Direct drive rotary table with heat dissipation function and thermal analysis method thereof

Abstract

This invention discloses a direct-drive turntable with heat dissipation function and its thermal analysis method, including a heat dissipation unit disposed inside the direct-drive turntable; the heat dissipation unit includes cooling pipes. A turntable model without oil channels is meshed to obtain a meshed model without oil channels; a turntable model with oil channels is meshed to obtain a meshed model with oil channels; transient simulation analysis of turntable heat conduction is performed on both models to obtain the turntable temperature field results with and without cooling pipes; temperature curves of the same monitoring nodes in the temperature field results of the two turntables are extracted to obtain the average temperature curves with and without cooling pipes; a comparative analysis of the average temperature curves with and without cooling pipes yields the analysis results of the heat dissipation of the turntable by the cooling pipes. This invention can overcome the limitation of only setting cooling structures in the stator and base, thereby achieving all-round and efficient heat dissipation of the direct-drive turntable.

Description

Technical Field

[0001] This invention relates to the field of rotary table thermal analysis, and more specifically to a direct-drive rotary table with heat dissipation function and its thermal analysis method. Background Technology

[0002] Large-size direct-drive rotary tables are widely used in aerospace, precision manufacturing, and heavy machinery industries. Their core advantages lie in zero transmission backlash, high positioning accuracy, and fast response speed. However, large-size direct-drive rotary tables are typically equipped with high-power motors, resulting in a bulky yet compact structure. During operation, in addition to the significant copper loss heat generated by the stator windings, electromagnetic losses (iron losses) between the rotor and stator, as well as frictional losses in the bearings, continuously generate heat. If this heat cannot be dissipated in time, the internal temperature of the rotary table will rise significantly, causing thermal expansion and deformation of components. This not only compromises the positioning accuracy and motion stability of the rotary table but, in severe cases, accelerates insulation aging, damages the motor and bearings, thereby reducing the equipment's lifespan and operational reliability.

[0003] Currently, the mainstream heat dissipation design for direct-drive rotary tables focuses on stationary components, primarily by arranging cooling pipes on the outside of the motor stator and within the base housing. While this traditional cooling structure can remove some heat generated on the stator side, its heat dissipation effect on rotating components (i.e., the rotary table surface and rotor) is very limited. Due to the air gap between the rotary table and the stator, and the high contact thermal resistance at the bearing connections, heat generated on the rotor side (such as rotor iron loss, conduction heat, and bearing friction heat) is difficult to dissipate effectively through the stator or base cooling system. This often results in the rotary table surface becoming a heat dissipation blind spot, making it highly susceptible to thermal deformation due to uneven temperature rise, directly affecting the machining accuracy of the workpiece.

[0004] Therefore, in order to solve the technical problem of many heat dissipation dead zones in the existing technology, a direct drive turntable with heat dissipation function and its thermal analysis method are needed to break through the limitation of setting traditional cooling structures only in the stator and base, so as to achieve all-round and efficient heat dissipation of the direct drive turntable. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to overcome the defects in the prior art and provide a direct drive rotary table with heat dissipation function and its thermal analysis method, which can break through the limitation of setting the traditional cooling structure only in the stator and the base, so as to achieve all-round and efficient heat dissipation of the direct drive rotary table.

[0006] The direct-drive rotary table with heat dissipation function of the present invention includes a heat dissipation unit disposed inside the direct-drive rotary table; the heat dissipation unit includes cooling pipes.

[0007] Furthermore, the cooling pipe is a spiral cooling pipe.

[0008] A method for thermal analysis of a direct-drive rotary table with heat dissipation function includes the following steps:

[0009] The direct-drive turntable without cooling pipes is taken as an oil-free turntable; the oil-free turntable is modeled to obtain the oil-free turntable model.

[0010] The oilless turntable model is meshed to obtain the oilless mesh model.

[0011] Construct a turntable model with cooling pipes, and use the turntable model with cooling pipes as a turntable model with oil passages;

[0012] Mesh the turntable model with oil passages to obtain a meshed model with oil passages;

[0013] Transient simulation analysis of turntable heat conduction was performed on mesh models without oil channels and mesh models with oil channels, respectively, to obtain the temperature field results of the turntable with and without cooling channels.

