FAST SIMULATION OF NON-STEADY STATE EMISSION PROBLEMS
**IN ENERGY CONVERSION**

Ph.A.J. Mees, E.H.P. Wolff, P.J.T. Verheijen, C.M. van den Bleek Department of Chemical Engineering

Delft University of Technology P.O. Box 5045, 2600 GA Delft

**The Netherlands**

ABSTRACT

Application of the Fast Fourier Transform (FFT) to the inversion
of Laplace transforms is a recent development in the solution of
**the equations describing the behavior of chemical reactors. Chen**
and Hsu (1987) used the Fast Fourier Transform for the
predic-tion of breakthrough curves of an isothermal fixed bed adsorber.
**The Fast Fourier Transform method has been further developed for**
**the calculation of breakthrough curves in non-isothermal**

adsorbers (Mees et al., 1989). The purpose of this work is the application of this method to a practical engineering problem involving parameter estimation.

**INTRODUCTION**

One of the topics of coal based energy production systems is the removal of sulfur compounds from reaction gases. High temperatu-re gas/solid sorption processes atemperatu-re being developed for both coal gasification and coal combustion to respectively remove hydrogen sulfide and sulfur dioxide.

The study of these processes deals with the modeling of non-steady state fixed bed percolation processes in order to gather sufficient information for the prediction and the design of the final sulfur removal unit. The calculation of breakthrough curves in these non-steady state reactor models and the estima-tion of the parameters involved from bench scale experiments is a cumbersome task due to a lack of adequate mathematical tools. Application of the Fast Fourier Transform (FFT) to the inversion of Laplace transforms is a recent development in the solution of the equations describing the behavior of these reactors. Chen and Hsu (1987) used the FFT for the prediction of breakthrough curves of an isothermal fixed bed adsorber. The method has further been developed by Mees et al. (1989) for the calculation of breakthrough curves in non-isothermal adsorbers.

The purpose of this paper is to apply the fast method to a practical engineering problem involving parameter estimation. Therefore the method will be incorporated in a simplex based parameter estimation procedure and be demonstrated by the D.U.T

Adsorption equilibrium:

Initial and boundary conditions are given by:

**(6)**

{7)
**(8)**

**(9**

### (I0

### (11

### (12)

### (13)

**(14)**

### (15)

(16) (17)**(18)**

For the solution method used, the model equations have to be Laplace transformed with respect to time (~) and space (x). These Laplace transformations can only be carried out if the equations are a linear function of concentrations, temperatures and their derivatives. The model equations (1)-(4) are linear functions of these variables, but the adsorption isotherm (5) will generally be non-linear. Therefore, the adsorption isotherm has to be represented by a linear function of the pore concen-tration and the temperature of the solid

Q = Ku.U. + Ko.O. (19)

In the case of an isothermal bed, the final stationary state determines Ku (Figure i). In the general case the constants Ku and Ko must be chosen in such a way that equation (19) calcula-tes the initial and final state correctly and gives an adequate

transforms with respect to time are calculated numerically by applying the Fast Fourier Transform. This method is described by Hsu and Dranoff (1987). The discrete inverse Laplace transform is given by the following formula:

exp(aj~T) N-I f(j~T)

-2T k=0 *F(a + ikn/T) .exp(i2njk/N)]*

**j = 0,1,2 ... N-I**

### ’ with &T=2T/N

T is half the time period considered and N is the number of points. ’a’ is the real part of the Laplace s-value and determi-nes the place where the line-integral of the inverse Laplace transform in the complex s-plane is calculated. The value of ’a’ must be greater than the real parts of the singularities of the Laplace-domain function.

By means of the program that has been developed based on these numerical methods concentration (U~ , Us and Q) and temperature

(el and 8s ) breakthrough curves are calculated in only a few minutes on a personal computer (PC). It has been shown before that by the application of this method, breakthrough curves can be calculated at a speed that is almost three orders of

magnitude larger than of conventional methods (Chen and Hsu, 1987).

SULFUR CAPTURE EXPERIMENTS

Fluidized bed combustion (FBC) is a technique for burning coal in a bed of sulfur accepting material (sorbent) to generate electric power and/or steam while meeting the emission standards for the oxides of sulfur and nitrogen. Uatural minerals such as limestone (CaCO3) and dolomite (CaCO~ .MgCOs ) are presently receiving most attention as sulfur capturing materials. However, synthetic regenerable sorbents are being studied as substitutes for the natural minerals with the prospect of producing a regenerable sorbent. In our experiments the SOz-reactivity of highly dispersed calcium-oxide on a ganuma-alumina support has been studied; due to its special texture this sorbent can already be regenerated at 8500C (Wolff et el. 1989). The sorbent is symbolized as CaO/gamma-AlzOs .

To investigate the sulfur capture behavior of the synthetic sorbent, experiments have been carried out in a 12 mm internal diameter fixed bed tubular reactor. The set-up consists of a gas mixing section, a reactor section and an analysis section. In the gas mixing section the fluidized bed combustor off-gas can be simulated by mixing nitrogen, oxygen and sulfur dioxide. This gas is introduced into the reactor which is kept at 850 0C and 1 bar. The sorbent material will catch the sulfur dioxide according to:

developed to minimize this objective function.

