Simultaneous integrated design for heat exchanger

Applied Thermal Engineering 128 (2018) 1510–1519

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Simultaneous integrated design for heat exchanger network and cooling water system Fuyu Liu a, Jiaze Ma a, Xiao Feng b,⇑, Yufei Wang a a b

State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, China School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China

h i g h l i g h t s A simultaneous methodology to integrate heat exchanger network and cooling water system. A novel stage-wise superstructure for the integrated design. A MINLP model based on economic performance for the integrated design. Better design obtained through the simultaneous methodology compared with the conventional one.

a r t i c l e

i n f o

Article history: Received 14 March 2017 Revised 1 August 2017 Accepted 21 September 2017 Available online 28 September 2017 Keywords: Cooling water system Heat exchanger network Simultaneous methodology MINLP

a b s t r a c t Heat exchanger network and cooling water system are two major elements of energy systems in processing plants. Such two subjects have a very close interaction with each other. However, most of current researches firstly synthesize heat exchanger network and then design cooling water system. This sequential methodology probably misses the optimum solutions, and results in some suboptimal designs from an overall perspective. To overcome this limitation of traditional methods, in present paper a simultaneous methodology is introduced to integrate heat exchanger network and cooling water system as a whole system. Unlike conventional approaches, the methodology treats cooling water as a special cold stream whose mass flow rate, initial and final temperatures are all unknown variables and require to be optimized. The methodology mainly makes use of a modified stage-wise superstructure that covers most possible configurations for integrating heat exchanger network and cooling water system. The mathematical optimization model corresponding to the superstructure is a mixed integer nonlinear programming (MINLP) problem. The total annual cost (TAC) is set as the objective function composed by utility cost, pumping cost, and capital cost of cooling tower and heat exchanger. An industrial case study is used to demonstrate the capabilities of the proposed methodology. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays the growing prices of fossil fuel and strict environment regulations seriously force industrial clusters to improve processing plants’ energy efficiency as much as possible. Therefore, industrial energy systems, like heat exchanger networks, cooling water systems, rankine cycles and so on, have attracted high attentions from both academic and industrial practices [1]. Among these energy systems in a processing plant, a heat exchanger network and a cooling water system are two major elements which have very close relationships to energy consumption [2]. Such two subjects have been widely and deeply studied in the past few years ⇑ Corresponding author. E-mail address: [email protected] (X. Feng). https://doi.org/10.1016/j.applthermaleng.2017.09.107 1359-4311/Ó 2017 Elsevier Ltd. All rights reserved.

and various kinds of design approaches have been proposed with extensive application to industrial cases. 1.1. Synthesis of heat exchanger network The concept of heat exchanger network synthesis is initially proposed by Masso and Rudd [3] in 1970s. According to NúñezSerna and Zamora [4], the methodologies for heat exchanger network synthesis could be classified into two categories, sequential and simultaneous approaches. In principle, the sequential methodology decomposes heat exchanger network synthesis into several sub-problems. One well-known sequential methodology for heat exchanger network synthesis is Pinch Technology (PT) proposed by Linnhoff and Hindmarsh [5]. They utilized a minimum temperature difference to find the bottlenecks for energy savings which

F. Liu et al. / Applied Thermal Engineering 128 (2018) 1510–1519

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Nomenclature Sets NH NC NK

set of hot streams, NH = {i | i = 1, 2 . . . I} set of cold streams, NC = {j | j = 1, 2 . . . J} set of stages, NK = {k | k = 1, 2 . . . K}

Parameters F heat capacity flow rate, kW/°C inlet temperature of hot streams i, °C Tini outlet temperature of hot streams i, °C Touti Tinj inlet temperature of cold streams j, °C Toutj outlet temperature of cold streams j, °C q density, kg/m3 cp specific heat capacity, kJ/°C kg l viscosity, pa s j thermal conductivity, W DTmin minimum temperature difference, °C X upper bound for heat load, °C C upper bound for temperature difference, °C Lt tube pitch of heat exchanger, mm Dtin internal diameter of the tube, mm Dtout external diameter of the tube, mm De equivalent diameter of the tube, mm CU hot utility cost, $/kW y Af annual factor for capital cost. Pe price of electricity, $/kW h

