Developing 1-Dimensional Transient Heat Transfer Axi-Symmetric MM to Predict the Hardness, Determination LHP and to Study the Effect of Radius on E-LHP of Industrial Quenched Steel Bar

© 2012 Elmaryami and Omar, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Developing 1-Dimensional Transient Heat Transfer Axi-Symmetric MM to Predict the Hardness, Determination LHP and to Study the Effect of Radius on E-LHP of Industrial Quenched Steel Bar


Introduction
A study of the constitution and structure of all irons and steel must first start with the ironcarbon equilibrium diagram and the steel part of the phase diagram as shown in Figure 1.
Iron: is a general word used to describe metals that have pure iron as their main constituent.
Most iron wares around us are not made of chemically pure iron but are alloys, the most important of which is Carbon. Carbon is a major factor in understanding the difference between Iron, Steel and Cast iron. Adding some carbon to chemically pure iron makes steel. If the quantity of carbon is increased, cast iron will be obtained.
Steel: is the most commonly and widely used metallic material in today's society as shown in Figure 2. It is one of the most signification engineering material and can be classified plain carbon steel and alloy steel.
Steel is an alloy of iron and carbon. The amount of carbon dictates whether the steel is hard or tough. Adding Carbon makes the iron harder. Steels can be hardened by heat treatment.
Carbon in steel may be present up to 2.03 percent. Steels with carbon content from 0.025 percent to 0.8 percent are called hypo-eutectoid steel. Steel with a carbon content of 0.8 percent is known as eutectoid steel. Steels with carbon content greater than 0.8 percent are called hyper-eutectoid steel. There are three major categories of steel which are as follows:  [23,24]. Many of the basic features of this system influence the behaviour of even the most complex alloy steels. The iron-carbon diagram provides a valuable foundation on which to build knowledge of both plain carbon and alloy steels.
Alloy Steels: are basically carbon steels with certain chemical elements added to improve the properties of the metal for specific applications or end products. Alloying elements include carbon, copper, sulphur, manganese, phosphorus, nickel, molybdenum, boron and chromium, and the resulting material is called an alloy steel as in the sample AISI-SAE 8650H, which will be studied in detail here. The 1-D mathematical model was developed to: i. Predict the hardness. ii. Determine E-LHP. iii. Study the effect of radius on E-LHP. TTT Diagram: T (Time) T (Temperature) T (Transformation) diagram is a plot of temperature versus the logarithm of time for a steel alloy of definite composition as shown in Figure 3. It is used to determine when transformations begin and end for an isothermal (constant temperature) heat treatment of a previously austenitized alloy. TTT diagram indicates when a specific transformation starts and ends and it also shows what percentage of transformation of austenite at a particular temperature is achieved. In Figure 3 the cooling rates A and B indicate two rapid cooling processes. In this case curve A (water cooled) will cause a higher hardness than the cooling rate B (sea water cooled). The end product of both cooling rates will be martensite. Cooling rate B is also known as the Critical Cooling Rate, which is represented by a cooling curve that is tangent to the nose of the TTT diagram. Critical Cooling Rate is defined as the lowest cooling rate which produces 100% Martensite while minimizing the internal stresses and distortions. Many of the basic features of this system influence the behaviour of even the most complex alloy steels. For example, the phases found in the simple binary Fe-C system persist in complex steels, but it is necessary to examine the effects of alloying elements have on the formation and properties of these phases. The iron-carbon diagram provides a valuable foundation on which to build knowledge of both plain carbon and alloy steels in their immense variety [1]. Quenching is physically one of the most complex processes in engineering, and very difficult to understand. Simulation of steel quenching is thus a complex problem [2,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Quenching is a heat treatment usually employed in industrial processes in order to control mechanical properties of steels such as toughness and hardness [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. 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 [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. 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,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Quenching of steels is a multi-physics process involving a complicated pattern of couplings among heat transfer, because of the complexity, coupled (thermal-mechanical-metallurgical) theory and non-linear 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) which will be used in this study [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. A larger diameter rod quenched in a particular medium will obviously cool more slowly than a smaller diameter rod given with a similar heat treatment [4].
It will be more important to know the lowest hardness point E-LHP (node E) once the radius of the quenched steel bar increases, in other words the E-LHP will be lower than the hardness on the surface (node 4), thus the radius of the bar will be inversely proportional to the hardness at E-LHP. 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 [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The heat transfer analysis in this study will be carried out in 3-dimensions. The three dimensional analysis will be reduced into a 1-dimensional axisymmetric analysis to save cost and computing time [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. This is achievable because in 1-dimensional axisymmetric conditions, there is no temperature variation in theta (Ɵ) and (z)-direction for 1-D as shown in Figure 4 (a)(b) and Figure 5, the temperature deviations is only in (r)-direction [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The Galerkin weighted residual technique will be used to derive the mathematical model to predict the hardness at any point (node) of the heat treated quenched steel bar. Therefore [E-LHP] can be calculated where it's exactly at half the length at the centre of the bar as shown in Figure 1. Experimentally, measurement of E-LHP is an almost impossible task using manual calculation techniques and furthermore the earlier methods only used hardness calculated at the surface (node 5) as shown in Figure 4 (a)(b) and Figure 5. This surface hardness is higher than E-LHP and this has negative consequences which can lead to the deformation and failure. In this chapter, 1-D line (radius) element will be used.
Quenching of steels in general has been and continues to be an important commercial manufacturing process for steel components. It is a commonly used heat treatment process employed to control the mechanical properties of steels. In this chapter hardness in specimen points was calculated by the conversion of calculated characteristic cooling time for phase transformation t8/5 to hardness [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Temperature histories must be performed to obtain more accurate transformation kinetics; an adequate tool has been produced for investigating the impact of process history on metallurgy and material properties. Mathematical modelling of axisymmetric industrial heat treated quenched steel bar on the finite element method has been developed to predict temperature history then the hardness at any point (node) even inside the bar can be determined and also the effect of the radius on the temperature history subsequently on LHP can be studied. The temperature history needs to be properly understood in order to efficiently produce high quality components.
It is clear that the first point (node) will be completely cooled after quenching (surface node) because it is located on the surface in contact with the cooling medium, then the other points (nodes) on the radial axis to the centre accordingly will be cooled and the last point will be completely cooled after quenching (centre node) [15,34].
Thus the maximum hardness will be on the surface node subjected to fast cooling, then the hardness will decrease from the surface node on the radial axis to the centre node of the quenched steel bar. Consequently the lowest hardness point of the quenched steel bar will be detected at the centre node [15,34].
The lowest hardness point (E-LHP) should be expected inside the heat treated quenched steel bar at the half of the length of the bar centre (centre node). To prove this experimentally is an almost impossible task using manual calculation techniques.
It will be more important to know the E-LHP (centre node) when the radius of the quenched steel bar increases because the lowest hardness point will be lower than the hardness on the surface (surface node). This means that increasing the radius of the bar is inversely proportional to LHP (centre node), while the hardness at the surface (surface node) will be the same [15,34].
No published information are available till date on this aspect. This work represents a contribution towards understanding of steel behaviour at elevated temperature during quenching at the LHP (centre node) of the steel bar. The results of this study may prove useful to obtain the hardness of the lowest point of the steel bar in order to reach the maximum benefit against the deformation and failure of the component [15,34].

