Bo Liu ,Yufei Wang, *,Xiao Feng

1 State Key Laboratory of Heavy Oil Processing,China University of Petroleum,Beijing 102249,China

2 School of Chemical Engineering &Technology,Xi’an Jiaotong University,Xi’an 710049,China

Keywords:Circulating cooling water system Uncertainty Chance constrained programming Design Optimization Simulation

ABSTRACT Recent research on deterministic methods for circulating cooling water systems optimization has been well developed.However,the actual operating conditions of the system are mostly variable,so the system obtained under deterministic conditions may not be stable and economical.This paper studies the optimization of circulating cooling water systems under uncertain circumstance.To improve the reliability of the system and reduce the water and energy consumption,the influence of different uncertain parameters is taken into consideration.The chance constrained programming method is used to build a model under uncertain conditions,where the confidence level indicates the degree of constraint violation.Probability distribution functions are used to describe the form of uncertain parameters.The objective is to minimize the total cost and obtain the optimal cooling network configuration simultaneously.An algorithm based on Monte Carlo method is proposed,and GAMS software is used to solve the mixed integer nonlinear programming model.A case is optimized to verify the validity of the model.Compared with the deterministic optimization method,the results show that when considering the different types of uncertain parameters,a system with better economy and reliability can be obtained(total cost can be reduced at least 2%).

1.Introduction

Circulating cooling water systems (CCWSs) are widely used in the field of process industry for waste heat removing.In a refinery,cooling water normally account for 70%-80%water usage,so large amount of power are consumed to transport cooling water from cooling tower to coolers.Therefore,it is of great importance to optimize CCWSs to reduce power consumption.

Recent research of CCWSs design mainly focuses on improving energy performance by optimizing the various components of the system,including cooling tower [1],pump system [2] and cooler network [3],etc.The interaction between the components were also involved,e.g.cooling tower and coolers network[4].The most popular methodology used to optimize CCWSs are Pinch based approach[5],mathematical programming[6]and the combination of the both [7].For retrofit problem,Picón-Nú?ezet al.[8] introduced new coolers into existing systems by integrating them in parallel branches.Zhuet al.[9]constructed an effective evaluation index system to evaluate the energy saving status of the CCWS.At the same time,a three-step energy-saving debottlenecking diagnostic method [10] is proposed for retrofit problem.In addition to the above description,air coolers were integrated with CCWS[11] to share the heat load between water cooler and air cooler.

The above studies are all designed or retrofitted under deterministic circumstance.However,many parameters change all over the operation period,such as ambient temperature,air humidity,and flow rate,etc.These factors are seldom considered in previous works.The uncertain parameters have a great impact on many systems,including geological systems [12],petroleum systems [13],and chemical systems [14].For the cooling systems widely used in chemical industries,the different sensitivities of the operating of air cooling and water cooling to uncertain factor indicates that the duty distribution between water cooling and air cooling should vary over the operation period to obtain optimal operation.

To describe uncertainty,Pistikopoulos [15] classified uncertain parameters into four categories and modelled general problems of uncertainty.For some commonly used uncertain optimization methods,Grossmannet al.[16] pointed out their development problems,and optimized some cases to introduce corresponding solutions.Based on the related research of the last decade,Chenet al.[17] focused on describing the development,application,and prospect of three uncertain optimization methods,namely,robust optimization,stochastic programming,and chance constrained programming.Among these uncertain optimization methods,stochastic programming is extremely difficult to calculate and requires methods to reduce the computation difficulty of solving[18].Robust optimization is too conservative [19].Chance constrained programming can overcome the conservativeness of robust optimization to a certain extent and is relatively easy to model and solve [20].

