a two-dimensional finite element model for kiln-drying of refractory concrete

$: A Two-Dimensional Finite Element Model for Kiln-Drying of Refractory Concrete$
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A Two-Dimensional Finite Element Model for Kiln-Drying of Refractory ConcreteZhen-Xiang Gong a & Arun S. Mujumdar aa Department of Chemical Engineering McCill University., Montreal, Quebec, H3A 2A7,CanadaPublished online: 10 May 2007.

To cite this article: Zhen-Xiang Gong & Arun S. Mujumdar (1995): A Two-Dimensional Finite Element Model for Kiln-Drying ofRefractory Concrete, Drying Technology: An International Journal, 13:3, 585-605

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DRYING TECHNOLOGY. 13(3), 585-605 (1995)

A TWO-DIMENSIONAL FINITE ELEMENT MODEL FOR KILN-DRYING OF REFRACTORY CONCRETE

Zhen-Xiang Gong and Amn S. Mujumdar Department of Chemical Engineering

McCill University, Montreal, Quebec, Canada H3A 2A7

Key Words and Phrases: Drying; Permeability; Diffusion Model; Pore Steam Pressure; Drying Schedule; Spalling.

ABSTRACT

A two-dimensional finite element model is developed for kiln-drying of refractory concrete. Using this model simulations are carried out for refractory concrete castings of different thicknesses to investigate the effects of drying schedules on Dore steam pressure and moisture removal. Simulation results from both one- and two-dimensional models are compared and discussed. On the basis of the simulation mults new realistic drying schedules are suggested.

INTRODUCTION

Refractory concrete is often prefabricated into pieces of castings to build

linings of high temperature installations used, for example, in the iron and steel

industry and the petrochemical industry. These prefabricated concrete castings

conlain extremely fine pores which hold large amounts of water. They must be

Copyright O 1995 by Mvos l Dckkcr, Inc.

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586 GONG AND MUJUMDAR

dried fully before putting them into service to avoid destruction by explosive

spalling when exposed to elevated temperatures.

The causes of explosive spalling are numerous. Early research [ S , 6, 71 on

this problem indicates that many factors, intricately related, are indirectly

influential. Excessive water addition during fabrication, curing below 20 "C, very

fine cement and cement content over 10% all may reduce the permeability or the

strength of the refractory concrete, leading to an increased explosive tendency.

Prefabricated refractory concrete castings are generally dried in a kiln. In the

dryingeprocess, heating of refractory concrete castings causes the moisture to

evaporate in the pore and therefore, produces pore steam pressure. This pressure

is thought to be both the driving force for moisture transfer and the direct cause

of explosive spalling. During drying, rapid heating rates of the castings may lead

to excessive build-up of pore steam pressure, resulting in explosive spalling; slow

heating rates not only take longer production time, but also is more energy-

consuming. Prediction of the pore steam pressure, therefore, is of practical

significance.

Prediction of the pore steam pressure requires a detailed understanding of the

heat and mass transfer process during drying. Based on a diffusion theory, Bazanl

and Thonguthai [I. 121 developed a mathematical model to describe the process.

They solved this model by the finite element method for the failure analysis of

concrete reactor vessels in accidents or concrete structures subjected to fire [2,

31. Dhatt el al. [ I l l also utilised a finite element model to calculate the

temperature and pore steam pressure responses to various heating rates for a one-

dimensional axisymmetrical installation. Gong et al. [4] developed a finite

element model to simulate the drying of refractory concrete by convection and

volumetric heating. Gong et al. [I31 also simulated the field drying processes of

refractory concrete installations and proposed a set of rules for drying schedules

according to the thicknesses of refractory concrete. The first attempt to simulate

the kiln-drying process of refractory concrete was carried out by Gong and

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KILN-DRYING OF REFRACTORY CONCRETE 587

Mujumdar [9]. They set up a one-dimensional finite element model and simulated

different time-temperature schedules. New reasonable drying schedules for the

kiln-drying of refractory concrete were suggested on the basis of the simulation

results.

