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Generalized performance maps for variable-speed residential heat pumps.


Ducted unitary heat pumps of the type used in this study have been well studied and much literature is available documenting their function. Several models are currently used to simulate the performance of fixed-compressor-speed heat pumps in building energy simulation programs. The ASHRAE secondary toolkit bypass model uses polynomials and a bypass factor approach derived from experimental data for direct expansion coils (Brandemuehl et. al 1993). The DOE 2.1 RESYS routine (Winkelmann et al 1993) is widely used for modeling single-speed systems.

The literature on building energy simulation of variable-compressor-speed heat pumps is less extensive than for single-speed systems. A curve fit based model was implemented in EnergyPlus to simulate a split variable refrigerant flow (VRF) system in a small commercial building in a Shanghai climate (Zhou 2006). However, this approach is not general for any size equipment and is not appropriate for general system simulation studies. Version 7.1 of EnergyPlus approximates a continuously-variable-compressor speed heat pump using a multispeed model that allows up to 4 discrete compressor speeds (DOE 2012). The multispeed model requires ratings data at each compressor speed, but such data is often unavailable. An empirical model developed by Cheung (2011) for modeling mini-split systems is more generic and requires less input data. Ecotope Inc. (2011) details the experiments on 2 heat pumps used to develop Cheung's model. EnergyPlus's VRF Heat Pump Model uses a similar approach of adjusting the performance at a rated condition for part load ratio and temperatures (DOE 2012). Cheung's model and the EnergyPlus VRF model are the basis for the generalized performance maps developed in this study. The resulting models are useful for general studies involving assessments of variable-speed heat pump performance.


The data used for heat pump mapping was generated using a detailed component based simulation program provided by the manufacturer of the heat pumps. Mapping was carried out for a family of three similar residential split systems with central air distribution and the correlations were implemented as components for an hourly building simulation program. The three units all have SEER 20 ratings and nominal cooling capacities of 3, 4 and 5 tons (10.6, 14.1 and 17.6 kW). In the simulation used to generate 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 indoor conditions. The refrigerant charge level was determined at 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 with condenser exit subcooling of 10[degrees]F. The charge level determined in cooling mode was used for heating mode operation.

The cooling simulations were performed on each unit with ambient temperatures from 67 to 115 [degrees]F (19.4-46.1[degrees]C) with 5 [degrees]F (2.8[degrees]C) increments from 70[degrees]F (21.1[degrees]C) and indoor wet bulb temperatures from 49.4 to 77 [degrees]F (9.8-25[degrees]C) with indoor dry bulb temperature from 65 to 85[degrees]F (18.3-29.4[degrees]C). The heating simulations had ambient temperatures of 20 to 62 [degrees]F (-6.7-16.7[degrees]C) in 10 [degrees]F (5.6[degrees]C) increments and indoor dry bulb temperatures from 55 to 75 [degrees]F (12.8 - 23.9[degrees]C). Although the heat pumps' compressors are continuously variable, simulations were carried out at 3 distinct compressor speeds in cooling and 4 in heating. Each unit was simulated with airflow rates that were a function of compressor speed.


Cheung (2011) suggested an empirical model to predict the behavior of a ductless split heat pump system in both cooling and heating under different loading conditions and environments. Equations 1-8 describe a modified version of Cheung's model. Chueng's model was intended to model the performance of a single, split heat pump system and was modified to account for differences in equipment sizing by including performance at rated conditions in the equations.

[] = [,rat][f.sub.0]([V.sub.a,in]) (1)

[Q.sub.max] = [Q.sub.max,rat][f.sub.1]([], [T.sub.out]) (2)

[W.sub.max] = [W.sub.max,rat][f.sub.2]([], [T.sub.out]) (3)

[]/[[]([V.sub.a,in]) + [W.sub.max]] = [f.sub.3](Q/[Q.sub.max], [V.sub.a,in]/[V.sub.a,in,max]) (4)

[f.sub.0]([V.sub.a,in]) = [a.sub.0] + [a.sub.1][V.sub.a,in]/[,max] + [a.sub.2][([V.sub.a,in]/[V.sub.a,in,max]).sup.2] (5)

[f.sub.1]([],[T.sub.out]) = [b.sub.0] + [b.sub.1][T.sub.amb] + [b.sub.2][T.sub.amb.sup.2] + [b.sub.3][] + [b.sub.4][] + [b.sub.5][][T.sub.amb] (6)

[f.sub.0]([V.sub.a,in]) = [c.sub.0] + [c.sub.1][T.sub.amb] + [c.sub.2][T.sub.amb.sup.2] + [C.sub.3][] + [c.sub.4][] + [c.sub.5][][T.sub.amb] (7)

