Jimin Wang*,Shen Lan,Tao Chen,Wenke Li,Huaqiang Chu

School of Energy and Environment,Anhui University of Technology,Ma'anshan 243002,China

Keywords:Aluminum holding furnace linings Reaction engineering Transport processes Numerical simulation Combination optimization Simulated annealing

ABSTRACT To reduce heat loss and save cost,a combination decision model of reverb aluminum holding furnace linings in aluminum casting industry was established based on economic thickness method,and was resolved using simulated annealing.Meanwhile,a three-dimensional mathematical model of aluminum holding furnace linings was developed and integrated with user-defined heat load distribution regime model.The optimal combination was as follows:side wall with 80 mm alumino-silicate fiber felts,232 mm diatomite brick and 116 mm chamotte brick；top wall with 50 mm clay castables,110 mm alumino-silicate fiber felts and 200 mm refractory concrete；and bottom wallwith 232 mm high-alumina brick,60 mmclay castables and 68 mmdiatomite brick.Lining temperature from high to low was successively bottom wall,side wall,and top wall.Lining temperature gradient in increasing order of magnitude was refractory layer and insulation layer.It was indicated that the results of combination optimization of aluminum holding furnace linings were valid and feasible,and its thermo-physical mechanism and cost characteristics were reasonably revealed.

1.Introduction

The present production of original non-ferrous metals barely meets national economic construction.Therefore,it is very important that non-ferrous scrap metals be effectively recovered.Up to now,the yields of secondary non-ferrous metals have already increased to 10 million tons,which is nearly three times in the past ten years.Secondary aluminum is very close to secondary non-ferrous metals processing industry.Due to energy shortages and global competition,the ultimate goal of aluminum casting industry is to reduce energy consumption and improve product quality.Metallurgical furnaces are essential equipments of secondary aluminum industry,which involve melting furnaces,holding furnaces,treating furnaces,and annealing furnaces.Aluminum holding furnaces play an important role in aluminum industry.It is a remarkable fact that a large amount of energy may be consumed in aluminum holding furnaces,which accounts for about 10%of total energy consumption during aluminum processing procedure.The linings are an important element of the furnace structure,which to significant degree depend on the cost,the life and the energy requirement of the equipment.Energy-saving of aluminum holding furnaces is closely related to refractory manufacturing technology,furnace structure design and kiln architecture construction.

Zhang et al.[1]discussed the slag skull forms and comes off in blast furnace at different temperatures,and revealed the in fluences of cooling plate with various water mass flows on furnace linings temperature,which may guide lining structure design.Yao et al.[2]established a mathematical model of temperature field of blast nozzle and throat of converter linings in different operating periods,and the reason that caused blast nozzle and throat being easily damaged was pointed out and expounded in detail.Zhou et al.[3]established a 3D comprehensive computational fluid dynamics(CFD)model to specifically simulate the blast furnace hearth which included both the hot metal flow and the conjugate heat transfer through the refractories.Yeh et al.[4]focused specifically on the effects of the gas temperature,the geometric thickness of the cooling stave,the slag layer thickness and the material and diameter of the sensor bar by a conjugate heat transfer model,including the copper stave and sensor bar.Regrettably,the effects of lining thickness on thermo-physical mechanism and cost characteristics were not evaluated in these literatures.To reduce heat loss,Wang et al.[5]conducted heat transfer analysis for different furnace lining structures of steel rolling furnaces and tunnel kilns.Unfortunately,heat storage loss and refractory investment were neglected,and only continuous working furnaces were considered.Wang et al.[6]carried out combination optimization and numerical simulation of aluminum melting furnace linings.It was noted that the used algorithm during optimization process is only valid in small data set and the results were a little arbitrary.

Moreover,simulated annealing(SA)[7]is a generic probabilistic metaheuristic for global optimization problem of locating a good approximation to the global optimum of a given function in a large search space.It is often used when the search space is discrete(e.g.,all tours thatvisita given setofcities).For certain problems,simulated annealing may be more efficient than exhaustive enumeration provided that the goal is merely to find an acceptably good solution in a fixed amount of time rather than the best possible solution.It has been widely used for the optimization of various fields such as mine planning[8],pipeline network design[9],cognitive radio system[10],multi-manned assembly line balancing[11],tube hydro-forming process[12],photovoltaic parameter identification[13],multiple-layer micro-perforated panels[14],and distillation column with intermediate heat exchangers[15].

