design and control of heat exchangers with bypasses … · transfer area. in terms of design...

DESIGN AND CONTROL OF HEAT EXCHANGERS WITH BYPASSES

By

Miss Arpaporn Somsuk

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree

Master of Engineering Program in Chemical Engineering

Department of Chemical Engineering

Graduate School, Silpakorn University

Academic Year 2014

Copyright of Graduate School, Silpakorn University

การออกแบบและควบคุมเครืองแลกเปลียนความร้อนทมีีบายพาส

โดย นางสาวอาภาภรณ์ สมสุข

วทิยานิพนธ์นีเป็นส่วนหนึงของการศึกษาตามหลกัสูตรปริญญาวศิวกรรมศาสตรมหาบัณฑิต สาขาวชิาวศิวกรรมเคมี ภาควชิาวศิวกรรมเคมี

บัณฑิตวทิยาลยั มหาวิทยาลัยศิลปากร ปีการศึกษา 2557

ลขิสิทธิของบัณฑิตวิทยาลยั มหาวทิยาลัยศิลปากร

The Graduate School, Silpakorn University has approved and accredited the Thesis title of "Design and control heat exchangers with bypasses" submitted by Miss Arpaporn Somsuk as a partial fulfillment of the requirements for the degree of Master of Engineering in Chemical Engineering

.........................................................................

(Assistant Professor Panjai Tantatsanawong, Ph.D.) Dean of Graduate School ............./.............../.............

The Thesis Advisor Veerayut Lersbamrungsuk, D.Eng.

The Thesis Examination Committee

...................................................Chairman (Tarawipa Puangpetch, Ph.D.) ............./.............../.............

...................................................Member (Chaiyapop Siraworakun, D.Eng.) ............./.............../.............

...................................................Member (Assistant Professor Sirirat Wacharawichanant, D.Eng.) ............./.............../.............

...................................................Member (Veerayut Lersbamrungsuk, D.Eng.) ............./.............../.............

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55404205 : MAJOR : CHEMICAL ENGINEERING KEY WORDS : BYPASS FRACTION/ FLEXIBILITY/ HEAT EXCAHNGER/ INTEGRATION OF DESIGN AND CONTROL ARPAPORN SOMSUK : DESIGN AND CONTROL OF HEAT EXCHANGERS WITH BYPASSES. THESIS ADVISOR : VEERAYUT LERSBAMRUNGSUK, D.Eng. 57 pp.

Design and control of heat exchangers with bypasses when inlet temperatures and stream flow rates as disturbances has been conducted and proposed in this research. In general, design parameters for this system are bypass fraction and heat transfer area. However, when flexibility or ability to reject disturbance is considered as control criterion, the important design parameter is only heat transfer area. The feasible operating range of target temperature is bounded by the values of inlet temperatures and outlet temperatures given at maximum heat duty (or zero bypass fraction) condition. The range is larger for heat exchanger with high heat duty or large heat transfer area. Hence, flexibility of heat exchanger is determined by heat transfer area. In terms of design (economic), small heat exchanger is required while in terms of control large heat exchanger is preferred. These criteria of design and control should be traded-off. For simplicity, the design and control can be viewed as choosing smallest (cheapest) heat exchanger that can satisfy flexibility requirement. Heat exchanger is modeled using a single cell model concept. The feasibility region of the system is generated by considering approach temperatures and feasibility of target temperature. The results showed that when inlet temperatures are considered as disturbances, the generated feasible region is convex. Hence, the flexibility test problem can be simplified as checking the feasibility of corner points or vertices of disturbances. However, when flow rates as disturbances are generated feasible region is non-convex shape. A method for integration design and control of heat exchanger is proposed and implemented with two case studies.

Department of Chemical Engineering Graduate School, Silpakorn University Student’s signature …………………………… Academic Year 2014 Thesis Advisor’s signature ……………………………

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คาํสาํคญั : สัดส่วนบายพาส/ ความหยดืหยุน่ได/้ เครืองแลกเปลียนความร้อน/ การออกแบบและ ควบคุม

อาภาภรณ์ สมสุข : การออกแบบและควบคุมเครืองแลกเปลียนความร้อนทีมีบายพาส.อาจารยที์ปรึกษาวทิยานิพนธ์ : ดร. วีรยทุธ เลิศบาํรุงสุข. 57 หนา้.

งานวจิยันีไดท้าํการศึกษาและเสนอการออกแบบและควบคุมเครืองแลกเปลียนความร้อนทีมีบายพาสเมือสิงรบกวนคืออุณหภูมิขาเขา้และอตัราการไหล โดยทวัไปพารามิเตอร์ของการออกแบบสําหรับระบบนีคือสัดส่วนของการบายพาสและพืนทีการถ่ายเทความร้อน อย่างไรก็ตามเมือพิจารณาความสามารถในการควบคุมจากความหยืดหยุ่นได้หรือความสามารถในการกาํจดัสิงรบกวน พารามิเตอร์ของการออกแบบทีสําคญัมีเพียงพืนทีในการแลกเปลียนความร้อน ช่วงดาํเนินการทีเหมาะสมของอุณหภูมิเป้าหมายนนัขึนอยู่กบัค่าของอุณหภูมิขาเขา้และค่าอุณหภูมิขาออก กรณีการถ่ายเทความร้อนสูงสุดหรือสัดส่วนของบายพาสเป็นศูนย ์โดยช่วงจะกวา้งขึนเมือพนืทีแลกเปลียนก็มีเพิมขึน ดงันนัความหยดืหยุน่ไดข้องเครืองแลกเปลียนความร้อนถูกกาํหนดดว้ยพืนทีแลกเปลียนความร้อน ในเชิงของการออกแบบหรือทางเศรษฐศาสตร์ต้องการเครืองแลกเปลียนความร้อนทีมีขนาดเล็กแต่ในทางของการควบคุมตอ้งการพืนทีแลกเปลียนทีมีขนาดใหญ่ ดงันนัจึงเป็นปัญหาทีควรจะพิจารณาทงัการออกแบบและควบคุมควบคู่กนั ปัญหาดงักล่าวอาจมองใหง่้ายไดโ้ดยเลือกเครืองแลกเปลียนความร้อนขนาดเล็กทีสุดโดยทีการควบคุมเป็นไปตามตอ้งการ แบบจาํลองแนวคิดเซลล์แบบหนึงหน่วยถูกนาํมาใช้จาํลองเครืองแลกเปลียนความร้อน ขอบเขตความเป็นไปได้ของระบบถูกสร้างผ่านค่าการพิจารณาค่าผลต่างอุณหภูมิเขา้ใกลแ้ละค่าอุณหภูมิเป้าหมาย ผลการวิเคราะห์พบว่าเมืออุณหภูมิขาเขา้เป็นสิงรบกวนขอบเขตความเป็นไปไดจ้ะเป็นคอนเวกช์ จึงทาํใหก้ารทดสอบความเป็นไปไดเ้ป็นเพียงการตรวจสอบของสิงรบกวนแต่เมือมีอตัราการไหลเป็นสิงรบกวนจุดมุมของสิงรบกวน แต่เมืออตัราการไหลเป็นสิงรบกวนขอบเขตความเป็นไปไดจ้ะไม่เป็นคอนเวกช์ วิธีการสําหรับอินทิเกรตการออกแบบและควบคุมเครืองแลกเปลียนความร้อนไดถู้กเสนอขึนและทดสอบกบักรณีศึกษา 2 กรณี

ภาควชิาวศิวกรรมเคมี บณัฑิตวทิยาลยั มหาวทิยาลยัศิลปากร ลายมือชือนกัศึกษา……………………………………. ปีการศึกษา 2557 ลายมือชืออาจารยที์ปรึกษาวิทยานิพนธ์……………………………

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Acknowledgements

The author wishes to express my sincere gratitude and appreciation to my

advisor, Dr. Veerayut Lersbamrungsuk for his valuable suggestions, stimulation,

useful discussions throughout this research and devotion to revise this thesis. In

addition, the author would also grateful to Dr. Tarawipa Puangpetch, the chairman,

Dr. Chaiyapop Siraworakun and Assistant Professor Dr. Sirirat Wacharawichanant the

members of the thesis committee. The author would like to thank Silpakorn

University graduate school thesis grant for financial support.