In this paper, a two-dimensional finite element model is developed for the

kiln-drying of refractory concrete. Simulation results are obtained for

prefabricated refractory concrete castings of different thicknesses. After the

results are carefully analyzed and compared with those from the one-dimensional

model [9], appropriate drying schedules are suggested for refractory concrete

casting according to the thicknesses of the concrete castings.

MATHEMATICAL MODEL

The governing equations of the heat and mass transfer process in refractory

concrete following Bazant and Thonguthai [I21 are as follows:

JT JW a J P J T J P J T pc--c -= -C -(--+--)+ at " J I w g ~ X J X a y a y

where W is the free water content; W, is the water liberated by dehydration

during heating; a is relative permeability (in mls); g is gravity acceleration (9.806

mls2); P is the pore steam pressure; t is time; x and y are spatial coordinates; p

and C are the mass density and the isobaric specific heat of the concrete,

respectively; C. is the evaporation heat of free water; C. is the specific heat of

water; k is the thermal conductivity of the concrete; and T is temperature. This

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588 GONG AND MUlUMDAR

model for drying is only an approximate one, however i t is considered suitable

for the engineering application under study.

Since W is a function of both temperature T and pore steam pressure P (W

= W(P,T)) we can write:

aw - a w a p awar +--. m aP at ar m

Substitution of equation 3 into equations I and 2 yields

where

aw aw JW JW A =- A = - C - A =-, A = C-C - . a p ' 2 o a p s 3 aT 4 " ar (6)

The boundary conditions are as follows:

for moisture:

P = P , ; (8)

and for temperature:

ar -k- =B, (T-Tem)+C,B,(P-P,J. an

(9)

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where B. and B, are convection transfer coefficients for moisture and heat,

respectively; T, and P, are ambient temperature and ambient steam partial

pressure, respectively; P. and T, are the pore steam pressure and temperature at

the boundary, respectively; n is the outward normal to the boundary.

FINITE ELEMENT FORMULATION

After the spacewise discretization of equations 4 and 5, subject to the

convective boundary conditions (equations 7 and 9) is accomplished using

Galerkin's method (101, the following semi-discrete matrix system is obtained:

in which

In the preceding equations, nel is the element number; the superposed dot denotes

differentialion with respect to time; C , (m=l , 2, 3, 4), K, and K, are

submatrices; F, and F2 are subvectors. The coefficients in the submatrices and

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subvectors are calculated according to the following equations:

(16)

11 a aNI aNJ aNI aNJ K, =// -(--+--) w+/ B, N , N l d r , (17)

O ' S au au I-

aN, aNJ aN aNJ JNJ K;=// [ K - - + ~ - ) + A J ~ - + A # ~ - I drdy " t r a u ibr au

F:=/, (B , T,+Ca By PJ NI d r

where N, and N, are shape functions of an element; and

It should be indicated that no special treatment is required for the convection

terms in equation 2 since the contribution of these terms is less important to the

heat transfer than the conduction terms.

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Discretization of the time derivatives in equation 11 is most often achieved

with a finite difference technique. In this work, a predictor-corrector scheme [8]

is used.

Predictors:

Correctors:

In the preceding equations, subwrip't n denotes the nth time step, superscript i

designates the ith iteration; y is an integral parameter which controls the accuracy

and stability of the time integration, i. = 0 - I ; arid a t is time step. In the

computations of this work, y=0.75;

Substitution of equations 26 and 27 into equation l l yields

where

The solution procedure is as follows:

(1) At the beginning of each timk step, calculate {U4+,) and {Ijo.+,)

according to equations 24 and 25.

(2) With the two starting vectors, solve equation 28 for (nl>,+,).

(3) Update {u~',,,) and {U"'.,,) by equations 27 and 26.

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GONG AND MUJUMDAR

(4) If the convergence condition is met (here, we compare the Cartesian

norms of {nfii.+,) and {R) to some selected constants TOLl and

TOLZ), set

and

if not, using {U'+'.+,) and {u'+'.+,) as starting vectors, go to step (2)

and undertake the next iteration.