In this paper,a combination optimization model of aluminum holding furnace linings was established based on economic thickness method,and implemented by simulated annealing.According to numerical heat transfer and computational fluid dynamics,a three-dimensional mathematical model of aluminum holding furnace linings was developed and integrated with user-defined heat load distribution regime model.The optimal lining scheme and its temperature distribution law were obtained using a CFD-SA combined method.It provided a combination solution strategy of furnace linings,thermo-physical mechanism and cost characteristics for developing new refractories integrated both with advantages of high-temperature protection and energy saving.

2.Combination Optimization of Holding Furnace Linings

2.1.Refractory selection principle

Traditionally,furnace lining thickness is determined by three calculation approaches:maximum heat loss,maximum outer wall temperature and economical thickness.The first two ideas involve heat loss and wall temperature,but do not consider construction cost and heat price.Therefore,the thickness is not reasonable.In the present work,economic thickness method regards heat loss expense and investment expense of furnace linings,which may obtain actual thickness.As is well-known,if furnace linings are thicker,there is little heat loss and more for both heat storage loss and investment expense.Thus,there is an optimum combination and best thickness for furnace linings.According to the geometry features of aluminum holding furnaces,the following hypotheses for heat transfer calculation of furnace linings are made:

(1)Furnace linings are assumed to be one dimensional static heat transfer.That is to say,heat flux does not vary with time and heat flux merely transfers along normal direction of isothermal surface.

(2)Thermal conductivity of furnace linings is supposed to be constant and is determined by average material temperature.

(3)There is good contact between the materials and they have the same temperature at the interface.

Liquid aluminum or magnesium steam inside aluminum holding furnaces should not permeate the refractories,which also need to resist abrasion and thermalshock.Generally,high-alumina materials are suitable forthe refractories in contactwith liquid aluminum.Forhigh purity aluminum alloy,mullite brick or corundum brick is employed based on acidic slag.Chamotte brick,clay castables or plastic refractory is used in other parts of the furnaces.The top of the furnace with refractory concrete is employed to improve its tightness and integrity.

The rules of furnace lining are as follows[16–20]:

(1)For fire brick with mortar joint or diatomite brick,the horizontal and vertical sizes are multiples of 116 mm and 68 mm,respectively.

(2)Side wall is composed of light and heavy refractories,whose thickness is 40–200 mm and 200–300 mm,respectively.

(3)The refractories oftop wallare the same as thatofside wall.Nevertheless,the thickness of light refractories is 50–150 mm,and 200–250 mm for heavy refractories.Based on design principle offurnace lining and initiallining combination,the refractories are selected as follows:

(1)Side wall:chamotte brick(A),diatomite brick(B),aluminosilicate fiber felts(C),and clay castables(D)

(2)Top wall:refractory concrete(E),clay castables,and aluminosilicate fiber felts

(3)Bottom wall:high-alumina brick(F),clay castables,diatomite brick,and high-alumina castables(G).

2.2.Lining optimization model with simulated annealing

According to heat transfer analysis of three-layer slab,the optimum economic thickness is determined by combination optimization as follows:

where S is the thickness set of refractories,N is the refractories total number,i,p,m and k are the refractory numbers,and j,q,n and l are the thickness numbers of the i th,p th,m th and k th refractories,respectively.T0is the inner wall temperature,Tfis the environment temperature, π(i), π(p), π(m)and π(k)are the thickness of total number of the i th,p th,m th and k th refractories.ek,λk,ck,Tkand ρkare the construction investment,the thermal conductivity,the heat capacity,the temperature and the density of the k th refractory,respectively.τ is the working time,f is the work period,and the working system is theoretically classified into three categories,i.e.,40 weeks,6 days and 16 h.In the present work,16 hour working system is adopted.η is the heat price,1.58 × 10-7CNY·J-1.

In the present work,furnace lining economic thickness belongs to the NP(nondeterministic polynomial time)-complete class of problems in combinatorial optimization.For these problems,there is a very effective practicalalgorithmcalled simulated annealing.The name and inspiration come from annealing in metallurgy,a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects.Both are attributes of the material that depend on its thermodynamic free energy.

In contrast to Monte Carlo learning only one weight or bias is changed at a learning cycle.This change is accepted or rejected.One of the advantages of simulated annealing is that learning does not get stuck in local minima and readily connects to highly sophisticated simulation codes,but SA is overkill for problems where there are few local minima and repeatedly annealing with a very slow Metropolis schedule,especially if the cost function is expensive to compute.

