CFD-based optimization of a transient heating process in a natural gas fired furnace using neural networks and genetic algorithms

Accepted Manuscript CFD-based optimization of a transient heating process in a natural gas fired furnace using neural networks and genetic algorithms Rene Prieler, Markus Mayrhofer, Christian Gaber, Hannes Gerhardter, Christoph Schluckner, Martin Landfahrer, Markus Eichhorn-Gruber, Günther Schwabegger, Christoph Hochenauer PII: DOI: Reference:

S1359-4311(17)36141-0 https://doi.org/10.1016/j.applthermaleng.2018.03.042 ATE 11930

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

22 September 2017 8 February 2018 12 March 2018

Please cite this article as: R. Prieler, M. Mayrhofer, C. Gaber, H. Gerhardter, C. Schluckner, M. Landfahrer, M. Eichhorn-Gruber, G. Schwabegger, C. Hochenauer, CFD-based optimization of a transient heating process in a natural gas fired furnace using neural networks and genetic algorithms, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/j.applthermaleng.2018.03.042

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CFD-based optimization of a transient heating process in a natural gas fired furnace using neural networks and genetic algorithms Rene Prieler

a,*

, Markus Mayrhofer a, Christian Gaber a, Hannes Gerhardter a,

Christoph Schluckner a, Martin Landfahrer a, Markus Eichhorn-Gruber b, Günther Schwabegger b, Christoph Hochenauer a a

Institute of Thermal Engineering, Graz University of Technology, Inffeldgasse 25/B, A-8010 Graz, Austria

b

IBS-Institut für Brandschutztechnik und Sicherheitsforschung GmbH, Petzoldstraße 45, 4020 Linz, Austria

* Corresponding author. Address: Institute of Thermal Engineering, Graz University of Technology, Inffeldgasse 25/B, A-8010 Graz, Austria. Tel.: +43 316 873 7810. E-mail address: [email protected]

Highlights 

CFD-based investigation of a transient heating process



CFD simulation of the transient heating process by a steady-state consideration



Application of neural networks and response surface method to predict the system response



Optimization of the heating characteristic using a multi-objective genetic algorithm

Abstract In the present study a transient heating process in a natural gas fired test furnace, used for fire resistance tests of construction and building materials, was investigated by computational fluid dynamics (CFD). To ensure the reproducibility of a fire resistance test, the thermal exposure of the tested fire safety material has to be homogeneous and, thus, the temperature distribution is of high importance. For that purpose, a CFD-based optimization of the transient heating process was carried out using different optimization algorithms. Based on the furnace setup, parameters with a potential to improve the temperature distribution were identified and used for the optimization procedure. CFD results were used to create system response surfaces, which represent the temperature distribution in the furnace as a function of the chosen design parameters. The system response was approximated by neural networks and genetic algorithms, and represents the basis for the optimization. Since the duration of the transient process was 35 minutes, the calculation time of the gas phase combustion and heat

transfer is high. Therefore, a novel CFD-based approach was used to investigate and improve the process by converting the transient heating problem to steady-state cases. A comparison of the initial and the optimized furnace configuration showed an improved temperature distribution, where the maximum temperature difference in the furnace at the measurement position was decreased from approx. 200 K to 162 K. This approach showed that the transient simulation can be optimized, and further used for other applications where a transient simulation is computationally too demanding.

1 Introduction Nowadays, the optimization of different industrial processes by experimental investigations is replaced or aided by computational methods. In firing systems and heating processes computational approaches can reveal an in-depth look into the process, especially when measurements are difficult (e.g. high temperature processes in the steel, cement or glass industry). Based on the knowledge of all transport phenomena, chemical reactions etc. in a process as well as their dependence on the operating conditions, the considered unit operation can be optimized without expensive test runs. computational fluid dynamics (CFD) is often used in industry as well as in the academic field of research for the numerical prediction of fluid flow, species transport, heat transfer etc. It was successfully used in the past to predict and optimize processes in a wide range of applications (e.g. cement industry [1], oxy-fuel combustion [2], solid oxide fuel cells [3]–[5], reactor design for ion transport membranes [6]). Since the main focus of this work is the consideration of a transient heating process by steadystate CFD simulations, two similar publications have to be mentioned. Prieler et al. [7] used a CFD approach to simulate the gas phase combustion in a reheating furnace for steel billets. They proposed an iterative solution procedure not only to calculate the fluid flow and heat transfer in the gas phase, but also the transient heating characteristic of the steel billets simultaneously. The solution procedure developed was applied to optimise the furnace with the main goal of decreasing the fuel consumption. The model was able to predict the temperature in the furnace and the product quality of the steel during the heating process. The same model was later used by Landfahrer et al. [8] to simulate the temperature of steel tubes in a reheating process. The results were compared to measured temperatures of the gas phase and the steel tubes, revealing a high accuracy of this CFD approach. However, no optimization procedure was applied in [7] and [8]. Despite the high level of detail, which a CFD simulation can provide, the consideration of the computational demand is important for the use in industrial scale unit operations. Therefore, numerical approaches, such as artificial neural networks (NN) or genetic algorithms (GA), which treat the considered process as a “black box”, can counteract the high computational demand. These methods use algorithms capable of identifying the conjunction between the operating conditions (input parameters of the “black box”) and the process output (e.g. temperature of the steel billets). The basic processes within the unit operation, such as chemical reactions, fluid flow, etc., are neglected. Hence, the numerical method describes the reaction of a system based on the input parameters in a general way. However, the NN has to

“learn” the response of the system output(s) due to input changes before they can be used to predict and optimize the process. Besides the optimization, a further application of NNs is the prediction of the process failure and the usage in process/robot manipulator control systems. Adedigba et al. [9] used NNs to carry out a dynamic failure analysis of a chemical process. A recent study published by Jin et al. [10] summarized the potential of different designs of the NN for their application in robot control systems. Since the main focus of the present paper is the optimization of a reactive and energetic system, the literature research was aimed at this field of research using NNs and GAs. Many studies were carried out in the past using these techniques to predict the system response in conjunction with the input parameters. An additional study focusing on the Fenton process for leachate treatment was performed by Biglarijoo et al. [11] using a NN as well. A CFD-based design optimization of wastewater disinfection reactors with the help of an artificial neural network was performed by Wei et. al. [12]. In this investigation the results from the CFD simulations, which generate the required input and output data, were used for the “learning algorithm” of the neural network. The special application of the neural network for the optimization of combustion systems was reviewed in the work of Kalogirou [13]. Additionally, this paper included the optimization procedure using NN as well as the possibility of using neural networks to control the complex behavior of combustion systems. Kalogirou [14], [15] summarized the possible application of neural network in energy applications. Especially their capability for solving very complex technical problems (e.g. in a steam generator and renewable energy systems) using NNs was presented in these works. Almonacid et al. [16] reviewed the use of NNs for modelling concentration photovoltaic devices and to predict the electrical characterization. Another statistical approach to predict the processes in a system is the response surface method (RSM) (described in section 4c). A system response surface can be created in several ways, such as polynomial data fit. To increase the accuracy of the RSM, a high number of datasets should be available to create the response surface. Therefore, it is useful to apply other techniques, such as NNs, which can predict the system output depending on the input and, subsequently, enhance the datasets obtained by experiments or numerical simulations. The application of the RSM is, similar to the NN, not restricted to a special field of research or processes. Thus, the following examples from various scientific problems are given: Yadav et al. [17]. They have shown an application of an artificial neural network in conjunction with a RSM to analyze the effect of different process variables, such as pulp density, oil dosage, agglomeration time and particle size in a coal oil agglomeration process. The NN and RSM were capable of predicting the ash rejection and organic matter recovery. Tang et al. [18] investigated a transonic NASA

