Study on Optimal Operation of Heat Pump Drying System Throughout the Entire Drying Process Based on the Material Drying Characteristics

Abstract

This study investigates the application of Heat Pump Drying (HPD) technology for drying agricultural products, aiming to address the practical inefficiency of HPD systems, which stems from the lack of an optimized operational strategy throughout the drying process. This study develops a mathematical model for a closed-loop HPD system. Tomato slices were selected as the research subject, and hot air-drying experiments were performed to determine their drying characteristics. The mathematical model was then used to simulate the effect of material moisture content fluctuations on HPD system performance during drying. Based on these drying characteristics, an optimal operational strategy was proposed. The results show that dynamically adjusting parameters such as evaporation temperature and air bypass ratio during different drying stages can significantly improve the system’s Specific Moisture Extraction Rate (SMER) and facilitate energy-efficient operation throughout the drying process. The average SMER values of the HPD system under the optimized strategy were 2.59 kg ∙ kW⁻¹∙h⁻¹ and 3.46 kg ∙ kW⁻¹∙h⁻¹ at drying temperatures of 60 °C and 80 °C, respectively. Additionally, the optimized operation reduced total electrical consumption by 31.60% and 32.87% compared to the constant evaporation temperature mode.

1. Introduction

The drying of agricultural products is a critical aspect of agricultural production and processing, but its high energy consumption has hindered the sustainable development of agriculture [1,2,3]. Heat pump drying (HPD) technology, known for its energy-saving and high-efficiency benefits, has been extensively applied in agricultural product drying [4,5,6,7,8,9]. As the demand for energy conservation and carbon reduction grows, enhancing the efficiency of heat pump drying technology, exploring its energy-saving potential, and improving overall energy efficiency have become key research priorities in agricultural science.

Numerous studies have focused on enhancing the energy efficiency of HPD systems [10,11,12,13]. These studies investigate the effects of key drying parameters, such as drying temperature [10] and air velocity [14], on HPD system performance through experiments and simulations. For instance, Zlatanovic et al. [10] investigated the effect of air recirculation rate on HPD system performance, systematically analyzing the variation trends of the Specific Moisture Extraction Rate (SMER), Specific Energy Consumption (SEC), Heat Pump Dryer Efficiency (HPDE), and Moisture Extraction Rate (MER). Zhu et al. [11] investigated the effects of drying temperature on the heat pump drying characteristics and system performance of Auricularia auricula. Their study found that the SMER of the system was maximized at a drying temperature of 55 °C, while the rehydration ratio was highest at 50 °C, leading to the best product quality. Singh et al. [12] developed an HPD system that combined closed and open circulation modes, significantly improving the system’s SMER by optimizing tray design and controlling air recirculation. Erbay et al. [14] optimized the operating parameters for olive leaf drying, determining that the optimal drying temperature was 53.43 °C with an air velocity of 0.64 m∙s⁻¹. Other studies have also focused on enhancing HPD system performance by optimizing system processes and improving components. Erbay et al. [15] conducted an exergy analysis and identified the condenser and compressor as key components affecting HPD performance, emphasizing the importance of their optimized design for improving system efficiency. Shen et al. [16] proposed a two-stage compression HPD system that operates in a two-stage compression mode at an ambient temperature of 0 °C and in a single-stage cycle mode at 20 °C, meeting the drying requirement of 70 °C. Liu et al. [17] investigated the drying performance of a closed HPD system with varying air flow ratios. Experimental results indicated that the bypass air duct strategy effectively reduced drying time and energy consumption. Using the bypass air duct, the time required to reduce the material’s moisture content to 20% was reduced by 30%, while energy consumption decreased by 15%. Cheng et al. [18] proposed an HPD system incorporating an external condenser and air bypass technology, improving system efficiency and operational robustness. At a drying temperature of 45 °C, its SMER increased by 17.3% compared to the baseline HPD system.

