Parametric modelling of domestic air

Underwood CP, Royapoor M, Sturm B.
Parametric modelling of domestic air-source heat pumps.
Energy and Buildings 2017, 139, 578-589.
© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
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Parametric modelling of domestic air-source heat pumps
C.P. Underwood1, M. Royapoor2, B. Sturm3
1. Faculty of Engineering and Environment, University of Northumbria, UK
2. Sir Joseph Swan Centre for Energy Research, Newcastle University, UK
3. Postharvest Technologies and Processing Group, Kassel University, Germany
A new domestic air-source heat pump model is proposed which can be parameterised either
from field data, experimental data or manufacturers’ standard rating data. The model differs
from the much more prevalent system-side regression models for these types of heat pumps
in that it operates on refrigerant-side variables. This makes it more suited to detailed
performance analyses of heat pumps in service. Because both field data and manufacturers’
data can be used for parameterising the model, it can be used to investigate problems
associated with the building performance gap where a heat pump is used. A new type of
efficiency-based compressor model is developed which enables compressor performance to
be directly compared with alternatives and a model of defrosting is included by introducing a
new defrost discounting term. Results tested using data from a field monitoring site and from
a laboratory installation show good predictive behaviour by the model especially at low
source air temperatures. When the model is fitted using manufacturer’s standard rating data, a
performance gap is evident when compared with the model fitted to field data but the gap is
generally nominal.
Air-source heat pump; heat pump modelling; compressor modelling; defrosting; performance
List of symbols
f (…)
Model-fitting parameters
Coefficient of performance
(a function of)
Defrost discount factor
Refrigerant enthalpy at evaporator outlet (kJkg-1)
Refrigerant enthalpy at compressor outlet (kJkg-1)
Refrigerant enthalpy at condenser outlet (kJkg-1)
Refrigerant mass flow rate (kgs-1)
Evaporating pressure (kNm-2)
Condensing pressure (kNm-2)
Average (steady) condenser heat output (kW)
Defrost-adjusted average condenser heat output (kW)
Compression ratio (= Pc / Pe)
Seasonal coefficient of performance
Seasonal performance factor
Seasonal performance factor including heat pump auxillary loads
External air temperature (oC)
Evaporating temperature (oC)
Condensing temperature (oC)
Condenser water outlet temperature (oC)
Compressor displacement volume (m3s-1)
Average (steady) overall compressor plus auxillary power (kW)
Auxillary power (kW)
Average compressor isentropic work (kW)
Condenser thermal approach (K)
Evaporator thermal approach (K)
Compressor isentropic efficiency
Compressor volumetric efficiency
Compression index of an ideal gas
Refrigerant density at compressor inlet (kgm-3)
1  2 cost function of RMSE values
1  2 vector of target optimisation goals
1  2 vector of inputs
1  2 vector of outputs
1  10 vector of model fitting parameters
1  10 vector of model fitting parameter lower bounds
1  10 vector of model fitting parameter upper bounds
1. Introduction
Most domestic air-source heat pumps in Europe sink to hot water heating systems and
generally consist of second-generation technologies. Typically, this is characterised by the
use of a hydro-fluoro-carbon working fluid, scroll compressors, compact brazed-plate
condensers, mechanical expansion device and conventional fin and coil evaporators. Controls
usually consist of fixed speed thermostatic control over the compressor and defrosting is
effected through the use of reverse-cycling changeover controls. These heat pumps are
becoming well established in Europe and north America and are expected to enjoy increased
use in other countries in the coming years. In the UK for example, a sharp increase in their
use is envisaged for domestic space heating in the coming years in order to address the twin
aims of reducing carbon emissions as well as to diversify away from an over-dependence on
gas for space heating [1].
Recent field trials of domestic air-source heat pumps in the UK have revealed seasonal
performance factors of the order of 2.5 [2,3]. Gleeson and Lowe conducted a detailed
comparison of UK and European field trials for domestic heat pumps which concluded that
UK performances are generally inferior to those observed in other European countries [4].
The reasons mentioned for better European performances include minimal use of back-up
heating; the use of compensating heating control; higher quality heating systems in terms of
components and controls; a low usage of domestic hot water (at higher temperatures) and
higher house insulation standards. Hence there is a significant ‘performance gap’ [5]
concerning UK heat pumps in service.