Considering furnace linings with three layers from the thickness set of refractories,in this paper,the approach is based on a coding procedure that makes use of an integer number series to represent and manipulate different thickness.Two integers Nrand Nsare used to denote different refractories,respectively.In applying a SA algorithm,each new neighboring structure needs to be generated in a random way.Due to the proposed approach using two integers Nrand Nsto represent thickness variables of the problem,the generation of a neighboring solution can be simply achieved by random number generator.When a thickness code has been generated,that is the configuration of furnace linings is given,the second step is to optimize the parameters to obtain a minimum total cost.In this study,the cost includes construction investment and heat loss expense.It should be pointed out that the installation of furnace linings must meet the requirement of the maximum allowable service temperature.

The simulated annealing algorithm takes random walks through the problem space,looking for points with low energies.In these random walks,the probability of taking a step is determined by the Metropolis algorithm as follows:

where ΔE is the energy difference and T is the current temperature.

The temperature T is initially set to a high value and a random walk is carried outatthattemperature.Then the temperature is lowered very slightly according to a cooling schedule.In this paper,the schedule that uses both the time of system and energy scales is known as constant thermodynamic speed annealing.The cooling rate is determined as follows:

where σEis the standard deviation of the energies,?τis the estimate of relaxation time,v is the thermodynamic speed and T is the current temperature.

Ideally,the system would like to reach equilibrium at each temperature.This implies that the ensemble is sufficiently close to thermal equilibrium.Therefore,the temperature can be further lowered.The criterion is to test for each time whether

and to lower the temperature if the condition is satis fied.

c is the preset constant,σEis the standard deviation of the energies,N is the ensemble size and ＜＜E＞＞ is the positive fluctuation of the ensemble average energy.

The stop temperature Tstopis usually set adaptively by specifying a number of steps Nstopsuch that if the energy does not change in the last Nstopsteps,then it is time to stop the algorithm.

In this paper,using an ensemble means sharing search among several random walkers and picking the best result obtained at the end of the runs,furnace lining expense problem should be minimized.Behaviorally constant application data includes maximum allowable service temperature,lining thickness,refractory type,walker's number,preset constant,max steps,acceptance ratio,thermodynamic speed,last time steps,and relative error.The flowchart of aluminum holding furnace linings based on simulation annealing is shown in Fig.1.

A random walk of duration τ performed by N independent random walkers is considered as defined as:

Fig.1.Flowchart of aluminum holding furnace linings based on simulation annealing.

2.3.Optimization result analysis

As shown in Fig.2,the algorithm reaches to the stop temperature about 200 iteration numbers,and the best-so-far(BSF)energy remains unchanged,which is 37950,27164 and 2.0205×105for top wall,side wall,and bottom wall,respectively.The optimum results are obtained using simulation annealing as listed in Table 1.

From Table 1,the total expense before optimization is 461941CNY ·m-2,yet 268803CNY ·m-2after optimization.Thus,it saves 193138CNY·m-2every year,which indicates that combination optimization of aluminum holding furnace linings based on simulated annealing is effective.The total expense is decreased by 69.23%,70.04%,and 15.24%for side wall,top wall and bottom wall,respectively.Owing to fixed inner wall temperature,the higher the wall temperature is,more heat loss would be.That is to say,the total expense of continuous operating furnaces is usually heat-loss-based expense.

Fig.2.Optimizing procedure of holding furnace linings.

Table 1 Optimizing results of holding furnace linings

Expense stack bars of aluminum holding furnace linings are shown in Fig.3.For any wall,it is apparently seen from Fig.3 that the construction expense is the lowest expense and it may be trivial.After optimization,the heat storage loss expense drops 83.47%,77.16%and 12.79%for side wall,top wall and bottom wall,respectively.Among them,the heat storage loss expense of second layer decreases to 90.47%,76.22%and 12.5%for side wall,top wall and bottom wall,respectively.In addition,the heat loss expense increases 53.64%,decreases 34.64%and keeps constantforside wall,top walland bottomwall,respectively.Therefore,the total expense is determined by the heat storage loss for periodic working furnaces.These results are in agreement with Ref.[21].

To make a comparison between the proposed algorithmic approach and an exhaustive search method with small data set as that suggested by Wang[6],a synthesis problem considering aluminum melting furnace linings is solved using the approach.To achieve this,the design is modified accordingly by adopting the same design considerations,the main cost data and the same utility conditions as what was done by Wang.Both the optima obtained by using SA approach and the results provided by Wang show that their cost is identical approximately and refractory combination also is similar,as shown in Fig.4,which illustrates the reliability of the developed algorithmic approach.

