Optimization of a Simple AmmoniaâWater Absorption Refrigeration

Ind. Eng. Chem. Res. 2009, 48, 1957–1972

1957

Optimization of a Simple Ammonia-Water Absorption Refrigeration Cycle by Application of Mixed-Integer Nonlinear Programming Luz Marıá Cha´vez-Islas and Christopher L. Heard* Instituto Mexicano del Petro´leo, Eje Central Lazaro Cardenas Norte No. 152, Col. San Bartolo Atepehuacan, Delegacioń GustaVo A. Madero, 07730 Me´xico D.F., Mexico

Optimization of an ammonia-water absorption refrigeration system, minimizing operating and annualized capital costs, is described. A mixed-integer nonlinear programming (MINLP) model, with the inclusion of discontinuous functions for capital costs for the main components of the system, is presented. The minimum total annualized cost is achieved with the simultaneous optimization of seven design variables. Practical limits of the design variables are established for the purpose of generating design parameters in accordance with the type of system studied. The model is applied to two examples. The analysis considers two types of heat rejection media: cooling water and air. The results obtained indicate that the selection of cooling medium is dependent on the refrigeration level that is required. The optimum configuration and operating conditions do not correspond to the best process integration and the coefficients of performance, and irreversibilities are high. The design variables with the highest impact on the objective function are dependent on the heat rejection medium characteristics and the process requirements. A sensitivity analysis of the effects of the evaluation criteria on the minimization of operating and capital costs is presented. Introduction The ammonia-water absorption refrigeration system (AWRS) is the oldest active refrigeration technology. After suffering a decrease in its use during the middle of the last century, there has been renewed interest during the past three decades. This new interest has grown simultaneously with the increasing awareness of environmental and energy issues that must be addressed in the present and near future. The main advantage of these systems is that they can be driven by low-grade heat, which constitutes an efficient way of generating useful refrigeration from waste or inexpensive heat. In addition, substances that occur naturally in the environment are used as the working pair: water and ammonia. Its low maintenance requirements, reliability, and long working life make it highly compatible with any sustainable development program. It is such that it can be an alternative to mechanical vapor compression refrigeration systems in a range of applications when a suitable source of driving heat is available. From the above, it may be seen that absorption refrigeration systems form part of the energy systems that will enjoy increasing demand in the future. However, in the ambit of research, it is still an active field and it cannot be regarded as a technology that has been completely developed. Thus, it is necessary to make an additional effort to improve its design and operation to satisfy the increasing demand and promote its more-extensive use in industrial and commercial applications.1 In the literature, the implications of design variables such as the evaporator purge (blowdown), the reflux ratio in the distillation column, the approach temperatures in the heat exchangers, and operating criteria for the components that comprise the ammonia-water absorption refrigeration cycle have been examined individually. For example, the decrease in evaporator pressure is caused by water accumulation in the liquid phase in the same evaporator, as a consequence of the traces of water in the refrigerant supplied from the generator system. This increase in water concentration in the evaporator liquid phase * To whom correspondence should be addressed. Tel.: +52 55 91758446. Fax: +52 55 91758258. E-mail: [email protected].

can be alleviated by a liquid purge (blowdown), which can be expressed as a mass fraction of the refrigerant vapor flow rate2 at the evaporator exit. On the other hand, it has been observed that, as the liquid refrigerant subcooler efficiency increases, the overall coefficient of performance increases.3 Also, when the boiling point of the refrigerant in the evaporator is increased and/or the exit temperature of the absorber is reduced, and/or the economizer efficiency is increased, and/or the condenser exit temperature is decreased, the cycle efficiency increases and the entropy generation is reduced. It has been reported that the generator is the key part of the absorption refrigeration cycle, and, for that reason, it is fundamental to take great care in its design.4 The operating pressure of this component is a function of the heat rejection medium temperature at the condenser entrance (air or cooling water) and the approach temperature in the same condenser. This pressure should not be excessively high, because of the high purchase cost of the column and the higher operating costs of high pressures. Another consideration is that of the concentration of the ammonia-rich solution supplied to the generator; this is independent of the cycle driving heat temperature. It is dependent solely on the process conditions in the evaporator and absorber. If the reflux ratio in the column increases with respect to the minimum reflux ratio, the number of equilibrium stages in the distillation column is reduced; however, the areas of the condenser and reboiler are increased, as well as heating Table 1. Economic Parameters concept

value

comments and references

7884 h per annum from Pineda19 12.5% based on CETES, Bank of Mexico20 inflation 3% from Bank of Mexico21 amortization 5 years from Polley and Haslego22 cooling water cost 0.0242 US$/m3 from PEMEX23 electric power cost 0.0663 US$/kWh from PEMEX23 cost index for equipment 1305 from Loh et al.6 1061.9 from Loh et al.6 538.7 from ref 24 394 from ref 24 exchange rate, Mexican pesos 10.9033667 from Bank of Mexico to U.S. dollars operation man-hours interest rate

10.1021/ie800828w CCC: $40.75  2009 American Chemical Society Published on Web 01/09/2009

1958 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009

Figure 1. Regions of the function for shell and tube heat-exchanger costs.