The concept of chance constrained programming(CCP)was first proposed by Charnes and Cooper[21].CCP is a stochastic programming method with a known probability distribution of uncertain parameters.Because the worst case may occur,the decision may not meet the constraints,and the decision variables may violate the requirements of the constraints to a certain extent,confidence levels [22] are introduced to describe the extent of constraint violations,so that the system can operate normally.CCP models are generally solved by converting the objective function and constraints into corresponding deterministic equivalents,based on given confidence levels and probability distribution functions.Geng and Xie [23] conducted a relatively comprehensive review of the solutions to the CCP problem,and summarized them into scenario methods,sampling average approximation methods,and robust optimization related methods.These methods mainly solve the model through three ways,namely Monte Carlo simulation[24],linearization [25],relaxation and approximation [20].The CCP method is used to solve uncertainty problems in water resources management [26],micro-grid system scheduling [27],etc.This paper proposed a new algorithm based on the Monte Carlo method to cope with the CCWS related uncertainty problems for it is a commonly used method [24,28,29].

The uncertain conditions have significant impact on the design and operation of cooling water system,especially for the heat duty distribution of water and air cooling.Considering these factors,this work proposes an optimization model considering the uncertainty of the CCWS.The CCP method is used to model the system.Multiple uncertain parameters with different distribution functions are considered simultaneously.A Monte Carlo based algorithm is designed according to the model to reduce the computational difficulty and obtain optimal results.The cooler network configuration is optimized and the total cost is minimized simultaneously.The best heat load distribution method between air cooling and water cooling is also obtained.Therefore,the innovations and contributions of this article mainly include the following aspects:

i.Considering the simultaneous optimization of multiple uncertain parameters in the CCWS,rather than taking the average value of the parameters for optimization.

ii.All the elements included in a CCWS are optimized simultaneously.

iii.A simple algorithm based on Monte Carlo method is designed.

iv.The relation between system reliability and economy under different confidence level is explored.

2.Problem Statement

2.1.Introduction of CCWS structure

A typical circulating cooling water system consists of a cooler network unit,a pump system unit,and a cooling tower unit.Given the cooling target,the hot streams are cooled to their target temperature by cooling water in water coolers.The used hightemperature cooling water returns to the cooling tower.And there is a part of blowdown and evaporation loss in the cooling tower,so it is necessary to make up water.

This paper studies a system that considers air coolers and series-parallel cooling water network,as shown in Fig.1.In this work,the supply temperature,target temperature and heat capacity of hot streams are given.The hot streams are first cooled by air coolers and then by water coolers.The used cooling water may be reused in another water cooler so that the total flow rate of cooling water can be lower.

In industry,air temperature is changed along with season and weather,and process stream parameters are often changed due to production fluctuation.In this work,air temperature and flow rate of hot streams are considered as uncertain parameters and can be described as different probability density distribution functions.These factors have significant influences on the system configuration,especially on the heat load distribution of air cooling and water cooling.The capacity of air coolers and cooling tower can be weakened when the air temperature is high.The change in the flow rates of the hot streams means that the total heat load is changing all the time.The cooling load that the CCWS needed to provide is changing consistent with this trend.

The objective function is to find the cooling water network configuration with minimum total cost while the multi-parameters uncertainty is taken into consideration.The optimal cooling water network considers the series-parallel structure of the water coolers to reduce the total flow rate of cooling water,which contains integer problems.In addition,the objective is a function of multiple uncertain variables.The model is a complex nonlinear model.Therefore,the optimization problem is a MINLP (mixed integer nonlinear programming)problem.The CCP method is used to build the model,and the MINLP problem is solved with the proposed Monte Carlo based algorithm.

Some assumptions are applied in this work:

? In addition to the selected uncertain parameters,the other parameters are assumed to be constant.

? In order to avoid fouling,the maximum return water temperature of the cooling water is 55 °C.

? The pump supply head is only affected by the largest head pressure of the cooler network.

2.2.Introduction of the CCP method

Chance Constrained Programming refers to a situation where the constraints contain random variables.Before decision making,all the random variables should be predicted in advance.Considering that the decision may not meet the constraints when an adverse situation occurs,it is necessary to adopt a principle that allows a decision to be made that violates constraints to a certain extent[22].Therefore,a certain confidence level is given to restrict such violation.