Understanding the drying kinetics of agricultural products is essential for linking material-level behavior to system-level operation. Due to the high initial moisture content of tomatoes, their drying mainly occurs in the falling-rate period, and the moisture transfer is strongly influenced by drying temperature, air velocity, and slice thickness. Several semi-empirical thin-layer models (e.g., Page, Lewis, Henderson-Pabis, and Midilli-Kucuk) have been applied to describe the moisture-ratio evolution of tomato slices [19,20,21]. Among these, the Page model provides a good balance between fitting accuracy and model simplicity, making it suitable for coupling drying kinetics with heat pump drying simulations and for designing dynamic operation strategies that respond to changing moisture loads.

However, most existing studies primarily focus on enhancing the performance of HPD systems under steady-state conditions, while fewer investigations address the dynamic variations in temperature and humidity during the drying process and their specific effects on system performance, especially dynamic optimization strategies informed by the drying kinetics of materials. In practical drying operations, the moisture content of materials fluctuates in complex patterns over time, resulting in continuous changes in the required cooling and heating loads. Therefore, investigating variable operating strategies for heat pumps to adapt to these load fluctuations is crucial for optimizing the overall performance of HPD systems throughout the drying process. This issue is urgent and forms the central focus of this study, which specifically employs experimentally determined drying kinetics of tomato slices to inform and validate dynamic HPD operation strategies.

2. Problem Statements

This study focuses on the HPD System with air drying in a closed loop, the most widely used type in industrial applications [22]. Its advantages include the lack of gas purification equipment, absence of exhaust emissions that pollute the environment, and independence from ambient temperature constraints. As shown in Figure 1, the heat pump drying system comprises an air loop and a heat pump cycle, with R134a used as the working fluid in the heat pump cycle. The state changes in the heat pump cycle and the air loop are illustrated in Figure 2a,b, respectively.

In the heat pump cycle, the condenser temperature is fixed to maintain stable temperature and humidity conditions at the drying chamber inlet. The refrigerant enters the compressor in a superheated state at point 2 after passing through the regenerator. The refrigerant then exits the compressor at point 3 and enters the condenser in a superheated state. After condensation, the working fluid re-enters the regenerator before undergoing isenthalpic expansion in the expansion valve (5–6), transitioning from condenser pressure to evaporator pressure. Finally, the refrigerant enters the evaporator, absorbs heat from the air, evaporates, and exits the evaporator at point 1. The evaporation temperature varies depending on the drying time.

In the air loop, air enters the drying chamber at point 1a, undergoes an isenthalpic humidification process, and reaches state 2a at the outlet. Subsequently, a portion of the moist air is directed into the evaporator for dehumidification and cooling, where its temperature drops below the dew point at state 2a’ before exiting the evaporator at point 4a. This cooled air is then mixed with another portion of the air that bypassed the evaporator, and the combined stream reaches state 5a. The mixed air at state 5a then enters the condenser, where it is heated to the required temperature (5a–1a), completing the air-drying cycle.

A key issue with the HPD system in practical applications is that it operates at low efficiency for most of the drying cycle. The main reason is that, although the drying characteristics of the material change over time, the HPD system lacks precise control strategies and operates only under rated conditions. The specific details are as follows.

On the heat demand side (drying chamber), the drying characteristics of the material change over time, typically in three periods: preheating, constant rate, and falling rate [23]. In the preheating period, the material is exposed to a high-temperature environment, causing its temperature to gradually rise. During the constant rate period, the drying rate remains relatively stable. In the falling rate period, the drying rate gradually decreases until drying is complete.

