heat transfer direct firing furnace

Modeling and Parametric Studies of Heat Transfer in a Direct-Fired Continuous Reheating Furnace

K.S. CHAPMAN, S. RAMADHYANI, and R. VISKANTA

A mathematical system model of a direct-fired continuous reheating furnace has been developed. The furnace is modeled as several well-stirred gas zones with one-dimensional (l-D) heat conduction in the refractory walls and two-dimensional heat transfer in the load. The load travels either parallel or counterflow to the combustion gases which enter the furnace throughout its length. The convective heat-transfer rate to the load and refractory surfaces is calculated using existing correlations from the literature. Radiative heat exchange within the furnace is calculated using Hottel's zone method by considering the radiant energy exchange between the load, the combustion gases, and the refractories. The nongray characteristics of the combustion gases are considered by using a four-gray gas model to treat the mixture as a radiatively participating medium. An extensive parametric investigation has been completed to determine the furnace design and operating characteristics that lead to optimum fumace efficiency. The parametric investigations included in this paper study the effects of the load and refractory emissivities and the height of the combustion space on the thermal performance of the furnace.

I. INTRODUCTION

THIS paper presents a mathematical system model of a direct-fired continuous reheating furnace. The model was developed to identify the design and operating parameters that significantly affect furnace performance. The details of the model are provided, followed by the results from a parametric investigation of some of the more important furnace parameters. The paper culminates with a summary of the major conclusions found during the study. Since the load moves steadily and continuously through the furnace, calculations are performed on a steady-state basis, but many details of the procedure are similar to those of a previously developed transient model for a batch reheating furnace. ~l~

Understanding of combustion kinetics, chemically re- acting flow, turbulent transport processes, and heat transfer in industrial and commercial furnaces and equipment is incomplete, even though natural gas has been used as a fuel for many decades. In the past, design of direct-fired natural gas fumaces and combustion systems has been mainly empirical. The complex physical and chemical processes involved in flames have, until more recently, eluded detailed analytical description. During the last two decades, the challenge of increasing productivity, improving thermal performance, and reducing combustion- generated pollution have attracted considerable attention from researchers and designers, leading to significant progress in science-based engineering design, t2,3,41

Phenomenological mathematical models, based on fundamental principles of fluid mechanics, thermodynamics, and heat transfer, have proven useful as tools in developing a basic understanding of heat transfer in furnaces. However, the computer resources required by

K.S. CHAPMAN, Assistant Professor, is with the Department of Mechanical Engineering, Kansas State University, Manhattan, KS 66506. S. RAMADHYANI, Associate Professor, and R. VISKANTA, Professor, are with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907.

Manuscript submitted June 7, 1990.

these fundamental models make them prohibitively ex- pensive for the furnace designer. System models, which are based on the phenomenological models but include substantial simplifying assumptions, provide design and operating information relatively quickly and inexpen- sively, enabling the designer to explore a wide range of options for improving system performance. Heating processes, including product reheat and heat treatment, in furnaces such as that shown in Figure 1 can benefit from the application of system modeling techniques. System models can be used to predict the thermal efficiency and fuel consumption or the design of equipment and con- struction materials required to achieve an optimum thermal efficiency and heating rate. This information is valuable in estimating the operating and capital costs of a new heating system. Additionally, system models en- able the quality of the heated product to be assessed by predicting both its temperature uniformity and temperature history as it is heated.

Radiative heat transfer from the flame and combustion products and from the refractory surfaces to the load is usually the dominant mode of heat transfer in high- temperature direct-fired natural gas combustion systems. Since radiative heat transfer can represent a large per- centage of the energy transferred to the load, there is a need to develop an understanding and modeling capa- bility to predict heat transfer in multidimensional combustion systems. Such systems contain selectively absorbing and emitting gases, such as CO2, H20, CO, and CH4, and possibly soot and dust particles. There- fore, predictions based on the assumption of a gray participating medium have limited accuracy, and some way of accounting for nongray effects is usually neces- sary. [5'6] The problem is complicated by the need to es- timate the temperature and concentration of the emitting species (gaseous and particulate, if any). Rigorous treatment of the problem is cumbersome and requires significant computer resources. The compromise is development of relatively simple radiative heat-transfer models, followed by a test of their adequacy against either

METALLURGICAL TRANSACTIONS B VOLUME 22B, AUGUST 1991--513

experimental measurements or the predictions from so- phisticated analytical models.