Mathematical model
Three dimensional heat transfers can be analyzed using one dimensional axisymmetric element as shown in Figure 4

Methodology of building the F. E. Model in details
The temperature distribution inside the cylindrical steel bar at thermal equilibrium will be calculated. These are special classes of three-dimensional heat transfer problem: i. Geometrically axisymmetric.
ii. Each thermal load is symmetrical about an axis.
The finite element method is applied to the one-dimensional cylindrical coordinates heat transfer problem.
The finite element formulation is developed with the Garlekin Weighted-residual method.
The appropriate working expressions of the conductance matrix, capacitance matrix and thermal load matrix are derived in details.
The time dependent solution is obtained by applying the Backward Difference Scheme.

Meshing the engineering problem of the domain
Since the modelling work is on one-dimensional axisymmetric elements then line element has been selected in this study. Let us consider a cylindrical chromium steel bar as shown in Figure 4 (a) which had been heated and then submerged in water.
The linear temperature distribution for an element (radius) line, T is given by: Where, T (R) = nodal temperature as the function of R a1 and a2 are constants. R is any point on the (radius) line element.

Shape function of 1-D axisymmetric element
The shape functions were to represent the variation of the variable field over the element. The shape function of axisymmetric 1-Dimensional line (radius) is element expressed in terms of the r coordinate and its coordinate are shown in Figure 6; They are derived to obtain the following shape functions [9][10][11][12][13][14][15][16][17][18][19][20][21][22]; Thus the temperature distribution of 1-D radius for an element in terms of the shape function can be written as: Eq. (3) can also be expressed in matrix form as: Thus for 1-dimensional element we can write in general: Where Ψi and Ψj represent the nodal values of the unknown variable which in our case is temperature. The unknown can also be deflection, or velocity etc.