Most of all, the author would like to express my highest gratitude to Somsuk’s

family who always pay attention to through these years for suggestions and their

wills. The most success of graduation is devoted to my parents.

Finally, I wish to thanks my friends, member of graduate school of Chemical

Engineering on Silpakorn University for furtherance and support.

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Tables of Contents

Page

English abstract .................................................................................................. d

Thai abstract ....................................................................................................... e

Acknowledgements ............................................................................................ f

List of tables ....................................................................................................... h

List of figures ..................................................................................................... j

Chapter

1 Introduction ........................................................................................... 1

2 Theory …………………….................................................................... 3

Bypass for heat Exchangers ..................................................... 3

Model of heat exchanger …….................................................. 4

Flexibility .................................................................................. 5

Convexity and its applications …………………........................ 6

Feed-effluent heat exchanger .................................................... 13

3 Lierature reviews ................................................................................... 15

Heat exchanger ......................................................................... 15

Integration design and control ................................................... 16

Feed-effluent heat exchanger .................................................... 17

4 Methodology .......................................................................................... 18

Dynamic model of heat exchanger ........................................... 18

Flexibility of heat exchanger .................................................... 19


Feed-effluent heat exchanger ..................................................... 20

5 Results and Discussion .......................................................................... 22

Cell model of heat exchangers .................................................. 22

Flexibility of heat exchangers ………………………………….26


Case study: Heat exchanger with bypass …................................ 34

Case study: Feed-effluent heat exchanger ................................. 35

6 Conclusions ........................................................................................... 38

Bibliography …………………………………………………………………… 39

h

Chapter Page

Appendix

Appendix A : Rearrange equations ……………………………………… 43

Appendix B : Hessian matrix ……………………………………………. 46

Appendix C : Nomenclature …………………………………………….. 54

Appendix D : International proceeding …………………………………. 56

Biography ………………………………………………………………………… 57

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List of Tables

Tables Page

1 Relations ship between the character of f(x) and state of H(x) ............... 12

2 Lists of parameters of heat exchanger used for simulation ..................... 22

3 Lists parameters used for the simulation when inlet temperatures as dis-

turbances …………………………………………….................. 28

4 Feasibility test for the case A = 5m2.......................................................... 30

5 Feasibility test for the case A = 5, 8 and 10 m2......................................... 30

6 Lists parameters used for the simulation when stream flow rates as dis-

turbance ...................................................................................... 31

7 Parameter used for the simulation in case study1 ……………................ 35

8 The value of bypass fraction at corner point of disturbance………... 35

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List of Figures

Figures Page

1 Cell model of heat exchanger ............................................................... 4

2 Local optimal due to objective function ............................................... 7

3 local optimal due to feasible region ....................................................... 7

4 Convex and nonconvex sets .................................................................. 7

5 Convex and concave functions of one variable....................................... 10

6 illustration of a convex set formed by a plane f(x) = k outing a convex

function......................................................................... 11

7 A) independent heating and cooling, B) FEHE process ....................... 13

8 Cell arrangement for a single-pass shell-and tube heat exchanger.......... 15

9 Sketch of tank model ............................................................................ 18

10 Heat exchanger with bypass .................................................................. 19

11 A heat integration with FEHE .............................................................. 21

12 Cell model of heat exchanger ................................................................ 22

13 Diagram inlet and outlet temperature in hot and cold side (1-cell) ....... 24

14 Diagram inlet and outlet temperature in hot and cold side (2-cell) ....... 24

15 Heat exchanger with bypass ................................................................... 25

16 Relationship between bypass fraction and outlet temperature on hot side 25

17 Feasible region of heat exchanger with 5 m2 .......................................... 28

18 Feasible region of heat exchanger with 5,8 and 10 m2 ........................... 29

19 Feasible region of heat exchanger with 5 m2 …………………............... 31

20 Feasible region of heat exchanger with 50 m2 ........................................ 32

21 The proposed algorithm for design and control for calculation ............... 33

22 Heat exchanger with bypass……………………………………………… 34

23 FEHE of DME process ........................................................................... 36

24 Feasible region of heat exchanger with 27.51 m2 ................................... 36

25 Feasible region of heat exchanger with 500 m2 ……….…….................. 37

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CHAPTER 1

INTRODUCTION

1.1 Statement and significance of the problems

Heat exchangers are major equipments used for heat recovery in chemical industries. Energy is transferred from hot streams which need to be cooled to cold streams which need to be heated. Most of them usually require a control scheme to maintain an outlet temperature at one side of outlet streams. Bypassing is one method widely used when flow rates of hot and cold streams are set by upstream or downstream process objectives. Nevertheless, poor design of the system can make the temperature control difficult.

In design of the system of heat exchangers with bypasses, heat exchanger is a distributed system whose dynamics can be described by a set of partial differential equations due to its temporal and spatial variation. Because solving a set of partial differential equations (PDEs) is difficult, a concept of lumped cell-based models is usually used to simplify the model that is a set of ordinary differential equations (ODEs).

Important design parameters of the heat exchanger with bypasses include heat transfer area and nominal fractional bypass. Controllability is one of the most important aspects of chemical process operability. Some other aspects, such as flexibility and stability, should be considered simultaneously when assessing the controllability of a chemical process (Chen et al., 2010). In this work, we focus on flexibility or ability to reject disturbances. In terms of control, one usually requires large heat transfer area and fractional bypass. However, larger heat transfer area results in higher equipment cost. Hence, one can consider this problem as design and control problem.

This work focuses on design and control of heat exchanger with bypasses. A design method to trade-off between economic design and control objectives for the system will be proposed and implemented to a case study of feed-effluent heat exchanger (FEHE).

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1.2 Research objective

The objective of the research is to propose a systematic method for design and control of heat exchangers with bypasses.

1.3 Scopes of research

The scopes of this work are listed below:

1. Develop dynamic model of heat exchangers with bypasses for design and control observation.

2. Design parameters of the system include heat transfer area and nominal bypass fraction.

3. Steady state controllability will be considered. 4. Case studies e.g. feed-effluent heat exchanger (FEHE), will be used to show

the implementation of the proposed method.

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CHAPTER 2

THEORY

This chapter provides some backgrounds necessary for understanding this research.

2.1 Heat exchangers

A heat exchanger is a piece of equipment built for efficient heat transfer from one medium to another. The media may be separated by a solid wall, so that they never mix, or they may be in direct contact. They are widely used in space heating, refrigeration refineries, natural gas processing, and sewage treatment. The classic example of a heat exchanger is found in an internal combustion engine in which a circulating fluid known as engine coolant flows through radiator coils and air flows past the coils, which cools the coolant and heats the incoming air, air conditioning, power plants, chemical plants, petrochemical plants, petroleum. There are two primary classifications of heat exchangers according to their flow arrangement. In parallel-flow heat exchangers, the two fluids enter the exchanger at the same end, and travel in parallel to one another to the other side. In counter-flow heat exchangers the fluids enter the exchanger from opposite ends. The counter current design is the most efficient, in that it can transfer the most heat from the heat (transfer) medium due to the fact that the average temperature difference along any unit length is greater. In a cross-flow heat exchanger, the fluids travel roughly perpendicular to one another through the exchanger. For efficiency, heat exchangers are designed to maximize the surface area of the wall between the two fluids, while minimizing resistance to fluid flow through the exchanger. The exchanger's performance can also be affected by the addition of fins or corrugations in one or both directions, which increases surface area and many channel fluid flow or induces turbulence. The driving temperature across the heat transfer surface varies with position, but an appropriate means temperature can be defined. In most simple systems this is the “log mean temperature difference” (LMTD). Sometimes direct knowledge of the LMTD is not available and the NTU method is used.