Few system models have been developed that include detailed treatment of radiative heat transfer and heat transfer within the load and refractories. KohlgriJber tT~ developed a continuous reheating furnace model capable of determining detailed temperature profiles in the load. The treatment applied to the combustion products, however, was simplified by prescribing the gas temperature as a function of the distance through the furnace. Ra- diation heat transfer was calculated using a mean-beam length technique, with the radiative characteristics of the combustion gases modeled as gray. Tucker and Lorton Isl used the zone method to predict radiative heat transfer in a reheating furnace and investigated the effect of nongray combustion products. They calculated the gas temperature in each gas zone by conducting a zonal energy balance. They did not, however, accurately couple the temperature distribution in the load and refractories with heat transfer from the combustion gases. The interested reader is referred to the topical report by Viskanta and Ramadhyani t91 for a recent thorough review of furnace system models.

The present paper describes a system model that treats heat transfer in the load and refractories simultaneously with radiative and convective heat transfer in the combustion space. The spectral radiative characteristics of the combustion gases are taken into account through the use of a weighted sum-of-gray-gases model.

II. CONTINUOUS H E A T I N G FURNACE M O D E L

The mathematical models for the combustion gases, furnace walls, and load are developed in this section. The assumptions used to develop the model are listed first, followed by derivation of the model equations.

The well-stirred furnace model used to simulate the thermal performance of the furnace is shown in Figure 1. The main assumptions and features of the model are the following:

(a) the furnace consists of several gas zones along the axis of the furnace; (b) the gas compostition throughout each gas zone is uniform and equal to that of the combustion products without dissociation; (c) the gas temperature is calculated from the heat balance on each gas volume and includes the effect of the combustion products moving through the furnace; (d) the wall surface temperatures are calculated from one-

Flue +

Fu:/Air /Qa. ~j~

~ " 0 ................. 0 ......... ~" "5" .....................

Fig. 1 - - S c h e m a t i c of the heat exchange occurring within a direct- fired continuous furnace.

dimensional (I-D) heat conduction equations for each zone; (e) radiation heat exchange is calculated using Hottel and Cohen's zone method tl~ using a sum-of-gray-gases model; (f) the convective heat-transfer coefficient varies with combustion gas velocity and temperature; (g) the load is divided into several subzones within each gas zone to accurately model the axial temperature gra- dients in the load; and (h) the bottom of the load is assumed to be adiabatic, i.e., symmetric about the center plane.

A. Gas Energy Balance

Hottel and Cohen's zone method tl~ provides an ac- curate procedure for analyzing radiative heat transfer in combustion systems where significant changes in gas temperature and composition and in load temperature or emissivity are present. The gas volume and surface areas are subdivided into smaller zones which are assumed to be isothermal. Energy balances are then made on each gas and surface zone separately. In general, if there are M gas zones and N surface zones, there will be a total of M + N energy balance equations.

Neglecting energy stored in the gas, the gas energy balance is written as

ABe + Ha. i + ncp,i_ 1 = aLoss,i + Hcp.,(T_,p) [1]

where QLoss,i represents the energy tranferred from the combustion products to the load and refractories, and Hcp,,-1 accounts for the enthalpy transport due to the flow of combustion products from one gas zone to the next. Referring to Figure 2, which shows the zones used to model the continuous furnace,

L

QLOSS,,----- E [akflk(Zg ,` -- Zs,k)] k=l

[)-4 o,o:4 + u (GIG: Tg,i - Ts,y)

+ (c,sj r . . - s : a l r : . ) [21

The first term on the right side of Eq. [2] denotes convective heat transfer from the combustion gases to surface zone k. The second and third terms represent radiative

Hf,i. I Hf,~ HU+ I

i ~ ' i ' H ( T~il [/I I 2 ................. i-I f i+I ................. " ' - - J ~ HExH

*l : : :1 : l ILoadll:: r I

L Fig. 2 - - S c h e m a t i c showing the zones and control volumes used to model the continuous furnace.