There is a need for a better understanding of how these systems perform in practice when
matched to alternative house construction, heating system design, hot water usage, system
control and operation, and user behaviour. To assist with this requires better modelling of the
performance of these heat pumps in service as informed by the new field trial data that is
beginning to become available [2,3]. Historically, deterministic models of heat pump
behaviour have been developed but these require a high level of detailed information about
the heat pump and its control to make them feasible. Many existing heat pumps systems can
be categorised as ‘information poor’ in which certain key data are either unavailable due to
commercial sensitivity or are uncertain. This has led to ‘black box’ modelling or boundary
variable modelling in which heat pump behaviour is fitted to field observations of ambient air
temperature and, optionally, heating system temperature. These approaches require much less
information but result in a lack of predictive detail that might otherwise be used to help
diagnose the extent of under-performance and what might be done about it. (For a general
review of approaches to modelling heat pump systems generally, see [6].) This work attempts
to address these limitations by developing a new type of heat pump model that can be readily
fitted to field monitoring data or manufacturers standard rating data so that heat pump
performance revealing performance gaps and other departures from expected operational
behaviour can be identified.
In this work a new model is developed that can be readily fitted to average performance data
and/or data arising from standard rating information. At the heart of this model is a new
compressor model which can be fitted with a relatively small number of parameters and is
able to capture the (varying) compressor isentropic efficiency. Because the model can be
fitted to field data as well as standard rating data, it is possible to compare expected heat
pump performance with actual heat pump performance in the field. The objectives of this
work are summarised as follows:
To develop a domestic air-source heat pump model that can be fitted to a variety of
data sources with sufficient detail to enable both heat pump and compressor
performance in service to be determined
To student defrost activity of heat pumps in service and incorporate adjustments to
the model to take account of defrosting
To test the model against field monitoring and laboratory data
To compare a model fitted to field monitoring data with a model fitted to standard
rating data in order to quantify the heat pump performance gap
To compare the results of the two alternative models with recent results reported
from extensive field monitoring
2. Review
Numerous attempts have been made to fit simple regression models to heat pump field data
based on boundary variables only [7,8,9,10]. Gupta and Irving [7] describe the development
of a regression model giving seasonal performance factor (SPF) as a function of temperature
lift (heating ‘sink’ temperature minus source temperature) for a variety of source types
including air. However the model is based on field results observed in Switzerland which, as
Gleeson and Lowe have noted [5], have exhibited higher performances than in the UK which
raises issues about the robustness of this model when used in UK applications. Also, it is not
clear as to whether and how defrosting has been accounted for in the air-source heat pumps.
Murphy et al. [8] develop a 5-parameter regression model of an air-source heat pump with
ambient air temperature and heating system return water temperature as independent
variables. The model is incorporated in a general building thermal model and results are
found to be within field trial data. Tabatabaei et al. [9] compare the accuracy of first, second,
third and fourth-order polynomial models of air-source heat pump SPF with just external air
temperature as the independent variable by fitting to field monitoring data from a number of
sites in Belgium. They conclude that best results arise from the higher-order models. Touchie
and Pressnail [10] fit simple linear models to experimental data to predict the CoP versus
external air temperature and compare the results with manufacturer’s data. (The experimental
test set up used a warm buffer zone to mitigate the effects of very cold source air in extreme
winter conditions.) Results show that the test data CoP values vary with the same slope as the
manufacturer’s data but the CoP values themselves are much lower for a number of reasons.
In one of few models explicitly considering part-load performance, Madonna and Bazzocchi
[11] develop a simple model of reversible air-to-water heat pumps by using field data to
derive adjustments to a maximum coefficient of performance value which is based on a
modified Carnot performance. Further fitted terms allow for part-load and defrost operation.
The model is integrated with a building simulation tool and applied to buildings in 3
contrasting Italian cities.
In a more detailed parametric approach, Jin and Spitler [12] describe detailed parameterextraction from manufacturers’ catalogue data for water-to-water heat pumps using scroll
compressors. The model leads to detailed performance data but is dependent on
manufacturers data which itself is usually generated by a model (i.e. their results are not
verified against field or laboratory data).
Guo et al. [13] use grey system theory to model air-source water heating systems. This
method is suited to time series observations with uncertainty and involves smoothing the data
to reduce randomness and then fitting a differential equation based ‘grey model’ to the
smoothed data. Reasonably representative models can be obtained from situations where data
are limited in quantity, incomplete or uncertain but they lack insightful data about detailed
plant performance.
Kinab et al. [14] develop a very detailed steady-state model of a reversible air-to-water heat
pump which explicitly calculates the compressor efficiency and so can be used to compare
compressor performances. However the compressor regression model is parameterised
through 18 fitted coefficients (9 each for power and mass flow rate). Results with
experimental data for an R410A heat pump are good and a detailed consideration is given to
the treatment of defrosting in the model. In one of few detailed studies on defrosting of airsource heat pumps, Qu et al. [15] report on experimental studies carried out on a 6.5kW heat
pump with multiple circuits. Results show the transients in key variables during defrosting
and conclude that defrosting is completed more quickly in up-steam circuits than in downstream circuits. In a follow-on article [16] a detailed semi-empirical model of defrosting is
developed which confirms that the protracted down-stream circuit defrosting behaviour is
attributed to delays caused by flowing water from melted ice coming from upstream circuits.
