# Neural-Accelerated Dynamic modeling of heat pumps

Shri Balaji Padmanabhan, Mohamed Tahar Mabrouk, Bruno Lacarrière

## ► To cite this version:

Shri Balaji Padmanabhan, Mohamed Tahar Mabrouk, Bruno Lacarrière. Neural-Accelerated Dynamic modeling of heat pumps. Applied Thermal Engineering, 2025, 274, pp.126653. 10.1016/j.applthermaleng.2025.126653. hal-05086722

HAL Id: hal-05086722

https://hal.science/hal-05086722v1

Submitted on 27 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

Research paper

# Neural-Accelerated Dynamic modeling of heat pumps

Shri Balaji Padmanabhan *, Mohamed Tahar Mabrouk ID *, Bruno Lacarrière ID

IMT Atlantique, Department of Energy Systems and Environment, GEPEA UMR CNRS, 6144, F-44307, Nantes, France

## ARTICLE INFO

**Keywords:**

Heat exchanger

Heat pump

Hybrid modeling

Numerical modeling

Machine learning

## ABSTRACT

In response to escalating climate challenges and the transition to renewable energy, integrating heat pumps with other energy systems has gained significant attention. Fast and accurate dynamic models for heat pump are essential in predicting its transient thermal behavior and simulating its interactions with other energy systems, and effectively controlling and optimizing its performance in real time. Among the heat pump's components, heat exchangers involve complex heat transfer phenomenon and contributes to majority of the dynamics of the heat pump. Among the common numerical approaches used in the dynamic modeling of heat exchangers, the finite-volume method is well-recognized for its robustness and high accuracy. However, it is often outpaced in terms of computational efficiency compared to other methods. This paper introduces novel Neural-Accelerated Dynamic (NAD) models for heat pump's heat exchangers, integrating the strengths of numerical modeling and machine learning. The NAD model's architecture comprises of simplified heat exchanger model and a deep neural network, which are dynamically strongly coupled, achieving a computationally efficient model while maintaining high precision. Furthermore, in validation, the NAD models demonstrated excellent computational performance, achieving an average error below 0.4% and operating on average 271 times faster than the state-of-the-art finite-volume model. This highlights the capability of the NAD model to rapidly and precisely approximate complex heat transfer interactions in heat exchangers of the heat pump, making it particularly suitable for model control applications, optimization tasks, and long-duration simulations.

## 1. Introduction

Heat pumps are highly versatile energy system for space heating and cooling, and water heating with wide range of applications, such as refrigeration and air conditioning, chemical processing, solar and thermal power plants and many others. To optimally control and optimize the operation of the heat pumps in real time in response to its own thermal behavior and its coupled energy systems, faster and accurate dynamic models are vital. Heat exchangers (condenser/evaporator) are crucial components of heat pumps which enables efficient heat transfer between fluids or between a fluid and a solid. Furthermore, heat exchangers are also crucial integral components of diverse vapor compression systems. The prominent dynamics of these systems come from the heat exchangers. Accurate dynamic modeling of heat exchangers is crucial for developing dynamic models of vapor compression systems. However, developing such accurate and fast models of the condenser and evaporator components is highly challenging due to the compressible nature of the working fluid and the phase change process occurring inside the heat exchangers.

As for numerical modeling of heat exchangers, the common approaches used in solving the governing equations are Lumped Parameter (LP) method [1–3], Moving Boundary (MB) method [4–6], and Finite Volume (FV) method [7,8]. In LP method, the heat exchanger model is simplified with aggregated thermal properties and lumped state variables which represents the entire heat exchanger. This assumption makes the solution process for the LP model very fast, but it sacrifices lot of accuracy, especially when dealing with compressible working fluids, it is difficult to capture the phase change. In the MB method, the heat exchanger is partitioned into three zones: vapor, two-phase, and liquid. These zones have moving boundaries and each zone is modeled using its respective governing equations. This approach has reasonable accuracy, but a significant challenge arises in accurately determining the boundary of the phase change making it slower than the LP model but its computation time still compatible with control applications. Both LP and MB methods compromise accuracy rendering them inefficient for accurate model control. In FV method, the heat exchanger is divided into *n* number of control volumes and the governing equations are integrated over each control volume. This method allows to better capture heat transfer and phase change of

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: shri-balaji.padmanabhan@imt-atlantique.fr (S.B. Padmanabhan), mohamed-tahar.mabrouk@imt-atlantique.fr (M.T. Mabrouk).</footnote>
the working fluid. The FV model stands out for its high robustness, versatility, and superior accuracy compared to LP and MB models. But, the FV model is computationally expensive when compared to other models [9], making it less suitable for real-time modeling and control applications.

<table><thead><tr><th colspan="2">Nomenclature</th></tr></thead><tbody><tr><td>A</td><td>Area (m<sup>2</sup>)</td></tr><tr><td>α</td><td>Heat transfer coefficient (Wm<sup>2</sup>K<sup>-1</sup>)</td></tr><tr><td>C<sub>p</sub></td><td>Specific heat capacity (Jkg<sup>-1</sup>K<sup>-1</sup>)</td></tr><tr><td>ρ</td><td>Density (kgm<sup>-3</sup>)</td></tr><tr><td>h</td><td>Enthalpy (Jkg<sup>-1</sup>K<sup>-1</sup>)</td></tr><tr><td>L</td><td>Heat exchanger length (m)</td></tr><tr><td>M</td><td>Mass (kg)</td></tr><tr><td>ṁ</td><td>Mass flow rate (kgs<sup>-1</sup>)</td></tr><tr><td>P</td><td>Pressure (Pa)</td></tr><tr><td>δQ̇</td><td>Volumetric heat transfer rate (Wm<sup>-3</sup>)</td></tr><tr><td>Q̇</td><td>Heat transfer rate (W)</td></tr><tr><td>S</td><td>Heat transfer area (m<sup>2</sup>)</td></tr><tr><td>Φ</td><td>Thermal power (W)</td></tr><tr><td>T</td><td>Temperature (K)</td></tr><tr><td>t</td><td>Time (s)</td></tr><tr><td>x</td><td>spatial coordinate (m)</td></tr><tr><td>N</td><td>Compressor speed (rps)</td></tr><tr><td>V<sub>s</sub></td><td>Swept Volume (m<sup>3</sup>)</td></tr><tr><td>η</td><td>Efficiency (%)</td></tr><tr><td>cr</td><td>Clearance ratio (-)</td></tr><tr><td>γ</td><td>Specific heat ratio (-)</td></tr><tr><td>C<sub>d</sub></td><td>Correction coefficient (-)</td></tr><tr><td>xr</td><td>Quality of refrigerant (-)</td></tr><tr><td>vo</td><td>Expansion Value Opening (%)</td></tr><tr><td colspan="2"><strong>Subscripts</strong></td></tr><tr><td>in</td><td>Inlet</td></tr><tr><td>out</td><td>Outlet</td></tr><tr><td>p</td><td>Primary fluid</td></tr><tr><td>s</td><td>Secondary fluid</td></tr><tr><td>w</td><td>Primary tube wall</td></tr><tr><td>c</td><td>Compressor</td></tr><tr><td>ev</td><td>Expansion valve</td></tr><tr><td>is</td><td>Isentropic</td></tr><tr><td>v</td><td>Volumetric</td></tr><tr><td colspan="2"><strong>Superscripts</strong></td></tr><tr><td>~</td><td>Equivalent variables</td></tr></tbody></table>

There have been efforts made to develop pure machine learning models for the heat exchanger to reconcile the trade-off between computation time and accuracy. In the modeling of heat exchangers, machine learning models have been developed specifically to predict the outlet temperature of hot and cold fluids [10–13]. Machine learning has also been used to improve the accuracy of physical models by predicting the thermodynamic properties of refrigerants [14–16] and key parameters such as the Nusselt number [17–19], heat transfer coefficient [20–22], pressure drop [11,23,24], friction factor [19,25,26], and fouling factors [27,28]. Even though the machine learning models provide faster predictions with reasonable accuracy, one of the primary disadvantage is that it lacks validation of the underlying physics, which decreases drastically their generality and explainability, making trust-worthiness of their predictions very low. Some of its other limitations include data dependency and overfitting, computationally intensive training, and low interpretability [29]. These challenges underscore the need for alternative approaches that can balance computation speed and accuracy effectively while validating the physics.

