American Journal of Engineering Research (AJER) 2018 
        American Journal of Engineering Research (AJER) 
e-ISSN: 2320-0847 p-ISSN: 2320-0936 
  Volume-7, Issue-8, pp-242-253 
  www.ajer.org 
Research Paper                                                                                                        Open Access 
 
 
Numerical Modelling of Thermal Distribution Control in a Furnace Y
 
1 1,2 1 1 1
Petinrin M. O., * Ajide O. O., Dare A. A., Oyewola O. M. and Ismail AO.S.R  
1
Department of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria 
2
 Department of Mechanical Engineering, Howard University, 2300 6th RStreet NW, 
Washington, DC, USA, 20059. IB 
*
Corresponding author  L
Email: olusegun.ajide@howard.ed
Alternate email:ooe.ajide@gmail.coNu  m 
Mobile: 2408987579, +2348062687126 A
 
ABSTRACT: Application of control to heat treatment processDes helps to achieve the desired mechanical 
properties of materials but improper controller design is a major problem causing short lifespan of components 
of locally made furnaces. In this study, the numerical controAl of the temperature distribution within a furnace 
cavity was carried out using COMSOL Multiphysics and Simulink. Six sensor points within the furnace cavity 
(with and without specimen) were selected and each  pIoBint was consecutively used to observe the time response of the sensor to the desired temperature. ThFe results from the time response analysis indicated uneven temperature distribution within the furnace with points located at the corners of the furnace recording the highest temperature rise while points at Othe centre of the furnace or within the specimen having the lowest temperature. Thus, the best position for a  sensor is at any corner of the furnace to protect the components of the furnace from damage. 
 
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KEYWORDS:Thermal controIllTer, sensor, Furnace, Heating element, Heat flux  
------------------------------S----------------------------------------------------------------------------------------------- ---------- Date of SubmissionR: 19-07-2018                                                                            Date of acceptance: 24-07-2018 ------------------E----------------------------------------------------------------------------------------------------------- ----------  V I. INTRODUCTION IHeat treatment is highly crucial in manufacturing industry for the development of the state-of-art materials which in turn can influence the quality of machineries production. It has been used as a veritable tool 
for Nimproving physico-mechanical, thermal and chemical properties of materials. Laudable attempts have been Umade on works that focuses on the influence of heat treatment on properties of materials. Daramola et al. [1] carried out investigations to study the effects of heat treatment on the mechanical properties of rolled medium carbon steel. The steel was heated to the austenizing temperature of 830 ºC and then 
water quenched. The obtained results showed that the steel developed has an outstanding improvement in tensile 
strength, impact strength and ductility which makes it suitable for structural applications.The production of 
superconducting radio frequency (SRF) cavities is a technology that has already been globally accepted [2]. It 
was further remarked that, one of the main procedures during the fabrication of SRF cavities is the heat 
treatment at     600 ºC for 10 h or 800 ºC for 3 h in ultra-high vacuum (UHV). It was found that heat treatment is 
required to reduce the mechanical stress in the cavities after fabrication.In the work of Sidhu [3], tremendous 
yield strength (greater than 1.7 GPa) and good deformability wereobserved in nanobainitic  steel after 
isothermal heat treatment at a low temperature of 250 ºC for a relatively short duration of 24 hours. 
 