rotor. They used the RSM to perform a design study based on 24 parameters. Lü et al. [19] carried out a reliability-based design optimization for a rock tunnel support system, where the shotcrete thickness and installation position were used as parameters. Ghafarzadeh et al. [20] optimized the wastewater treatment using ultrasonic waves. It was found by the RSM that the best results can be achieved at an ultrasound power of 625 W. Xu et al. [21] used the RSM to predict the thermal cloaking performance with multiple parameters (e.g. number of layers of the multi-layer material, thermal conductivity etc.). An entropy generation analysis was used to study this process. The methods used to calculate the system output in conjunction with the operating conditions (RSM, NN, etc.) described above, can be used for an optimization procedure/algorithm, such as a genetic algorithm. Elsayed and Lacor [22] used the RSM and NN for the optimization procedure of a cyclone. The multi-objective optimization was done using the desirability density function. In contrast to this optimization procedure, the multi-objective genetic algorithm (MOGA) (described in section 4d) was successfully used in the past. Bhatti et al. [23] used the RSM and the NN for the electrocoagulation of copper in wastewater, created by experimental data, to describe the system response. It was found that the NN is more accurate than the RSM and was, therefore, used for the MOGA. A further study concerning heat exchangers with fold helical baffles done by Wang et al. [24] was published in 2018. They used experimental data to create the system’s RSM and used it for the MOGA. Uebel et al. [25] performed also CFD simulations of a quench reactor to increase the H 2/CO ratio in the syngas. For the optimization the multi-objective genetic algorithm was supplied with the simulation results. In this work approximately 200 designs were simulated leading to three optimized configurations. An increase of the H2/CO ratio by 94 % was found. It can be seen from the literature that techniques, such as RSM, NN or genetic algorithms, can be used in several applications. They can be combined not only to predict the system response, but also optimize the operating conditions. In contrast to the reported studies above, only a limited number of CFD simulations, which are the basis to create a NN and are also used for a genetic aggregation approach, were carried out in the present work. Both methods are used to create the RSM separately. Based on the two RSMs, the MOGA was used to optimize the considered process. In the academic literature many papers were published dealing with process optimization under steady-state conditions using NNs or genetic algorithms, however, research focusing on transient combustion processes are rare. The present work investigates the optimization of a transient combustion and heating process in a fire resistance test furnace. Fires, either caused

by technical errors or inadvertence, can spread explosively and the danger they impose on humans and the environment must not be underestimated. Thus, it is very important to design safety equipment and building materials that are capable of curtailing or extinguishing such fires. Therefore, fire resistance tests are necessary to provide knowledge about the resistance of materials and structures against fires. Several experimental investigations were already done in the past for different materials and structures (e.g. [26], [27]). For the development of new products, many experiments have to be performed to achieve the desired standard on fire resistance. This procedure increases the development costs, since a high number of test subjects are destroyed during the testing. Thus, CFD offers a good possibility to reduce costs and time by carrying out this test procedure in a virtual space. A further problem of the existing test configuration, which was investigated in this work, is the inhomogeneous temperature distribution in the testing furnace. Due to this fact, hot or cold spots occur in the gas phase of the furnace leading to a significant deviation from the average heat flux to the tested object. As a consequence, parts of the tested construction or material receive a higher or lower heat flux and thermal stresses. Therefore, an optimisation of the temperature distribution was required. Since the temperature distribution was the main issue in this study, CFD simulations of the gas phase combustion and heat transfer in the solids (e.g. walls) had to be done. In previous studies of Prieler et al. [28] and Mayr et al. [29] an overview of the simulation procedure for large-scale natural gas fired furnaces is presented. Boundary condition, solver settings and the influence of different combustion, radiation and turbulence models were tested. The basic idea to consider a transient heating process with steady-state simulations was proposed by Prieler et al. [28] for the transient heating of steel billets. Since there is a lack of optimization of transient combustion and heating processes using NNs and GAs, this study is focusing on the investigation of a fire resistance test furnace, where the furnace, the gas phase and the tested fire protecting material are heated up from ambient temperature. The entire transient heating process was simulated to reveal the high deviation of maximum and minimum temperature in the furnace. To avoid an inhomogeneous temperature distribution, different optimization techniques were used for this purpose. The optimization was done as described below: 

The transient process was investigated for three different (discrete) parts of the transient case by steady-state CFD simulations with varied operating conditions.



Based on the CFD simulations input (operating conditions) and output (temperature in the furnace) data were generated for each time step.



NNs and genetic aggregation were used in combination with the RSM and multiobjective genetic algorithm (MOGA) to define the “best” operating conditions leading to an improved temperature distribution.

Subsequently, the optimized parameters were applied in the range of the considered time step for the second transient simulation (optimized configuration) and the results were compared to the initial state. Since the complete consideration of the transient process performed by CFD, including various parameter modification, would require substantial computational time, this study shows the transformation of a transient process to a steady-state. This leads to a significant reduction of the calculation time and offers the possibility to generate input and output data used for the NN and GA. The presented approach, described above, can generally be applied for all transient processes with low calculation time and without considering the entire transient process in detail.

2 Experimental setup – Fire resistance test furnace The investigated furnace is used for testing fire safety products and civil construction materials regarding their response (thermal conduction and deformation) to fire. In Fig. 1 the 3D model of the test furnace, including an overview of all components, can be seen. Furthermore, a detailed view of the burner is shown. The setup of the considered fire resistance test is displayed in Fig. 2, where fire safety glass is used as the test specimen. The dimension of the furnace was 4.5 x 4 x 1.25 m and the thickness of the side and rear wall was 0.3 m. The thickness of the bottom and the ceiling of the furnace were 0.345 and 0.23 m, respectively. All walls of the furnace consist of bricks and fibre insulation panels. As a simplification, the walls were modelled as a homogeneous body with equal material properties. The values for the thermal conductivity, density and the specific heat capacity of the walls, which were used for the CFD simulation, are displayed in Table 1. Furthermore, in this table the material properties of the test specimen and the mounting frame, which holds the position of the fire safety glass in the front wall, are given. The furnace is heated by four natural gas burners with a total fuel input between 450 and 600 kW, which is equally distributed to the burners. The control unit of the furnace provides a constant fuel to air ratio determining an O2 concentration of 7 % per volume in the flue gas (wet). To achieve a better temperature distribution, the burners are equipped with baffle sheets, which should provide a good circulation of the hot flue gas. Although the fire safety glass consists of many thin layers of glass and fire protecting soluble glass (e.g. sodium

silicate), it was assumed to be a homogeneous material with the material properties from Table 1. For the testing process the test specimen was fixed in a frame, which was sealed against the environment to prevent air leakage.

Fig. 1. CAD model of the investigated testing furnace (left) and the burners (right).

Fig. 2. Experimental setup of the considered fire resistance test of fire safety glass: Initial state (left); after fire resistance test (right).