2.1.1 Bypass for heat exchanger

An efficient, technological solution for controlling the heat transfer performance of a heat exchanger is to install on process streams a bypass circuit, whose flow rate is determined in order to keep the outlet temperature of process fluids at target value. Control on outlet temperature is often necessary when working conditions undergo transients and modifications.

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Many methods are used for controlling temperatures in heat exchanger systems. Direct manipulation of the flow rate of either the hot or the cold stream is most often used when that stream is a utility (cooling water, steam, hot oil, or refrigerant). When the flow rates of both streams are set by process requirements, heat-exchanger bypassing is widely used. A portion of one of the streams (either hot or cold) is sent through the heat exchanger, and the remainder is bypassed around the exchanger. The temperature of the mixed stream is controlled by valves in each path. This system provides very tight temperature control, since the dynamics of blending a hot stream and a cold stream are very fast. Installing a bypass on the heat exchanger is one of the most efficient solutions for its control and operability. 2.1.2 Model of heat exchanger

Due to the spatial variation of temperatures along the flow direction, heat exchanger can be described by a set of partial differential equations (PDEs). However, to simplify the calculation, a concept of cell-model (Varbanov et al., 2011) can be used , that is, considering heat exchangers as a series of mixing tanks as shown in Figure 1.

Figure 1 Cell model of heat exchanger.

The PDEs can be transformed as a set of ordinary differential equations (ODEs) as follow:

ihocohp

ihohih

h

i

ho TTVC

UATTVv

dtdT (1)

icohocp

icocic

i

co TTVC

UATTVcv

dtdT (2)

Equations 1 and 2 are energy balances of hot and cold side of a cell unit in a heat exchanger. If the number of cells increases, amount of variables and equations will also increase and these can affect driving force and accuracy of the model.

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2.2 Flexibility and convexity

2.2.1 Flexibility

In chemical process (CP) design, some design specifications must always be met. Examples of these specifications are (a) safety, (b) ecological, and (c) performance. For specifications that are cast as constraints, we distinguish between hard and soft constraints. Specifically, a violation of hard specifications is not allowed. In this paper, we will assume that all constraints are hard. The satisfaction of design specifications is complicated by the presence of uncertainties in the models that are used in the design. Some sources of uncertainty are: (a) Inherent inaccuracies of coefficients in the mathematical models. (b) Changes in some of the coefficients in the mathematical models during the chemical process operation (for example rate constants, heat and mass transfer coefficients). (c) Variations in some of the parameters (e.g. temperature, flow rates, species concentrations) associated with external streams during the chemical process operation.

Controllability is one of the most important aspects of chemical process operability. Some other aspects, such as flexibility and stability, should be considered simultaneously when assessing the controllability of a chemical process. In this work, we focus on flexibility or ability to reject disturbances. Grossmann and Struab (1991) proposed two types of flexibility analysis problem:

a) The feasibility problem: Determines if a given design can be feasibly operated over the considered range of uncertainty.

b) The flexibility index problem: Evaluates a measure to quantify the maximum range of uncertainty that the design can be feasibly operated.

Halemane and Grossmann (1983) formulated two problems of the flexibility analysis of chemical process. The first problem is the Feasibility Test, which is formulated, in the following form

0)(d (3)

),,(maxminmax)( zdfd jJjZzT (4)

where χ(d) is a feasibility function (measure) for a given d, d is a vector of design variables, z is a vector of control variables, Z is a region of admissible values of control variables, and T is the domain for the uncertain parameters t such that T ={θ : θ L ≤ θ ≤ θU }, J = (1, . . . ,m). If χ(d) ≤ 0 then feasibility of chemical process operation can be guaranteed for all θ.

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If χ(d) > 0 then the design is infeasible at least for some values of θ. The reduced process constraints

,0),,( tzdf j mj .....,,.........1 (5)

are obtained from the original mathematical models

0),,,( tzxdh (6)

0),,,( tzxhg (7) by explicitly solving for x, the vector of state variables. Here, dim h = dim x. Equation (6) describe the states (i.e. material and energy balances), while the inequalities in (7) are design specifications.

2.2.2 Convexity and its applications (Edgar et al., 2001)

The concept of convexity is useful both in the theory and applications of optimization. We first define a convex set, then a convex function, and lastly look at the role played by convexity in optimization.

A set of points (or a region) is defined as a convex set in n-dimensional space if, for all pairs of points x1 and x2 in the set, the straight-line segment joining them is also entirely in the set. Figure 2 illustrates the concept in two dimensions. A mathematical statement of a convex set

For every pair of points x1 and x2 in a convex set, the point x given by a linear combination of the two points

21 1 xxx , 10 (8)

is also in the set. The convex region may be closed (bounded) by a set of functions, such as the sets A and B in figure 4 or may be open (unbounded) as in figures 4. Also, the intersection of any number of convex set is a convex set.

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Figure 2 Local optimal due to objective function.

Source: Thomas F. Edgar, David M. Himmelblau and Leon S. Lasdon Optimization of chemical processes, 2nd ed. (McGraw-Hill, 2001), 121-127.

Convex function

Next, let us examine the matter of a convex function. The concept of a convex Function is illustrated in figure 5 for a function of one variable. Also shown is a concave function, the negative of a convex function. (If f(x) is convex, -f(x) is concave.) A function f(x) defined on a convex set F is said to be a convex function if the following relation holds

2121 11 xfxyfxxf (9)

Figure 3 Local optimal due to feasible region.


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where is a scalar with the range 0 ≤ ≤ 1. If only the inequality sign holds, the function is said to be not only convex but strictly convex. ( If f (x) is strictly convex, -f (x) is strictly concave.) figure 5 illustrates both a strictly convex and a strictly concave function. A convex function cannot have any value larger than the values of the function obtained by linear interpolation between x, and x, (the cord between x, and x, shown in the top figure in figure 5). Linear functions are both convex and concave, but not strictly convex or concave, respectively. An important result of convexity is If f (x) is convex, then the set kxfxR (10)

is convex for all scalars k. The result is illustrated in figure 6 in which a convex quadratic function is cut by the plane f(x)= k. The convex set R projected on to the x1 - x2 plane comprises the boundary ellipse plus its interior.

Figure 4 Convex and nonconvex sets.

Source: Use by permission: Optimization of chemical processes, 2nd edition, McGraw-Hill, 2001.

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The convex programming problem

An important result in mathematical programming evolves from the concept of convexity. For the nonlinear programming problem called the convex programming problem

Minimize: xf

Subject to: 0xgi (11)

in which (a) f(x) is a convex function, and (b) each inequality constraint is a convex function (so that the constraints form a convex set), the following property can be shown to be true. The local minimum off (x) is also the global minimum. Analogously, a local maximum is the global maximum off (x) if the objective function is concave and the constraints form a convex set. Role of convexity

If the constraint set g(x) is nonlinear, the set

0xgxR (12)

is generally not convex. This is evident geometrically because most nonlinear functions have graphs that are curved surfaces. Hence the set R is usually a curved surface also, and the line segment joining any two points on this surface generally does not lie on the surface.

