Comprehensive Performance Modeling and Evaluation Method for Machine-Tool Thermal Control Plates Based on an Equivalent Thermal Resistance Network

Abstract

To address the coupled challenge of heat-transfer enhancement and energy consumption in machine-tool temperature control plates under high-flow-rate conditions, a comprehensive performance evaluation method based on an equivalent thermal resistance network is developed. By introducing heat-transfer power, equivalent total thermal resistance, and a coefficient of performance (COP), the thermal performance and energy cost are quantitatively characterized. Building upon established thermal resistance modeling approaches, the method provides a systematic framework for performance evaluation. The effects of inlet flow rate and heat-source temperature are investigated using CFD under consistent conditions, and experimental validation is conducted. The results show that increasing the flow rate enhances heat transfer but exhibits diminishing returns, while the rapidly increasing pressure drop reduces energy efficiency. Increasing the heat-source temperature mainly improves heat-transfer power by strengthening the temperature difference, with a limited impact on thermal resistance. Good agreement among theoretical, numerical, and experimental results confirms the validity and engineering applicability of the proposed method.

1. Introduction

In precision and ultra-precision machining processes, thermal deformation of machine-tool structures is one of the key factors affecting machining accuracy and long-term stability. Previous studies have shown that the geometric errors induced by thermal deformation of machine-tool structures can account for approximately 40–70% of the total machining error [1,2,3,4]. Therefore, implementing effective temperature control on key structural components is an important means of enhancing the thermal stability and machining accuracy of machine tools. For components such as spindles and feed systems, where internal flow channels can be arranged, internal circulation cooling is typically employed to achieve direct temperature control, whereas for structural components such as columns, which are inconvenient for the machining of internal flow channels, external temperature control devices are commonly used for indirect temperature regulation, among which temperature control plates are typical heat exchange units. The heat-transfer capability of temperature control plates and the associated flow-induced energy consumption directly affect the effectiveness of thermal error suppression and the overall energy efficiency of the system [5,6,7,8].

In recent years, extensive research has been conducted on thermal management and heat-transfer enhancement of temperature control structures. In the field of active thermal control, Zhang et al. [9] combined PID control with neural networks to achieve adaptive temperature regulation of spindles, while Zheng et al. [10,11] proposed active temperature control strategies based on real-time feedback mechanisms to reduce thermal errors. Liu et al. [12] further introduced intelligent optimization algorithms to improve thermal balance performance. These approaches mainly focus on regulating temperature fields through control strategies.

In terms of heat-transfer enhancement, structural optimization of flow channels has been widely studied. Salem et al. [13] designed spiral flow channels to significantly improve heat-transfer efficiency, while Zhang et al. [14] proposed cosine-shaped channels based on field synergy theory, achieving notable enhancement in heat-transfer performance. Wang et al. [15] developed parallel serpentine channel configurations to improve temperature uniformity and reduce the pressure drop. However, such structural optimization methods often lead to increased flow resistance. Similarly, turbulence-inducing structures have been extensively investigated. Feng et al. [16] and Li et al. [17] demonstrated that vortex-generating structures can effectively enhance convective heat transfer, while Kong et al. [18] optimized baffle configurations under different Reynolds numbers. Nevertheless, these approaches generally introduce additional pressure loss, which adversely affects system energy efficiency [19].

At the modeling level, thermal resistance network methods and thermodynamic optimization theories have been widely applied to heat-transfer systems. Chen et al. [20] established a coupled thermal model based on entropy generation minimization, while Zhou et al. [21] optimized heat-exchanger performance using minimum entropy production principles. Minaei et al. [22] developed a hybrid heat capacity–thermal resistance model for predicting heat-transfer performance. Wang et al. [23,24] further proposed local conduction–convection thermal resistance networks to evaluate the contributions of different heat transfer mechanisms. In addition, Wang et al. [25] analyzed the influence of geometric parameters on dissipation thermal resistance and constructed a multi-objective optimization model based on entransy dissipation theory, achieving improved comprehensive performance compared with traditional design criteria.

Despite these efforts, existing studies are often conducted separately in terms of structural optimization, heat-transfer enhancement, and performance evaluation. For machine-tool temperature control plates, the availability of a systematic modeling approach that integrates heat-transfer capability and energy consumption characteristics for engineering-oriented calculation and comparison remains limited.