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Talabi et al. [4] examined the effects of heat treatment processes on the mechanical properties of as-
cast Al-4% Ti alloy samples. The outcome of the study showed that as-cast Al-4% Ti alloy produced significant 
effect on the rigidity and strain characteristics of the alloy. Significant improvement in the ductility of the 
samples was recorded in the tempered sample. The mechanical properties of Al-Si-Mg cast aluminum alloys 
such as A356 and A357 used for engine applications do not only depend on casting process, but highly 
influenced by heat treatment [5].To understand the influence of microstructural change on mechanical properties 
of Haynes 282, tensile test was performed at room temperature on the heat treated samples [6]. The gamma 
prime morphology was observed to change from cuboidal to spherical and then to bimodal in three different heat 
treatment conditions. The strength of the material is affected by the size and shape of the cuboidal precipitates, 
while the ductility at room temperature was reasoned to be affected by interconnected morphology of the 
carbides at the grain boundaries. Super duplex stainless steels were subjected to solution annealing heYat 
treatment at 1050 °C for two hours followed by water quenching in a study carried out by DavanagReri [7]. 
Results obtained showed an improvement in the mechanical and chemical characteristics of the alloys. This was 
attributed to the formation of secondary phases such as sigma (σ), Chi (χ), secondary austenite (χA2), carbides, 
nitrides and intermetallics originating from the cast microstructure which dissolved at higRh heat treatment temperatures. It was also observed that water quenching assisted enormously in preventiBng the reoccurrence of precipitation thereby keeping large quantities of solute atoms in solid solution.Tanwer [I8] investigated the effect of heat treatment (annealing, normalizing and quenching) on the mechanical properties of stainless steel. 
Although the three heat treatment techniques adopted in the study showed a general improvement in elongation, 
the effect of normalizing was found to be most significant. More remarkable effLorts have been made on the 
impact of heat treatment in respect of material properties modification [9-12].  
Heat treatment which can be described as the controlled heating Nand cooling of material in order to 
deliberately modify or alter its properties is usually carried out in furnaces. Rajan et al. [13] described heat 
treatment furnace as a heating chamber with a refractory or laggedA enclosure. It contains charge and has 
capacity for retaining heat that is measurable and controllable. There are so many types of furnaces and can be 
grouped into three which are bath, roller-hearth and strand-type fuDrnaces [14]. The high cost of procuring these 
furnaces is attracting enormous concern. As part of researchers’ efforts, a diesel fired heat-treatment furnace 
using locallysourced materials has been developed [15]. TesAts carried out in order to evaluate the performance 
of the furnace showed heating rate of 61.24 ºC/m in Bto attain a pre-set temperature of 900 ºC and a fuel consumption rateless than 1.41 L/h.An attempt was Imade by Ukoba et al. [16] to develop a heat treatment furnace locally. The developed furnace has a mFaximum temperature capacity of 880 ºC in the heating zone and 21 ºC temperature reading at the surface of the external casing after a period of 90 minutes. The furnace was 
found to heat treat both ferrous, non-ferrOous metals and their alloys. 3 kg capacity electric induction furnace with a power rating of 2500 W for heat treatment of ferrous and non-ferrous alloys was developed byAnaidhuno 
and Mgbemena [17]. The furnace wYhich was made from mild steel sheet was monolithically lined with fire clay refractories and designed to attain a temperature of 1200 ºC on the automatic control panel. Performance evaluation showed that the temTperature of the furnace was maintained at 1200 ºC over a period exceeding 2000 seconds.  
There is growinSg neIed for control of heat and temperature distributions in heat treatment furnace. This is because heat treatment furnaces are sensitive equipment and slight deviations from normal or designed 
operating conditionRs can attract enormous damages. Heat transfer simulation within heating furnaces has been considered to Ebe of great significance for prediction and control of furnace performance [18]. Several other efforts have been made in this respect.A numerical method was used by Türkmen [19] for determination of temperaturVe distribution in the furnace. The results obtained were utilized for optimization of gas circulation in the furnIace system under examination. Yang et al. [20] investigated thermal performance of the furnace and heaNting process of the metal pieces. The furnace was used to heat treat dredging pumps and impellers to obtain the required microstructure and mechanical properties (through stress relief, annealing, hardening and Utempering). Simulation provides a useful tool for predicting the temperature evolution within the metal pieces during the heat treatment. The outcome of the study indicated an overall energy balance that can be considered as relatively low energy efficiency of the furnace. To improve energy efficiency, design changes were made 
with the model. This reduces amount of excess air which in turn also save a lot of energy and thus reduces 
production cost. 
Depree et al. [21] used a 3D model to investigate the temperature distribution of steel strip that is being 
annealed and the furnace thermocouple probes. The simple model could predict furnace and strip temperatures. 
Overall, it has a very short solution time and has been found suitable for rapid simulation of alternative furnace 
operating conditions. The outcome of the study has potentials for optimizing heat treatment quality, plant 
throughput and energy consumption. Numerical modelling of heat transfer effects arising during heat treatment 
of single-crystal nickel-based superalloys used for turbine blades in jet engines has been carried out [22]. 
Control of thermal characteristics during processing is required in order to avoid incipient melting and to attain 
 
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properties suitable for service. The authors employed modelling for prediction of the temporal evolution of the 
temperature distribution inside the treated component, calculated heat transfer coefficients and analyzed the 
homogeneity of heat transfer.  
Although significant efforts have been made on the application of control to heat treatment processes, 
controller design is one of the major problems causing short lifespan of furnace components and thus mitigating 
against its local development. This study focuses on the numerical control of the temperature distribution within 
a furnace cavity with respect to sensor location. Each of the selected locations was consecutively explored for 
monitoring the time response of the sensor to the desired temperature. The outcome of this investigation could 
be found useful in design modifications of furnace cavity. 
 