The parameters to be determined are physical properties, such as diffusion constants. These are all positive and finite, so the parameter variation was not done with steps, but with fractions of the parameter by taking the natural logarithms of the

parameters to be optimized. Comparing the two approaches showed that the latter was faster in convergence, and did not require additional coding to limit the search area to positive values of the parameters.

Finally, a sensitivity analysis was performed by estimating the error in each parameter through its contribution to the

curvature of the surface of the sum of squares as a function of the parameters. It should be realized that this only gives an estimate of the accuracy.

In the sulfur capture example, the measured curves appeared to be only the very first part of the complete breakthrough curves. Because the method used for the calculation of the breakthrough curves can only calculate the complete curves, so, until the stationary, situation, by far the biggest part of the model curves can not be used. The consequence of this redundant calculation is that, if we want to calculate the model curve at a sufficient number of points for which a comparison with the experimental curve is possible, the number of points that has to be used for the complete curve reaches the limits of what is possible with this program on a PC.

**RESULTS**

The method of modeling, using a combination of the Laplace transform for the differential equations and a FFT for the numerical calculations, together with a simplex parameter optimization procedure has been applied to the sulfur capture case described above. Because no temperature effects are involved in the sulfur system chosen, a simplified version of the model could be used. This contains only five parameters: the axial dispersion coefficient (De) , the external film diffusion coefficient (Kf) , the pore diffusion coefficient (Dp) , the adsorption coefficient (Ku) and the scaling factor.

It was expected that film diffusion was a non-limiting process. Therefore, Ks was kept at a constant, high, value. It was also expected that these measurements could give an estimate for the adsorption capacity of the sorbent in situ. Optimizations showed that this parameter and Da are highly correlated, so that for Ko a value was substituted obtained from measurements of the CaO content of the sorben~. The remaining three-parameter problem was solved (Figure 2, Table 2).

robust compromise was found by taking it equal to the sampling interval at which the contour integral of the inverse Laplace integral was calculated. Without these precautions systematic inaccuracies in the model curves appear at the beginning of the curve, making the optimization meaningless.

The method as a whole proved to be robust: the optimization procedure could be started at conditions differing by orders of magnitude and still leading to the same result. Fitting the three breakthrough curves simultaneously resulted in an extra stability. Table 2 illustrates the results of the curves, when f±tting them separately.

The calculation was done on an 8 MHz, IBM compatible FC with a mathematical coprocessor, requirin~ 40 seconds per experimental curve. An optimization typically took l0 to 50 steps in order to reach a relative deviation of only 10-6 in the sum of squares.

**DISCUSSION ~D CO~{CLUSIONS**

It was found that a parameter optimization can indeed be performed together with the solution of the differential

equations via the Laplace Transform coupled with a FFT. However, in the isothermal situation as discussed here, the limits of what is practical on a PC or AT has been reached. The method as such proved to be robust and reliable.

The mathematical framework for the non-isothermal case is also given in this paper. The calculation of one curve requires a factor twenty or more CPU time (Mees et al., 1988). In that case, concentration and temperature breakthrough curves have to be measured. Because the non-isothermal model has ii parameters

(or 12, if scaling is free}, it is clear that this has to be done, using considerably faster hardware.

Finally, the method proved especially useful in simulating
*various reactor designs; starting from the process parameters as*
given in Table 2 and for example varying the reactor length or
the sorbent particle diameter, each simulation curve could be
produced within a minute.

ACKNOWLEDGEMENT

The financial support of the Com~ission of the European Communi-ties (contract E~I3F-0014-~4L(GDF)) and of the Netherlands Agency for Energy and Environment, ~OVEM (contract 20.35-016.30) is greatly acknowledged

The concentrations in the solid are directly found:

Io.=osh~o-sinhlo + sinhlo _{~}

Bl(m) ~" (A-2)

**= Ku . Us + K0 . s**

NOTATION

c~ compound concentration in the fluid

c6 compound concentration in the pores per m~ fluid co incoming fluid concentration

Cpf heat capacity of the fluid Cp. heat capacity of the solid

D. axial mass dispersion coefficient of the fluid Dp diffusion coefficient in the pores

hf film heat transfer coefficient

hw~ heat transfer coefficient, fluid to wall hw~ heat transfer coefficient, solid to wall -~H adsorption heat

k~ film mass transfer coefficient L reactor length

q adsorbed concentration per m~ solid R reactor radius

rp radial place in the particle R, particle radius

t time

T~ temperature of the fluid

T~ temperature of the incoming fluid T= temperature of the solid

Tw temperature of the wall v intrinsic fluid velocity

z axial place in the reactor

mol/m~
mol/m~
mol/m=
J/(k~ K)
J/(kg K)
m~ /s
m~ /s
J/(m~ s K)
J/(m~ s K)
J/(m~ s K)
J/mol
m/s
m
mol/m=
m
m
m
**s**
K
K
K
K
m/s
m
qreek symbols:

porosity of the bed

axial heat dispersion coefficient of fluid axial heat conduction coefficient of solid fluid density

solid density per m~ particles

J/(m s K)
J/ (m s K)
kg/m~
*kq/m~*