was also called as pinch point. Heat exchanger network synthesis is thereby divided into designing two sub-networks above and below the pinch points. Papoulias and Grossmann [6] and Cerda et al. [7] respectively developed a transshipment and a transportation model, in which heat exchanger network synthesis was implemented as a sequentially mathematical programming problem. Energy saving target, number and area of heat exchanger are sequentially optimized step by step. However, these sequential methods may not find the optimum solution, since energy saving and capital investments are not trade off simultaneously. The other methodology for heat exchanger network synthesis is simultaneous approach that considers various factors holistically like utility expenses, pumping cost and capital investments of heat exchangers. Such methodology normally makes use of mathematical programming models to optimize an objective function subject to several heat and mass constraints. For example, Yee and Grossmann [8] studied heat exchanger network and established a stage-wise superstructure for it. The corresponding mathematical model named Synheat Model is a mixed integer nonlinear programming (MINLP) model. Since Synheat Model can cover most possible configurations for heat exchanger networks, it has been deeply studied in several previous literatures, such as Short et al. [9] and Zhang et al. [10]. Nevertheless, the aforementioned studies were mostly performed ignoring pressure drop which directly affects the capital and operation cost of driving pump. So Souza et al. [11] considered this key factor for simultaneous synthesis of heat exchanger networks. The cost statistic in their researches shows that pumping cost possesses a considerable percentage of the total costs. Some good reviews on simultaneous heat exchanger network synthesis can be found in recent works like Zhang et al. [12] and Lv et al. [13]. 1.2. Synthesis of cooling water system Cooling water system is another important element of energy systems in processing plants. It removes most waste heat rejected

Hy

annual operation time, h

Variables t q dt qu dtu A Au DP Q M Mu Md tmu R tinw toutw Pumping

stream temperature at the end of stage, °C heat load of a heat exchanger, kW temperature difference of a heat exchanger, °C heat load of a heater, kW temperature difference of a heater,°C area of a heat changer or cooler, m2 area of a heater, m2 pressure drop, Pa pump power, W mass flow rate of stream, kg/s mass flow rate of blow-down water, kg/s mass flow rate of make-up water, kg/s temperature of make-up water, °C temperature range, °C supply temperature of cooling water, °C target temperature of cooling water, °C annualized pumping cost, $/y

Binary variable z existence of a heat exchanger or cooler zu existence of a heater

from hot process streams, thus the flow rate of cooling water is particularly large as well as its capital investments and operation costs. Early attempts to cooling water system synthesis were carried out through graphic targeting methods. For instance, Kim and smith [14] initially developed a PT method for cooling water system synthesis by focusing on the systems’ components from an overall aspect. Due to the highly close interactions between cooling water networks and cooling tower performance, it is particularly necessary to take account into all components as a whole. Continuously, Kim et al. [15] extended their approach to cooling water system synthesis for effluent flow-rate reduction. Their method can rearrange some coolers from parallel to series configurations, rather than increasing the mass flow-rate of cooling water. However, their researches were mostly based on graphic targeting tools that cannot addressed capital investments and operation cost simultaneously. Thus the optimum solutions may be missed and suboptimal designs may be obtained accordingly. Actually, there exist various costs in the synthesis of cooling water systems, such as pumping costs, capital costs of cooling towers and heat exchangers. Unlike graphic methods, mathematical programming methods consider these costs simultaneously and holistically. Because of this point, many researchers suggested mathematical programming methodology for the synthesis of cooling water system. Kim and Smith [16] presented a mathematical optimization model for the retrofit of cooling water systems. An automated method based on mathematical programming was explored to find guidelines for the debottlenecking of cooling water systems. But the cooling towers and cooling water networks in their research were studied separately and this may result in suboptimal solutions. Thereby, Majozi and Moodley [17] treated cooling towers and cooling water network as a whole. A new MINLP model was presented for synthesizing cooling water systems wherein multiple towers were used to remove the waste heat on process streams to the environment. More reviews on cooling water system synthesis can be found in several papers like Sun et al. [18] and Ma et al. [19].