To address the coexistence of heat-transfer enhancement and energy consumption constraints encountered by machine-tool temperature control plates under high-flow-rate conditions, this study establishes a comprehensive performance modeling and evaluation method oriented toward engineering calculation and comparative analysis. First, based on the concept of equivalent thermal resistance, an equivalent thermal resistance network is constructed to form a heat-transfer power calculation framework, within which performance indicators such as the heat-transfer power, equivalent total thermal resistance, and energy efficiency ratio are uniformly defined, and a comprehensive performance evaluation factor is introduced to achieve an integrated characterization of heat-transfer enhancement and flow resistance penalty. Second, in combination with numerical simulation methods, the effects of the inlet flow rate and heat-source conditions on the heat transfer and flow characteristics of temperature control plates are analyzed, key dimensionless parameters are extracted, and correlated expressions are established. Finally, a comprehensive experimental platform is developed to validate the proposed method, providing a basis for operating-condition matching and engineering application.

2. Experimental and Numerical Simulation Methods

2.1. Experimental Platform and Testing Methods

To obtain data on the heat-transfer performance and flow characteristics of the temperature control plate under different operating conditions, a comprehensive performance-testing experimental platform for the temperature control plate was established. The experimental platform mainly consists of a multi-loop differential active temperature control system, a temperature acquisition system, and a constant-temperature electric heating device; its overall configuration is shown in Figure 1. The multi-loop differential active temperature control system is used to provide stable coolant circulation conditions, with a circulation flow-rate adjustment range of 0–10 L/min and a circulation temperature range of 15–30 °C. The outlet temperature of the coolant is regulated by a PID control strategy, while the flow rate is precisely controlled by a throttling valve. The pressure loss between the inlet and outlet of the temperature control plate is measured using pressure gauges with an accuracy class of 0.4.

The temperature acquisition system consists of a LabVIEW platform, an NI data acquisition card, and PT100 temperature sensors, enabling real-time measurement of coolant temperatures at different locations along the flow channels, as well as at the inlet and outlet, with a sampling frequency of 1 Hz. Heat-source loading is realized using a constant-temperature electric heating plate, with a temperature control range from ambient temperature +5 to 420 °C and a maximum output power of 3000 W. Considering that the measurement accuracy of the temperature sensors is ±0.15 °C, the surface temperature of the heating plate is uniformly set to 50 °C during the experiments to reduce the influence of relative error on the experimental results. The structural configuration and geometric characteristics of the temperature control plate are illustrated in Figure 2. The temperature control plate consists of a heat-transfer plate and a flow plate, with the coolant flowing through the internal channels to achieve forced convective heat transfer. The main geometric parameters of the temperature control plate are listed in

Table 1. Temperature control board size parameter table.

Model Parameter	Value
Length (L, m)	1.784
Heat Transfer Plate Thickness (δ, m)	0.008
Overall Thickness (H, m)	0.028
Overall Width (W, m)	0.071

.

2.2. Establishment of the Numerical Simulation Model

To establish a computable numerical model and ensure consistency with the experimental operating conditions, reasonable simplifications are applied to the internal flow and heat-transfer processes of the temperature control plate, and the following basic modeling assumptions are adopted: the flow within the channels is steady; the thermophysical properties of the coolant are treated as constants; the coolant is assumed to be an incompressible fluid; the fluid temperature at each cross-section perpendicular to the main flow direction is represented by its bulk-average value for engineering modeling purposes; the heat-transfer plate and the flow plate are considered as an integrated structure, and the contact thermal resistance between them is neglected; the channel walls are assumed to be hydraulically smooth; the ambient temperature is maintained at a constant value of 20 °C; and the effect of thermal radiation heat transfer within the channels is neglected. Axial heat conduction along the flow direction is neglected, as the heat-transfer process is dominated by forced convection under the investigated operating conditions, where convective transport is significantly stronger than axial conduction.

Based on the above assumptions, an equivalent computational model of the temperature control plate is established, and its structural configuration is shown in Figure 3. During the numerical calculation, the heat-transfer process in the temperature control plate can be regarded as a forced convection problem in internal channels, and the continuity equation, momentum equation, and energy equation are solved in a coupled manner using Ansys Fluent 2022r2 software. The standard k–ε model is adopted due to its robustness and computational efficiency for engineering heat-transfer problems; it has been shown to provide reasonable predictions in similar systems with complex flow paths [26]. The temperature control plate is made of aluminum alloy, and No. 2 spindle oil is selected as the cooling medium; its thermophysical properties at 20 °C are listed in

Table 2. Material—main parameters of the medium.

