Synthesis of indirect work exchange networks considering both isothermal and adi

Yu Zhuang,Linlin Liu,Lei Zhang,Jian Du*

Institute of Chemical Process Systems Engineering,School of Chemical Engineering,Dalian University of Technology,Dalian 116012,China

Keywords:Work exchange networks Transshipment model Adiabatic process Exergy analysis Isothermal process Work cascade

ABSTRACT In this paper,an efficient methodology for synthesizing the indirect work exchange networks(WEN)considering isothermal process and adiabatic process respectively based on transshipment model is first proposed.In contrast with superstructure method,the transshipment model is easier to obtain the minimum utility consumption taken as the objective function and more convenient for us to attain the optimal network configuration for further minimizing the number of units.Different from division of temperature intervals in heat exchange networks,different pressure intervals are gained according to the maximum compression/expansion ratio in consideration of operating principles of indirect work exchangers and the characteristics of no pressure constraints for stream matches.The presented approach for WEN synthesis is a linear programming model applied to the isothermal process,but for indirect work exchange networks with adiabatic process,a nonlinear programming model needs establishing.Additionally,temperatures should be regarded as decision variables limited to the range between inlet and outlet temperatures in each sub-network.The constructed transshipment model can be solved first to get the minimum utility consumption and further to determine the minimum number of units by merging the adjacent pressure intervals on the basis of the proposed merging methods,which is proved to be effective through exergy analysis at the level of units structures.Finally,two cases are calculated to confirm it is dramatically feasible and effective that the optimal WEN configuration can be gained by the proposed method.

1.Instruction

Energy is a major concern in the 21st century,whose worldwide demand is predicted to rise by 57%during 2004–2030[1].The total primary energy consumption is predicted to arise gradually all over the world in the past few decades,where the industrial sector was the largest consumer of energy accounting for over 20%[2].As a result,the increase of energy efficiency is of vital importance in transformation processes due to its dominating responsibility for a large portion of expenditures and decisive actions on environmental aspects.The main reasons to develop techniques for sustainable energy utilization with high efficiency are the global greenhouse effect and the increasingly costly energy because of the rapid reduction in the available fossil fuels[3].In other words,it is particularly critical to conserve energy in industrial plants.

The two most common forms of energy in these plants are heat and work.In spite of the fact that work is much more costly and has higher energy quality than heat,far more extensive studies have been conducted to optimize the heat exchange networks(HEN)than work exchange networks(WEN),the former of which has been widely adopted to recover more thermal energy in actual production[4].Additionally,it has been demonstrated that HEN is critical to promote the decrease of gas emissions and fossil fuels consumption,as reducing energy consumption has great relations with the improvement of heat transfer.Thereby,optimal HEN can be attained consisting in promotion for thermal integration of the whole system through an effective network design,in thermodynamic and even economic terms including minimal number of heat exchangers and minimum utilities consumption[5].

Despite considerable efforts to synthesize HEN achieving the outstanding results,the other form of energy widely used in chemical plants,such as work,is rarely paid attention to,particular in the aspect of how to integrate work efficiently.In oil refineries,cryogenic processes such as the production of liquefied natural gas(LNG)[6]and synthetic processed like methanol and ammonia synthesis[7],it is vitally significant to take the responsibility for considerable energy consumption by handling pressure,where some streams need work for compression while others can undergo expansion to produce work[8].However,very few papers have been published to describe the work integration and work has been poorly explored in process synthesis during the actual operation.Since work exchanger networks,as an important part of energy recovery systems,will have significant influence on energy consumption in process systems,it is significantly meaningful to integrate work between high-pressure(HP)streams and low-pressure(LP)streams.Furthermore,it would also be possible to realize the integration of both heat and work simultaneously to further conserve energy in the same network.

Work sources and work sinks can exchange work via direct or indirect work exchangers.Liu et al.[9]reported that the direct work exchanger was mainly composed of a pair of combined operating piston pumps.The mechanical energy can be transferred from work sources to work sinks directly with 100%recovery efficiency of a piston pump in theory.Nevertheless,it increases over reliance on the inlet and outlet pressures of streams that possess highly nonlinear relationship with work quantity.In addition,direct work exchangers may stay unstable during operation and conduce to bad performance of the system.In contrast,the indirect work exchangers can remain with higher stability and stronger operating performance.Hence,this paper mainly focuses on the use of indirect work exchangers.

