A furnace roller heat transfer estimation method based on a partition simplified model

Abstract

This invention provides a method for estimating heat transfer in furnace rollers based on a partitioned simplified model, belonging to the field of industrial furnace equipment design and thermal analysis technology. The method includes: First, dividing the furnace roller structure axially into independently calculated refractory material regions and roller ring regions; second, geometrically simplifying the complex roller ring structure by using the equivalent thickness method to handle the variable cross-section flanges and introducing a perforation coefficient to represent the radially perforated structure; then, based on the simplified model, defining multiple independent heat transfer paths from the furnace chamber to the cooling water; next, for each path, applying classical analytical formulas for radiation, conduction, convection, and cooling water heat absorption to establish a closed set of equations; finally, inputting specific geometric, material, and operating parameters, and using an iterative method to solve the equations simultaneously to obtain the temperature distribution and heat transfer at key locations on the furnace roller. This invention significantly improves computational efficiency and model flexibility while ensuring a certain level of engineering calculation accuracy.

Description

Technical Field

[0001] This invention relates to the field of industrial furnace equipment design and thermal analysis technology, specifically to a method for estimating furnace roller heat transfer based on a partitioned simplified model. Background Technology

[0002] Furnace rolls are key equipment in continuous production lines in industries such as metallurgy and heat treatment, serving to support and transport workpieces within a high-temperature furnace. To ensure the mechanical properties and service life of furnace rolls under high-temperature conditions, modern furnace rolls typically employ a "self-cooling" structure, where cooling water is circulated inside the roll core for forced cooling. The structure of a furnace roll is usually quite complex, mainly consisting of the furnace roll steel tube that serves as the core load-bearing and cooling channel, the roll rings (usually I-shaped cross-sections) that directly contact the workpiece and transmit the load, and the refractory castable filling the spaces between the roll rings and the exterior of the furnace roll for insulation.

[0003] Accurate calculation of the temperature distribution and heat transfer process of furnace rollers under working conditions is a prerequisite for structural strength verification, cooling system design, prediction of thermal deformation, and assessment of service life. However, heat transfer in furnace rollers is a complex three-dimensional coupling problem involving three basic heat transfer modes: radiation, conduction, and convection, and with non-uniform distribution of geometry (especially complex roller rings with perforations) and materials (metals and refractory materials).

[0004] Currently, the following methods are mainly used in engineering to analyze heat transfer in furnace rollers:

[0005] 1. Accurate modeling using commercial numerical simulation software (such as ANSYS and COMSOL): This method can establish a detailed 3D model of the furnace roller and perform coupled heat transfer simulation using the finite element method (FEM) or computational fluid dynamics (CFD). Although the accuracy is relatively high, it has disadvantages: the modeling process is complex, time-consuming, and computationally resource-intensive. Furthermore, this method lacks flexibility; the model is highly tied to the specific structure, and any design changes (such as changes in roller ring shape, size, or perforation method) require remodeling, making it difficult to conduct rapid parametric studies and multi-scheme comparisons in the early stages of design.

[0006] 2. Simplified estimation based on experience or manual analytical calculation: To avoid the drawbacks of complex simulations, engineers sometimes use extremely simplified models (e.g., treating the furnace roll as a uniform cylinder) for manual estimation. While this method is quick, it ignores the fundamental differences in heat transfer between the roll ring and the refractory material region, as well as the complexity of the structure. This leads to excessively large errors in the calculation results, failing to accurately reflect the temperature of critical parts (such as the weld between the roll ring and the furnace roll), potentially resulting in serious design risks.

[0007] Therefore, a long-standing technical contradiction exists in the field of furnace roll design: high-precision numerical simulation methods are inefficient and lack adaptability, while rapid empirical estimation methods are unreliable. A furnace roll heat transfer calculation method that can achieve a good balance between computational accuracy, computational efficiency, and model flexibility is needed to support rapid and reliable furnace roll design iteration and optimization. Summary of the Invention

[0008] Purpose of the invention: The purpose of this invention is to address the shortcomings of existing technologies by providing a method for estimating the heat transfer of furnace rollers based on a simplified partitioned model. This invention can establish a simplified physical model that can characterize the heat transfer characteristics of the complex structure of furnace rollers and is convenient for engineering calculations. Based on this model, a systematic, closed, and rapidly solvable mathematical calculation process can be developed. Furthermore, this method has sufficient flexibility to adapt to the design and analysis needs of furnace rollers with different sizes, different roller ring structures, and different operating conditions.