Property	Aluminum Alloy	No. 2 Spindle Oil
Density (kg/m3)	2719	854
Thermal Conductivity [W/(m·K)]	170	0.14
Specific Heat Capacity [J/(kg·K)]	871	1935.2
Dynamic Viscosity (pa·s)	-	0.00314

.

With respect to boundary condition settings, a velocity inlet boundary is applied at the channel inlet, while a pressure outlet boundary is used at the outlet. The inlet temperature is uniformly set to 20 °C, and the inlet flow rate ranges from 4 to 20 L/min, which is converted into the corresponding inlet velocity according to the inlet cross-sectional area. The heat-transfer surface is specified as a constant-temperature boundary, with temperatures of 30 °C, 35 °C, 40 °C, 45 °C, and 50 °C. The external convective heat-transfer coefficient is set to 5 W/(m²·K), corresponding to natural convection in air, which typically ranges from 5 to 10 W/(m²·K). A lower-bound value is adopted to provide a conservative estimate of external heat dissipation, and the ambient temperature is 20 °C. A contact thermal resistance is introduced at the interface between the heat-transfer plate and the heat source, with a value of 0.05 K/W. During the calculation, convergence is ensured by monitoring residuals, as well as key physical quantities such as inlet and outlet temperatures and pressures.

To systematically analyze the effects of different operating conditions on the heat-transfer and flow characteristics of the temperature control plate, the operating conditions are divided into two categories. The first category examines the effect of the inlet flow rate: under an inlet temperature of 20 °C and a heat-source temperature of 50 °C, the inlet flow rate is varied from 4 to 20 L/min. The second category investigates the effect of the heat-source temperature: under an inlet temperature of 20 °C and an inlet flow rate of 8 L/min, the heat-source temperature is varied from 30 to 50 °C. Prior to the formal experiments, pre-tests are conducted by setting the heat-source temperature to 20 °C to eliminate temperature-rise interference caused by friction between the coolant and the temperature control plate; each test condition is maintained for no less than 10 min after reaching steady-state heat transfer.

3. Comprehensive Performance Evaluation and Modeling Method for the Temperature Control Plate

3.1. Definition of Performance Evaluation Indicators

To comprehensively evaluate the heat-transfer capability and energy consumption penalty of the temperature control plate under a unified criterion, a comprehensive performance evaluation indicator system is constructed in this study from three perspectives—namely, heat-transfer output, thermal resistance, and energy efficiency. The heat-transfer power (Q), actual thermal resistance (Rtotal) and coefficient of performance (COP) are selected as the core evaluation metrics, and a performance evaluation criterion (PEC) is introduced to quantify the relative variation between heat-transfer enhancement and resistance penalty. The heat-transfer power (Q) is used to characterize the actual heat-transfer capability of the temperature control plate under given operating conditions, and its calculation depends on the inlet and outlet temperatures of the coolant and the mass flow rate, as expressed by the following equation:

$$Q=c_{p}V\rho (T_{\mathit{out}}-T_{in})$$

(1)

where c_p is the specific heat capacity of the coolant under constant pressure, ρ is the coolant density; V is the volumetric flow rate; and T_out and T_in are the outlet and inlet temperatures of the coolant, respectively.

To eliminate the influence of temperature difference and flow-rate variations on heat-transfer performance under different operating conditions, an equivalent total thermal resistance (R_total) is introduced, by which the heat transfer process is transformed from an output quantity into a structure- and operating condition-related heat-transfer resistance, which is defined as follows:

$$R_{total}={\displaystyle \frac{T_w-T_f}{c_{p}V\rho (T_{\mathit{out}}-T_{in})}}$$

(2)

where T_w is the average surface temperature of the heat source and T_f is the average temperature of the coolant.

While evaluating the heat-transfer capability, the associated flow energy consumption penalty must also be considered [27,28,29,30,31,32]. The COP is used to quantify the heat-transfer benefit per unit of energy consumption and is defined as the ratio of heat-transfer power to pumping power, namely

$$COP={\displaystyle \frac{Q}{P_{\mathit{pump}}}}={\displaystyle \frac{Q}{\Delta PV}}$$

(3)

where ΔP is the pressure drop across the temperature control plate.

To avoid drawing one-sided conclusions based solely on either heat transfer or flow resistance, a PEC is introduced to provide a normalized assessment of the comprehensive performance under different structural configurations and operating conditions, which is expressed as

$$PEC={\displaystyle \frac{Nu/N{u}_0}{{(f/f_0)}^{\frac{1}{3}}}}$$

(4)

where N_u is the average Nusselt number in the flow channels; f is the average dimensionless friction factor in the channels; and Nu₀ and f₀ are the average Nusselt number and average dimensionless friction factor under the reference condition, respectively.