As a consequence, the problem

Minimize: xf Subject to: 0xgi mi .,,.........1

0xhk nrk ,.......,1 (13) may not be a convex programming problem in the variables x1, . . . , xn if any of the functions hk(x) are nonlinear. This, of course, does not preclude efficient solution of such problems, but it does make it more difficult to guarantee the absence of local optima and to generate sharp theoretical results

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Figure 5 Convex and concave functions of one variable.


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Figure 6 Illustration of a convex set formed by a plane kxf outing a convex function.


In many cases the equality constraints may be used to eliminate some of the variables, leaving a problem with only inequality constraints and fewer variables. Even if the equalities are difficult to solve analytically, it may still be worthwhile solving them numerically. This is the approach taken by the generalized reduced gradient method

Although convexity is desirable, many real-world problems turn out to be nonconvex. In addition, there is no simple way to demonstrate that a nonlinear problem is a convex problem for all feasible points. Why, then is convex programming studied? The main reasons are

1. When convexity is assumed, many significant mathematical results have

been derived in the field of mathematical programming. 2. Often results obtained under assumptions of convexity can give insight into

the properties of more general problems. Sometimes, such results may even be carried over to nonconvex problems, but in a weaker form.

For example, it is usually impossible to prove that a given algorithm will find

the global minimum of a nonlinear programming problem unless the problem is convex. For nonconvex problems, however, many such algorithms find at least a local minimum. Convexity thus plays a role much like that of linearity in the study of

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dynamic systems. For example, many results derived from linear theory are used in the design of nonlinear control systems. Determination of convexity and concavity

The definitions of convexity and a convex function are not directly useful in establishing whether a region or a function is convex because the relations must be applied to an unbounded set of points. The following is a helpful property arising from the concept of a convex set of points. A set of points x satisfying the relation

1xxHxT (14)

is convex if the Hessian matrix H(x) is a real symmetric positive-semidefinite matrix. H(x) is another symbol for 2f(x), the matrix of second partial derivative of f(x) with respect to each xi

xfHxH 2 (15)

The status of H can be used to identify the character of extreme. A quadratic

form Q(x) = XTHX is said to be positive-definite if Q(x) > 0 for all x # 0, and said to be positive-semidefinite if Q(x) 2 0 for all x # 0. Negative-definite and negative-semidefinite are analogous except the inequality sign is reversed. If Q(x) is positive-definite (semidefinite), H(x) is said to be a positive-definite (semidefinite) matrix. These concepts can be summarized as follows: 1. H is positive-definite if and only if xTHx is > 0 for all x ≠ 0 . 2. H is negative-definite if and only if xTHx is < 0 for all x ≠ 0. 3. H is positive-semidefinite if and only if xTHx is ≥ 0 for all x ≠ 0. 4. H is negative-semidefinite if and only if xTHx is ≤ for all x ≠ 0. 5. H is indefinite if X~HX < 0 for some x and > 0 for other x.

It can be shown from a Taylor series expansion that if f(x) has continuous second partial derivatives, f(x) is concave if and only if its Hessian matrix is negative-semidefinite. For f(x) to be strictly concave, H must be negative-definite. For f(x) to be convex H(x) must be positive-semidefinite and for f(x) to be strictly convex, H(x) must be positive-definite.

For a multivariate function, the nature of convexity can best be evaluated by examining the eigenvalues of f(x) as shown in Table 1 We have omitted the indefinite case for H, that is when f(x) is neither convex or concave.

13

Table 1 Relationship between the character of f(x) and the state of H(x)

xf xH All the eigenvalue of

xH are

Strictly convex positive-definite 0

Convex positive-semidefinite 0

Concave negative-semidefinite 0

Strictly concave negative-definite 0

2.3 Feed-Effluent heat exchanger (FEHE)

Figure 7 A) Independent heating and cooling, B) Feed-Effluent heat exchanger (FEHE) process.

Source: Luyben “A feed-effluent heat exchanger/reactor dynamic control laboratory experiment”, Chemical Engineering Education (2000): 56-73.

Feed-effluent heat exchanger (FEHE) used for pre-heating reactors feed. Feed preheating unit can be a steam-heated heat exchanger or furnace, depending on the temperature level required. Cooling of the reactor effluent is usually required, and this can be done steam generation or using cooling water. The use of independent utility

14

stream for preheating and cooling makes the control problem easy because there is no interaction (Figure 7A).

However, it is quite inefficient from a capital-investment and energy standpoint. Separate heating and cooling heat exchanger are required, which increases capital investment in heat transfer area. In industrial applications, the hot reactor effluent is used to preheat the cold reactor feed. The resulting decrease in heat transfer area means lower capital investment.

Figure 7B shows a FEHE coupled with an adiabatic exothermic reactor. The heat of reaction produces a reactor effluent temperature. That is higher than the temperature of the feed stream to the reactor. Hence, heat can be recovered from the hot stream leaving the reactor (Luyben, 2000).

15

CHAPTER 3

LITERATURE REVIEWS

This chapter consists of three parts including the literature related to heat exchanger, controllability of chemical process and feed-effluent heat exchanger.

3.1 Heat exchangers

Varbanov et al. (2011) described cell-based models can be used represent model of heat exchanger. The equation of cell model is simple and the approach offers a form outline and modeling flexibility. The main assumption is that each side of streams consists of a series of ideal mixing tank and shown in Figure 8. Hence, equation (16)-(18) are energy balances for a cell derive from figure 8.

Figure 8 Cell arrangement for a single-pass shell-and-tube heat exchanger

Source: Petar S. Varbanova, Jiˇri J. Klemeˇs and Ferenc Friedler “Cell-based dynamic heat exchanger models – Direct determination of the cell number and size”, Computers and Chemical Engineering 35 (2011): 943-948.

wHOHpHCELLH

CELLCELLHHOHI

CELLH

HHO TTCV

ATT

Vv

dtdT

,,

,

,

(16)

COwCpCCELLC

CELLCELLCCOCI

CELLC

CCO TTCV

ATT

Vv

dtdT

,,

,

,

(17)

COwwpww

CELLCELLCwHO

wpww

CELLCELLHw TTCVA

TTCVA

dtdT

,

,

,

, (18)

16

Jogwar et al. (2009) focused on high duty counter-current heat exchangers characterized by high Staton number, in which the dynamics can be describes by stiff first order hyperbolic partial differential equations (PDEs). The non-stiff reduced model that can capture important dynamic feature of the heat exchanger is proposed.

Heo et al. (2011) analyzed a class of high duty counter-current heat exchangers exhibit multi-time scale dynamic. The dynamic model can be described by two-first order of partial differential equations (PDEs).

Dobos et al. (2009) studies two objective: the first is created dynamic model with cell model with can represent the main characteristics of a district heating network. The second is design a non-linear model predictive controller to satisfy the heat demand of the consumer in heat exchanger networks.

Manenti (2011) proposed numerical model based on mass, momentum and energy balance, has been developed on the specific case of a shell and tube waste heat boiler exchanger.

Luyben (2011) studies heat exchanger with bypass control. Effect of design parameter are range of heat exchanger area, circulation rates and valve design pressure drop that are interested in finding the maximum heat removal in each case or, equivalently, the maximum feed flow rate to the reactor while still maintaining the 400 K reactor temperature. Case study, the exothermic liquid-phase reaction of aniline with hydrogen to form cyclohexyl amine (CHA) is carried out in a jacketed CSTR reactor.

3.2 Integration between design and control

Chen et al. (2011) proposed controllability analysis of chemical processes which is one of operability as well as flexibility or other aspects. However, methodology of controllability analysis can be classified two main sets: linear and non-linear approaches. And proposed methodology for improving controllability analysis which is the optimization methods. Multi-objective optimization is one method that guideline to solving integration between design and control.