2.1 Transient heating process – Fire resistance test During a fire resistance test, the entire furnace and, accordingly, the test specimen is heated up from ambient temperature following a pre-defined time-dependent temperature trend. This temperature trend is displayed by the blue line in Fig. 3 (left). This line represents the average temperature measured by all the thermocouples in the furnace. In the furnace 12 thermocouples (T1 – T12) were arranged in a distance of 10 cm from the test specimen. A detailed overview of the temperature measurement positions as well as the burner positions are displayed in Fig. 4. Each individual thermocouple it is placed on a metal sheet and insulated by a plate of ceramic insulation (see Fig. 3 (right)). According to the standard, the tolerance of the measured average value of the temperature is quite high and has to be within a range of +/- 100 K (see Fig. 3 (left)). Therefore, the reproducibility in such fire resistance tests is questionable. While the resistance test is running, the test specimen is exposed to the hot gases of the 4 burners. The deformation and the temperature of the fire safety glass are observed (not relevant in this study), and allow a statement whether a fire safety class is accomplished or not. The duration of the fire resistance test was 35 minutes. A detailed description of the time-dependent temperature characteristic, the allowed limits and all required boundaries for the test are given by standard [30]. To obtain a close accordance between the measured average temperature and the pre-defined temperature trend, a homogeneous temperature distribution in the furnace is desired. Therefore, the optimization techniques were used to achieve this goal. Since each thermocouple is shielded by the ceramic insulation plate resulting in a response delay of the measured temperature compared to the gas temperature, the entire thermocouple (metal sheet and ceramic insulation) was modelled. As a simplification, they were designed as homogenous bodies with constant material properties (see Table 1). In Fig. 5 the fuel input and the measured temperatures at T1 to T12 are displayed.

Fig. 3. Time-dependent temperature trend for the fire resistance test (left); Thermocouple (right).

Fig. 4. Position [units in mm] of the burner and flue gas exit (left); Position [units in mm] of the 12 thermocouples in the furnace (right).

Fig. 5. Measured fuel input and temperatures in the furnace during the test.

3 CFD simulation To predict the fluid flow within the furnace the Reynolds-averaged Navier-Stokes equations were solved (see Eqs. (1) and (2)). In this study a double precision pressure-based solver was used and the flue gas was assumed to be incompressible. The density fluctuations caused by the temperature gradients were calculated by the ideal gas law. In the simulation, the operating pressure was set to 1013.25 mbar. Pressure-velocity coupling and pressure interpolation were done by the SIMPLE and PRESTO! algorithm, respectively. The spatial discretization of the transport equations was done by the second order upwind scheme. Only the radiative heat transfer was spatially discretized with the first order upwind scheme because of the linear propagation of the radiative heat transfer. (1) (2) In Eq. (2) the Reynolds stresses

were modelled using the realizable k-epsilon

model proposed by Shih et al. [31]. This model was already tested for jet flames by Prieler et al. [32] and compared to the predicted values from Reynolds stress model. It was found that this model showed close accordance to the measured temperatures and the results from the RSM. The realizable k-epsilon model, as a 2 equation approach, needs only two additional transport equations for the turbulent kinetic energy

3.1

and the dissipation rate .

Combustion and radiation modelling

To predict the gas phase combustion within the furnace, the steady laminar flamelet model (SFM) was used, assuming that the fuel and the oxidant enter the reaction zone separately (non-premixed flame). With this approach the main reaction zone (near stoichiometric mixture) can be represented by a number of small laminar counter-flow diffusion flamelets. The advantage of this approach is that the thermochemical state in such a laminar flamelet can be related to a single parameter, called the mixture fraction defined in Eq. (3). In this equation,

denotes the elemental mass fraction of the element . The variables

represent the elemental mass fraction of the element [34].

and

in the fuel and the oxidant [33],

(3) Thus, the temperature , species concentrations

and density

at each point in the counter-

flow diffusion flamelet can be determined before the CFD simulation, solving the flamelet equations given in Eqs. (4) to (6) related to the mixture fraction space. In the flamelet equations

is the scalar dissipation rate,

enthalpy of the species ,

is the reaction rate of species

is the strain rate and

,

is the

is the density of the oxidizer at the inlet.

Here, the scalar dissipation rate can be seen as the inverse of the diffusion time or the residence time of the fluid. When the scalar dissipation is equal to zero, chemical equilibrium has been reached. At higher dissipation rates, the difference to the chemical equilibrium increases. The scalar dissipation varies with the mixture fraction space, and has to be modelled according to Eq. (6) [35]. The results of these calculations are stored in look-up tables. Therefore, detailed chemical kinetics can be considered without solving the complex flow field in the furnace. In this study a skeletal reaction mechanism proposed by Peeters [36] involving 17 species and 25 reversible reactions was used. This mechanism was already tested for air-fuel and oxygen enriched combustion by Prieler et al. [32]. (4)

(5)

(6)

Since the fluid flow within the furnace is not laminar, as assumed in the flamelet approach, a turbulence-chemistry interaction model has to be applied to couple the instantaneous values for temperature, species concentrations and density with the Favre-averaged values from the transport equations for mass, momentum etc. This was done using a probability density function (PDF). The PDF represents the relationship between the instantaneous value of a fluctuating scalar

and the averaged value

In this equation, is in state

as shown in Eq. (7). for a non-adiabatic case.

is the presumed PDF, which can be seen as the fraction of time the fluid

in the fluctuating fluid flow. For the presumed PDF, a so-called -function was

used. Average values

, e.g. temperature, were calculated by Eq. (7). For the integration of

, the mixture fraction variance

[37] is required with

.

(7) The chemistry calculation as well as the integration of the PDF were carried out before the CFD simulation. The results for the averaged values related to the mixture fraction and the mixture fraction variance were stored in look-up tables. Therefore, time expensive calculations of the reaction mechanism can be avoided during the CFD simulation. Average scalars

for temperature and species concentration in a turbulent flame can be achieved by

two additional transport equations for the mean mixture fraction variance and

(see Eq. (8) and (9)). In the transport equations,

and the mixture fraction

is turbulent viscosity, and

,

are model constants with values of 0.85, 2.86 and 2.0, respectively. (8) (9)

The radiative heat transfer in the furnace is predicted by solving the radiative transport equation (RTE) given in Eq. (10). In Eq. (10), the position and the direction vector, coefficient, function and

is the refractive index,

is the radiation intensity depending on

is the absorption coefficient,

is the scattering

is the direction of scattered radiation,

is the phase

the solid angle. The second term represents optical thickness, and calculates

the loss of radiation intensity due to scattering and absorption in the gas phase. Since radiative heat transfer is only considered in the gas phase without any particles, the refractive index and scattering coefficient were set to values of 1 and 0 s-1. (10) In this study the RTE was solved using the discrete ordinates model (DOM) [38], [39]. For each direction

(angle), a RTE is solved. The discretization of the angles (number of angles)

has a distinct effect on the calculation time. In this work, each octant was discretized with 4 x 4 solid angles leading to an overall number of 128 directions for the radiative heat transfer. For modelling the radiative properties of the flue gas the weighted sum of grey gases (WSGG) model was employed in conjunction with parameters published by Smith et al. [40].

3.2 Boundary conditions and numerical grid In the CFD simulation, pure methane was used as fuel instead of natural gas and air as oxidant with 79 % N2 and 21 % O2 per volume. All inlets were modelled by mass-flow inlets with the hydraulic diameters of 0.062 m (fuel) and 0.1 m (air), respectively. Furthermore, the turbulent intensity at the inlets was set to a value of 10 %. Due to the fuel-lean conditions, an oxygen concentration of 7 % per volume in the flue gas was present. Methane and air were fed to the furnace with 25 °C. At the outer surface of the furnace walls the heat transfer coefficient of 10 W/(m²*K), an ambient temperature of 25 °C and emissivity of the surface of 0.5 was defined. The heat flux to the test specimen was calculated by coupling the fluid and solid zone at the interface, which had an emissivity of 0.55. For the numerical grid, a mesh consisting of polyhedrons with 772,545 cells, which includes the walls, the gas phase, the mounting frame and the fire safety glass, was used. The grid independency test was done using different types of cells (polyhedrons, hexahedrons and tetrahedrons) and mesh sizes within a range of approximately 310,000 and 1,350,000 cells. Steady-state simulations with the different meshes were carried out. It was found that a mesh with 772,545 polyhedrons is able to predict the same temperatures in the furnace as the finer grid with 1.35 million cells.