An effective alternative to purely machine learning-based or purely physics-based models is the integration of both physics and machine

<figure><img src="image_3.png" /><figcaption>Fig. 1. PH Diagram Illustrating the Heat Pump's Vapor Compression Cycle.</figcaption></figure>

learning models together, blending the benefits of both approaches. This approach, however, remains largely unexplored in dynamic modeling applications for heat exchangers. The machine learning and physics models can be put together in different ways like in series or in parallel [30]. In both architectures, the physics based and machine learning models may function independently or with one model taking precedence. The key consideration when combining physics-based and machine learning models is achieving seamless integration, ensuring strong dynamic coupling between the sub-models, allowing each to complement and enhance the other's strengths. Additionally, the hybrid approach should enable the models to adapt and improve easily for new scenarios and discover additional dynamics if needed. One of the promising methodologies for this is the Universal Differential Equations (UDE) framework [31], which is when neural networks are part of Ordinary Differential Equations. Traditionally, UDEs are applied in scenarios where limited knowledge of the underlying physics could restrict model accuracy and generalizability. UDEs address this by learning the unknown part which contributes to the dynamics from data and coupling it with physics-based model. This modeling paradigm is used across many domains [32–34], yet building a UDE tailored to a specific application demands in-depth knowledge and understanding of that domain.

This paper introduces an accelerated model for heat exchanger called Neural Accelerated Dynamic (NAD) model that integrates a universal approximator into a simplified model in accordance with the UDE framework. Furthermore, a novel training procedure is proposed which enforces the simplified model to generate its own training data using the complex model results. This training set is used to train the machine learning part of the model. Specifically, here UDE framework is applied to accelerate the complex dynamics of heat exchanger modeling, an approach that, to the authors' knowledge, has not been previously explored in the literature. This has been done in an innovative and distinct way, achieving a perfect balance between computational efficiency and accuracy while largely preserving the interpretability of the full physical model by validating the dominant physics. Additionally, this paper focuses on achieving acceleration without sacrificing the generality of the heat pump's heat exchanger modeling and this choice is motivated by the complex phenomena occurring within the heat pump's heat exchanger. To evaluate the NAD model, its performance is compared against a state-of-the-art finite volume model for heat exchangers.

The paper is organized as follows: Section 2, summarizes the state-of-the-art in dynamic modeling of heat pump components, primarily heat exchangers, with a focus on solving the governing equations using the finite volume method. Section 3, discusses the simplified model that forms the foundation for the NAD model, elaborates the structure, the training procedure, and the generation of training data for the machine learning component for NAD model. In Section 4, the NAD model is validated against the FV model for different validation simulations, and its performance is showcased and discussed.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Heat Pump Layout.</figcaption></figure>

## 2. Dynamic modeling of heat pump

The heat pump operates on a vapor compression cycle (Fig. 1). Its dynamic model incorporates detailed mathematical formulations of essential components, including the compressor, condenser, evaporator, and expansion valve (Fig. 2), to simulate transient thermal performance and optimize energy transfer. The compressor and expansion valve have relatively very fast dynamics when compared to the heat exchangers. Also since the NAD model is specifically for heat exchangers, in this study, compressor and expansion valve are modeled as static models, while detailed dynamic modeling is applied exclusively to the heat exchangers, which predominantly govern the system's overall transient behavior.

### 2.1. Compressor

In a static model of the compressor [35], the outlet mass flow rate (Eq. (1)) of the refrigerant can be expressed as following,

$$ \dot{m}_c = N_c V s_c \rho_{in} \eta_v \quad (1) $$

where the volumetric efficiency is calculated as,

$$ \eta_v = 1 + cr_c - cr_c \left( \frac{P_{out}}{P_{in}} \right)^{\frac{1}{\gamma}} \quad (2) $$

The outlet enthalpy of the refrigerant Eq. (3) from the compressor is calculated using the isentropic compression relation.

$$ h_{out} = h_{in} + \frac{h_{out, is} - h_{in}}{\eta_{is}} \quad (3) $$

### 2.2. Expansion valve

The expansion valve considered in this study is a combination of a convergent-divergent nozzle and an abrupt diffuser. The mass flow rate through the expansion device is given by Eq. (4) and the dimensionless correction coefficient $C_d$ is determined using an empirical correlation, which is a function of the valve opening and the operating conditions of the heat pump. The detailed formulation for $C_d$ corresponding to the refrigerant used in this study and the mass flow rate expression are referenced from [36].

$$ \dot{m}_{ev} = C_d A_{ev} \sqrt{2 \rho_{r,in} (P_{p,in} - P_{p,out})} \quad (4) $$

### 2.3. Condenser and evaporator

The condenser and evaporator are heat exchangers operating under different thermal conditions. These two components are modeled using the same mathematical model based on conservation principles, encompassing mass and energy conservation for the primary fluid, as well as energy conservation within the primary fluid's tube and the secondary fluid. The following assumptions are considered while modeling the dynamics of the heat exchanger:

(i) The physical variables of the primary and secondary fluids are considered homogeneous across the cross-sectional area, allowing the system to be modeled as one-dimensional.

(ii) The pressure drop in both primary and secondary fluids resulting from changes in momentum and viscous friction are not modeled.

(iii) The axial heat conduction in the fluids is negligible when compared to the heat transfer between fluids and tube, so it is not considered.

(iv) The separating wall is considered thin enough to disregard heat conduction. Only heat accumulation is considered.

(v) The heat exchanger is considered adiabatic, with no heat exchange with its environment.

Based on these assumptions, the heat exchanger is modeled as a one-dimensional system featuring two fluid channels of arbitrary cross-sectional areas A, heat transfer area S and volumes V for the primary fluid, the secondary fluid, and the wall which separates the fluids. This approach remains general by focusing solely on these three fundamental volumes (primary fluid, secondary fluid, and the intervening wall), ensuring applicability to various heat exchanger designs in a 1D finite-volume framework. The following modeling is applicable for the scenario where the primary fluid is compressible and the secondary fluid is incompressible. The mass conservation equation (Eq. (5)) and energy conservation equation (Eq. (6)) for the primary fluid, and the energy conservation equation for the separating tube wall (Eq. (7)) and secondary fluid (Eq. (8)) are as following [37],

$$ A_p \frac{\partial \rho_p}{\partial t} + \frac{\partial \dot{m}_p}{\partial x} = 0 \quad (5) $$

$$ A_p \frac{\partial(\rho h - P)_p}{\partial t} + \frac{\partial(\dot{m} h)_p}{\partial x} + \delta \dot{Q}_{pw} = 0 \quad (6) $$

$$ (A \rho C_p)_w \frac{\partial T_w}{\partial t} - \delta \dot{Q}_{pw} + \delta \dot{Q}_{ws} = 0 \quad (7) $$
$$ (A \rho C_p)_s \frac{\partial T_s}{\partial t} + (\dot{m}_{in} C_p)_s \frac{\partial T_s}{\partial x} - \delta \dot{Q}_{ws} = 0 \quad (8) $$

The conservation equations are discretized and transformed into ordinary differential equations using the finite volume method. Given its superior accuracy compared to moving boundary and lumped parameter models, the finite volume method [9] is chosen as the discretization technique in this study.