Sharifzadeh (2013) review integration of process design and control that can be classified two main: sequential/iterative design and control approach and integration design and control approach. The sequential methods have a yes/no attitude to the problem while the integrated design and control methods incorporate some control aspects into the process design.

Floudas (2001) presented for addressing flexibility test and flexibility index problems in a rigorous and efficient way. The approach is based on a convexification/relaxation of the feasible region, coupled with convex underestimation schemes within a branch and bound global optimization framework.

17

3.3 Feed-effluent heat exchanger

Luyben (2012) presented feed-effluent heat exchanger with case study is the production of dimethyl ether from methanol. The case study is without and with furnace. Hence, the results show new process configuration and a new control structure have been developed and tested. Dynamic advantages in terms of robustness to large disturbances have been demonstrated. In addition reduction of furnace energy consumption at design and lower throughputs has been achieved.

Jogwar et al. (2008) analyzed the energy dynamics of process networks comprising of chemical reactor and a feed-effluent heat exchanger (FEHE). Using singular perturbation analysis, show in case of tight energy integration, the energy dynamics of the network evolves over two times scale, with enthalpy of individual unit evolving in the fast time scale and overall network evolving in the slow time scale.

Luyben (2000) presented design and control for gas phase reactor/recycle processes with reversible exothermic reactions which feature important trade-offs among the reactor size, recycle flowrate, and reactor inlet temperature.

18

THi-1 THi+1 THi

TCi TCi+1 TCi-1

CHAPTER 4

METHODOLOGY

4.1 Dynamic model of heat exchanger Heat exchanger can be described as distributed parameter system. In unsteady

state condition, the model is represented by a set of partial differential equations (PDEs) with respect to temporal and spatial variation. However, solving PDEs is difficult especially when solving a group of heat exchangers. Therefore, cell-based model is more popular because of its simplicity. In lumped cell-based model (Mathisen et al., 1994; Varbanov et al., 2011), a heat exchanger is divided as several compartments or mixing tanks. This concept simplifies an original partial differential equation to a number of ordinary differential equations (ODEs) that are easier to solve. When a bypass is installed, the model to explain dynamic model behavior can be divided into two parts. The first part concerns dynamic model of heat exchanger. The other concerns dynamic model of mixer between the outlet stream of heat exchanger and the bypass stream.

4.1.1 Cell model in heat exchanger Figure 9 Sketch of cell-model of heat exchanger.

A simple heat exchanger cell is defined as two perfectly stirred tanks,

exchanging heat only with each other through a dividing wall. This type of arrangement is illustrated in Figure 9. The equations for one cell model of heat exchanger can be written as, (19)

)()(,

cohohphh

hhohi

h

hho TTCV

ATT

Vv

dtdT

19

(20) 4.1.2 Cell model in heat exchanger with bypass.

When a bypass is installed (assumingly on hot side, Figure 10), the model equation (14) is slightly modified by including the bypass fraction (f). Then, an additional equation for stream mixing is required. The model of heat exchanger with bypass can be written as follows,

Figure 10 Heat exchanger with bypass.

hochp

hohih

hho TTVC

UATTV

vfdt

dT 1 (21)

chocp

ccic

cc TTVC

UATTVv

dtdT

(22)

hihoh fTTfT 1 (23) 4.2 Flexibility of heat exchanger

Flexibility is one of the most important aspects of chemical process operability. Flexibility is defined as ability to accommodate uncertainties over a range of uncertain parameters. For the system considered in this research, the flexibility design will be regarded as design of heat transfer area and nominal bypass fraction to make the system able to handle a given set of disturbances.

),c

()( hococpc

ccoci

c

cco TTCVA

TTVv

dtdT

20

4.3 Integration of design and control

In general, integration of design and control can be considered as a multi-objective optimization with economic design and control objective. To solve this problem there are two approaches: sequential/iterative and simultaneous approaches. In sequential approach, an economic objective might be firstly considered where then a controllability objective will be later solved or checked if the controllability target can be met. In case that controllability target cannot be met, some modification will be made and the two-step will be repeated until the target is satisfied. In simultaneous approach, the two objectives including economic and controllability objectives will be solved simultaneously in a single optimization problem. However, note that this requires weighting factors to convert the two objectives into the same unit. A controllability objective considered here is flexibility. Chen et al. (2010) noted that one may solve the problem by considering controllability objective as constraint to be satisfied as follows,

min J1 s.t. J2 ≥ a f1 = 0 f2 ≥ 0 (24)

where J1 is economic design objective while J2 consider control objective, a is parameter of index control, f1 and f2 represents equality and inequality constraints.

4.4 Feed-Effluent heat exchanger (FEHE)

After a design and control method for heat exchanger with bypasses is proposed, it will be implemented with a case study of feed-effluent heat exchanger as shown in Figure 11. FEHE is a heat exchanger used to transfers the heat available from the hot effluent stream from chemical reactor to the cold reactor feed stream. From the figure, the reactor is usually operated in adiabatic mode and the inlet temperature of the reactor is usually kept at a desired value to ensure stability and productivity of the system. In general, a bypass is required as to manipulate the system.

21

Figure 11 A heat integration with FEHE

Source: Sujit S. Jogwar, Michael Baldea and Prodromos Daoutidis, “Dynamics and control of reactor-feed effluent heat exchanger networks” (paper presented at meeting of the American control conference, Seattle, Washington, USA, June 11-13, 2008).

22

CHAPTER 5

RESULTS AND DISCUSSION

In this chapter, we firstly describe dynamic modeling of heat exchange via the concept of cell-model. After that a flexibility problem of the system will be proposed and then will be used as control objective for trading-off with economic design objective in which a design and control method for the heat exchanger with bypasses will be proposed. Finally, the proposed method will be implemented with a case study of feed-effluent heat exchanger.

5.1 Cell model of heat exchangers

5.1.1 Effect of number of cells in model of heat exchangers

Due to the spatial variation of temperatures along the flow direction, heat exchanger can be described by a set of partial differential equations (PDEs). Glemmestad (1997) shown that discretizing partial differential equations in the spatial direction may lead to exactly the same set of equations as when the mixing tank concept is used as basis (or set of ordinary differential equations).

Figure 12 Cell model of heat exchanger.

Heat exchanger is modeled as a set of ordinary differential equations (ODEs) that is easier to solve than the original set of PDEs. The main assumption is that each side of streams consists of a series of ideal mixing tanks as shown in Figure 12.

The following assumptions are made for heat exchanger model:

1) Density of fluid is constant 2) Specific heat capacity of fluid is constant

23

3) Heat transfer coefficients are constant and flow independent 4) Phase of fluid is not changed

To illustrate the effect of number of cells, parameters of the heat exchanger system in Table 2 will be used.

Table 2 List of parameters of the heat exchanger used for the simulation.

The cell model of heat exchanger can be derived by performing energy balances on hot and cold sides of ith cell as shown in equations (25)-(26). Note that there are two equations for describing the dynamics of one cell. For the model with n cells, there will be 2n state variables and 2n ordinary differential equations. Increasing the number of cells can improve accuracy but requires more computation efforts.

ihocohp

ihohih

h

i

ho TTVC

UATTVv

dtdT

(25)

icohocp

icocic

i

co TTVC

UATTVcv

dtdT

(26)

Parameter Value U 6.81 J/min.cm2.°C Vh 10000 cm3/min

vc 10000 cm3/min

Cp,h 4.187 J/g.°C

Cp,c 4.187 J/g.°C

h 1 g/cm3

c 1 g/cm3

Vh 50000 cm3

Vc 50000 cm3

A 5000 cm2

Thi 70 °C

Tci 35 °C

24

T

T

The temperature results from using 1 and 2 cells are shown in Figures 13 and 14. Note that T of 1-cell model overestimates T of 2-cell model. However, if considering the amount of heat exchanger or outlet temperature on hot side (assuming that the output temperature on hot side has a target), the result of 1-cell model underestimates the 2-cell model. Based on this observation, although the result from 1-cell is not accurate as the result from 2-cell, the 1-cell model will still be used in the flexibility analysis in the next section to keep the flexibility problem easier to solve. Some tricks may be required to guarantee the design result (as will be discussed in the next section).