[f.sub.3] = [d.sub.0] + [d.sub.1]Q/[Q.sub.max] + [d.sub.2][V.sub.a,in]/[V.sub.a,in,max] + [d.sub.3][(Q/[Q.sub.max]).sup.2] + [d.sub.4][([V.sub.a,in]/[V.sub.a,in,max]).sup.2] + [d.sub.5](Q/[Q.sub.max])([V.sub.a,in]/[V.sub.a,in,max]) (8)

The variables [] and [V.sub.a,in] are the indoor unit power consumption in Watts and the indoor volumetric air flow rate, respectively [Q.sub.max] is the heating or cooling capacity and [W.sub.max] is the outdoor unit power consumption at [Q.sub.max] [] is the total system power consumption in Watts. [T.sub.tatal] is the unit's maximum indoor air volumetric flow rate. [] is the indoor air entering wet bulb for cooling operation and indoor air entering dry bulb temperature in [degrees]F for heating operation is the outdoor air entering dry bulb temperatures in [degrees]F. Parameters subscripted with rat are values at a rated condition for normalization. The rated condition for heating operation is 50[degrees]F (10.0[degrees]C) ambient dry bulb temperature and 70[degrees]F (21.1[degrees]C) indoor dry bulb temperatures with maximum compressor and indoor fan speeds. The rated condition for cooling operation is 95[degrees]F (35[degrees]C) ambient dry bulb temperature and 66.7[degrees]F (19.3[degrees]C) indoor wet bulb with maximum compressor and indoor fan speeds.

Sensible Heat Ratio. Equations 2 and 6 make use of indoor inlet wet bulb temperature and only provide total cooling capacity for wet coils. However, it is also necessary 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 bypass factor approach (Brandemuehl et. al 1993), which involves solution of Equations 9-12 for apparatus dew point condition given inlet conditions and outlet enthalpy determined from the total capacity model along with [NTU.sub.rat] and [V.sub.a,in,max] For a dry coil, the toolkit model will predict a sensible cooling capacity greater than the total capacity and iteratively solves for an apparatus dew point humidity ratio that gives a unity SHR. 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.

SHR = [h([T.sub.evap,in],[[omega].sub.adp]) - [h.sub.adp]]/[[h.sub.evap,in] - [h.sub.adp]] (9)

[h.sub.adp] = [h.sub.evap,in] - [[h.evap,in] - [h.sub.evap,out]]/[1 - BF] (10)

BF = [e.sup.-NTU] (11)

NTU = [NTU.sub.rat]/[([V.sub.a,in]/[V.sub.a,in,max])] (12)

Regression Method. The regression coefficients: a's, b's, c's and d's in Equations 5-8 as well as [NTU.sub.rat] in Equation 12 were found for all units in both heating and cooling mode using optimization based regression. Regression was performed to minimize the percentage deviation between the actual and predicted parameter values. For example, [NTU.sub.rat] is found by minimizing the sum of the square of the relative residuals between the actual SHRs and those found from evaluating Equation 9 with different guesses of [NTU.sub.rat]. In order to simplify the models for use in building simulations, the data from each heat pump and operating mode (heating or cooling) was aggregated and generalized models for the heat pump family were fit using Equations 1-12. To judge the loss in accuracy from using a generalized mapping approach, individualized fits for each heat pump were also generated.


Figure 1 contains parity plots comparing the [] predictions found using generalized mapping in heating and cooling mode. Figures 2, 3 and 4 compare the generalized map predictions for [Q.sub.max], [W.sub.max], [], respectively for heating and cooling operation. Figures 5 shows a parity plot for [] using individualized fits for each unit. Figures 6 shows a parity plot for SHR prediction found using the generalized map's [NTU.sub.rat].

The parity plots in Figure 1 show reasonable agreement between predicted and actual values of [] There are relatively few data points because the indoor unit power only depends on relative air flow and not on ambient or room conditions. A higher order model than the quadratic of Equation 5 might provide more accurate results. However, since indoor unit power is small compared to compressor power, the effect of this deviation on total power and COP is small except for the lower compressor speeds. Errors in indoor unit power prediction will generally have a larger effect for cooling mode than heating mode since the compressor power is lower for cooling.

Figures 2-3 show good agreement between predicted and actual values of both [Q.sub.max], and [W.sub.max] for the generalized maps with agreement well below 5% for all data points. Each cluster of data points in Figure 2a corresponds to one of the 5 ambient dry bulb temperatures used for simulating the heat pumps in heating mode. The spread within the clusters is due to different indoor temperatures because all other factors were kept constant. Figure 3b shows similar clustering to Figure 2a for the same reasons.

Figure 4 shows parity plots for [W.sub.max] which is predicted well but shows the effects of the errors in Figures 1-3. The cooling mode parity plot Figure 4a shows larger errors than that for heating mode, reflecting the larger errors in [W.sub.max] for cooling mode. The clusters in Figure 4a correspond to the various compressor speeds used to generate performance data. In Figure 5, applying individualized sets of regression coefficients for each heat pump reduces the error in [W.sub.max] to below 10% for most of the data points. Figure 6 illustrates the SHR's goodness of fit. The individual units' NTUrat were similar and the SHR of all units is well predicted with one value of [NTU.sub.rat].