3.3 Transient CFD simulation of the entire process Two transient simulations of the considered fire resistance test, presented in section 2, were carried out: 

Initial state of the furnace configuration to reveal the temperature distribution in the furnace (see section 5.1).



Transient simulation with the optimized furnace configuration (see section 5.4).

For the transient simulation the time step sizes were chosen according to Table 2. Due to the fact that the highest temperature gradients in the gas phase occur at the beginning of the heating process, the first two minutes were considered with a lower time step size.

3.4 Steady-state simulation for optimization The existing fire resistance test is an unsteady heating process. Therefore, complete transient simulations of the process with different configurations are computationally demanding and not reasonable for large scale furnaces. Thus, the transient simulation was reduced to simple

steady-state simulation cases by subdividing the process in three “discrete” parts, which were defined by the inner wall temperature of the furnace. Therefore, the steady-state simulations were carried out on the same mesh as the transient cases, but at the same time neglecting the solid walls, the mounting frame and the test specimen. The interface between the solids is now defined by a fixed temperature, representing the testing progress. The simulations of the first part were done with a fixed inner wall temperature of 100 °C, which stands for the range between ambient condition and 200 °C of the wall temperature. During the experiment the inner wall temperature was observed. It was found that the inner wall temperature of 200 °C was reached after 18 seconds. This means that the first part has a duration of 18 seconds. For the second part, the inner wall temperature ranging from 200 to 400 °C was defined, which is corresponding to a time range between 18 and 60 seconds. This seems to be a small time range compared to the complete test duration. Since the main gradients on the temperature in the furnace are present at the beginning (see measured temperatures in Fig. 5), the inner wall temperature is quite constant after the first minute reaching a final level of approximately 700 °C. Therefore, the used subdivisions can be seen as sufficient for the test case. The defined wall temperatures, the corresponding temperature ranges and the simulation times are displayed in Table 3. With this approach the optimization of the transient process is possible by a steady-state consideration with lower computational demand. As a consequence, the optimization algorithms have to be applied for each part to accomplish the optimized configuration for each part. To determine the level of temperature uniformity in the furnace, the temperature deviation at each temperature measurement position is considered as given in Eq. (11). In this equation, is the temperature at the position .

(11)

4 Optimization procedure In the academic literature, various methods of optimization algorithms are reported to be helpful for a wide range of applications (e.g. [41], [42]). In the present study, a transient process, which was subdivided in three “discrete” parts (see section 3.4), had to be optimized. Therefore, the optimized parameters (operating conditions) had to be identified for each part. Three parameters, presented in section 4.1, were defined for the considered fire resistance test

furnace. The main goal was to achieve a high temperature uniformity in the entire furnace defined by the deviation from the average temperature at the measurement positions. The procedure carried out for each of the three “discrete” parts, was done according to the following points presented in Fig. 6. Two methods (NN and GA) were applied for the optimization.

Fig. 6. Optimization scheme using NN and GA.

a) Generation of datasets (input variables and output variables/temperatures): This can be done by experiments or simulations. In this study CFD simulations were carried out and the temperatures at the measurement positions were calculated depending on the operating conditions for each “discrete” part. The operating conditions, which were used, are presented in section 4.1. b) Neural networks/Genetic aggregation: Two methods were applied to describe the system/furnace and predict values for the temperatures in correlation to the operating conditions, which were not directly considered by CFD. This was the NN (see section 4.2) and the genetic aggregation (see section 4.3). Hence, it is possible to estimate the temperatures at the measurement positions for each operating condition within the chosen parameter range. c) Response surface method (RSM): It was proposed by Box and Wilson [43] to determine the optimum in chemical investigations. Box later described other possible applications of the RSM [44]. The RSM shows the relationship of one or more variables (input parameters/operating conditions) of a system to the output parameters

(T1 – T12). The response surfaces are generated by the NN and genetic aggregation approaches, which can describe the system response based on the input parameters. Such a response surface for two variables and one output can be displayed by a surface plot (e.g. Fig. 21). d) Multi-objective optimization: The considered system represents a multi-objective problem with 12 temperature measurement positions. This means that the optimized conditions for one measurement position may not be appropriate for the others. For such problems, evolutionary algorithms are well established and commonly used [45]– [47]. In this case, the multi-objective genetic algorithm (MOGA) proposed by Fonseca and Fleming [48] was applied to find the optimum parameters for each “discrete” part of the transient process. A short description of the working principle of the MOGA is presented and depicted in Fig. 7 [49]: 

The algorithm starts with a random generation of an initial population (input parameters/operating conditions).



The genetic algorithm of MOGA generates a new population (input parameters) from the initial state. For each iteration, the population of the previous run was used for the generation of the new population (number of operating conditions: 2000). The procedure of a genetic algorithm to create new population has already been described in detail in the literature (e.g. [50], [51]) and is further explained in section 4.3.



The new population is updated and the temperatures are determined based on the response surfaces created by the NN and genetic aggregation. The target of the optimization was to minimize the deviation of all temperatures from the average temperature in the furnace, which is defined in Eq. (11).



In the next step, the convergence criteria, defined by minimizing

, was checked

considering the Pareto efficiency. The Pareto front displays the operating conditions where one parameter cannot be changed for minimizing the temperature deviation at one position without at the same time increasing the deviation on another position in the furnace. For this study, a Pareto percentage of 75 % was fixed, which means that 1500 operating conditions of the population have to fulfil the requirements of the Pareto front. 

In the last step, the stopping criterion is considered by comparing the number of the present iteration step with the maximum defined number of iterations (fixed value of 800).

Fig. 7. Scheme of the optimization algorithm using MOGA [49].

4.1 CFD simulations – Operating conditions For the steady-state CFD simulations, which are the basis for the further optimization algorithm, three varying parameters were defined (see Fig. 8). These input parameters are the mass flow-rate of natural gas of the two burners at the top furnace, the angle of the baffles

and

and at the bottom

of the

as well as the furnace depth . In Table 4, the

chosen input parameters and values for the CFD simulations are given. The values for the variables were changed separately. For the variation of the fuel input, 25 CFD simulations were carried out considering only one “discrete” part leading to an overall number of simulations of 75. To investigate the effect of the baffle angle, 9 simulations (overall: 27) were required and for the furnace depth 5 (overall: 15). Eventually, 117 steady-state CFD simulations were done to generate the input and output data (temperature deviation) for the optimization procedure.

Fig. 8. Input parameters: Mass flow-rates natural gas (left), angle of the baffles (middle) and furnace depth (right).

4.2 Neural network (NN) for response surface The theory and topology of NNs is already available in the literature and their possible applications are well described (e.g. [52]). Therefore, only a brief introduction is given in this section. In 1943, McCulloch and Pitts [53] proposed the idea that the nervous activity can be calculated using propositional logic, which was the basis for the development of neural networks. A neural network can be seen as a number of neurons connected in different ways. a) Single neuron: A single neuron is displayed in Fig. 9. The variables

represent the input signal to the

neuron. Depending on their significance, the input signals are weighted by factors resulting signal output

is summarized as given in Eq. (12). With the activation function

. The , the

of the single neuron is formed. Since McCulloch and Pitts were focused on the

nervous activity, the activation function can be represented by a step function (“all-or-none”). In this study a hyperbolic tangent function was used for threshold represents the minimum value of the

Fig. 9. Scheme of a single neuron.

in each neuron (see Eq. (13)). The

-signal to activate the neuron.