Figure 13 Diagram of inlet and outlet temperature in hot and cold side (1-cell model).

Figure 14 Diagram of inlet and outlet temperature in hot and cold side (2-cell model).

Tem

pera

ture

(°C

)

Time

70.00

45.83 59.16

35.00

Tem

pera

ture

(°C

)

Time

70.00

63.59

35.00

47.41

47.82 57.18

25

5.1.2 Effect of bypass fraction

Figure 15 Heat exchanger with bypass.

When a bypass is installed (assumingly on hot side, Figure 15), the model equation (25) is slightly modified by including the bypass fraction (f). Furthermore, an additional equation for stream mixing is required. The model of heat exchanger with bypass can be written as follows,

hochp

hohih

hho TTVC

UATTV

vfdt

dT 1 (27)

chocp

ccic

cc TTVC

UATTVv

dtdT (28)

hihoh fTTfT 1 (29)

Figure 16 Relationship between bypass fraction and outlet temperature on hot side (Th,out).

26

When outlet temperature on hot side has desired target, the bypass will be manipulated to meet it. Figure 16 shows the relationship between bypass fraction and outlet temperature on hot side (Th,out). Note that the operating range of outlet temperature is bounded between the temperature at maximum heat duty (fully close of bypass or bypass fraction = 0) and hot side inlet temperature (fully open of bypass or bypass fraction = 1). Note also that this means that the feasibility of temperature target can be simplified by considering only whether the temperature at maximum heat duty is lower than the desired target.

5.2 Flexibility of heat exchanger

As described in the previous section, the model of heat exchanger to be used further will be considered only the case of no bypass fraction or maximum heat duty. Because flexibility analysis is usually considered at steady-state, the 1-cell model of heat exchanger can be simplified as follows,

0chhhphihp TTUATFCTFC (30)

0chccpcicp TTUATFCTFC (31)

For heat exchanger, the flexibility can be considered in terms of satisfaction of approach and target temperatures as follows,

minTTT cih (g1)

minTTT chi (g2)

etth TT arg (g3)

From the results in section 5.1.1, the 1-cell model overestimates approach temperature and underestimates the amount of heat exchange. This means that the satisfaction of the equations (g1) and (g2) when using 1-cell model cannot guarantee the satisfaction of the real heat exchanger. To handle this problem, one needs to provide a safety factor to Tmin. In consideration of satisfaction of equation (g3), 1-cell model can guarantee this satisfaction for the real heat exchanger.

27

011

1hicpcihp

cphpcphp

TFCUA

TFCFCFCFCFC

UA

minTTTFC cihicp

011

1cicphihp

cphpcphp

TFCUA

TFCFCFCFCFC

UA

minTTTFC hicicp

011

1arg etthicphicphpcihp

cphpcphp

TTFCTFCFCUA

TFCFCFCFCFC

UA

5.2.1 Inlet temperatures as disturbance (effect to flexibility)

Inequalities g1, g2 and g3 define feasible region of a heat exchanger under

variation of disturbances or uncertainty. The feasible region under the case of inlet

temperatures as disturbances can be given using equations 30-31 to eliminate Th and

Tc from inequalities g1-g3 to make these inequalities as function of Tci and Thi

resulting inequalities 32-34.

(32)

(33)

(34)

The parameters used for the illustration of flexibility analysis of heat exchangers is given in Table 3. Further assuming that the disturbance windows are ± 5 oC and ± 10 oC for Tci and Thi, respectively. Under these condition, feasible region can be generated as shown in Figure 17-18 for heat exchanger with area = 5, 8 and 10 m2.

28

Table 3 Parameters used for the simulation when inlet temperatures as disturbances.

Figure 17 Feasible region of heat exchanger with A = 5 m2.

Parameters Values FCp,h 697.47 W/ºC FCp,c 1336.537 W/ºC

U 850 W/m2/ºC Tmin 1 ºC Ttar 60 ºC

Nominal Tci 30 ºC Nominal Thi 90 ºC

A 5 m2

Feasible region

Disturbance window

29

Figure 18 Feasible region of heat exchanger with A=5, 8, and 10 m2.

Figure 17 plots the feasible region of heat exchanger under inlet temperatures

as disturbances. The heat exchanger with A = 5 m2 is infeasible, that is, the heat exchanger is not able to reject disturbances in some regions on the top-right of the disturbance window.

An attempt to make the heat exchanger feasible is to increase the heat transfer area. Figure 18 shows feasible regions when heat transfer areas are 5, 8, and 10 m2. Note that the feasible region is larger for higher heat transfer area. The heat exchangers with A = 8 and 10 m2 have ability to reject the given disturbances.

As described in Figure 16, the feasible operating range of target temperature is bounded by outlet temperature given at maximum heat duty condition. The range is larger for heat exchanger with higher duty or larger heat transfer area. Therefore, heat exchanger with higher heat transfer area is more flexible.

As can be seen from Figures 17 and 18, the feasible region tends to be convex shape. The convexity condition is checked and shows that the inequalities 32-34 are convex functions forming a convex set. Detail for checking the convexity is shown in the Appendix B.

Feasible region

Disturbance window

A=5m2

A=10m2

A=8m2

30

The important implication of the convexity of the inequalities 32-34 is that the flexibility test problem can be reduced as checking the feasibility of the corner points or vertices of disturbance windows without the need of feasible region plot.

Table 4 shows that the feasibility test of the heat exchanger with A = 5 m2 at the four vertices of given disturbances. At the vertex Tci = 35oC and Thi = 100oC, the inequality (h3) cannot be satisfied and hence the heat exchanger is not flexible. The test of the heat exchangers with A = 8 and 10 m2 are shown in Table 5. The results show that these heat exchangers are flexible.

Table 4 Feasibility test for the case A = 5 m2.

Table 5 Feasibility test for the case A = 8, 10 m2.

Vertices Satisfaction of inequalities

Thi Tci (g1) (g2) (g3)

25 80

25 100

35 80

35 100

Vertices Satisfaction of inequalities

Thi Tci (g1) (g2) (g3)

25 80

25 100

35 80

35 100

31

5.2.1 Stream flow rates as disturbances (effect to flexibility)

This section discusses the effect of stream flow rates to flexibility of the heat exchanger. By using equations 30-31 to make the inequalities g1-g3 as function of stream flow rates, (FCp)c and (FCp)h, the feasible region (under parameters in Table 6) can be generated as shown in Figures 19-20 for heat exchanger with area = 5 and 50 m2.

Table 6 Parameters used for the simulation when stream flow rates as disturbances


Figure 19 plots the feasible region of heat exchanger under flow rates as disturbances. The heat exchanger with A = 5 m2 is infeasible. The heat exchanger is

Parameters Values Nominal FCp,h 700 W/ºC Nominal FCp,c 1400 W/ºC

U 850 W/m2/ºC Tmin 5 ºC Ttar 60 ºC Tci 30 ºC Thi 90 ºC A 20 m2

Disturbance window

Feasible region (FC

p)h

(FCp)c

32


not able to reject disturbances in some region on the top-left of the disturbance window.