In indirect work exchangers,namely, single-shaft-turbine-compressor(SSTC)units,energy is exchanged in two steps:the pressure energy of work sources is converted to mechanical energy through turbines at first and further converted to the pressure energy of work sinks through compressors[10],as illustrated in Fig.1.The notion of SSTC is a straightforward extension of a gas turbine manipulating a compressor occupying a common shaft.To achieve the continuous operation of indirect work exchangers,a high-pressure stream in the turbine rotates the shaft further to drive the compressor pressurizing the low-pressure stream,which can also be generalized to involve multiple turbines with HP streams running multiple compressors with LP streams sharing a common shaft[11,12].Constructing a network configuration for exchanging work in this manner would be called “indirect work exchange network synthesis”.This is direct and useful extension of the well-known heat exchange network synthesis.Although it is very similar to HENS,surprisingly,only seldom papers have developed a systematic procedure to exchange work between HP and LP streams.

Shin et al.[13]proposed a mixed integer linear programming(MILP)to optimize boil-off gas compressor operations targeting the minimization of total average energy consumption in an LNG receiving and regasification terminal.Likewise,an optimization and framework with a combination of MILP model and stochastic formulation was presented by DelNogal et al.[14–16]to integrate the power system and refrigeration process based on the previous paper published by them where an MILP model was introduced driver and power plants election using the opinion of superstructure for multistage compressors.However,their purpose is to arrange compressor stages rather than to design the whole networks.

In addition,Aspelund et al.[17,18]presented a heuristic graphical method by utilizing the pressure-exergy to minimize energy consumption under sub-ambient condition on the basis of the Extended Pinch Analysis and Design(ExPAnD).By means of the optimization of compression and expansion work for process streams and the work needed to produce essential cooling utilities,this method shows great potential for energy requirement conservation.However,since compressors and turbines are used separately,no mention is referred to the use of combinations with these pressure manipulation devices running on a common axis,where only the aspects on the exergy analysis of the system were evaluated as well.Beside this,a heuristic-based method could not offer the most reasonable network as the number of devices is ignored.Therefore,there is a need for a comprehensive structural optimization approach for indirect work exchange network synthesis as presented in this paper.

Fig.1.Structure of indirect work exchanger.

M.S.Razib et al.[19,20]proposed a superstructure for the WEN configuration and developed a mixed integer non-linear programming(MINLP)to minimize the total annualized cost(TAC)for a constant speed of the single shaft on 2-stream SSTC units.Furthermore,In the paper published by Onishi et al.[21,22],a superstructure for synthesizing HEN simultaneously is proposed with the adjustment of pressure levels of streams taken into account to improve heat integration,where several configuration possibilities including compressors,turbines and valves are discussed at a goal of minimal the TAC of the network.However,the pressure operating equipment was considered independently,such as stand-alone turbines and compressors.Later,they introduced a novel multi-stage superstructure to optimize WEN configuration with heat integration simultaneously at a constant speed of the single shaft for SSTC,also with the goal of TAC.

In respect to the exergy analysis,the exergy composite curves were adopted to explore the potential for ORC(organic Rankine cycle)process improvements[23].Then O.Ozgener et al.[24]evaluated the exergy performance of pressure reduction stations with turbo expanders to improve the potential of the system.Nevertheless,in these works the exergy analysis is aiming at the single heat exchanger or the single ORC to recover more energy and decrease the exergy loss which is mainly focusing on the temperature exergy,not relating to the pressure exergy that is involved in the turbines,compressors and work exchangers.

In what follows,based on the literature researches as mentioned before,study on the integration of WEN with multiple streams using transshipment model in the indirect work exchanger has not been reported.Hence,we employ the transshipment model with work cascadestaken into account and formulate the synthesis of work exchanger network as MINLP in adiabatic process together with exergy analysis.Then an example is used to demonstrate the benefits of this method.