[0014] Temperature curves of the same monitoring node in the temperature field results of the turntable with and without cooling pipes were extracted respectively to obtain the average temperature curves of the turntable with and without cooling pipes.

[0015] By comparing and analyzing the average temperature curves of the turntable with and without cooling pipes, the analysis results of the cooling pipes on heat dissipation of the turntable are obtained.

[0016] Furthermore, the oil-free turntable model is meshed to obtain an oil-free mesh model, specifically including:

[0017] Using ANSA software, the entire turntable is divided into n independent parts along geometric features; each independent part has a separate PID.

[0018] For the first n-1 regular geometric regions, a high-quality hexahedral mesh is generated using the MAP command;

[0019] For the nth complex transition region encompassing irregular contours and the intersection of multiple features, an adaptive tetrahedral mesh is used for filling to adapt to the complex boundary.

[0020] Finally, the Paste command is used to achieve seamless bonding of the interface nodes of each region, and a complete turntable mesh model is finally constructed.

[0021] Furthermore, in terms of dimensional control, the grid transitions smoothly from the inside to the outside of the turntable, with dimensions ranging from... mm gradually increased to mm, and control the grid aspect ratio at Within.

[0022] Furthermore, a turntable model with cooling pipes is constructed, specifically including:

[0023] In UG software, draw a spiral line to create a cooling pipe guide line; at the starting point of the spiral line, create a sketch to draw a rectangular cross-section of the pipe; use the sweep command to obtain the cooling pipe, and use Boolean operations to obtain a turntable model with the cooling pipe.

[0024] Furthermore, the turntable model with oil passages is meshed to obtain a meshed model with oil passages, specifically including:

[0025] In ANSA software, the Cut command is used to divide the center of the turntable model with cooling pipes into a separate PID.

[0026] The Coons command is used to fill the cut areas and then fill the model with a tetrahedral mesh; the mesh size is controlled within... Approximately mm;

[0027] The paste command is used to glue together several nodes in contact area, ultimately generating a complete mesh model of the turntable with cooling pipes.

[0028] Furthermore, transient simulation analysis of turntable heat conduction was performed on mesh models without oil passages and mesh models with oil passages, specifically including:

[0029] The thermal parameters, physical property parameters, and boundary conditions of the two mesh generation models were set respectively;

[0030] For mesh generation models with oil passages: use keywords to define convection boundary conditions, select the turntable surface that is in direct contact with the coolant, and set the convection heat transfer coefficient and coolant temperature to complete the forced convection heat transfer settings between the coolant and the contact surface.

[0031] The heat conduction transient simulation was performed on the two mesh generation models after the settings were configured using a solver, and heat conduction transient simulation analysis files for the turntable with and without cooling pipes were obtained respectively.

[0032] Furthermore, the convective heat transfer coefficient is the forced convective heat transfer coefficient between the coolant in the cooling pipe and the contact surface of the turntable;

[0033] The forced convection heat transfer coefficient is calculated using the following method:

[0034] Based on the cross-sectional area and flow velocity of the cooling pipe, the wetted perimeter of each section is calculated; then the Reynolds number is used to determine the flow state of the coolant, and the friction coefficient and Nusselt number are calculated according to the correlation formula; finally, the forced convection heat transfer coefficient between the coolant and the contact surface of the turntable is obtained by combining the thermal conductivity.

[0035] Furthermore, the forced convection heat transfer coefficient is determined according to the following formula. :

[0036] ;

[0037] in, For Nusselt numbers; ; Friction factor; It is the Reynolds number; It is a Prandtl number; The thermal conductivity of the fluid; ; ; The characteristic velocity of the fluid is taken as the average velocity across the cross section when flowing inside the pipe. A qualitative measure of geometric features; The kinematic viscosity of the fluid; This refers to the cross-sectional area of ​​the cooling pipe. It is a wetted period.

