Developing 1D MM of Axisymmetric Transient Quenched Chromium Steel to Determine LHP

The modelling of an axisymmetric industrial quenched chromium steel bar AISI-SAE 8650H based on finite element method has been produced to investigate the impact of process history on metallurgical and material properties. Mathematical modelling of 1-dimensional line (radius) element axisymmetric model has been adopted to predict temperature history and consequently the hardness of the quenched steel bar at any point (node). The lowest hardness point (LHP) is determined. In this paper hardness in specimen points was calculated by the conversion of calculated characteristic cooling time for phase transformation t8/5 to hardness. The model can be employed as a guideline to design cooling approach to achieve desired microstructure and mechanical properties such as hardness. The developed mathematical model is converted to a computer program. This program can be used independently or incorporated into a temperature history calculator to continuously calculate and display temperature history of the industrial quenched steel bar and thereby calculate LHP. The developed program from the mathematical model has been verified and validated by comparing its hardness results with commercial finite element software results.


Introduction
Quenching is a heat treatment usually employed in industrial processes in order to control mechanical properties of steels such as hardness [1]. The process consists of raising the steel temperature above a certain critical value, holding it at that temperature for a specified time and then rapidly cooling it in a suitable medium to room temperature [2]. The resulting microstructures formed from quenching (ferrite, cementite, pearlite, upper bainite, lower bainite, and martensite) depend on cooling rate and on chemical composition of the steel [3].
Quenching of steels is a multiphysics process involving a complicated pattern of couplings among heat transfer, because of the complexity, coupled (thermal-mechanicalmetallurgical) theory, and nonlinear nature of the problem, no analytical solution exists; however, numerical solution is possible by finite difference method, finite volume method, and the most popular one-finite element method (FEM) [4]. During the quenching process of the steel bar, the heat transfer is in an unsteady state as there is a variation of temperature with time [5]. The heat transfer analysis in this paper will be carried out in 3-dimensions. The three dimensional analysis will be reduced into a 1-dimensional axisymmetric analysis to save cost and computer time [4,6,16]. This is achievable because in axisymmetric conditions, there is no temperature variation in the theta (θ) direction and in (z) direction, the temperature deviations are only in (r). The Galerkin weighted residual technique is used to derive the mathematical model. In this paper, 1-dimensional line (radius) element will be developed.

Mathematical Model
The temperature history of the quenched cylindrical steel bar at any point would be calculated; three-dimensional heat transfer can be analyzed using one dimensional axisymmetric elements as shown in Figures 1, 2, and 3.
The linear temperature distribution for an element (radius) line, T is given by:  where T(R) = nodal temperature as the function of R, a 1 and a 2 are constants, R is any point on the (radius) line element.

Shape Function of the Axisymmetric Triangular Element.
The shape functions were to represent the variation of the field variable over the element. The shape function of axisymmetric 1-dimensional line (radius) element expressed in terms of the r coordinate and its coordinate as shown in Figure 4; which are derived to obtain the following shape functions as shown: Thus the temperature distribution of 1D radius for an element in terms of the shape function can be written as: where [S (r) ] = [ Si Sj ] is a row vector matrix and is a column vector of nodal temperature of the element. Equation (3) can also be expressed in matrix form as: Thus for 1-dimensional element we can write in general: where Ψ i and Ψ j represent the nodal values of the unknown variable such as in our case temperature also the unknown can be deflection, velocity, and so forth. The two length natural coordinates L 1 and L 2 at any point P inside the element are shown in Figure 5 from which we can write: Since it is a one-dimensional element, there should be only one independent coordinate to define any point P. This is true even with natural coordinates as the two natural coordinates L 1 and L 2 are not independent, but are related as: The natural coordinates L 1 and L 2 are also the shape functions for the line element, thus:

