An Innovative E-LHP Software to Study the Effect of Radius on (LHP) During Transient Heat Transfer Quenched Alloy SCM440-Steel Bar

— The finite element method (FEM) has been used to model an axisymmetric industrial quenched alloy steel, SCM440, in order to examine the effects of process history on material properties and metallurgy. The temperature history and subsequently the hardness of the heat-treated steel bar at any node are determined mathematically using the 1-Dimensional element axisymmetric model. Calculated is the lowest hardness point (LHP), which will be at the center and will depend on the radius. By converting the computed characteristic cooling time for phase transformation t 8/5 , which equals (t 5 - t 8 ), to hardness, the hardness (HRC) in specimen points was calculated in this work. To obtain desired mechanical qualities including hardness, toughness, and microstructure, the model can be used as a design tool to develop a cooling strategy. A computer software based on the developed mathematical model was created. The computer program can be used independently or integrated into a temperature history program called (E-LHP-software) to continuously calculate and display the temperature history of a heat-treated quenched steel bar and thereby determine its lowest hardness point and analyze and study the effect of radius on LHP. By contrasting its hardness results with those of the experimental work, the generated program from the mathematical model (MM) has been confirmed and validated. The comparison shows how trustworthy the suggested model is.


I. INTRODUCTION
Steels are quenched (quickly chilled, such as in water), which is a multi-physics process involving an intricate web of heat transport couplings. There is currently no analytical solution due to the problem's complexity, linked (thermalmechanical-metallurgical) theory, and non-linear character. However, the most popular method, the finite element method (FEM), the finite volume method, and the finite difference method can all be used to solve numerical problems [1]. The heat transfer is temporary.
The investigation of heat transmission in this manuscript will be done in three dimensions. To save money and computing time, the 3-D analysis will be condensed into a 1-D axisymmetric analysis [2]- [7]. This is possible because, as shown in Fig. 1, Fig. 2 The mathematical model was created using the Galerkin weighted residual method (MM). This project will construct a 1-Dimensional element to determine the lowest harness point and investigate the impact of four different alloy steel SCM440 radii on temperature history and LHP, water cooled.
II. MATHEMATICAL MODEL It will be possible to determine the temperature history of the industrial quenched cylindrical alloy steel SCM440 bar at any node. As illustrated in Fig. 1, Fig. 2, and Fig. 3 [2], [8]- [11], 1-D axisymmetric elements can be used to evaluate 3-D heat transfers.

A. Detailed Description of the Finite Element Model (FEM) Construction Process
When thermal equilibrium is reached, the temperature distribution within the industrial quenched cylindrical steel bar will be calculated.
The several types of 3-D heat transfer problems include: i. Geometrically axisymmetric, to start. ii. Every thermal load has axisymmetrical behavior. As seen in Fig. 1, Fig. 2, and Fig. 3, 1-D axisymmetric elements can be used to study this 3-D heat transfer problem [2], [12]- [15]. In order to solve the 1-D cylindrical coordinates heat transfer problem, the finite element technique (FEM) is used. The Garlekin Weighted-residual approach is used to create the finite element formulation. It is thoroughly deduced how to get the relevant working expressions for the thermal load matrix, capacitance matrix, and conductance matrix.
Using the Backward Difference Scheme (BDS) yields the time-dependent answer.

1) Integrating the domain's engineering challenge
Meshing of the domain is done based on the type of element chosen, and since this mathematical model is for 1-D axisymmetric elements, the line element has been chosen. Consider the cylinder-shaped alloy steel SCM440 bar in Fig.  1 that was heated before being immersed in water to quench it. For an element (radius) line, T, the linear temperature distribution is given by: (1) where a1 and a2 are constants and T(R) is the nodal temperature as a function of R. Any point on the radius is R (element).    3. The boundary at node j [5] for element 4 and the selected 4 elements with 5 nodes in an axisymmetric one-dimensional line (radius) element from the domain.

2) One-dimensional axisymmetric element's shape function (SF)
The variation of the field variable over the element is referred to as the shape functions. Fig. 4 depicts the shape function of an axisymmetric one-dimensional radius (element) stated in terms of the R coordinate and its coordinate, from which the following shape functions [1] are obtained.
In terms of the (SF), the temperature history (distribution) of a 1-D radius for an element can be expressed as (3).
Thus for 1-dimensional element we can write in general: (5) where and stand for the nodal values of our work's temperature-related unknown variable. The unknown could also be deflection, velocity, etc.