In relation to dynamic modelling of air-source heat pumps, Ji et al. [17] develop a dynamic
model of a multi-functional heat pump which includes regression-fit models for both
refrigerant flow rate and power consumption fitted to manufacturer’s data. The fitted
equations each involve 6 coefficients (12 in all). Kelly and Cockcroft [18] develop a secondorder dynamic model with parameters fitted from experimental data for domestic air-source
heat pumps and then used to compare with results from field monitoring. Results are good at
low ambient air temperatures but the model tended to over-predict performance at higher
temperatures though this was brought into good alignment when temperature compensation
was disabled which helped to reveal inappropriate operation in the field.
In summary, though significant progress has been made in relation to air-source heat pump
modelling informed by field monitoring data, the resulting models are either too simplistic
and cannot be used for detailed equipment performance evaluation, or are too complex and
require extensive detailed information for their derivation. What is needed is a model of
intermediate complexity which can be fitted reliably from limited data. This is addressed in
the work reported here.
3. Parametric modelling
The key component behaviour to capture in a compression-cycle heat pump is that of the
compressor. Performances of scroll compressors expressed in terms of isentropic efficiency,
ηi, versus compression ratio, Rp, are shown in Fig. 1 for 3 alternative compressor
manufacturers (denoted here as ‘A’, ‘B’ and ‘C’ to retain commercial anonymity), two
alternative working fluids (zeotrope R407C and near-azeotrope R410A) and condensing
temperatures of 50–55oC. The characteristics show a rise component and a decay component
separated by a peak efficiency which occurs at low compression ratio. Exponential growth
and decay functions are found to describe this behaviour very well indeed with just 4
parameters. The resulting model is given by Eq. (1) in which a…d form fitting parameters:
i 
  
1  exp  d  Rp  c  
a exp b Rp  c
Results of the fitted models are superimposed as solid lines in Fig. 1 (the manufacturers’ data
points are shown as crosses). An excellent fit is achieved with a goodness of fit statistic of
nearly 1 in all cases. This model has the advantages of requiring far fewer fitting coefficients
than other models reported in the literature (i.e. [14,17]), as well as leading to a definitive
compressor performance variable – the isentropic efficiency – which can be directly
compared with alternative compressor types.
Fig. 1. Efficiency performances of 3 alternative scroll compressors
(Legend: solid lines – fitted models, crosses – manufacturers’ data points)
The efficiency can be used in conjunction with the classical analytical model of a
compressor, Eq. (2):
 1
mr Pe   Pc  
Wi 
   1
suc   1   Pe 
The mass flow rate delivered by the compressor is obtained from the volumetric efficiency,
Eq. (3):
v 
A reasonable assumption for the variation in volumetric efficiency is a linear function of
compression ratio, Eq. (4), [19]:
v  e  fRp
(in which e and f are fitting coefficients).
The overall heat pump electrical power is then obtained from Eq. (5):
W  i  Waux
Waux in Eq. (5) is the auxillary power consumed by the heat pump and is assumed to be
constant. For an air-source heat pump this will mainly consist of the power drawn by the
source air fan but will also include minor electrical losses associated with the power
connections, wiring loom and control circuits.
Note that at:
 0 , the compression ratio at the peak isentropic efficiency can be located.
It is possible to show that this can be expressed by Eq. (6):
max(i )
cd  ln adab
Therefore the maximum isentropic efficiency can be obtained by applying the value for Rp
from Eq. (6) in Eq. (1).
The electrical load consumed by the heat pump can be fully described by 8 model
parameters: a…f, Vd and Waux. Note that Vd is the leading parameter for heat pump capacity.