(12) (13) b) Neural network: In Fig. 10, the basic principle of a neural network is displayed (feed-forward network). It consists of several neurons arranged in an input layer, one or more hidden layers and an output layer. The input parameters are introduced to the network by the input layer. Each connection between the neurons in the network is weighted according to their importance on the output variable as shown before. There is a considerable number of types of neural networks described in literature, which would extend the framework of this paper. Detailed information can be found for example in [52]. In the present study a feed-forward network, as shown in Fig. 10, was used with one hidden layer containing 8 neurons, one input (5 neurons) and one output layer (12 neurons). A feed-forward network transports the incoming signal from one layer to the next one only in one direction. In contrast, recurrent neural networks can also transport the signal to the previous layers or use the output of a neuron again as an input. In the input layer, 5 signals are introduced to the network (2 baffle angles, 2 fuel inputs and the furnace depth). The outputs represent the 12 temperature measurement positions in the furnace. The number of neurons in the output and input layer are fixed based on the number of parameters. Although it was shown in previous studies that the optimum number of hidden nodes should be

, where

is the number of the hidden nodes [54], the optimum number of neurons in the hidden layer was determined by trial and error. It was found that the neural network containing 8 neurons in the hidden layer showed the best results. c) Training algorithm: Before the NN can predict the output (temperature) as a function of the input variables, it has to be “trained”. The training of a neural network is done in the present case to achieve the best prediction of the output parameters in close accordance to the data from the CFD simulation (temperature). During the training procedure, the neural network is adapted in several ways: (i) Variation of the number of hidden layers and neurons in the layers, (ii) adding and removing connections between the neurons, (iii) adaptation of the weighting factors. In literature, there are several methods reported to train neural networks capable of considering multi-dimensional problems, such as the gradient descent method, Newton’s method, conjugate gradient method etc. In this study, the training of the neural network was done by

changing the weighting factors with the Levenberg-Marquardt algorithm [55], [56], which is based on the least-squares method.

Fig. 10. Principle of a neural network.

d) Training and validation data: To train the NN, not all of the data gained by the 117 CFD simulations were used. Only 94% of the data were applied to create the NN and the rest was used for the validation. That means that after the network was created, using 94 % of the CFD data, the NN was used to predict the temperature in the furnace for the other 6 % of the tested cases. The results from the CFD simulation and from the NN were compared. The deviation was determined by the root mean square error of all temperatures observed in the furnace (12 temperatures in 117 simulations), which was 5.4 K. Since the error tolerance of the thermocouples is +/- 4 K, according to the standard [30], the created response surfaces using the NN was found to be sufficient.

4.3 Genetic aggregation (GA) for response surface The second method to create the response surfaces was the genetic aggregation. The description in this section is based on [49], where information on the sub-models can be found as well. Further information can be found in [57], [58]. The genetic aggregation utilizes four different meta-models (sub-models) to generate a response surface: (i) polynomial regression; (ii) non-parametric regression; (iii) kriging; (iv) moving least squares; (v) linear basis

function; (vi) support vector regression. Other types of sub-models are available, but were not used in the resulting ensemble resulting ensemble weighting factors

. Each sub-model generates a response surface

is created from the individual response surfaces (see Fig. 11). In Fig. 11 the variable

, and the

and the appropriate

stands for the total number of

response surfaces from the sub-models. The sub-models and their settings are not explained in detail in this paper, but more information can be found in [49].

Fig. 11. Resulting response surface due to the combination of the sub-models.

Since each sub-model allows the generation of a response surface

with different settings, a

genetic algorithm was used to find the best response surfaces. The genetic algorithm is shown in Fig. 12 which is the same procedure used by the MOGA in section 4d. The procedure has to be done for each inner wall temperature. After the generation of the initial response surfaces with different type and setting, the evaluation and selection procedure starts. In the evaluation/selection step, the predicted temperatures from the response surfaces are compared to the design points. For this purpose, the root mean square errors (RMSE) (see Eq. (14)) and the RMSE based on cross-validation (PRESSRMSE) (see Eq. (15)) were used to determine the best response surfaces with its settings. In Eq. (14) and Eq. (15),

and

are the number of

simulations and the number of response surfaces from the sub-models used. The variable stands for the simulation , simulation and

represents the output parameter (temperature) of the

is the result given by ith response surface, which was created without the

data from the jth simulation. It has to be mentioned that the weighting factors are calculated by an approach from Viana et al. [57]. The best response surfaces are selected for further consideration in the algorithm. The cross-over step compares the types of the response surface (for example kriging or nonparametric regression etc.). If two response surfaces are of the same type, the settings are

mixed. This means that a part of the settings of the first response surface (e.g. kernel type of the kriging method) is transported to the second one. Hence, the settings (“genes”) are mixed. If the response surfaces are of different types, the new response surface will be a linear combination of them. The mutation randomly changes a setting of the response surface type. If the response surface is created as a combination of different types, a setting of both types is changed. For example, a resulting response surface for the temperature measurement position T11 and an inner wall temperature of 300 °C with its sub-models, weighting factors, the RMSE and the PRESSRMSE is presented in Table 5. It can be seen that the resulting response surface for T11 is leading to an RMSE of 0.7737 K for all operating conditions at 300 °C wall temperature. For each temperature measurement position in the furnace (12) and inner wall temperatures (3), a response surface was created by the algorithm. The algorithm stops after 12 iterations, which was found to be sufficient (trial and error – similar to the NN). The results from the CFD and the response surface from the GA were compared, leading to a root mean square error of all temperatures observed in the furnace (12 temperatures in 117 simulations) of 6.6 K.

Fig. 12. Procedure of the genetic algorithm.

(14)

(15)

5 Results and discussion This section is subdivided into three main parts. In the first part, the proposed CFD simulation model of the fire resistance testing furnace will be validated. In section 5.1, the transient simulation of the entire experiment will be carried out, and the predicted temperatures at the measurement positions are compared to the measured data. The parameters used for this simulation are given in Table 4 where the used values are marked in red. The second part (sections 5.2 and 5.3) is dealing with the optimization procedure of the initial furnace configuration. It will reveal the optimization potential of the chosen parameters to achieve a more homogeneous temperature distribution. The third part, presented in section 5.4, shows the transient simulation of the experiment with the optimized operating conditions and compares the temperature distribution in the furnace with the initial configuration.