3) Natural area coordinate
Utilizing the natural length coordinates and their connection to the (SF) via the integral of the Galerkin solution simplification.   Given that it is a 1-D element, each point P should be defined by a single independent coordinate. This is true even for natural coordinates since L1 and L2 are connected rather than independent because of:

B. Create Equations for Each Domain Element
By using the conservation of energy on a differential volume cylindrical segment, it has been possible to derive the equation of heat transfer in axisymmetric 1-D radius elements [1]. as depicted in Fig. 6.
Mathematically, the differential volume term can be simplified to represent the unsteady state heat transport within the component during quenching [5], [13]; the heat conduction equation is developed and provided by: (9) kr, kθ, kz -heat conductivity coefficient in r, θ and in zdirection respectively, W/m°C; T -temperature, °C; q -heat generation, W/m 3 ; ρ = mass density, kg/m 3 ; c -specific heat of the medium, J/kg•K; t -time, s.

C. The Assumption
As seen in Fig. 1, Fig. 2, and Fig. 3, there is no temperature variation in the vertical axis (Z-direction) due to the axsymmetric conditions in 1-D radius (element). Hence, we may write, Since Fig. 1, Fig. 2, and Fig. 3 make it plain that the temperature history (distribution) along the radius will remain the same if the radius moves with angle (θ), 360°, the axsymmetric scenarios similarly result in no temperature fluctuation in the angle (θ-direction). Therefore, (11) The rate at which energy is transformed from chemical, electrical, electromagnetic, or nuclear forms into thermal energy inside the volume of the system is known as the thermal energy generation rate; however, in this study on heating and cooling, no heat creation has been considered. Therefore, q = 0 When (8) is made simpler, it becomes: (12) additionally referred to as residual or partial differential equation.

D. Formulation of the Galerkin Weighted Residual Method (GWRM)
The Galerkin residual for a 1-D radius (element) in a transient heat transfer can be determined from the derived equation for heat conduction by integrating the transpose of the shape functions (SF) times the residual (R), which reduces the residual to zero: where, -the shape function matrix transposed. and -the residual contributed by element (e) to the final system of equations. After derivation, simplification, and reorganization, we obtain: (15) The term A in the conductance and thermal load matrix is the term for heat convection. The heat conduction terms in Term B are a part of the conductance matrix. The capacitance matrix is a component of term C, which is the transient equation.
When we finally formulate the conductance matrix for terms A, B, and C in the r-direction, we obtain:

E. Term A (Heat Convection)
Term A1 (the convection term) contributes to the conductance matrix): (16) Term A2 (Convection is a term that affects the thermal load matrix): (17) Term B (Contributing to the conductance matrix is the conduction term): (18) Term C (The Capacitance Matrix benefits from heat that is stored): (19) F. Create the Element Matrix (EM) for the Global Matrix (GM) Each element's capacitance, conductance, and thermal load matrices have been generated, and the global, capacitance, conductance, and thermal load matrices as well as the global of the unknown temperature matrix have been built. Each and every finite element analysis requires the assembly of these components (FEA). Constructing these elements will result into the following finite element equation: (20) As seen in Fig. 1, 2, and 3, is the conductance matrix due to conduction (Elements 1 to 4) and heat loss through convection at the element's boundary (Element 4 node 5). T stands for temperature in degrees Celsius at each node, C for capacitance as a result of the transient equation (heat stored), and for temperature rate at each node in degrees Celsius per second.
: the heat load resulting from internal heat generation and convection heat loss at the element's border (element 4 node 5) (element 4 node 5).

G. Euler's Method (EM)
We shall be able to calculate the nodal temperatures as a function of time using two point recurrence formulas. The rate of change in temperature and the temperature distribution at any node of the industrial steel bar will be calculated using Euler's approach, often known as the backward difference scheme (BDS) [5], [16]- [19]. If the time step is not equal to zero (t 0) and the derivative of temperature (T) with respect to time (t) is written in the reverse manner, we have: Therefore, the matrix becomes: !""""""#"""""" $ % % &""""""'"""""" ( All of the right-hand side of (20), including t = 0 for which the initial condition applies, is fully known at time t.
As a result, using the temperature from the previous period, the nodal temperature for a subsequent time can be calculated. The alloy steel SCM440's significant mechanical properties, such as hardness, toughness, strength, etc., may be determined if the temperature distribution is understood.