and heat rejection requirements. Another relevant aspect is that, for low heat source temperatures, lower heat rejection temperatures are required, to allow more flexibility in the design process.4 The design of the absorption refrigeration cycle is dependent on the refrigeration level required by the process stream to be cooled, the pressure and temperature conditions of the heat rejection, and the driving heat media. In the same way, it is dependent on the values adopted for the previously mentioned design variables, of which there are an innumerable series of possible combinations to satisfy the same refrigeration service. Herein, a simultaneous optimization of approach temperature in the absorber, generator, condenser, liquid refrigerant subcooler and reboiler, as well as the evaporator purge and the economizer efficiency, in conjunction with the reflux ratio minimizing the annualized operating and capital costs. A mixed-integer nonlinear programming (MINLP) model developed by Cha´vez-Islas and Heard,5 with the inclusion of discontinuous capital cost functions (multiple regions of equipment size) for the main system components, was used. The functions generated for the majority of the main components of the refrigeration cycle were based on data generated using the ICARUS process evaluation program.6 For the remainder of the equipment (the distillation column), information that has been reported by Seider et al.7 was used. Upper and lower practical limits were established for the design variables, for the purpose of generating parameters in agreement with the type of system under study. The modified MINLP model was applied to the case studies reported by Cha´vez-Islas and Heard.5 A sensitivity analysis is presented for evaluation criteria such as coefficient of performance and exergy efficiency, as well as the cost optimization, as the design parameters were freed one by one to become variables until all were optimized simultaneously. Two different heat rejection media were considered: cooling water and air. Previous Studies Fernańdez-Seara and Sieres8 analyzed the effects of ammonia purification, as well as those of evaporator liquid carryover and purge in AWRSs with flooded evaporators. To this end, they developed a mathematical model that took into account the distillation column efficiency and the liquid carryover and purge parameters. By means of heat-exchanger efficiencies, they quantified the irreversibilities in the absorber, the economizer,

and the liquid refrigerant subcooler, as well as the ammoniarich solution pump. The model permitted the simulation and analysis of the operation of the system. The analysis was performed by varying the distillation column parameters, the evaporator carryover and purge, and the system operating conditions; each one was varied separately while keeping the remaining parameters constant. The same authors9 described and quantified the negative effects of residual water content in the refrigerant based on the design variables that determine the efficacy of the purification process. The efficacy was analyzed in terms of the efficiency of both the rectification and stripping sections (design parameters) of the distillation column, in conjunction with the reflux ratio and the generator temperature (operating conditions). The implications of the design parameters and operating conditions were presented and discussed; these parameters and conditions included the concentration of the distillate vapor, the key operating parameters (such as evaporator and condenser pressure), the concentrations of the refrigerant-rich and refrigerant-poor solutions, and mass flow rates, as well as the liquid quality at the evaporator exit and the coefficient of performance of the system. The analysis covered the operating conditions with heat rejection to cooling water and air, as well as applications to air conditioning and refrigeration by changing the evaporation temperature. The analysis was performed by varying the design parameters and the operating conditions one at a time and keeping the remaining parameters constant. Artificial neural networks (ANNs) have also been applied to the analysis of the functioning of AWRSs. Sencan10 presented a new formulation based on a neural network model to determine the cycle coefficient of performance and the flow ratio. The estimation of these two criteria was dependent on the temperatures in the generator, evaporator, condenser, and absorber, as well as the concentrations of the refrigerant-rich and refrigerantpoor solutions. Equations derived from the neural network were used to calculate these parameters. The advantages claimed for the neural network included speed, simplicity, and the capacity to learn from examples. In the case of this work, the conditions used to teach the network were limited and, therefore, it is to be expected that the range of conditions over which it could provide reliable results would be equally limited. To increase the range of applicability of a neural network, it is necessary to teach it over a wide range of conditions with carefully selected examples. Bulgan11 presented the optimization of a single-stage AWRS by simulating various cases. A theoretical model was developed for the refrigeration cycle where the criterion was the maximization of the coefficient of performance for various evaporator, condenser, and absorber temperatures. An analysis of the degrees of freedom to calculate the number of free design parameters was conducted. Some of these were determined by the heat rejection medium and the refrigeration requirements. The remaining variables were used to maximize the coefficient of performance of the system. The cycle that was considered had no rectifier, and the nonequilibrium generator is modeled by using a coefficient. The economizer and liquid refrigerant subcooler were modeled with the effectiveness method. Expressions were used to model the pressure drop from the evaporator to the absorber and from the generator to the condenser. A computer program was written to facilitate the optimization study. The functions and subroutines used Newton-Raphson numerical methods and direct substitution to calculate thermodynamic properties,

Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1959 Table 2. Cooling Water (CW) Design Parameters and Variable Ranges: Case 1 concept

1A(CW)

1B(CW)

1C(CW)

1D(CW)

1E(CW)

1F(CW)

1G(CW)

purge subcooler approach (K) economizer efficiency reflux ratio, (Lr/R)min reboiler approach (K) absorber approach (K) condenser approach (K)

0.01-0.1 8.3 0.865 1.051 5.6 5.6 5.6

0.01-0.1 2.8-11.1 0.865 1.051 5.6 5.6 5.6

0.01- 0.1 2.8-11.1 0.8-0.9 1.051 5.6 5.6 5.6

0.01-0.1 2.8-11.1 0.8-0.9 1.02-1.25 5.6 5.6 5.6

0.01-0.1 2.8-11.1 0.8-0.9 1.02-1.3 2.8-11.1 5.6 5.6

0.01-0.1 2.8-11.1 0.8-0.9 1.02-1.25 2.8-8.3 2.8-11.1 5.6

0.01-0.1 2.8-11.1 0.8-0.9 1.02-1.25 2.8-11.1 2.8-11.1 2.8-8.3

a

a

Expressed in terms of mass fraction.

