ĐẠI HỌC QUỐC GIA TP HỒ CHÍ MINH TRƯỜNG ĐẠI HỌC BÁCH KHOA NGUYỄN CHÍ THUẦN NGHIÊN CỨU ỔN ĐỊNH BÌNH PHẢN ỨNG HÓA HỌC PHÁT NHIỆT DÙNG NHIỆT ĐỘNG LỰC HỌC CHUYÊN NGÀNH: KỸ THUẬT HÓA HỌC MÃ SỐ CHUYÊN NGÀNH: 60520301 LUẬN VĂN THẠC SĨ TP HỒ CHÍ MINH – THÁNG 01 NĂM 2018 CƠNG TRÌNH ĐƯỢC HỒN THÀNH TẠI TRƯỜNG ĐẠI HỌC BÁCH KHOA –ĐHQG TP HCM Cán hướng dẫn khoa học: TS Hoàng Ngọc Hà PGS TS Nguyễn Quang Long Cán chấm nhận xét 1: TS Ngô Thanh An Cán chấm nhận xét 2: TS Lý Cẩm Hùng Luận văn thạc sĩ bảo vệ Trường Đại học Bách Khoa, ĐHQG Tp HCM ngày 29 tháng 01 năm 2018 Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm: (Ghi rõ họ, tên, học hàm, học vị Hội đồng chấm bảo vệ luận văn thạc sĩ) PGS TS Phan Minh Tân TS Ngô Thanh An TS Lý Cẩm Hùng TS Lê Xuân Đại TS Nguyễn Thành Duy Quang Xác nhận Chủ tịch Hội đồng đánh giá luận văn Trưởng Khoa quản lý chuyên ngành sau luận văn sửa chữa (nếu có) CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA ĐẠI HỌC QUỐC GIA TP.HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM Độc lập - Tự - Hạnh phúc NHIỆM VỤ LUẬN VĂN THẠC SĨ Họ tên học viên: Nguyễn Chí Thuần MSHV: 7140198 Ngày, tháng, năm sinh: 30/5/1989 Nơi sinh: Bến Tre Chuyên ngành: Kỹ thuật hóa học Mã số : 60520301 I TÊN ĐỀ TÀI: Nghiên cứu ổn định bình phản ứng hóa học phát nhiệt dùng nhiệt động lực học II NHIỆM VỤ VÀ NỘI DUNG: Nghiên cứu tổng quan tài liệu, phương pháp tiếp cận nghiên cứu Cơ sở nhiệt động lực học phương pháp ổn định Lyapunov Nghiên cứu xây dựng mơ hình tốn thiết bị khảo sát tính tốn trạng thái dừng Nghiên cứu ổn định/ không ổn định trạng thái thiết bị Tính tốn mơ kiểm chứng kết dùng phần mềm Matlab & Simulink III NGÀY GIAO NHIỆM VỤ : 10/7/2017 IV.NGÀY HOÀN THÀNH NHIỆM VỤ: 03/12/2017 V CÁN BỘ HƯỚNG DẪN : TS Hoàng Ngọc Hà PGS TS Nguyễn Quang Long Tp HCM, ngày tháng năm 20 CÁN BỘ HƯỚNG DẪN (Họ tên chữ ký) CHỦ NHIỆM BỘ MÔN ĐÀO TẠO (Họ tên chữ ký) TRƯỞNG KHOA….……… (Họ tên chữ ký) i LỜI CẢM ƠN Lời cho gửi lời cảm ơn sâu sắc chân thành đến TS Hoàng Ngọc Hà - người thầy dìu dắt tơi suốt thời gian thực luận văn Trong trình thực luận văn, dù có trải qua khó khăn với tình cảm, hướng dẫn, động viên giúp đỡ thầy giúp vượt qua hoàn thành luận văn Những lời dạy thầy ghi nhớ động lực để phấn đấu, khơng để hồn thành luận văn mà việc mà làm công việc sống Tôi xin chân thành cảm ơn PGS.TS Nguyễn Quang Long, người thầy tiếp thêm cho tơi nguồn động lực để hồn thành luận văn Những lời dạy dỗ, lời động viên ân cần sâu sắc thầy tiếp thêm nghị lực động lực giúp tiếp cận hồn thành luận văn Tơi xin cảm ơn Khoa Kỹ thuật Hóa học, Trường Đại học Bách Khoa – ĐHQG Tp Hồ Chí Minh giúp tơi hồn thành luận văn Xin gửi lời cảm ơn đến quý thầy cô, đồng nghiệp, anh chị em, người thân quen động viên, hỗ trợ tinh thần, kỹ điều kiện để tơi hồn thành luận văn Đặc biệt, xin cảm ơn ba mẹ ni nấng, chăm sóc dạy dỗ nên người Tình yêu thương tin yêu ba mẹ nguồn sáng, động lực giúp vượt qua khó khăn việc ii TĨM TẮT Trong kỹ thuật hóa học, nhiệt động lực học đóng vai trò trung tâm cho việc nghiên cứu khảo sát biến đổi hệ q trình hóa học Hai biểu diễn xem xét nhiệt động lực học biểu diễn lượng biểu diễn entropy Một mặt, thay đổi trạng thái (ví dụ nhiệt độ, áp suất nồng độ cấu tử…) liên quan trực tiếp đến biến đổi lượng entropy hệ trình hóa học Kết nguyên lý thứ thứ hai nhiệt động lực học chi