Equipment Mapping Coefficients

From the generalized mapping, [NTU.sub.rat] was found to be 2.026. The remaining regression coefficients for the generalized equipment maps are shown in Table 1 and Table 2.

Table 1. Generalized Mapping Coefficients for Equations 5 and 8

Operation         0            1            2            3

cooling    a  5.3984E-01  -2.0139E+00   2.4778E+00           --

           d  5.7084E-01   1.3615E+00  -2.4475E+00  -2.3323E+00

heating    a  5.7895E-01  -2.1548E+00   2.6101E+00           --

           d  1.6199E-01   1.0517E+00  -7.8999E-01   1.3384E+00

Operation         4            5

cooling    a          --           --

           d  1.0010E+00   2.8457E+00

heating    a          --           --

           d  1.1627E+00  -1.9262E+00

Table 2. Generalized Mapping Coefficients for Equations 6-7

Operation  b      0            1            2            3
Mode              [-]      [[degrees]   [[degrees]   [[degrees]
                          [F.sup.-1]]  [F.sup.-2]]  [F.sup.-1]]

cooling    b  6.0700E-03   3.5645E-03  -1.7664E-05   1.5758E-02

           c  5.8151E-01  -2.3241E-03   5.7481E-05  -1.3087E-03

           b  4.4500E-01   1.2039E-02   2.7980E-05   3.3760E-04

heating    c  4.9289E-01   8.2779E-04   7.1494E-06  -1.6981E-04

Operation  b       4            5
Mode          [[degrees]   [[degrees]
              [F.sup.-2]]  [F.sup.-2]]

cooling    b   7.1319E-05  -8.7988E-05

           c  -1.5824E-05   4.3542E-05

           b  -8.2954E-06  -2.8511E-05

heating    c   4.5548E-05   5.0398E-05


Generalized maps were developed for a family of 3 similar variable-speed-compressor heat pumps that can be used in building energy simulation. These generalized maps work well but are somewhat less accurate than maps generated using custom data for each heat pump. However, the generalized performance maps are valuable when only rating point data is available. Generalized maps can also be used when data is only available for one size of heat pump but simulations must be conducted for a similar but different sized unit. The accuracy of the generalized maps decreases for operating conditions further away from the ratings conditions and care should be taken not to extrapolate beyond the data range and capacities used to generate the generalized maps.


The authors thank Trane Inc. for supporting this research project.


T = temperature

a-d = regression coefficients

COP = coefficient of performance

SHR = sensible heat ratio

W = power consumption

NTU = number of transfer units

f = correction factor

h = moist air enthalpy

BF = bypass factor

Q = capacity

V = volumetric flowrate across indoor unit heat exchanger

[omega] = humidity ratio


a = air

adp = apparatus dew point

in = indoor return air

max = maximum

out = outdoor ambient air

rat = at rating condition

wb = wet bulb

0-5 = regression coefficient


Brandemuehl, M. J., Gabel, S. and Andresen, I. 1993. HVAC 2 Toolkit: A Toolkit for Secondary HVAC System Energy Calculations. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.

Cheung, H., Nyika, S. and Braun, J. E. 2011. Performance Testing of Ductless Heat Pumps: Progress Report for February. West Lafayette. Indiana: Herrick Laboratories, Purdue University.

DOE. 2012. EnergyPlus Engineering Reference. CA: US Department of Energy.

Ecotope, Inc (2011). Ductless Heat Pump Impact & Process Evaluation: Lab-Testing Report, Northeast Energy Efficiency Aliance Report No. E11-225. Seattle, WA: Ecotope, Inc.

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.

Zhou, Y.; Wu, J.; Wang, R.; Shiochi, S. 2006. Module Development and Simulation of the Variable Refrigerant Flow Air Conditioning System under Cooling Conditions in Energyplus. Shanghai: The International Conference for Enhanced Building Operations (ICEBO),HVAC Technologies for Energy Efficiency. 4.1-2.

Simbarashe Nyika

Student Member ASHRAE

Seth O. Holloway

Student Member ASHRAE

W. Travis Horton, PhD


James E. Braun, PhD, PE


Simbarashe Nyika and Seth O. Holloway are graduate research assistants at Purdue University's Herrick Laboratories, West Lafayette, Indiana. W. Travis Horton is an assistant professor in Civil Engineering at Purdue University. James E. Braun is a professor in Mechanical Engineering at Purdue University.
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Title Annotation:DA-13-C002
Author:Nyika, Simbarashe; Holloway, Seth O.; Horton, W. Travis; Braun, James E.
Publication:ASHRAE Transactions
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2013
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