Pis the form of probability.α,β are the given confidence levels.f(x,θ)is a function of the decision variablexand uncertain parameter θ.When the probability thatf(x,θ) is not greater thanis not less than α,and the probability that the other constraints are satisfied is not greater than β,the objective function is to find the minimumthat satisfies these requirements.

Fig.1. The improved CCWS configuration.

The premise of solving the CCP model is that the probability density distribution function of the uncertain parameters is known and can be fitted according to historical data.Typical distributions include normal distribution,uniform distribution,Cauchy distribution,etc.The confidence level α and β are artificially determined value,which are greater than 0 and less than 1.When they are equal to 1,the model is equivalent to directly solving the minimum value of the function.Constraints are not allowed to be violated,and all possible worst-case effects are considered.When the confidence level value is closer to 1,the probability of constraint violation becomes smaller,indicating that the final design is good in reliability but bad in economy.The value of the confidence level can be artificially specified or determined according to the correspondence between the confidence level and the final optimized value of the objective function.

Only CCP models with specific forms can be transferred to the corresponding deterministic equivalents for solving,in most cases the models are difficult to transform directly and require new algorithms to solve.From the above discussion,it can be known that the CCP model is easy to establish and can achieve the optimal target with a certain probability.And the key to solve the model is to convert it into the corresponding approximate deterministic equivalent,and to design an algorithm that can reduce the computational difficulty.Therefore,a CCP model is established and the Monte Carlo-based algorithm is designed to avoid the complex transforming process.

3.Modeling and Algorithm Design

As mentioned earlier,CCWS can be divided into four parts,air coolers unit,water coolers unit,pump unit,cooling tower unit.They are modeled separately as follows.

3.1.The objective function

The objective function of the system is to minimize the total cost,and the total cost is the sum of the costs of the various parts,and,as shown in Eq.(2).

whereCuis the uncertain total cost with all the uncertain parameters and cannot be solved directly.Ca,Cw,Cp,Ctare the total cost of the air cooler,the water cooler,the pump,the cooling tower,respectively.

3.2.Constraints

3.2.1.Air cooling process

In CCWSs,the hot stream may be first cooled by an air cooler,the air is heated and the temperature of the hot stream is reduced.The cost of the air cooler section includes equipment costs and operating costs.

whereCacrepresents the capital cost,Caorepresents the operating cost.

The equipment cost is mainly related to the heat exchange area of the equipment.

In the formula,Afis the annual average cost factor,cais a model constant.iis number of the hot stream.It is worth noting that each hot stream can only be cooled by one water cooler and one air cooler.So the water cooleriis the water cooler that cools the hot streami,the air cooleriis the air cooler that cools the hot streami.Therefore,Aa(i) in Eq.(4) is the area of the air cooleri.

The heat exchange area of an air cooler can be calculated from its heat load,the approximation method is used to obtain logarithmic mean of temperature [30].

Qa(i) is the heat load of air cooleri,the calculation method of the average temperature comes from literature [30].ΔTai(i) is the heat transfer temperature difference between the hot stream and air at the entrance of the air cooleri.ΔTao(i) is the heat transfer temperature difference between the hot stream and air at the exit of the air cooleri.h(i)is the heat transfer coefficient of hot streami.

Andhais the heat transfer coefficient of air.vnfis the actual face velocity of air cooler,vfis the face velocity of air cooler,and their relation is shown in Eq.(6) [31].

Tais the air temperature,and the symbol～indicates that the parameter is uncertain.

Each hot stream is connected to at most one air cooler.The heat load of air cooleriis equal to the heat lost of hot streami.

Thi(i) is the initial temperature of the hot streami,Thao(i) is the temperature of the hot streamiafter being cooled by air,Tao(i) is the air temperature at the outlet of the air cooleri,ΔTminis the minimum heat transfer temperature difference.Fa(i)is the air flow rate in air cooleri.Fh(i) is the flow rate of hot streami,which is another set of uncertain parameters considered in this work.

The operating cost of an air cooler refers to the cost of consumed electricity.