514-- VOLUME 22B, AUGUST 1991 METALLURGICAL TRANSACTIONS B

by an empirical equation I131 based on measurements:

heat exchange between gas zone i and any other gas or surface zone j.

The enthalpy of the combustion products was calculated from

H~p = ~ Cp.i aT + Ah~i Ni [31 i "

The temperature-dependent molar specific heats were approximated by the functions cited in Van Wylen and Sonntag t121 and are valid for temperatures between 300 and 2500 K. The molar flow rates, Ni, appearing in the foregoing equation depend on the actual air-to-fuel ratio, the stoichiometry of the reaction, the fuel composition, and the molar flow rate of the fuel and were calculated by conducting an atom balance on the generalized combustion equation for a hydrocarbon fuel:

CxHy + YO 2 + 3.76YN2 "--> NcoCO + Nco:CO2

+ NH~oH20 + NN~N2 + No:O2 [4]

The convective heat-transfer coefficient for a surface exposed to the combustion gases was calculated

experimental

r . ~ 0 , 7 5 hDeq_f|mgO~q I ~ u u - ~ L A~g _1

[51

where C is 0.283, 0.175, or 0.142 for crown, end, or side firing, respectively.

The direct exchange areas, gisj and sisj, were calculated using the Monte Carlo statistical method, tll,14~ The method was used because it is computationally efficient for a small number of zones and yields good accuracy relative to the computational effort. The total exchange areas, GiSj and SiSj, were then calculated using the procedure suggested by Hottel and Sarofim. mJ When the weighted sum-of-gray-gases approach was used to eval- uate the radiation characteristics of the combustion gases, the directed-flux exchange areas were calculated from the total exchange areas by using a four-gray gas model: t5,61

4 4

S~S~ : 2 An,Ti(SiSi)n GiS~ : E Bn,Tj(rs,j ) (GiSj)n n = l n = l

[6]

B. Load and Refractory Heat-Transfer Model

Assuming 1-D heat conduction in the wall and the load, the heat conduction equation is written as

pcVL - [71 Oy Ox \ Ox /

where x is the unit normal into the surface exposed to the combustion gases, and y is the coordinate along the furnace axis. This equation is applied to the ftLmace walls by setting the advection term to zero. The boundary condition on the inside surface of the wall as well as on the load exposed to the combustion products is

OTi = q~ = QffA~ [8] - k i Ox inner surface

where the total (convective plus radiative) heat-transfer rate is given by

Qi aihi(Tg T, i) + 2 ' , cr(G~iT4,k > = -- __ SiGkZ4,i) k

~- 2 - - " " ' 4 4 _ ) 4 , tr(SjSiTsj S,SjT, i) [9] J

Heat transfer from the outside of the furnace wall to the ambient is by convection and radiation and is expressed a s

m k s i - -

' O X outer surface

= h i < L , i - rAMB)

+ eeo.(T4,i 4 -- T AMB) [10]

An adiabatic wall (bound~_y condition) can be obtained from Eq. [10] by setting hi = eR = 0.

C. Method of Solution

The combustion gas temperatures were calculated by solving the gas zone energy balance given by Eq. [2]. This was accomplished using the iterative Newton- Raphson method. ~aS~ The temperature distributions in the walls and load were determined by solving the energy equation, given by Eq. [7], which was discretized using the fully implicit control volume approach described by Patankar. tl6j The boundary heat flux in Eq. [8], which is the result of radiative and convective heat exchange with the combustion gases and radiative heat exchange with the other furnace interior surfaces, is nonlinear. Therefore, the heat flux was linearized using a first order Taylor series expansion of the heat flux about the surface temperature.J16]

The foregoing numerical method was used to calculate the temperature distribution in each furnace wall by ap- plying the model to each wall separately, assuming the only interaction between the walls was due to radiative heat exchange. Complete details of the solution method are provided by Chapman et al. t171

The individual submodels (combustion products, furnace walls, and load) were integrated into the furnace system model. Due to the thermophysical property de- pendence on the temperatures assumed for the walls and load, the calculation of the wall and load temperatures was an iterative process. The iterations were continued until the temperatures in all of the furnace walls (and the load) changed by less than a prespecifled error tolerance. The furnace model flow diagram is shown in Figure 3.