Table 3. Air-Cooling (AC) Design Parameters and Variable Ranges: Case 1 concept

1A(AC)

1B(AC)

1C(AC)

1D(AC)

1E(AC)

1F(AC)

1G(AC)


0.01-0.1 2.8 0.865 1.05 5.6 22.2 22.2

0.01-0.1 2.8-8.3 0.865 1.05 5.6 22.2 22.2

0.01-0.1 2.8-8.3 0.8-0.88 1.05 5.6 22.2 22.2

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 5.6 22.2 22.2

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 4.4-8.3 22.2 22.2

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 4.4-8.3 19.4-25 22.2

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 4.4-8.3 19.4-22.2 22.2-23.3

a

a


Table 4. Cooling Water (CW) Design Parameters and Variable Ranges: Case 2 concept

2A(CW)

2B(CW)

2C(CW)

2D(CW)

2E(CW)

2F(CW)

2G(CW)


0.01-0.1 2.8 0.865 1.05 4.4 5.6 5.6

0.01-0.1 2.8-8.3 0.865 1.05 4.4 5.6 5.6

0.01-0.1 2.8-8.3 0.8-0.88 1.05 4.4 5.6 5.6

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 4.4 5.6 5.6

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 2.8-10 5.6 5.6

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 2.8-10 5.6-11.1 5.6

0.01-0.1 2.8-8.3 0.7-0.9 1.02-1.25 2.8-10 5.6-11.1 2.8-11.1

a

a


Table 5. Air-Cooling (AC) Design Parameters and Variable Ranges: Case 2 concept

1A(AC)

1B(AC)

1C(AC)

1D(AC)

1E(AC)

1F(AC)

1G(AC)


0.01-0.1 2.8 0.865 1.075 2.8 22.2 22.2

0.01-0.1 2.8-8.3 0.865 1.075 2.8 22.2 22.2

0.01-0.1 2.8-8.3 0.8-0.9 1.075 2.8 22.2 22.2

0.01-0.1 2.8-8.3 0.8-0.9 1.03-1.25 2.8 22.2 22.2

0.01-0.1 2.8-8.3 0.8-0.9 1.03-1.25 2.8-8.3 22.2 22.2

0.01-0.1 2.8-8.3 0.8-0.9 1.03-1.25 2.8-8.3 19.4-25 22.2

0.01-0.1 2.8-8.3 0.8-0.9 1.03-1.25 2.8-8.3 19.4-25 21.7-25

a

a


specific heats, and coefficients of performance. The optimization variables were generator temperature (equivalent to the reboiler temperature) and the mole fraction of ammonia in the refrigerant (distillate from the generator). Bayramoglu and Bulgan12 analyzed a small-scale AWRS with a simple process scheme and a low-temperature heat source. They applied short-cut design cost model methods in an approximate optimization.13 The correlations of Guthrie14 were used to calculate the installed cost of the equipment. Even in 1994, when the paper was written, these correlations were somewhat dated. The installed costs were translated into annualized values using a factor of 1/3 per annum and estimating the costs of auxiliary services. It was assumed that the equipment sizes are continuous and that there were no regions where construction materials change according to equipment operating conditions. The objective of the sensitivity analysis was to minimize the total annual cost, which is comprised of capital and operating costs. In this approximate optimization, a base case was designed, after which one optimization or design variable was changed slightly, leaving the remaining parameters constant. The incremental change in the total annualized cost then was divided by the incremental change in the selected variable. Eleven variables were chosen to limit the number of

variables to those of greatest impact on the total annualized cost: heat source temperature, condenser approach temperature, reflux ratio, ammonia solution concentration, and temperature increase in the evaporator. Bulgan15 presented a study based on low-temperature heat sources in the range of 85-115 °C for a single-stage ammonia-water absorption refrigeration cycle. A single-pass reboiler was used on the column, and complete condensation for the reflux and top product stream (ammonia at better than 99%) was performed. A model was programmed, which was used to perform studies of the effects of design parameters that are important for the functioning of the system. The criteria of operation that were considered included the coefficient of performance and the flow ratio. The system model consisted of a mass and energy balance for the cycle. The design variables that were restricted by the environment and the design options included the driving heat source temperature, the cooling water temperature, and the refrigeration temperature. Other variables were chosen to eliminate iterative calculations as much as possible. The effect of one design parameter (flow ratio, ammonia-rich solution concentration, driving heat source temperature) was evaluated when that parameter was changed while the other parameters were kept constant at their reference value.


Figure 2. Case 1: Trends of evaporator purge (top) and ammonia purity (bottom) as parameters are converted to design variables.

Figure 3. Case 1: Trend of distillation column cost as parameters are converted to design variables.