Natural area coordinate
Using the natural length coordinates and their relationship with the shape function by simplification of the integral of Galerkin solution: The two length natural coordinates L1 and L2 at any point p inside the element are shown in Figure 7 [9, 10,19]. 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 L1 and L2 are not independent, but are related as: The natural coordinates L1 and L2 are also the shape functions for the line element, thus:

The assumption made in this problem was
For axisymmetric situations one dimensional line (radius) element, there is no variation of temperature in the Z-direction as shown in Figure 4 (a)(b)and Figure 5. This is because we have 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 E-LHP which is at [node 1], where it is the last point that will be cooled. This gives the maximum advantage to make our assumption safer, because it is the last point that will be affected by convection and radiation from the front and back cross section of the steel bar.
Therefore we can write, 0 T z For axisymmetric situations, there is no variation of temperature in the θ-direction, because it is clear from Figure 4 (a)(b) and Figure 5 that the temperature distribution along the radius will be the same if the radius moves with an angle θ, 360 o .
The thermal energy generation rate q  represents the rate of the conversion of energy from electrical, chemical, nuclear, or electromagnetic forms to thermal energy within the volume of the system. The conversion of the electric field will be studied with details in the 2 nd part of our research, however in this manuscript no heat generation has been taken into accounted.
Therefore, q  = 0 After simplifying, the Eqn. (13) 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 can be obtained by integration the transpose of the shape functions times the residual which minimize the residual to zero becomes; Where, [S] T = the transpose of the shape function matrix { } ( ) e ℜ = the residual contributed by element (e) to the final system of equations.

Chain rule
The term 1 and 2 of Eqn. (17) can be re-arranged using the chain rule which states that; By substitute Eqn. (18) in to Eqn. (17), get Term A is the heat convection terms and contributes to the conductance and thermal load matrix. Term B is the heat conduction terms and contributes to the conductance matrix. Term C is the transient equation and contributes to the capacitance matrix. Where, Note: term (1) and term (3) contributed to the conductance matrix since they contain the unknown temperature {T}. Term (2) and (4) contributed to thermal load matrix as Tf is the known fluid temperature. Term (3) and term (4) heat radiation very important if our heat treatment is Annealling [cooling in the furnace] or Normalizing [cooling in air or jet air], but it can be ignored if the cooling is quenching in water, sea water or oil as in this work.
From earlier explanations, derivation and after simplification we can formulate the conductance matrix in the r-direction for term B finally we get: Term B (the conduction term) contributes to the conductance matrix ( ) Similarly, term C, the unsteady state (transient) which contributes to the Capacitance Matrix becomes: Term C (heat stored) contributes to the Capacitance Matrix

Term A1 -Contributes to Conductance Matrix
Term A1 (the convection term) contributes to the conductance matrix

Term A2 -Contributes to thermal load Matrix
Term A2 (the convection term) contributes to thermal load matrix
Constructing these elements will result into the following finite element equation: Where:

Euler's method
Two points recurrence formulas will allow us to compute the nodal temperatures as a function of time. In this paper, Euler's method which is known as the backward difference scheme (FDS) will be used to determine the rate of change in temperature, the temperature history at any point (node) of the steel bar [27,28,32].
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 (∆t ≠ 0), we have; With; Ṫ = temperature rate (°C/s); T (t) = temperature at t s (°C); T (t-∆t) = temperature at (t-∆t) s, (°C) ∆t = selected time step (s) and t = time (s) (at starting time, t = 0) By substituting the value of {Ṫ} into the finite element global equation, we have that; Finally, the matrices become; ( ) From Eqn. (27) 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 low carbon steel bar can be obtained such as hardness and strength.

Calculation of 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 chromium steel AISI 8650H. The cylindrical chromium steel bar has been heated to 850°C. Then being quenched in water with Tsea water = 32°C and the convection heat transfer coefficient, hsea water = 1250 W/m2•°C.
The temperature history for the selected nodes of the cylindrical chromium steel AISI 8650H after quenching is being identified in Figure 9 and Figure 10.
The cylindrical bar was made from chromium steel AISI 8650H, with properties as seen below [26].
We have convection in element 4 at node j (5) only as shown clearly in Figure 5 and Figure 9.
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) of the quenched steel bar. As an example the transient state temperature distribution results of the selected five nodes from the center [W1] to the surface [W5] of the quenched chromium steel AISI 8650H which were computed as shown in Figure 10 with dimension as in Figure  9. [26].

Calculating the Jominy distance from Standard Jominy distance versus cooling time
Cooling time, tc 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.
In this chapter the standard Table [Cooling rate at each Jominy distance (Chandler, H., 1998)] will be used. Then Jominy distance of nodes W1 to W5 will be calculated by using the data from [Cooling rate at each Jominy distance (Chandler, H., 1998)] [33]. The final results shown in Table 1, where the Rate of Cooling, ROC, was defined as; 500 C 800 C 800 C-500 C 800 C-500 C ROC

Predict the hardness of the quenched steel bar
The HRC of chromium steel AISI 8650H can be calculated by using the relation between the J-Distance and the HRC from the Practical date Handbook, the Timken Company 1835 Duebex Avenue SW Canton, Ohio 44706-2798 1-800-223, the final results shown in Figure 11 & Table 1: Table 1. Cooling time, Cooling rate, Jominy distance and HRC for the nodes W1 to W5, sea water cooled by 1-D mathematical model when the radius = 12.5 mm.

