# Importance of the reconciliation method to handle experimental data in refrigeration and power cycle: application to a reversible heat pump/organic Rankine cycle unit integrated in a positive energy building

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## Abstract

Experimental data is often the result of long and costly experimentations. Many times, measurements are used directly without (or with few) analysis and treatment. This paper, therefore, presents a detailed methodology to use steady-state measurements efficiently in the analysis of a thermodynamic cycle. The reconciliation method allows to correct each measurement as little as possible, taking its accuracy into account, to satisfy all constraints and to evaluate the most probable physical state. The reconciliation method should be used for multiple reasons. First, this method allows to close energy and mass balances exactly, which is needed for predictive models. Also, it allows determining some unknowns that are not measured or that cannot be measured precisely. Furthermore, it fully exploits the collected measurements with redundancy and it allows to know which sensor should be checked or replaced if necessary. An application of this method is presented in the case of a reversible HP/ORC unit. This unit is a modified heat pump which is able to work as an organic Rankine cycle by reversing its cycle. Combined with a passive house comprising a solar roof and a ground heat exchanger, it allows to get a positive energy building. In this study case, the oil mass fraction is not measured despite its strong influence on the results. The reconciliation method allows to evaluate it. The efficiency of this method is proven by comparing the error on the outputs of steady-state models of compressor and exchangers. An example is given with the prediction of the pinch-point of an evaporator. In this case, the normalized root mean square deviation (NRMSD) is decreased from 14.3 to 4.1 % when using the reconciliation method. This paper proves that the efficiency of the method and also that the method should be considered more often when dealing with experimentation.

## Keywords

Reconciliation method Experimental analysis Reversible heat pump/organic Rankine cycle## Abbreviations

## Nomenclature

*A*Expander exchange area, m

^{2}*c*Reconciled variable

*C*Specific heat capacity, J/(K.kg)

*h*Specific enthalpy, J/(kg)

*m*Number of constraints

- \(\dot{m}\)
Mass flow rate, kg/s

*n*Number of measured variables

*NRMSD*Normalized root mean square deviation

*P*Pressure, bar

- \(\dot{Q}\)
Heat flow rate, W

*t*Temperature, °C

*u*Measured variable

*U*Expander heat exchange coefficient, W/(m

^{2}K)*w*Weight function

- \(\dot{W}\)
Power, W

*x*Fraction

*z*Unmeasured variable

## Greek symbols

- α
Lagrange function

- γ
Redundancy level

- Δ
Difference

- λ
Lagrange multiplier

*φ*Minimization function

*σ*Standard deviation

*ρ*Density, kg/m

^{3}

## Subscripts and superscripts

- amb
Ambient

- cd
Condenser

- el
Electrical

- ex
Exhaust

- exp
Expander

- ev
Evaporator

*i*Index of the measured/reconciled variable

*j*Index of the constraint

*k*Index of independent variables

*m*Mean

- min
Minimum

- max
Maximum

- meas
Measured

- min
Minimum

- oil
Oil

- pred
Predicted

*p*Constant pressure

*r*Refrigerant

*s*Index of measurement

- su
Supply

*w*Water

## Introduction

Numerical values are always affected by random errors plus gross errors (error that cannot be explained with statistical distribution). Gross errors are outliers (process leaks and malfunction) or bias (systematical offset). This paper presents the application of a mathematical tool, called the reconciliation method (RM). The latter is recommended to obtain reliable information about the studied process but gross errors have to be identified and eliminated before the procedure. This technique is used since 1961 in chemical engineering [1]. In 1980, the reconciliation method was applied to adjust material balancing of mineral processed data [2]. Later, Weiss and Romagnoli used this tool to better determine the regeneration cycle time of a reactor in an industrial case study [3]. Heyen and Kalitvebtzeff developed a RM optimization to reduce energy use in production plants [4]. Placido and Loureiro study the placement of new instruments to improve the estimation accuracy in ammonia plant units [5]. Schladt and Hu developed a rigorous model to estimate concentrations in a distillation column trough the reconciliation method [6]. In 2008, Lid and Skogestad [7] used the RM method to assess the optimal operation of a catalytic naphtha reformer. Despite the proven performance of the method, few authors use it in refrigeration systems. In 2007, Bruno et al. applied the method to a hybrid solar/gas single/double effect absorption chiller [8]. In 2013, Martinez-Maradiaga et al. used the method for absorption refrigeration system to obtain performance calculations that are in agreement with the laws of conservation [9]. In 2015, an optimization of redundant measurements location for thermal capacity of power unit steam boiler using data reconciliation method is performed [10]. Finally, a data reconciliation based framework for integrated sensor and equipment performance monitoring in power plants is provided by [11].