A compromise must be made between the cost of the column and the efficiency of the plant, because the reflux ratio adversely affects the coefficient of performance, as well as benefiting costs, by reducing the number of trays needed in the column to maintain a given refrigerant composition. Because the economy of the process is as important as the efficiency, the author recommended that a balance be struck by imposing a minimum acceptable coefficient of performance while optimizing for a maximum rate of return. The author considered that thermo-

dynamic efficiency and economic performance had the same weight. In industrial practice, this is unlikely to be the case, except that regulations, laws, or taxes (e.g., carbon taxes) have an impact on the allowable or economically optimum design. The analysis of Koc et al.4 has shown that the use of a lowtemperature heat source puts certain strict limits on the design of an AWRS, which could be avoided through the use of sufficiently cold cooling water or by considering other process schemes. In addition, the simple refrigeration cycle was

Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1961

Figure 4. Case 1: Trends of liquid refrigerant subcooler (top) and reboiler (bottom) approach temperatures as parameters are converted to design variables.

economically designed through a careful optimization of the process using simplified models for the equipment, process restrictions, and research into possible improvements from a base case. This simulation study investigated the effects of design variables (driving heat source temperature, ammoniarich solution composition, and the approach temperatures in the absorber, condenser, and evaporator) on the cooling water consumption, electric power consumption, and the heat-transfer areas. For this, a base case design was established and then each variable was changed within practical limits while the remaining parameters were held constant at the base case values. However, the authors concluded that, to overcome the driving heat source lower temperature limit, changes in the process flow scheme would be necessary. Note that this study did not consider the use of a liquid purge from the kettle-type flooded evaporator, which is a basic design option when one must use a lowtemperature heat source. The disjunctive programming techniques applied to discontinuous cost functions were developed in the work of Tu¨kay and Grossmann.16,17 The first of these two papers16 showed that an algorithm of generalized disjunctive programming was rigorous in the handling of discontinuities in complex cost functions, as well as being robust and efficient for process flow diagram optimization problems. The complex cost functions were defined over intervals of equipment size, operating pressures, and operating temperatures. Their respective disjunctions could be used to formulate capital costs in optimization models of fixed flow diagrams or for superstructures in process systems. In addition, the authors showed that the formulation of the convex hull produces a MINLP model for fixed process configurations that

could be successfully solved. This is contrary to the case of optimizing the process scheme configuration, which produced a generalized disjunctive programming model with disjunctions in itself. A new algorithm was proposed to resolve this problem, whose main characteristic involved the selection of MINLP model subproblems. They applied these developments to examples. The second publication17 was directed toward the optimization of process models that considered discontinuous cost functions with fixed charges and were defined by some ranges of equipment size. An application of the disjunctive programming approximation based on the formation of the convex hull of disjunctions and the disjunctive branch-and-bound (B&B) technique with linear underestimators. A theoretical comparison between the formulation of the convex hull, the big-M model, and the model with linear underestimators was presented. The proposed solution methods were proven with examples of heat exchange networks, process flow design, and the synthesis of a petrochemical complex. It was shown that the convex hull of disjunctions gave the best overall behavior of the proposed techniques, because it had a tendency to be computationally efficient and produce robust reliable results. The basis of the these two publications was reported previously by Tu¨rkay and Grossmann.18 This paper proposed a MINLP model based on the logic and solution structure for the synthesis of process networks. The generalized disjunctive programming model was supported by expression of restrictions through propositions and logical disjunctions, with which the nonlinear programming subproblems only require the equations that are relevant to the existing units.


Figure 5. Case 1: Trends of adsorber (top) and condenser (bottom) approach temperatures as parameters are converted to design variables.

Figure 6. Case 1: Trend of economizer effectiveness as parameters are converted to design variables.

Disjunctive models allow redundant restrictions to be eliminated and avoid the solution of nonlinear restrictions in the case of zero flows. Another advantage of the disjunctive model was that it allowed the elimination of the restrictions for nonexistent process units, thus reducing difficulties with nonconvexities. In addition, because the size of the problem

was reduced, the solution time of the nonlinear progamming subproblems was also reduced. Problem Statement Given a simple ammonia-water absorption refrigeration cycle, finding the best operating conditions to minimize the


Figure 7. Case 1: Trend of reflux ratio as parameters are converted to design variables.

Figure 8. Case 1: Trend in total annualized cost as parameters are converted to design variables.

Figure 9. Case 1: Coefficient of performance as parameters are converted to designed variables.

combined capital and operating costs is desirable. In addition, the degree of compatibility with this economic criteria and those of thermal integration (such as coefficient of performance, efficiency, and cycle reversibility criteria) is of interest. To this end, a fixed process scheme was considered, and practical limits were established for each design variable, which included evaporator purge, heat exchanger approach temperatures (liquid

refrigerant subcooler, absorber, reboiler, and condenser), economizer efficacy, and the ratio between the minimum and real reflux rates. As well as the aforementioned, nonlinear capital cost correlations for the main equipment items were used where the functions can be continuous or discontinuous. The MINLP reported by Cha´vez-Islas and Heard was applied with the inclusion of economic functions and necessary additional


Figure 10. Case 1: Irreversibilites as parameters are converted to design variables.

Figure 11. Case 2: Trends in evaporator purge (top) and ammonia purity (bottom) as parameters are converted to design variables.

preliminary design functions for these economic functions (the thickness and weight of the column). Basis of the Modified MINLP Model The basis of the modified MINLP model consists of two parts: technical and economic. The former has been described in detail in Cha´vez-Islas and Heard,5 whereas the economic part can be described as follows. However, in the distillation column, the capital cost is dependent on the weight of the vessel shell and headers. Thus, additional characteristics of the column are determined: design pressure, shell diameter, shell thickness, length between tangents, and weight.