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1.3. The work in this paper Based on the aforementioned literature reviews, we can learn that most of existing researches study heat exchanger networks and cooling water systems individually or sequentially. Firstly, in heat exchanger network synthesis, the initial and final temperatures of cooling water are assumed as constants. Then, an optimal network for heat recovery is well-established and only the hot process streams with coolers will be sub-sequentially considered in the cooling water system synthesis. Clearly, this traditional methodology is essentially a sequential approach that cannot design the heat exchanger network and cooling water system simultaneously. In fact, the inlet and outlet temperatures of the cooling water are variables that should be determined by the stream data and relevant cost parameters. Nevertheless, most current studies treat such two temperatures as fixed constants which accordingly results in suboptimal solutions. To our knowledge, designing heat exchanger networks and cooling water systems simultaneously has not been addressed in existing literatures. In this paper, an integrated methodology is proposed for simultaneous synthesis of both a heat exchanger network and a cooling water system. The methodology treats the cooling water as a special cold stream with unrestricted flow rate, unknown supply and target temperatures. In this way, the cooling water stream can exchange heat with all hot process streams. Heat exchanger network and cooling water system are thereby simultaneously integrated as a whole. The parameters of cooling water, like mass flow rate, supply and target temperatures, can be optimized automatically. Our methodology makes use of a stage-wise superstructure to cover most possible network structures for simultaneous synthesis of the heat exchanger network and cooling water system. The mathematical model is a MINLP model, in which TAC is set as the objective function including utility cost, pumping cost, capital expenses of cooling tower and heat exchangers. This paper is organized as follows. Problem statement Section describes the proposal and some definitions. Next, methodology Section presents a new superstructure representation for simultaneous synthesis of a heat exchanger network and a cooling water system. Then, mathematical formulation Section details the model formulations. Case study Section shows an industrial example and some conclusion is drawn finally.

2. Problem statement The problem addressed in this study is briefly stated as follows. Given a set of hot and cold process streams that need be cooled or heated from their initial temperatures to final temperatures. Each stream has its own data, including heat capacity flow rate (F), specific heat capacity (cp), film heat transfer coefficient (h), viscosity (l), thermal conductivity (j), density (q), initial and target temperatures (Tin and Tout). As defined in conventional heat exchanger network synthesis, these process streams could exchange heat with each other. For a cold process stream, if the heat recovery within the heat exchanger network cannot satisfy its heating demand, an available hot utility are required to continuously heat it to its target temperature. Similarly, for a hot process stream, if the heat recovery within the heat exchanger network cannot satisfy its cooling demand, an cooler is needed to cool it to its target temperature. Noted that the surplus heat on all hot process streams is cooled by cooling water, so only cooling water coolers are used in this work. Like cold process streams mentioned above, the cooling water may extract heat from hot process streams. As its mass flow rate, supply and target temperatures are unknown, the cooling water

can be seen as a special cold stream. Its flow rate, supply and target temperatures have great impacts on the integrated design for the heat exchanger network and cooling water system. For instance, if the supply temperature is too high, too much cooling water is needed, and so the capital and operation cost of the cooling water system will be increased. On the other hand, if the target temperature is too high, the heat exchanged between process streams may be reduced and the operation cost of the heat exchanger network may be high. From this view, it can be seen that the heat exchanger network and cooling water system have close interconnection between each other. So we intend to treat the heat exchanger network and cooling water system as a whole, and our objective is to synthesize the network considering various costs, like utility costs, pumping costs and the capital costs of cooling tower and all heat exchanger. To make the problem easy to be solved, some basic assumptions are given as follows. (1) The supply temperature (Tin) and target temperature (Tout), heat capacity flow rate (F), film heat transfer coefficient (h), specific heat capacity (cp), viscosity (l), thermal conductivity (j) and density (q) of all process streams are all constants. (2) The available utilities for cold process streams are all placed at the extreme end of the streams. (3) The surplus heat on all hot process streams is cooled by cooling water. (4) All process streams are liquid fluids, and hot streams and cold streams are respectively transported in shell side and tube side of 1–1 counter current heat exchangers. (5) The inlet and outlet temperatures (Tinu and Toutu), cost data of the hot potential utilities (Cu) for all process streams should be given. (6) Other parameters for piping cost, pumping cost, investments of cooling tower and capital cost of heat exchangers, need to be given. 3. Methodology In this section, a simultaneous methodology is proposed to integrate a heat exchanger network and a cooling water system as a whole system. The methodology makes use of a novel stage-wise superstructure modified from the previous studies of heat exchanger networks. The superstructure is presented in Fig. 1, where the cooling water is treated as a special cold stream, and extracts heat from hot process streams. In this way, the heat exchanger network and cooling water system are converted to be a whole system since the cooling water stream has integrated these two energy systems together. Clearly, the superstructure in Fig. 1 can model most possible heat exchange among the cooling water, hot utilities, hot and cold process streams. Note that the possible hot utilities are located at the extreme ends of cold process streams in Fig. 1. Moreover, on entering any stage, every stream is split into multiple substreams. On leaving any stage, all splits of each stream are mixed or combined isothermally to reform the original stream. Several index sets are defined as follows. NH = {i | Stream i is a hot process stream}, NC = {j | Stream j is a cold process stream}, NK = {k | stage k for heat exchanger network}. Let I = Card [NH], J = Card [NC] and K = Card [NK] where Card[X] refers to the cardinality of set X and K = max {I, J}. 4. MINLP formulations In Fig. 1, the superstructure has K + 1 stages (k = 1, 2, 3 . . . K, K + 1) for a representative hot process stream i and a cold process stream j. As in the existing literatures about heat exchanger