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II. MATHEMACTICAL MODELLING R
Geometrical Model 
In this analysis, two-dimensional representation of a muffle furnace cavity and a spAecimen was 
considered with assumption that the heating element was just at the surface of the insulating material. The 
detailed dimension of the furnace cavity is as given in Fig. 1. It was assumed the furnace caviRty was filled with 
air and the specimen to be heat-treated was a medium carbon steel (MCS) while the material of insulation was 
the fire brick. The thermo-physical properties of these furnace constituents are as shLownI in
B Table 1 [23,24]. 
 
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Fig. 1 : TO
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he dimension of the furnace cavity 
 
Table 1: The Thermo-physical pTropYerties of the constituents of the furnace Properties MCS Air Fire Brick 
3
Density (kg/m ) SI 7850 0.6964 - 2Thermal Conductivity (W/m K) 52.5 0.0407 - 
Specific Heat (J/kgKR) 485 1030 - 
EmissivityV E 0.12 - 0.75  
 NIModelling of Furnace Cavity without Thermal Controller In order to simulate transient heat distribution in the furnace, the governing heat equation is given as U Tcp .kT   0          (1) t
where, T is the temperature, k is the thermal conductivity, ρ represents the density, cp is the specific heat 
capacity at constant pressure and t is time [23,25].The initial temperature of the furnace cavity was set to 400 K. 
Two cases were considered to check the effect of the specimen on the temperature distribution within the 
furnace, and they are: 
i. By assuming a constant temperature from the heating element. This can only be achieved if the furnace 
temperature is under control, therefore the temperature at the wall of the furnace is at a fixed 
temperature, Tw = 600 K 
ii. By taking the power input, P of the heating element to be 4000W, therefore the boundary condition at 
the furnace wall is set to be 
 
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n.kT   q  GT 4          (2) 
where q  is the heat flux, ε is the emissivity of material, G is defined to be irradiation on the surface and σ is the 
-8 2 4
Stefan-Boltzmann constant which is 5.670x10  W/m K  
The heat flux was determined from the power input and furnace surface area, A as  
P
q              (3) 
A
Also, the irradiation is calculated from 
J T 4
G            R(4)Y 1And as illustrated in Fig. 2 for radiation exchange between elemental surfaces with elemental areas on 
each surface, dAi and dAj from arbitrarily oriented surfaces Ai and Aj, the radiosity, which accounAts for all the 
rate of radiant energy leaving surface i to surface j is 
q Ri j
J           IB  (5) i AiFij
while net rate at which radiation leaves surface i as a result of its interaction with j 
 Lq  qdA            (6) i j i
the fraction of the radiation leaving surface i intercepted by surface j knowNn as the view factor is determined 
from [23,25] 
1 cosi cos j A
Fij    dAidAj         (7) A R2i A A Di j
To ensure heat energy balance between air in the furAnace cavity and the specimen, we have 
na.kaTa ns.ksTs   0    IB      (8)  
F 
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Fig. 2: Radiation exchange between elemental surfaces [23] 
 
The two-dimensional computational domain was discretised using unstructured triangular finite-
element meshes. The number of mesh elements are 1047 and 885 for computational model with and without the 
specimen respectively (Fig. 3). The governing heat equation was solved with a finite element software code, 
COMSOL Multiphysics using UMFPACK solver for a computational time of 60 s and 2000 s at interval of 0.1 
s. 
 
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(a) (b) R
 
Fig. 3: The finite element discretised model (a) 1047 elements (b) 885 elementRs A 
 Modelling of Furnace Cavity with Thermal Controller 
The time response analysis of the temperature distribution in the furnace undBer the influence of a 
thermal controller was carried out in Simulink. With the help of the robust interactive eInvironment of Simulink 
for non-linear dynamic problems, the complexity that could be involved in  coLmpleting the simulation in COMSOL Multiphysics was reduced. The location point of the sensor was selected within COMSOL furnace 
model for every simulation run in Simulink. There were six points selected and each was consecutively used to 
observe the time response of the sensor to the desired temperature (FiAg. 4)N. With reference to the bottom left-hand corner of the furnace cavity, the points are: PT1 (15 mm, 130 mm), PT2 (15 mm, 20 mm), PT3 (125 mm, 100 mm), PT4 (125 mm, 20 mm), PT5 (235 mm, 130 mm) and PDT6 (235 mm, 20 mm). Each model exported from COMSOL to Simulink was implemented as an S-function. This function is solved within Simulink using 
COMSOL solvers [26]. 
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SI
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R   E Fig. 4: The probe points for sensor location 
 ITVhe block diagram of the furnace control system is shown in Fig. 5, which includes the furnace, the heaNting element, an electrical relay, a comparator and a typical thermocouple sensor. Also, the representation of the system in Simulink is as shown in Fig. 6 with the S-function block containing the furnace model being Urenamed as COMSOL Multiphysics Subsystem. From this figure, the “Flux_State” indicates the state of the heating element, its value is one when the measured temperature, “Temp” is less than the desired temperature 
but returns zero when the measured temperature is greater than the desired temperature. 
 
