupled and it is possible to capture the effect of temperature changes completely with data containing fixed airflow rate. These assumptions are reasonable as the airflow rate can be set by the heat pump control system independently of the temperature and the data show a similar effect of temperature changes across different airflow rates. To maintain consistency with the previous step of fixing [f.sub.cap,m] equal to 1 when [[??].sub.a, in] is equal to [[??].sub.a, in, max], the airflow correction factor coefficients [X.sub.1] and [Y.sub.1] must satisfy Equation 16, which is an equality constraint. This constraint was applied by substitution, as shown in Equations 17-20. The resulting Equation 20 is a modification of Equation 1, effectively constraining the value of [f.sub.cap,m] to be equal to 1 when [[??].sub.a, in] is equal to [[??].sub.a, in, max]:

[Y.sub.1] + [X.sub.1] = 1 (16)

[f.sub.cap, m] = [X.sub.1] + [Y.sub.1]([[[??].sub.a,in]]/[[??].sub.a,in,max]) (17)

[f.sub.cap,m] = (1 - [Y.sub.1]) + [Y.sub.1]([[[[??].sub.a,in]]/[[??].sub.a, in max]]) (18)

[f.sub.cap,m] - 1 = [Y.sub.1]([[[[??].sub.a,in]]/[[??].sub.a,in,max]] - 1) (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[Y.sub.1] was found by linear least-squares regression, minimizing the sum of the squared residuals as shown in Equation 21. [X.sub.1] was then found by substituting [Y.sub.1] into the constraint, Equation 16. The COP correction factor coefficients were found using the same procedure as the capacity correction factor coefficients.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

The rated NTU that produced the best SHR estimation was found by minimizing Equation 22, the sum of the square of the residuals between the actual SHRs and those found from evaluating the expression in Equation 7. A Newton's method minimization routine was used to find the rated NTU since SHR is a nonlinear function of rated NTU.

Error = [summation][([SHR.sub.predicted] - [SHR.sub.data]/[SHR.sub.data]).sup.2] (22)

Dual-Speed Heat Pump Models

For dual-speed units, the same approach outlined for single speed units was used. The high compressor speed was treated as a separate unit from the low compressor speed operation, resulting in two sets of cooling coefficients and two sets of heating coefficients for each unit.

GENERALIZED HEAT PUMP MODEL RESULTS

It was observed that the correction factors for single-speed units were similar for given indoor and outdoor conditions. The same was true for the dual-speed units. To simplify the models for use in building simulations, the data for each heat pump family (single- and dual-speed compressors) and operating mode (heating or cooling) was aggregated and aggregate models for each heat pump family were fit using Equations 1-10. The unit's rating data was also applied to the ASHRAE secondary toolkit's (Brandemuehl et al. 1993) direct-expansion cooling model and the DOE-2 model (Winkelmann et al. 1993) for heating with default coefficients. A comparison of the correction coefficients developed in this study with the established models is helpful for deciding if heat pump efficiency changes over time necessitate an update to the established models.

Having a single heat pump model for a family of units allows approximate simulation of other similar equipment when the only data available is performance at rating conditions. It also enables precise and consistent sizing of equipment that is needed when performing parametric studies using building simulations.

Figures 1 through 4 use parity plots to compare the prediction accuracy for capacity and power consumption of using a generalized model for each family against the accuracy of using the default correction factor coefficients. The dimensionless values on the axes of Figures 1 through 3 are the absolute values of capacity and power normalized against the values at the rating condition. Figures 1 through 3 show the single-speed heat pump results while Figure 4 shows the results for dual-speed heat pumps. The SHR predictions for cooling conditions are shown in Figure 5. For clarity, the parity plots show every other data point.

For the single-speed heat pump family (Figures 1-3), the results for the updated generalized mapping show good agreement with the predicted capacity and power consumption, having a maximum power prediction error of 7.2% in cooling mode and maximum capacity prediction error of 10.7%. In contrast, use of the default parameters associated with the ASHRAE toolkit and DOE-2.1 RESYS models shows poorer performance for this family of units, particularly for power predictions. Figure 1 shows larger prediction errors for the default parameters for cooling capacity occur at lower outdoor ambient conditions of 75[degrees]F to 80[degrees]F (23.9[degrees]C to 26.7[degrees]C). However, capacity limitations are usually not encountered at these low ambient conditions and the model would perform well under normal cooling operation. Figure 2 shows that the default RESYS routine map does not take the indoor temperature variation into account, reducing accuracy when indoor temperature is well above or below the 70[degrees]F (21.1[degrees]C) rating condition. However, most homes have heating setpoints above 60[degrees]F (15.6[degrees]C), where the default map is more accurate.