The evaporating and condensing temperatures are obtained with reference to a heat exchange
thermal approach or ‘pinch’. In its simplest terms, the pinch point of a heat exchanger is the
difference in temperature between the hot fluid and cold fluid where it is a minimum (i.e.
usually at an inlet/discharge connection). For the condenser this will be the difference in
temperature between the condensing temperature and the water outlet temperature (i.e. the
heating water flow to the heating system which will usually be specified as the controlled
heating temperature value). For the evaporator this will be the difference between the source
air outlet temperature and the evaporating temperature but, for convenience in the present
work, the difference between the source air inlet temperature (the ambient air temperature)
and the evaporating temperature will be used. Thus the evaporator pinch will include the
temperature drop experienced by the air (typically 3-5K for domestic heat pumps). These
pinch temperature differentials form two further model parameters, Tc and Te which are
used to obtain the evaporating and condensing temperatures as follows:
Tc  Tcwo  Tc
Te  Tao  Te
Note that, in this work, refrigerant pressure losses, evaporator superheating and condenser
sub-cooling are neglected. It would be an easy matter to include these extra variables but they
would each involve one further fitting parameter for what would likely be only a small
improvement in model accuracy.
The condensing and evaporating pressures may now be obtained from refrigerant properties:
Pc,e  f Tc,e 
Remaining properties including the refrigerant enthalpy at condenser outlet (hence evaporator
inlet), the refrigerant enthalpy at evaporator outlet and the refrigerant suction gas density are
similarly obtained from refrigerant properties:
H cro,ero  f  Pc,e 
suc  f  Pe 
The enthalpy at compressor discharge (i.e. condenser inlet) is now obtained from Eq. 12:
H cri  H ero 
W  Waux
The refrigerant properties were obtained using NIST REFPROP 8 [20] by direct interpolation
as described in [21]. The target heat pump heat output (condenser load) can now be found:
Qc  mr  H rci  H rco 
Finally, the performance of the heat pump for a defined set of boundary conditions is the
coefficient of performance, CoP. For performance over the extended time horizon of a
heating season, the seasonal coefficient of performance, SCoP, is applicable and the
corresponding seasonal performance based on observations from field or test data is the
seasonal performance factor, SPF. In this work, only the essential auxillary power usages are
included in these performance metrics (i.e. the source air fan and power lost in wiring and
control circuits within the heat pump itself). The heating water pump and any top-up heating
are excluded. This means that the results presented here refer to the heat pump only. The
seasonal performance factor so-defined is sometimes referred to as SPFH2 (further details of
the various definitions and performance boundaries for heat pumps can be found, for
example, in [4]). The coefficient of performance and seasonal coefficient of performance are
defined as follows (Eq. (14) and Eq. (15)):
CoP  c
SCoP 
Qc 
4. Model fitting
From the previous Section, a heat pump model can be defined from two inputs, I. Ten model
fitting parameters, P, require to be found, and there are two target outputs variables, O.
I  Tao Tcwo 
P  a b c d
Vd Waux
O  Qc W 
Te 
Model fitting was initially carried out using two data sets: a field monitoring set and a
laboratory data set. This would then enable comparisons to be made regarding a heat pump
in-service with ‘real-world’ social user interventions and a heat pump operating under
defined conditions free from social interaction.
The field monitoring data set was obtained from an air-source heat pump installation in a
1970s mid-terrace house in Leeds, UK. The house is occupied by a family of 5 and has 3 bed
rooms and has been upgraded with cavity wall insulation, roof insulation, double-glazed
UPVC windows). New heating radiators were installed with the heat pump and sized to
operate at lower circulating temperatures (45oC). The heat pump also connects to a baseload
coil in a hot water tank (which has a top-up electric immersion heater).
The laboratory air-source heat pump is also of domestic-scale and though larger, has many
technical details in common with the field installation. It is connected to 4 low-grade heating
radiators (sized to operate nominally at 40oC) which are used to heat the local laboratory
Details of the two heat pumps are given in Table 1.
Table 1
Details of the test heat pumps
Field installation
Laboratory installation
Rated power (A7-W35)
Fixed-speed scroll
Fixed-speed scroll
Main control
Heating water thermostat
Heating water thermostat
Brazed plate
Brazed plate
Fin and coil
Fin and coil
Expansion device
Mechanical TEV
Mechanical TEV
Defrosting method
Reverse cycle
Reverse cycle
Rated heat (A7-W35)
* Rated according to standard BS EN 14511-2:2004 at an ambient (source) air temperature of
7oC and a condenser water outlet (heating water supply) temperature of 35oC.
Data capture from the two sites consisted of heat meters (optionally reporting individual
temperatures and flow rate), current and voltage transducers. A separate measurement of
power factor at the laboratory site was found to be 0.924 and no such measurement was made
at the field site and so a value of 0.95 was assumed. Data was integrated (or averaged from
more frequent sampling as appropriate) and reported at 5-min intervals for the field test site
and at 3s intervals for the laboratory site. (This meant that full defrost transients could be
constructed from the latter.) The data set from the field test site was recorded over the main
part of one heating season (not the complete heating season) starting in November 2009 and
ending in April 2010. For the laboratory site, numerous shorter files of data were obtained
and combined.