### 2.3.1. Finite Volume model

In the Finite Volume (FV) model, the heat exchanger is divided into n number of equally sized control volumes. Fig. 3 illustrates the discretized, one-dimensional arbitrary fluid channels separated by a wall for counter flow. In the discretization process, two types of variables are used: cell variables and node variables. The node variables are indicated by a “*” superscript. The Finite Volume (FV) formulation detailed here is in accordance with [9,38]. By integrating Eqs (5)–(8) over the control volumes results in the following set of coupled Ordinary Differential Equations for kth control volume.

$$ a_k \frac{dP_p}{dt} + b_k \frac{dh_{p,k}}{dt} = \dot{m}_{p,k-1}^* - \dot{m}_{p,k}^* \quad (9) $$

$$ c_k \frac{dP_p}{dt} + d_k \frac{dh_{p,k}}{dt} = \dot{m}_{p,k-1}^* h_{p,k-1}^* - \dot{m}_{p,k}^* h_{p,k}^* - \dot{Q}_{pw,k} \quad (10) $$

$$ (M C_p)_w \frac{dT_w}{dt} = \dot{Q}_{pw,k} - \dot{Q}_{ws,k} \quad (11) $$

$$ (M C_p)_s \frac{dT_s}{dt} = \dot{m}_{s,in} C_p s (T_{s,k+1}^* - T_{s,k}^*) + \dot{Q}_{ws,k} \quad (12) $$

where the coefficients are defined as,

$$ \begin{aligned} a_k &= V_{p,k} \left( \frac{\partial \rho_{p,k}}{\partial P_p} \right)_{h_{p,k}} & b_k &= V_{p,k} \left( \frac{\partial \rho_{p,k}}{\partial h_{p,k}} \right)_{P_p} \\ c_k &= V_{p,k} \left[ h_{p,k} \left( \frac{\partial \rho_{p,k}}{\partial P_p} \right)_{h_{p,k}} - 1 \right] & d_k &= V_{p,k} \left[ h_{p,k} \left( \frac{\partial \rho_{p,k}}{\partial h_{p,k}} \right)_{P_p} + \rho_{p,k} \right] \\ \dot{Q}_{pw,k} &= S_{pw,k} \alpha_{pw} (T_{p,k} - T_{w,k}) & \dot{Q}_{ws,k} &= S_{ws,k} \alpha_{ws} (T_{w,k} - T_{s,k}) \end{aligned} $$

In this study, an upwind scheme is employed, where the outlet value of a variable from a control volume is determined by the average value of that variable in the control volume immediately upstream. This approach ensures that the value at the outlet reflects the conditions in the subsequent control volume, accounting for the flow direction. The intermediate mass flow rates of refrigerant are not primary variables of interest and therefore they are algebraically eliminated from the refrigerant equations to reduce the number of variables [9]. This is done by substituting mass flow rate leaving kth cell from Eq. (5) into Eq. (6). Then, from Eq. (5), a recursive formula is created which relates $\dot{m}_{p,k}$ to $\dot{m}_{p,in}$. A modified version of the energy balance equation for the kth cell is obtained by applying this recursion formula in the substituted energy balance which is as follows,

$$ \left[ (c_k - a_k h_{p,k}) - (h_{p,k} - h_{p,k-1}) \sum_{j=1}^{k-1} a_j \right] \frac{dP_p}{dt} + (d_k - b_k h_{p,k}) \frac{dh_{p,k}}{dt} + (h_{p,k} - h_{p,k-1}) \sum_{j=1}^{k-1} \left( b_j \frac{dh_{p,j}}{dt} \right) = \dot{m}_{p,in} (h_{p,k} - h_{p,k-1}) - \dot{Q}_{pw,k} \quad (13) $$

The above equation is applied to the first n-1 control volumes and yields a system of n-1 independent equations, which do not explicitly involve the intermediate mass flow rates. The new mass and energy equations of the refrigerant are as follows,

$$ \left( \sum_{j=1}^{n} a_j \right) \frac{dP_p}{dt} + \sum_{j=1}^{n} \left( b_j \frac{dh_{p,j}}{dt} \right) = \dot{m}_{p,in} - \dot{m}_{p,out} \quad (14) $$

$$ \left( \sum_{j=1}^{n} c_j \right) \frac{dP_p}{dt} + \sum_{j=1}^{n} \left( d_j \frac{dh_{p,j}}{dt} \right) = \dot{m}_{p,in} h_{p,in} - \dot{m}_{p,out} h_{p,out} - \sum_{j=1}^{n} \dot{Q}_{pw,j} \quad (15) $$

In total, for n control volumes, there are 3n+1 equations and 3n+1 unknowns. The system of equations in matrix form is given as,

$$ \vec{A} * \vec{X} = \vec{B} \quad (16) $$

$$ \text{with } \vec{X} = [P, h_{p,1}, \dots, h_{p,n}, T_{w,1}, \dots, T_{w,n}, T_{s,1}, \dots, T_{s,n}] $$

<table><caption>**Table 1**
Design parameters and operating conditions of the heat pump.</caption><thead><tr><th>Design parameters and dimensions</th><th>Values</th></tr></thead><tbody><tr><td>Nominal condenser power (kW)</td><td>150</td></tr><tr><td>Nominal evaporator power kW</td><td>50</td></tr><tr><td>Overall heat transfer area - condenser side (m²)</td><td>5.46</td></tr><tr><td>Overall heat transfer area - evaporator side (m²)</td><td>3.12</td></tr><tr><td>Inlet temperature of secondary fluid - condenser side (K)</td><td>290.0 - 295.0</td></tr><tr><td>Inlet temperature of secondary fluid - evaporator side (K)</td><td>290.0 - 295.0</td></tr><tr><td>Inlet mass flow rate of secondary fluid - condenser side (kgs⁻¹)</td><td>0.8 - 1.0</td></tr><tr><td>Inlet mass flow rate of secondary fluid - evaporator side (kgs⁻¹)</td><td>1.1 - 1.3</td></tr><tr><td>Compressor speed (rpm)</td><td>3000 - 3250</td></tr><tr><td>Expansion Valve opening - Full opening is 1.0</td><td>0.45 - 0.55</td></tr></tbody></table>

### 2.4. Numerical resolution

In the heat pump model, the static equations for the compressor and expansion value are strongly coupled with the discretized dynamic FV ODEs for condenser and evaporator, and the coupled system is solved in Julia using the DifferentialEquations.jl [39] library's ODE solver. In this study, R410 A is selected as the primary refrigerant, while water serves as the secondary fluid for both the condenser and evaporator. The flow between the primary and secondary fluids follows a counter-flow configuration. The heat pump design parameters and operating conditions are mentioned Table 1. As for time integration algorithm, Rosenbrock-W method of order 2/3 is used. This algorithm is particularly suited for this problem because of its stiffness-robust error estimation capability which enables accurate adaptive time-stepping and effectively handle highly stiff problems. In the literature, it has been stated that the optimal number of control volumes for the FV model of the condenser and evaporator ranges from 15 to 20 [37,40]. Therefore, in this study, 20 CV is used as the discretization size for the heat exchanger models. For both fluids in the condenser and evaporator, corresponding nominal constant values for the heat transfer coefficient were used. As for the refrigerant's thermal properties, cubic spline interpolation functions built from thermodynamic library data [41,42] within the operating range are used to compute them dynamically rather than directly calling the library. This approach, examined widely in the literature [43–46]. Furthermore, the use of cubic spline interpolation functions enhances computational speed, eliminates overhead from thermodynamic library wrappers, API translation delays, and data type conversions problems.