Figures 3 and 4 show that there is a much greater range of errors in power predictions when using the default correlations, especially for the dual-speed heat pump. The capacity and power prediction accuracy of the generalized map for the dual-speed heat pumps is comparable to that for the single-speed units. However, the accuracy of using the default parameters for the power model is significantly worse than for the single-speed case, as shown in Figure 4. Figure 4, like Figure 2, shows the effect of the heating mode RESYS model not accounting for indoor temperature effect. Although not shown, high-compressor-speed results were similar to those for low speed. Figure 5 shows that predictions of SHR in cooling are similar for the updated and default parameter models.

Tables 1 and 2 compare statistics regarding accuracy of models obtained using individualized coefficients as well as the generalized map developed for this study and the default toolkit model. The last two rows of Table 1 and most of the high-compressor-speed cases in Table 2 show a negative coefficient of determination for the toolkit model power predictions with default parameters, making the prediction for the 4 and 5 ton (14.1 and 17.58 kW) dual-speed units in heating mode worse than simply using the mean value for all predictions. One reason for this is that the capacity and COP prediction errors combine, resulting in a large error in power prediction. The default RESYS routine map was developed from single-speed heat pumps, and applying it to dual-speed heat pumps reduces the accuracy.

Equipment Mapping Coefficients and Characteristics

Tables 3 and 4 contain the capacity and COP correction factor coefficients, respectively, for the generalized mapping described in this study. The coefficients require the temperatures in Equations 1-5 to be given in [degrees]F.

Figure 6 shows the variation of the temperature-dependent capacity and COP correction factors, [f.sub.cap,t] and [f.sub.cop,t], for single-speed heat pumps in cooling mode. The default coefficients underpredict the capacity at the lower ambient temperature conditions, reflecting the same lower accuracy at low ambient conditions as Figure 1. The influence of indoor wet-bulb temperature on capacity is similar for both models, with higher wet-bulb temperatures giving higher total cooling capacity.

Figure 7 shows the variation of [f.sub.cap,t] and [f.sub.cop,t] for single-speed heat pumps in heating mode. The main distinction from the cooling mode models is that the RESYS model (Winkelmann et al. 1993) does not account for variation of indoor temperature in predicting capacity and COP. This accounts for the horizontal spread of predictions in the parity plot for capacity in Figures 2 and 4. The default coefficients underpredict capacity at low ambient conditions, indicating that newer equipment on the market has better low-temperature performance than equipment in the past. The predictions based on default coefficients and the updated generalized map predictions are most different at ambient temperatures further away from the rating condition.

Figure 8 shows that the single-stage heat pump generalized map cooling mode airflow correction factors, [f.sub.cap, m] and [f.sub.cop, m], are similar to the original ASHRAE toolkit model (Brandemuehl et al. 1993) correction factors. Indoor unit airflow has a smaller influence on COP and capacity than the indoor and outdoor air temperatures, as shown by the smaller range of the airflow correction factor values. The generalized map airflow correction factors are consistently higher than those given by the default coefficients. The generalized map heating mode airflow correction factors are similarly close to those of the original ASHRAE toolkit model.

Figure 9 shows the dual-stage heat pumps' airflow correction factors, [f.sub.cap,m] and [f.sub.cop,m], in heating mode. Unlike the RESYS default coefficients, the updated generalized map predicts a decrease in capacity with increasing airflow. This counterintuitive trend was seen in individual data for each dual-speed heat pump operating in heating mode. Figure 10 illustrates the issue for the high-compressor-speed cases.

For many of the individual cases, the heating capacity increases with decreasing indoor airflow, particularly at the higher flows. One of the reasons is that the condensing temperature and pressure increase to compensate for a lower heat transfer conductance with decreasing airflow. In some operating ranges, the larger pressure rise across the compressor leads to increased compressor power that causes a higher heat rejection requirement for the condenser. However, results for the 5 ton (17.6 kW) unit given in Table 5 show that the compressor power increase accounted for only a portion (87 Btu/h[26W] outofthe456Btu/h[134W]) of the increase in net heating capacity when airflow decreased from 2080 to 2010 scfm (1.179 to 1.139 kg/s).