Checks with instrument manufacturers used at both sites confirmed maximum measurement
uncertainties of  0.2K (temperature),  3% of reading for both current and voltage and  2%
of reading for flow rate. A calculation of equal-odds compound errors with respect to average
condenser heat output rates was found to be  6.6% for the field test installation and  12.2%
for the laboratory installation. For power, average compound errors were found to be 
4.26% (field site) and  4.22% (laboratory site). The corresponding compound uncertainties
with respect to the average CoP values were  7.8% (field site) and  12.9% (laboratory site).
The relatively high compound uncertainty for the heat rate at the laboratory site was due to
operating this plant with a relatively high flow rate and low heating temperature differential
(usually < 4K) which magnified the impact of the relatively high uncertainties in temperature
measurement in the final heat values. Full details of the uncertainty analysis procedure can be
found in the Appendix.
Raw results were first processed using the following procedure:
The data from each site were collected into one file (initially as time-series data)
and all random spikes and null reporting rows (i.e. when the heat pump was off)
were removed (see Note 1, below)
All rows containing data during defrost events (evidenced by abnormally low and
negative heat rates during phases of steady operation and during low ambient
temperatures) were removed (separate files containing these data were retained for
later use – see Section 5)
The data were formed into a matrix of n rows and 4 columns – the first two
columns containing the input data (Tao, Tcwo) and the other two rows containing the
target values (Qc, W)
The data were ranked with respect to the Tao values and then averaged using a
moving-average resulting in a smaller data set with a reasonable spread (and
increment) in Tao values (see Note 2, below)
Note 1:
When the heat pump switches on or off during a logging interval, unrealistic results (mainly
power) arise because the variables are evolving from zero within the sampling time interval.
Including these results would lead to distortions in the data as far as steady-state heat pump
performance is concerned. Therefore these initial switch-on and switch-off time rows of data
were discarded.
Note 2:
The purpose of this work was to develop a steady-state heat pump model responsive to
variations in boundary conditions but giving results that are essentially static with respect to
time. Because the data available from the two data logging systems were essentially dynamic
(i.e. logged at either 3s time intervals in the case of the laboratory heat pump and reported at
5min intervals in the case of the field trial heat pump) the data were averaged across wider
time increments so as to eliminate transient noise such as the response to load disturbances.
The sampling interval for averaging was found to have little effect on results for intervals
greater than several minutes which suggested that this approach for steady-state modelling
was sound. This was confirmed by the quality of results obtained by model-fitting to just one
set of data – that of manufacturers’ rated performance data which will be evident later.
A multiple objective function optimisation algorithm was used to determine the 10 parameter
values of the heat pump model stated formally as follows:
C  Goal
min C subject to:  P  P
 PP
The cost function, C, was formed from the root-mean-square error values between measured
(target) and predicted values of the condenser heat output and heat pump power consumption.
The Goal vector was set at [0.05 0.1] equivalent to less than 3% of the rated heat and power
values of the smaller heat pump (in practice much smaller values were achievable). Initial
estimates of the parameter set, P, were taken from the average values of the manufacturers’
compressor models fitted to data in Fig. 1, from typical values in [19] for the compressor
volumetric efficiency model parameters, and deduced from the manufacturers’ data on the
two heat pumps for the remaining parameters. The lower and upper bound parameter sets, Pand P+, were initially set at 10% below and above the nominal parameter values respectively.
These margins were then increased until solutions were returned within (rather than at) the
bounded values. (The Matlab optimization function ‘fgoalattain’ was used for the above.)
It was found that a very good model could be fitted simply with reference to a single row of
inputs and target values. The single row of data for each heat pump was obtained by
averaging all of the data in the originally processed set of data. The originally processed set
of data was then used as model validation data. Note that the randomness caused by defrost
events was dealt with by removing all data rows containing defrost transients so that the
model fitting was carried out using defrost-free data. The data including defrost was however
retained for later use (the treatment of defrosting in the heat pump model is dealt with in
Section 5).
The averaged model fitting rows of data for the two heat pumps were as follows:
Field data site:
I = [Tao Tcwo] = [3.10 44.90]; O = [Qc W] = [4.90 2.30]
Laboratory data site: I = [Tao Tcwo] = [5.93 42.38]; O = [Qc W] = [7.74 3.36]
The fitted model parameters for the two heat pumps are given in Table 2.