Among them,Pfa(i) is the fan electricity power consumption[32]of air cooleri,peis the electricity power price,tis the operating time,ΔPais the pressure drop of air cooler fan,Va(i) is the volume flow rate of air in air cooleri,fris the air cooler friction factor,Nbis number of bundles in air coolers,Gmaxis the maximum mass velocity rate,Gis the mass velocity rate,ρa(bǔ)is the density of air.From Eqs.(3)-(13),the total cost of air cooling can be calculated.The total cost is affected by uncertain parameters,namely air temperature and hot stream flow rates.

3.2.2.Water cooling process

3.2.2.1.Water cooling cost.Only investment costs are considered in the cost of the water cooler.

wherea,bare constants of water coolers,Aw(i) is the area of water cooleri.

Like air coolers,the heat exchange area of water coolers is determined by the heat load,the temperature can be calculation by the Chen approximation [30].

In the above formula,Qw(i)is the heat load of the water cooleri,Tho(i)is the target temperature of hot streami,Two(i)is the temperature of the cooling water at the outlet of the cooleri,andTwi(i)is the inlet temperature of the cooleri.Fin(i)is the water flow rate of the cooleri.AndAw(i)in Eq.(16)is the area of water cooleri,hwis the heat transfer coefficient of water,ΔTwo(i) is the temperature difference between the hot stream and the cooling water at the outlet of the water cooleri,ΔTwi(i) is the temperature difference between the hot stream and the cooling water at the outlet of the water cooleri.

3.2.2.2.Cooling water network.In order to increase the temperature of the return water,increase the efficiency of the cooling tower,and reduce water flow rate,a series-parallel structure of cooling water network is considered [33].Z(i,j) is a binary variable that can describe the connection relationship between different water coolers.iandjrepresent two different coolers.WhenZ(i,j)is equal to 1,cooleriis connected to coolerj.WhenZ(i,j) is equal to 0,cooleriand coolerjare not connected.

The coolerican only connect to one coolerjat most,as shown in Eq.(19),and can only be connected to one coolerjat most,as shown in Eq.(20).The cooler cannot be connected to itself,as shown in Eq.(21).

Therefore,the inlet water flow rate and temperature of the cooleriare either equal to the flow rate and temperature of the fresh water from the cooling tower,or equal to that of the outlet water of the coolerjconnected to it.

Tcwis the cooling water temperature from the cooling tower,Fin(i) is the water flow rate of the cooleri,andFcw(i) is the water flow rate from the cooling tower to the cooleri,Ftis the total water flow rate.

When uncertain parameters exist,the structure of the cooling water network may change,and the cost is different from the results under certain conditions.

3.2.3.Pressure head of pumps

The pump transports water from cooling tower to individual water coolers.

whereCpoandCpcare the operation cost and capital cost,respectively.Both of them are determined by the pressure drop of the cooler network.

cfpandcpare constant.ΔPpis the maximum pressure head in all branches of the cooler network [34].

Pp(i) is the pressure head of the cooleri,ρ is the density of the water,and ηpis the pump efficiency,Kpis conductivity,hpis the film transfer coefficient.

3.2.4.Cooling tower

Model equations related to cooling towers are mainly quoted from literature [7,35].The total cost (Ct) of the cooling tower includes operating costCtoand capital costCtc[35].

Ctofis the cost of the cooling tower fan.

The air flow rate of cooling tower fan is expressed as the following form [36]:

The evaporation flow rate of the cooling tower is a function of total flow rate and the humidity difference of inlet and outlet air[35].Eis the amount of water evaporation,woandwiare the air humidity of inlet and outlet cooling tower fan separately:

cfis the fan factor,Fcais the air flow rate of the cooling tower fan,and ηtfis the efficiency of the cooling tower fan.MaandMware molecular weight of air and water.Pais the atmospheric pressure.

The water vapor pressurePsis a function of the average inlet and outlet temperature [37].

Fm,Tm,pmare the flow rate,temperature and price of makeup water respectively,tis the annual operation time,πcis the cycle of concentration [7].

Fbis the flow rate of blowdown water.