The remaining increase in capacity is due to the indoor fan. For the 5 ton unit, the indoor fan power increased by 424 Btu/h (124 W) when airflow changed from 2080 to 2010 scfm, as shown in Figure 11a. The increased fan power is added to the refrigerant-side capacity to give a higher net capacity. Figure 11a indicates that there is a peak in the fan power with respect to changes in airflow. For each compressor speed of the dual-stage units, the airflow was varied by changing the external static pressure (ESP) while holding the fan speed constant. A typical fan power curve (Dayton 2013) for a fixed-speed fan is shown in Figure 11b. To decrease the airflow, the system resistance curve is shifted to the left, leading at first to an operating point with higher fan power. However, as the system resistance curve is shifted further to the left to give lower airflow, the fan power decreases after going past the peak in the fan power curve. The linear fit used for [f.sub.cap,m] does not capture the peak in the fan power curve but instead gives a monotonic trend. However, the correction factor is small (less than 3%) and the model is sufficiently accurate.

CONCLUSIONS

The updated generalized single- and dual-compressor-speed heat pump maps provide an update to the ASHRAE toolkit (Brandemuehl et al. 1993) and RESYS (Winkelmann et al. 1993) models and correlate the data within approximately 10% for both capacity and power consumption. These maps were more accurate than the widely used ASHRAE toolkit model with default coefficients and DOE 2.1's default model at predicting the performance of the families of heat pumps under study. In particular, the default maps performed poorly at predicting the power consumption for dual-speed heat pumps in heating operation. The accuracy of the default correlations decreases for operating conditions further away from the rating conditions, and custom coefficients should be considered for these off-design conditions when appropriate performance data is available. Defrost operation was not modeled, and the mapping results in this paper may not be appropriate when there is a significant buildup of ice that may occur at low ambient conditions.

Updating the model to reflect modern equipment on the market after a period of 30 years since the original RESYS routine is useful to building designers, government regulators, electrical utility companies, and HVAC equipment manufacturing firms. These stakeholders can make more informed choices when selecting space-conditioning equipment than they would have been able to using the published SEER and (Heating Seasonal Performance Factor) HSPF performance ratings or the originally published models' default coefficients. The updated generalized maps can also be used in building energy simulations.

ACKNOWLEDGMENTS

The authors thank Ingersoll Rand Inc. for supporting this research project.

NOMENCLATURE a through f = regression coefficients BF = bypass factor COP = coefficient of performance f = correction factor h = specific enthalpy NTU = number of transfer units [??] = heating or cooling capacity [R.sup.2] = coefficient of determination SHR = sensible heat ratio T = temperature [??] = volumetric air flow rate across indoor unit heat exchanger [??] = power consumption X, Y = regression coefficients [omega] = humidity ratio Subscripts 1 through 6 = regression coefficient a = air adp = apparatus dew point amb = ambient air cap = capacity correction factors cop = COP correction factors data = based on heat pump performance data evap = evaporator in = indoor return air m = mass flow rate max = maximum predicted = value calculated from model rat = at rating condition t = temperature based correction factor wb = wet bulb

REFERENCES

AHRI. 2008. AHRI Standard 210/240-2008, Performance Rating of Unitary Air-Conditioning and Air-Source Heat Pump Equipment. Arlington, VA: Air-Conditioning, Heating, and Refrigeration Institute.

Brandemuehl, M. J., S. Gabel, and I. Andresen. 1993. HVAC 2 Toolkit: A Toolkit for Secondary HVAC System Energy Calculations [CD]. Atlanta: ASHRAE.

Brandemuehl, M.J., and M. Wassmer. 2009. Updated energy calculation models for residential HVAC equipment. Report ASHRAE 1197 RP. Atlanta: ASHRAE.

Cutler, D., J. Winkler, N. Kruis, and C. Christensen. 2012. Improved modeling of residential air conditioners and heat pumps for energy calculations. Report NREL/TP5500-56354. Golden, CO: National Renewable Energy Laboratory.

Dayton. 2013. Fan Curves. Dayton Fans. www.solutionsforair.com/dayton_content/pdfs/FanCurves.pdf

DOE. 2011. Units to be tested. Code of Federal Regulations, 10 CFR 430.24(m). Washington, DC: Government Printing Office.

Holloway, S.O. 2013. An annual performance comparison of various heat pumps in residential applications. Master's Thesis, Purdue University, West Lafayette, IN.