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Plant HENs Stage k = 2

Stage k = 1 H1

C1

C2

H2

Evaporaton

Cooling water system Make up Blew dwon Fig. 1. The superstructure for the heat exchanger network and cooling system.

network synthesis, the first stage k = 1 is for the heat transfers between cold process streams and the potential hot utilities. The other intermediate stages (k = 1 through k = K) are for the heat transfers between hot and cold process streams. Meanwhile, the cooling water stream also can exchange heat with every hot process stream which is particularly different from current heat exchanger network researches. Our MINLP model for the superstructure in Fig. 1 involves five primary decisions: (1) existence of heat exchanges, (2) selection of stream splits, (3) the duties and areas of all heat exchangers, (4) the flow rate of cooling water and (5) the supply and target temperatures of cooling water. In the following context, we formulate mathematical programming models for the heat exchanger network and cooling tower respectively. 4.1. Heat exchanger network model In Fig. 1, it can be found that the network for heat exchange between hot and cold streams is very similar to the stage-wise superstructure for heat exchanger network synthesis in conventional researches. For a representative hot process stream i, it can exchange heat with all cold process streams and the cooling water stream at every stage. Similarly, a cold stream j can exchange heat with all hot process streams and the hot utility. Thus, the overall heat balances are needed to ensure sufficient heating or cooling of all process streams that are shown in Eqs. (1) and (2),

X XX qi;j;k þ qw i6I i;k ¼ F i ðTini Tout i Þ j6J k6K

XX qi;j;k þ quj ¼ F j ðToutj Tinj Þ j 6 J i6I k6K

ð1Þ

k6K

ð2Þ

where q is the heat load of the heat exchanger between process streams, qu is the heat duty of the hot utility and qw is the heat exchanged between the cooling water stream and hot process streams. Fig. 1 also shows each hot process stream enters the superstructure at the first stage (k = 1) and leaves from the superstructure at the last stage (k = K). So, its initial and finial temperatures (ti,1 and ti,K) are respectively equal to its supply and target temperatures. Likewise, the initial and final temperatures (tj,K and tj,1) of a cold stream to and from the superstructure should be equal to its supply and target temperatures (Tinj and Toutj). For these, we need Eqs. (3)–(8).

ti;1 ¼ Tini

i6I

ð3Þ

ti;K ¼ Touti

i6I

ð4Þ

tj;1 ¼ Toutj

j6J

ð5Þ

tj;K ¼ Tinj w

j6J

ð6Þ

tin ¼ tw K

ð7Þ

toutw ¼ t w 1

ð8Þ

where the variable t denotes the temperature of hot and cold streams at each stage, tinw denotes the inlet temperature of the cooling water stream, and toutw denotes the outlet temperature of the cooling water stream. Fig. 2 shows the relevant variables in the superstructure, wherein isothermal mixing assumption is supposed. Only the overall balances shown in Eqs. (9)–(11) are needed for process streams and the cooling water stream. Here, Mw is the

1514


bound for its temperature difference. The equations are feasible only if the binary variable is equal to zero, otherwise, the equations will be not active.

Stage k t i ,k

t i ,k + 1

dti;j;k 6 t i;k tj;k þ ð1 zi;j;k Þ Ci;j

dti;j;kþ1 6 t i;kþ1 t j;kþ1 þ ð1 zi;j;k Þ Ci;j

Fj w

dtuj 6 Tinuj tj;1 þ ð1 zuj Þ Cuj

Mw tw k

t k +1 tj ,k +1

w

w w dti;k 6 t i;k tw k þ ð1 zi;k Þ Ci

t j ,k

i 6 I; k 6 K

ð9Þ

j 6 J; k 6 K

ð10Þ

j6J

i6I w M w cpw ðt w k t kþ1 Þ ¼

X qw i;k

k6K

ð11Þ

As previous studies on heat exchanger network synthesis, utility heaters are all placed at the extreme end of cold process streams. Thus, Eq. (12) is utilized to compute the heat loads of the heater.