A. History of Temperature Calculation
To simulate the outcomes of the temperature history with respect to time in unsteady state heat transfer of the heattreated alloy steel SCM440, the established mathematical model (MM) is programmed using MATLAB. The temperature of the cylindrical industrial quenched alloy steel SCM440 bar is 850 °C. Then being cooled in water with Twater = 32°C and hwater = 5000 W/m 2 .°C (for convection heat transfer coefficient) [2], [17], [20]- [24].
The [E-LHP-software] is used to do a sensitivity analysis with the input data and boundary conditions to determine the temperature history at every point (node) of the quenched steel bar. As an illustration, Fig. 8 and 9 illustrate the computed unsteady state temperature distribution findings for the selected five nodes from the core [W1] to the surface [W5] of the quenched alloy steel SCM440 bar. In a similar manner, when the radii 25, 50, and 100 mm are determined, the temperature history of the five selected nodes is obtained. Fig.  10, Fig. 11, and Fig. 12 illustrate the ultimate outcomes, accordingly.   .022

B. Lowest Hardness Point (LHP) Calculation 1) Figuring out the cooling time (tc)
Due to the fact that the characteristic cooling period, significant for structural transformation for the majority of structural steels, is the time of cooling from 800 to 500°C (time t8/5) [9]- [12], [25]- [29], there are two important temperatures [800 o C and 500 o C] to consider during quenching [13], [30]- [34]. (27) We can calculate the time it takes for node W1 to reach 800 o C (t800) and the time it takes for node W1 to reach 500 o C from Fig. 9 when the radius is 12.5 mm (t500).
Thus, the node W1's cooling time is tc (t8/5): The same method was used to compute the cooling time tc for nodes W2 to W5, with the final results displayed in Table I.

2) Typical Jominy distance (Jd) vs cooling time (tc) curve for calculating the Jominy distance
To obtain the correspondent Jominy distance, cooling time, tc (t8/5) will now be placed into the Jominy distance (Jd) against cooling time, tc (t8/5) curve. Jominy distance can also be estimated in Microsoft Excel using polynomial regression and polynomial expressions.
The conventional Table [Cooling rate at each Jominy distance can be used too [17].
Following the calculation of the Jominy distance between nodes W1 and W5, the final findings are displayed in Table I,  where: The definition of ROC, or the Rate of Cooling;

3) Estimate the hardness of the alloy steel SCM440 bar after quenching
Using the relationship between the J-Distance and the HRC as described in [9], [12], the hardness (HRC) can be computed. The findings are displayed in Table I and in Fig. 13a, 13b, 13c, and 13d for radii of 12.5, 25, 50, and 100 mm, respectively. Fig. 13. HRC distribution along WW cross section for the selected nodes from the center to the surface at half the length at the center of the alloy steel SCM440 bar for four different radii: a) radius =12.5 mm; b) radius =25mm; c) radius = 50mm; d) radius =100 mm respectively.
Following that, Fig. 14 shows clearly how radius affects LHP of alloy steel SCM440 when the radius is 12.5 mm, 25 mm, 50 mm, or 100 mm.

IV. MODEL VERIFICATION IN MATH
Whereas the lowest hardness point (LHP) can be calculated using a mathematical model (MM), but it cannot be calculated using experimental work, the developed mathematical model has been validated by calculating the hardness on the surface, for both the mathematical model and experimentally as shown in Fig. 15. However, a strong agreement between both results conformed thereby this served as a validated to the experiment, the comparison showing that the reliability of the proposed model indicated that the calculation of the lowest hardness point (LHP) at is correct. Whereas it was discovered as indicated in Fig. 13a and Table II

V. CONCLUSIONS AND RESULTS
In order to calculate the temperature history in specimens with four different radii cylindrical geometry of 12.5, 25, 50, and 100 mm, respectively, and to determine the lowest hardness point (LHP) as well as to study the effect of the radii on temperature history and E-LHP, a 1-D mathematical model of unsteady state quenched alloy steel SCM440 has been developed.
Our findings demonstrate that the nodes (points) on the surfaces cool more rapidly than the nodes at half the length in the center because, respectively, the tc of nodes W5 to W5555 is lower than that of nodes W1 to W1111. This implies that distinct mechanical characteristics, like as hardness, will exist. The outcomes demonstrated that the surface node will be the first to reach room temperature following quenching where the hardness on the surface nodes will be higher than the hardness on the center nodes.
The lowest hardness point (LHP), which is at half the length at the center and is represented in Fig. 9 to 12, is the final point that will totally cool after quenching because it is not in contact with the cooling medium, followed by the other points (nodes) on the radial axis to the center, respectively.
Knowing the lowest hardness point (LHP) will become more crucial after the quenched steel bar's radius is huge since the E-LHP will be low at that time. In other words, LHP will be lower than the surface hardness, which indicates that increasing the bar's radius will cause E-LHP to decrease, as illustrated in Fig. 14.
Using manual calculation methods, it is nearly hard to calculate the experimental E-LHP. Moreover, older approaches exclusively used surface hardness calculations, which are higher than LHP, which has detrimental effects and can cause bending, deformation, and component failure.