3.2. Equivalent Thermal Resistance Network and Heat-Transfer Power Model

During operation of the temperature control plate, heat is transferred from the heat source to the coolant flowing within the channels through the structure of the temperature control plate, and the heat-transfer process involves multiple mechanisms, including solid conduction, interfacial contact heat transfer, and fluid convective heat transfer [33,34,35,36]. To model and analyze the heat-transfer capability under a unified framework, this process is equivalently represented by a thermal resistance network composed of multiple thermal resistances connected in series, as shown in Figure 4. Similar thermal resistance-based modeling approaches have been widely used in heat-transfer analysis of multilayer structures, demonstrating their effectiveness in describing coupled conduction and convection processes [37]. According to the heat-transfer path, the thermal resistances involved in the heat-transfer process of the temperature control plate mainly include the contact thermal resistance between the heat source and the temperature control plate (R_contact) the solid conduction thermal resistance within the temperature control plate (R_conduction), the convective heat-transfer thermal resistance on the coolant side (R_convection), and the heat capacity resistance corresponding to the fluid heat absorption process (R_heat). The heat flow rate (Q) is identical through all heat-transfer stages, and each thermal resistance component can be calculated separately.

The solid conduction thermal resistance is determined by the material thermal conductivity and the structural dimensions, and its expression is given by

$$R_{conduct}={\displaystyle \frac{\delta }{\lambda LW}}$$

(5)

where λ is the thermal conductivity of the temperature control plate’s material in W/(m·K) and L is the total length of the flow channels in the temperature control plate.

The convective heat-transfer thermal resistance is determined by the flow state and the heat-transfer coefficient, and its expression is given by

$$R_{convection}={\displaystyle \frac{1}{2h(W_1+\eta H_1)L}}$$

(6)

where η is the fin efficiency for convective heat transfer on the side walls of the temperature control plate and h is the average convective heat-transfer coefficient within the flow channel in W/(m²·K).

The fluid heat-capacity resistance reflects the ability of the coolant to absorb heat, and its expression is given by

$$R_{heat}={\displaystyle \frac{1}{\rho W_1{H}_{1}v{c}_p}}$$

(7)

$$R_{contact}={\displaystyle \frac{R_{cA}}{A}}$$

(8)

where v is the coolant flow velocity (m/s), R_cA is the area-specific contact thermal resistance between the mating surfaces, and A is the contact area between the heat-transfer plate of the temperature control plate and the heat source.

Accordingly, the equivalent total thermal resistance of the temperature control plate can be expressed as

$$R_{total}=R_{contact}+R_{conduct}+R_{convection}+R_{heat}.$$

(9)

Under given heat-source temperature and inlet temperature conditions, the heat-transfer power is jointly determined by the driving force of the temperature difference and the equivalent total thermal resistance, and its expression is given by

$$Q={\displaystyle \frac{T_w-T_f}{R_{total}}}.$$

(10)

Further analysis indicates that the convective heat-transfer thermal resistance and the fluid heat capacity resistance vary with flow rate and can be regarded as power functions of the flow rate, whereas the contact thermal resistance and the solid conduction thermal resistance are mainly determined by structural and material parameters and can be treated as constants under operating conditions where the flow rate is the single independent variable. Therefore, the total thermal resistance can be approximately expressed in an empirical functional form of the flow rate as

$$R=a{V}^{-b}+c.$$

(11)

To incorporate the effect of the coolant progressively absorbing heat and warming up along the flow direction, a discretized in-line integration method is adopted by dividing the heat-transfer channel into $n$ micro-elements, and an equivalent thermal resistance model is established for each micro-element, thereby yielding the expressions for the coolant temperature and the cumulative heat-transfer power at an arbitrary position as

$$\left\{\begin{array}{l}{R}_{conduct,i}=n{R}_{conduct}\\ R_{convection,i}=n{R}_{convection}\end{array}\right..$$

(12)

The equivalent thermal resistance of each micro-element is expressed by the following equation:

$$R_{total,i}=R_{contact}+n{R}_{conduction}+n{R}_{convection}+R_{heat}$$

(13)

3.3. Quantitative Relationship Between Inlet Flow Rate and Heat-Transfer Performance

The heat-transfer performance of the temperature control plate is governed by the equivalent thermal resistance network, in which the dominant variation under different operating conditions arises from the convective heat-transfer resistance. Therefore, accurate determination of the convective heat-transfer coefficient is essential for predicting the overall thermal resistance and heat-transfer power.