## 3. Neural accelerated dynamic modeling for the heat exchanger

### 3.1. Simplified model

The NAD model proposed in this paper is based on a simplified model for the heat exchanger which is formulated as following. In the simplified model, rather than dividing the heat exchanger into n number of control volumes as in FV model, entire heat exchanger is considered as a single control volume. Thus, the governing equations are integrated over entire length L of the heat exchanger. This comprehensive integration naturally leads to FV ODEs Eq. (9) – Eq. (12) for just one control volume. Here, the state variables can be seen as lumped variables, capturing the overall variation of each respective variable across the entire heat exchanger with respect to the inlet and outlet conditions. These lumped variables are referred as equivalent state variables, denoted with “-”.

$$ \bar{a} \frac{d \tilde{P}_p}{dt} + \bar{b} \frac{d \tilde{h}_p}{dt} = \dot{m}_{p,in} - \dot{m}_{p,out} \quad (17) $$

$$ \bar{c} \frac{d \tilde{P}_p}{dt} + \bar{d} \frac{d \tilde{h}_p}{dt} = \dot{m}_{p,in} h_{p,in} - \dot{m}_{p,out} h_{p,out} - \tilde{Q}_{pw} \quad (18) $$

$$ (MCp)_w \frac{dT_w}{dt} = \tilde{Q}_{pw} - \tilde{Q}_{ws} \quad (19) $$

$$ (MCp)_s \frac{dT_s}{dt} = \dot{m}_{s,in} C_p s (T_{s,in} - T_{s,out}) + \tilde{Q}_{ws} \quad (20) $$
<figure><img src="image_5.png" /><figcaption>Fig. 3. Discretized heat exchanger.</figcaption></figure>

where the coefficients are defined as,

$$
\begin{align*}
\bar{a} &= V_p \left( \frac{\partial \bar{\rho}_p}{\partial \bar{P}_p} \right)_{\bar{h}_p} \\
\bar{c} &= V_p \left[ \bar{h}_p \left( \frac{\partial \bar{\rho}_p}{\partial \bar{P}_p} \right)_{\bar{h}_p} - 1 \right] \\
\bar{Q}_{pw} &= S_{pw} \alpha_{pw} (\bar{T}_p - \bar{T}_w) \\
\bar{Q}_{ws} &= S_{ws} \alpha_{ws} (\bar{T}_w - \bar{T}_s)
\end{align*}
$$

In order to solve the simplified model, dynamic outlet variables such as h<sub>p,out</sub> and T<sub>s,out</sub> are needed. Typically, some assumptions are made to determine the outlet variables, like in lumped parameter models, a typical approach is to consider a linear profile for the state variables across the heat exchanger [47].

3.2. NAD model

$$
\frac{du}{dt} = f(u, t) + nn(\theta, u) \qquad (21)
$$

The NAD model adopts the simplified model to represent the heat exchanger. In the simplified model, instead of making some assumptions on the variation profile of the state variables to approximate the outlet variables h<sub>p,out</sub> and T<sub>s,out</sub>, these outlet variables are dynamically predicted by the neural network. This dynamically coupled simplified model and neural network combination result in the Universal Differential Equations (UDE) [31] representation. The main idea of UDE is to use neural networks as universal approximators within differential equations as shown in Eq. (21), where u is the arbitrary dynamic state variable, f is known part of the physics and nn is neural network. This representation is generally used for model discovery [32–34], allowing the portion of physics contributing to the system’s main dynamics to be approximated by incorporating the neural network into the differential equations. However, the UDE representation can become computationally expensive for complex and large systems because the system of equations must be solved numerically during neural network training.

In this study we propose model acceleration using universal differ-
ential equations representation for heat exchanger dynamic modeling,
in which the certain part of complex physics from the detailed model
(FV model for heat exchanger) is forced into simplified model for heat
exchanger via neural networks, the resulting model is computationally
faster with very good precision compared to detailed model, while
validating the overall physics for each instance. In this case the outlet
values corresponding to the primary and secondary fluid from the
FV model are imposed into the simplified model. In general UDE
representation, the main dynamic variables (which are known) are
given as inputs to the neural networks and part which contributes to the
dynamics is found. However in NAD model, the part which to be forced

<figure><img src="image_6.png" /><figcaption>Fig. 4. The Structure of the NAD model.</figcaption></figure>

is known (from FV model outlets), but the resulting modified dynamics
of simplified model is unknown. To find out the equivalent/modified
simplified dynamics, we propose adaptive physics-guided data synthe-
sis process. This approach identifies the dynamics of the equivalent
variables corresponding to the already known outputs extracted from
the complex finite-volume model. The neural network is trained sepa-
rately using these training data, rather than being trained dynamically
alongside the resolution of the dynamical system, as done usually
in UDE framework. This technique ensures that the neural network
predicts the outlet variables based on the corresponding equivalent
state variables, enabling the new model to maintain robustness and
reduce overdependence on the neural network. Detailed aspects about
adaptive physics-guided data synthesis process are discussed in the later
part of the paper. By this distinct process, the UDE framework can be
used for model acceleration and can be extended to other applications.
A simple deep neural network is used to predict the outlet variables
and the updated simplified ODEs Eq. (17)–Eq. (20) corresponding to
the NAD model are given as following,

$$
\bar{a} \frac{d \tilde{P}_p}{dt} + \bar{b} \frac{d \tilde{h}_p}{dt} = \dot{m}_{p,in} - \dot{m}_{p,out} \quad (22)
$$

$$
\bar{c} \frac{d \tilde{P}_p}{dt} + \bar{d} \frac{d \tilde{h}_p}{dt} = \dot{m}_{p,in} h_{p,in} - \dot{m}_{p,out} nn_1 - \tilde{Q}_{pw} \quad (23)
$$

$$
(MCp)_w \frac{d\tilde{T}_w}{dt} = \tilde{Q}_{pw} - \tilde{Q}_{ws} \qquad (24)
$$
<figure><img src="image_7.png" /><figcaption>Fig. 5. Workflow of adaptive physics-guided data synthesis process.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Schematics of the neural network.</figcaption></figure>

$$ (MCp)_s \frac{dT_s}{dt} = \dot{m}_{s,in} C_{ps}(T_{s,in} - nn_2) + \tilde{Q}_{ws} \quad (25) $$

where, $nn_1$ is neural network's first output corresponding to $h_{p,out}$ and $nn_2$ is neural network's second output corresponding to $T_{s,out}$. The above equations can be simplified and rewritten as,