Some authors predict unmeasured values (flowrate, oil fraction…) simply by minimizing the sum of the residue of each component [12]. A more complete and accurate method taking into account measurements’ redundancy and accuracy of sensors exists: the reconciliation method corrects each measurement as little as possible, taking its precision into account (assuming a Gaussian distribution around the measured value), to satisfy all constraints and to evaluate the most probable physical state [9]. Redundancy is obtained by having two sensors measuring the same variable and/or variable that can be obtained through balance equations (heat balance, residue, mass balance, thermodynamic state of equilibrium…). This redundancy allows correcting measurements while non-redundant measurements will remain untouched. The RM method does not correct data to better fit a model but simply imposes constrains (physical laws) to improve the dataset intrinsic quality.

Reconciliation method should be used for multiple reasons. First, without this method, it is impossible to close energy and mass balances exactly, which is needed for predictive models. Also, it allows determining some unknowns that are not or that cannot be measured precisely (oil fraction, refrigerant mass flow rate…). Moreover, it fully exploits the collected measurements with redundancy. Finally, it allows to know which sensor should be checked or replaced if necessary.

*c*

_{ i }) (and eventually additional unknowns) based on the measured values (

*u*

_{ i }) and on their standard deviation (

*σ*

_{ i }) with regard to a certain number of constraints (

*ψ*) by minimizing (2) with Lagrange formalism (

*λ*is a Lagrange multiplier).

### Validation of reconciliation

Data reconciliation is based on two main assumptions. On the one hand, most influent physical phenomena should be correctly described. The first assumption is reached using the validation of measurements. The validation of measurements is achieved by checking heat balances on exchangers, on compressors and on expanders, cross-checking of pressures…

On the other hand, it assumes a Gaussian distribution of the errors. This needs to eliminate gross error (outliers). In this paper, a Kriging method (or Gaussian process regression) is used in this aim [13]. Other advanced methods exist to treat gross error in data reconciliation: Fair, Welsch, Hampel, Cauchy, logistic, Lorentzian, and Quasi-Weighted Least Square, for example [14, 15, 16].

*w*

_{ i }) should be evaluated (3) to give the confidence level of the correction.

*γ*, the degree of freedom. The degree of freedom is equal to the number of reconciled variables minus the number of constraints (=the redundancy level). For example, the confidence level of the RM with a redundancy level of 5, a weight of 1.145 and 21 measured variables is 95 %.

### Global methodology

## Description of the study case

The reconciliation method is applied in the case of a reversible heat pump/organic Rankine cycle (HP/ORC) unit. This unit is a modified heat pump that is able to work as an ORC by reversing its cycle. The test bench is fully described with all its components and sensors by Dumont et al. [17, 18].

## Application of the method to the case study

### Model: assumptions and constraints

*T*

_{amb}) and heat transfer coefficient between the expander and the ambient (

*U*)] because they play an important role in the heat balance of the expander.

### Optimization function, derivatives and redundancy level

In this case study, the number of unknowns is equal to 29: there are 21 measurements to reconcile, plus two additional variables (expander heat transfer coefficient and ambient temperature), plus the oil fraction and 5 Lagrange multipliers (2). 29 equations are needed: the 5 physical constraints (4, 5, 7–9), 23 equations resulting from the partial derivatives regarding each reconciled variable to minimize (2) and 1 additional coming from the minimisation of (1). The solution is computed with EES solver coupled with the Coolprop library [19] (which allows to evaluate derivatives of thermodynamic properties). The redundancy level is simply equal to the number of constraints in this case, 6.

### Methodology

- 1.
The corrected values are imposed to be equal to measurements (guess value).

- 2.
The standard deviation for each sensor is computed following sensor datasheet.

- 3.
The weights are evaluated through Eq. 3.

- 4.
The confidence in the reconciliation method is evaluated (see “Validation of reconciliation”).