Ris the temperature difference between cooling tower inlet temperatureTciand outlet temperatureTco.

Apis the temperature difference between cooling tower outlet temperatureTcoand air wet bulb temperatureTwb.

Qtis the heat load of cooling tower.cpis the specific heat of water.

3.3.The CCP model

In this work,the original objective function is rewritten into the form of chance constrained programming.BecauseTaiandFh(i)are uncertain parameters that obey different distributions,all the variables related to them in the foregoing are uncertain,so that the original model cannot be directly solved.The important point is that except the objective function containing chance constraints,which can be partially violated,other constraints must be strictly satisfied.

Given a confidence level α,the objective function Eq.(2)can be rewritten as:

The objective function becomes to find a minimum,subject to Eq.(45):

The corresponding constraint becomes that the probability thatis greater than all possible values ofCuis not less than α.Cucan be regarded as a function of air temperature and hot stream flow rate,so Eq.(45) can be rewritten as Eq.(46).

The probability distribution function of the uncertain parameter is known,so the value ofCucan be calculated by multiple integrals.

p(·) represents the probability distribution function.

3.4.A Monte Carlo based algorithm

A new optimization framework base on Monte Carlo simulation is proposed to avoid the complex transformation process.The algorithm is shown in Fig.2.And the details are as follows:

a.Establish a chance constrained programming model.

b.Collect and fit the historical data of uncertain parameters to obtain the corresponding probability distribution function.

c.GenerateNsets of random numbers based on the parameter’s probability distribution function.

d.Bring the above random numbers into the model and calculate the corresponding total costCuofNgroups.

e.Sort the total cost ofNgroups in order.

f.According to the set confidence level,select the total cost that meets the requirements.

Fig.2. The framework for the Monte Carlo based algorithm.

Table 1 Hot stream parameters

g.Recalculate the total cost according to the uncertain parameters obtained when calculating the total cost in step f.This total cost is thethat meets the probability requirements.

h.Change the value of the confidence level αiand repeat the step c,d,e,f and g to get a series of optimal solution.

i.According to the calculation results in steps f,g,h,the rule that the optimal solution changes with the confidence level is obtained,and the most reasonable confidence level and the corresponding solution are selected as the final optimal solution.

4.Case Study and Result Analysis

In this section,a case from Maet al.[33]is used to show the priority of our work.The initial case only considers system design and optimization under certain fixed conditions.In this case,we consider the optimization in the following situations:

Fig.3. The optimal system configuration of initial case.

Firstly,the optimization results of the original case under nominal conditions are obtained.Secondly,consider the optimization of the ten hot streams with uncertain flow rates at the same time.Thirdly,consider the optimization where the ambient temperature is changing all the time.Finally,consider the optimization when the hot stream flow rate and the ambient temperature vary simultaneously.

4.1.Case 1:Initial case

The optimal design of the case in the literature is obtained when the air temperature is 25°C,the cooling tower outlet temperature is 20°C and the characteristic parameters of hot streams are fixed.The same operating parameters are selected in Case 1.The parameters of the hot streams are shown in Table 1.The purpose is to obtain the optimal CCWS design under general conditions.

The optimal cooling configuration is shown in Fig.3.Air coolers and water coolers share the total cooling task.The initial temperatures of these hot streams are relatively high,and the temperature difference between them and air is sufficient to complete part of the cooling task.Therefore,most of the hot streams are first cooled by the air cooler and then cooled by the water cooler to complete the heat exchange target.The use of air coolers reduces the need for circulating water compared to systems that only use water coolers.

With the series water cooler structure,the water return temperature is increased,and the amount of circulating water is reduced.In addition,the final outlet temperature of the water cooler is all approximately 55°C.This reason is that there are many hot streams whose target temperature is greater than 80°C or even higher,so it is possible to provide enough heat transfer driving force for heating limited cooling water to its upper bound (55 °C).

In this case,9 air coolers are used,which is less than the number of water coolers.And water coolers are configured in series structure,the total cost of water-cooling part is relatively low,resulting in a high proportion of water-cooling load and a low proportion of air-cooling load.And only the economics of the design is considered,the resource and environmental problems caused by water consumption are not involved,so the water-cooling part shares more cooling duty.