5.1 Simulation of the reference test To validate the CFD simulation model, a transient simulation was performed. Additionally, the used CFD model and numerical settings were already tested, and validated for lab-scale and industrial scale furnaces under steady-state conditions by Prieler et al. [7], [32], [28]. In Fig. 13, the predicted and measured temperatures at T2 and T11 are displayed for the entire duration of the test. The solid blue and red lines represent the measured temperatures while the dashed lines display the corresponding simulated values. Only 2 measurement positions were displayed because of the same trend observed at all positions. The measured data shows that the heating rate is high at the beginning of the test. Within 5 minutes, the gas temperature in the furnace rises from ambient temperature up to 600 – 700 °C. As a consequence, the high heating rate is leading to an inhomogeneous temperature distribution in the gas phase, which

has to be minimized. After 5 minutes, the heating rate is decreasing while the temperature increases until the end of the experiment where it reaches approximately 200 to 300 K. Upon comparison of the measured and the simulated data, it can be seen that at the beginning of the experiment (4 minutes), the measurement is in close accordance at T2 and T11. This means that the CFD model can predict the heating rate in the furnace. In Fig. 14 the comparison of the temperature measurement and the simulation result is displayed after 4 minutes. At the bottom of the furnace (T1 to T6) the accordance is very close. In the upper part, the simulation slightly over-predicts the temperature. This can be explained by the numerical setting used in this study. During the generation of the PDF for the combustion modelling, compressibility effects were neglected. Thus, the buoyancy of fluid flow is too high in the simulation, which is leading to a mean deviation between measurement and simulation of 22 K in the entire furnace after 4 minutes. In addition, a slight deviation of the angle of the baffles can have a significant effect on the calculated temperature, especially T2 and T11 where the flames are directed at the baffles. After 5 minutes the heating rate decreases. During this period of the fire resistance test, the simulation calculates lower temperature than in the experiment. Here, the mean deviation of 38 K can be observed after 20 minutes (see Fig. 14). Similar to the data after 4 minutes, the buoyancy effect is still overpredicted, which can be determined at the measurement positions T10 to T12 that represent the highest simulated temperatures (except T2). It can be further observed that the temperature distribution is more homogeneous than at the beginning of the test in both the simulation and the experiment. The mass flow-rate of fuel is increased after 19 minutes. This point can be seen in Fig. 13 by a slight increase of the heating rate. Until the end of the experiment, the measured temperature trend is in good accordance with the experimental data. Considering the results in Fig. 14 after 35 minutes (end of the experiment), nearly all predicted data at measurement points are within the measurement uncertainty of +/- 4 K of the thermocouples used. The mean temperature deviation at the end of the experiment was 7 K at a temperature level of approximately 900 °C. These results suggest that the CFD approach is capable not only calculating the temperatures in furnaces under steady-state conditions, as mentioned above, but also the transient heating process in fire resistance tests.

Fig. 13. Measured and simulated temperatures at T2 and T11 during the experiment.

Fig. 14. Measured and calculated temperatures at all measurement positions after 4, 20 and 35 minutes.

As a criterion for the temperature distribution in the furnace, the maximum temperature difference (considering all measurement positions) was chosen and defined by Eq. (16). A comparison between the experimental and the simulated values for the initial configuration is shown in Fig. 15. (16) The maximum temperature difference in the furnace was detected within 5 minutes from the beginning of the experiment. In this time a maximum value of 200 K (simulation) and 150 K (measurement) was found. The maximum temperature difference at the beginning is higher in the CFD simulation due to the neglected compressibility effects during the generation of the PDF for combustion modelling. A further reason for this variance between measurement and CFD result is based on the slight deviation on the baffle angle in the experiment as already

mentioned before. For instance, the contour plot of the temperature in the furnace representing the state of the system after 60 seconds is shown in Fig. 16. Since the buoyancy is forcing the hot flue gases to the top of the furnace, there is a cool region at the bottom (see red circle). After 7 minutes, the simulated temperature difference shows a close accordance to the experimental value. Since the inner wall temperature increases, the radiative heat flux affects the temperature distribution in the furnace, which is leading to a temperature difference of approximately 50 K. The main goal in the next sections is to identify the optimum parameters by using steady-state simulations.

Fig. 15. Maximum temperature difference in the furnace (only considering the measurement positions).

Fig. 16. Contour plot of the temperature in the furnace after 60 seconds.

5.2 Steady-state simulation – Case studies For the optimization procedure, 117 datasets (input and output) have been created by CFD simulations. However, before the steady-state simulations could be used for the case studies, it had to be determined whether a steady-state simulation can calculate the temperature in close accordance to the transient simulation. For this purpose, the results of the reference transient simulation shown in section 5.1 were used, where the average inner wall temperature was 300 and 500 °C, respectively. These results were compared to the steady-state simulations of the initial configuration used in the reference case. In order to determine whether the results from the steady-state simulation can represent the transient case sufficiently, the calculated temperature at all measurement positions for an inner wall temperature of 300 and 500 °C are displayed in Fig. 17. At an inner wall temperature of 300 °C, it can be seen that the temperature difference between the hottest and coldest point is lower in the steady-state simulation than in the transient (see red and blue lines). Since the inner wall temperature is fixed with one value in the steady-state simulation, the gas temperature is more homogeneous. In the transient simulation, the wall temperature is a result of the simulation, which is not evenly distributed across the entire inner wall. Therefore, the calculated temperature in the transient simulation shows a higher difference between the minimum and maximum value. Nevertheless, the temperatures in Fig. 17 are in good accordance between the transient and steady-state results at an inner wall temperature of 300 °C. Similar results can be found at an inner wall temperature of 500 °C. At both temperature levels the mean deviation between the temperatures predicted by the transient and steady-state simulations was 18 K (at 300 °C) and 4 K (at 500 °C). Thus, it can be concluded that the steady-state simulations can be used for an appropriate prediction of a specific time step in the transient case.

Fig. 17. Calculated temperatures in the furnace for the steady-state and transient case (inner wall temperatures: 300 and 500 °C).

5.2.1 Effect of the furnace depth In Fig. 18, the simulated temperatures in the furnace are displayed for a varying furnace depth. The solid lines in this figure represent the results for an inner wall temperature of 100 °C, which consider the first 18 seconds of the experiment. The results show that a furnace depth of 1.25 m or lower is leading to a similar temperature distribution, detectable by the solid red and blue lines. From the lowest temperature at T2 to the highest one at T10 and T12, respectively, there is a maximum temperature difference of approximately 150 K. In contrast, when the furnace depth is increasing, the temperature distribution is getting more homogeneous. A furnace depth of 1.85 m (solid green line) shows the lowest temperature difference at an inner wall temperature of 100 °C. In that case the maximum temperature difference is 50 K lower than with the initial configuration. The same effect can be seen with an inner wall temperature of 500 °C, which is depicted by the dashed lines. For each furnace depth, the homogeneity of the gas temperature is improved at a higher inner wall temperature. Since the wall temperature is increasing during the fire resistance, the radiative heat transfer in the furnace is improved and is further leading to a lower temperature gradient.

Fig. 18. Calculated temperatures in the furnace depending on the furnace depth for an inner wall temperature of 100 and 500 °C.

5.2.2 Effect of the baffle angle In Fig. 19, the temperatures of the initial angle of the baffles are displayed by the blue solid (inner wall temperature of 100 °C) and dashed (inner wall temperature of 500 °C) lines. In this figure, for example, these temperatures were compared to another configuration, which represented the best configuration of all 117 CFD simulations. Here, the angle of the baffles of the top burners is reduced to a value of 15°. It can be determined that the initial configuration of the baffles, with an angle of 45° (bottom and top), shows a close accordance to this version. In the case of an inner wall temperature of 100 °C, both simulations showed a maximum temperature difference of approximately 100 K between the thermocouples at the bottom (T2 and T3) and at the top (T11 and T12). When the fire resistance test is in progress and the wall temperature is increasing, the heat transfer is dominated by radiation (see dashed lines representing a wall temperature of 500 °C). The heat transfer from the solid bodies (walls) is leading to a maximum temperature difference of approximately 55 K.

Fig. 19. Calculated temperatures in the furnace depending on the angle of the baffle for an inner wall temperature of 100 and 500 °C.