2.Problem Statements

The problem in this study to be solved is as follows:

Given a set of gaseous streams at high pressure and low pressure with known mass flows,inlet pressure and outlet pressure,inlet temperature and outlet temperature,as well as utilities for work(mechanical energy)etc.,a network of work exchangers,compressors and expanders is designed in such a way that the utility consumption is minimized combined with the minimal number of units by determining the equipment configuration and its corresponding operating conditions.

In addition,the maximum compression/expansion ratio is also provided.The main objective is separately to synthesize an initial WEN with the minimum utility consumption and to obtain the optimal WEN configuration with the minimal number of units through work recovery between HP and LP streams,utilizing turbines and compressors running on a common shaft on the basis of the transshipment model.Compression ratio,expansion ratio,inter-stage pressure,inter-stage temperature,energy requirement and number of units are variables in the synthesis of this work exchange networks.

For simplicity of solving the built systematic mathematical model,the following assumptions are made:

(1)All streams are in ideal gas phase without phase transition.

(2)Only isothermal and adiabatic reversible compression/expansion is considered.

(3)The work transfer efficiency is constant in different pressure intervals.

(4)All the compressors and turbines for operating alone or in the SSTC are single-stage and centrifugal.

(5)Heatexchange among process streams is negligible excepta final heater or cooler considered.

(6)Only in adiabatic process can we consider the temperature constrains.

(7)Pressure drops and heat losses in all heaters and coolers are zero.

3.Model Formulations

3.1.Division of pressure intervals

The first step to establish the transshipment model is to divide all the pressures of streams into several intervals,marked as SNk.Since no pressure difference constraint(also referred to as driving force)is demanded for the work exchange between high-pressure streams and low-pressure streams via the SSTC,which differs from the synthesis of HEN with necessary temperature difference for heat-transfer in heat exchangers considered,a new strategy is proposed for the division of pressure intervals in accordance with the maximum compression ratio CRmaxand the maximum expansion ratio ERmax.This strategy for the division of pressure intervals guarantees that one high-pressure stream matches with one low-pressure stream through one SSTC unit.The pressure intervals,also called sub-networks,consist of two pressure endpoints of all the streams and the constructed intermediate pressure,which are arranged in a sequence pressure values from high to low.The intermediate pressure of all streams is expressed by Eqs.(1)–(2).

Obviously,the constructed intermediate pressure must be limited within a range of inlet and out let pressure of the corresponding streams,as provided by Eqs.(3)–(4).

Clearly,the pressure intervals–namely sub-networks–can be obtained by exerting these equations and the aforementioned method.

3.2.The structure and formulation of transshipment model

As depicted in Fig.1,we only consider the possibility of coupling a compressor with a turbine manipulated by a common shaft to exchange work between HP streams and LP streams.It is thus apparent that to get that the pressure driving force is not essential for exchanging work through the SSTC in the absence of pressure difference constraints.Hence,the essence of work exchange between high-pressure streams and low-pressure streams for SSTC is analogous to the threshold problem of HEN,which means that only one kind of utility(pressurization utility or depressurized utility)is used for the synthesis of WEN.In each pressure interval,the inlet and outlet pressure of all streams are limited between the minimum and maximum value of the inlet and outlet pressure in the WEN.Moreover,the pressure relationship in these sub-networks is presented by Eqs.(5)–(8),where Eqs.(5)and(6)indicate the pressures of HP streams decrease gradually which should also be restricted to the supply and target pressures while LP streams undergo compression in the increasing trend of pressures in addition to limiting within the range of inlet and outlet pressures as expressed by Eqs.(7)and(8).

Regarding to the isothermal process,under a significantly idealized condition,temperatures of all streams keep unchanged when streams flow in and outside the turbines,compressors and the SSTC.Specifically,the temperatures of all streams are equal to their respective inlet temperatures no matter which pressure interval they enter or drain out of.Moreover,for ideal gas,the relationship between pressure and volume in isothermal process can be represented as Eq.(9).