phối vận động xu hướng tiến triển trạng thái trình Mặt khác, tượng đặc trưng (truyền vận, truyền khối động học phản ứng) diễn hệ giải thích mơ hình hóa sở nhiệt động lực học Dựa vào đặc tính này, chúng tơi đề xuất xây dựng sở cho phép nghiên cứu toán ổn định trạng thái cân dừng hệ dùng phương pháp Lyapunov Phản ứng hoá học phát nhiệt hợp nước xúc tác axít 2-3- epoxy-1- propanol tạo glycerol thiết bị khuấy trộn hoạt động với nhiều trạng thái cân dừng sử dụng để minh hoạ kết phát triển luận văn iii ABSTRACT In chemical engineering, thermodynamics plays a central role for studying the changes of the states of chemical process The two key representations considered in thermodynamics are the energy and entropy representations of the system On the one hand, the evolution of the system states (such as the temperature, pressure and concentrations of species…) is directly linked to the energy and entropy transformations As a consequence, the first and the second principles of thermodynamics allow to predict the evolution of the system states On the other hand, typical phenomena (such as heat and mass transfers, transport phenomena and reaction kinetics) can be explained and modelled by thermodynamics On this basis, we propose a novel approach, which is combined with Lyapunov stability theory and can be further considered for the stability analysis of steady states (called also stationary equilibrium points) of the dynamical system A liquid phase reactor modelled with the CSTR (continuous stirred tank reactor) in which the acid-catalyzed hydration of 2-3-epoxy-1propanol to glycerol subject to steady state multiplicity takes place is used to illustrate the results iv LỜI CAM ĐOAN CƠNG TRÌNH ĐƯỢC HỒN THÀNH TẠI PHỊNG THÍ NGHIỆM TRỌNG ĐIỂM ĐIỀU KHIỂN SỐ VÀ KỸ THUẬT HỆ THỐNG (DCSELAB), TRƯỜNG ĐẠI HỌC BÁCH KHOA, ĐẠI HỌC QUỐC GIA TP HỒ CHÍ MINH Tơi xin cam đoan cơng trình nghiên cứu riêng tơi hướng dẫn TS Hoàng Ngọc Hà Nội dung kết nghiên cứu luận văn trung thực chưa cơng bố hình thức trước Những thông tin, liệu thu thập từ nhiều nguồn khác để phục vụ cho việc thực luận văn có ghi rõ phần tài liệu tham khảo Nếu phát có gian lận tơi xin hồn tồn chịu trách nhiệm nội dung luận văn Trường Đại học Bách khoa – ĐHQG TP Hồ Chí Minh khơng liên quan đến vi phạm tác quyền, quyền tơi gây q trình thực (nếu có) TP Hồ Chí Minh, tháng 01 năm 2018 NGUYỄN CHÍ THUẦN v MỤC LỤC LỜI CẢM ƠN i TÓM TẮT ii ABSTRACT iii LỜI CAM ĐOAN iv MỤC LỤC v DANH MỤC BẢNG vii DANH MỤC HÌNH viii CHƯƠNG TỔNG QUAN 1.1 Đặt vấn đề 1.2 Nhiệt động lực học 1.2.1 Hệ nhiệt động 1.2.2 Nguyên lý thứ nhiệt động lực học 1.2.3 Nguyên lý thứ hai nhiệt động lực học 1.3 Phần mềm Matlab Simulink mô 1.4 Các nghiên cứu công bố nước 1.5 Nhận xét chung CHƯƠNG THIẾT LẬP MƠ HÌNH CỦA HỆ PHẢN ỨNG 2.1 Những giả thiết ban đầu 2.2 Phương trình cân vật chất 10 2.3 Phương trình cân lượng 11 2.4 Mơ hình thiết bị khuấy trộn hoạt động liên tục cho phản ứng hợp nước xúc tác axít tạo glycerol từ 2,3-epoxy-1-propanol 13 2.5 Nhận xét chung 14 CHƯƠNG CƠ SỞ VÀ PHƯƠNG PHÁP NGHIÊN CỨU ỔN ĐỊNH HỆ PHẢN ỨNG 16 3.1 Ổn định Lyapunov 16 3.1.1 Khái niệm ổn định Lyapunov 16 3.1.2 Trạng thái cân dừng 16 3.1.3 Phương pháp Lyapunov trực tiếp 17 3.2 Nhắc lại số khái niệm nhiệt động lực học 18 3.3 Xây dựng hàm lưu trữ ổn định hệ thống 19 3.3.1 Độ sẵn có