Based on the procedure described above, a two-dimensional finite element

computer code, called DRY-RC was developed with FORTRAN 77. Using this

code, simulations were carried out for the kiln-drying process of refractory

concrete.

RESULTS AND DISCUSSION

Suppose that initially a casting has a uniform temperalure of 25 "C and

relative humidity of PIP. (25 T)=90% in the pore. When dried in a kiln, the

casting is exposed to the hot air. To guarantee a good quality and possibly shon

drying time and to avoid explosive spalling, the surface temperature of the

castings is required to increase in accordance with a prescribed time-temperature

schedule determined empirically.

A long casting can be approximared by a two-dimensional model. A square

section as illustrated in Figure 1 is taken as the model geometry. The four sides

of the square can be approximated as having the same temperature and moisture

boundary conditions (in practice, there is some difference for different sides).

Such a simplification is made not because the numerical model can not simulate

the non-uniform temperature and moisture boundary but because of the lack of

needed information since there is no experimental data available about the non-

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KILN-DRYING OF REFRACTORY CONCRETE

Figure I Model Geometry

uniformity. Due to the symmetry of the geometry and the boundary conditions,

we can use only one half of a quarter of the square as the computational domain.

The finite element mesh is demonstrated in Figure 2. Sides AB and CA are

insulated for heat as well as moisture. The tempenture of side BC is increased

according to a prescribed time-temperature schedule and the moisture flux on this

side can be described by equation 7. The moisture transfer coefficient B, and the

ambient steam partial pressure P, are taken to be I .Ox lOb slm and 2850 Nlm2

respectively.

The following physical properties are assumed constant:

~ = 2 2 0 0 kglml, C = 1100 Jlkg K.

k=1.67 Wlm K, C,=4100 Jlkg K.

C, is calculated from the following equation:

I

3.5x10'(374.15-7)' ( T ~374.15 'C ) (33) 0 ( T >374.15 'C )

The permeability a is strongly dependent on moisture and temperature and is

computed according to the following empirical formula [2,12,14]:

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GONG AND MUJUMDAR

Figure 2 Finite Element Mesh

in which a, is the permeability at initial temperature To, in this work a,= IO-'' mls

when T,=25 T and this value of a, is used for all the subsequent computations.

h is the relative humidity in the pore, h=PIP,(T); and

1 f,(T)= exp [ 2700 (--- I 11.

273+To 273+T

aWIBP and aWlaT are obtained from the state equation, W=W(P,T), which

is as follows [2,12,14]:

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in which W . and W, are the anhydrous cement content and saturation water

content at 25 "C per m3 of concrete, respectively (in this model, W,=300 kglm',

Wo=lOO kglm'); W, and W, are the free water content when h=0.96 and 1.04,

respectively; m(T) is a coefficient related to temperature,

The central difference scheme is employed for the calculation of aWlaP and

JWlJT.

A, is calculated with the aid of the following equation:

aw, - aw, a T A =---- a a r m

in which aWdaT is obtained from the dehydration curve of Figure 3 [ I l l

Computation of A,, and A, is as follows:

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~r<.l,,~,<~,.;tL,,,~v ( *'c: )

Figure 3 Dehydration Curve

in which nen is the node number of an element.

lsoparametric bilinear quadrilateral elements are utilized in this model. A total

of 300 elements with a time step of 30 seconds are used and the convergence

condition is selected to be TOLI=0.005 and TOLZ=O.O05 for all the

computations.

Computations are carried out for two thicknesses of concrete castings, one is

L=200 mm and the other is L=400 rnm (see Figure I). The surface temperature

of each casting is raised according to a prescribed time-temperature schedule. The

prescribed time-temperature schedules are presented in TABLE I and TABLE 2.

These two schedules were suggested by Gong et a1 [91 for concrete slabs with

thicknesses of 200 mm and 400 mm respectively.