On the heat supply side (heat pump cycle), the ideal operation of the heat pump should dynamically adjust its load in response to variations in heat demand. There are two adjustment methods: first, activating the bypass on the evaporator side to reduce airflow, thereby lowering the thermal load and power consumption of system components; second, controlling the expansion valve opening to decrease the evaporation temperature, ensuring that the moist air cools below the dew point for effective dehumidification. Theoretically, as drying progresses, the evaporator-side air bypass ratio and evaporation temperature can be dynamically adjusted to meet the changing drying and dehumidification demands. However, in most cases, such optimization remains unrealized due to the lack of variable operating strategies. In other words, due to the complexity and variability of material drying characteristics, operators find it challenging to determine the optimal operating strategy using heuristic and empirical methods. When the supplied heat exceeds the required amount, excess energy results in low efficiency. Conversely, when the supplied heat is insufficient, the drying quality of the material may be compromised. As a result, operators are often forced to maintain the heat pump at rated conditions, supplying excess heat to ensure drying quality, leading to significant energy waste. Specifically, when the heat pump system operates continuously under rated conditions during the early drying stage, the high dehydration rate of the material results in high relative humidity at the drying chamber outlet. In this case, the air can be easily cooled below the dew point in the evaporator, enabling effective dehumidification. However, as drying progresses into later stages, the moisture removal rate declines, leading to air cooling without effective dehumidification and consequently resulting in energy waste. As a result, the performance of the HPD system deteriorates markedly, as indicated by a reduced SMER.

Therefore, this study aims to explore the optimal operating strategy for an HPD system based on material drying characteristics, providing guidance for operators to implement variable load control and maximize the system’s SMER. From this perspective, a mathematical model of the HPD system is developed, and a parametric analysis of the bypass ratio and evaporation temperature is conducted. The influence of evaporation temperature and air bypass ratio on the system’s SMER at different drying stages is identified, leading to optimal operating parameters for each stage. Ultimately, a comprehensive optimal operating strategy for the entire drying process of the HPD system is established.

3. Mathematical Model

3.1. Numerical Simulation

This study utilizes Aspen Plus to simulate the closed heat pump drying (HPD) system. Previous studies by Zhang [24], Yang [25], and others have shown that Aspen Plus is well-suited for simulating heat pump systems, with most deviations between simulated and experimental results remaining below 10%.

To accurately represent the process flow shown in Figure 1, appropriate modules were selected from Aspen Plus to establish the process model, as shown in Figure 3.

Table 1. Modules of different components.

Component	Module
Condenser	HeatX
Evaporator	HeatX + Flash2
Regenerator	HeatX
Expansion valve	Valve
Compressor	Compr
Drying chamber	Heater + FSplit
Mixer	MIXER
Cooler	Heater

details the module selection for each component of the HPD system. The system is divided into two units: the heat pump cycle and the air loop. Considering the discrepancies between the actual HPD process and the simulation, several simplifications and assumptions were applied in the Aspen Plus model:

1.

All unit modules of the heat pump system operate under steady-state conditions [26];

2.

Pressure drops in heat exchangers and heat losses in the drying chamber are neglected [27];

3.

Working fluid leaving the evaporator and condenser is saturated vapor and saturated liquid, respectively [28];

4.

The isentropic efficiency of the compressor is assumed to be 75% [29].

The simulation involves H₂O and air in the air circulation loop, and R134a as the refrigerant in the heat pump cycle. The Peng–Robinson (PR-BM) equation of state was used to evaluate the thermodynamic properties of components in the air loop, as it is suitable for non-polar or weakly polar mixtures. Thermophysical properties in the heat pump cycle were determined using the REFPROP method [30]. Aspen Plus simulations follow a sequential modular approach, solving each unit operation in a specified sequence. For recirculating streams, iterative calculations are performed until convergence is achieved. The Wegstein method, widely recognized for its speed and reliability, is used for tear stream convergence. The maximum number of iterations is set to 30, with stream A5 specified as the tear stream.

In the simulation, the inlet air conditions of the drying chamber were kept constant by coordinating the compressor frequency and the air bypass ratio. When the evaporation temperature changed, the compressor speed was adjusted to vary the system pressure ratio and condensation temperature, while the bypass ratio was modified to stabilize the inlet air temperature and humidity. This control strategy reflects the practical operation of a variable-frequency heat pump dryer. For instance, at a drying temperature of 60 °C, the circulating air flow rate was fixed at 400 kg · h⁻¹, and the relative humidity of the drying air was maintained at 15%. The condenser delivers high-temperature air at 60 °C, which removes moisture from the material in the drying chamber. This process is simplified in the model by mixing the stream WIN in the Chamber, with the mass flow rate of WIN determined by the drying rate. The heat required for drying is supplied by the refrigerant in the heat pump system. The condensation temperature of the refrigerant varies with the drying rate of the material, while a constant minimum temperature difference of 10 °C is maintained in the condenser to ensure effective heat exchange. The compressor operates at the saturation pressure corresponding to the condensation temperature. A heat regenerator is installed at the condenser outlet to reduce throttling losses. The pinch point in the regenerator occurs between stream H4 and stream H2, with the pinch temperature difference set to 10 °C. Similarly, a pinch temperature difference of 10 °C is specified for the evaporator. An air bypass is introduced on the evaporator side, with the air bypass ratio adjusted based on the evaporation temperature. The expansion valve operates at the saturation pressure corresponding to the evaporation temperature.