The convective heat-transfer coefficient is determined through the Nusselt number. Due to the complex S-shaped flow channels with multiple bends and flow disturbances, conventional empirical correlations for straight channels are not directly applicable. Therefore, the Nusselt number is obtained based on experimentally derived correlations.

First, the hydraulic diameter of the flow channel is defined as

$$D_h={\displaystyle \frac{4A}{U}}={\displaystyle \frac{2W_1{H}_1}{W_1+H_1}}.$$

(14)

Based on the hydraulic diameter and the volumetric flow rate, the Reynolds number in the channel is calculated as

$$Re={\displaystyle \frac{v{D}_h}{\mu }}={\displaystyle \frac{V{D}_h}{60000W_1{H}_1\mu }}={\displaystyle \frac{V}{30000\mu (W_1+H_1)}}$$

(15)

where $v$ is the average flow velocity, $\mu$ is the dynamic viscosity of the fluid, and $\rho$ is the fluid density.

In general, the Nusselt number can be expressed as a power-law function of the Reynolds number and Prandtl number. Considering that the temperature variation in the present study is relatively small, the Prandtl number can be treated as approximately constant. Therefore, the Nusselt number is simplified as a function of the Reynolds number:

$$Nu=CR{e}^n$$

(16)

Based on the Nusselt number, the convective heat transfer coefficient is obtained as

$$h={\displaystyle \frac{Nu\cdot {\lambda }_1}{D_h}}$$

(17)

where λ₁ is the thermal conductivity of the coolant.

From the analysis of the thermal resistance network, it can be observed that both the convective thermal resistance and the fluid heat-capacity resistance can be expressed as power-law functions of the flow rate, whereas the contact thermal resistance and conduction thermal resistance are mainly determined by structural parameters and remain nearly constant under varying flow-rate conditions. Therefore, the equivalent total thermal resistance can be simplified as a function of the flow rate:

$$R_{total}=f(V)$$

(18)

To account for the temperature rise of the coolant along the flow direction, the channel is discretized into multiple control volumes, and the local thermal resistance is evaluated for each segment. The overall heat-transfer performance is then obtained through integration along the flow direction.

Through the above formulation, a quantitative relationship between the inlet flow rate and heat-transfer performance is established, linking flow conditions to thermal behavior via the Reynolds number, Nusselt number, and convective heat-transfer coefficient.

3.4. Flow Resistance and Energy Efficiency Model

On the basis of heat-transfer performance modeling, the flow resistance and flow state are key factors characterizing the energy consumption penalty and heat-transfer enhancement mechanisms [38,39,40,41]. In this section, the internal flow characteristics of the temperature control plate are described from two perspectives: pressure loss and turbulence intensity. When the coolant flows through the channels of the temperature control plate, pressure loss is generated due to wall friction and structural discontinuities. To uniformly evaluate the energy consumption penalty under different structural configurations and operating conditions, the total pressure loss is decomposed into frictional (along-the-path) pressure loss and local pressure loss.

The frictional pressure loss is calculated using the Fanning equation, and its expression is given by

$$\Delta P_1={\lambda }_f{\displaystyle \frac{L\rho v^2}{2d}}$$

(19)

where λ_f is the friction factor during the flow process.

Under laminar flow conditions, the friction factor (λ_f) is related to the Reynolds number (Re) and its expression is given by

$${\lambda }_f={\displaystyle \frac{64}{Re}}.$$

(20)

By combining the definition of the Reynolds number, a simplified expression for the frictional (along-the-path) pressure loss of the temperature control plate can be obtained as

$$\Delta P_1={\displaystyle \frac{\mu Lv}{2\rho d^2}}.$$

(21)

According to the structural configuration of the temperature control plate, the local pressure loss during the flow process mainly originates from geometric discontinuities such as inlet contraction, channel bending, and outlet expansion, and it can be expressed as

$$\Delta P_2=\zeta {\displaystyle \frac{\rho v^2}{2}}$$

(22)

where ζ is the local loss coefficient, which is related to the geometry and dimensions of the corresponding local resistance region.