An attempt to make the heat exchanger feasible is to increase the heat transfer area. Figure 20 shows feasible regions when heat transfer areas is 50 m2. Note that the feasible region is larger for higher heat transfer area. However, the heat exchangers with A = 50 m2 can reject the given disturbances.

As can be seen from Figures 19 and 20, the feasible region is not obvious to be convex or nonconvex shape. The convexity condition is checked here again like performing in the previous section. The result showed that the region is nonconvex. In this case, the shape of disturbance is a rectangle that can be check the corner point was enough. But if the disturbance window is other shape that this mean the flexibility test problem cannot be simplified as checking the corner point as the case of inlet temperature as disturbances.

5.3 Integration of design and control

As shown in the previous section, larger heat transfer area improves flexibility (control objective) but results in higher equipment cost (economic design objective). These two objectives should be traded-off and this results in a multi-objective optimization problem. A procedure for design and control of heat exchanger is proposed as shown in Figure 19. Alternatively, the procedure can also be formulated as the optimization problem P1.

(FC

p)h

(FCp)c

Disturbance window

Feasible region

33

Figure 21 The proposed algorithm for design and control for calculation.

For heat exchanger, the design objective is heat transfer area and the controllability objective is flexibility. Thus the optimization problem can be written:

Start

Give data and desired outlet temperature

Determine heat exchanger area(A)

Min u

s.t. g1 ≤ u g2 ≤ u g3 ≤ u

Check flexibility

End

Yes

No

Increase A

34

Optimization Problem P1:

Minimize A s.t. min u s.t. g1 ≤ u g2 ≤ u

g3 ≤ u where Tci, Thi {vertices of disturbances} (P1)

The trading-off between design and control objectives can be given by solving the problem P1. However, the solution is quite intuitive, that is, choosing the smallest flexible heat exchanger. As shown in Figure 18 that the heat exchanger with A = 8 m2 is better because this is the smallest (and cheapest) one that is flexible under given disturbances.

5.4 Case study

The proposed method will be implemented with two cases studies as follows,

5.4.1 Case study 1 : Heat exchanger with bypass

Figure 22 Heat exchanger with bypass

The configuration of heat exchanger in the first cases study is shown in Figure 22. A bypass is installed on the hot-side. Parameter used for the design is shown in Table 8. Assuming disturbances are ± 5oC of inlet temperatures. Aspen Plus is used in this case study for modeling of heat exchanger. Using the proposed method by performing optimization in Aspen Plus, the optimal area given is 30 m2. To check if this designed area can reject the given disturbances, design specification is used to find if there is a value of bypass fraction to meet the target temperature (65 oC). As shown in Table 8, the designed heat exchanger can tolerate all disturbances.

35

Table 7 Parameters used for the simulation in case study 1

Table 8 The value of bypass fraction at corner point of disturbance

5.4.2 Case study 2 : Feed-effluent heat exchanger (FEHE)

The case study is taken from Luyben (2012). The case concerns the production of dimethyl ether (DME) from methanol. The vapor-phase reaction is exothermic and reversible.

OHDMEMeOH 22

Parameters Values Nominal f 0.265

U 850 W/m2/ºC Tmin 1 ºC Ttar 65 ºC Tci 30 ºC Thi 85 ºC vH 5 kg/s

vc 3 kg/s

Point Tci,Thi( ºC) Bypass fraction

V1 90,35 0

V2 80,35 0.6062

V3 80,25 0.7064

V4 90,25 0.4698

nominal 85,30 0.2656

36

REACTOR 425 K 662 K 541 K

F = 295.2 kmol/h

95 mol%MeOH

5 mol%H2O

F = 295.2 kmol/h

38.68 mol%DME

17.48 mol%MeOH

43.84 mol%H2O

Figure 23 Feed-effluent heat exchanger of DME process.

Figure 24 Feasible region of heat exchanger with A = 27.51 m2.

The disturbance in this case study is ± 75 K of inlet temperature of cold stream (nominal value = 425 K). The operating range of hot stream given by running the simulation in Aspen Plus with the flowsheet in Figure 23 is ± 60 K (nominal value = 662).

FEHE

Inle

t tem

pera

ture

on

hot s

trea

m, T

hi (K

)

Disturbance window

Feasible region

Inlet temperature on cold stream, Tci (K)

37

Figure 24, the cold stream to the heat exchanger is saturated vapor at 425 K and 14 atm coming from the vaporizer. The desired reactor inlet temperature is 541 K, so the cold exit stream from the FEHE must be heated to 541 K. The hot reactor exit stream that enters the hot end of the FEHE is at 662 K. Using an overall heat-transfer coefficient of 0.17 kW/K/m2 and required heat transfer area is 27.51 m2.

Figure 25 plots the feasible region of heat exchanger under inlet temperatures as disturbances. Note that disturbance window are get the value of simulation process from Aspen plus program that define range of Tci are simulate to find range of Thi. The heat exchanger with A = 27.51 m2 is feasible, the window is also included in the plot. It is clearly that the heat exchanger is not able to reject disturbances in region.

Figure 23 plots the feasible region of heat exchanger under inlet temperatures as disturbances. To see whether under given disturbance windows (Tci = 425±75 K and Thi = 662±60 K) the heat exchanger with A = 27.51 m2 is feasible, the window is also included in the plot. It is clearly that the heat exchanger is not able to reject disturbances in some region.

Figure 24 shows feasible regions when heat transfer areas is 500 m2. Note that the feasible region is smaller for higher heat transfer area. The heat exchangers with A 500 m2 have not ability to reject the given disturbances.

As described in Figure 24, the target temperature is 541 K that point must be in feasible region. However, red line is means a target temperature in equation 26 which is the problem not able conditions. Thus, solving is decrease target temperature lower 541 K that is bad to solve because temperature reaction effect to product and conversion etc. Luyben (2012) proposed methodology for this problem that the system add furnace before inlet reactor.


rnac

Inle

t tem

pera

ture

on

hot s

trea

m, T

hi(K

)

Inlet temperature on cold stream, Tci (K)

Disturbance window

Feasible region

38

CHAPTER 6

CONCLUSIONS

Design and control of heat exchanger with bypass when flexibility considered as control objective has been conducted. The disturbances considered are inlet temperatures and stream flow rates. An important design parameter is heat transfer area because larger heat exchanger can accommodate broader range of uncertainties than the smaller one. With one’s intuition, the design and control of the system can be viewed as choosing smallest (cheapest) heat exchanger that can satisfy flexibility requirement.

The results also showed that when inlet temperatures as disturbances, the flexibility test problem can be simplified as checking the feasibility of corner points or vertices of disturbances.

Although increasing heat transfer area tends to improve flexibility, in many cases the system cannot be feasible. This may be because insufficient of available heat or feasibility of approach temperature, etc. Furthermore, in practice, heat exchangers are usually operated as a network, i.e. the problem on flexibility of heat exchanger networks should be considered to make the system more controllable.