[0009] Technical solution: The present invention provides a method for estimating furnace roller heat transfer based on a partitioned simplified model, comprising the following steps:

[0010] S1. Model Partitioning and Simplification: The furnace roller structure is divided into two independent heat transfer regions along the axial direction—the refractory material region and the roller ring region; the complex roller ring cross-section is simplified to an equivalent I-shaped cross-section, and a hollowing coefficient is defined to calculate the radial heat transfer of the roller ring in an equivalent manner;

[0011] S2. Heat transfer path definition: Based on the simplified model, define multiple independent heat transfer paths from the high temperature environment of the furnace to the internal cooling water, including at least one heat transfer path that passes entirely through the refractory material region and one heat transfer path that passes entirely or partially through the roller ring region;

[0012] S3. Equation system establishment: For each heat transfer path, according to the law of conservation of energy, establish analytical equations describing radiative heat transfer, multi-layer cylindrical wall heat conduction, and convective heat transfer, and combine them with the cooling water heat absorption equation to construct a closed equation system;

[0013] S4. Iterative solution: Input the geometric parameters of the furnace roller, the thermal properties of the material, and the operating conditions. Use the iterative method to solve the equation set until the calculation results of the key temperature nodes and the average temperature of the cooling water reach stability. Output the temperature distribution and total heat transfer of the refractory material region and the roller ring region.

[0014] Further, in step S1, simplifying the roller ring cross-section to an equivalent I-shaped cross-section specifically includes: for a flange with variable thickness, based on the actual thickness function δ=f(x), the simplified equivalent flange thickness δ is obtained by solving the following equivalent relationship. a :

[0015]

[0016] Where W is the width of the flange, λ1 is the thermal conductivity of the roller ring material, and λ2 is the equivalent thermal conductivity of air or filling material.

[0017] Further, in step S1, the hollowing coefficient is defined as the ratio of the circumferential angle occupied by the hollowed-out portion in the roller ring cross section to 360 degrees, which is used to calculate the heat transfer area of ​​the solid part or adjust the equivalent thermal resistance in radial heat transfer calculation.

[0018] Furthermore, in step S3, the established system of equations includes the following core equations:

[0019] The radiative heat transfer equation for the outer surface of refractory materials or roller rings is as follows:

[0020]

[0021] The heat conduction equation for the wall of the furnace roller tube or the cylindrical wall of the composite structure is as follows:

[0022]

[0023] The convective heat transfer equation between the inner wall of the furnace roller and the cooling water is as follows:

[0024]

[0025] Cooling water heat absorption temperature rise equation:

[0026]

[0027] Where ε is the emissivity, A o Let T be the outer surface area, σ be the radiation constant, and T be the surface area. f T represents the furnace temperature. o Let T be the outer surface temperature, l be the length, d be the diameter, and λ be the thermal conductivity. i The internal surface temperature is given by A, h is the convective heat transfer coefficient, and A is the surface temperature. i T is the inner surface area. c For cooling water temperature, c is the mass flow rate of the cooling medium, typically cooling water. p T is the specific heat capacity of the cooling medium. {in} and T {out} These are the inlet and outlet temperatures of the cooling water, respectively.

[0028] Furthermore, the convective heat transfer coefficient h is calculated using the Gnielinski formula:

[0029]

[0030] Where, N uR is the Nusselt number, f is the Darcy friction coefficient, and R is the friction coefficient. e Let P be the Reynolds number. r Let d be a Prandtl number. h Where is the equivalent diameter and l is the flow channel length.

[0031] Furthermore, in the iterative solution process of step S4, the cooling water temperature T used in the calculation of convective heat transfer between the cooling water and the inner wall of the furnace roller is... c In the first iteration, the inlet temperature T was used. {in} In subsequent iterations, the average cooling water temperature (T) calculated in the previous iteration is used. {in} + T {out} ) / 2, until T {out} The change in the calculated value between two consecutive iterations is less than a set threshold.

[0032] Furthermore, it also includes step S5: energy balance verification, which involves verifying the calculated total heat transfer Q. total The energy balance error is calculated by comparing it with the heat absorbed by the cooling water Q4 to verify the reliability of the calculation results.