$$ \frac{d \tilde{P}_p}{dt} = \frac{\tilde{d}(\dot{m}_{p,in} - \dot{m}_{p,out}) - \tilde{b}(\dot{m}_{p,in} - \dot{m}_{p,out}) nn_1 - \tilde{Q}_{pw})}{\tilde{a}\tilde{d} - \tilde{b}\tilde{c}} \quad (26) $$

$$ \frac{d \tilde{h}_p}{dt} = \frac{-\tilde{c}(\dot{m}_{p,in} - \dot{m}_{p,out}) + \tilde{a}(\dot{m}_{p,in} - \dot{m}_{p,out}) nn_1 - \tilde{Q}_{pw})}{\tilde{a}\tilde{d} - \tilde{b}\tilde{c}} \quad (27) $$

$$ \frac{d \tilde{T}_w}{dt} = \frac{\tilde{Q}_{pw} - \tilde{Q}_{ws}}{(MCp)_w} \quad (28) $$

$$ \frac{d \tilde{T}_s}{dt} = \frac{\dot{m}_{s,in} C_{ps}(T_{s,in} - nn_2) + \tilde{Q}_{ws}}{(MCp)_s} \quad (29) $$

The neural network is trained to capture the variation profile of the state variables from the FV results, and predicts the outlet values. In the NAD model, the simplified model and the neural network are strongly coupled as shown in Fig. 4 and the NAD model is dynamically solved using a numerical ODE solver just like the FV model. During the resolution of the NAD model, for each instance of time, given the equivalent variables values and primary inputs, the neural network effectively approximates the outlet values and give it back to the simplified model. These equivalent variables act as hidden states, providing the dynamic memory that captures and retains information based on what has happened previously. The NAD model requires solving only 4 ODEs, whereas the FV model necessitates solving $3n + 1$ ODEs with n being number of control volumes. This disparity in the number of equations and the less stiff nature of the simplified ODEs provide the NAD model with a significant computational advantage over the highly stiff FV model.

### 3.3. Adaptive physics-guided data synthesis process for neural network

As previously mentioned, training the NAD model requires knowledge of the equivalent state variables dynamics. This is achieved through a novel adaptive physics-guided data synthesis process, outlined as follows, with the workflow for generating this training data illustrated in Fig. 5. First given the primary inputs, the heat pump FV model is solved numerically using ODE solver and the FV solution is obtained. From the FV solution, the refrigerant outlet enthalpy $h_{p,out}$ and water outlet temperature $T_{s,out}$ are extracted for both condenser and evaporator. Then, the heat pump simplified model is solved using an ODE solver to approximate the solution of the equivalent state variables for both condenser and evaporator given these outlet values. At this stage, the ODE solver is forced to produce the solution of the equivalent

state variables which satisfies the FV outlet solutions in both condenser and evaporator. In the data generation process, key inputs such as the inlet conditions of the secondary fluid in the condenser and evaporator, compressor speed, and expansion valve opening are randomly varied. These variations help develop and validate a robust model capable of handling extreme scenarios. The bounds for these input variations are specified in Table 1. Data is generated through 50 simulations, each for 5000 s time span, during which inputs are continuously varied within the defined ranges. Once the solution of the equivalent state variables and the FV outlet solutions are obtained, the data is used to train the neural network.

### 3.4. Neural network

Fig. 6 illustrates the schematics of the neural network. A simple deep neural network (DNN) is used in the NAD model. For condenser and evaporator NAD model, separate DNNs are used. For both, similar neural network architecture used in this study, which comprise an input layer, 2 hidden layers with 30 neurons each, and an output layer. Rectified Linear Unit (ReLU) activation functions were applied after each layer except input and output layer to introduce non-linearity. Adam optimizer with a very small decaying learning rate is used to train the neural network. The network architecture and the hyperparameters were meticulously selected after extensive grid search and cross-validation to enhance the model's performance. For the condenser's DNN, in addition to the primary inputs, the compressor outlet conditions are also provided as inputs. This enhances the accuracy of both the condenser side and the overall model. Training neural networks for the NAD model is highly efficient, requiring minimal time and computational resources. This is mainly because of the way the NAD model works and the adaptive physics-guided data synthesis process. Since, here the DNNs only learns the relation between the equivalent states and outlet values considering the phase change occurring inside the heat exchangers. The dynamics is taken care by the numerical ode solver. The equivalent state variables given by the ODE solver depending on the initial condition and past value of the respective variable, these equivalent variables acts as hidden states. This enables the DNN to maintain context and capture temporal dependencies, making it possible to learn patterns over time with a simple neural network, thereby eliminating the need for a complex recurrent neural network which is the case for pure data driven models. Furthermore, when compared to recurrent neural networks, simple deep neural networks offer lower inference times and very easy to train. This aspect holds particular significance, as the neural network has to predict outlet values for each instance of time. The low training and inference time of the deep neural network serves to minimize the
overall computational costs associated with solving the NAD model. In this study, the DNNs for the NAD model are trained on a single-core CPU, achieving the training time of just around 0.02 s per epoch with a batch size of 16 for data from 50 simulations, over a total of 1000 epochs. Training on a GPU would further accelerate the process, significantly reducing the computational time.

## 4. Results and discussion

The performance of the heat pump NAD model is extensively evaluated using multiple validation simulations to assess its computational efficiency, accuracy and robustness. The heat pump NAD model is numerically solved using ODE solver and the choice of the time integration algorithm is crucial. The NAD models are found to perform well with most of the stiff and non-stiff time integration methods. Despite exhibiting only small stiffness, stiff solvers tend to provide stable and accurate solutions, especially when dealing with varying time scales and ensuring robustness in the presence of mild stiffness. Providing a good balance between accuracy and computational cost, the Trapezoidal method is found to be the most efficient for the NAD heat pump model. The performance of the heat pump NAD model is compared against the state-of-art heat pump FV model with 20 CVs. For the heat pump FV model, the Rosenbrock-W method of order 2/3 is found to be the best choices.

### 4.1. Results and error analysis

To validate the heat pump NAD model performance against FV model, 5 different sets of simulations are performed with time spans 5000, 10000, 15000, 20000, 30000 s respectively. In these validation simulations, the primary inputs of condenser and evaporator, as well as the compressor and expansion valve inputs are varied randomly, as illustrated in Figs. 7 and 8. These figures correspond specifically to the first validation scenario, where the heat pump NAD model is tested under extreme operating conditions. For the 1st and 2nd validation simulations, the dynamics of the secondary fluid outlet temperature for both condenser and evaporator side of the heat pump from the FV and NAD heat pump models are compared in Figs. 9 and 10 respectively.

Regarding the local dynamics of the condenser and evaporator of the heat pump, as shown in Figs. 9 and 10, the solutions for the secondary fluid outlet temperatures from the NAD and FV heat pump numerical models align closely, following similar dynamics. Although the NAD model exhibited minor discrepancies in certain cases, its overall performance remained highly accurate, effectively capturing system dynamics and demonstrating strong reliability across varying operating conditions. Notably, all training simulations were conducted for a fixed duration of 5000 s. However, during testing, the model was evaluated far beyond this time span, effectively performing temporal extrapolation by predicting system behavior beyond the observed data range, this is done to showcase its robustness and stability. The NAD model demonstrated exceptional consistency even over extremely long simulation windows, maintaining stability without any signs of divergence, which case seen in Fig. 10 where simulation time span is 10000 s, and in the accuracy comparison for simulations with longer time span than this, which can be found in later part. This is a significant advantage over purely data-driven recurrent neural network models, which may tend to drift or become unstable over extended simulations, making the NAD model far more reliable for long-term predictions. In contrast, purely data-driven models often lack the inherent stability and reliability to handle such large simulation spans, as they do not benefit from the stabilizing influence of physical constraints. This robust performance is attributed to the tightly integrated design of the simplified physics and neural network architecture. The simplified model ensures that the neural network predictions are grounded in physical principles while getting the information about the state of the fluids in heat exchangers, enabling stable and accurate simulations over extended time span.