In addition,with regard to the adiabatic process,the temperatures of streams will be changed when the streams are pressurized or depressurized.Hence,all the streams are taken as ideal gas,the temperature constraints of which are stated by Eq.(10).

Eqs.(11)and(12)can be derived from the Eq.(10)combined with the state equation of ideal gas,which indicates the relationship between temperatures and pressures of all streams where the outlet temperatures will have a tendency to increase over the increment of the outlet pressures.

In each pressure interval of the transshipment model,all the streams outlet temperatures should be limited to a range of the lower and upper bounds,as expressed by Eqs.(13)and(14),

where the temperature lower bound is taken from the minimum temperature of all streams while the maximum temperature of streams denotes the temperature upper bound.Hereby,the temperature lower bound is set to 273 K and the temperature upper bound is equal to 700 K.

With the avoidance of excessively high or low temperature of streams in each pressure interval,the coolers and heaters should be added to the work exchange network so that the temperature can be restricted to the above normal range represented in Eqs.(13)and(14),by cooling or heating the corresponding streams.Due to no consideration of heat integration involved in the work exchange network synthesis,thus the coolers and heaters should be located at the end of the final pressure interval in order to attain the target temperature of the high-pressure or low-pressure streams.

Consider WLPj,kas the compression work of the low-pressure streams consumed on a SSTC in the pressure interval k,and WPU,kas the compression work of LP streams consumed by stand-alone compressors.Similarly,WHPi,kand WEU,kcan be recognized as the energy generated by turbines associated to the SSTC and energy provided by utility turbines,respectively,to high-pressure streams in the pressure interval k of transshipment model.Hence,the transshipment model is depicted in Fig.2.

Fig.2.Work exchange in pressure interval k.

The corresponding required(produced)work by compressors(turbines)is given by the following expressions.Eqs.(15)and(16)are used for computing work in isothermal process while work in adiabatic process can be obtained as represented by Eqs.(17)and(18).As reported by Couper et al.[25],the expansion work and the compression work should also be limited within a range of a lower and an upper bound,expressed as follows.

Additionally,the global energy balance should be taken into account in each pressure interval.As previously stated,the turbines associated to the SSTC and alone-stand turbines produce work,while the SSTC and stand-alone compressors consume work.Then the total generated work plus the residue work derived from the previous pressure interval must be equal to the total work of compression consumed in each compressor with the addition of excess work transferring the subsequent pressure interval.In this context,the total energy balance in each pressure interval is expressed by Eq.(23).

In conclusion,as it is discussed above,to synthesize the initial work exchange network,the objective function in this study is to minimize the total pressurized and depressurized utility of WEN according to the following expression:

where the first and the second item mean the pressurization utility and the depressurization utility,respectively.

3.3.Merging of the adjacent pressure intervals

Although the minimum utilities can be obtained based on the proposed transshipment model,an excess of units have to be utilized in order to recover more work,which results in a complex work exchange network.Therefore,it is essential to reduce the number of units with utility consumption unchanged.And then a new strategy,merging of adjacent pressure intervals,is proposed to solve this problem.A stream existing in any of the original pressure intervals having been merged is contained in the new interval.All the properties of these streams remain the same while the pressure drop ratio in the new interval is enlarged.However,this new pressure drop ratio must be equal to or less than the required maximum compression/expansion ratio.Furthermore,the work recovery between work sources and work sinks does not change,which indicates no redundant utility is needed.

The parts of all the above streams in the merged interval make up a sub-network which will be optimized independently.As for the manipulation of indirect work exchangers with no need for any pressure constraints,which is distinct from the demand of a direct work exchanger,the work value sum of the high-pressure streams is evidently still equal to that of low-pressure streams in the novel interval.In other words,the work demanded by LP streams can be exactly provided by HP streams in the sub-network and no pressurization or depressurization utility is needed.Moreover,the merging process can also be recognized as an optimized variable in the proposed method.

Then to make a detailed explanation about the merging process,a merging vector which consists of binary variables,is presented to describe the merging position over all the intervals,as depicted in Fig.3.