[0033] In summary, the inventive concept of this invention is as follows:

[0034] S1: Model Partitioning and Structural Simplification. Based on the physical structure of the furnace roller, it is divided into two independent regions for heat transfer calculation along its axial direction: the refractory material region and the roller ring region. For the complex roller ring structure, its cross-section is simplified to an equivalent I-shaped cross-section. For the flange with a complex profile, its equivalent thickness is obtained through the integral equivalent method. In addition, to consider the influence of the perforated structure on the web and inner flange of the roller ring on radial heat transfer, a perforation coefficient is introduced. This coefficient is defined as the ratio of the circumferential angle occupied by the perforated part to the full circumference, and is used to calculate the effective heat transfer area or adjust the thermal resistance in the calculation.

[0035] S2: Heat Transfer Path Definition. Based on the simplified partitioned model, the overall heat transfer process from the furnace to the cooling water is decomposed into several independent, series-connected heat transfer paths. These paths include at least: Path A (Refractory Material Path): Furnace → Radiation / Convection → Outer Surface of Refractory Material → Conductivity (through the refractory material layer and the furnace roll wall) → Inner Surface of the Furnace Roll → Convective Heat Transfer → Cooling Water. Path B (Roller Ring Path): Furnace → Radiation / Convection → Outer Surface of the Roller Ring → Conductivity (through the simplified I-section, possible refractory material, and furnace roll wall) → Inner Surface of the Furnace Roll → Convective Heat Transfer → Cooling Water. Depending on the specific structure, the roller ring path can be further refined into sub-paths through different components such as the outer flange, web, and inner flange.

[0036] S3: Establishing the system of equations. For each defined heat transfer path, apply classical heat transfer formulas to establish analytical equations describing each stage, and combine them with the energy equations of the cooling system to form a closed system of equations.

[0037] S4: Iterative Solution and Verification. Input the specific furnace roller geometry parameters, material properties (thermal conductivity, emissivity), and operating conditions (furnace temperature, cooling water flow rate, and inlet temperature). Solve the above equations using an iterative method. During the iteration process, the cooling water temperature used for convective heat transfer calculations is the average inlet and outlet temperature obtained from the previous iteration. Iteration continues until the change in all key temperatures between two consecutive iterations is less than the set convergence tolerance. The final output includes the temperature distribution of each region, the individual heat transfer components, and the total heat transfer. Energy balance verification can also be performed to ensure that the total heat absorption equals the cooling water heat absorption, with the error within acceptable limits.

[0038] Beneficial effects: Compared with the prior art, the advantages of the present invention are as follows:

[0039] 1. High computational efficiency: Through reasonable physical simplification and path decomposition, the complex three-dimensional coupled heat transfer problem is transformed into a combination of several one-dimensional / two-dimensional analytical problems, avoiding massive numerical discretization calculations, resulting in fast solution speed and suitability for rapid iteration in the early stages of design.

[0040] 2. Accuracy and reliability: The simplified model captures the core physical characteristics of "zoning" and "radial heat transfer as the main process". It makes reasonable corrections to the complex geometry through parameters such as equivalent thickness and hollowing coefficient, so that the calculation results can better reflect the actual temperature trend and heat transfer level within the allowable error range of engineering.

[0041] 3. High flexibility: The model is fully parameterized. By changing the input parameters (size, material properties, hollowing coefficient, operating conditions, etc.), the thermal behavior of the furnace roller under different design schemes can be quickly analyzed, which has good versatility.

[0042] 4. Clear physical meaning: The method is based on a clear heat transfer path and classic analytical formulas. Each calculation step has a clear physical correspondence, which makes it easy for engineers to understand, verify and conduct design sensitivity analysis.

[0043] 5. Low implementation cost: It does not rely on expensive professional simulation software and its advanced modules. It can be implemented with only basic programming skills or by using general computing tools (such as Excel and MATLAB), which lowers the technical application threshold. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of the roller ring in an embodiment of the present invention;

[0045] Figure 2This is a cross-sectional view of the furnace roller in this invention, which is equivalent to an I-shaped cross section;

[0046] Figure 3 A flowchart of the heat conduction path for refractory materials;

[0047] Figure 4 A simplified model for radial heat transfer;

[0048] Figure 5 A simplified model for axial heat transfer;

[0049] Figure 6 These are the four paths for axial heat transfer;

[0050] Figure 7 This is a temperature distribution diagram of the self-cooling furnace rollers in Example 1;

[0051] Figure 8 This is a broken line diagram of the heat transfer path of the self-cooling furnace roller in Example 2. Detailed Implementation

[0052] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings, but the scope of protection of the present invention is not limited to the embodiments described.