4.2.Case 2:Optimization under uncertain hot streams flow rates

The fluctuating flow rates data is fitted using uniform distribution functions (Fh(i)～U(Fh0(i)-5,Fh0(i)-5).The ten hot streams’changing mode are not completely consistent.The heat load varies with the flow rate.To study the effect of flow rates alone,the ambient temperature is still set to be 25 °C.

The optimized cooling network configuration is obtained when the value of confidence level is 0.8,the specific parameters and settings are shown in Table 2.The optimized network structure is similar to that in Fig.3,indicating that uncertain flow rate of hot streams in this case does not have great impact on the configuration of cooling system.The reason is that in this case,the flow rate of some hot streams increases,and some decreases.The overall effect is that the total heat load does not change significantly.Therefore,the deviation of uncertain flow rate is not that large.

4.3.Case 3:Optimization under uncertain ambient temperature

In this case,the ambient temperature is considered as uncertain parameter.By fitting historical data,the normal distribution function of ambient temperature is obtained asTa～N(25,6).

The optimal network configuration of the system considering temperature difference has changed significantly.The results are obtained at a confidence level of 0.8.All the hot streams are first cooled by the air coolers.Three pairs of water coolers are connected in series,and the other four water coolers are connected in parallel.Compared with case 1,the total area of the air cooler is increased,and that of the water cooler is reduced in this case.The connection method of the water cooling part is also changed.All the details are shown in Table 2.

The area of both water cooler and air cooler are increased accordingly.Meanwhile,the increase in air cooling area is lower than water cooling area.The first reason is that the increase of air temperature (5 °C) is lower than cooling water supply temperature(about 7°C).The second reason is that the temperature of hot stream in air cooler is much higher than that in water cooler,and due to the small temperature difference in water cooler,area is more sensitive to the temperature difference.In order to ensure the lowest total cost of the system and a lower cooling water flow rate,the cooling duty of the system is achieved through more air cooling and less water cooling.Moreover,in some branches,the outlet temperature of cooling water is too high to continue to provide cooling load,so the number of series structure is lower.For the water coolers that still use reused cooling water,it can be found that the initial and target temperatures of hot streams in these coolers are all high.From the outlet temperature of all the cooling water branches,it can be found that the total cooling water flow rate is tend to minimum.

4.4.Case 4:Optimization under uncertain ambient temperature and uncertain flow rates

When both the flow rates of hot streams and the ambient temperature change over all the time,the designed system is more in line with the real operating condition.Consider that the flow rates obey the uniform distribution of case 2 in Section 4.2 and the ambient temperature obeys the normal distribution of Case 3 in Section 4.3.

Fig.4 is the optimized result when the confidence level is 0.8.This structure is with 10 air coolers and four pairs of water coolers in series.It can be found that the connection method of the water coolers is changed,and the total area of the air cooler and watercooler in this case is increased.In this case,the air temperature is about 31 °C and the cooling tower outlet temperature is 28.3 °C,which is higher than other cases.

Table 2 Parameters of the optimal network configuration for the 4 cases (α=0.8)

Fig.4. The optimal configuration under uncertain Ta and uncertain Fh(i) (α=0.8).

Fig.5. The heat load distribution of air cooling and water cooling.

4.5.Discussion

4.5.1.Comparison for optimal configuration

Table 2 shows the calculation results of the optimal network configuration parameters for the four cases.The difference in the optimal structure is mainly reflected in the number of air coolers,the connection method of water coolers,and the total water flow rate of the cooling tower.The specific differences and reasons have been described in above sections.The main difference for the results is caused by the changing heat load distribution methods of the water cooling and air cooling under different uncertain conditions.

The difference in heat load distribution among the four cases can be clearly seen in Fig.5.In Case 1 and Case 2,the air temperature and cooling water temperature are relatively low,so the water-cooling part accounts for a larger proportion.In Case 3 and Case 4,both the air temperature and the water temperature rise,and the water temperature rises by a large extent,so the proportion of air-cooling increases.