The basis of the economic parameters considered are shown in Table 1. The year 2006 was used for the operating and capital costs in the study. The Marshall & Swift indices and Chemical Engineering indices were used to bring the equipment costs from a given year to the year 2006. The average annual inflation rate for 2006 was used, together with the interest rate and the amortization period, to transform the capital cost to constant annual payments, using eq 1:

[

K1 ) K where

icomb(1 + icomb)n (1 + icomb)n - 1

]

(1)


icomb ) iint + iinflation + (iintiinflation) Note that the low-pressure steam that is considered as the driving heat source for the system has no price, because it is waste heat. Because the optimization and applications considered are for the chemical, petrochemical, or petroleum refining industries, only shell and tube heat exchangers were considered, in accordance with the conservative design practices of these industries. For the various equipment sizes, the capital costs of the shell and tube heat exchangers, the air-cooled exchangers, and the triplex solution pump are taken from Loh et al.6 These authors used the ICARUS process evaluation program

(version 5) to generate a series of point costs for a range of heat exchangers. The costs generated correspond to installed equipment costs, because ICARUS estimates costs from first principles (i.e., it “builds” the equipment from a proprietary database of raw materials, services, man-hours, transport, packaging, unpacking, installation, inspection, paint, foundations, structural steel, etc.) to attain an installed cost. In the case of air-cooled heat exchangers, this, of course, includes the fans, motors, their electrical installation, etc. As previously mentioned, the FOB cost of the distillation column is a function of its weight, which includes nozzles, inspection

Figure 12. Case 2: Trend in distillation column cost as parameters are converted to design variables.

Figure 13. Case 2: Trends in liquid refrigerant subcooler and reboiler approach temperatures as parameters are converted to design variables.


Figure 14. Case 2: Trends in absorber (top) and condenser (bottom) approach temperatures as parameters are converted to design variables.

Figure 15. Case 2: Trend in economizer effectiveness as parameters are converted to design variables.

flanges, supports, etc. To this cost, an installation factor for vertical vessels was applied, as reported by Seider et al.7 This value includes man-hours and materials. A cost for this is included, because the column is considered to have sieve trays. The aforementioned information was converted to nonlinear correlations of capital cost. The shell and tube heat exchanger correlations were applied to the refrigerant subcooler, the evaporator, the absorber, the economizer, the

reboiler, and the condenser, being valid for the absorber and condenser in the case of using cooling water for heat rejection. Otherwise, correlations for air-cooled heat exchangers were used for these two components. CAIITC ) RITCAITCβITC

(2)

The correlation for the installed capital cost of the shell and tube heat exchangers is shown as eq 2 and has five regions


Figure 16. Case 2: Trend in reflux ratio as parameters are converted to design variables.

Figure 17. Case 2: Trend in the total annualized cost as parameters are converted to design variables.

Figure 18. Case 2: Trend in the coefficient of performance as parameters are converted to design variables.

for separate equipment sizes (the differing coefficients and ranges of application can be seen in eq 6). For the air-cooled exchangers, the range required is characterized with four regions, all of which are described by eq 3 for the respective coefficients and ranges of applicability. For the capital cost of the solution pump, there are two size regions (eq 4), whereas the distillation column has one region that covers

all the common sizes for this application (see eqs 5, 8, 9, and 10). The column was considered to be made of carbon steel (the preferred construction material for AWRSs), and, thus, the materials factor (FM) is unity. CAIE-air ) RE-airAE-airβE-air

(3)


CAIR-Pmp ) RR-PmpPR-PmpβR-Pmp CAIColumn ) 4.16(FMCV + CPL) + NTFNTCBT

(4) (5)

where CV ) 4.1171062WColumn (1.9 tonnes e WColumn e 5.4 tonnes) 0.59057607

CPL ) 0.48412045DColumn0.7396LColumn0.70684 (0.9 m e DColumn e 6.4 m; 3.7 m e LColumn e 12.2 m) 2.25 FNT ) 1.0414NT CBT ) 0.112367 exp(0.57053806DColumn) (0.6 m e DColumn e 4.9 m) In addition, an analysis of the sensitivity to cooling water costs is presented for subcases 1G and 2G, where its cost is increased by 25%, 50%, and 75% over the basis price.

Disjunctive programming of the capital costs of the shell and tube heat exchangers16-18 results in eq 6:

[

YITC-1 CAIITC ) 10.811AITC 9.3 e AITC e 37.2 YITC-1

0.145

[

][ ][ [

] ]

∨ CAIITC ) 9.241AITC0.198 ∨ 37.2 < AITC e 92.9 YITC-1

CAIITC ) 3.369AITC0.414 ∨ CAIITC ) 0.156AITC0.91 ∨ 92.9 < AITC e 557.4 557.4 < AITC e 836.1 YITC-1 CAIITC ) 0.232AITC0.838 836.1 < AITC e 3716.1

]

(6)

Equation 7 shows the application of the convex hull for each of the disjunctions in eq 6: 5

CAIITC )

∑ CAI

ITC-1

i)1 4

Formulation of the Modified MINLP Model Discontinuous capital cost functions for various regions of equipment size were incorporated into the MINLP model of Cha´vez-Islas and Heard.5 The disjunctive programming approximation of Tu¨kay and Grossmann17,18 was applied. The discontinuities, with respect to equipment size, are modeled with disjunctions that are converted to mixed-integer restrictions using the convex hull formulation for each disjunction. The selection of the correct equipment size intervals is modeled with binary variables. The configuration of the system does not change and corresponds to a simple cycle. The basic concept of the formulation with the convex hull is to decompose the heat-transfer areas (heat exchangers) into power demand x (pump) and the capital cost c in variables xi and ci, respectively, for each size region I ∈ D (where D is the number of intervals of equipment size) and select the appropriate region with the binary variables yi. This also allows natural representation of the discontinuities in the real cost functions. Annualized Capital Costs. The heat-transfer areas specified in the mass and energy balance for the refrigerant subcooler, evaporator, absorber, economizer, reboiler, and condenser are disaggregated into five area variables, which are identified with each region of the shell and tube heat-exchanger capital cost function, as shown in Figure 1. The same occurs with the equipment cost.