 
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Fig. 5: The block diagram of the furnace control system 
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ADB   Fig. 6: The furnace  coIntrol system in Simulink  
III. RESUFLTS AND DISCUSSION The temperature distributions within the furnace at 60 s are as shown in Fig. 7. From this figure, the minimum 
and maximum temperatures are 558 K and 600 K; 1007 K and 1175 K; 404 K and 600 K; and 401 K and 1175 
K, respectively. Contrary to the distribut ioOns of the temperature contours observed in Fig. 7(a) and Fig. 7(b), the 
lowest temperature points in Fig. 7Y(c) and Fig. 7(d) were found to be within the specimen. This is because the specimen has lower thermal difTfusivity, which is a measure of the ability of a material to conduct heat energy to its ability to store the same enIergy, than the air in the furnace cavity. S
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(a) (b) 
 
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(c) (d) R
 
Fig. 7: Temperature contour distribution within the furnace (a) fixed wall temperature without specimAen (b) wall 
heat flux without specimen (c) fixed wall temperature with specimen (d) wall heat flux wi
 Rth specimen. 
The plots of temperature rise at the probe points for a period of 60 s in the COMSOL modBel are shown in Fig. 8 
to Fig. 11. The plots for points PT2, PT5 and PT6 were not shown in Fig. 8 and FLig. I9 due to their symmetry with point PT1 without the specimen. Despite the differences in the temperature rise at corresponding points in 
Fig. 8 to Fig. 11, similar trends were observed in Fig. 8 and Fig. 9, and Fig. 1N0 and  Fig. 11; this is partly because the air convection was assumed to be negligible. Thus, the ratio at which heat is conducted to heat absorption by the air molecules is constant. 
 
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U Fig. 8: The rise in temperature at probe points for fixed wall temperature without specimen  
 
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Fig. 9: The rise in temperature at probe points Dfor wall heat flux without specimen A
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Fig. 11: The rise in temperature at probe poinDts forA
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 wall heat flux with specimen 
 
The temperature rises for a period of 2000 s (33 min, 20 s) at all the points when the specimen is within the 
furnace and fixed furnace wall temperature of 600 K is as shoAwn in Fig. 12. It is obvious from the figure that the temperatures at 2000 s are very close to the wall temperaBture, and the minimum temperature obtained was found within the specimen at 591 K. This suggests a simi laIr temperature distribution if the furnace heating had been controlled to a temperature of 600 K.  F
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Fig. 12: The rise in temperature within a period of 2000 s for fixed wall temperature with specimen 
 
In the instance of removing the control components in Simulink, the results obtained agreed with the 
results from COMSOL Multiphysics as indicated in Fig. 13 for PT3 when the furnace cavity with the specimen 
was subjected to heat flux from heating element. Also, the rise in temperature with the relay output when the 
heating of the cavity with specimen was under control and the sensor was located at probe point PT3 for 100 s is 
as depicted in Fig. 14. The relay output indicated one when the desired temperature is below 600 K and 
 
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indicated zero when this temperature rises above 600 K. This implies that the heating element would be 
switched on the output is one and switched off when the output is zero. 
 
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Fig. 13: The rise in temperature at point PT3 in COMSOL MAultiphysics and Simulink for wall heat flux with specimen 
 
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Fig. 14: The controlled temperature distribution and relay output PT3 for wall heat flux with specimen 
 
 
The time responses at all the probe points, each at a simulation run, when the furnace cavity with the 
specimen was subjected to heat flux from heating element are as depicted in Fig. 15. Placing the sensor at points 
PT1, PT2, PT5 and PT6 there were very short rise time with higher overshoot while the longest rise time was 
recorded at PT3. Hence, to protect the insulating material, heating element, sensor or any other component of 
the furnace from possible damage, the best position of the sensor is at any corner of the furnace. 
 
 
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Fig. 15: The controlled temperature distribution at PT3 for wall heat f luLx with specimen 
. 
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IV. CONCLUSIONS 
The distributed parameter model of a heat-treatment furnace has beenA simulated successfully with COMSOL 
Multiphysics and Simulink. The results indicated that spatial temperatures at any particular time are not the 
same. Sensor positioning at any point close to the wall of the furDnace will help to prolong the life span of the 
heating element, sensor and insulating material. 
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