Table 2. Parameters of the HPD system under specific working conditions.

Parameter	Value	Unit
Drying Temperature	60	°C
Drying Relative Humidity	0.15	%
Circulating Air Pressure	101.3	kPa
Circulating Air Flow Rate	400	kg ∙ h−1
Drying Rate	3.085	kg ∙ h−1
Condensation Temperature	59.95	°C
Compressor Pressure	1679.75	kPa
Compressor Isentropic Efficiency	0.75	-
Evaporation Temperature	12	°C
Air Bypass Ratio	0.241	-
Refrigerant Mass Flow Rate	70.07	kg ∙ h−1

summarizes the operating parameters for the HPD system.

It should be noted that the evaporation temperature in Table 2 refers to the nominal initial condition used to establish the baseline operating point of the HPD system. In the subsequent stage-wise optimization analysis (Section 4.2), the evaporation temperature was treated as a dynamic variable determined by the simulation. For each drying stage, the material drying rate obtained from experiments was used to define the air humidity load, and the Aspen Plus model was solved iteratively to identify the evaporation temperature that maximizes the SMER.

The Aspen Plus simulation was developed based on a steady-state thermodynamic model of a closed heat pump dryer. The system consists of a compressor, condenser, expansion valve, and evaporator connected to an air-circulation loop. Each component was modeled using fundamental energy balance equations:

$$Q_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}=\left(h_4-h_3\right)Q_{\mathrm{r}134\mathrm{a}}$$

(1)

$$W_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}}=\left(h_3-h_2\right)Q_{\mathrm{r}134\mathrm{a}}$$

(2)

$$Q_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}=\left(h_1-h_6\right)Q_{\mathrm{r}134\mathrm{a}}$$

(3)

where $Q_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ (kW) is the condenser heat load, $W_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}}$ (kW) is the compressor power, $Q_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ (kW) is the evaporator heat load, and $Q_{\mathrm{r}134\mathrm{a}}$ (kg · h⁻¹) is the refrigerant mass flow rate. $h_1$, $h_2$, $h_3$, $h_4$ and $h_6$ are the specific enthalpies (kJ · kg⁻¹) of refrigerant R134a at the respective state points.

The expansion valve was modeled as an isenthalpic throttling process:

$$h_6=h_5$$

(4)

The air-circulation loop was closed by coupling the outlet humidity and temperature from the drying chamber to the evaporator inlet. The mass and energy balances for the circulating air streams were satisfied as

$$d_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{o}\mathrm{u}\mathrm{t}}=d_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{i}\mathrm{n}}+{\displaystyle \frac{M_{\mathrm{w}}}{Q_{\mathrm{v}}}}$$

(5)

$$h_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{o}\mathrm{u}\mathrm{t}}=h_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{i}\mathrm{n}}$$

(6)

where $d_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{o}\mathrm{u}\mathrm{t}}$ (kg water∙kg⁻¹ dry air) represents the absolute humidity of the air at the drying chamber outlet, $d_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{i}\mathrm{n}}$ (kg water ∙ kg⁻¹ dry air) represents the absolute humidity of the air at the drying chamber inlet, $M_{\mathrm{w}}$ (kg ∙ h⁻¹) is the material drying rate, $Q_{\mathrm{v}}$ (kg ∙ h⁻¹) is the air flow rate, and $h_{\mathrm{d}\mathrm{r}\mathrm{y},\mathrm{o}\mathrm{u}\mathrm{t}}$ (kJ ∙ kg⁻¹) represents the enthalpy of the air at the drying chamber outlet.