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 Figure 12(a) and Figure 12(b); Figure 12(a) shows the temperature distribution just before the steel bar becomes completely cooled and Figure 12(b) shows the temperature distribution at the moment that the entire steel bar becomes completely cooled after 1175s.
The temperature time graph from the ANSYS analysis is depicted as shown in Figure 13; From the graphs shown in Figure 10 by mathematical model and Figure 13 by ANSYS, it can be clearly seen that the temperature history of the quenched steel bar has the same pattern. The heat transfer across the steel bar is uniform. From Figure 13 the cooling time, Jominy-(a) (b) distance and consequently the hardness of the quenched Chromium steel 8650H at any point (node), even the lowest hardness point (E-LHP) is determined by ANSYS too, the final results shown in Table 2 and Figure 14.   From the above results it was found that in the mathematical model for the 1st node with W1 in the center, we found that HRC = 51.688. While by ANSYS for the same node A1, it was found that HRC = 50.359. And for the nodes on the surfaces W5 and A5, it was found that HRC = 54.506 and 54 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 E-LHP will be at half the length at the centre of the quenched industrial Chromium steel bar.

Effect of the radius on the temperature history
Where there is strong agreement between the mathematical model and the Ansys simulation results and thereby the result is validated where, the comparison indicated reliability of the proposed model thereby; we will apply the mathematical model to study the effect of radius on the temperature history and on E-LHP. By the same way the temperature history of the 5 selected nodes has been obtained when the radii 18.75, 25, 37.5 and 50 mm is determined.
The final results are shown in Figure 15, Figure 16, Figure 17, and Figure 18 respectively.   It is clear from the results that the nodes on the surfaces cooled faster than the nodes at half the lengths at the centres in other words the nodes on the surfaces will be the 1st to be completely cooled after quenching because it is in the contact with the cooling medium then the other points (nodes) on the radial axis to the centre respectively while the last point that will be completely cooled after quenching will be at half the length at the centre because the cooling time tC of nodes W5, W5-12, W55, W5-23 and W555 less than tC of nodes W1, W1-12, W11, W1- 23 and W111, respectively. A larger diameter rod quenched in a particular medium will obviously cool more slowly than a smaller diameter rod. where Figure 10, Figure 15, Figure  16, Figure 17, and Figure 18 showed that when the radii equals 12. . Based on the above results we expect that if the radius is 100 mm, 3000 sec is required to reach the fluid temperature. Figure 18. Graph of temperature history along WW cross-section when the radius = 50 mm from MATLAB program.

Effect of the radius on E-LHP
As explained above and from Figure 15, Figure 16, Figure 17, and Figure 18 of the temperature history the hardness when the radii equals 18.75, 25, 37.5 and 50 mm is determined, the final results shown in Figures 19, Table 3, Figure 20, Table 4, Figures 21, Table 5, Figure 22 and Table 6.        From the above results it was found that the hardness on the surfaces nodes will be higher than the hardness on the centres nodes as shown in Figures 11, 19 It is clear from the results as shown in Figure 23 and Table 7 that increasing the radius of the bar inversely proportional with E-LHP, then it's more important to know E-LHP once the radius of the quenched steel bar is large because this has negative consequence which can result of the deformation and failure of the component. Table 7. Effect of radius on E-LHP of transient heat transfer axi-symmetric industrial quenched chromium steel AISI 8650H, sea water cooled by 1-D mathematical model. From the above explanation we can say that the developed 1-Dimensional mathematical modelling in this chapter solved design problems in transient heat transfer axi-symmetric industrial quenched chromium steel 8650H bar, where experimental calculation of E-LHP is an almost impossible task using manual calculation techniques. Also the earlier methods only used hardness calculated at the surface, which is higher than E-LHP.

Conclusion
A mathematical model of steel quenching has been developed to compute E-LHP of the quenched Chromium steel 8650H at any point (node) in a specimen with cylindrical geometry and to study the effect of radius on E-LHP. 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 [W5, W5-12, W55, W5-23, W555 and D55] respectively cools faster than the nodes on the center [W1, W1-12, W11, W1-23, W111 and D11] because tCW5, tCW5-12, tCW55, tCW1-23, tCW555 and tCD55 less than tCW1, tCW1-12, tCW11, tCW1-23, tCW111 and tCD11, this means that the mechanical properties will be different such as hardness where the hardness on the surfaces nodes [W5, W5-12, W55, W5-23, W555 and D55] will be higher than the hardness on the center nodes [W1, W1-