YITC-1

AITC )

∑A

ITC-i

i)1

CAIITC-1 ) 10.811AITC-10.145 CAIITC-2 ) 9.241AITC-20.198 CAIITC-3 ) 3.369AITC-30.414 CAIITC-4 ) 0.156AITC-40.910 CAIITC-5 ) 0.232AITC-50.838 9.3yITC-1 e AITC-1 e 37.2yITC-1 37.2yITC-2 < AITC-2 e 92.9yITC-2 92.9yITC-3 < AITC-3 e 557.4yITC-3 557.4yITC-4 < AITC-4 e 836.1yITC-4 836.1yITC-5 < AITC-5 e 3716.1yITC-5 5

∑y

ITC-i ) 1

(7)

i)1

The cost function for air-cooled heat exchangers (four regions) and the solution pump (two regions) is applied in the same way. The distillation column capital cost correlation applies to the weight range of 1.9-5.4 tonnes and has only one correlation region. However, because the capital cost correlation is expressed in terms of column weight, the expressions for the

Figure 19. Case 2: Trend in irreversibility as parameters are converted to design variables.


column wall thickness and design pressure needed to be included in the modified MINLP model, because both values are required in the column weight calculation. For the column wall thickness, in accordance with ASME codes for pressure vessels, the formulation in eq 8 was used: ts )

14.4946PdDColumn 28.9892SE - 17.39352Pd

(8)

where Pd and S are given in units of bar and the values of DColumn and ts are given in meters. The values for S and E are 1034.86814 bar (15 000 psi) and 0.85, respectively.7 The design pressure correlation is shown in eq 9: Pd ) exp[3.06685193 + 0.924521593 + ln(P0) + 0.0015655(ln{P0})2] ⁄ 14.4946003 (9) where Pd and P0 are given in units of bar. The column weight correlation is shown in eq 10: WColumn ) [π0.3048(DColumn + ts)(3.2808399LColumn + 2.624672DColumn)tsF] ⁄ 1000 (10) where the values of DColumn, ts, and LColumn are given in meters, F is the density of steel (F ) 7849.2 kg/m3), and the value of WColumn is given in tonnes. When the annualized capital cost expressions were incorporated into the model, the indices to correlate the costs to U.S. dollars in the year 2006 were also applied. Objective Function The objective function expression was composed of the sum of the annualized capital and operating costs, which were minimized within the model. When the cooling water heat rejection medium case was considered, the objective function took the form of eq 11: CTAAE )

[

ITC

∑

]

IMS2006 CAIR-Bom + CAIi + IMS1998 i)1

CE2006 CAIColumn + COAEFAEHHoper + COEEPR-BomHHoper CE2000 (11) Otherwise, when air-cooled heat rejection exchangers are considered, the objective function took the form of eq 12: CTAair )

[

ITC

∑

IMS2006 CAIR-Bom + (CAIi)i*3,5 + IMS1998 i)1 E-air

∑ CAI

]

j

j)1

+

CE2006 CAIColumn + CE2000

[

COEEHHoper PR-Bom +

E-air

∑P

Vent-j

j)1

]

(12)

Application to Case Studies, Results, and Analysis of the Same The case studies are the same as those described by Cha´vezIslas and Heard.5 However, in the optimization phase, a sensitivity analysis of the total annual cost, cycle coefficient of performance, and overall irreversibility or exergy, relative to the progressive change of fixed parameters into design variables, was performed. That is, parameters were freed to take variable values within their application interval. Initially, the evaporator purge was freed as a design parameter, releasing others until reaching the simultaneous optimization of the seven previously

Table 6. Effect of Cooling Water Cost on Design Variables: Case 1 Scenarios (1G) item

COAE

1.25 COAE

1.5 COAE

1.75 COAE

purgea subcooler ∆t (K) economizer effectiveness reflux ratio, (Lr/R)min reboiler ∆t (K) absorber ∆t (K) condenser ∆t (K)

2.025 13.6 0.791 1.25 9.7 9.0 2.8

2.025 13.3 0.791 1.25 9.8 9.0 2.8

2.025 11.4 0.791 1.25 9.8 8.9 2.8

2.025 10.2 0.811 1.25 9.7 8.9 2.8

a

Expressed in terms of mass percent.