3.2. Evaluation Criterion

The primary performance metric for the HPD system is SMER (kg · kW⁻¹ · h⁻¹) [31,32], which indicates the amount of moisture removed from the material per unit of energy consumed. It is defined as the ratio of the moisture removal rate from the material to the energy consumed by the heat pump drying system.

$$SMER={\displaystyle \frac{M_{\mathrm{w}}}{W_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}}}}$$

(7)

To assess the economic benefits of the system, energy savings during the drying process are quantified by the amount of standard coal saved, denoted as $∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$ (kg ∙ h⁻¹). $∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$ represents the difference in standard coal consumption between conventional hot air drying and the HPD system under identical drying conditions.

$$∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}=C_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}-C_{\mathrm{h}\mathrm{p}\mathrm{d}}$$

(8)

$$C_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}=Q_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}{\lambda }_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$$

(9)

$$C_{\mathrm{h}\mathrm{p}\mathrm{d}}=W_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}}{\lambda }_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$$

(10)

where $C_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}$ (kg ∙ h⁻¹) denotes the equivalent amount of standard coal required for hot air drying, $C_{\mathrm{h}\mathrm{p}\mathrm{d}}$ (kg ∙ h⁻¹) denotes the equivalent amount of standard coal consumed by the HPD system, and ${\lambda }_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$ (kg · kW⁻¹ · h⁻¹) is the conversion factor used to translate electricity consumption into equivalent standard coal consumption. The conversion factor for electricity is based on the Chinese national standard and is 0.32469 kg · kW⁻¹ · h⁻¹ [33].

To evaluate the environmental benefits of the system, the CO₂ reduction, denoted as $∆{CO}_2$ (kg ∙ h⁻¹) is selected as the evaluation metric.

$$∆{CO}_2=∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}{E}_{{\mathrm{C}\mathrm{O}}_2}$$

(11)

where $E_{{\mathrm{C}\mathrm{O}}_2}$ (kg ∙ kg⁻¹) represents the CO₂ emissions per kilogram of standard coal, with a value of 2.493 kg · kg⁻¹.

4. Case Study

This paper presents a case study on the optimization of HPD system operation using tomato slices as the drying material. Specifically, the drying characteristics of tomato slices were first determined through hot air-drying experiments. Based on the acquired drying characteristics, a mathematical model was used for parameter analysis and optimization of the HPD system, aiming to maximize SMER.

4.1. Tomato Drying Experiments

The tomatoes used in the experiment were purchased directly from a supermarket, selecting those with a diameter of 80 ± 10 mm. The tomatoes’ surfaces were cleaned to remove dirt. After cleaning, the tomatoes were placed in a room-temperature environment to drain excess moisture from the surface, then sliced into pieces with a thickness of 5 ± 0.5 mm. The surface moisture of the slices was absorbed with filter paper before setting them aside as experimental samples. The experiment was conducted using a high-temperature electric hot air drying oven (China Zhengzhou Ansheng Scientific Instrument Co., Ltd., Zhengzhou, China, Model 101-2AB), with a working chamber size of 550 × 550 × 450 mm and a temperature range of RT + 10~250 °C. Sample mass measurements were performed using an electronic balance (Model JJ1523BC) with a maximum capacity of 1520 g and accuracy of 0.01 g.

The tomato slices were evenly placed in Petri dishes with labeled parameters and placed in the drying oven for the drying experiment. Two drying temperatures (60 °C and 80 °C) were set, with airflow velocity kept constant at 3.0 m∙s⁻¹. During the drying process, samples are periodically extracted from the dryer to measure their weight. This procedure must be completed within 15 s. Under the 60 °C drying condition, samples were removed and weighed every 5 min for the first 20 min to accurately capture the transient changes in sample mass. Thereafter, the sampling interval was extended to 20 min to minimize disturbances caused by frequent handling. Drying was considered complete when the change in sample mass was less than 0.5% of the initial value. Under the 80 °C condition, samples were weighed every 5 min during the first 20 min and subsequently every 10 min until the same drying-end criterion was met. Each drying experiment is conducted in triplicate to minimize random errors, and the drying kinetics are determined by averaging the results of the three trials. Figure 4 shows the appearance of tomato slices at different drying times, illustrating the gradual shrinkage and color change during the drying process.