Develop Equation for All Elements of the Domain.
Derivation of equation of heat transfer in axisymmetric one-dimensional line (radius) elements by pplying the conservation of energy to a differential volume cylindrical segment as shown in Figure 6, The transient heat transfer within the component during quenching can mathematically be described by simplifying the differential volume term; the heat conduction equation is derived and given by The assumptions made in this problem were: (i) for axisymmetric situations one dimensional line (radius) element, there is no variation of temperature in the Z-direction as shown in Figures 1, 2, and 3. Because we already assumed that in steel quenching and cooling process of the steel bar is insulated from convection at the cross section of the front and back.
It means that we have convection and radiation at one node only which is on the surface (node 5), in our research we focus to calculate LHP which is at (node 1), where it is the last point will be cooled, this give the maximum advantage to make our assumption more safe, because it is the last point which will effect by convection and radiation from the front and back cross-section of the steel bar therefore we can write, (∂T/∂z = 0), (ii) for axisymmetric situations, there is no variation of temperature in the θ-direction, because it is clear from Figures 1, 2, and 3 that the temperature distribution along the radius will be the same if the radius move with angle θ, 360 • therefore, (∂T/∂θ = 0), (iii) the thermal energy generation rateq represent the rate of the conversion of energy from electrical, chemical, nuclear, or electromagnetic forms to thermal energy within the volume of the system. Such as conversion is the electric field which will be studied with details in the 2nd part of our research, however in this manuscript no heat generation therefore,q = 0.
After simplifying, (11) becomes and also known as residual or partial differential equation

Galerkin Weighted Residual Method Formulation.
From the derived heat conduction equation, the Galerkin residual for 1-dimensional line (radius) element in an unsteady state heat transfer by integration the shape functions times the residual which minimize the residual to zero becomes where [S] T = the transpose of the shape function matrix, { } (e) = the residual contributed by element (e) to the final system of equations 2.5. Chain Rule. The terms 1 and 2 of (16) can be rearranged using the chain rule which states that; Note that term (1) and term (3) contributed to the conductance matrix since they contains the unknown temperature {T}. Terms (2) and (4) contributed to the thermal load matrix as T f is the known fluid temperature. Term (3) and term (4) heat radiation are very important if our heat treatment is annealing (cooling in the furnace) or normalizing (cooling in air or jet air), but can be ignore (neglect) if the cooling is quenching in water as in our paper. From earlier explanations, derivation and after simplifying we can formulate the conductance matrix in the rdirection for Term B finally we get: Term B (the conduction term) contributes to the conductance matrix: Similarly, Term C is the unsteady state (transient) which contributes to the Capacitance Matrix, Journal of Metallurgy 5 Term C (heat stored) contributes to the Capacitance Matrix: Term A (heat convection): (i) Term A 1 : contributes to conductance matrix, Term A 1 (the convection term) contributes to the conductance matrix: (ii) Term A 2 : contributes to thermal load Matrix: Term A 2 (the convection term) contributes to thermal load matrix

Construct the Element Matrices to the Global Matrix.
Assemble the global, conductance, capacitance, and thermal load matrices and the global of the unknown temperature matrix for all the elements in the domain, that is, the element's conductance, capacitance and thermal load matrices have been derived. Assembling these elements is necessary in all finite element analysis.
Constructing these elements will result into the following finite element equation:

Euler's Method.
Two point recurrence formulas will allow us to compute the nodal temperatures as a function of time. In this paper, Euler's method which known as the backward difference scheme (BDS) will be used to determine the rate of change in temperature, the temperature history at any point (node) of the steel bar [3]. If the derivative of T with respect to time t is written in the backward direction and if the time step is not equal to zero (Δt / = 0), we have that, Finally, the matrices become From (29) all the right hand side is completely known at time t, including t = 0 for which the initial condition apply. Therefore, the nodal temperature can be obtained for a subsequent time given the temperature for the preceding time.
Once the temperature history is known the important mechanical properties of the steel bar can be obtained such as hardness and strength.

Calculation the Temperature History.
The present developed mathematical model is programmed using MATLAB to simulate the results of the temperature distribution with respect to time in transient state heat transfer of the industrial quenched steel bar. The cylindrical chromium steel bar has been heated to 850 • C. Then being quenched in water with T water = 32 • C, the water convection heat transfer coefficient, h water = 5000 W/m 2 · • C.
The temperature history at any point (node) of the cylindrical steel bar after quenched is being identified in Figures 6 and 7. The cylindrical bar was made from chromium steel 8650H, with properties as mentioned below.
Thermal capacity, ρc (J/m 3 · • C):  (4) are the capacitance matrices due to transient [unsteady state] in 1-dimensional line (radius) element: We have convection in element 4 at node j(5) only as shown clearly in Figures 3 and 7.
With the input data and boundary conditions provided, a sensitivity analysis is carried out with the developed program to obtain the temperature distribution at any point (node) From Figure 7 we can determine the time taken for node W 1 to reach 800 • C, By the same way the time taken for node W 1 to reach 500 • C is t 500 • C = 13.075 sec. So the Cooling time t c for node W 1 ; t c = t 500 • C − t 800 • C = 13.075 − 4.830 = 8.245 sec.
For nodes W 2 to W 5 , the cooling time t c calculated by the same way, the final results shown in Table 1. Table 1 shows the cooling time t c and the rate of cooling ROC.