Note that the element in the merging vector has a one-to-one correlation with the pressure intervals.To model the existence of merging,0–1 variables are defined to represent that a value 1(such as mi)means that the corresponding two adjacent pressure intervals(PIiand PIi+1)need to be merged while a value 0 signifies no requirement of merging in the relevant position.Furthermore,if both miand mi+1have been assigned the value 1,then the PIi,PIi+1and PIi+2will be merged into one pressure interval and the rest can also be deduced by analogy.From Fig.2 it can be seen that the scale of the merging vector is one less than the number of the pressure intervals.Clearly,these binary elements are considered to be decision variables to find the optimum configuration,the initial values of which should first be assigned randomly and then be optimized targeting the minimum number of equipment.

Finally,it is significantly important to note that the intervals which contain utility streams are similarly included in this merging strategy since compression or expansion work between HP and LP streams is not affected by the utility which is also a pressurization or depressurization process via compressors orturbines.That is to say,merging these pressure intervals will not bring in a disturbance of the use of utility without any error to be created.

Fig.3.Merging vector.

Fig.4.The exergy analysis of expanders and compressors.

3.4.Exergy analysis of work-exchange units

From the perspective of the second law of thermodynamics,exergy is the driving force for the work exchange,the loss of which promotes the work exchange between work sources and work sinks.Consequently,it is necessary to analyze the feasibility of work exchange utilizing the exergy concepts and exergy economics.As the maximum exergy loss has great relations with the maximum output work,the exergy loss analysis could better reflect the performance of the work exchangers.

As for ideal gas,exergy is defined as Eq.(25)and the exergy loss via single work-exchange equipment should be defined as Eq.(26).

The exergy analysis model is established for the compressors,turbines and work exchangers,as shown in Figs.4 and 5.Based on the properties of each work-exchange device,the corresponding exergy balance expressions of the compressors and turbines are given by Eqs.(27)and(28),respectively.In addition,Eq.(29)denotes the exergy balance expression of the single work exchanger,namely,the SSTC.

According to the above exergy analysis models,the thermodynamic feasibility and rationality of the ultimately optimal work exchange networks could be verified by calculation,which is in the light of the second law of thermodynamic.

4.Case Studies

In this section,two cases from literatures are supplied to demonstrate the feasibility and ability of the presented method in isothermal process and in adiabatic process,respectively.The corresponding comparison with solutions obtained by other authors is listed after each case.

4.1.Case 1

In this example taken from M.S.Razib et al.[20],the WEN is designed to allow work integration between three high-pressure streams(HP1,HP2,and HP3)and two low-pressure streams(LP1,LP2),the problem data of which are presented in Table 1.The last row at each column in this table lists the available energy provided by the respective HP and the required energy consumed by the corresponding LP.For simplification of computing,we set some variables equal to const in this case as follows:

All the pressure values of streams are divided into 7 pressure intervals and then the above model is employed to solve each subnetwork taking the minimum utility as the objective function including 124 variables and 111 constraints.Hence,the work cascade diagram can be obtained,illustrated as Fig.6,according to the established linear programming model solved by simplex method to search local optima in each sub-network.After that,each match between high-pressure streams and low-pressure streams should be gained by analysis of all the sub-networks.Thus the corresponding initial work exchanger network configuration is shown as Fig.7.

Fig.5.The exergy analysis of indirect work exchanger.

Fig.6.The work cascade diagram for the case 1.

From the Fig.7 it can be seen that there exist up to twenty- five units including fifteen turbines,one compressor and nine SSTCs in the initial work exchanger network configuration,which results in a large redundancy at the number of units.Based on this,it is essential to optimize the initial work exchanger network by reducing the amount of work-exchange equipment according to the presented strategy of merging the adjacent pressure intervals.The optimal work exchanger network configuration is shown in Fig.8.It can be found that the number of work-exchange equipment is 8 less in the network obtained that that in initial one,which also leads to 6 less in comparison with that reported by M.S.Razib.