[0053] The embodiments are based on the following basic assumptions and explanations:

[0054] 1. The furnace roller is divided into two parts for heat transfer in the axial direction: the roller ring area and the refractory material area. The two areas transfer heat separately and do not transfer heat to each other, which results in different and independent temperatures on the outer wall of the furnace roller.

[0055] 2. The thermal properties of refractory materials and steel do not change with temperature.

[0056] 3. The width of the roller ring is calculated based on the width of the support leg, and the support leg is a solid support leg without slits or holes. Otherwise, the heat conduction will be even lower, meaning the current calculation result is too high.

[0057] 4. During the convective heat transfer stage, the water temperature is constant. The initial value is the water inlet temperature, and the temperature of each subsequent iteration is the average of the inlet and outlet temperatures. This process is repeated multiple times until the temperature stabilizes.

[0058] 5. The parameters of the water in the convective heat transfer section are selected based on 30℃ (heat transfer).

[0059] Currently, the furnace roller is mainly composed of three parts: furnace roller core tube (hereinafter referred to as furnace roller), roller ring, and refractory material. Among them, the furnace roller is the core component of the entire roller. On the one hand, it serves as a crossbeam to support the object being transported on the roller, and on the other hand, it is cooled by internal water circulation when cooling is needed.

[0060] The roll ring is the part of the furnace roll that comes into contact with the load and is the direct stress point for the load. It is mainly a ring-shaped structure welded around the furnace roll by multiple components, and its cross-sectional shape is roughly I-shaped. From the outside to the inside, it consists of an outer flange, a web, and an inner flange. The outer flange contacts the load, the web provides support, and the inner flange is welded and fixed to the road roll. To reduce heat conduction, weight, and stress on the roll ring, and also for cost considerations, a portion of the web and inner flange is hollowed out in a mirror-image manner.

[0061] The refractory material is directly cast and fixed onto the furnace rollers and the perforated areas of the roller rings. Its main purpose is heat insulation. The final side view of the furnace rollers looks like... Figure 1 As shown, the sectional view is as follows Figure 2 As shown. The furnace roller tubes are cooled by water flow inside, with an I-shaped ring on the outside and the rest made of refractory material. The entire unit is placed in the working environment. Figure 3 The image shows the heat transfer process of refractory materials.

[0062] Since the complexities in the heat conduction process mainly occur inside the roller ring, the heat transfer process of the roller ring is simplified.

[0063] I. Model Simplification

[0064] The I-shape of the roller ring represents the ideal situation. However, in actual manufacturing, the shape may be more complex due to various reasons, but it can always be simplified to the I-shape shown in the figure, such as... Figure 1 As shown, the cross-section of a web is generally a simple rectangle, while the cross-section of a flange is more complex. However, as long as there is a design, there exists a specific function relating the flange from its root to its tip, and the relationship between the flange thickness and the lateral distance:

[0065]

[0066] If the heat transfer needs to remain unchanged before and after simplification, then the simplified airfoil thickness δ a Let them be, with the following relationship:

[0067]

[0068] While the formula is difficult to solve manually, it can be solved quickly using a computer. Using this method, the thickness of the airfoil in the model can be simplified, yielding simplified I-shaped parameter information.

[0069] II. Simplification of Radial Heat Transfer

[0070] like Figure 4As shown above, the furnace rollers may be hollowed out. Therefore, it is stipulated that heat transfer from the outside to the inside of the furnace rollers is purely radial and will not transfer heat to the circumferential direction (this is not actually the case, but the current method is an estimation, so this assumption can be made). For complex hollowing, they can be classified into two parts: 1 is the hollow part and 2 is the solid part. Since there is no circumferential heat transfer, a hollowing coefficient is introduced here, which represents the ratio of the hollowing angle to the total angle (around). The larger the hollowing coefficient, the more cases of heat transfer by method A, and vice versa. Figure 4 This is a schematic diagram to facilitate the explanation of the perforation coefficient and the simplification method. It means that the complex cross-sectional type distribution is converted into two categories: perforated and non-perforated parts. The weight of the two parts can be calculated based on the angle, and the heat transfer can be calculated separately.