nhiệt động lực học 19 3.3.2 Hàm lưu trữ nhiệt động lực học 20 vi 3.4 Nhận xét chung 22 CHƯƠNG MÔ PHỎNG KẾT QUẢ VÀ NHẬN XÉT 23 4.1 Điều kiện để thực mô 23 4.2 Khảo sát trạng thái cân dừng hệ phản ứng 24 4.3 Khảo sát tính ổn định/không ổn định trạng thái cân dừng hệ 26 CHƯƠNG KẾT LUẬN VÀ HƯỚNG PHÁT TRIỂN 35 5.1 Kết luận 35 5.2 Hướng phát triển 35 TÀI LIỆU THAM KHẢO 36 PHỤ LỤC 39 PHỤ LỤC A: DỮ LIỆU NHẬP CÁC THÔNG SỐ MÔ PHỎNG VÀO MATLAB 39 PHỤ LỤC B: CHƯƠNG TRÌNH TÍNH VÀ VẼ GIẢN ĐỒ VAN HEERDEN CỦA HỆ PHẢN ỨNG DÙNG MATLAB 41 PHỤ LỤC C: MÔ PHỎNG SIMULINK CỦA HỆ PHẢN ỨNG 42 PHỤ LỤC D: CÁC CÔNG TRÌNH CƠNG BỐ 46 vii DANH MỤC BẢNG Bảng 4.1: Thông số nhiệt động hệ thống 23 Bảng 4.2 Điều kiện vận hành hệ mô 23 Bảng 4.3 Ba trạng thái cân dừng hệ phản ứng 26 Bảng 4.4 Điều kiện ban đầu cho mô 27 23rd Regional Symposium on Chemical Engineering (RSCE2016) Innovation in Chemical Engineering towards the linkages among education, academia, industry 27-28 Oct 2016 in Vung Tau City, Vietnam [20] Alvarez J., Alvarez-Ramírez J., Espinosa-Perez G., Schaum A., Energy shaping plus damping injection control for a class of chemical reactors Chem Eng Sci 66(23):6280-6286, 2011 [21] Farschman C.A., Viswanath K.P., Ydstie B.E., Process systems and inventory control AIChE Journal 44(8):1841-1857, 1998 [22] Khalil H.K., Nonlinear systems Prentice Hall, 3rd edition, 2002 [23] Callen, H.B., Thermodynamics and an 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Vleeschhouwer P.H.M., Fortuin J.M.H., Theory and experiments concerning the stability of a reacting system in a CSTR AIChE Journal 36:961–965, 1990 [31] Hoang H., Couenne F., Jallut C., Le Gorrec Y., Thermodynamics-based stability analysis and its use for nonlinear stabilization of CSTR Computers & Chemical Engineering 58(11):156-177, 2013c [32] Nguyen T., Nguyen L., Hoang H., A revisit on dissipation and its relation to irreversible processes Proc DREMD-1 Can Tho, Vietnam pp 128-138, 2016 487 A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity Thuan C Nguyen Ha N Hoang *,1,2 University of Technology, VNU-HCM, 268 Ly Thuong Kiet Str., Dist 10, HCM City, Vietnam Duy Tân University, 254 Nguyen Van Linh Road, Da Nang, Vietnam *e-mail: ha.hoang@hcmut.edu.vn Following the second law of thermodynamics, the entropy is always created in irreversible processes such as reacting systems, etc Under certain operating conditions, the reaction system can be operated with multiple steady states (also called the steady state multiplicity behavior) This behavior is considered for the illustration of the stability analysis of all possible steady states by Lyapunov methods using thermodynamics More precisely, a novel symmetric storage function (or Lyapunov function candidate) is proposed on the basis of the so-called (non-symmetric) thermodynamic availability function The acid-catalyzed hydration of 2-3-epoxy-1-propanol to glycerol subject to steady state multiplicity is used for further technical developments The results are discussed with the inclusions of the simulations Keywords : Dynamical systems, multiplicity, Lyapunov function, stability, control, thermodynamics INTRODUCTION In chemical engineering, the reaction kinetics plays a central role as the source ”A theory is the more impressive the generating the abnormal dynamical