플라스틱 제품의 전과정 평가를 위한 수학적 모델링 및 해석

송현섭, 문광식, 현재천

고려대학교 화학공학과

Mathematical Modeling and Analysis for the Life-Cycle Assessment

of Plastic Materials

Hyun-Seob Song, Kwang-seek Moon and Jae Chun Hyun

Dept. of Chemical Engineering, Korea University

Introduction

Life-cycle assessment(LCA) is defined as an objective process evaluating the environmental burdens associated with certain products, processes or activities including the identification and quantification of energy and materials used and wastes produced therefrom[1]. Generally, an LCA study is performed following the four substages of 1) goal definition and scoping, 2) inventory analysis 3) impact assessment and 4) interpretation. The present study deals with the mathematical modeling of the LCA process with a focus on the inventory analysis. The resulting model can be used for many purposes including the comparison of various alternative methods to handle waste management problems. The particular example of PET bottles has been selected for LCA study in this report.

Goal definition and scoping

Our goal in an LCA study of plastic materials is to find ecologically optimal waste management routes for used plastic materials and wastes. For PET bottles, the following waste management routes are considered possible:

Closed-loop recycling	-  recycle as polymer to PET bottle production process

Open-loop recycling	-  recycle as polymer to PET carpet production process

Chemical recycling	-  recycle as a monomer by solvolysis

or recycle as fuels by pyrolysis

Thermal recycling	-  recycle as energy by incineration

Disposal	-  landfill

Fig. 1 shows the system boundary illustrating the above alternative waste management routes of PET bottles. By adjusting the values of the following parameters in Fig. 1, we can easily represent each alternative route or any combinations of them. Some of the results are shown in Table 1.

c = collection ratio

r = closed-loop recycle ratio

□ = open-loop recycle ratio

f = ratio of recycle as polymer

w = ratio of incineration to pyrolysis

Table 1. Various alternatives with different parameter sets.

Alternatives	c	r	f	w	□
L L+I L+P L+D L+R L+O	0 0<c<1 0<c<1 0<c<1 0<c<1 0<c<1	* 0 0 1 1 1	* * * 0 1 *	* 1 0 * * *	* * * 0 0 1

L=landfill, I=incineration, P=pyrolysis, D=depolymerization

R=reprocessing, O=open-loop recycle, *=not applicable

Mathematical modeling

Now that the overall flow diagram within the system boundary illustrating the alternatives is established, we proceed to obtain the process energy required and emissions released for each block of Fig. 1. All the values of the energy and emissions are taken from the open literature data such as BUWAL, APME, PEMS, SimaPro, etc. The energy needed for the collection of PET bottles has been assumed to be of the following functional form:

Pc = p c/(1-c) (1)

This is based on the fact that as the collection ratio approaches unity (100% collection), the required energy becomes infinite[2]. The parameter "p" in Eq. (1) can be determined from the specific data, e.g., Pc = Pc* at c=c* which can be obtained from the real collection data, e.g., governments statistical data. In our particular example of PET bottles recycling in Korea, the value of the parameter, p, has been determined to be approximately 1.0. Fig. 2 shows the curves of Eq. (1) when p varies. When the collection ratio is less than 0.5, which is the case in real situations, the modeling errors are considered to be negligible.

The final form of the equations for the total energy consumed and some of emissions released during the life-cycle of PET bottles is then given as follows:

E = 13960 - 808.8c - 3674rc - 3077□rc - 595.6f(1-□)rc - 400.8w(1-r)c + 60c2/(1-c) (2)

C = 616.3 + 84.4c - 287.9rc - 177.8□rc - 64.4f(1-□)rc + 11.5w(1-r)c + 4.7c2/(1-c) (3a)

N = 1.42 + 0.09c - 0.48rc - 0.40□rc - 0.18f(1-□)rc - 0.53w(1-r)c + 0.08c2/(1-c) (3b)

where

E=energy[MJ], C=CO2[kg], N=NOx[kg]

and

0≤c<1, 0≤r≤1, 0≤□≤1, 0≤f≤1, 0≤w≤1 (4)

The above equations have the same functional form because the required energy and the released emissions for each block are based on the same functional unit of PET bottles. Using the model equations we can easily compare the performance of the different waste management alternatives by adjusting the relevant parameters set to specific values and then further seek optimum conditions producing best energy and emissions results.

Optimization

The most environmentally favorable route can be found solving optimization problems where the equation (2) or (3) is taken as an objective function to be minimized with adjustable parameters. Most mathematical models used in LCA study are linear to be amenable to linear tools like linear programming[3]. However, this is not the case for the nonlinear systems of the present study. The nonlinear form of Eq. (2) and (3) warrants the use of nonlinear programming such as GRG or SQP to solve the constrained nonlinear optimization problem[4]. The optimum parameter values obtained from nonlinear programming are different depending on which objective function we choose. Table 2 shows the results when different objective function was chosen.

Table 2. Optimum parameter sets.

Obj.	c	r	□	f	w	Minimum
E	0.889	1.0	1.0	*	*	10,092[MJ]
C	0.848	1.0	1.0	*	*	467.3[kg]
N	0.613	1.0	1.0	*	*	1.21[kg]

One of the possible methods to combine these different optimal results is to select one from Eqs. (2) and (3) as the objective function while the others treated as constraints. For example, as shown below, we can take the energy function as the sole objective function and CO2 and NOx emissions as constraints,

J = E (5)

subject to

C ≤ CSET  and  N ≤ NSET (6)

where the values of CSET and NSET could be given by government regulations or determined from impact assessment analysis.

Conclusions

A mathematical model for the overall system encompassing the life-cycle of plastic materials has been developed based on the material and energy flow diagram adopting PET bottles as an example. A simple model of energy for collection has been chosen reflecting the qualitative nature of collection energy term which satisfies the two limiting cases of c=0 and c□□1. The obtained mathematical models for total energy and emissions can be used for the comparison of various alternatives and more importantly for finding the strategy for the optimum management of plastic wastes. The results and methodology adopted in this study for PET bottles can be readily applied to a wide range of waste management problems for plastic materials.

Acknowledgement

The work was supported by Korea Science and Engineering Foundation under Grant 96-0502-07-01-3.

References

1. A Technical Framework for Life Cycle Assessments, Society of Environmental Toxicology and Chemistry(SETAC) (1991)

2. Boustead, I., "Principles of Plastics Recycling," Society for the Promotion of Life-cycle Assessment Development(SPOLD) (1994)

3. Azapagiz, A. and R. Clift, "Life Cycle Assessment and Linear Programming - Environmental Optimisation of Product System," Computers Chem. Engng., 19, Suppl., s229 (1995)

4. Reklaitis, G.V., A. Ravindran and K.M. Ragsdell, Engineering Optimization: Methods and Applications, Wiley-Interscience, New York (1983)

Fig. 1. A schematic diagram for life-cycle of PET bottles.

Fig. 2. Collection energy curves of Eq. (1) with the various

values of the parameter "p".