4.5.2.Comparison for cost components

Fig.6 shows the cost curves varying with the confidence level for the four cases.

The confidence level represents the probability that the constraint needs to be satisfied.The larger the value,the more likely it is to get the optimal solution in the worst case.Fig.6(a) is a graphical display of the changing costs for Case 2 with different confidence levels.In this case,the confidence level does not have great impact on the costs when uncertainty of the flow rate is considered.The reason is that at a certain level of confidence,the flow rates of some hot streams are increased and others are decreased,so the fluctuation of total heat load is not that significant.Fig.6(b)shows the various costs for Case 3.Almost all the cost increases with the increasing confidence level.On the contrary,only the cost of water coolers declines with increasing confidence level.It can be seen that the total cost increases sharply after the confidence level exceeds 0.9,which indicates that the optimal solution is much more conservative.Therefore,when the decision makers finally choose the appropriate confidence level,they can consider a value below 0.9 to maintain an economic system.In Fig.6(c),the total cost,cooling tower cost,pump cost and air cooler cost all increase as the confidence level increases.Only the cost of water coolers are almost constant,since with the increased water temperature,the use of water coolers is decreased,but the operating cost of the cooling tower is significantly increased

Fig.6. The costs comparison under different conditions for the 4 cases.

Fig.7. Cost comparison for the 4 cases at the same confidence level (α=0.8).

It can be seen from the Fig.6(d)that the total cost of Case 1 is a constant,and the total cost of other cases increases with the increase of the confidence level.In this figure,it can be found that with the increased confidence,at some point,TAC is slightly decreased.The reason is that the value of uncertain parameters are set randomly in each case.For instance,for uncertain flow rate,the total heat duty of a case with lower confidence level is high,then the TAC may be higher.It can be also found that when the confidence level is very high,the TAC of Case 3 and Case 4 are significantly higher than the other cases.The reason is that when the confidence level is very high,the system must design to accommodate very extreme air temperature condition,leading to a very high TAC.But for Case 1 and Case 2,air temperature is set to be constant.

In order to further compare the uncertainty optimization effects considering different parameters,the results of the above four case optimizations at the same confidence level are compared.The related result is shown in Fig.7.The total costs of the four cases are 9.06 × 105USD,8.29 × 105USD,9.73 × 105USD,7.27 × 105USD,respectively.The costs of the four components in the system and the operating costs and capital costs of each part are compared separately.Case 1 has the highest total cost because the ambient temperature and the flow rate of the hot stream remain unchanged.So with the uncertain flow rates and varying temperature,Case 4 has the lowest total cost.The cost of air coolers has changed significantly,showing an increasing trend.The cost of cooling tower and water coolers also varies greatly.Because these structures are directly affected by temperature and flow rates.

5.Conclusions

In this work,uncertain factors are considered in the circulating cooling water system,which has not been investigated in previous research.Ambient temperature and hot stream flow rate are set to be uncertain parameter in this design.In order to obtain a reliable and economical system,chance constrained programming is used to establish an MINLP model.A new Monte Carlo based algorithm is designed to reduce computational difficulty.GAMS software is used to implement the model.

This article gives the relationship among temperature,flow rates and CCWS network configuration.From the results,it can be found that the uncertainty of hot stream flow rate do not have great impact on the system.The air cooling is increased when considering the uncertainty of ambient temperature,and the water cooling is decreased when the uncertain flow rate is taken into consideration.Under the same confidence level,the optimization considering multiple uncertain parameters is much better than other optimization schemes.And about 20% of the total cost can be reduced compared with deterministic design.The effect of confidence levels on the system has also been studied.When the confidence level becomes high,the worst condition tends to emerge,so the system reliability tends to be higher,but the economic benefit may be poorer.The schemes using the CCP method can balance the relationship between system reliability and operational economic.Decision makers can choose an appropriate value of confidence level accordingly.