greater the simplicity of its premises, the behavior of the reaction system (e.g non- more varied the kinds of things that it minimum phase behavior, limit cycle or relates and the more extended the area of multiple steady states, etc (Bayer et al its classical 2011, Hoang et al 2013a)) On the other a hand, applicability thermodynamics Therefore have made deep the reaction be framework of modeled theory of universal content which I am thermodynamics (Dammers and Tels 1974, convinced, the Tarbell 1977, Ydstie and Alonso 1997) In applicability of its concepts, that it will that respect, the use of thermodynamics never been overthrown.” (Albert Einstein, for the stability analysis and control design 1949) of the areas of chemical the can impression on me It is the only physical within within kinetics reaction networks is Thuan C Nguyen and Ha N Hoang considered (Eberard et al 2007, Ederer et function candidate used for the stability al 2011, Hoang et al 2012, Hoang and analysis is then shown The liquid phase Dochain 2013a, 2013b, García-Sandoval et acid-catalyzed hydration of 2-3-epoxy-1- al 2015, 2016, Georgakis 1986, Rodrigues propanol to glycerol modeled with CSTR et al 2015) The Continuous Stirred Tank subject to the steady state multiplicity Reactors (CSTRs) belong to a typical class behavior is given in section From this, of nonlinear dynamic systems described the by Ordinary Differential Equations ODEs dynamics is developed Section ends the (Luyben 1990) Several applications of paper with conclusions nonlinear control methods to CSTRs have been developed in the literature (see e.g., stability analysis of the system Throughout the paper, the following notations are used: Viel et al 1997, Farschman et al 1998, •  is the set of real numbers Alvarez-Ramírez • m and n are the positive integers and Morales 2000, Hangos et al 2001, Antonelli and Astolfi 2003, Hudon et al 2008, Favache and Dochain 2010, Alvarez et al 2011, Bayer et al 2011, Hoang et al 2011, 2013a, 2013b, • R can be either the (ideal) gas constant or thermodynamic storage function • t can be either time or matrix transpose Ramírez et al 2013, Hoang and Dochain 2013b) In (Viel et al 1997, Favache and THE LYAPUNOV STABILITY THEORY Dochain (LYAPUNOV'S SECOND METHOD FOR 2010), the steady state multiplicity behavior is considered for the STABILITY OR LYAPUNOV’S control design of the unstable CSTR METHOD) DIRECT This paper focuses on the stability analysis of all possible steady states of the Let us consider a nonlinear dynamical acid-catalyzed hydration of 2-3-epoxy-1- system which is affine in the control input propanol u and whose dynamics are given by the to glycerol by Lyapunov methods using thermodynamics derived following set of ODE’s: directly from the second principle For that purpose, we first review the basis of thermodynamics and then remind how to obtain the availability function Consequently, this allows us to construct a Lyapunov function candidate (LFC) usable for the stability analysis This is the main contribution of this work The paper is organized as follows The Lyapuvov stability theory is briefly introduced in section In section 3, thermodynamics and its properties associated to the single-phase reaction mixture are represented A Lyapunov dx(t ) •  x(t )  f ( x(t ))  g ( x(t )) u, x(t  0)  x0 dt (1) where x  x(t )   n is the state vector, f ( x(t ))   n represents the smooth nonlinear function with respect to g ( x(t ))   nm x, is the input-state map and u   m is the control input A positive continuous function V : n   is called a Lyapunov function candidate for the stability analysis at a desired state xe  (i.e, at the origin) if 10 A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity and only if the three following conditions using the Gibbs’ relation (Callen 1985): are met (Khalil, 2002): n dH  TdS   i dNi i) V  x(t )   0, x(t )  ii) V  x (t )   0, x (t )  (2) • iii) V  x(t )   0, x(t ) xe  is said to be stable Moreover, • V x(t )   0, x(t )  then xe  if is (globally) stable Remark 1: If V  x (t )   0, x(t )  , then V is From (3), since the absolute temperature T >0 we have: Then, the state (locally) (3) i 1 positive definite If V  x(t )   0, x(t )  , then V is positive semi-definite Remark 2: If x    V (x)   then V is said to be radically unbounded In this dS  n  i dH   dNi T i 1 T (4) As the entropy S is also an extensive variable, it is a homogenous function of degree of (H, Ni), we get by using the Euler’s theorem (Callen 1985, Hoang et al 2009): S (H , Ni )  n  i H  Ni T i 1 T (5) • case, if V  x(t )   0, x(t )  then xe  is asymptotically stable Equivalently, one can write the variation of the entropy as: THERMODYNAMICS AND THERMODYNAMIC STORAGE FUNCTION S ( Z )  wt Z  w  w( Z )  S ( Z ) Z (6) The Overview of Thermodynamics Let us consider a reaction mixture composed of n (active) chemical species In thermodynamics, the system variables are divided into extensive variables (such as the internal energy U, the entropy S, the volume V, and the molar number Ni) and intensive variables (such as the temperature T, the pressure p, and the chemical potential µi) The variation of the internal energy U (under isobaric conditions, the enthalpy H defined as H = U + pV can then be used instead of the internal energy U) is directly derived from the variation of the extensive variables where: −𝜇1 −𝜇2 −𝜇 𝑡 , , … , 𝑛) 𝑇 𝑇 𝑇 𝑇 (𝐻, 𝑁1 , 𝑁2 , … , 𝑁𝑛 )𝑡 𝑤=( , 𝑍= , and (7) Thermodynamic Availability Based on the second law of thermodynamics, in case of homogenous thermodynamic systems, the entropy function S(Z) is concave with respect to its arguments Z (Ruszkowski et al 2005, Hoang et al 2011) The strict concavity of the entropy function S with respect to the extensive variable Z induces that the Thuan C Nguyen and Ha N Hoang 11 Fig 1: Entropy with respect to Z intersection between the tangent plane   A( Z * , Z )  S ( Z )  wt ( Z *  Z )  S ( Z * )  (10) A( Z * , Z )  0, Z *  Z ; and A( Z , Z )  (11) and the entropy surface S(Z) reduces to a point Such a situation is depicted in Figure (Alonso and Ydstie 2001) It can be shown that the non-negative function (Ydstie and Alonso 1997),  availability function A( Z , Z * ) is not symmetric, i.e A( Z , Z )  A( Z , Z )  t Also, it is worth noting that the * A(Z , Z * )  S (Z * )  w* (Z  Z * )  S (Z )  * (8) Thermodynamic Storage Function where Z* is a fixed reference state (for example, the desired set point for the control design), is the so-called thermodynamic availability (Keenan 1951) From a mathematical point of view, the thermodynamic availability A( Z , Z * ) is the distance between the entropy S(Z) and the tangent plane passing through Z* As a Let us define the following nonnegative potential function: ( Z , Z * )  A( Z , Z * )  A( Z * , Z ) (12) It is shown that when Z  Z * then R( Z , Z * )  since A( Z , Z * )  (from Eq (9)) and A( Z * , Z )  (from Eq (11)) From Eq (6), Eq (8) and Eq (10), Eq consequence, one concludes: (12) becomes: (9) A( Z , Z )  0, Z  Z ; and A( Z , Z )  * * * * R( Z , Z * )  ( w  w* )t ( Z  Z * ) (13) Similarly, it is shown that A( Z , Z ) is * the distance between the entropy S(Z) and Where: the tangent plane passing through Z In 1 −𝜇1 𝜇1∗ −𝜇2 𝜇2∗ −𝜇𝑛 𝜇𝑛∗ 𝑡 𝑤 − 𝑤∗ = ( − ∗ , + ∗, + ∗,…, + ∗) 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 other words, we have: 𝑍 − 𝑍 ∗ = (𝐻 − 𝐻 ∗ , 𝑁1 − 𝑁1∗ , 𝑁2 − 𝑁2∗ , … , 𝑁𝑛 − 𝑁𝑛∗ )𝑡 12 A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity Let us state the following proposition Proposition  R Z, Z * The storage function  has the following properties: The • where R is the (ideal) gas constant and non-negative    RZ , Z   , * *       we have: A Z , Z  Consequently, we derive:  the intensive variable vector   NA w    R ln   NA  NB  Proof From Eq (9), Eq (11) and Eq (12), * vector and *  variable Z  ( N A , N B ) We can check easily that The symmetry R Z , Z *  R Z * , Z • extensive t definiteness R Z, Z  A Z, Z  * the  t and, S m ( Z )  wt Z Z *  ( N A* , N B* ) t be the reference Let A Z * , Z  R( Z , Z * )  A( Z , Z * )    NB  ,  R ln       N A  NB  state In this case, the availability is derived Or we have R( Z , Z * )  A( Z , Z * )  In other words, from (12) and (13), it follows     that R Z , Z *  R Z * , Z since: using Eq (8) as follows:  NA N A*  N B*   A( Z , Z * )  RN A ln  N A*   N A  NB  NB N A*  N B*  RN B ln  N B*  NA  NB R( Z , Z )  ( w  w) ( Z  Z ) * * t * and R( Z , Z )  ( w  w ) ( Z  Z )  * R( Z , Z * ) is given by considering Eq (13):  NA N A*  N B*   R( Z , Z * )  R N A  N A* ln  N A*   N A  NB  Finally, * * * t * *  Let us consider the following example the sake of illustration of the thermodynamic concepts CSTR involving one exothermic irreversible chemical reaction A A B B Assume that the reaction solution is ideal and under isobaric and isochoric conditions The mixing entropy Sm of the reaction mixture can be expressed as follows (Hoang et al 2012):  NA   NB    RN B ln   S ( Z )   RN A ln   N A  NB   N A  NB  m The * B  geometric representation (16) of R( Z , Z ) (15) is given in Figure * Example We consider a jacketed homogeneous   NB N A*  N B*    R N B  N ln  N B*   N A  NB The latter completes the proof  (15) Consequently, the storage function  ( w*  w)t ( Z *  Z )  ( w  w* )t ( Z  Z * )  R( Z , Z * ) for     (14) As a consequence of Proposition 1, R( Z , Z * ) is a natural Lyapunov candidate For the dynamical stability analysis, it remains to check the sign condition of dR( Z , Z * ) (see Section for the Lyapunov dt stability theory), i.e.: dR( Z , Z * ) 0 dt (17) Thuan C Nguyen and Ha N Hoang 13 Fig 2: The thermodynamic storage function R( Z , Z ) with Z  (15, 15) * * Hence, the function R( Z , Z * ) is strongly related to the availability function t H C3 H 6O2  H 2O  C3 H 8O3        (18) A( Z , Z * ) (see Eq (12)) In some instances, this function is strictly convex and usable where the subscripts 1, and