Compared with the solutions of Razib et al.[20]with a complex superstructure method,a better result was found by the proposed method.The details of the comparison with the literature are shown in Table 2.This table illustrates that both utility consumption and the number of work-exchange units are reduced,the former of which even has a 25.5%decrease to the solution reported by the literature.Consequently,according to the present method,a better configuration can be obtained by combination of the pressure intervals and the heuristic strategies proposed.

To sum up,as a consequence of the above analysis,the optimal results of case 1 plenarily demonstrate the high efficiency of the proposed method.

4.2.Case 2

Case 2,also taken from Razib[20],has three high-pressure streams(HP1,HP2and HP3)and two low-pressure streams(LP1,LP2),where Table 3 lists their various parameters,properties,and pressure–temperature targets.

In the calculation,these variables are still set to const according to Eq.(30).Afterwards,the adiabatic exponent should be derived from Eqs.(11)and(12).Combining work quantity with the adiabatic exponent,the product of molar flow rates and gas constant can be obtained by Eq.(31).

Fig.7.The initial work exchanger network for case 1.

Fig.8.The optimal work exchanger network for case 1.

Table 2 Comparison of the solution for case 1

where the left hand side of this equation represents the thermal internal energy while the reversible shaft work is expressed by the right hand side of the above equation.Because there is no heatloss in adiabatic process,the thermal internal energy should be equal to the reversible shaft work according to first law of thermodynamics.

Table 3 Stream data of case 2

Additionally,all the variables in the established model are nonnegative.Subsequently,the whole system is divided into 7 pressure intervals according to streams pressures and then with the above model employed,each sub-network is solved by taking the minimum utility consumption as the objective function including 152 variables and 181 constraints.Thus,the work cascade can be obtained,illustrated as Fig.9.After that,each match between high-pressure streams and low pressure streams should be gained by analysis of all the sub-networks.Thus the corresponding initial work exchanger network configuration is shown as Fig.10.

Fig.9.Work cascade of case 2.

Fig.10.The initial work exchanger network for case 2.

However,more than 30 units are needed to complete the work exchange in the initial WEN.Hence,according to the proposed strategy with the purpose of reducing the number of units,the corresponding optimal work exchange networks configuration is illustrated as Fig.11.

The minimum utility consumption is 1709.51 kW,less than 2289.9 kW of M.S.Razib,while the number of units is also slightly smaller than that in the literature,shown as Table 4.Additionally,the exergy loss of all units has been calculated to analyze the feasibility of the proposed method.Hereby,calculating the exergy loss of the compressor,turbine and SSTC marked as“red”in the Fig.10 is taken for example,respectively,based on the proposed exergy analysis models shown as Figs.4 and 5,the quantity of exergy loss of which is respectively 0.34 kJ,0.60 kJ and 43.94 kJ.Obviously,the exergy loss can almost be neglected due to the tiny values,which indicates that the available energy of HP1and HP2is mainly converted to work.However,the SSTC has more exergy loss just certifying the existed lower energy conversion efficiency in the indirect work exchangers,which also verifies the validity and the thermodynamic feasibility of the proposed method.

Fig.11.The optimized work exchange network of case 2.

Table 4 Comparison of the final solution of the case with literature

5.Conclusions

In this paper,a sequential optimization methodology for indirect work exchange network synthesis with isothermal process and adiabatic process based on the transshipment model was first proposed by introducing division of pressure intervals according to the maximum compression/expansion ratio and putting forward adjacent pressure interval merging principles for optimizing WEN configuration.Furthermore,the exergy analysis model is established for the compressors,turbines and work exchangers(SSTC)to verify the thermodynamic feasibility and rationality of the ultimately optimal work exchange networks based on the second law of thermodynamics.Two examples from literature have been studied and their satisfactory solutions justify the feasibility of the proposed method.As the efficiency of compressors(turbines)has a strong nonlinear relation with the pressure and temperature of streams,the influence of this efficiency is neglected in this paper.However,further studies should consider the effect of compressors(turbines)efficiency on the work load in addition to the pressure and temperature of streams,which will lead to the immense changes in the exergy of work units.In addition,the focus of the study can be on the design of the optimal indirect work exchange networks with the objective of the minimum exergy loss of whole networks.

Nomenclature