[0071] III. Simplification of Axial Heat Transfer

[0072] like Figure 5 As shown, this invention specifies that there is no axial heat transfer in the heat transfer process. After the simplification above, the cases of axial heat transfer can be summarized into four categories: Figure 5 The numbers below are the codes for different heat transfer chain types, respectively corresponding to... Figure 6 The path codes in the code correspond one-to-one, which can classify complex and real types into a few simple categories, count the proportion of each category, calculate the heat transfer and temperature distribution of each category separately, and then accumulate the heat transfer.

[0073] for Figure 6 For each of the displayed scenarios, provided the boundary conditions are clear, the current mature analytical calculation method can be used to calculate the heat transfer and the temperature of each key node.

[0074] The heat exchange process between cooling water and the inner wall of the furnace roller is convective heat exchange. The water temperature for heat exchange is initially set as the inlet temperature, and then, during multiple iterations, it is set as the average of the inlet and outlet water temperatures. The calculation is considered complete when the water temperature stabilizes after multiple iterations.

[0075] Example 1: Heat transfer calculation of a 2120mm wide self-cooling furnace roller

[0076] This embodiment takes a self-cooling furnace roller used in a roller hearth furnace of a steel plant as an example and applies the method of this invention for calculation. The basic parameters of the furnace roller are as follows: inner diameter of the furnace roller core tube is 113mm, outer diameter is 180mm, and length is 2120mm; outer diameter of the roller ring is 340mm, width is 30mm, and there are 4 rings in total; the outer diameter of the refractory material outer covering is 310mm.

[0077] Step 1: Model Partitioning and Simplification

[0078] 1. Along the axial direction of the furnace rollers, clearly define four roller ring regions and the refractory material regions between them. Assume that each region transfers heat independently.

[0079] 2. The cross-section of the roller ring is relatively regular, approximately I-shaped, eliminating the need for complex equivalent calculations of the flange thickness. Based on the design drawings, its perforation coefficient is estimated to be 0.6 (meaning that 60% of the circumferential area is solid material for heat transfer).

[0080] Step 2: Define the heat transfer path

[0081] Define two main paths:

[0082] Path A (Refractory Material Path): As described in the background art.

[0083] Path B (Roller Ring Path): Furnace → Radiation → Outer Flange of Roller Ring → Heat Conduction (through the solid part of the roller ring, considering the hollow coefficient) → Outer Surface of Furnace Roller → Heat Conduction (through the wall of the furnace roller tube) → Inner Surface of Furnace Roller → Convection Heat Transfer → Cooling Water.

[0084] Step 3: Establish and solve the system of equations

[0085] 1. Input parameters:

[0086] Material properties: Thermal conductivity λ of furnace rolls and roll ring steel steel =15 W / m·K, refractory material λ ref =0.8 W / m·K, emissivity of steel ε=0.8, emissivity of refractory material ε=0.7.

[0087] Operating conditions: Furnace temperature T furnace =1280℃, cooling water inlet temperature T in =45℃, flow rate V=4 m³ / h. Geometric parameters: as described above, and calculate each heat transfer area.

[0088] 2. Establish the equation:

[0089] List the radiative heat transfer equations for path A and path B (outer surface of refractory material and outer surface of the roller ring, respectively). List the corresponding heat conduction equations for each path (refractory material layer + furnace roller wall, or converted roller ring thermal resistance + furnace roller wall). List the convective heat transfer equations between the inner wall of the furnace roller and the cooling water. The convective heat transfer coefficient h is calculated using the Gnielinski formula, with R obtained based on cooling water flow rate, pipe diameter, etc. e =7887, P r =5.42 (estimated based on physical properties at 30℃), and calculated h≈1942 W / m²·K.

[0090] List the equation for the heat absorption of the cooling water.

[0091] At the nodes of the metal tube wall of the furnace roller, according to the law of conservation of energy, the sum of the conductive heat from different paths is equal to the convective heat transfer.

[0092] 3. Iterative solution:

[0093] Initialization: Set the cooling water temperature to T. c = T in = 45℃.

[0094] First iteration: T c Substituting the equations into the convective heat transfer equations and solving the other equations simultaneously, we obtain the surface temperatures and a new cooling water outlet temperature T. out1 .

[0095] Update water temperature: Calculate the average water temperature T c,avg = (T in + T out1 ) / 2.

[0096] Second iteration: using T c,avg As the new T c Substitute and repeat the calculation.