stand as a Lyapunov function for open loop for the stability analysis or closed loop control involved in Eq (18) The reaction rate per -1 -1 mass unit (i.e., mo l.k g s ) of the reaction design three active chemical species is given by (Vleeschhouwer et al 1988, THE LIQUID PHASE ACID – CATALYZED HYDRATION OF – – EPOXY – – PROPANOL TO GLYCEROL Hoang et al 2013a):  rm  k c H  e  Ta T (19) c1 Let us consider an irreversible reaction, namely the liquid phase acid-catalyzed hydration of 2-3-epoxy-1-propanol to glycerol taking place in a CSTR (continuous stirred tank reactor) For this reaction system, oscillations or unstable states have been experimentally shown (Heemskerk et al 1980, Rehmus et al 1983, Vleeschhouwer et al 1988, Vleeschhouwer and Fortuin 1990) The stoichiometry follows: equation is written as where cH  , c1 , k0 and Ta stand for the molar concentrations of H+ and 2-3epoxy-1-propanol per mass unit, the  constant  kinetic and the activation temperature, respectively The system is fed with the inlet total mass flow rate q in containing a mixture of 2-3-epoxy-1propanol, water and sulfuric acid where the mass fraction of the sulfuric acid is much smaller and negligible compared to 14 A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity the others The reaction system is • Fi , Fi , hi operated with the mass maintained to be flow rate, the inlet molar enthalpy equals the outlet total mass flow rate, and the molar enthalpy, respectively; q  q ) in • ci and ci in the modeling purposes, and hi are the inlet molar flow rate, the outlet molar constant (i.e., the inlet total mass flow rate For in in the are the concentrations per mass unit of species i in the inlet following hypotheses are made and (A1) The fluid mixture is assumed to be the reaction system, respectively; ideal, incompressible and under isobaric • M is the total mass; conditions • Q is an extra term accounting for (A2) The heat flow rate coming from the possible mechanical dissipation and • jacket Q J is modelled by the following mixing effects expression: Note also that in an ideal mixture, the • (20) Q J   (TJ  T ) where  is the heat exchange coefficient and TJ is the jacket temperature (A3) The specific heat capacities c *p ,i are assumed to be constant enthalpy of species i can be expressed as follows (Sandler 1999): hi (T )  c *p ,i (T  Tref )  hi ,ref       T   N *    si ,ref  R ln  n i si (T )  c p ,i ln    Tref     N j   j 1       (22) Mathematical Model of the System and Steady States Analysis energy where Tref , hi ,ref and si ,ref are the reference balances, the reaction system dynamics is values calculated at standard conditions given as follows (Vleeschhouwer et al Hence the energy balance given in (18) 1988, Hoang et al 2013a): can Using the material and be rewritten temperature  dN1 in in in  dt  q c1  qc1  rm M  F1  F1  rm M   dN  q in c in  qc  r M  F in  F  r M 2 m 2 m  dt   dN   qc3  rm M   F3  rm M  dt  dH •    Fi in hiin   Fi hi  Q J  Q  dt i i where: T by in terms of the considering the hypothesis of local equilibrium (De Groot and Mazur 1962) and the expression of the total enthalpy and entropy as follows (21) (Sandler 1999): n H   N i hi (23) i 1 n S   N i si i 1 (24) Thuan C Nguyen and Ha N Hoang 15 Fig 3: The Van Heerden diagram of the CSTR We