- 5.
Physical constraints are imposed and partial derivatives of Eq. 2 are computed and imposed equal to zero. At this step, guess values for corrected values have to be removed.

- 6.
Finally, the minimization of Eq. 1 allows to evaluate unmeasured variable(s).

## Results

### Reconciliation method

*φ*(Eq. 1), that should follow a Chi-square distribution [4]. In this case, the confidence of the reconciliation method reached a probability of 73 %.

Results from reconciliation method for one measurement point. Each measurement is detailed in Fig. 2

Measurement | Std. deviation | Original value | Reconciled value | Weight | Confidence |
---|---|---|---|---|---|

T1 (°C) | 0.5 | 16.02 | 16.02 | 0 | 1 |

T2 (°C) | 0.5 | 17.14 | 17.11 | 0.0549 | 1 |

T3 (°C) | 0.5 | 99.3 | 99.31 | 0.0265 | 1 |

T4 (°C) | 0.5 | 98.5 | 98.38 | 0.2181 | 0.9998 |

T5 (°C) | 0.5 | 63.14 | 63.11 | 0.0634 | 1 |

T6 (°C) | 0.5 | 34.53 | 34.65 | 0.2406 | 0.9997 |

T7 (°C) | 0.5 | 11.51 | 11.24 | 0.5438 | 0.9973 |

T8 (°C) | 0.5 | 31.54 | 31.82 | 0.5438 | 0.9973 |

T9 (°C) | 0.5 | 105 | 104.9 | 0.1635 | 0.9999 |

T10 (°C) | 0.5 | 83.68 | 83.76 | 0.1635 | 0.9999 |

T11 (°C) | 10 | 20 | 19.64 | 0.0070 | 1 |

P1 (bar) | 0.0625 | 8.325 | 8.325 | 0.01386 | 1 |

P2 (bar) | 0.1 | 28.45 | 28.56 | 1.075 | 0.9826 |

P3 (bar) | 0.06 | 28.59 | 28.56 | 0.6045 | 0.9963 |

P4 (bar) | 0.0625 | 8.608 | 8.599 | 0.1411 | 0.9999 |

DP1 (bar) | 0.0012 | 0.2738 | 0.2738 | 0 | 1 |

DP2 (bar) | 0.00075 | 0.06781 | 0.06781 | 0 | 1 |

M1 (g/s) | 0.000235 | 0.235 | 0.235 | 0.0158 | 1 |

M2 (l/s) | 0.02984 | 0.5968 | 0.5863 | 0.3521 | 0.9992 |

M3 (l/s) | 0.02495 | 0.499 | 0.5255 | 1.062 | 0.9831 |

W1 (W) | 0.25 | 2630 | 2630 | 0.0872 | 1 |

U [W/(m | 2 | 10 | 10.02 | 0.00863 | 1 |

ρ1 (kg/m | 24.25 | 1209 | 1213 | 0.4359 | 0.9985 |

### Improvement in the validation of semi-empirical models

*x*

_{pred}corresponds to the prediction of the model and

*x*

_{meas}corresponds to the measurement value (or to the reconciled value in the case of the RM).

Comparison of the normalized root mean square deviation of the prediction of semi-empirical models with and without the reconciliation method

Model | NRMSD without RM (%) | NRMSD with RM (%) | Improvement (%) |
---|---|---|---|

Evaporator pinch point | 14.3 | 4.1 | 10.2 |

Condenser pinch point | 10.1 | 8.7 | 1.4 |

Expander electrical power | 5.8 | 4.4 | 1.4 |

## Conclusion

Experimental data are often the result of long and costly experimentations. Many times, measurements are used directly without (or with few) analysis and treatment.

This paper presents a simple mathematical tool to threat and enhance the quality of measured data. This reconciliation method is described and a global methodology including validation of measurement, elimination of irrelevant points and validation of the reconciliation method is proposed.

The efficiency of the global methodology is proven with experimental data of a reversible HP/ORC unit. The normal root mean square deviation on model predictions is significantly lower when using reconciled values for model calibration. This proves the validity of the method.

The presented methodology is simple and fast to perform. More advanced methodologies exist but are more complex and require more computational time [12, 13]. Moreover, advanced physical phenomena such as oil solubility could be taken into account for more accurate results.

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