I l l . P A R A M E T R I C INVESTIGATION

There are several instances when additional insight can be obtained from a parametric study by transforming the model equations into suitable dimensionless forms. In the present study, the governing equations are highly nonlinear because (a) radiative heat transfer varies as the fourth power of temperature, (b) the radiative characteristics of the combustion gases are modeled as nongray, and (c) the thermophysical properties of the load, refractories, and combustion products are modeled as functions of temperature. Due to the nonlinearity of the

M E T A L L U R G I C A L T R A N S A C T I O N S B V O L U M E 2 2 B , A U G U S T 1991 - - 5 1 5

governing equations, the number of parameters cannot be significantly reduced. Therefore, additional insight cannot be obtained by making the equations dimensionless, and the governing equations are left in terms of physical parameters.

The important dimensionless groups that affect furnace performance were identified from dimensionless forms of the model equations. In view of the foregoing discussion, the procedure was simplified by treating the thermophysical properties as constants, i.e., independent of temperature. This treatment is justified since the sole purpose of this exercise is to identify the important dimensionless groups as opposed to presenting the governing equations and results of the parametric investigation in dimensionless form. groups were obtained:

Ai/AL

hTL,i,AL

&/kR > ) ) )

GkSi SiSj SLGk SiG~:

Gk==~'L' Gk:~' Gk=~L ' GkS;

The following dimensionless

XIdL T//Tf, i

, fid /& GSL o-T4.

i- b

dL/LL i,/fi

' R/O L na/I-b

VLdL/aL

Although there are several identifiable dimensionless groups, it is apparent that they are not all independent (i.e., the area ratios, Ai/AL, are related to the directed- flux exchange areas, the thermal conductivity ratio is related to the thermal diffusivity ratio, etc.). However, it is possible to identify a set of primitive variables that would be appropriate for a parametric investigation. These are shown in Table I along with the associated dimensionless groups. The primitive variables were selected so that the associated dimensionless parameters would be independent of each other. Variation of the load velocity, Vc, for instance, affects only the quantity VLdL/aL. If the load thermal diffusivity, aL, were chosen rather than the load velocity, then both, the quantity VLdL/aL and the ratio ag/aL, would be affected.

The parameters selected for inclusion in this report are the load and refractory emissivities and the furnace combustion height. Additional parametric studies are provided by Chapman et al.[17J The studies were conducted by varying each parameter from a baseline condition and observing the response of the furnace. In addition to the furnace efficiency, the variations of the load and refractory surface temperatures, the temperature of the combustion products, and the heat-transfer rate to the load were examined. The furnace efficiency was defined as

Heat transfer rate to the load

~7 = [111 Rate at which energy enters the furnace

The energy entering the furnace was entirely due to the fuel and did not take into account any air preheat, since the preheat was assumed to be from a furnace recuper-

I

I

- H

READ INPUTS I

SET UP GRIDS I

CALC. GAS TEMP. I

CALC. PROP. [ _ ~ (p, g, k, etc.)

CALC. BOUNDARY q SOURCE TERMS

GAS MODEL I

RADIATION MODEL ]

CONVECTION MODEL [

CALC. WALL & ~ ' t WALL MODEL I LOAD TEMP. DISTR.

CALC. ERROR I

OUTPUT DATA

UPDATE ] TEMP. ARRAY

Fig. 3--Flow diagram of the direct-fired continuous furnace model.

ator. The convective heat-transfer rate to the load was calculated by a correlation for a crown firing configuration, u2] and the load moved through the furnace counterflow to the combustion gases (from the right to the left in Figure 1). The baseline operating condition for these studies is shown in Table II. The fuel firing rate was reduced in the exit region of the furnace, as is typical in reheating furnaces, providing for a more uniform load temperature. However, for purposes of com- parison, one simulation was also performed in which the firing rate was uniform throughout the furnace.