Fig.3.Expense stack bars of holding furnace linings.

3.Numerical Simulation of Holding Furnace Linings

3.1.Physical model

Aluminumholding furnaces with three-layerrefractories are usually rectangular as shown in Fig.5.Two burners and the flue locate on the same side,and the furnace is divided into combustion space and liquid aluminum below.It has the pool depth of 650 mm,burners with horizontal distance between burners of 2050 mm are 405 mm away from the bath surface,and the height of the flue is 655 mm.In the present study,some assumptions are made to simplify the physical model:

(1)Liquid aluminum can be assumed to be stagnant,i.e.,natural convection(conv)inside melt bath may not be considered.Chemical reactions and mass transfer above the melt surface can be also negligible.Radiation(rad)and convection heat transfer at the melt interface are only taken into account.

(2)During holding phase,melt temperature is around 1017 K.Besides,total time is assumed to be 2.5 h.

(3)When liquid aluminum temperature exceeds 1017 K,burners begin to low fire.On the contrary,burners transform low fire into high fire.In addition,the amount of heat supplied of high fire is ten times that of low fire.

Fig.4.Comparison configurations of furnace linings with 40-week working system between the algorithm(a)and literature(b).

(4)The emissivity ofaluminum surface and inner walls is 0.33 and 0.8,respectively.

3.2.Mathematical model

The major function of holding furnaces is to make liquid aluminum be alloyed within reasonable temperature range.Alloying temperature of melt usually is kept at about 1017 K.During production process,outer walls emit heat to the environment by natural convection and radiation heattransfer.The change ofheatload is determined by heatload distribution regime model as described as follows:

where φ1is the heat load of heating phase,φ2is the heat load at high fire,φ3is the heat load at low fire,F(τ)is the fluctuation factor of heat load,τ1is the heating time,τ2is the holding time and τ3is the standing time.

In this work,the fuel is mixed and burnt with air inside holding furnaces,which provides chemical reaction heat to make up for heat loss.For combustion space,the unsteady three-dimensional conservation equations can be expressed as[22]

Where,v is the velocity vectors,Γφis the diffusion coefficient,ρ is the density,and Sφis the source term.While scalar φ equals 1,the above general equation was transformed into continuity equation.Momentum equations,energy equation,κ-ε turbulence model,and non-premixed combustion model were derived respectively when φ is velocity vectors,temperature,turbulent kinetic energy,turbulent dissipation rate and mixture fraction.

Fig.5.Geometry model of holding furnace with linings.

For thermal process inside holding furnaces,radiation heat transfer plays the dominantrole.Discrete transfer radiation model(DTRM)is chosen in most of the simulations in this study.This model is derived based on the assumptions thatscattering is relatively smallin comparison to absorption and emission,and the radiating gases are gray.The radiation transfer equation used by the DTRM is

where I is the radiation intensity,α is the absorption coefficient and σ is the Stefan–Boltzmann constant.In this model,the absorption coefficient α is calculated by the Weighted Sum of Gray Gases Model(WSGGM).

For metal pool and furnace linings according to the assumption(1),since there is no fluid flow inside the metalpool and the density in solid remains constant,the energy equation is reduced to an unsteady heat conduction equation without any source terms,

where λ is the thermal conductivity.

According to the assumption(1),in the current model the melt bath is regarded as a conducting solid.The gas–solid interface coupled equation is used to describe heat transfer through the bath surface.

whereλis the thermalconductivity of the solid,h is the localconvective heat transfer coefficient,Tfis the fluid temperature,Twis the temperature of the wall,ε is the solid emissivity,and σ is the Stefan–Boltzmann constant.

3.3.Initial condition and boundary condition

Generally,it is supposed that the tapping aluminum temperature from aluminum melting furnaces is about 1013 K and the pouring temperature is set approximately 1023 K.

During holding phase,aluminum temperature is kept constant by adjusting heatload and burners alternately fire high and low.Moreover,total time should meet alloying requirements.In the paper,the total time is about 2.5 h.The fuel and air inlet boundaries are as follows:

where Mairis air mass flow,and Ufuelis natural gas velocity.

Outerwallheattransfer may be assumed to be the third type boundary condition and total heat transfer coefficient is as follows:

where Δt is the temperature difference.For vertical and horizontal walls,C is 2.56 and 3.26,respectively,due to different natural convection modes.