therefore obtain (Luyben 1990): Table Kinetic Parameters Symbol (unit)   dT     N i c*p ,i     Fi in c*p ,i  T in  T  i  dt  i   Numerical value -1  (25)    r H rm M  Q cH  (kg.mol ) 3108 k0 (kg.mol-1.s-1) 86 10 8822 Ta (K) where  r H   h1 (T )  h2 (T )  h3 (T )   is the enthalpy of (exothermic) reaction Tables 1, and are extracted from   Table The CSTR Operating Conditions Symbol (unit) Numerical value (K) 298 2013a) give thermodynamic, kinetic and TJ (K) 298 operating parameters of the reaction q  q (kg.s-1) 0.46x10-3 F1in (mol.s-1) 0.0013 F2in (mol.s-1) 0.0200 M (kg) 75x10-3  (W.K-1) 0.4 Q (W) 8.75 Tref (K) 298 T (Vleeschhouwer et al 1988, Hoang et al mixture Table Thermodynamic Parameters Symbol (unit) c * p ,i (J.mol-1.K-1) hi ,ref (J.mol ) C3H6O2 (1) H2O (2) C3H8O3 (3) 128.464 75.327 221.9 -2.95050x105 -2.8580x105 -6.6884x105 -1 si ,ref (J.K-1.mol-1) in in Under the operating conditions imposed, as shown in Figure (Hoang et 316.6 69.96 247.1 al 2013a, Nguyen et al 2016a, 2016b), the system exhibits three stationary operating 16 A thermodynamic Lyapunov Approach to the Stability Analysis of a Nonlinear Irreversible Process Having Multiplicity Table The Numerical Values of the Three Steady States (Hoang et al 2013a) Symbol (unit) T (K) N1 (mol) Point P1 314.35 0.1723 3.2181 0.0470 323.60 0.1364 3.1822 0.0829 346.47 0.0469 3.0927 0.1724 Point P2 Point P3   N (mol)  N (mol)  dR( Z , Z * )  0, t dt Time (s) Fig 4: The time evolution of the storage function R( Z , Z ) with Z  P1 * * points denoted by P1 , P2 and P3 Table Table Initial Conditions for Simulations gives the numerical values of these three stationary operating points, which are C1 C2 C3 330 350 325 0.05 0.04 0.12 3 analysis of these three steady states is T (t  0) (K) N1 (t  0) (mol) N (t  0) (mol) given on the basis N (t  0) (mol) 0.1880 0.0835 0.0817 calculated using MATLAB In the next part, the dynamical stability of the availability function by considering the sign condition (17) This condition is indeed checked through the numerical simulations All Figures and show the time evolution of the storage function these are the main contributions of the R( Z , Z ) paper conditions It is shown that the dynamical Simulation Results For the sake of illustration, three different initial conditions are considered Their values given in Table are chosen close enough to the steady states * stability for analysis the different consists in initial deriving * dR( Z , Z )  The simulations results of dt Figures (with the initial condition C1 ) and (with the initial condition C ) show that P1 and P3 are stable Otherwise, the Thuan C Nguyen and Ha N Hoang 17 dR( Z , Z * ) dt  0, t Time (s) * Fig 5: The time evolution of the storage function R( Z , Z ) with Z  P3 * dR( Z , Z * )  0, t   dt Time (s) Fig 6: The time evolution of the storage function R( Z , Z ) with Z  P2 * point P2 is unstable since R( Z , Z ) does * not asymptotically tend to zero in Figure (with the initial condition C3 ) Figure presents the open loop phase plane The results presented here are satisfactory from both a qualitative and a quantitative point of view CONCLUSION In this paper, the stability analysis of * the acid-catalyzed hydration of 2-3-epoxy1-propanol to glycerol 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