Finally, because it was found to be possible to fit an adequate model to a single row of
averaged input and target data, it seemed appropriate to consider a similar parameter fitting
exercise but this time using the manufacturer’s standard rating data (see Table 1). This would
then lead to two alternative heat pump models for each site of test data – a ‘most optimistic’
or ‘standard’ model based on manufacturer’s data and an in-service model based on actual
From Table 1, the standard rating data (according to standard BS EN 14511-2:2004) for the
two heat pumps gives the following alternative rows of model-fitting data:
Field data site:
I = [Tao Tcwo] = [7 35]; O = [Qc W] = [5.92 1.64]
Laboratory data site: I = [Tao Tcwo] = [7 35]; O = [Qc W] = [10.20 2.83]
Again, the fitted model parameters for the two heat pumps are given in Table 2.
Table 2
Fitted parameter results for the heat pump models
0.739 0.179 1.655 8.400 1.090
0.359 0.0012
0.771 0.149 1.703 8.400 1.095
0.300 0.0013
0.660 0.177 1.153 8.400 0.915
0.359 0.0014
0.737 0.148 1.346 8.400 1.099
0.272 0.0015
* F-test = field test data; F-mnf = manufacturer’s rated test data (field test heat pump); L-test = laboratory test
data; L-mnf = manufacturer’s rated test data (laboratory heat pump)
5. Incorporating defrost
The initial model fitting was carried out using average seasonal performance data excluding
periods of defrost operation because the latter contains random disturbances in normal heat
pump operation and resulted in poor quality model fitting. This requires a model of defrosting
to be separately developed and included. A typical energy transient during reverse-cycle
defrosting was captured from the laboratory heat pump and is plotted in Fig. 2.
Fig. 2. Transient energy transfers during defrosting (laboratory heat pump)
Each defrost cycle lasts typically for about 5 minutes including periods of inactivity as the
changeover valve repositions and a period during which the compressor restarts and the
heating delivered drops to a negative value (i.e. heat is sourced from the heating system to
effect defrosting in the evaporator).
These defrost events were identified in the original test data sets and two data sets for each
heat pump were generated – one including defrost event data and the other in which defrost
event data were stripped out prior to averaging. Initial model fitting was carried out using the
latter whilst the former were retained for final model testing. Fig. 3 shows the final data sets
with and without defrosting. As implied by Fig. 2, Fig. 3 shows that defrosting results in a
significant fall in the average heat delivered by the heat pump with only a very small fall in
average power consumption. Also, these changes take effect at ambient air temperature below
approximately 7oC. Note that these findings are similar to those reported in [14].
Fig. 3. Heat output and power consumption with and without defrost event data
Therefore a simple and sufficient model of defrosting may be envisaged by discounting the
average heat output from the heat pump at low ambient temperatures whilst leaving the
average power consumption unchanged. A defrost discount factor, Fdf, is proposed in this
work which is defined as the average heat delivered including periods of defrosting divided
by the average heat delivered without. Results of this from both measured data sets were
found to be very similar and were therefore combined. They are plotted in Fig. 4. The
compound uncertainty in Fdf assuming average heat with and without defrost to be uncertain
with equal odds was estimated to be 15.6%. The defrost discount factor is noted to fall
reasonably sharply as ambient temperature fall below approximately 7oC and then levels off
at temperature below approximately 0oC. A simple model through the data therefore
comprises a linear region and a constant region. In this simple model, the impact of wet bulb
temperature on defrost intensity and frequency has not been considered. Indeed Kinab et al.
[14] reviewed two similar defrosting models one of which is based on ambient dry bulb
temperature and the other on ambient wet bulb temperature. They concluded that the two
provided similar results regarding average corrections to heating capacity during low ambient
temperatures when defrosting would occur. This seems reasonable since, at the low
temperatures involved, winter air states tend to be close to saturation at most times and this
means that dry bulb or wet bulb temperature may therefore be used alternatively to some
extent. In common with the work reported here, Kinab et al. [14] adopted the model based on
ambient dry bulb temperature which, whilst applied to CoP rather than Qc (as in the present
work), indicates results that are broadly in agreement with those reported here.
The proposed model is included in Fig. 4.
The model of defrost is applied by adjusting eqs. (13, 14, 15) as follows:
SCoP 
Qc,df  Fdf Qc
CoP  c,df
Qc,df 
Fig. 4. Proposed defrost discounting model fitted to measured data
6. Model testing and evaluation
The two models with parameters listed in Table 2 have been tested using the original test data
including defrost operating data. Also, included for comparison, are the results of ‘most
optimistic’ models of each heat pump using the parameters derived from the manufacturer’s
standard rating data for each heat pump (also listed in Table 2). The results are given in
figures 5-7. Results are plotted against two alternative independent variables – the external air
temperature and the overall plant temperature lift (i.e. the difference between the heating
water supply temperature and the source air temperature).