described design variables. The sensitivity analysis was performed for both heat rejection media (water and air). To this end, seven subcases were developed. In the first (1A or 2A), the evaporator purge was allowed a range of 0.01-0.1 mass fraction of the vapor flow in the evaporator. The second (1B or 2B) was the liquid refrigerant subcooler, and the third (1C or 2C) added the economizer efficiency. For the fourth subcase (1D or 2D), the reflux ratio was liberated, in addition to those variables previously mentioned. In the fifth subcase (1E or 2E), the reboiler approach temperature was added; in the sixth subcase (1F or 2F), the approach temperature of the absorber was freed. Finally, the condenser approach temperature was liberated in the seventh subcase (1G or 2G). In each subcase, the variables were optimized to minimize the annualized sum of the capital and operating cost. Tables 2-5 show the allowable ranges for each of the design variables in each subcase and the fixed values of those that remained as parameters. Tables 2 and 3 respectively correspond to water and air cooling for Case 1, and Tables 4 and 5 respectively correspond to water and air cooling for Case 2. Results Analysis. Case 1. In Figure 2, the evolution of the optimum evaporator purge rate as the design variables are freed one by one is shown. When cooling water is used as the heatrejection medium, the purge has a tendency to be less than that in the case of air cooling. This is as would be expected, because the lesser condensation temperature that is possible with water cooling would make the production of a higher-purity refrigerant with a given driving heat source feasible. However, it can be seen that the other design variables do have an effect on the optimum level of refrigerant purge, especially in the case of air cooling. As might be anticipated, the variations in refrigerant purity as the various design variables were released mirror the behavior of the purge rate. In the case of the annualized capital cost of the distillation column, there are appreciable variations as the design variables are liberated (see Figure 3). The rise in capital cost as the economizer efficiency was freed while the reflux ratio, absorber, and condenser approach temperatures were still fixed parameters shows the importance of this component to the overall cycle running costs. As the reflux ratio and the approach temperatures are added to the design variables to be optimized (see Figures 4 and 5), there was an appreciable reduction in the optimum efficiency of the economizer (see Figure 6). This tendency reduces the thermodynamic efficiency of the cycle as the internal integration of the system is reduced. When the reflux ratio is added to the optimization design variables, this increases and thus reduces the number of equilibrium stages in the column, slightly reducing its size (see Figure 7). The overall annualized cost (Figure 8) is progressively reduced as more design variables come into play. For the watercooled case, the main impact of optimization does not occur until all the design variables have been optimized simultaneously. However, in the air-cooled case, it can be observed

1970 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 Table 7. Impact of Cooling Water Costs on Total Costs: Case 1 Scenarios (1G) item

COAE

1.25 COAE

1.5 COAE

1.75 COAE

annual total cost (thousands of USD) annual operating cost (thousands of USD) annual capital investment cost (thousands of USD) increase in cost (%)

382.2 77.0 305.2

399.8 94.5 305.2 23

417.2 111.7 305.5 45

434.7 128.3 306.3 67

Table 8. Effect of Cooling Water Cost on Design Variables: Case 2 Scenarios (1G) concept

COAE

1.25 COAE

1.5 COAE

1.75 COAE

purge subcooler ∆t (K) economizer effectiveness reflux ratio, (Lr/R)min reboiler ∆t (K) absorber ∆t (K) condenser ∆t (K)

3.76 4.9 0.700 1.20 2.8 6.0 2.8

3.76 4.9 0.700 1.20 2.8 6.0 2.8

3.76 4.9 0.704 1.20 2.8 6.9 2.8

3.77 4.9 0.731 1.17 2.8 6.9 2.8

a

a

Expressed in terms of mass percent.

that there is a more gradual progression as more variables enter into the optimization. In the case of cycle coefficient of performance and irreversibilities (shown in Figures 9 and 10), the former in both the water-cooled and air-cooled cases has a tendency to decrease as the system becomes more optimized (for total annualized cost), whereas the irreversibilities increase. Case 2. Case 2 involves lower-temperature cooling duties than the previous case and, therefore, is more demanding of the design of the absorption refrigeration system. In both waterand air-cooled options, the evaporator liquid purge rate is almost unchanged, except when the reboiler approach temperature is liberated while still keeping the absorber and condenser approach temperatures as fixed parameters for the air-cooled case (see Figure 11). The purge rate is 2%-4% of the refrigerant vapor flow rate for this case, compared to 2%-3% in the watercooled option for the first case and 5%-7% for the air-cooled option. The corresponding ammonia purities in the column dome are also shown in Figure 11. Figure 12 shows how the annualized capital cost of the distillation column gradually decreases as the number of design variables included in the optimization increases. The optimized approach temperature in the liquid refrigerant subcooler is much greater than that used as a fixed parameter (see Figure 13). It can be observed that the optimized reboiler approach temperatures for both cooling media are almost the same. In the case of the approach temperatures in the absorber and the condenser (see Figure 14), the result is similar to the first case. However, in the second case, the approach temperature in the absorber is ∼5.6 K, instead of 8.3 K for the water-cooled option, reflecting the advantage of lower approaches when the refrigeration is required at a lower temperature. With regard to economizer efficiency, when this is optimized, the result is much less than that initially fixed as a parameter, although in the air-cooled case, when the absorber and condenser approach temperatures are also optimized, the optimum economizer approach temperature is higher (see Figure 15). In Figure 16, it can be observed that the optimum reflux