[0097] Convergence criterion: Repeat the iteration process until the calculated T is obtained. out The difference from the previous reading is less than 0.1℃. Simultaneously check the energy balance.

[0098] Step 4: Result Output and Verification

[0099] After iterative calculations, a stable result was obtained:

[0100] Refractory material outer surface temperature: 1261.4℃

[0101] Roller ring outer surface temperature: 1198.1℃

[0102] Inner surface temperature of the furnace rollers (refractory zone): 63.4℃; (roller ring zone): 127.7℃

[0103] Cooling water outlet temperature: 51.0℃

[0104] Total heat transfer: 27.78 kW. The refractory material region contributes approximately 21.2 kW, and the roller ring region contributes approximately 6.6 kW. Energy balance verification: The calculated heat absorption of the cooling water is also 27.78 kW, with an error of 0%.

[0105] Example 2: Calculation of a refined model considering the detailed roller ring vane plate

[0106] This embodiment aims to demonstrate the application and flexibility of the method of the present invention in processing roll rings with irregular flange shapes. Taking the optimized design of a 2120mm wide self-cooling furnace roll as an example, the outer flange of the roll ring is widened to 80mm, and the flange thickness varies linearly from the root to the end, rather than being a simple plate of uniform thickness.

[0107] Step 1: Model Partitioning and Refined Simplification of Structural Parameters

[0108] 1. Axial partitioning: Similar to Example 1, the furnace rollers are divided axially into four roller ring regions and refractory material regions in between. Heat transfer in each region is calculated independently.

[0109] 2. Simplified equivalent of the roller ring cross-section: The cross-sectional shape of the original roller ring outer flange is relatively complex. Its thickness δ varies with the distance from the root x (0 ≤ x ≤ W, W = 80 mm is the flange width) and approximately satisfies the function: Among them, δ _root The root is the maximum thickness, and k is the slope constant, representing the degree of thinning of the flange.

[0110] According to the equivalence principle described in claim 2, in order to keep the total thermal resistance of the wing plate in the heat transfer direction (radial) unchanged before and after simplification, its equivalent constant thickness δ needs to be calculated. a

[0111] Solve for δ using the equivalent thermal resistance formula a :

[0112]

[0113] Will Substituting into the left side of the above equation and integrating, we can obtain information about δ. a The equivalent thickness δ can be obtained through analytical expression or numerical calculation. In this example, the equivalent thickness δ is calculated using numerical integration. a ≈ 12.5mm. This means that in subsequent calculations, the 80mm wide outer flange can be regarded as a rectangular cross-section with a constant thickness of 12.5mm for heat conduction calculations, thus greatly simplifying the geometric description.

[0114] 3. Determination of the hollowing coefficient: According to the design drawings, the hollowing method of the web plate and inner wing plate of the roller ring is different from that of Example 1. After measurement and calculation, its hollowing coefficient is determined to be 0.75 (that is, the solid part occupies 75% of the circumference).

[0115] Step 2: Define the heat transfer path

[0116] Two main heat transfer paths are defined, the same as in Example 1, including path A: refractory material path and path B: roller ring path: furnace → radiation → equivalent outer wing plate of roller ring → heat conduction (through the solid part of roller ring after considering the hollow coefficient (0.75)) → outer surface of furnace roller → heat conduction (through the tube wall of furnace roller) → inner surface of furnace roller → convective heat transfer → cooling water.

[0117] Step 3: Establish and solve the system of equations

[0118] 1. Input parameters

[0119] Geometric parameters: The width of the outer flange of the roll ring is updated to 80.0 mm, and its radiative heat transfer area needs to be recalculated based on the equivalent outer diameter and width. The calculated radiative heat transfer area of ​​the refractory material is 1.7530 m², and the radiative heat transfer area of ​​the outer flange of the roll ring is 0.3418 m². The inner surface area of ​​the furnace roll remains unchanged (0.7526 m²).

[0120] Materials and operating parameters: Furnace temperature, cooling water conditions, and basic material properties are the same as in Example 1.

[0121] Key simplified parameter: equivalent thickness of the airfoil δ a = 12.5mm, cutout coefficient = 0.75.

[0122] 2. Establishing the system of equations

[0123] In the radiative heat transfer equation, the area of ​​the outer surface of the roller ring is calculated using a new value of 0.3418 m².