Table I. Primitive Variables Appropriate for a Parametric Investigation

(Also Shown Are the Related Dimensionless Groups)

Primitive Dimensionless Variable Parameter

f~ h/h dL, holding VLdL constant dffLL

(pc)R, (pck UR/~L rha AF

Ha = f(rair,in) H,~/Hf eg, eL, furnace height, AL directed flux

exchange area ratios VL VLdL/aL

516--VOLUME 22B, AUGUST 1991 METALLURGICAL TRANSACTIONS B

Table II. Base Furnace Configuration Parameters Used for the Parametric Study

pc k Thickness Material (kJ/m3K) [ls] (W/mK) t18} (cm) ei lS]

Load iron 3.77(T-300) 112(1 - 0.001T + 5 • 10-TT 2) 15 0.3 Refractory red clay brick 253.9 1 + 0.008(T-475) 50 0.8

Gas Properties p~(T) 119] = 4.1 :,< 10-7T ~ k g / m s k(T) llgl = 0 .00028T ~ W / m K

Other

TAMB = 300 K

Furnace Parameters

Dimensions Length = 20 m Height = 1 m Width = 2 m

AF = 20

p(T)t191 = 348T -l k g / m 3 Nongray (four-gray gas model)

/~ = 10 W / m 2 K

Air preheat = 800 K Firing rate:

0 < y < 4 m, 25y k W / m y > 4 m , 1 0 0 k W

Fuel = CH4

A. Variation of the Load Emissivity

Figure 4 shows the change in the load surface temperature with varying load emissivity. The remaining furnace parameters were as stated in the base furnace configuration (Table II). As eL is increased from 0.1 (al- most reflective) to 1.0 (radiatively black), larger amounts of radiant energy are absorbed at the load surface, re- suiting in higher load surface temperatures. Since the additional heat transferred to the load is removed from the combustion gases, lower gas temperatures are expected. The decreasing trend in the gas temperature, however, is only observed at the end of the furnace where the load enters (the right side, at a y coordinate of 20 m), as shown in Figure 5. The combustion gas temperature-in the exit region of the furnace (the left side, at a y coordinate of 0 m) displays the opposite trend, increasing with an increase in the load emissivity. These opposite trends are due to a combination of two circumstances: (a) as the load travels through the furnace, its surface temperature increases; and (b) the firing rate is decreased in the exit region of the furnace (Table II).

Figure 5 shows that the trends described above be- come more prominent as the load emissivity is changed from a reflective to a black surface. When the load surface is black, a large quantity of heat is transferred from the combustion gases to the load in the entrance region of the furnace, since the load is cold relative to the temperature of the combustion gases. This causes the load surface temperature to rapidly approach the temperature of the combustion gases, as shown in Figures 4 and 5. Near the exit end of the furnace, since the load surface temperature is close to the temperature of the combustion gases, the heat-transfer rate from the gases to the load is small. The result is that the combustion gas temperature remains relatively high in the exit region of the furnace for a high emissivity load.

As eL is decreased, the load absorbs less energy in the entrance region of the furnace. The observable trends in the entrance region are higher gas temperatures and lower load surface temperatures, as shown in Figures 4 and 5. At the exit end of the furnace, since the load surface temperature is not as high as it was for the case of the black surface, the heat-transfer rate from the gases to the

- , ~ el. =1"0

o - - _ ._/ / - - 0 . r =:

o i ............... "".?.~.~'.~'-:-.:~:7--'.._':L.~... ~,'. O:a'"::::::::~.;,"~f~.~Uniform... ':::::'. ~ ' . , '~ - , . ' ~ F i r ing Rate

Di " of Load Mov " ' ' : ,~ .

o | . ~ . ~ . ~ .

0 5 10 15 80 y ( m )

Fig. 4 - - L o a d surface temperature as a function of the load emissivity and distance through the furnace.

M [..,

I - - ~ . . 0.1

~L__I, 0 ~ .~.