The experimental results of moisture content change during the drying process are presented in terms of dimensionless moisture content $MR$, defined as

$$MR={\displaystyle \frac{M-M_{\mathrm{e}}}{M_0-M_{\mathrm{e}}}}$$

(12)

where $M$ is the dry basis moisture content, $M_0$ is the initial dry basis moisture content of the sample, and $M_{\mathrm{e}}$ is the equilibrium moisture content of the sample. $MR$ ranges from 0 to 1. The equation is defined as: $M=\left(m-m_{\mathrm{d}}\right)/m_{\mathrm{d}}$, where $m$ (g) is the sample mass, and $m_{\mathrm{d}}$ (g) is the mass of dry matter. When the drying time is relatively long, the equilibrium moisture content $M_{\mathrm{e}}$ approaches zero. Accordingly, in this study, $M_{\mathrm{e}}$ is assumed to be zero [34], allowing $MR$ to be simplified as follows:

$$MR={\displaystyle \frac{M}{M_0}}$$

(13)

Figure 5 illustrates the dimensionless water content of tomato slices over time at different drying temperatures. As shown in Figure 5, the dimensionless moisture content decreases with increasing drying time, and the rate of decline gradually slows. Higher temperatures result in a faster decrease in the dimensionless moisture content. At drying temperatures of 80 °C and 60 °C, the drying times to completion were 250 min and 400 min, respectively.

4.2. Optimized Operation of HPD System Under Variable Operating Conditions

4.2.1. Drying Characteristic Curves of Material Samples

The Page model was selected after comparison with other empirical thin-layer models such as Henderson–Pabis and Midilli–Kucuk, which have also been used for tomato drying. Among these, the Page model provided the most accurate description of the moisture ratio variation while maintaining a simple two-parameter form suitable for coupling with system-level simulation [35]. The parameters of the Page model were obtained using the nonlinear least squares (NLS) method to minimize the difference between the predicted and experimental moisture ratios. The regression analysis was performed using Origin 2022. The mathematical expression is

$$MR=\exp \left(-k{t}^n\right)$$

(14)

where $k$ (min⁻¹) and $n$ are the fitting parameters, and $t$ (min) is the drying time. The parameters $k$ and $n$ obtained from the fitting at different drying temperatures are listed in

Table 3. Fitted Parameters of Page Model.

Drying Temperature	k	n	R2	Sum of Squared Residuals
60 °C	0.00278	1.20002	0.99290	9.4417 × 10−4
80 °C	0.00333	1.27248	0.99465	6.3922 × 10−4

. The model fitting curves are plotted in Figure 5.

In the HPD system simulation, the material drying rate $M_w$ is an input parameter used to calculate the humid air state at the drying chamber outlet. $M_w$ can be expressed as the rate of decrease in material mass:

$$M_{\mathrm{w}}=m_{\mathrm{d}}{M}_0{\displaystyle \frac{\mathrm{d}MR}{\mathrm{d}t}}$$

(15)

Substituting Equation (14) into Equation (15) yields the calculation expression for $M_w$:

$$M_w=-kn{m}_{\mathrm{d}}{M}_0{t}^{n-1}\exp \left(-k{t}^n\right)$$

(16)

Figure 6 illustrates the variation in the dehydration rate $M_{\mathrm{w}}$ of tomato samples over time at different drying temperatures. As shown in Figure 6, the $M_{\mathrm{w}}$ of the tomato samples is relatively high at the beginning of the drying process. As drying progresses, the surface moisture of the material gradually decreases, and the material drying rate significantly declines.