Figures 4, 5 and 6. 7 exhibit the evolutions with time of the pore steam

pressure and temperature disaibutions along side AB. It can be observed from

Figures 4 and 5 that the peak of the pore steam pressure migrates from the outside

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TABLE I Time-Temperature Schedule for Concrete of 200 mrn Thickness

I L=ZW mm 1 Rate ("Clhr) ( Time (hr) Il

TABLE 2 Time-Tempemture Schedule for Concrete of 400 mrn Thickness

2 5 I I 3 O C

130 "C-130 OC

130 "C-600 "C

Total

2 0

0

20

5 .25

12

23.5

40.75

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GONG AND MUlUMDAR

DisLance from the Centre of the Square (x/L)

Figure 4 Pore Steam Pressure Distributions along Side AB for L=200 mm

Distance (!.om Lhe Centre of the Square (x /L)

Figure 5 Pore Steam Pressure Distributions along Side AB for L=400 mm

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Figure 6 Temperature Distributions along Side AB for L=200 mm

600

- 500 U - 400 u

300 m L ' :

100

Distance from The Centre of the Square (x/L)

I I I I

- 35 hr -

- -

- - 25 hr

8 hr 12 hr -

I IS hr - 4 hr

Figure 7 Temperature Distributions along Side AB for L=400 mm

0 ' ~ ' ~ ~ ~ " ~ ~ ~ ~ " ~ " " ~ ~ ~ ~ '

0.0 0.2 0.4 0.6 0.8 1 .O

Distance from the Centre of the Square (x/L)

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to the inside with the progress of the drying process and soon arrives at the centre

of the square section.

Figure 8 displays the comparison of the maximum pore steam pressure history

curves resulting from the two-dimensional (2-d) model and the one-dimensional

(I-d) model of reference [9] for concrete casting of 200 mm thickness. The solid

line is for the 2-d model, and the dashed line for the l-d model. The predicted

global maximum pore steam pressure from the 2-d model is 1.919 bar, almost

identical to that of the l-d model (1.902 bar). The predicted time of reaching the

first local maximum (also the global maximum), however, is shorter according

to the 2-d model than by the 1-d model. This is because heat transfer is faster in

the 2-d case than in the l-d case since more heated surfaces exist in the 2-d

model. The second local maximum pore steam pressure is much lower in the case

of 2-d than that of l-d model.

Figure 9 shows a comparison of the maximum pore steam pressure history

for concrete of 400 mm thickness. From this figure one can make the same

observation as those for Figure 8.

Since the l-d model of reference [9] was for concrete slabs, the section of the

castings must be long and wide enough to neglect the heal and mass transfer in

the length and width directions. In the 2-d model, a square section is selected to

be as the model geometry. These two cases can be taken as two extreme cases of

the possible geometry of long castings. The maximum pore steam pressures

produced with these two schedules (TABLES 1 and 2) are far from those inducing

explosive spalling. The highest pore steam pressure measured in heated concrete

has been reported to be about 8 arm [IZ]. This indicates the critical pore steam

pressure in concrete is about 8 atm. Therefore, it can be concluded that the two

schedules presented in TABLES I and 2 are appropriate for concrete castings of

less than or equal to 200 mm and 400 mm thicknesses respectively.

Figure 10 compares of the cumulative water release curves resulting from the

2-d and l-d models corresponding to Figure 8. The predicted water release is

higher according to the 2-d model than by the l-d model. This is because there

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Figure 8 Maximum Pore Steam Pressure History Curves for L=2W mm

Z-d model

Time (hour)

Figure 9 Maximum Pore Steam Pressure History Curves for L=400 mm

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602 GONG AND MUlUMDAR

-

m w

U .> 1.0 2-d model 1-d model

U 0 5 10 15 20 25 30 35 40 45 50

Time ( h o u r )

Figure 10 Cumulative Water Release Curves for L=200 mm

is larger specific surface for moisture removal with a casting of a square section

than that of a slab.

Figure 11 is the comparison of the cumulative water release curves for

concrete castings of 400 mm thickness. The same observation can be made as in

Figure 10.

It is noted that a long casting with a square section is easier for removal of

moisture than a slab, and the two schedules in TABLES I and 2 are safer for the

drying of non-slab like castings. This further indicates that the time-temperature

schedule suggested by [Y] is appropriate.