0 ~ ,~ t~ ~o y(m)

Fig. 5 - - C o m b u s t i o n gas temperature as a function of the load emissivity and distance through the furnace.


load is higher. This results in cooler gas temperatures relative to the case of the black surface and is com- pounded by the reduction in the fuel firing rate in the furnace exit region. Figure 6, which shows the heat flux on the load surface as a function of eL, confirms this argument.

The combustion gas and load surface temperatures for the case of uniform firing with eL = 0.3 are also shown in Figures 4 and 5. As expected, the load surface temperature resulting from the uniform firing rate is slightly higher in the furnace exit region relative to the nonuniform firing rate case. The gas temperature resulting from the uniform firing rate, shown in Figure 5, exhibits a small but definite decrease at the furnace exit. The decrease is not as pronounced as that for the nonuniform firing rate but is still attributed to a larger heat-transfer rate from the gases to the load which, in tum, results from the lower load emissivity. Consequently, the nonuniform firing rate emphasizes rather than causes the trends shown in Figures 4 through 6.

The furnace efficiency was found to increase rapidly for small increases in the load emissivity for near reflective surfaces. The increase in furnace efficiency as a function of increasing emissivity becomes less pronounced as the surface approaches a black surface. This trend is explained by referring to the load heat flux shown in Figure 6 and is due mainly to the increased heat flux in the entrance region of the furnace.

B. Variation of the Furnace Combustion Space

The height of the furnace combustion space was increased from 0.1 to 1.0 m. The remaining furnace parameters were as stated in the base furnace configuration (Table II).

Figure 7 shows the load surface temperature as a function of the combustion space height. As shown, the load surface temperature decreases as the height of the combustion space increases. This trend is due to the decrease in combustion gas temperature with the increase in the combustion height, shown in Figure 8. The temperature of the combustion gases decreases since the surface area of the refractory increases as the combustion height is increased. This increase in refractory surface area in-

~-. Combustion Height - - 0 . 1 m

- .............. 0.3 m

":'=='~ ~'~ ~" ~ / _ . . . . . . . O.Sm - ' - ~ ~ - 0 s m

/ - - - ,.Or

Decrees'rag Combustion ~ ~ Height

0 I

o 5 ,'o l~ 2o y (In)

Fig. 7--Load surface temperature as a function of the height of the combustion space and the distance through the furnace.

creases the radiative and convective heat transfer from the gases to the refractory side walls, leading to an increase in the amount of energy that is transferred through the refractory to the ambient.

Figure 9 shows the crown surface temperature as a function of the height of the combustion space. The crown surface temperature increases in the entrance region as the combustion height increases, while the opposite trend is observed in the exit region of the furnace. The entrance region trend is explained by noting that the load surface temperature is fairly insensitive to the combustion height in the entrance region, as shown in Figure 7. Therefore, the change in the crown surface temperature is primarily a function of the heat transferred from the combustion gases to the crown as the combustion height is increased. Since the combustion gases emit volu- metrically, the directed-flux exchange area between the gas volume and the crown surface zone increases as the combustion height increases. The gas temperature, however, slightly decreases in the entrance region as the combustion height increases (Figure 8). The actual emission of the gas is, therefore, controlled by two opposite effects. The volumetric effect causes an increase in the overall emission, while the decrease in gas temperature

0

0

0 , 5 ~

~ .S: ~;0: .................. ~:o.1

y (~)

Fig. 6 - - L o a d total heat flux as a function of the load emissivity and distance through the furnace.

Height=0.1 m

: ~ . . . . . . . . . . . ~ / / - 0 . 5 m m

m

-

;o l~ ~o y (m)

Fig. 8 - -Combus t ion gas temperature as a function of the height of the combustion space and the distance through the furnace.

518--VOLUME 22B, AUGUST 1991 METALLURGICAL TRANSACTIONS B

" Height=O.l m

Z L L - - . ~ ; 2 - ' T ' : : ........ , ~':'~..

5 l O 15 20

y ( m }

Fig. 9- -Crown surface temperature as a function of the height of the combustion space and the distance through the furnace.

causes the opposite effect. The decrease in the gas temperature is somewhat offset by the resulting increase in the gas absorption coefficient (the gas absorption coefficient is inversely related to the gas temperature). The final outcome is that the overall emission from the gas to the crown increases as the combustion height is increased, resulting in higher crown surface temperatures, as shown in Figure 9.