5.2.3 Effect of the mass flow-rates of natural gas Fig. 20 shows the effect of the natural gas input through the bottom and top burners. Again the solid lines highlight the conditions at the beginning of the experiment with a low wall temperature. As in the initial configuration an equally distributed natural gas input (240 kW bottom and top) to the burners was chosen (see red lines). Although the fuel input was not the same during the experiment, at least the distribution of the fuel was equal for all burners. The case studies have shown that the fuel input has to be decreased at the top of the furnace to 120 kW and, subsequently, increased at the bottom to 280 kW (see solid blue line). It was found that the lower fuel input at the top of the furnace decreases the temperature at T8 to T12 significantly, compared to the initial condition. Thus, the maximum temperature difference is lower. The same effect can be seen for higher wall temperatures, marked by the dashed lines. However, the temperature distribution is very similar to the optimized configuration when the wall temperature is high.

Fig. 20. Calculated temperatures in the furnace depending on the fuel input for an inner wall temperature of 100 and 500 °C.

5.3 Optimization using MOGA The case studies, presented in section 5.2, have already shown the optimization potential of the used parameters and will be a reference for the following procedures using NNs and GAs. The operating conditions can be easily improved by such case studies for a low number of parameters. However, solving multi-objective problems with a high number of parameters is still a time-consuming task if only the case studies are being considered. Therefore, the aforementioned optimization algorithms were applied for the present experimental setup and compared to the results of the case studies. The results of the 117 CFD simulations (input/operating conditions and output/temperatures), according to Table 4, were used to create the response surfaces. These response surfaces shows the correlation of the input parameters (fuel input, baffle angle and furnace depth) and the output (temperature deviation from the mean temperature defined in Eq. (11)). Additionally, the response surfaces have to be created for each “discrete” part of the experiment (100, 300 and 50 °C). For instance, the response surface for the sum of all temperature deviations

is highlighted in Fig. 21. The plots show the response surfaces

created with the NN (left) and GA (right) for an inner wall temperature of 500 °C. Here, only the dependence of the fuel input is given. Both methods suggest that the fuel input at the top burners should be minimized to achieve a homogeneous temperature distribution. Considering the region of the lowest fuel input at the top, two local minima can be detected. These are reached for a fuel input at the bottom of 120 kW and approximately 250 kW, which corresponds to the results in Fig. 20. Based on all response surfaces (all

dependent on all

input parameters), MOGA calculates the optimized operating conditions for each “discrete” part of the transient experiment.

Fig. 21. Response surfaces depending on the fuel inputs created by the NN (left) and GA (right) for an inner wall temperature of 500 °C.

The parameters found by the case studies in section 5.2 and MOGA (with NN and GA) are given in Table 6, except the furnace depth with its optimum at 1.85 m. It can be seen that all methods propose similar operating conditions. Nevertheless, the case studies show only one optimized operating point although several local minima exist. The identification of local minima for multi-objective issues is achievable with reasonable effort by the used techniques. Particularly, this was found for an inner wall temperature of 500 °C where both algorithms recommend two different operating conditions concerning the fuel input (see also Fig. 21). The following figures (Fig. 22 to Fig. 24) display the steady-state simulation for the different inner wall temperatures carried out with the optimized configurations from Table 6. The furnace depth was 1.85 m. In Fig. 22, the simulations with a wall temperature of 100 °C are compared to the initial configuration (see red line). It is highlighted that the maximum temperature difference (T2 and T8) is approximately 100 K for the improved conditions. In contrast, the initial configuration showed a maximum difference of about 150 K. Furthermore, the position of the maximum temperature is not at the top of the furnace, represented by T10 to T12, when the improved parameters are used. With these operating conditions the maximum temperature is at T8 which is in the middle of the furnace. With an increase of the inner wall temperature up to 300 °C, the maximum temperature difference with the improved operating conditions is decreasing to approximately 120 K (see Fig. 23). However, the temperature level is 50 K lower than with the initial configuration. This suggests that a lower heating rate would be beneficial, in particular at the beginning of the experiment. However, all efforts to improve the temperature distribution aside, it must be considered that the trend of

the mean temperature in the furnace during the fire resistance test has to be achieved within the tolerance (see Fig. 3). Thus, for the transient simulation, later presented in section 5.4, not the same fuel input will be used. Fig. 24 represents the CFD simulations for an inner wall temperature of 500 °C. The thermocouples at the centre of the furnace (T5, T8 and T11) detected temperature peaks. These peaks are decreased by the improvement of the operating conditions leading to a more homogeneous temperature profile in the furnace.

Fig. 22. Temperatures in the furnace at an inner wall temperature of 100 °C calculated with the initial configuration and the optimized parameters found by the case studies, neural network and genetic aggregation.

Fig. 23. Temperatures in the furnace at an inner wall temperature of 300 °C calculated with the initial configuration and the optimized parameters found by the case studies, neural network and genetic aggregation.

Fig. 24. Temperatures in the furnace at an inner wall temperature of 500 °C calculated with the initial configuration and the optimized parameters found by the case studies, neural network and genetic aggregation.

The case studies as well as the optimization algorithms were able to improve the temperature distribution in the furnace considerably. However, this statement can only be given for the steady-state simulations at the moment. Therefore, the optimized input parameters will be used for the transient simulation in section 5.4.

5.4 Furnace simulation with optimized parameters Based on the results of the previous sections, the optimized parameters for the new transient simulation of the fire resistance test are summarized in Table 7. Since the baffle angle and the furnace depth cannot be changed during the experiment, they were fixed with 45°/15° (bottom/top) and 1.85 m, respectively. It was examined in section 5.3 that the proposed fuel inputs might lead to a lower temperature level than the initial configuration (e.g. Fig. 23). This has to be avoided since the temperature trend of the mean temperature during the experiment has to be achieved within the tolerances given in Fig. 3. Subsequently, the fuel ratio between the bottom and top burners were calculated for each “discrete” part (see Table 7). To calculate the ratios, the suggested fuel inputs from the GA were used (see Table 6). The overall fuel input, which was measured in the experiment, is the basis for the distribution of the fuel input through the bottom and top burners. The resulting time-dependent fuel input for the optimized simulation is represented by the red line in Fig. 25. For comparison the overall fuel input of the standard configuration is given by the blue line and the corresponding fuel input for the bottom burners by the green dashed line. These lines highlight the evenly distributed fuel input through all burners in the initial configuration. For the improved

operating conditions, it is pointed out that more fuel is supplied to the furnace through the bottom burners. The ratio between the bottom and top burners is 50/50 only between 18 and 60 seconds.

Fig. 25. Transient boundary conditions for the fuel input.

In Fig. 26, the maximum temperature difference in the furnace, according to Eq. (16), is displayed for the measurement, the CFD simulation of the initial configuration and the optimized version. In particular, at the beginning of the test procedure the optimized configuration has a much lower temperature difference of approximately 162 K instead of 200 K (initial configuration). Additionally, the contour plots at 60 and 110 seconds in this figure determine that the cold region at the bottom of the furnace is rather distinct within the optimization. Subsequently, the temperature distribution is more homogeneous at the beginning. With continuation of the experiment, both temperature differences are decreasing significantly to levels of 50 K (optimized) and 75 K (initial). Between 5 and 20 minutes the temperature distribution is quite constant in both cases although the initial configuration showed a higher value of approximately 25 K. This can be explained by the contour plots at 720 seconds. Due to the lower baffle angle at the top, the hot flue gas is not directed to the thermocouples at the top, and the flue gas cannot gather in this region. Instead, a better circulation of the hot flue gas is achieved. After 20 minutes, the initial and optimized configurations predict similar temperature differences because of the increasing inner wall temperatures boosting the radiative heat transfer emitted by the solids.

Fig. 26. Maximum temperature difference in the furnace with the initial and optimized configuration (contour plots at 60, 110 and 720 seconds).