ð12Þ

Along the stages of the superstructure shown in Fig. 1, all hot and cold streams specify monotonic decrease in their temperatures which are shown in Eqs. (13)–(17).

t i;k P ti;kþ1

i 6 I; k 6 K

ð13Þ

t j;k P tj;kþ1

j 6 J; k 6 K

ð14Þ

t i;Kþ1 P Touti t j;1 6 Tout j w tw k P t kþ1

i6I

i 6 I; k 6 K

ð24Þ

i 6 I; k 6 K

ð25Þ

dti;j;k P DTmin i 6 I; j 6 J; k 6 K

ð26Þ

dtuj P DTmin j 6 J

ð27Þ

w

dti;k P DTmin i 6 I; k 6 K

ð15Þ

j6J

ð16Þ

k6K

ð17Þ

In Eqs. (18)–(20), we use binary variables z and zu to denote the existence of heat exchangers. Here, upper bound constraints are utilized to relate the heat loads of heat transfer units with the corresponding binary variables. In the following equations, X is the upper bound parameter for heat load that is not equal to zero when its binary variable is one, otherwise the variable will be set as zero to minimize the number of heat exchangers.

qi;j;k 6 zi;j;k minfXi ; Xj g i 6 I; j 6 J; k 6 K

ð18Þ

quj 6 zuj Xj

j6J

ð19Þ

w qw i;k 6 zi;k Xi

i 6 I; k 6 K

ð20Þ

Besides, Big-M constraints should be set to ensure the temperature differences hold if heat transfer occurs. The constraints can be written as Eqs. (21)–(25) in which the parameter C is the upper

ð28Þ

The area requirements for all heat exchangers are calculated in Eqs. (29)–(31), wherein the area directly depends on the heat load of heat exchangers. Chen approximation [20] for the logarithmic mean temperature difference is used in the following equations.

i6I

quj ¼ F j ðTouti tj;1 Þ j 6 J

ð22Þ

where, the variable dt is the temperature difference between hot and cold streams at each stage and dtu denotes the temperature difference of a heater. In addition, a minimal temperature difference DTmin is required for heat exchangers and the constraints are written as Eqs. (26)–(28).

mass flow rate of the cooling water stream which is a free variable and needs to be optimized.

X qi;j;k

i 6 I; j 6 J; k 6 K

ð23Þ

w

Fig. 2. The relative variables in the superstructure.

F j ðtj;k t j;kþ1 Þ ¼

ð21Þ

j6J

w w dti;kþ1 6 ti;kþ1 tw kþ1 þ ð1 zi;k Þ Ci

Fi

X F i ðti;k t i;kþ1 Þ ¼ qi;j;k

i 6 I; j 6 J; k 6 K

Ai;j;k

1 1 qi;j;k hi þ hj ¼ 0:3333 dt i;j;k dt i;j;kþ1 0:5 dt i;j;k þ dti;j;kþ1

i 6 I; j

6 J; k 6 K

ð29Þ

1 1 qhu j hj þ huj Auj ¼ 0:3333 j 6 J dtuj Tinuj tj;1 0:5 dtuj þ Tinuj tj;1

Aw i;k

h i 1 w 1 qw i;k hi þ ðh Þ ¼h i0:3333 w w w w dti;k dt i;kþ1 0:5 dti;k þ dt i;kþ1

i 6 I; k 6 K

ð30Þ

ð31Þ

where the variables A and Au represent the areas of the corresponding heat exchangers. According to Souza et al. [11], the general relationship between pressure drop in shell side (DPs) and tube side (DPt) depends on the film heat transfer coefficients and the heat exchanger areas that are written as Eqs. (32) and (33). Notice that all heat exchangers are 1–1 counter current shell and tube exchangers with known parameters. The parameters Ks and Kt are the coefficients in the tube and shell side that are calculated in Eqs. (34)– (36): 5:1

ð32Þ

3:5

ð33Þ

DPs ¼ Ks A hs DPt ¼ Kt A ht Ks ¼

1:3 67 Lt ðLt Dtout Þ Dt1:1 e ls 3:4 1:7 Dtout M s qs js cps

Kt ¼

1=2 0:0232:5 Dtin l11=6 Dtin t Dtout M t qt j7=3 cp7=6 t t

Dte ¼

4 L2t p Dt2out p Dtout

ls lr

lt lr

0:868 ð34Þ

0:63 ð35Þ

ð36Þ


where the index s and t denote shell and tube sides respectively, Dtin, Dtout and Dte are the internal, external and equivalent diameters of the tube, L is the tube pitch, and lr is the standard viscosity for water. Notice that the pressure drop of a process stream is linearly distributed according to the areas of the exchangers [11]. So, the pressure drops of process streams in heat exchangers can be expressed as Eqs. (37)–(39): 3:5

DPi ¼ Ksi hi

XX Ai;j;k þ Aw i;k

!

i6I

ð37Þ

j6J

ð38Þ

j6J k