Due to a solid base,equivalent heat transfer coefficient of bottom wall is as follows:

where

and

where λiis the thermal conductivity of the i th refractory,and δiis the thickness ofthe i th refractory.Considering the in fluence ofthe thickness of side wall,k is 0.96,1,and 1.08–1.1 respectively when the equivalent thickness of the refractories(B)is D/4,D/6,and D/8.D is the diameterof bottom wall or shortedge length ofrectangular furnaces.For round furnaces,square furnaces and rectangular furnaces,φ is 4,4.4 and 3.73,respectively.In the present study,k and φ are 1.08 and 3.73,respectively.

3.4.Implementation and CFD validation

ANSYS FLUENT software was employed to resolve lining temperature field.The hybrid programming method of FULENT UDF(userdefined function)and FLUENT Scheme was used to compute aluminum temperature and heat transfer coefficient.Save the results and exit FLUENT when the total time exceeds 2.5 h.The ANSYS FLUENT solution procedure is shown in Fig.6 and its explanations are given below[23]:

fuel_proc:Adjust heat load based on liquid aluminum temperature,i.e.,when aluminumtemperature exceeds 1017 K,burners fire high,and on the contrary,burners transform high fire into low fire.Therefore,the alloying process of liquid aluminum is accomplished.al_func:Calculate liquid aluminum temperature using volume-average method.After the total time reaches to process requirements,the results are saved and FLUENT software is exit.heat_transfer_coeff:Get total heat transfer coefficients of the outer walls of the furnace.

The multi-block grid is used in this simulation since there is a great difference between the sizes of the burner and furnace.Considering computing resource and accuracy of results,mesh independence test and time step independence test have been carried out.When the relative error of furnace temperature is less than 5%,the grid spacing and time step have no in fluence on the simulation results.Thus,226888 elements and 1.5 s are chosen for the computation.To validate reliability and accuracy of the model,heat balance test and numerical simulation have been used for 50 ton aluminum holding furnace in a company.Wall temperatures measured by infrared thermometer are compared with computational results,and the average value of all the measurement points is calculated accordingly,as shown in Table 2.Table 2 illustrates that numerical results are basically in good agreement with the experimental results,and the difference is within 8%.In most of these points,the CFD results match well with the measured data in both trends and magnitudes.The source of the errors may be due to the model hypotheses in the numerical calculation.However,it shows that the agreements are acceptable in both magnitudes and trends.Thus,the computational models were proved to be reliable and accurate.

3.5.Simulation process analysis of holding furnace linings

As is shown in Fig.7(a),the outer wall temperatures increase parabolically with the time.The outer wall temperature from high to low is successively bottom wall,side wall and top wall.Due to a solid base,the equivalent heat transfer coefficient using Eq.(15)is a little low.Therefore,the temperature of bottom wall is higher than other furnace walls.In addition,two burners and the flue are installed on side wall,where high-temperatureflue gas transfers heat to the refractories.As a result,the outer wall temperature of side wall is also higher than that of top wall.It can be seen from Fig.7(a)that the outer wall temperature of side wall stepwise increases with the time.It is found from Fig.7(b)thatduring the holding process the aluminum temperature remains unchanged and the curve looks like a horizontal line because alloying temperature is assumed to be 1017 K according to Eq.(11).Besides,the furnace temperature also keeps at about 1020 K.From Fig.7,the interval time between high fire and low fire is about 200 s.

Fig.6.ANSYS FLUENT solution procedure of holding furnace linings.

Table 2 Comparisons of simulation results and test values for holding furnace linings

According to the assumption(1),the significant heat transfer mechanism is thermal conduction in furnace linings,and thus the temperature in longitudinal-section and cross-section of the furnace linings shows hierarchical distribution in general,as shown in Fig.8(a).The inner wall temperature is further higher than the outer wall temperature except from that of the bottom wall.From Fig.8(b),there are different temperature gradients along the thickness direction due to different thermal conductivities of the refractories.For top wall,the temperature gradient from inner wall to outer wall is 1.15 K·mm-1,1.42 K·mm-1,and 2.43 K·mm-1.It can be also derived from Fig.8(b)that the gradient of side wall is successively 0.47 K·mm-1,1.63 K·mm-1and 2.97 K·mm-1.In terms of bottom wall,the temperature gradient along the thickness direction in order is 0.048 K·mm-1,1.03 K·mm-1,and 2.15 K·mm-1.Regardless of walls,it may be found that the order of the temperature descent rate is refractory layer( first layer)and insulation layer(second layer and third layer).The temperature gradient of different parts from high to low is top wall,side wall and bottom wall.