Calculating the Jominy Distance from Standard Jominy Distance versus Cooling Time Curve.
Cooling time, t c , obtained will now be substituted into the Jominy distance versus cooling time curve to get the correspondent Jominy distance. Jominy distance can also be calculated by using polynomial expressions via polynomial regression via Microsoft Excel.
The standard Table (cooling rate at each Jominy distance (Chandler, 1999)) can be used too [18].
Then Jominy distance of nodes W 1 to W 5 will be calculated, the final results shown in Table 2, where the rate of cooling, ROC, was defined by; Time (s) Temperature history of the selected 5 nodes

Predict the
Temperature ( • C) Node Hardness (HRC) Figure 9: Hardness distribution along WW cross-section for the nodes W 1 to W 5 from the centre to the surface, respectively, at half the length at the centre of the quenched steel bar.

Mathematical Model Verification
The same data input for the steel properties and boundary condition used in the mathematical model is applied to the ANSYS software to verify the temperature simulation results. The temperature distribution from the ANSYS analysis is depicted figuratively as shown in Figures 10(a) and 10(b). The temperature time graph from the ANSYS analysis is depicted as shown in Figure 11.
From the graphs shown in Figure 8 by mathematical model and Figure 11 by ANSYS, it can be clearly seen that the temperature history of the quenched steel bar have   the same pattern. The heat transfer across the steel bar is uniform. From Figure 11 the cooling time, Jominy-distance, and consequently the hardness of the quenched chromium steel bar at any point (node), even the lowest hardness point (LHP) is determined by ANSYS too, the final results shown in Table 4 and Figure 12. From our results we found that in the mathematical model for the 1st node with W 1 in the center, we found that HRC = 59.297. While in ANSYS for the same node A 1 , we found that HRC = 58.492.
And for the nodes on the surfaces W 5 and A 5 , it was found that HRC = 61.421 and 61.295 for the mathematical model and ANSYS, respectively. From the above, it can be seen that there is a strong agreement between both results. The difference between all the results of the mathematical model and the ANSYS simulations can be accounted due to the fact that the ANSYS software is commercial purpose, and thereby has some automated input data. But the developed mathematical model is precisely for a circular steel bar axisymmetric cross section. However, there is strong agreement between both results and thereby the result is validated, where the comparison indicated reliability of the proposed model.
Also the results showed that the node on the surface will be the 1st which completely cooled after quenching because it is in the contact with the cooling medium then the other nodes on the radial axis to the centre, respectively, and the last point will be completely cooled after quenching will be at half the length at the centre. Hence LHP will be at half the length at the centre of the quenched industrial chromium steel bar. It will be more important to know LHP once the radius of the quenched steel bar is large because LHP will be low, in other words, it will be lower than the hardness on the surface, that means increasing the radius of the bar inversely proportional with LHP.
LHP calculation experimentally is an almost impossible task using manual calculation techniques also the earlier methods only used hardness calculated at the surface, which is higher than LHP, which has negative consequence that can result to the deformation and failure of the component. Node Figure 12: Hardness distribution by ANSYS along AA cross-section for the nodes A 1 to A 5 from the centre to the surface, respectively, at half the length at the centre of the quenched steel bar.

Conclusion
A mathematical model of steel quenching has been developed to compute LHP of the quenched chromium steel bar at any point (node) in a specimen with cylindrical geometry. The model is based on the finite element Galerkin residual method. The numerical simulation of quenching consisted of numerical simulation of temperature transient field of cooling process. This mathematical model was verified and validated by comparing the hardness results with ANSYS software simulations. From the mathematical model and ANSYS results, it is clear that the nodes on the surface (W 5 and A 5 ), respectively, cools faster than the nodes on the center (W 1 and A 1 ) because t cW5 < t cW1 and t cA5 < t cA1 , this means that the mechanical properties will be different such as hardness where the hardness on the surface nodes (W 5 and A 5 ) will be higher than the hardness on the center nodes (W 1 and A 1 ).