Fig. 5 shows results of the two models compared with test data for heat output (Qc, kW). The
models fitted to test data predict the observed data very well in the case of the laboratory heat
pump and very well at lower external air temperatures (higher temperature lift) but not so
well at higher external air temperatures (lower temperature lift) for the field installation heat
pump. However, in the latter case there are insufficient data points in these regions to be
conclusive. The average root-mean-square error is 0.30kW (field heat pump) and 0.58kW
(laboratory heat pump) which is less than 8% of the mean observed heat output of both heat
pumps. In both cases, the manufacturer’s rated model predicts that slightly higher heat
outputs should be obtained from the field heat pump and significantly higher heat outputs
from the laboratory heat pump.
Fig. 6 gives results for power consumed (W, kW). The manufacturers’ rated models predict
power much closer to the models fitted using observed data. Again the laboratory heat pump
results agree generally well but the field heat pump models are tending to under-predict
power at high ambient temperatures (low temperature lift) though, again, there are
insufficient data points in this region. It is most likely that any divergence here is due to the
heat pump operating at persistent light load – the transient effects of this are not included in
the present model structure which forms an avenue of further work. For power, the average
RMS errors between the models fitted using average observed data and the observed data
including defrost is 0.063kW (field heat pump) and 0.27kW (laboratory heat pump) –
equivalent to approximately 3% and 8% of mean observed values respectively.
Fig. 7 shows the coefficient of performance predictions and observations. Again both models
perform best at low external temperatures (high temperature lift) though there are limited data
points where the models appear to perform less well. In all cases, models fitted to
manufacturers’ standard ratings suggest that higher performances (but not dramatically
higher) should have been expected from these heat pumps.
Fig. 5. Model predictions compared with test data – heat output
Fig. 6. Model predictions compared with test data – power
Fig. 7. Model predictions compared with test data – CoP
Fig. 8 shows the isentropic efficiencies of both heat pump compressors plotted against
temperature lift. Using Eq. (6) and Eq. (1) together with the parameters of Table 2, it is
possible to deduce the peak isentropic efficiencies based on the experimentally-fitted
compressor modelling parameters. These are 0.67 for the field heat pump and 0.60 for the
laboratory heat pump. However these results occur at low compression ratios at which the
low heating temperature values are likely to be impractical. Nonetheless, Fig. 8 confirms that
both compressors are always operating at compression ratios higher than those corresponding
to the peak efficiencies as shown in the typical compressor performance curves of Fig. 1.
This is an area that requires to be addressed in order to get the very best out of these small
Fig. 8. Compressor isentropic efficiency
7. Application and verification
In UK practice air-source heat pumps are usually assessed in conjunction with the
government’s ‘standard assessment procedure’ – SAP [22]. This can be determined from
results generated using the reference simulation model ‘BREDEM’ [23]. Currently,
BREDEM uses a constant effective mean SCoP value for domestic air-source heat pumps of
2.5. In fact, this has been verified by recent UK field trials of domestic air-source (and other)
heat pumps with average results of 2.45 [2] and 2.56 [3]. As a final analysis, the two fitted
heat pump models in the present work were applied to a range of heating system operating
temperatures and to the monthly average external air temperatures for the ‘Thames’ region
used in BREDEM-12 [23]. The monthly results were then further averaged to form annual
SCoP values for both heat pumps combined. Results plotted against heating system
temperature are shown in Fig. 9. The ‘most optimistic’ results here are those based on models
fitted to manufacturers’ standard rating data whereas the ‘typical’ results are those based on
models fitted to the test data used in the present work. The results confirm the recent UK field
trial averages for heating system temperatures up to 45oC [2,3] and also reveal a moderate
performance gap with respect to typical values given in manufacturers’ technical literature.
To achieve the much higher SCoP values of 3 or more (for example, reported in [4] based on
field trials in other European countries) will require consistently lower heating system
operating temperatures then we are accustomed to using in the UK.
Fig. 9. Overall results compared with alternative field data
8. Conclusions
A new domestic air-source heat pump model has been reported in this work. The model can
be parameterised from either field data or manufacturer’s standard rating data and differs
from most other regression-type heat pump models in that key heat pump performance
parameters such as compressor isentropic efficiency and heat exchange pinch are deduced
which enables comparisons between alternative systems to be made.