[0038] The beneficial effects of this invention are as follows: This invention discloses a direct-drive turntable with heat dissipation function and its thermal analysis method, which breaks through the limitation of setting traditional cooling structures only in the stator and base. It introduces cooling pipes inside the turntable, establishes a turntable cooling pipe model, and combines it with heat conduction simulation to analyze the heat dissipation effect of adding cooling pipe structure to the turntable, thereby improving the heat dissipation efficiency and operational stability of the turntable, and providing accurate and efficient technical support for the heat dissipation design of large-size direct-drive turntables. Attached Figure Description

[0039] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0040] Figure 1 A simplified model diagram of one side of the turntable;

[0041] Figure 2 A simplified schematic diagram of the other side of the turntable;

[0042] Figure 3 A schematic diagram of the mesh generation model for an oil-free turntable;

[0043] Figure 4 This is a schematic diagram of the spiral cooling pipeline oil circuit for the turntable.

[0044] Figure 5 This is a schematic diagram of a rotary table model with an oil circuit.

[0045] Figure 6 This is a schematic diagram of a mesh generation model for a rotary table with an oil passage.

[0046] Figure 7 This is a schematic diagram of the transient temperature field distribution of a rotary table model without an oil path.

[0047] Figure 8 This is a schematic diagram of the transient temperature field distribution of a turntable model with an oil circuit.

[0048] Figure 9 A schematic diagram of the temperature measurement node on the upper surface of the turntable;

[0049] Figure 10 A schematic diagram of the temperature curves at the temperature measurement nodes on the upper surface of the oil-free turntable model;

[0050] Figure 11 A schematic diagram of the temperature curves at the temperature measurement nodes on the upper surface of a turntable model with an oil circuit.

[0051] Figure 12 This is a schematic diagram comparing the average temperature curves of the surface temperature measurement nodes of the two models. Detailed Implementation

[0052] The present invention will be further described below with reference to the accompanying drawings, as shown in the figures:

[0053] This embodiment discloses a direct-drive rotary table with heat dissipation function, including a heat dissipation unit disposed inside the direct-drive rotary table; the heat dissipation unit includes cooling pipes.

[0054] The above-described design allows the cooling medium (such as cooling oil) to form an effective heat dissipation channel inside the turntable. Compared to traditional solutions that only install cooling structures on the outside of the stator or the base housing, this design is closer to heat-generating parts such as the rotor and turntable surface, enabling timely heat removal from rotating components and reducing heat dissipation blind spots on the rotor side and the turntable surface. Furthermore, this structure reduces the overall temperature rise and temperature gradient of the turntable, minimizes thermal deformation caused by uneven local temperature rise, and improves the thermal and structural stability of the turntable. This enhances the positioning and machining accuracy of the direct-drive turntable during precision machining, effectively improving heat dissipation efficiency and controlling thermal deformation. It is well-suited to the precise thermal design requirements of large-size direct-drive turntables in aerospace and other fields.

[0055] In this embodiment, the cooling pipe is a spiral cooling pipe. The spiral cooling pipe structure allows the cooling medium to flow continuously along a spiral path inside the turntable, effectively extending the flow path and increasing the heat exchange area, thereby improving the heat exchange efficiency between the cooling medium and the turntable structure. Furthermore, the spiral arrangement allows the cooling effect to be more evenly distributed inside the turntable, reducing localized temperature concentration, further minimizing the temperature gradient of the turntable, and improving the overall heat dissipation effect and thermal stability.

[0056] The present invention also relates to a method for thermal analysis of a direct-drive rotary table with heat dissipation function as described in the above embodiments, comprising the following steps:

[0057] The direct-drive turntable without cooling pipes is taken as an oil-free turntable; the oil-free turntable is modeled to obtain the oil-free turntable model.

[0058] The oilless turntable model is meshed to obtain the oilless mesh model.

[0059] Construct a turntable model with cooling pipes, and use the turntable model with cooling pipes as a turntable model with oil passages;

[0060] Mesh the turntable model with oil passages to obtain a meshed model with oil passages;

[0061] Transient simulation analysis of turntable heat conduction was performed on mesh models without oil channels and mesh models with oil channels, respectively, to obtain the temperature field results of the turntable with and without cooling channels.

[0062] Temperature curves of the same monitoring node in the temperature field results of the turntable with and without cooling pipes were extracted respectively to obtain the average temperature curves of the turntable with and without cooling pipes.

[0063] By comparing and analyzing the average temperature curves of the turntable with and without cooling pipes, the analysis results of the cooling pipes on heat dissipation of the turntable are obtained.