6 Conclusion In the present study, a CFD-based optimization of a fire resistance testing furnace was carried out by a steady-state consideration. The present setup of the fire resistance test furnace showed high temperature differences at the measurement positions which have to be avoided during a fire resistance test. The transient process was subdivided in three representative (“discrete”) parts which were considered by steady-state simulations instead of a fully transient approach. Two parts were considered at the beginning of the experiment with the highest temperature gradients and one in the middle of the transient process. Due to the timeefficient numerical approach, using a flamelet model to predict the gas phase combustion, the steady-state simulations were done with low computational demand. A total number of 117 CFD simulations of the entire furnace were performed with different operating conditions. These simulations were the basis for the optimization algorithms. Two different approaches were applied to optimize the “discrete” parts of the transient process: (a) multi-objective genetic algorithm using response surfaces created by neural networks and (b) multi-objective genetic algorithm using response surfaces created by genetic aggregation.

The optimization resulted in a new configuration of the furnace geometry and adaptation of the baffle angles which deflect the flame. Furthermore, the fuel supply to the burners was optimized by the algorithms. All approaches suggested similar furnace configurations and operating conditions to improve the temperature distribution. Using these furnace configurations and fuel distribution to the burners, the transient CFD simulation of the entire heating process showed a significant improvement of the maximum temperature difference at the measurement positions. Especially, at the beginning the maximum temperature difference was decreased from 200 to 162 K. After 20 minutes, radiation dominates the heat transfer and the temperature distribution has in both cases similar homogeneity. This study presents an optimization procedure for transient combustion and heating processes by a steady-state consideration of the problem. The steady-state approach constitutes a time efficient method instead of solving the entire time-dependent heating process using CFD which is computationally demanding. Based on the results, the used optimization algorithms will be applied in the future with a higher number of variables to change. Furthermore, several multi-objective functions will be considered as well. Additionally, the proposed approach using “discrete” parts is not limited to heating or combustion processes, and can also be applied for other transient processes where the computational demand of a full transient simulation is high.

Nomenclature ,

Absorption coefficient [1/m], strain rate [1/s] ,

Model parameters for the flamelet approach Specific heat capacity [J/(kg*K)] Furnace depth [m], jth simulation (design point)

, ,

,

Mean mixture fraction (instantaneous value), mixture fraction variance, Favre-averaged mean mixture fraction [-]

,

Enthalpy, mean enthalpy [J/kg] Radiation intensity [W/m²] Mass flow-rate [kg/s] ,

Number of response surfaces from the sub-models, number of Simulations Refractive index [-] Weighted signal (input for activation function) Output of a neuron Probability density function Root mean square error based on cross-validation Root mean square error

,

,

Direction of the radiation, direction of the scattered radiation, deviation of the temperature at position

,

Temperature [K, °C], time [s, min]

,

Velocity [m/s], velocity fluctuation [m/s] Velocity vector for x, y and z direction [m/s] Weighting factor

,

Coordinates

,

Species mass fraction [-], input signal for a neuron

,

,

Elemental mass fraction [-], response surface of a sub-model, resulting response surface using the sub-models and their weighting factors

Greek letters Turbulent viscosity [kg/(s*m)] Density [kg/m³] ,

,

Stefan-Boltzmann constant (5.670373*10-8 [W/(m²*K4)]), scattering coefficient [1/s], model parameter for the flamelet approach/turbulent

Schmidt number (0.85) , , ,

Phase function, instantaneous value of a scalar, average value of a scalar, angle of the baffles [°] Scalar dissipation [1/s]

,

Source term [J/(m³*s)] or [kg/(m³*s)], solid angle [rad]

Subscript , ,

Control variable Component k in the mixture Fuel Oxidant Ensemble/resulting response surface

Abbreviations CFD

Computational fluid dynamics

DOM

Discrete ordinates model

GA

Genetic algorithm

MOGA

Multi-objective genetic algorithm

NN

Neural network

PDF

Probability density function

RSM

Response surface model

RTE

Radiative transfer equation

SFM

Steady laminar flamelet model

WSGG

Weighted sum of grey gases model

Acknowledgments This work was financially supported by the Austrian Research Promotion Agency (FFG), “Virtuelle Bauteilprüfung mittels gekoppelter CFD/FEM-Brandsimulation” (project 857075, eCall 6846234).

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List of table captions Table 1

Material properties of the walls, the mounting frame and the test specimen for the numerical simulation.

Table 2

Time step sizes and number of time steps for the transient simulation.

Table 3

Temperature range, temperature boundary condition and considered time section for the steady-state simulation.

Table 4

Input parameters and their values for the CFD simulations (values in red represent the standard configuration).

Table 5

Best sub-models their weighting factors and settings for the resulting response surface created by the genetic algorithm (for T11 and 300 °C wall temperature).

Table 6

Optimized baffle angle and fuel input proposed by the case studies, NN and GA (used values in red).

Table 7

Optimized operating conditions for the transient simulation.

Table 1 Material properties of the walls, the mounting frame and the test specimen for the numerical simulation.

Density [kg/m³] Specific heat capacity [J/(kg*K)] Thermal conductivity [W/(m*K)]

Walls

Mounting frame

Thermocouple

Test specimen

450 733 0.2

700 800 0.2

1000 800 10

2500 700 0.43

Table 2 Time step sizes and number of time steps for the transient simulation. Time [min]

Time step size [s]

Number of time steps

0…1 1…2 2 … 35

1 2 5

60 30 396

Table 3 Temperature range, temperature boundary condition and considered time section for the steady-state simulation. Part

Temperature range of the inner wall [°C]

Boundary condition at the inner wall [°C]

Time [s]

1 2 3

25 … 200 200 … 400 400 … end

100 300 500

0 … 18 18 … 60 60 … 2100

Table 4 Input parameters and their values for the CFD simulations (values in red represent the standard configuration).

Fuel input [kW] Angle of the baffles [°] Furnace depth [m]

Parameter

Range of values

Number of CFD simulations

, , d

120, 160, 200, 240, 280 15, 30, 45 0.85, 1.05, 1.25, 1.45, 1.85

75 27 15

Table 5 Best sub-models their weighting factors and settings for the resulting response surface created by the genetic algorithm (for T11 and 300 °C wall temperature).

Sub-model

Weighting factor

Setting

Support vector regression

0.049601

Linear

Kriging

0.47212

Gaussian; Anisotropic; Full quadratic

Support vector regression

0.032714

Polynomial

Linear basis function

0.44557

Full quadratic; Inverse quadratic

RMSE [K]

PRESSRMSE [K]

0.7737

6.7571

Table 6 Optimized baffle angle and fuel input proposed by the case studies, NN and GA (used values in red). Angle of the baffles [°] (bottom/top) Wall temperature [°C]

Case studies

NN

GA

100 300 500

45/15 45/15 45/15

49.8/13.4 49.5/14.6 49.8/13.4

49.9/12.2 49.4/10.3 49.9/12.2

Fuel input of the bottom and top burners [kW] (bottom/top) Wall temperature [°C]

Case studies

NN

GA

100 280/120 248/124 300 120/120 121/126 500 240/120 252/121 and 121/121* * Two operating conditions were suggested by the algorithms

294/120 120/120 247/121 and 120/120*

Table 7 Optimized operating conditions for the transient simulation. Inner wall temperature [°C]

Time [s]

Baffle angle [°] (bottom/top)

Fuel input [%] (bottom/top)

Furnace depth [m] (bottom/top)

100 300 500

0 … 18 18 … 60 60 … 2100

45/15 45/15 45/15

71/29 50/50 67/33

1.85 1.85 1.85