Accordingly, the total pressure loss during the flow process in the temperature control plate is given by

$$\Delta P=\Delta P_1+\Delta P_2.$$

(23)

In addition to pressure drop, the turbulence intensity ($TI$) is used to characterize the level of velocity fluctuations in the flow, and it is defined as the ratio of the root mean square of velocity fluctuations to the mean velocity:

$$TI={\displaystyle \frac{u^{\prime }}{U}}\times 100\%={\displaystyle \frac{\sqrt{\frac{u_x^{\prime 2}+u_y^{\prime 2}+u_z^{\prime 2}}{3}}}{\sqrt{u_x^2+u_y^2+u_z^2}}}\times 100\%$$

(24)

where u′ is the root mean square of the fluid velocity fluctuations; U is the mean velocity of the coolant; $u_x^{\prime }$, $u_y^{\prime }$, and $u_z^{\prime }$ are the velocity fluctuations in different flow directions; and u_x, u_y, and u_z are the corresponding mean velocity components in each direction.

An increase in turbulence intensity is beneficial for enhancing convective heat transfer; however, it simultaneously intensifies vortex formation and flow separation, leading to increased energy consumption. Therefore, in this study, the pressure loss ($∆P$) and turbulence intensity ($TI$) are taken as important characteristic parameters to describe the flow penalty and flow state and are jointly used with heat-transfer indicators for comprehensive performance evaluation.

4. Results and Discussion

4.1. Validation of the Numerical Simulation Model

To verify the reliability of the established numerical model in predicting the heat-transfer and flow characteristics of the temperature control plate, a comparative analysis between experimental results and numerical simulation results is conducted under variable inlet flow-rate and heat-source temperature conditions. The experimental heat-transfer results under varying inlet flow rates are shown in Figure 5. As the inlet flow rate increases, the temperature difference between the coolant inlet and outlet gradually decreases, while the heat-transfer power continues to increase with a progressively diminishing growth rate; when the inlet flow rate is increased from 4 L/min to 10 L/min, the heat-transfer power is enhanced by approximately 33%. Meanwhile, the equivalent thermal resistance gradually decreases with increasing flow rate, indicating that an increase in the inlet flow rate can effectively reduce the convective heat-transfer resistance, thereby enhancing the overall heat-transfer performance of the temperature control plate.

To further verify the accuracy of the numerical model in predicting heat-transfer performance, the heat-transfer power obtained from simulations is compared with the experimental results, as shown in Figure 6. It can be observed that the overall trends of both experimental and simulated values with respect to the inlet flow rate are highly consistent, indicating that the established numerical model can reasonably capture the influence of inlet flow-rate variation on the heat-transfer performance of the temperature control plate. Quantitative comparison shows that the variation amplitude of the experimental heat-transfer power is slightly smaller than that of the simulation results, and the theoretically calculated values are generally lower than the simulated ones. This discrepancy is mainly attributed to the fact that, under practical operating conditions, the contact thermal resistance between the heat source and the temperature control plate is larger than the ideal contact condition assumed in the numerical model, thereby reducing the relative contribution of convective heat-transfer resistance variation to the total thermal resistance. Overall, the relative error between the simulation and experimental results is controlled within 15%.

The experimental results of the flow characteristics are shown in Figure 7. As the inlet flow rate increases, the pressure drop between the inlet and outlet of the temperature control plate rises significantly, exhibiting a pronounced nonlinear growth trend; meanwhile, the friction factor shows an overall decreasing trend with increasing flow rate, and its variation pattern is generally consistent with the numerical simulation results. It should be noted that the experimentally measured pressure-drop values are overall higher than the simulation results, with deviations of approximately 30–50%. The deviation in pressure drop between experimental and numerical results is relatively large, which is mainly attributed to the idealized assumptions adopted in the numerical model. In the simulations, the channel walls are treated as hydraulically smooth, and additional local losses are not considered, whereas in practical experiments, surface roughness and connection-induced local losses significantly increase flow resistance. These factors are difficult to quantify accurately and are not included in the present model, leading to higher measured pressure drops.

As shown in Figure 8, under different heat-source temperature conditions, the variation in equivalent thermal resistance is relatively small, indicating that the structure and flow state of the temperature control plate do not change significantly, and the increase in heat-transfer power is mainly attributed to the enhancement of the temperature-difference driving force, which is consistent with the assumptions of the equivalent thermal resistance model established in this study. To verify the applicability of the numerical model under varying heat-source temperature conditions, a comparative analysis between simulation results and experimental results is conducted for cases with different heat-source temperatures. The corresponding comparison results of heat-transfer power are presented in Figure 9. It can be observed that the overall trends of both simulation and experimental results with respect to heat-source temperature are consistent, and the relative errors are all controlled within 15%, indicating that the established numerical model can effectively capture the influence of heat-source temperature variation on the heat-transfer performance of the temperature control plate.