<table><caption>**Table 2**
Comparison of different performance metrics for secondary fluid side condenser and evaporator thermal power solutions: NAD models vs. FV heat pump model.</caption><thead><tr><th rowspan="3">Validation simulation</th><th colspan="4">NAD vs FV heat pump model</th></tr><tr><th colspan="2">Φ<sub>c</sub> (W)</th><th colspan="2">Φ<sub>e</sub> (W)</th></tr><tr><th>Error %</th><th>R<sup>2</sup></th><th>Error %</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>1</td><td>0.2827</td><td>0.8958</td><td>0.2762</td><td>0.9591</td></tr><tr><td>2</td><td>0.3999</td><td>0.9345</td><td>0.2188</td><td>0.9469</td></tr><tr><td>3</td><td>0.3622</td><td>0.9578</td><td>0.2551</td><td>0.9715</td></tr><tr><td>4</td><td>0.3295</td><td>0.9834</td><td>0.2336</td><td>0.9545</td></tr><tr><td>5</td><td>0.3035</td><td>0.9856</td><td>0.2257</td><td>0.9698</td></tr></tbody></table>

<table><caption>**Table 3**
Comparison of COP values between NAD and FV heat pump models.</caption><thead><tr><th>Validation simulation</th><th>COP (FV model)</th><th>COP (NAD model)</th><th>Difference (%)</th></tr></thead><tbody><tr><td>1</td><td>2.950</td><td>2.937</td><td>0.43</td></tr><tr><td>2</td><td>2.964</td><td>2.956</td><td>0.27</td></tr><tr><td>3</td><td>2.956</td><td>2.949</td><td>0.23</td></tr><tr><td>4</td><td>2.977</td><td>2.971</td><td>0.21</td></tr><tr><td>5</td><td>2.954</td><td>2.942</td><td>0.37</td></tr></tbody></table>

This indicates that the NAD heat pump model effectively captures the underlying physics of the heat exchanger, accurately reflecting its dynamic behavior and heat transfer processes. Specifically, the neural network has effectively learned the variation profile of state variables from the FV heat pump model solutions, enabling the simplified model to produce results that closely align with the highly accurate and complex FV heat pump model.

$$ \Phi_c = \dot{m}_{s,c,in} C p_s (T_{s,c,out} - T_{s,c,in}) \quad (30) $$

$$ \Phi_e = \dot{m}_{s,e,in} C p_s (T_{s,e,in} - T_{s,e,out}) \quad (31) $$

Furthermore, to comprehensively evaluate performance, an error analysis is conducted by comparing the performance metrics for quantities of interest between the NAD model and the FV heat pump models. In this context, the quantities of interest are the condenser and evaporator side thermal power, as given by Eqs. (30) and (31). Therefore, heat pump models are evaluated based on their performance with respect to these quantities of interest. The evaluation employs following metrics to assess the models' effectiveness,

$$ \text{Overall Error Percentage} = \left( \frac{1}{n} \sum_{i=1}^{n} \left| \frac{y_i - \hat{y}_i}{y_i} \right| \right) \times 100 \quad (32) $$

$$ R^2 score = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2} \quad (33) $$

where $y$ and $\hat{y}$ being actual value and approximated value respectively, with $n$ being number of samples. Performance metrics for the quantities of interest are used to evaluate the differences between the FV and NAD heat pump model results, and tabulated in Table 2, and mean Coefficient of Performance (COP) is tabulated in Table 3 for different validation simulations. For both quantities of interest, the NAD heat pump model demonstrated strong agreement with the FV model, effectively capturing system dynamics with high accuracy. Across all validation simulations, the NAD model consistently maintained reliable performance, reaffirming its accuracy and dependability in replicating the system's thermal behavior.

### 4.2. Computation time benchmarking

In this study, the code for all models are written with careful attention to type stability and memory allocation to ensure optimal performance. This guarantees a meaningful and accurate computational performance comparison between the FV and NAD models. The benchmarking of the computational performance of the models is conducted
<figure><img src="image_9.png" /><figcaption>Fig. 7. Dynamic variation of primary inputs conditions of condenser and evaporator for validation simulation 1.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Dynamic variation of inputs of compressor and expansion valve for validation simulation 1.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 9. Dynamics of secondary fluid outlet temperature in condenser and evaporator side for validation simulation 1 for time span 5000 s: FV vs. NAD heat pump model.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Dynamics of secondary fluid outlet temperature in condenser and evaporator side for validation simulation 2 for time span 10000 s: FV vs. NAD heat pump model.</figcaption></figure>

using Julia's BenchmarkTools.jl. The computation time taken by different models to perform different validation simulations, and amount of computational acceleration attained with the NAD heat pump model against FV model are tabulated in Table 4.

In terms of computation performances, NAD heat pump model demonstrated a significant advantage over the FV model. The NAD model is on average 271 times faster than FV model with interpolation functions. The FV model involves solving a large number of stiff ODEs. In contrast, the NAD models only solve 4 ODEs which exhibits mild stiffness, and require less memory allocation. These factors make the

NAD models significantly faster than the FV model. A potential limitation of the NAD model is its computational performance during very long simulations, as increasing the simulation duration can notably reduce the computational speed, but it is still several times faster than FV heat pump model.

### 4.3. Discussion

Overall, the NAD heat pump model performed exceptionally well against the FV model, demonstrating remarkable accuracy and computational efficiency. Despite the inherent complexities in modeling
<table><caption>**Table 4**

Computation time for different validation simulations of heat pump.</caption><thead><tr><th rowspan="2">Validation simulation</th><th colspan="2">Computation time (s)</th><th rowspan="2">Achieved speedup</th></tr><tr><th>FV model</th><th>NAD model</th></tr></thead><tbody><tr><td>1 - 5000 s</td><td>7.581</td><td>0.039</td><td>194x</td></tr><tr><td>2 - 10000 s</td><td>13.893</td><td>0.060</td><td>233x</td></tr><tr><td>3 - 15000 s</td><td>21.608</td><td>0.063</td><td>343x</td></tr><tr><td>4 - 20000 s</td><td>27.099</td><td>0.085</td><td>319x</td></tr><tr><td>5 - 30000 s</td><td>41.394</td><td>0.155</td><td>267x</td></tr></tbody></table>

the dynamics of heat exchangers, the NAD model effectively captured essential thermal characteristics and transient behaviors. Furthermore, the NAD model consistently satisfied the governing conservation equations at each time instance, reinforcing its reliability and physical fidelity. By rigorously upholding underlying physical principles while employing neural networks to predict outlet conditions, the NAD model establishes a high degree of confidence in its predictive capability.