Temperature distribution of different parts inside aluminum holding furnace linings is illustrated in Fig.9.There is high temperature near burners or flue,and thus the refractories with high working temperature are employed to resist temperature wear.It may be observed that the lining temperature is related to flue gas,which means that the high-temperature zone of linings is in accordance with the high-velocity zone of flue gas.The lining temperature of furnace corners is a little low.The lining temperature of bottom wall is successively 1014.9 K,899.98 K and 824.05 K.For side wall,the lining temperature from high to low is 982.28 K,796.72 K and 522.88 K.Moreover,in the case of top wall,the lining temperature from the inner layer to outer layer is 945.99 K,907.81 K and 650.66 K.In summary,the lining temperature from high to low is successively bottom wall,side wall,and top wall.

Fig.7.Thermal characteristics of production process of aluminum holding furnace versus time:(a)outer wall temperature；(b)aluminum temperature and furnace temperature.

Fig.8.Temperature distribution of aluminum melting furnace linings(a)cross section at z=-655 mm,z=-405 mm,y=1025,and y=-1025；(b)temperature gradient along refractory thickness direction.

4.Conclusions Remark and Future Work

Based on economic thickness method,combination optimization of different parts of holding furnace linings is performed by simulated annealing.The temperature field of holding furnace linings is achieved using a three-dimensional mathematical model integrated with user-defined model.This paper has presented an application of the parameter design of the CFD-SA integrated intelligence ideas in combination optimization of aluminum holding furnace linings.It not only enriches existing knowledge of thermal behaviors of metallurgical reaction engineering,but also provides a novel optimization strategy for other complicated systems such as chemical engineering.The corresponding results can also guide or improve industrial production practices.It demonstrates that a novel study framework with mathematical model,numerical simulation,intelligent integration,comprehensive decision,virtual reality and visualization may be successfully developed to explore thermo-physical mechanism and optimization strategy of process engineering.

Generally,the refractory damage is related to complex erosion mechanisms such as chemical erosion,thermal stress,and mechanical wear.To prolong the campaign life of the furnaces,the stress inside the furnaces should be as low as possible.Due to complicated structure and bad operation,it is impossible that the stress is achieved by instrument measurement.From the thermal–structural simulations,thermal stress mechanism should be incorporated into the mathematical model,especially for local corrosion at slag–metal interface with Marangoni effects,to critically discuss their implications on the design of the lining structure and clearly explain how it can prolong campaign life,which is currently under way.

Nomenclature

B equivalent thickness,mm

c preset constant

ckheat capacity,J·kg-1·K-1

D diameter or short edge length,m

dithickness of the i th refractories,mm

E energy

ΔE energy difference

＜＜E＞＞ average energy fluctuation

e construction investment,CNY·m-3

F fluctuation factor

f work period,s

h convective heat transfer coefficient,W·m-2·K-1

I radiant intensity,W·m-2·K-4

k modified coefficient

Mairair mass flow,kg·s-1

mod modulo

N refractory total number,ensemble size

Nr,Nsinteger number of refractories

p the sum of partial pressures of absorbing gases,N·m-2

q heat flux,w·m-2

RND randomize

S refractories thickness,mm

s radiation path length

Sφsource term

T temperature,K

T0inner wall temperature,K

Tffluid temperature,K/environment temperature,K

Twwall temperature,K

Δt temperature difference,K

Ufuelfuel velocity,m·s-1

v velocity vector/thermodynamic speed,m·s-1

x equivalent physical properties of refractories(various unit)

xiphysical properties of the i th refractory(various unit)

α absorption coefficient

αε,iemissivity weighting factors of the i th fictitious gray gas

Γφdiffuse coefficient

δirefractory thickness,mm

ε solid emissivity,estimate relaxation time,s

η heat price,CNY·J-1

θ1,θ2,θwinterface temperature,K

κiabsorption coefficient of the i th gray gas

λ,λkthermal conductivity,W·m-1·K-1

π(i),π(p),π(m),π(k)thicknesstotalnumber ofthe i th,p th,m th and k th refractory

ρ,ρkdensity,kg·m-3

σ Stefan-Boltzmann constant,W·m-2·K-4

σEenergies standard deviation

τ duration time/working time,s

τ1,τ2,τ3heating time,s；holding time,s；standing time,s

φ scalar(various unit)

φ1,φ2,φ3heat load(various unit)

Subscripts

i,p,m,k refractory number

j,q,n,l thickness number

Fig.9.Temperature distribution for different parts of aluminum holding furnace linings(a)bottom wall；(b)side wall；(c)top wall.