39

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40

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Appendix

Appendix A : Rearrange equations

43

hhihphhpc UATTFCTFCUAT

ccicpccph UATTFCTFCUAT

hicphicphpcihp

cphpcphp

h TFCTFCFCUA

TFCFCFCFCFC

UA

T 11

1

Rearrange Equations

From Equations 22 - 26

0chhhphihp TTUATFCTFC (22)

0chccpcicp TTUATFCTFC (23)

minTTT cih (24)

minTTT chi (25)

etth TT arg (26)

From Equation 22 rearrangement given Tc represented by Th can be written as:

(h1)

From Equation 23 rearrangement given Th represented by Tc can be written as:

(h2)

Substituting into h1 to h2:

Then, Th equation depend on Thi, Tci:

hhhphihpc TTFCTFCUA

T )()(1

cccpcicph TTFCTFCUA

T )()(1

),(1 cihih TTfT

44

cicpcicphphihp

cphpcphp

c TFCTFCFCUA

TFCFCFCFCFC

UA

T 11

1

011

1hicpcip

cphpcphp

TFCUA

TFCFCFCFCFC

UA

minTTTFC cihicp

011

1cicphip

cphpcphp

TFCUA

TFCFCFCFCFC

UA

minTTTFC hicicp

011

1arg etthicphicphpcihp

cphpcphp

TTFCTFCFCUA

TFCFCFCFCFC

UA

Substituting into h2 to h1: Then, Tc equation depend on Thi, Tci:

Substituting Th into Equation 24 can be written: (g1) Substituting Tc into Equation 25 can be written:

(g2)

Substituting Th into Equation 26 can be written:

(g3)

Equations g1, g2 and g3 are used the feasible region when temperature and flow rate are disturbances. But in case flow rates Tc and Th depend on (FCp)h and (FCp)c.

),(2 cihic TTfT

Appendix B : Hessian matrix

46

hichichcih

chch

h TTUA

T

UA

T 11

1

ciccichhih

chch

c TTUA

T

UA

T 11

1

AUch ,,,

011

1minTTTT

UAT

UA

cihichichcih

chch

Hessian Matrix

Inlet temperature as disturbance

From Equations 30 and 31

Where, pFC

Parameter: Variable : Thi, Tci (g1)

hcc

hchcUA

UAxf

xf 1

11:

1

121

12

0:21

12

xf

11

1:2

122

12

c

hchcUAxf

xf

0:22

12

xf

47

011

1minTTTT

UAT

UA

hiciccichhih

chch

11

1:12

1

121

12

c

hchcUAxx

fxxx

f

0:21

12

xxf

hcc

hchcUA

UAxx

fxxx

f 11

1:21

1

212

12

0:12

12

xxf

From equation f1 transform to Hessian matrix

21

12

21

12

1

xxf

xf

H

22

12

21

12

xf

xxf

Hence, H1 = 00

00

Find eigenvalue ( ) ; H1 = 00

00

2-0 =0

= 0,0 That result is convex function.

(g2)

48

hcc

hchcUA

UAxf

xf 1

11:

1

221

22

0:21

22

xf

11

1:2

222

22

h

hchcUAxf

xf

0:22

22

xf

chc

hchcUA

UAxx

fxxx

f 11

1:12

2

121

22

0:21

212

xxf

h

hchcUAxx

fxxx

f1

11:21

2

212

22

0:12

22

xxf


21

22

21

22

2

xxf

xf

H

22

22

21

22

xf

xxf

Hence, H2 = 00

00

= (0,0) That result is convex function.

49

011

1arg etthichichcih

chch

TTTUA

T

UA

(g3)

hchcUAxf

xf

11:

1

321

32

0:21

32

xf

h

hchcUAxf

xf

11:

2

322

32

0:22

32

xf

h

hchcUAxx

fxxx

f1

1:12

3

121

32

0:21

32

xxf

cch

hchcUA

UAxx

fxxx

f 11

1:21

3

212

32

0:12

32

xxf

50

ch ,

011

1minTTTT

UAT

UA

cihichichcih

chch


21

32

21

32

3

xxf

xf

H

22

32

21

32

xf

xxf

Hence, H3 = 00

00

= (0,0) That result is convex function.

Stream flow rate as disturbance Parameter: Tci, Thi, U, A

Variable:

(g1)

From equation f2 transform to Hessian matrix,

21

12

21

12

1

xxf

xf

H

22

12

21

12

xf

xxf

minmin1

121

1 111: TTUA

TTUA

TUA

Txf

xf

ccicichichi

0:21

12

xf

minmin2

122

12 111: TT

UATT

UAT

UAT

xf

xf

hcicichicci

0:22

12

xf

51

011

1minTTTT

UAT

UA

hiciccichhih

chch

minmin21

2

212

12 111: TT

UATT

UAT

UAT

xxf

xxxf

ccicichichi

min12

12 1 TTT

UAxxf

cihi

minmin22

2

121

12 111: TT

UATT

UAT

UAT

xxf

xxxf

ccicichicci

min12

12 1 TTT

UAxxf

cihi

Hence, H1 = min

10

TTTUA cihi 0

1minTTT

UA cihi

H1 = min

10

TTTUA cihi

0

1minTTT

UA cihi

The value of = nonconvex function.

(g2)


21

22

21

22

2

xxf

xf

H

22

22

21

22

xf

xxf

0:21

22

xf

, 0:2

2

22

xf

min12

22 1 TTT

UAxxf

cihi

, min

12

22 1 TTT

UAxxf

cihi

52

Hence, H2 = min

10

TTTUA cihi 0

1minTTT

UA cihi

H2 = min

10

TTTUA cihi

0

1minTTT

UA cihi

The value of = (is not defined sign) that show nonconvex function.


21

32

21

32

3

xxf

xf

H

22

32

21

32

xf

xxf

0:21

32

xf

, 0:2

2

32

xf

hitar TTUAxx

f 112

32

, hitar TT

UAxxf 1

12

32

Hence, H3 = hitar TT

UA1

0

0

1hitar TT

UA

H3 = hitar TT

UA1

0

0

1hitar TT

UA

The value of = (is not defined sign) that show nonconvex function.

Appendix C : Nomenclature

54

Nomenclature

Abbreviations

Thi = inlet temperature on hot stream Th = outlet temperature on hot stream Tci = inlet temperature on cold stream Tc = outlet temperature on cold stream Fh = mass flow rate of fluid in hot side Fc = mass flow rate of fluid in cold side Cp,h = specific heat capacity of the fluid hot steam Cp,c = specific heat capacity of the fluid cold stream U = overall heat transfer coefficient A = heat exchanger area Tmin = approach temperature Ttar = target temperature Tho = temperature on hot side before enter to mixer vc = volumetric flow rate in hot cell tank vh = volumetric flow rate in cold cell tank Vh = volume of the hot cell tank Vc = volume of the cold cell tank

Appendix D : International proceeding

56

International Proceeding

Arpaporn Somsuk, Veerayut Lersbamrungsuk, “Cell-based models of heat exchangers with bypasses” PACCON International Conference 2014, Khon Kean, Thailand, 8-10 Jan, 2014 (Poster Presentation)

Arpaporn Somsuk, Nattawat Petchsoongsakul, Ratikorn Phaethong, Veerayut Lersbamrungsuk, “Design and control of heat exchangers with bypasses when inlet temperatures as disturbances” TIChE International Conference 2014, Chiang Mai, Thailand, 18-19 Dec, 2014 (Oral Presentation)

57

Biography

Name: Miss Arpaporn Somsuk

Birth date: 15th July 1990

Place of birth: Suphan Buri, Thailand

Nationality: Thai

Religion: Thai

Address: 17 M. 7 T. Khao-Phra A. Doembangnangbuat, Suphan Buri, Thailand, 72120. Tel. 081-1997596

Contact: [email protected], Tel. 0811997596

Education:

2008 High school certificate from Sa-Nguan Ying School

2011 Received the degree of the Bachelor of Engineering (Chemical

Engineering), Faculty of Engineering and IndustrialTechno logy, Silpakorn University, Nakorn Pathom, Thailand

2014 Futher studied in the degree of the master of Chemical Engineering at graduate school, Faculty of Engineering and Industrial Technology, Graduate School, Silpakorn University, Thailand.

design and control of heat exchangers with bypasses … · transfer area. in terms of design...

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