[0064] In this embodiment, the oilless rotary table can be modeled using the software LS-prepost. Specifically, this includes: ignoring all screws, vents, oil holes, and other small structures, treating them all as solids; omitting fillets and chamfers in the model, as well as small holes that do not significantly affect the analysis results; and treating grooves, threaded holes, etc., as solids, ultimately resulting in the model shown below. Figure 1 as well as Figure 2 The turntable model shown.

[0065] In this embodiment, given the structural complexity of the turntable model, a discretization strategy based on hierarchical transition and hybrid units can be used to mesh the turntable model.

[0066] The oil-free turntable model is meshed to obtain an oil-free mesh model, specifically including:

[0067] Using ANSA software, the turntable was divided into 8 independent parts along its geometric features; each independent part has its own PID.

[0068] For the first 7 regular geometric regions (such as ribs, cylindrical surfaces, etc.), a high-quality hexahedral mesh is generated using the MAP command to ensure the accuracy of the heat conduction analysis;

[0069] For the 8th complex transition region, which encompasses irregular contours and the intersection of multiple features, an adaptive tetrahedral mesh is used for filling to adapt to the complex boundary.

[0070] Finally, the Paste command is used to achieve seamless bonding of the interface nodes in each region, ultimately constructing a complete turntable mesh model, such as... Figure 3 As shown, it contains a total of 3,368,300 individual elements, including 1,639,903 hexahedral elements, 24,744 pentahedral transition elements, and 1,703,653 tetrahedral elements.

[0071] In terms of size control, the grid transitions smoothly from the inside to the outside of the turntable, with dimensions ranging from... mm gradually increased to mm, and control the grid aspect ratio at To avoid the negative impact of mesh distortion on calculation accuracy, the range is kept within a certain limit.

[0072] In this embodiment, a turntable model with cooling pipes is constructed, specifically including:

[0073] In UG software, draw a spiral line to create the cooling pipe guide line; at the starting point of the spiral line, create a sketch and draw the rectangular cross-section of the pipe; use the sweep command to obtain the cooling pipe, and use Boolean operations to obtain the turntable model with the cooling pipe, such as... Figure 4 as well as Figure 5 As shown, where, Figure 5 The arrow marks the location of the spiral cooling pipe.

[0074] For the turntable model with oil passages, a layered transition strategy is also adopted; that is, the turntable model with oil passages is meshed to obtain a meshed model with oil passages, specifically including:

[0075] In ANSA software, the Cut command is used to divide the center of the turntable model with cooling pipes into a separate PID.

[0076] The Coons command is used to fill the cut areas and then fill the model with a tetrahedral mesh; the mesh size is controlled within... Approximately mm;

[0077] The paste command is used to glue together several nodes in contact area, ultimately generating a complete mesh model of the turntable with cooling pipes, such as... Figure 6 As shown, the mesh model contains 1,029,655 hexahedral elements, 15,156 pentahedral transition elements, and 3,986,579 tetrahedral elements (a total of 5,031,390 individual elements).

[0078] In this embodiment, transient simulation analysis of turntable heat conduction is performed on mesh models without oil passages and mesh models with oil passages, respectively, specifically including:

[0079] The thermal parameters, physical properties, and boundary conditions of two different mesh generation models are set respectively; the thermal parameters (thermal conductivity, specific heat capacity), physical properties (elastic modulus, Poisson's ratio, density), and boundary conditions of the turntable model can be set in LS-prepost software.

[0080] For meshed models with oil passages: In LS-prepost software, use the keyword *BOUNDARY_CONVECTION_SET to define convection boundary conditions, select the turntable surface that is in direct contact with the coolant, and set the convection heat transfer coefficient and coolant temperature in HMULT to complete the forced convection heat transfer settings between the coolant and the contact surface.

[0081] The convective heat transfer coefficient is the forced convective heat transfer coefficient between the coolant and the contact surface of the turntable in the cooling pipe. The heat exchange between the turntable and the coolant is a forced convective heat transfer within the pipe. The coolant flows in the spiral rectangular groove of the turntable's cooling jacket. The geometry of the spiral rectangular groove can be expanded into an equivalent oil pipe with a rectangular interface. Different flow regimes of the cooling oil in the pipe exhibit different heat transfer characteristics, and the formulas used to calculate the heat transfer coefficient also differ. Therefore, the Reynolds number must first be calculated. The flow regime is determined, and then the appropriate formula is selected for calculation.