ratio is somewhat higher than that used as a fixed parameter, which would result in less equilibrium stages in the column. In this case, the total annualized operating and capital costs are lower with a water-cooled system rather than the air-cooled option, resulting from the optimization in case 1 (see Figure 17). The effect of including more design variables in the optimization has a clear impact on total costs and shows the importance of including all those considered. Under the fully optimized condition, it is also apparent that this does not represent the most thermodynamically efficient design (see Figures 18 and 19). Sensitivity to Cooling Water Pricing. As an example of how the optimization result can be affected by a change in one of the operating costs for the system, a series of optimizations with varying cooling water costs were performed. It was found that, for Case 1, the following design variables underwent significant changes: the liquid refrigerant subcooler approach temperature (reduced with rising cooling water cost), economizer efficiency (increase with rising cooling water cost), and capital cost (small rise with rising cooling water cost) (see Tables 6 and 7). The total annualized operating cost for the water-cooled option in Case 1 increases considerably, reinforcing the advantage of air cooling in this case. In Case 2, the following design variables show changes due to the cooling water price: evaporator liquid purge (small increase at the highest cooling water cost), economizer efficiency (increases with increasing cooling water cost), and the column reflux ratio (reduced as the cooling water cost increases) (see Tables 8 and 9). The annualized capital cost of the system is only slightly affected by the increase in cooling water costs; however, the operating costs and, therefore, the total annualized cost increases significantly with rising cooling water costs. Even so, the total annualized cost increase does not increase to a value greater than that of the fully optimized air-cooled system. Conclusions The modified mixed-integer nonlinear programming (MINLP) model for the simulation of a simple ammonia-water absorption cooling cycle with discontinuous cost functions for the main equipment items has been shown to provide a practical means of optimizing the design variables for an ammonia-water absorption refrigeration cycle. The two example cases studied show that all of the design variables included in the optimization have a significant impact on the resulting design and total annualized cost of such a system. To attempt such an optimization by means of parametric studies of the impact of the design variables included would be

Table 9. Impact of Cooling Water Costs on Total Costs: Case 2 Scenarios (1G) item

COAE

1.25 COAE

1.5 COAE

1.75 COAE

total annualized cost (thousands USD) annualized operating cost (thousands USD) annualized capital investment cost (thousands USD) increase in cost (%)

322.9 42.3 280.6

331.9 51.3 280.6 21

340.9 60.2 280.7 42

349.8 68.2 281.6 61


impractical, because of the vast number of combinations of parameters required and the discontinuous nature of many of the cost functions. Ammonia-water absorption refrigeration is capable of providing reliable, cost-effective industrial and commercial refrigeration services, using a variety of heat sources and heat sinks, the cost of which is strongly dependent on the location of the plant. The plant construction materials are mostly carbon steel and the working fluids are commonly available, inexpensive, and natural constituents of the atmosphere. This observation means that these systems must be optimized for each case and a robust MINLP model can be an efficient tool to enable such an optimization. Acknowledgment The authors would like to acknowledge the financial support of the Instituto Mexicano del Petro´leo Postgraduate programme and the Mexican National Council for Science and Technology, and the invaluable technical advice and encouragement of Dr. Ignacio E. Grossmann. Supporting Information Available: Formulation of the convex hull for the cost functions for air-cooled heat exchangers and the solution pump (two regions), to determine the cost functions of each type of equipment, is shown. (PDF) This information is available free of charge via the Internet at http:// pubs.acs.org. Nomenclature A ) heat-transfer area CAI ) annual capital investment cost (thousands USD/yr) CBT ) base cost of perforated plates (thousands USD/yr) CE2006 ) Chemical Engineering equipment cost index to 2006 CE2000 ) Chemical Engineering equipment cost index to 2000 COAE ) operating cost of cooling water COEE ) operating cost of electricity CTAAE ) annual total cost in the case of cooling water (thousands USD/yr) CTAair ) annual total cost in the case of air (thousands USD/ yr) CPL ) added cost for platforms and ladders (thousands USD/yr) Cv ) cost of the vertical column, based on the weight of the shell and two heads, including nozzles, manholes, a skirt and internals (not including plates) (thousands USD/yr) E ) fractional weld efficiency FM ) material of construction factor for pressure vessels FBM ) bare module factor for columns; FBM ) 4.16 FNT ) factor that is dependent on the number of trays FAE ) cooling water mass flow rate HHoper ) operating time iint ) discount rate iinflation ) annual inflation rate icomb ) effective combined discount rate IMS2006 ) Marshall & Swift equipment cost index to 2006 IMS1998 ) Marshall & Swift equipment cost index to 1998 K1 ) annualized periodic payment of equipment K ) present value of equipment investment cost n ) amortization period NT ) number of trays PR-Bom ) pump power demand (kW) PVent-j ) power of air cooler fan j S ) maximum allowable stress shell material at the design temperature

WColumn ) distillation column weight (tonne) Y ) logical variable Greek Letters R ) annual cost correlation coefficient β ) annual cost correlation exponent F ) density of carbon steel; F ) 7849 kg/m3 Subscripts ITC ) shell and tube heat exchanger ITC-1 ) subcooler ITC-2 ) evaporator ITC-3 ) absorber ITC-4 ) economizer ITC-5 ) condenser ITC-6 ) reboiler E-air ) air cooler R-Bom ) triplex pump Column ) distillation column i ) regions of the capital cost function of the shell and tube heat exchangers EE ) electricity

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(22) Polley, G.; Haslego, Ch. Using plate exchangers in heat recovery network, part 2. CEP Mag. 2002, 98, 48. (23) Gerencia de comunicacioń social y Superintendencia de fuerza y services principales de la Refinerıá “Ingeniero Antonio M. Amor (RIAMA)”, PEMEX, Salamanca, Gto., Mexico. E-mail correspondence. (Information obtained under Mexican Freedom of Information legislation, July 17, 2007.) (24) Chemical Engineering Plant Cost, Economic Indicators. Chem. Eng. 2007, 113 (August), 80.

ReceiVed for reView May 23, 2008 ReVised manuscript receiVed October 23, 2008 Accepted November 24, 2008 IE800828W

Optimization of a Simple AmmoniaâWater Absorption Refrigeration

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