The model can be parameterised from a small amount of data – mean source air temperature,
mean heating system temperature and refrigerant type as inputs; corresponding mean heat
output and power consumption as target data. Thus it is possible to parameterise the model
from either average seasonal field trial results or from standard rating data with the advantage
that it is possible to consider both typical (field data) and most optimistic (rating data)
performance scenarios from which considerations into building ‘performance gap’ might be
The model includes a new type of fitted compressor model which leads directly to the
compressor isentropic efficiency. Results presented here confirm the relatively low efficiency
at which these small compressors operate at in the field compared with larger commercial
plant. A model which includes an allowance for defrosting is incorporated which ensures that
it tracks performance at low external air temperatures with good accuracy.
Predictive results of heat output and power consumption by the model give root-mean-square
errors over test data that are generally significantly less than 10% of the mean test data
values. The model has also been verified to provide results that are in broad agreement with
recent field monitoring results from UK domestic installations. The model was noted to be
less able to predict performance at higher external air temperatures most likely due to
transient part-load performance behaviour at these conditions. However this is more strongly
evident in the case of the field test heat pump than in the case of the laboratory heat pump.
Because the model reported here is essentially a steady-state model, it will tend to predict
increasing heating capacity as the source temperature (and, hence, suction pressure) increases
since a greater refrigerant flow rate will be delivered by the compressor. In practice, the heat
pump will tend to operate intermittently in these conditions and this will result in a reduction
in performance. Remedies for this include thermal buffering (i.e. heat storage in series with
the heat pump) and continuous operation with a variable speed compressor drive. None of
these details have been included in the model reported here but they have been reported
extensively elsewhere (e.g. [21]). In summary, the model reported here will be most suited to
application in which the heat pump capacity is well matched to the heating demand
throughout the operating cycle.
The overall advantages of a parametric model of the kind reported here are twofold: It can be
fitted to limited data sets; and it can be used to reveal detailed parameters on the refrigerant
side of the plant such as compressor efficiency and heat exchange thermal approaches. There
are three specific areas in which the reported model has advantages over the more commonly
reported boundary variable type model:
a) It may be fitted to manufacturer’s rated data according to BS EN 14511:2:2004 to
describe optimum performance over the heating season which can then be crossmatched to actual performance of the heat pump in the field in order to help identify
potential areas of improvement.
b) It may be used where field monitoring data are available but incomplete. In these
circumstances, the model may be used to fill in gaps of missing performance data in
order to inform longer term performance issues and running costs (e.g. in relation to
Renewable Heat Incentive payments).
c) It can be used to evaluate alternative compressor and heat exchanger performances
and characteristics.
Avenues for further work are as follows:
Design and introduce one or more additional parameters to enable the model to
fully describe part load performance (which tends to occur when the heat pump is
enjoying higher source temperatures – the effects are, to some extent, cancelling)
Further testing and evaluation using some of the new field data which is starting to
become available (e.g. [3]) and the development of strategies to troubleshoot
design and installation flaws at specific sites
Further analysis of the defrost model to consider the impact of other influencing
variables such as ambient humidity
Extension of the heat pump model to include other system components such as
thermal storage and top-up heating
Development of a variable speed compressor/fan model
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Calculation of uncertainties
The measured results from both heat pumps averaged over all data are given in Table A1.
Table A1
Average value (field trial)
Heating mass flow rate
Heating flow temperature
Heating return temperature
Power supply voltage
Power supply current
Power factor
Average value (lab rig)
The compound uncertainty in heat output, uQ, from each heat pump arising from individual
uncertainties in the two temperatures, T1, T2, and the mass flow rate, m, can be estimated
from the following:
uQ  
 
q 2
um m
 uT1 T
 
 uT2 T
From the data in Table A1 for the field trial heat pump:
 18.855kJkg-1;
 0.93kWK-1 and:
 0.930kWK-1
Applying Eq. (A1) leads to a compound uncertainty of 0.276kW with respect to the mean
heat measurement (6.6% of the mean measured value).
The same calculation for the laboratory heat pump gave a result of 0.90kW (12.2% of the
mean measured value).
Using a similar approach, compound uncertainties for power were found to be 0.096kW
(field heat pump) and 0.14kW (laboratory heat pump) corresponding to 4.26% and 4.22%
of the mean measured values respectively.
Compound uncertainties in heat pump CoP are estimated using the mean heat and power
uncertainty values calculated above using the following:
uCoP  
uQ CoP
  u
CoP 2
W W
 Q 1
 Q
 
  2
Q  W  W
W  W 
For the field trial heat pump, the compound uncertainty in CoP based on measured mean heat
and power values of 4.186kW and 2.251kW respectively will be:
uCoP  
0.276 2   0.0964.186  0.146
This is 7.8% of the mean measured CoP value for this heat pump. The same calculation
using the data for the laboratory heat pump gives a result of 0.288 which is 12.9% of the
mean measured CoP.