4.2.2. System Simulation and Optimization of Operating Parameters

Based on the previously obtained drying characteristic curves, this case performs parameter analysis and optimization of the HPD system at different drying stages, starting from the beginning of drying with a time interval of 20 min. Specifically, the condensation temperature of the heat pump cycle was kept constant (i.e., drying chamber inlet temperatures of 80 °C and 60 °C), while the evaporation temperature and bypass ratio were adjusted to accommodate changes in the drying characteristics (i.e., material drying rate). To meet these conditions, the evaporation temperature, cycle flow rate, and air bypass ratio of the heat pump system varied accordingly.

From all the simulation results, the representative operating conditions at 60 °C (drying times of 40 min, 80 min, 120 min, 160 min, 200 min, 240 min, 280 min, 320 min, 360 min) and at 80 °C (drying times of 40 min, 60 min, 80 min, 100 min, 120 min, 140 min, 160 min, 180 min, 200 min) were selected and plotted as representatives in Figure 7 and Figure 8, respectively.

For the drying temperature of 60 °C, Figure 7 shows the curves of system SMER variation with evaporation temperature at different drying stages. The SMER shows an initial increase followed by a decrease as the evaporation temperature rises, with a peak occurring in a certain intermediate evaporation temperature range. This indicates that, for the majority of the drying time, there exists an evaporation temperature that maximizes the SMER, representing the system’s optimal operating state.

For a drying temperature of 80 °C, Figure 8 shows the curves of system SMER variation with evaporation temperature at different drying stages. In the early drying stage (as shown in Figure 8a), due to the high moisture content of the material, the system’s dehumidification demand is high, and the SMER increases as the evaporation temperature rises. As the drying process progresses (as shown in Figure 8b–i), with the material moisture content gradually decreasing, the trend of SMER shifts. In the subsequent drying stages, the variation trend of SMER is like that shown in Figure 7.

Based on the results, it can be concluded that throughout the drying process, the HPD system should operate at the optimal evaporation temperature at each stage, along with the corresponding air bypass ratio, compression ratio, refrigerant flow rate, and air flow rate to achieve maximum SMER.

Figure 9 illustrates the variation curves of the optimal evaporation temperature and the optimal air bypass ratio on the evaporator side of the HPD system throughout the entire drying process, shown in Figure 9a,b, respectively. Together, these two curves constitute the operational profile for optimal performance of the HPD system, representing the optimal operating strategy. The optimal evaporation temperature remains constant initially, then gradually decreases as the material’s drying rate decreases, eventually stabilizing. At a drying temperature of 60 °C, the optimal evaporation temperature decreases from 12.5 °C at the start of drying to 1 °C, while at 80 °C, it decreases from 29 °C to 17 °C. Higher drying temperatures result in higher optimal evaporation temperatures and a greater range of change in the optimal evaporation temperature.

Figure 10 presents the SMER curve of the HPD system over the entire drying process under the optimal operating strategy. Combining Figure 9 and Figure 10 shows that the SMER value is influenced by both evaporation temperature and the material’s drying rate. At a constant evaporation temperature, SMER varies with the material’s drying rate. Figure 10 shows that SMER initially increases and then decreases. Overall, higher drying temperatures result in greater SMER values for the system. In the later stages of drying, SMER decreases significantly with time, and the rate of decrease is higher at higher drying temperatures.

For the conditions in this case, the average SMER values at 60 °C and 80 °C are 2.59 kg ∙ kW⁻¹ ∙ h⁻¹ and 3.46 kg ∙ kW⁻¹ ∙ h⁻¹, respectively. At these drying temperatures, the SMER values at the end of drying decrease by 50.7% and 51.9%, respectively, compared to those at the maximum drying rate.

4.2.3. Analysis of Energy-Saving Effects of Optimized Operation

In the intermittent drying process, as the material’s moisture content decreases, the optimal evaporation temperature for the HPD system also changes. When the HPD system operates under fixed conditions, its performance in the later stages of drying significantly deviates from the optimal state. To quantify the energy-saving benefit of the proposed dynamic operation strategy, a reference baseline known as the “constant evaporation temperature mode” was defined. In this mode, the HPD system operates under rated steady-state conditions without dynamic control. The evaporation temperature was kept constant at 14 °C for the 60 °C drying condition and 29 °C for the 80 °C condition, corresponding to the initial nominal values. The air bypass channel was completely closed (air bypass ratio = 0), and all circulating air passed through the evaporator during the entire drying process.