The NAD heat pump model meets the essential requirements of fast and accurate simulation, making it suitable for optimization, planning, and control frameworks. Since the NAD model closely replicates the behavior of the FV model, which is a thoroughly validated and widely accepted numerical approach with proven alignment to experimental data [9,48,49], it inherently benefits from the established credibility of the FV model. Consequently, the NAD model is anticipated to perform reliably in practical scenarios, indicating its strong potential for real-world application and integration into advanced control and optimization systems.

Training NAD models using experimental data and evaluating their performance under conditions beyond constant heat transfer coefficients remain open directions for future research. Generally, if experimental data are available, a common procedure would be to first fit numerical models of the static components, such as compressors and expansion valves, to experimental data. And for the heat exchangers, training data for the neural network can initially be generated using the adaptive physics-guided data synthesis process applied to synthetic data from the FV numerical model. Subsequently, another set of training data can be generated using the same adaptive approach but based on experimental data, assuming that inlet and outlet measurements of the heat exchangers are available. During the neural network training, higher weight is given to the data obtained from experimental measurements compared to the synthetic data. This hybrid training strategy significantly reduces the volume of experimental data required, while still capturing the key physical dynamics, thereby improving the generalizability and robustness of the NAD model predictions. Furthermore, a key advantage of the NAD model is its ability to perform continuous learning from real-time data after deployment.

## 5. Conclusion

In this study, an innovative neural accelerated model for the heat exchanger (condenser and evaporator) are proposed for the heat pump modeling framework. NAD model couples a simplified dynamic model for heat exchanger with a neural network as universal approximator, leveraging the strengths of numerical and machine learning models ensuring robustness and reliability. The dynamic NAD heat pump model showcased good performance in terms of both accuracy and speed compared to the state-of-the-art FV model. During the validation stage, the NAD model achieved an average acceleration of 271 times, compared to the FV model. It has demonstrated excellent accuracy compared to the FV model, with errors of approximately less than 0.4%, and an R² score of around 0.95, underscoring their high level of accuracy and reliability. Furthermore, the combination of simplified physical model and machine learning makes the NAD model more robust and interpretable. Since the physics is validated for each instance while solving the NAD models, it instills a sense of trust and enhances the

model's explainability, which are the great strengths of the NAD model. As for future directions, one potential approach could involve replacing the simplified heat exchanger model's ODE for separating wall with modified heat transfer properties. This adjustment aims to achieve results consistent with those obtained when solving for the wall, even without explicitly solving its ODE. Such a modification could further enhance the computational efficiency of the NAD model. Overall, these findings highlights that the NAD model is not only significantly reduce computation time but also maintain excellent precision while validating considerable physics, making them highly compatible with model control applications, optimization studies and long time period simulations.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

This work was carried out under the auspices of the ValaDoE chair at IMT Atlantique in partnership with Télécom Paris and Mines Saint Etienne, and was supported by Enedis, Région Pays de la Loire, Nantes Métropole and Akajoule.

## Data availability

Data will be made available on request.

## References

[1] José A.R. Parise, Simulation of vapour-compression heat pumps, Simulation 46 (2) (1986) 71–76, Sage Publications Sage CA: Thousand Oaks, CA.

[2] Braun James Edward, Methodologies for the Design and Control of Central Cooling Plants, The University of Wisconsin-Madison, 1988.

[3] Hui Jin, Jeffrey D. Spitler, A parameter estimation based model of water-to-water heat pumps for use in energy calculation programs, ASHRAE Trans. 108 (3) (2002) American Society of Heating, Refrigeration and Air Conditioning Engineers, Inc.

[4] Grald Eric W, MacArthur J. Ward, A moving-boundary formulation for modeling time-dependent two-phase flows, Int. J. Heat Fluid Flow 13 (3) (1992) 266–272.

[5] Xiang-Dong He, Sheng Liu, Haruhiko Asada, Moving-interface model of two-phase flow heat exchanger dynamics for control of vapor compression cycle, in: The 1994 International Mechanical Engineering Congress and Exposition, 1994, pp. 69–75.

[6] NBOL Pettit, M Willatzen, L. Ploug-Sørensen, A general dynamic simulation model for evaporators and condensers in refrigeration, part II: Simulation and control of an evaporator, Int. J. Refrig. 21 (5) (1998) 404–414.

[7] MacArthur JW, Grald EW, Prediction of cyclic heat pump performance with a fully distributed model and a comparison with experimental data, ASHRAE Trans. 93 (1987) Honeywell Corporate Systems Development Div. Minneapolis, MN (US).

[8] Rossi, TM, Braun JE, A real-time transient model for air conditioners, in: Proc. 20th International Congress of Refrigeration, vol. 743, Sydney, 1999.

[9] Bendapudi. Satyam, E. Braun, A. Groll, A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers, Int. J. Refrig. 31 (8) (2008) 1437–1452.

[10] Ajay Kumar Gupta, Pankaj Kumar, Ranjit Kumar Sahoo, Anoop Kumar Sahu, Sunil Kumar Sarangi, Performance measurement of plate fin heat exchanger by exploration: ANN, ANFIS, GA, and SA, J. Comput. Des. Eng. 4 (1) (2017) 60–68, Oxford University Press.

[11] Emad MS El-Said, Mohamed Abd Elaziz, Ammar H. Elsheikh, Machine learning algorithms for improving the prediction of air injection effect on the thermohydraulic performance of shell and tube heat exchanger, Appl. Therm. Eng. 185 (2021) 116471, Elsevier.

[12] Mohammad Hassan Shojaeefard, Javad Zare, Amir Tabatabaei, Hassan Moham-adbeigi, Evaluating different types of artificial neural network structures for performance prediction of compact heat exchanger, Neural Comput. Appl. 28 (2017) 3953–3965.

[13] Uguz Sinan, Osman Ipek, Prediction of the parameters affecting the performance of compact heat exchangers with an innovative design using machine learning techniques, J. Intell. Manuf. 33 (5) (2022) 1393–1417.
[14] Liang-Hui Zhi, Peng Hu, Long-Xiang Chen, Gang Zhao, Viscosity prediction for six pure refrigerants using different artificial neural networks, Int. J. Refrig. 88 (2018) 432–440.

[15] Neng Gao, Xuehui Wang, Yongmei Xuan, Guangming Chen, An artificial neural network for the residual isobaric heat capacity of liquid HFC and HFO refrigerants, Int. J. Refrig. 98 (2019) 381–387.

[16] Xuehui Wang, Ying Li, Yuying Yan, Edward Wright, Neng Gao, Guangming Chen, Prediction on the viscosity and thermal conductivity of hfc/hfo refrigerants with artificial neural network models, Int. J. Refrig. 119 (2020) 316–325.

[17] Yinjie Ma, Cheng Liu, E Jiaqiang, Xianwen Mao, Zhenhuan Yu, Research on modeling and parameter sensitivity of flow and heat transfer process in typical rectangular microchannels: From a data-driven perspective, Int. J. Therm. Sci. 172 (2022) 107356.

[18] Raffaele L. Amalfi, John Kim, Machine learning-based prediction methods for flow boiling in plate heat exchangers, in: 2021 20th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), IEEE, 2021, pp. 1131–1139.

[19] Changgui Xie, Gongxing Yan, Qiong Ma, Yasser Elmasry, Pradeep Kumar Singh, AM Algelany, Makatar Wae-hayee, Flow and heat transfer optimization of a fin-tube heat exchanger with vortex generators using response surface methodology and artificial neural network, Case Stud. Therm. Eng. 39 (2022) 102445.