In summary, the numerical simulation results show good agreement with the experimental results in terms of the variation trends of both heat-transfer performance and flow characteristics, with errors remaining within an acceptable range. This indicates that the established numerical model exhibits high predictive reliability and can be used for mechanistic analysis of heat-transfer and flow characteristics of temperature control plates under different operating conditions.

4.2. Effect of Inlet Flow Rate on Heat-Transfer and Flow Characteristics

A temperature control plate with a channel width of 40 mm and a height of 12 mm is selected as the reference structure. Numerical simulations are carried out by considering the inlet flow rate and heat-source temperature as independent variables. The temperature-field, pressure-field, and flow characteristic parameters are extracted to investigate the influence of the inlet flow rate on the heat-transfer and flow behavior of the temperature control plate.

The temperature-field distributions under different inlet flow rates are presented in Figure 10. The coolant temperature increases progressively along the flow direction, exhibiting the typical characteristics of forced convection in internal channels. As the inlet flow rate increases, the overall temperature level of the coolant decreases, and the inlet–outlet temperature difference is significantly reduced. This is attributed to the increased mass flow rate of coolant participating in heat transfer per unit of time, which leads to a lower temperature rise under the same heat-transfer power.

The variations in heat-transfer performance indicators with the inlet flow rate are shown in Figure 11. The inlet–outlet temperature difference decreases monotonically with increasing flow rate, while the heat-transfer power increases significantly within the range of 4–10 L/min and approaches a plateau near 14 L/min, with an overall increase of approximately 35% compared to the 4 L/min condition. Beyond this point, only minor fluctuations are observed. Meanwhile, the Nusselt number continues to increase but with a gradually decreasing growth rate, indicating that the enhancement effect of flow intensification on convective heat transfer tends to saturate in the high-flow-rate regime. The equivalent thermal resistance decreases with increasing flow rate, reflecting a reduced contribution of convective resistance to the total thermal resistance; however, its rate of decrease gradually diminishes. These results indicate that enhancing heat transfer solely by increasing the inlet flow rate exhibits a clear characteristic of diminishing marginal returns.

Based on the observed trend that heat-transfer enhancement gradually levels off, subsequent structural comparisons and parametric analyses are primarily focused on the inlet flow-rate range of 4–10 L/min.

The inlet flow-rate range of 4–20 L/min is selected to cover typical operating conditions of machine-tool thermal control systems. For temperature-field visualization (Figure 10), a representative subset of 4–10 L/min is adopted, as it is sufficient to clearly illustrate the evolution of temperature distribution patterns. The full range is used in Figure 11 for quantitative analysis of performance variations.

Figure 12 illustrates the pressure-field distribution inside the temperature control plate under inlet flow rates of 4–10 L/min. The contour plots show pronounced pressure discontinuities in the inlet and outlet regions of the channels, and as the inlet flow rate increases, the overall pressure drop rises significantly, with particularly steep pressure gradients observed in the turning regions. The velocity-vector distributions (Figure 13) indicate that under higher flow-rate conditions, obvious velocity-gradient variations occur at the bends, accompanied by vortex formation and local recirculation zones, which further intensify local energy losses.

The turbulence intensity distributions are shown in Figure 14. Under low-flow-rate conditions, the flow remains relatively stable with low turbulence intensity; as the inlet flow rate increases, the overall turbulence intensity rises and is especially pronounced in the turning regions, as well as the inlet and outlet areas. This indicates enhanced flow disturbance, which is beneficial for boundary-layer renewal, but simultaneously introduces higher energy consumption penalties.

The variations of flow and energy-efficiency indicators are comprehensively summarized in Figure 15. The inlet–outlet pressure drop increases approximately in a quadratic manner with increasing inlet flow rate, while the friction factor decreases as the flow rate increases, reflecting the characteristic variation of the flow resistance coefficient with increasing Reynolds number.

From the perspective of comprehensive performance, the PEC increases monotonically with increasing inlet flow rate, showing an enhancement of approximately 66% at 20 L/min compared with the 4 L/min condition. This indicates that, within a certain range, the contribution of flow intensification to heat-transfer enhancement remains dominant. In contrast, the COP decreases monotonically with increasing inlet flow rate. This behavior is attributed to the fact that the pressure drop increases much more rapidly than the heat-transfer power, resulting in a significant rise in pumping power consumption. Although the heat-transfer power continues to increase, its growth rate is substantially lower than that of the pressure drop, leading to a continuous reduction in energy efficiency.