LBNL. 1981. DOE-2, Version 2.1A. Berkeley, CA: Lawrence Berkeley National Laboratory.

Winkelmann, F.C., B.E. Birdsall, W.F. Buhl, K.L. Ellington, E. Erdem, J.J. Hirsch, and S. Gates. 1993. DOE-2 Supplement (Version 2.1E). Berkeley, CA: Lawrence Berkeley National Laboratory.

Yang, H., and H. Li. 2010. A Generic Rating-Data-Based (GRDB) DX Coils Modeling Method. HVAC&R Research 16(3):331-53.

DISCUSSION

Kelly Kissock, Professor, University of Dayton, Dayton, OH: How much do your results vary with other manufacturers? Simbarashe Nyika: We cannot say for sure how our results compare with heat pumps from other manufacturers in the industry as we did not have access to this level of detailed data from other suppliers. However, we anticipate the difference is small. We consulted the recent study by NREL (Cutler et al. 2012) of 260 cooling-only and 200 reversible ducted heat pumps rated between SEER 13 and SEER 21 based on manufacturer performance tables and found that it was possible to use a single model to describe the performance of heat pumps from various manufacturers with the same SEER rating.

Cutler, D., J. Winkler, N. Kruis, and C. Christensen. 2012. Improved modeling of residential air conditioners and heat pumps for energy calculations. Report NREL/TP5500-56354. Golden, CO: National Renewable Energy Laboratory. www.nrel.gov/docs/fy13osti/56354.pdf. Hugh Henderson, Principal, CDH Energy Corporation, Cazenovia, NY: Your comparison to the ASHRAE secondary toolkit using the 1982 coefficient was not the best baseline. I would suggest using the EnergyPlus DX coil coefficient, which has been updated.

Simbarashe Nyika: At the time of writing the paper, the EnergyPlus Engineering Reference guide (DOE 2014) referenced the 1993 ASHRAE toolkit, which in turn references the 1982 coefficients. As of July 21, 2014, the EnergyPlus reference guide still shows the same information. The engineering reference did not list any default coefficients and asks for these as input from the user.

DOE. 2014. EnergyPlus Engineering Reference. Washington, DC: U.S. Department of Energy. http:// apps1.eere.energy.gov/buildings/energyplus/pdfs /engineeringreference.pdf.

Simbarashe Nyika

W. Travis Horton, PhD

Member ASHRAE

Seth O. Holloway

Associate Member ASHRAE

James E. Braun, PhD, PE

Fellow ASHRAE

Simbarashe Nyika is a cooling development engineer at Whirlpool Corporation, Benton Harbor, MI. Seth O. Holloway is a systems engineer at Ingersoll Rand, Tyler, TX. W. Travis Horton is an assistant professor of civil engineering and James E. Braun is a professor of mechanical engineering in the Ray W. Herrick Laboratories School of Mechanical Engineering at Purdue University, West Lafayette, IN.