Nomenclature

Aa(i) area of air cooleri,m2

Afannual average cost factor

Aw(i) area of water cooleri,m2

Aptemperature difference between cooling tower outlet temperature and air wet bulb temperature,°C

a,bconstants of water coolers

Catotal cost of air coolers,USD

Caccapital investment cost of air coolers,USD

Caooperating cost of air coolers,USD

Cpcost of the pump,USD

Cpooperation cost of the pump,USD

Cpccapital cost of the pump,USD

Cttotal cost of the cooling tower,USD

Ctooperating cost of cooling tower,USD

Ctcinvestment cost of cooling tower,USD

Ctofcost of the cooling tower fan,USD

Cuuncertain total cost,USD

caa constant of air coolers

cfcooling tower fan factor

cfp,cpconstant of pump

cp,cpaThe specific heat of water and air,kJ·kg-1·°C-1

Eamount of water evaporation,kg·s-1

Fa(i) air flow rate in air cooleri,kg·s-1

Fcaair flow rate of the cooling tower fan,kg·s-1

Fcw(i) water flow rate from the cooling tower to the cooleri,kg·s-1

Fh(i) flow rate of hot streami,kg·s-1

Fin(i) water flow rate of the cooleri,kg·s-1

Fmflow rate of makeup water,kg·s-1

Fttotal water flow rate,kg·s-1

frair cooler friction factor

Gmass velocity rate

Gmaxmaximum mass velocity rate

h(i) heat transfer coefficient of heat streami,kW·m-2·°C-1

haheat transfer coefficient of air,kW·m-2·°C-1

hpfilm transfer coefficient,kW·m-2·°C-1

hwheat transfer coefficient of water,kW·m-2·°C-1

inumber of heat streams,air coolers and water coolers

jdifferent coolers fromi

Kpconductivity,W·m-2·°C-1

Ma,Mwmolecular weight of air and water

Nbnumber of bundles in air coolers

Paatmospheric pressure,Pa

Pswater vapor pressure,Pa

peelectricity power price,USD·kW-1·h-1

pmprice of makeup water,USD·t-1

Pfa(i) fan electricity power consumption of air cooleri,kW

Pp(i) pressure head of water cooleri,Pa

ΔPapressure drop of air cooler fan,Pa

ΔPpmaximum pressure head in all branches of the cooler network,Pa

Qa(i) heat load of air cooleri,kW

Qtheat load of cooling tower,kW

Qw(i) heat load of the water cooleri,kW

Rtemperature difference between cooling tower inlet and outlet temperature,°C

Taambient temperature,°C

Tao(i) air temperature at the outlet of the air cooleri,°C

Tcwcooling tower outlet temperature of cooling water,°C

Tmtemperature of makeup water,°C

Thi(i) initial temperature of the hot streami,°C

Thao(i)air coolers’ outlet temperature of the hot streami,°C

Tho(i)target temperature of hot streami,°C

Two(i) coolerioutlet temperature of the cooling water,°C

Twi(i) inlet temperature of water cooleri,°C

TciTcocooling tower inlet temperature and outlet temperature,°C

Twbair wet bulb temperature,°C

toperating time,h

ΔTai(i) heat transfer temperature difference at the entrance of air cooleri,°C

ΔTao(i) heat transfer temperature difference at the exit of air cooleri,°C

ΔTwo(i) the temperature difference at the outlet of water cooleri,°C

ΔTwi(i) the temperature difference at the inlet of the water cooler,°C

ΔTminminimum heat transfer temperature difference,°C

Va(i) volume flow rate of air in air cooleri,m3·s-1

vfface velocity of air cooler,m·s-1

vnfactual face velocity of air cooler,m·s-1

wo,wiair humidity of inlet and outlet cooling tower fan

Z(i,j) a binary variable that describes the connection relationship between different water coolers

ηppump efficiency

ηtfefficiency of the cooling tower fan

πccycle of concentration

ρ density of the water,kg·m-3

ρa(bǔ)density of air,kg·m-3

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Financial support from the National Natural Science Foundation of China (22022816,22078358) are gratefully acknowledged.