Numericai experiments found that the pore steam pressure produced in the

drying process is greatly influenced by the permeability and heat conductivity,

especially the permeability, of the concrete. The permeability and heat

conductivity vary with different types of refractory concrete. Therefore, when the

concrete is a different type, a different time-temperature schedule should be

prescribed.

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- -

-

-

-

- 9-d model - - - I-d model

-

I . . , . L

0 10 20 30 4 0 50 GO 70 80 DO

Time (hour)

Figure I I Cumulative Water Release Curves for L=400 mm

CONCLUDING REMARKS

A two-dimensional finite element model was developed for prediction of the

moisture removal rate and the variation of maximum pore steam pressures for

refractory concrete. Schedules suggested by [9] with a one-dimensional model

were further tested by the two-dimensional model and found appropriate. The

two-dimensional model is more accurate to predict the geometry effects on the

drying process. Experimental verification is required before putting the two time-

temperature schedules into industrial use. The finite element code DRY-RC can

deal with any arbitrary cross-section as long as it can be modeled as a two

dimensional problem with known boundary conditions.

REFERENCES

1. Bazant, Z. P. and Najjar, L.J., 1972, Nonlinear Water Diffusion in Nonsaturated Concrete, Materials and Suuctures: Research and Testing, Vol. 5, No. 25, pp. 3-20.

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2. Bazant, Z.P. and Thonguthai, W., 1979, Pore Pressure in Heated Concrete Walls: Theoretical Prediction, Magazine of Concrete Research, Vol. 31, No. 107, pp. 67-76.

3. Bazant, Z.P., Chern, J.C. and Thonguthai, W., 1981, Finite Element Program for Moisture and Heat Transfer in Heated Concrete, Nuclear Engineering and Design, Vol. 68, pp. 61-70.

4. Gong, Z.X., Song, B. and Mujumdar, A.S., 1991, Numerical Simulation of Drying of Refractory Concert, Drying Technology, Val. 9 , No. 2, pp. 479- 500.

5. Crowley, M.S. and Johnson, R.C., 1972, Guidelines for lnslalling and Drying Refractory Concrete Linings in Petroleum and Petrochemical Units, American Ceramic Society Bulletin, Val. 51, No. 3, pp. 226-230.

6. Giuen, W. H. and Han, L.D., 1961, Explosive Spalling of Refractory Castables Bonded with Calcium AluminateCement, American Ceramic Society Bulletin, vol. 40, No. 8, pp. 503-510.

7. Smith, K., 1990, Refractory Heat Dry Considerations, Hydrocarbon Processing, January, pp. 57-59.

8. Hughes, T.J.R., Pister, K.S. and Taylor, R.L., 1979, Implicit-Explicit Finite Elements in Nonlinear Transient Analysis, Comput. Meths. Appl. Mech. Engrg. Val. 17/18, pp. 159-182.

9. Gong, Z. X. and Mujumdar, A.S., 1993, A model for Kiln-Drying of Refractory Concrete, Drying Technology, Val. 11, No. 7, pp. 1617-1 639.

10. Zienkiewicz, O.C., 1977, The Finite Element Method, McGraw-Hill, New York.

11. .Dhatt. G.. Jacquemier, M. and Kadje, C., 1986. Modelling of Drying Refractory Concrete, Drying' 86 (edited by A.S. Mujumdar), Vol. 1, pp. 94- 104.

12. Bazant, Z.P., and Thonguthai, W., 1978, Pore Pressure and Drying of Concrete at High Temperature, Proceedings of the American Society of Civil Engineers, Vol. 104, No. EMS, pp. 1059-1079.

13. Gong, Z.X., Zhang, G.S. and Mujumdar, A.S., 1992, Prediction of Pore Pressure of Refractory Concrete Produced by Firing, Drying' 92 (edited by A.S. Mujumdar), Pan B, pp. 1780-1789.

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14. Song, B . , 1990, Finite Element Analysis of Heat and Mass Transfer in Porous Bodies, Master thesis, Tianjin Institute of Light Industry, Tianjin, China.

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a two-dimensional finite element model for kiln-drying of refractory concrete

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