Table 1. Cooling and Heating Mode Accuracy Comparisons of the Individual, Generalized, and Default Models for Single-Speed Heat Pumps Capacity, Mode ton 1.5 C 2 C 2.5 c 3 c 3.5 c 4 c 5 c 1.5 H 2 H 2.5 H 3 H 3.5 H 4 H 5 H Individual Coefficients Capacity, SHR, Total Capacity Power [R.sup.2] [R.sup.2] Power, Max Max [R.sup.2] Error, Error, % % 0.9963 0.9653 0.9957 2.9 3.8 0.9869 0.9677 0.9831 4.5 3.7 0.9913 0.9717 0.9926 3.3 2.8 0.9955 0.9526 0.9969 3.2 2.6 0.9821 0.9704 0.9876 4.2 4.1 0.9961 0.9583 0.9991 2.2 1.1 0.9961 0.9736 0.9811 3.3 5.4 0.9997 -- 0.9372 1.6 5.1 0.9994 -- 0.9563 1.5 5.2 0.9997 -- 0.9836 1.0 2.8 0.9999 -- 0.9920 1.2 1.8 0.9997 -- 0.9625 1.3 3.9 0.9999 -- 0.9929 0.7 2.4 0.9991 -- 0.9122 1.3 5.9 Generalized Family Coefficients Capacity, SHR, Total Capacity Power [R.sup.2] [R.sup.2] Power, Max Max [R.sup.2] Error, % Error, % 0.9699 0.9319 0.9807 8.6 3.6 0.9821 0.9507 0.9763 6.0 4.2 0.9890 0.9543 0.9688 4.3 5.2 0.9896 0.9483 0.9732 4.6 4.2 0.9835 0.9572 0.9672 4.3 7.2 0.9897 0.9553 0.9822 3.3 5.2 0.9812 0.9712 0.9758 4.9 6.9 0.9767 -- 0.9164 10.7 6.8 0.9867 -- 0.8688 7.9 9.6 0.9913 -- 0.8646 6.0 10.0 0.9928 -- 0.9028 5.1 7.9 0.9928 -- 0.8434 5.2 7.4 0.9932 -- 0.8780 4.8 6.9 0.9860 -- 0.8720 6.7 7.2 ASHRAE Toolkit Default Capacity, SHR, Total Capacity Power [R.sup.2] [R.sup.2] Power, Max Max [R.sup.2] Error, Error, % % 0.8717 0.9425 0.7776 15.6 13.2 0.9174 0.9591 0.7614 13.1 16.1 0.9084 0.9638 0.7692 12.6 15.2 0.8955 0.9394 0.7590 13.7 15.7 0.9516 0.9653 0.6178 9.2 24.1 0.9288 0.9511 0.6279 10.5 22.9 0.8950 0.9675 0.7497 11.3 21.3 0.9465 -- 0.1918 13.5 13.2 0.9405 -- 0.7119 14.2 9.6 0.9306 -- 0.7394 14.9 9.2 0.9266 -- 0.6151 15.2 10.9 0.8357 -- 0.2068 19.4 18.1 0.8564 -- -0.4986 17.6 19.2 0.8071 -- -0.1090 19.4 20.9 Table 2. Cooling and Heating Mode Accuracy Comparisons of Individual, Generalized, and Default Models for Dual-Speed Heat Pump Family Capacity, Mode Comp. ton Speed 2 C low 2 C high 3 c low 3 c high 4 c low 4 c high 5 c low 5 c high 2 H low 2 H high 3 H low 3 H high 4 H low 4 H high 5 H low 5 H high Individual Coefficients Capacity, SHR, Total Capacity Power [R.sup.2] [R.sup.2] Power, Max Max [R.sup.2] Error, % Error, % 0.9932 0.956 0.9914 4.2 4.3 0.9943 0.945 0.9718 4.0 6.1 0.9991 0.977 0.9974 1.3 3.3 0.9961 0.964 0.9773 2.7 19.5 0.9987 0.955 0.9940 1.9 4.5 0.9967 0.938 0.9730 2.1 6.4 0.9986 0.954 0.9953 1.3 4.9 0.9973 0.931 0.9791 2.1 11.6 0.9979 -- 0.9752 2.6 5.5 0.9979 -- 0.9193 2.9 7.3 0.9999 -- 0.9959 1.5 1.2 0.9962 -- 0.8915 2.9 20.8 0.9991 -- 0.9722 3.8 5.0 0.9968 -- 0.8815 3.8 10.2 0.9993 -- 0.9788 4.5 6.8 0.9964 -- 0.8903 3.9 13.7 Generalized Family Coefficients Capacity, SHR, Total Capacity Power [R.sup.2] [R.sup.2] Power, Max Max [R.sup.2] Error, % Error, % 0.9954 0.9530 0.9879 3.5 7.0 0.9925 0.9321 0.9514 3.9 7.3 0.9967 0.9506 0.9834 2.2 6.2 0.9916 0.9220 0.9259 2.5 8.3 0.9987 0.9477 0.9960 2.0 6.1 0.9966 0.9243 0.9726 2.3 8.9 0.9981 0.9451 0.9962 2.4 4.9 0.9971 0.9193 0.974 3.5 13.9 0.9965 -- 0.9215 5.9 7.7 0.9957 -- 0.7867 4.9 6.7 0.9988 -- 0.9158 2.0 5.3 0.9959 -- 0.7785 3.0 9.5 0.9992 -- 0.9796 3.9 3.9 0.9981 -- 0.923 5.5 11.0 0.9985 -- 0.9723 4.0 5.7 