The exit region trend in the crown surface temperature is explained in a manner similar to the entrance region trend. Figure 8 shows that the gas temperature in the exit region of the furnace is greatly reduced as the combustion height is increased. Therefore, the decrease in gas emission due to the decrease in the gas temperature is not offset by the increase in the directed-flux exchange area as it was in the entrance region of the furnace. The result is a decrease in the crown surface temperature as the combustion height is increased.

Figure 10 shows the load heat flux distribution as a function of the combustion height. The heat flux tends to decrease slightly as the combustion height is increased. This trend is explained by noting that the temperature of the combustion gases decreases as the furnace height is increased, resulting in a decrease in the convective and radiative flux from the gas to the load. As

f ~-- 0.1 m

fo ,~ ~o y(m)

Fig. 10--Load total heat flux as a function of the height of the combustion space and the distance through the furnace.

the combustion height is increased, the radiative heat transfer from the combustion gases becomes large relative to the radiative heat transfer from the refractory due to the relative sizes of the directed-flux exchange areas. Therefore, the combustion gases tend to control the heat flux to the load, and since the gas temperature decreases, so does the load heat flux. The furnace efficiency shows a slightly decreasing trend as the combustion height increases, which is expected in view of the results of Figure 10.

C. Variation of the Refractory Emissivity

The refractory emissivity was varied from 0.01 (al- most reflective) to 1.0 (radiatively black). The remaining furnace parameters were as stated in the base furnace configuration (Table II).

Figures 11 and 12 show the crown surface and combustion gas temperatures as functions of the refractory emissivity. The crown surface temperature increases as the refractory emissivity increases, since the crown surface absorbs more of the incident radiative energy at higher eR. As the refractory emissivity increases, radiative energy incident on the refractory from the gas is absorbed rather than reflected back into the gas. A portion of the absorbed energy is then re-emitted by the refractory at

~ _ _ - ~ ~ t.o

. . . . . . . . . . . . . . / F 0.5 .............. 0.5

. . . . . . .

i i t 0 5 10 15 2 0

y(m) Fig. 11 --Crown surface temperature as a function of the refractory emissivity and distance through the furnace.

o

. . . . . . . . . . . .

~ E I I = O . O I

-~......-?-,,~ 0.3

0 10 15 20

y (m)

Fig. 12--Combustion gas temperature as a function of the refractory emissivity and distance through the furnace.


/ / o.ol .............. 0.1

0.8 LO

C

~, ,~ 1~ ~o y (m)

Fig. 1 3 - - L o a d total hea t f lux as a funct ion of the refractory emis- s iv i ty and d is tance through the furnace.

the refractory surface temperature, which is less than the temperature of the combustion gases. Since the re-emitted energy is spectrally continuous, only a portion of it is absorbed by the combustion gases, while the remaining portion is transmitted through the transparent bands of the combustion gases to the surface of the load. There- fore, the gas temperature decreases with an increase in eR, as shown in Figure 12.

Figure 13 shows the load heat flux as a function of the refractory emissivity. The heat flux is highest for a reflective refractory and fairly insensitive to eR as the refractory becomes black. The only exception is in the entrance region of the furnace. The heat flux reaches a minimum as the refractory emissivity is increased from 0.01 to 0.1, followed by a gradual increase as the emissivity is further increased toward a black surface. The entire difference, however, is quite small, as shown in Figure 13.

The load surface temperature was fairly insensitive to the refractory emissivity and was highest when the refractory was reflective, reached a minimum for eR = 0.3, and gradually increased as the refractory emissivity ap- proached the black condition. In general, however, the load surface temperature was invariant with respect to the refractory emissivity.

The furnace efficiency was maximum for the reflective refractory, which is obvious from the results shown in Figure 13. A minimum was reached for a refractory emissivity of approximately 0.3, followed by a gradual increase with increasing eR. The entire range of effi- ciencies, however, is small, accounting for only a few percent. This small range is explained by the fact that the furnace is long, causing radiative heat transfer to be dominated by the combustion gases. The efficiency trend was remarkably similar to the results obtained by Chapman et al. m for a batch reheating furnace. This indicates that the minimum furnace efficiency at eR = 0.3 may be more of a general result as opposed to a result that is limited to this furnace configuration.