Therefore, the variation of different performance indicators reflects the combined influence of flow enhancement and energy consumption under varying flow conditions. From an engineering perspective, the selection of the inlet flow rate should be determined according to specific application requirements and performance priorities. A higher flow rate is favorable for achieving stronger heat-transfer performance, whereas a moderate flow rate is more suitable for maintaining better energy efficiency. This provides practical guidance for optimizing operating conditions of machine-tool temperature control plates.

To verify the accuracy of the heat-transfer power calculation framework, the Nusselt numbers are extracted from the simulation results and fitted to obtain the correlation with the Reynolds number, as shown in Figure 16. This correlation is then incorporated into the equivalent thermal resistance model to calculate the theoretical inlet–outlet temperature difference and heat-transfer power, which are subsequently compared with the numerical simulation results, as shown in Figure 17. The results demonstrate that the variation trends of the two are highly consistent, with numerical deviations not exceeding 10%, thereby confirming the applicability of the established heat-transfer power calculation model within this flow-rate range.

4.3. Effect of Heat-Source Temperature on Heat-Transfer Performance

With the inlet flow rate held constant, the effect of heat-source temperature variation on the heat-transfer performance of the temperature control plate is further analyzed. The temperature-field distributions under different heat-source temperature conditions are shown in Figure 18. As the heat-source temperature increases, the overall temperature of the coolant within the channels rises, and under the 50 °C condition, the temperature variation rate along the flow direction is the largest, indicating a significant enhancement of the heat-transfer driving force.

The variations of heat-transfer performance indicators with heat-source temperature are presented in Figure 19. It can be observed that both the inlet–outlet temperature difference of the coolant and the heat-transfer power increase approximately linearly with increasing heat-source temperature, while the convective heat-transfer coefficient shows no significant variation and the equivalent thermal resistance fluctuates slightly around a stable value. This indicates that under conditions of a relatively small coolant temperature rise and insignificant variations in thermophysical properties, the heat-source temperature enhances the heat-transfer power mainly by increasing the temperature-difference driving force, whereas the heat-transfer resistance characteristics of the temperature control plate remain essentially unchanged. These results indirectly validate the basic assumption of the equivalent thermal resistance model that the thermal resistance is primarily determined by the structural configuration and flow state. Within the investigated heat-source temperature range, the heat-transfer power can be approximately considered to be proportional to the heat-source temperature.

5. Conclusions

Aiming at the application requirements of high heat transfer power and low energy consumption for machine-tool temperature control plates, this study proposes a comprehensive performance evaluation method with heat-transfer power, equivalent total thermal resistance, and coefficient of performance (COP) as the core metrics. A heat-transfer power calculation model based on an equivalent thermal resistance network is established, enabling unified modeling and comparative analysis of temperature control-plate performance under different operating conditions.

• Increasing the inlet flow rate enhances heat-transfer performance and reduces the equivalent thermal resistance. However, the increment in heat-transfer power gradually diminishes, while the pressure drop increases approximately in a quadratic manner and the COP decreases accordingly. This indicates that further improvement in heat transfer performance by simply increasing the flow rate is constrained by energy-efficiency limitations.
• Increasing the heat-source temperature primarily enhances heat-transfer power by strengthening the temperature-difference driving force, whereas the equivalent thermal resistance remains nearly unchanged. This suggests that heat-source temperature mainly affects the thermal driving potential rather than the intrinsic heat-transfer characteristics.
• When incorporating the Nusselt number–Reynolds number correlation derived from numerical simulations into the proposed model, the deviation between theoretical predictions and simulation results is within 10%. Experimental results further demonstrate that, within the inlet flow-rate range of 4–10 L/min, the heat-transfer power increases by approximately 33%, with simulation–experiment discrepancies below 15%. The deviation in pressure drop is mainly attributed to surface-roughness effects in practical flow channels.

Overall, the results verify the accuracy and engineering applicability of the proposed comprehensive performance evaluation method and provide a theoretical basis for operating-condition optimization and structural design of machine-tool temperature control plates. In future work, the effect of channel surface roughness will be incorporated into the model to further improve the prediction accuracy of flow resistance and heat-transfer performance, particularly under practical engineering conditions involving non-ideal surface characteristics.