[0082] Therefore, the forced convection heat transfer coefficient is calculated using the following method:

[0083] Based on the cross-sectional area and flow velocity of the cooling pipe, the wetted perimeter of each section is calculated; then the Reynolds number is used to determine the flow state of the coolant, and the friction coefficient and Nusselt number are calculated according to the correlation formula; finally, the forced convection heat transfer coefficient between the coolant and the contact surface of the turntable is obtained by combining the thermal conductivity.

[0084] The forced convection heat transfer coefficient is determined according to the following formula. :

[0085] ;

[0086] in, For Nusselt numbers;

[0087] For turbulent convective heat transfer inside a rectangular tube, the Gnielinski correlation is used: ;

[0088] Friction factor; It is the Reynolds number; It is a Prandtl number; The thermal conductivity of the fluid;

[0089] ; ; The characteristic velocity of the fluid is taken as the average velocity across the cross section when flowing inside the pipe. A qualitative measure of geometric features; The kinematic viscosity of the fluid; This refers to the cross-sectional area of ​​the cooling pipe. For a rectangular oil passage with a wetted perimeter, the width is... Gao Wei ,but .

[0090] It should be noted that the above The calculation is applicable to , In this case, the friction factor can be approximated by the turbulence formula: hour, .

[0091] The heat conduction transient simulation was performed on the two mesh generation models after the settings were configured using a solver, resulting in heat conduction transient simulation analysis files for the turntable with and without cooling pipes. Specifically, in the LS-prepost software, the solver keywords *termination time*, *solver method*, *thermal time step*, *thermal solver*, and *nonlinear solver* were set, and the result output was set to global data output to ensure complete acquisition of simulation data for each region of the turntable. The solver software APDL was then started to perform the calculation, resulting in the turntable heat conduction transient simulation analysis result file (d3plot format).

[0092] Open the d3plot result files of the turntable model without cooling pipes and the turntable model with spiral cooling pipes in the numerical simulation post-processing software, as shown below. Figure 7 as well as Figure 8 As shown; subsequently, the nodal temperature and deformation curves of the central region of the turntable surface for each scheme were extracted in the "History" module; to reduce data errors, 16267 feature nodes were selected for analysis for each scheme (see...). Figure 9 , 10 、11);

[0093] Figure 12The average temperature curves of the two models are obtained by averaging the measured characteristic node temperature curves. From the temperature data at each time point in the figure, it can be clearly seen that the heat dissipation effect is continuously and significantly improved after adding water-cooled cooling pipes to the direct-drive turntable: at 2500s, the average surface temperature of the turntable with cooling pipes is 25.5℃, which is 0.9℃ lower than 26.4℃ without cooling; as time goes on, the temperature difference between the two gradually increases, reaching 1.6℃, 2.3℃, and 2.8℃ at 4500s, 6500s, and 8500s, respectively; at the simulation endpoint of 10000s, the temperature of the turntable with cooling pipes stabilizes at 26.7℃, while the temperature of the turntable without cooling pipes rises to 29.9℃, and the temperature difference further increases to 3.2℃.

[0094] This indicates that the cooling pipes not only continuously suppressed the temperature rise rate of the turntable throughout the entire simulation cycle, but also significantly reduced its steady-state operating temperature, effectively improving the thermal stability of the turntable, and demonstrating a clear and positive heat dissipation effect.

[0095] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A direct drive rotary table with heat dissipation function, characterized in that: It includes a heat dissipation unit disposed inside the direct drive turntable; the heat dissipation unit includes cooling pipes.

2. The direct drive rotary table with heat dissipation function according to claim 1, characterized in that: The cooling pipes are spiral cooling pipes.

3. A method for thermal analysis of a direct-drive rotary table with heat dissipation function as described in any one of claims 1-2, comprising the following steps: The direct-drive turntable without cooling pipes is taken as an oil-free turntable; the oil-free turntable is modeled to obtain the oil-free turntable model. The oilless turntable model is meshed to obtain the oilless mesh model. Construct a turntable model with cooling pipes, and use the turntable model with cooling pipes as a turntable model with oil passages; Mesh the turntable model with oil passages to obtain a meshed model with oil passages; Transient simulation analysis of turntable heat conduction was performed on mesh models without oil channels and mesh models with oil channels, respectively, to obtain the temperature field results of the turntable with and without cooling channels. Temperature curves of the same monitoring node in the temperature field results of the turntable with and without cooling pipes were extracted respectively to obtain the average temperature curves of the turntable with and without cooling pipes. By comparing and analyzing the average temperature curves of the turntable with and without cooling pipes, the analysis results of the cooling pipes on heat dissipation of the turntable are obtained.