IV. CONCLUSIONS

The mathematical model of a continuous reheating furnace presented in this paper provides an efficient

method of identifying the effects of design and operating parameters on the furnace performance. These findings should be qualified by mentioning that the results are based on a particular furnace design and operating condition. In addition, the calculations were performed on the gray basis for the radiation properties of surfaces and with Hottel's sum-of-gray gases model for the combustion products. The major conclusions are summarized as follows.

1. The furnace efficiency becomes large as the refractory surface becomes radiatively reflective. From a practical point of view, however, a nearly reflective refractory surface is not possible with conventional refractories. Even a coating would soon lose its reflective characteristics due to the harsh operating environment. Unless reflective refractory surfaces can be maintained, a nearly black surface is recom- mended. The effect of the refractory emissivity was not as important in the continuous furnace as it was in the batch reheating furnace. [11

2. The furnace efficiency is decreased by increasing the height of the furnace combustion space above the load. The increased losses to the environment more than offset the increased volumetric emission from the combustion gases to the load.

3. The furnace efficiency is increased substantially by making the load approach a radiatively black surface. This can be easily accomplished by coating the load surface with a material such as graphite.

4. This system model of a continuous reheating furnace showed its usefulness in facilitating the design of a new furnace or in optimizing the operating parameters of an existing furnace.

The computer program used to generate the results in this paper may be obtained by contacting the authors.

N O M E N C L A T U R E

Parameters

A A F A,,,r~

Bn.rj

c Cp

Deq dL G~Si

)

GjS, H _at-it h

area (m 2) air-to-fuel ratio by mass coefficient used in the weighted sum-of-gray- gases model coefficient used in the weighted sum-of-gray- gases model specific heat for a solid (J/kg K) specific heat at constant pressure (J/kmole K) hydraulic diameter (m) load thickness (m) gas to surface radiation total exchange area (m 2) gas to surface directed-flux radiation exchange area (m 2) enthalpy flow rate (kW) rate of heat release by combustion (kW) internal average heat-transfer coefficient (W/m2K) external average heat-transfer coefficient (W/m2K) heat of formation of substance i (J/krnole K)

520- -VOLUME 22B, AUGUST 1991 METALLURGICAL TRANSACTIONS B

k rh N

Nu QL

QLOSS

q S, Sj

) S, Sj

T

V x , y Y

thermal conductivity (W/mK) mass flow rate (kg/s) mole number molar flow rate (kmole/s) average furnace Nusselt number heat-transfer rate (kW) heat loss from the combustion gases, defined by Eq. [2] (kW) heat flux (kW/m 2) surface to surface total radiation exchange area (m 2) surface to surface directed-flux radiation exchange area (m 2) temperature (K) reference temperature where the specific heat is known (K) velocity (m/s) coordinates, as shown in Figure 2 molar oxygen-to-fuel ratio

Greek Parameters

E /x

P

or

emissivity viscosity (N s/m 2) density (kg/m 3) furnace efficiency defined by Eq. [11] Stefan-Boltzmann constant

Subscripts

AMB ambient a air conv convection cp combustion products

f fuel g gases i species index number i , j , k zone index number in inlet quantity L load R refractory

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of the Gas Research Institute, Chicago, IL (Contract No. 5086- 260-1293).

REFERENCES

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pp. 1035-46. 9. R. Viskanta and S. Ramadhyani: Report No. GRI-88/0154, The

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14. R. Siegel and J.R. Howell: Thermal Radiation Heat Transfer, 2nd ed., McGraw-Hill, Inc., New York, NY, 1981.

15. K. Atkinson: Elementary Numerical Analysis, John Wiley and Sons, Inc., New York, NY, 1985, pp. 68-70.

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19. F.H. White: Viscous Fluid Flow, McGraw Hill, Inc., New York. NY, 1974, pp. 27-33.


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