Figure 11a,b display the SMER versus drying time curves for the HPD system under both the optimized operation and constant evaporation temperature modes. The results indicate that under the optimized operation mode, the system’s SMER increases significantly throughout the drying process, particularly in the later stages. The optimized operation effectively prevents a significant decrease in SMER. At drying temperatures of 60 °C and 80 °C, the HPD system’s total electrical consumption was reduced by 31.60% and 32.87%, respectively, demonstrating the energy-saving effect of the optimized operation.

Based on Equations (5) and (8), the values of $∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$ and $∆{CO}_2$ for the HPD system are calculated to further verify the synergistic advantages of the optimized operating strategy in both economic and environmental benefits. At a drying temperature of 60 °C, $∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$ under the optimized operating mode reaches 23.07 kg · h⁻¹, marking an 8.67% increase compared to 21.23 kg · h⁻¹ under the constant operating condition. Meanwhile, $∆{CO}_2$ increases from 52.94 kg · h⁻¹ to 57.50 kg · h⁻¹. When the drying temperature is raised to 80 °C, $∆S_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}$ under the optimized operation reaches 17.87 kg · h⁻¹, reflecting a 9.70% improvement over 16.29 kg · h⁻¹ under the constant operating condition. Meanwhile, $∆{CO}_2$ increases from 40.60 kg · h⁻¹ to 44.55 kg · h⁻¹. These results indicate that the HPD system, when operated under the optimized strategy, improves energy utilization efficiency while simultaneously exhibiting favorable economic and environmental performance.

The optimized operating strategy proposed in this study enhances system performance while maintaining constant temperature and humidity during drying, thereby preventing drastic fluctuations in the drying environment caused by varying in operating parameters. This strategy contributes to the stability of the drying process and ensures consistent product quality. Moreover, by identifying the drying characteristics of a specific material and applying the modeling and optimization method proposed in this study, the optimal operating strategy for the material in an HPD system can be developed, demonstrating strong engineering adaptability and practical applicability.

5. Conclusions

This study analyzes the dynamic performance changes in the HPD system at various drying stages through a mathematical model and experimental measurements, and proposes an optimal operating strategy based on the material’s drying characteristics. The main conclusions are as follows:

• Necessity of optimized operation: In practical applications, the HPD system often operates inefficiently due to the lack of dynamic control strategies. Dynamic control of the optimal evaporation temperature can significantly improve system performance, particularly in the later drying stages. Optimized operation effectively prevents a substantial decrease in SMER.
• Optimal operation strategy: Based on a case study, an optimal operation strategy for the entire drying process of the HPD system, tailored to the material’s drying characteristics, is proposed. During drying, the system should dynamically adjust the evaporation temperature and air bypass ratio in response to changes in the material’s drying rate. The simulated results show that optimal operation significantly improves the system’s SMER and reduces the total energy consumption during drying.
• Significant energy-saving effects: At drying temperatures of 60 °C and 80 °C, the HPD system with the optimized operating strategy reduced the total electrical consumption by 31.60% and 32.87%, respectively, compared to the constant evaporation temperature mode.

This study provides a theoretical basis and practical guidance for the efficient operation of HPD systems, contributing to energy conservation and carbon reduction in agricultural drying, and promoting the widespread application of HPD technology in drying agricultural products.

6. Limitations

This study has several limitations that should be noted. First, the drying chamber and heat pump cycle were modeled separately under steady-state assumptions, without considering transient coupling effects. Second, the control of evaporation temperature and air bypass ratio was idealized, and the dynamic response of the control system was not analyzed. Third, the experimental validation was conducted only for tomato slices, which may limit the general applicability of the proposed strategy to other materials. Future work will address these aspects through coupled transient modeling and extended experimental validation.