4. The method for thermal analysis of a direct drive rotary table with heat dissipation function according to claim 3, characterized in that: The oil-free turntable model is meshed to obtain an oil-free mesh model, specifically including: Using ANSA software, the entire turntable is divided into n independent parts along geometric features; each independent part has a separate PID. For the first n-1 regular geometric regions, a high-quality hexahedral mesh is generated using the MAP command; For the nth complex transition region encompassing irregular contours and the intersection of multiple features, an adaptive tetrahedral mesh is used for filling to adapt to the complex boundary. Finally, the Paste command is used to achieve seamless bonding of the interface nodes of each region, and a complete turntable mesh model is finally constructed.

5. The method for thermal analysis of a direct-drive rotary table with heat dissipation function according to claim 4, characterized in that: In terms of sizing control, the grid transitions smoothly from the inside to the outside of the turntable, with dimensions ranging from... mm gradually increased to mm, and control the grid aspect ratio at Within.

6. The method for thermal analysis of a direct-drive rotary table with heat dissipation function according to claim 3, characterized in that: Construct a turntable model with cooling pipes, specifically including: In UG software, draw a spiral line to create a cooling pipe guide line; at the starting point of the spiral line, create a sketch to draw a rectangular cross-section of the pipe; use the sweep command to obtain the cooling pipe, and use Boolean operations to obtain a turntable model with the cooling pipe.

7. The method for thermal analysis of a direct-drive rotary table with heat dissipation function according to claim 3, characterized in that: The model of the turntable with oil passages is meshed to obtain a meshed model with oil passages, specifically including: In ANSA software, the Cut command is used to divide the center of the turntable model with cooling pipes into a separate PID. The Coons command is used to fill the cut areas and then fill the model with a tetrahedral mesh; the mesh size is controlled within... Approximately mm; The paste command is used to glue together several nodes in contact, ultimately generating a complete mesh model of the turntable with cooling pipes.

8. The method for thermal analysis of a direct-drive rotary table with heat dissipation function according to claim 3, characterized in that: Transient simulation analysis of turntable heat conduction was performed on mesh models with and without oil passages, respectively, including: The thermal parameters, physical properties, and boundary conditions of the two mesh generation models were set respectively; For mesh generation models with oil passages: use keywords to define convection boundary conditions, select the turntable surface that is in direct contact with the coolant, and set the convection heat transfer coefficient and coolant temperature to complete the forced convection heat transfer settings between the coolant and the contact surface. The heat conduction transient simulation was performed on the two mesh generation models after the settings were configured using a solver, and heat conduction transient simulation analysis files for the turntable with and without cooling pipes were obtained respectively.

9. The method for thermal analysis of a direct-drive rotary table with heat dissipation function according to claim 8, characterized in that: The convective heat transfer coefficient is the forced convective heat transfer coefficient between the coolant in the cooling pipe and the contact surface of the turntable. The forced convection heat transfer coefficient is calculated using the following method: Based on the cross-sectional area and flow velocity of the cooling pipe, the wetted perimeter of each section is calculated; then the Reynolds number is used to determine the flow state of the coolant, and the friction coefficient and Nusselt number are calculated according to the correlation formula; finally, the forced convection heat transfer coefficient between the coolant and the contact surface of the turntable is obtained by combining the thermal conductivity.

10. The method for thermal analysis of a direct-drive rotary table with heat dissipation function according to claim 9, characterized in that: The forced convection heat transfer coefficient is determined according to the following formula. : ； in, For Nusselt numbers; ; Friction factor; It is the Reynolds number; It is a Prandtl number; The thermal conductivity of the fluid; ; ; The characteristic velocity of the fluid is taken as the average velocity across the cross section when flowing inside the pipe. A qualitative measure of geometric features; The kinematic viscosity of the fluid; This refers to the cross-sectional area of ​​the cooling pipe. It is a wetted period.

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