0.9979 -- 0.854 5.7 17.7 ASHRAE Toolkit Default Capacity, SHR, Total Capacity Power [R.sup.2] [R.sup.2] Power, Max Max [R.sup.2] Error, % Error, % 0.908 0.948 0.536 13.0 30.6 0.924 0.940 0.591 11.6 25.8 0.933 0.966 0.438 10.7 31.3 0.941 0.953 0.515 9.7 25.3 0.920 0.947 0.515 11.8 29.8 0.920 0.932 0.698 11.6 21.9 0.926 0.945 0.515 11.1 29.0 0.929 0.923 0.673 10.7 21.0 0.959 -- 0.004 11.6 20.7 0.861 -- -0.466 19.4 25.2 0.936 -- -0.702 14.3 23.4 0.830 -- -0.228 19.9 23.1 0.946 -- -0.567 13.6 23.1 0.863 -- -0.430 18.1 21.6 0.925 -- -0.768 15.2 22.9 0.823 -- -0.664 19.3 20.5 Table 3. Generalized Capacity Mapping Coefficients for Single- and Dual-Speed Heat Pump Families Family Mode Comp. a b c Speed Single C normal 1.3275E-01 1.7770E-03 -8.6000E-06 Speed H normal 5.3097E-01 1.2256E-02 4.3500E-05 Dual C low 3.7534E-02 4.2490E-03 -2.2000E-05 Speed high 1.1396E-01 3.0590E-03 -1.6000E-05 H low 3.8622E-01 1.3718E-02 4.3900E-05 high 5.2554E-01 1.2121E-02 4.1300E-05 Comp. d e f Speed normal 1.5663E-02 7.7700E-05 -9.8000E-05 normal -1.0200E-03 -8.5000E-06 -2.7000E-05 low 1.3278E-02 9.5900E-05 -8.8000E-05 high 1.3301E-02 8.5200E-05 -8.4000E-05 low 1.5310E-03 -1.8000E-05 -4.5000E-05 high -9.0000E-04 -9.6000E-07 -3.6000E-05 Comp. X Y NTU Speed Rated normal 0.82527 0.17473 1.76 normal 0.88873 0.11127 low 0.76666 0.23334 1.69 high 0.73154 0.26846 1.43 low 1.00497 -0.00497 -- high 1.05631 -0.05631 -- Table 4. Generalized COP Mapping Coefficients for Single- and Dual-Speed Heat Pump Families Family Mode Comp. a b c Speed Single C normal 9.2942E-01 -1.6480E-02 9.5900E-05 Speed H normal 1.0746E+00 1.9370E-02 1.1700E-05 Dual C low 9.1881E-01 -1.9900E-02 1.3700E-04 Speed high 8.6967E-01 -1.3330E-02 6.7200E-05 H low 1.1991E+00 2.5049E-02 2.9200E-05 high 1.1714E+00 1.8446E-02 -4.0000E-06 Comp. d e f Speed normal 3.1921E-02 6.6500E-05 -2.6000E-04 normal -1.1150E-02 4.6200E-05 -1.4000E-04 low 3.6696E-02 1.5800E-04 -3.8000E-04 high 2.9190E-02 5.9900E-05 -2.3000E-04 low -1.6650E-02 8.2100E-05 -2.1000E-04 high -1.2440E-02 4.3000E-05 -1.1000E-04 Comp. X Y Speed normal 0.94381 0.05619 normal 0.77239 0.22761 low 0.58442 0.41558 high 0.45184 0.54816 low 0.54052 0.45948 high 0.50783 0.49217 Table 5. Effect of Varying Airflow at 30[degrees]F (- 1.1[degrees]C) Ambient and 70[degrees]F (21.1[degrees]C) Indoor Temperatures on Heating Performance for the 5 ton (17.6 kW) Unit External Indoor Indoor Static Pressure, Airflow, Fan Power, in. [H.sub.2]O (Pa) scfm (kg/s) Btu/h (kW) 0.1 (24.9) 2080 (1.179) 2730 (0.800) 0.3 (74.7) 2010 (1.139) 3155 (0.925) 0.9 (224.2) 1750 (0.992) 2969 (0.870) Compressor Indoor Outdoor Power, Capacity, Capacity, Btu/h (kW) Btu/h (kW) Btu/h (kW) 10026 (2.938) 43485 (12.74) 35278 (10.339) 10113 (2.964) 43941 (12.88) 35271 (10.337) 10434 (3.058) 43682 (12.80) 34998 (10.257)

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Author: | Nyika, Simbarashe; Horton, W. Travis; Holloway, Seth O.; Braun, James E. |
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Publication: | ASHRAE Transactions |

Article Type: | Report |

Date: | Jul 1, 2014 |

Words: | 6350 |

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