[20] Liwei Zhou, Deepak Garg, Yue Qiu, Sung-Min Kim, Issam Mudawar, Chirag R. Kharangate, Machine learning algorithms to predict flow condensation heat transfer coefficient in mini/micro-channel utilizing universal data, Int. J. Heat Mass Transfer 162 (2020) 120351.

[21] Guangya Zhu, Tao Wen, Dalin Zhang, Machine learning based approach for the prediction of flow boiling/condensation heat transfer performance in mini channels with serrated fins, Int. J. Heat Mass Transfer 166 (2021) 120783.

[22] Xiao Zheng, Romuiao Yang, Qifan Wang, Yuchao Yan, Yu Zhang, Jiahong Fu, Zhentao Liu, Comparison of GRNN and RF algorithms for predicting heat transfer coefficient in heat exchange channels with bulges, Appl. Therm. Eng. 217 (2022) 119263.

[23] Adrián Mota-Babiloni, Alejandro Montanez Barrera, Juan Manuel Barroso-Maldonado, A.F. Bedoya-Santacruz, Correlated-informed neural networks: a new machine learning framework to predict pressure drop in micro-channels, 2022.

[24] Giovanni A Longo, Giulia Righetti, Claudio Zilio, Ludovico Ortombina, Mauro Zigliotto, J. Steven Brown, Application of an artificial neural network (ANN) for predicting low-GWP refrigerant condensation heat transfer inside herringbone-type brazed plate heat exchangers (BPHE), Int. J. Heat Mass Transfer 156 (2020) 119824.

[25] Matthew T Hughes, Sarah M Chen, Srinivas Garimella, Machine-learning-based heat transfer and pressure drop model for internal flow condensation of binary mixtures, Int. J. Heat Mass Transfer 194 (2022) 123109.

[26] AN Skrypnik, AV Shchelchkov, Yu F Gortyshov, I.A. Popov, Artificial neural networks application on friction factor and heat transfer coefficients prediction in tubes with inner helical-finning, Appl. Therm. Eng. 206 (2022) 118049.

[27] Saleh Hosseini, Amith Khandakar, Muhammad EH Chowdhury, Mohamed Ar-selene Ayari, Tawsifur Rahman, Moajjem Hossain Chowdhury, Behzad Vaferi, Novel and robust machine learning approach for estimating the fouling factor in heat exchangers, Energy Rep. 8 (2022) 8767–8776.

[28] Seyit Ahmet Kuzucanlı, Ceren Vatansever, Alp Emre Yaşar, Ziya Haktan Karad- eniz, Assessment of fouling in plate heat exchangers using classification machine learning algorithms, in: CLIMA 2022 Conference, 2022.

[29] Junjia Zou, Tomoki Hirokawa, Jiabao An, Long Huang, Joseph Camm, Recent advances in the applications of machine learning methods for heat exchanger modeling—a review, Front. Energy Res. 11 (2023) 1294531.

[30] Maja Rudolph, Stefan Kurz, Barbara Rakitsch, Hybrid modeling design patterns, J. Math. Ind. 14 (1) (2024) 3.

[31] Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, Alan Edelman, Universal differential equations for scientific machine learning, 2020, arXiv preprint arXiv: 2001.04385.

[32] Jishnu Ayyangatu Kuzhiyil, Theodoros Damoulas, W. Dhammika Widanage, Neural equivalent circuit models: Universal differential equations for battery modelling, Appl. Energy 371 (2024) 123692.

[33] Zhilu Lai, Charilaos Mylonas, Satish Nagarajaiah, Eleni Chatzi, Structural identi-fication with physics-informed neural ordinary differential equations, J. Sound Vib. 508 (2021) 116196.

[34] Fernando Arrais RD Lima, Carine M Rebello, Erbet A Costa, Vinícius V Santana, Marcellus GF de Moraes, Amaro G Barreto Jr., Argimiro R Secchi, Maurício B de Souza Jr., Idelfonso B.R. Nogueira, Improved modeling of crystallization processes by universal differential equations, Chem. Eng. Res. Des. 200 (2023) 538–549.

[35] R.N.N. Koury, L. Machado, K.A.R. Ismail, Numerical simulation of a variable speed refrigeration system, Int. J. Refrig. 24 (2) (2001) 192–200.

[36] Chasik Park, Honghyun Cho, Yongtaek Lee, Yongchan Kim, Mass flow charac- teristics and empirical modeling of R22 and R410a flowing through electronic expansion valves, Int. J. Refrig. 30 (8) (2007) 1401–1407.

[37] Satyam Bendapudi, James E Braun, Eckhard A. Groll, Dynamic modeling of shell-and-tube heat-exchangers: Moving boundary vs. finite volume, 2004.

[38] Quoilin Sylvain, Sustainable energy conversion through the use of organic rank- ine cycles for waste heat recovery and solar applications, 2011, ULiège-Université de Liège.

[39] Christopher Rackauckas, Qing Nie, Differentialequations.jl-a performant and feature-rich ecosystem for solving differential equations in Julia, J. Open Res. Softw. 5 (1) (2017) Ubiquity Press.

[40] Adriano Desideri, Bertrand Dechesne, Jorrit Wronski, Martijn Van den Broek, Sergei Gusev, Vincent Lemort, Sylvain Quoilin, Comparison of moving boundary and finite-volume heat exchanger models in the modelica language, Energies 9 (5) (2016) 339, MDPI.

[41] Eric W Lemmon, Ian H Bell, ML Huber, M.O. McLinden, NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP, version 10.0, in: National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, 2018.

[42] Ian H Bell, Jorrit Wronski, Sylvain Quoilin, Vincent Lemort, Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp, Ind. Eng. Chem. Res. 53 (6) (2014) 2498–2508, http://dx.doi.org/10.1021/ie4033999.

[43] J.M. Corberán, J. González, D. Fuentes, Calculation of refrigerant properties by linear interpolation of bidimensional meshes, in: Proceedings of the IRR Interna-tional Conferences on Commercial Refrigeration: Thermophysical Properties and Transfer Processes of Refrigerants, 2005.

[44] Joseph Al Khoury, Rabih Al Haddad, Ghiwa Shakrina, Christelle Bou Malham, Haytham Sayah, Chakib Bouallou, Maroun Nemer, A fast method for the calculation of refrigerant thermodynamic properties in a refrigeration cycle, Int. J. Refrig. 157 (2024) 98–108.

[45] Christopher Laughman, Yiming Zhao, Daniel Nikovski, Fast refrigerant property calculations using interpolation-based methods, 2012.

[46] Lixiang Li, Jesse Gohl, John Batteh, Christopher Greiner, Kai Wang, Fast calculation of refrigerant properties in vapor compression cycles using spline-based table look-up method (SBTL), in: The American Modelica Conference, Cambridge, MA, USA, 2018, pp. 77–84.

[47] Arturo Zavala-Río, Raúl Santiesteban-Cos, Reliable compartmental models for double-pipe heat exchangers: An analytical study, Appl. Math. Model. 31 (9) (2007) 1739–1752.

[48] Bendapudi Satyam, E. Braun, A. Groll, Dynamic model of a centrifugal chiller system—model development, numerical study, and validation, ASHRAE Trans. 111 (1) (2005).

[49] Javier Bonilla, Sebastian Dormido, Francois E. Cellier, Switching moving bound- ary models for two-phase flow evaporators and condensers, Commun. Nonlinear Sci. Numer. Simul. 20 (3) (2015) 743–768.