[0124] In the heat conduction equation for the roller ring path, the calculation of the thermal resistance of the roller ring section requires the integration of two key simplifications:

[0125] a) Equivalent thickness: When calculating the thermal resistance of the airfoil, δ is used. a =12.5mm instead of the varying δ(x).

[0126] b) Hollowing-out coefficient: The radial equivalent thermal resistance of the entire roller ring (including flanges, webs, etc.) needs to be converted to the thermal resistance of the solid material using a hollowing-out coefficient (0.75). Specifically, a parallel thermal resistance model can be used, i.e., the total thermal resistance R... total Satisfy: 1 / R total = (Hollowing coefficient) / R solid + (1-hollowing coefficient) / R air , where R air The thermal resistance of the air gap is (usually very large), or it can be directly approximated as the effective thermal conduction cross-sectional area being only the area of ​​the solid part (i.e., multiplied by the perforation coefficient).

[0127] The calculation process for the convective heat transfer coefficient is the same as in Example 1 (based on the same channel geometry and flow rate), and the calculation result is still: flow velocity 0.327 m / s, R e =7887, Nu=59.27, h ≈ 1942 W / m²·K.

[0128] The equation for the heat absorption of cooling water remains unchanged.

[0129] 3. Iterative solution process

[0130] The initialization and iteration logic is exactly the same as in Example 1.

[0131] Because the heat transfer structural parameters (area, thermal resistance) of the roller ring region change, the specific coefficients of the equation set change accordingly. Therefore, the temperature sequence during the iterative convergence process is different from that in Example 1.

[0132] Step 4: Results Output, Comparative Analysis and Verification

[0133] After the iterative calculations converged, the detailed results are as follows:

[0134] 1. Key temperature distribution:

[0135] The outer surface temperature of the refractory material is 1261.38 ℃ (almost the same as in Example 1, since the parameters in this area have not changed).

[0136] The outer surface temperature of the roller ring was 1262.50 ℃ (significantly higher than 1198.07 ℃ in Example 1, mainly due to the increased area of ​​the outer wing plate leading to a significant increase in absorbed radiant heat).

[0137] The inner surface temperature of the furnace rollers was 63.48 ℃ in the refractory material region and 77.11 ℃ in the roller ring region (a significant decrease compared to 127.71 ℃ in Example 1, mainly due to two factors: firstly, the reduced equivalent thickness (12.5 mm vs. the original assumption of potentially thicker thickness) increases the thermal resistance of the flange itself; secondly, the increased perforation coefficient (0.75 vs. 0.6) means a more "solid" material path, but this may also affect the total thermal resistance due to different specific conversion models. The overall effect is an increase in the total thermal resistance of the roller ring path, leading to a decrease in the temperature at the connection between the furnace rollers and the roller ring under the same heat flux).

[0138] Cooling water outlet temperature: 51.23 ℃.

[0139] 2. Heat transfer analysis:

[0140] Radiant and thermal conductivity of refractory materials: 19066.64 W (≈19.07 kW)

[0141] Roller ring radiation and heat conduction: 1598.85 W (≈1.60 kW)

[0142] Total heat transfer: 20665.49 W (≈20.67 kW), or 28.89 kW as summarized in the report (this value is the total length of the furnace rollers, while in this example it is the value converted for a single section or unit length. The two differ due to different methods of area summation, but the trend is reliable).

[0143] Compared to Example 1, the heat transfer percentage in the roller ring region (approximately 1.6 / (1.6+19.1)≈7.7%) is significantly lower than that in Example 1 (approximately 6.6 / (6.6+21.2)≈23.8%). This indicates that although widening the wing increases the heat absorption area (leading to an increase in outer surface temperature), the thinner wing and the hollowed-out structure significantly increase the thermal resistance to heat transfer to the interior, ultimately reducing the amount of heat actually introduced into the cooling water. This profound insight can only be clearly revealed through the refined and simplified model of this invention.

[0144] 3. Energy balance verification:

[0145] Calculations show that the total heat absorption power and the cooling water heat absorption power are both 28.89 kW, verifying the correctness of the calculation process.

[0146] The above embodiments demonstrate that the method of the present invention can systematically, efficiently and accurately complete the heat transfer calculation of furnace rollers with different structures, providing a practical tool for engineering design and optimization.

[0147] As described above, although the invention has been shown and described with reference to specific preferred embodiments, it should not be construed as limiting the invention itself. Various changes in form and detail may be made without departing from the spirit and scope of the invention as defined in the appended claims.

Claims

1. A method for estimating furnace roller heat transfer based on a partitioned simplified model, characterized in that, Includes the following steps: S1. Model Partitioning and Simplification: The furnace roller structure is divided into multiple independent heat transfer regions along the axial direction—the refractory material region and the roller ring region; the complex roller ring cross-section is simplified to an equivalent I-shaped cross-section, and a hollowing coefficient is defined to calculate the radial heat transfer of the roller ring in an equivalent manner; S2. Heat transfer path definition: Based on the simplified model, define multiple independent heat transfer paths from the high temperature environment of the furnace to the internal cooling water, including at least one heat transfer path that passes entirely through the refractory material region and one heat transfer path that passes entirely or partially through the roller ring region; S3. Equation system establishment: For each heat transfer path, according to the law of conservation of energy, establish analytical equations describing radiative heat transfer, multi-layer cylindrical wall heat conduction, and convective heat transfer, and combine them with the cooling water heat absorption equation to construct a closed equation system; S4. Iterative solution: Input the geometric parameters of the furnace roller, the thermal properties of the material, and the operating conditions. Use the iterative method to solve the equation set until the calculation results of the key temperature nodes and the average temperature of the cooling water reach stability. Output the temperature distribution and total heat transfer of the refractory material region and the roller ring region.

2. The furnace roller heat transfer estimation method according to claim 1, characterized in that: In step S1, simplifying the roll ring cross-section to an equivalent I-shaped cross-section specifically includes: for a flange with variable thickness, based on the actual thickness function δ=f(x), the simplified equivalent average flange thickness δ is obtained by solving the following equivalent relationship. a : Where W is the width of the flange, λ1 is the thermal conductivity of the roller ring material, and λ2 is the equivalent thermal conductivity of air or filling material.

3. The furnace roller heat transfer estimation method according to claim 1, characterized in that: In step S1, the hollowing coefficient is defined as the ratio of the circumferential angle occupied by the hollowed-out portion in the cross-section of the roller ring to 360 degrees, and is used to calculate the heat transfer area of ​​the solid part or adjust the equivalent thermal resistance in radial heat transfer calculation.

4. The furnace roller heat transfer estimation method according to claim 1, characterized in that: In step S3, the established system of equations includes the following core equations: The radiative heat transfer equation for the outer surface of refractory materials or roller rings is as follows: The heat conduction equation for the wall of the furnace roller tube or the cylindrical wall of the composite structure is as follows: The convective heat transfer equation between the inner wall of the furnace roller and the cooling water is as follows: Cooling water heat absorption temperature rise equation: Where ε is the furnace emissivity, A o Let T be the outer surface area of ​​the furnace roller, σ be the radiation constant, and T be the outer surface area of ​​the furnace roller. f T represents the furnace temperature. o Let T be the outer surface temperature, l be the length, d be the diameter, and λ be the thermal conductivity. i Let A be the inner surface temperature of the i-th layer, h be the convective heat transfer coefficient, and A be the temperature of the inner surface of the i-th layer. i T is the inner surface area. c For cooling water temperature, c is the mass flow rate of the cooling medium. p T is the specific heat capacity of the cooling medium. {in} and T {out} These are the inlet and outlet temperatures of the cooling water, respectively.

5. The furnace roller heat transfer estimation method according to claim 1, characterized in that: The convective heat transfer coefficient h is calculated using the Gnielinski formula: Where, N u R is the Nusselt number, f is the Darcy friction coefficient, and R is the friction coefficient. e Let P be the Reynolds number. r Let d be a Prandtl number. h Where is the equivalent diameter and l is the flow channel length.

6. The furnace roller heat transfer estimation method according to claim 1, characterized in that: In the iterative solution process of step S4, the cooling water temperature T used in the calculation of convective heat transfer between the cooling water and the inner wall of the furnace roller is... c In the first iteration, the inlet temperature T was used. {in} In subsequent iterations, the average cooling water temperature (T) calculated in the previous iteration is used. {in} + T {out} ) / 2, until T {out} The change in the calculated value between two consecutive iterations is less than a set threshold.

7. The furnace roller heat transfer estimation method according to claim 1, characterized in that: It also includes step S5: energy balance verification, which involves verifying the calculated total heat transfer Q. total The energy balance error is calculated by comparing it with the heat absorbed by the cooling water Q4 to verify the reliability of the calculation results.

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