Simulation of Industrial Processes for Control Engineers by Philip J. Thomas • ISBN: 0750641614 • Publisher: Elsevier Science & Technology Books • Pub. Date: July 1999 Foreword by Prof. Dr Ing. Dr. h.c. mult. Paul M. Frank, Gerhard-Mercator-Universit~it, Duisburg, Germany Mathematical modelling and simulation are of funda- mental importance in automatic control. They form the backbone of the analytical design methodology for open-loop and closed-loop control systems. They rep- resent the first step that a control engineer has to take when he has the task of designing a control system for a given plant. Not only is the analytical model an essential part of the design method, it is also indispens- able in the analysis of the resulting control concept. On the one hand, it is needed for the analysis of stabil- ity and robustness of the control system, on the other hand it is used for the (nowadays exclusively digital) computer simulation of the plant in order to perform an online check of the resulting electronic controller within the closed-loop control systems. Besides this, mathematical modelling and simulation play an increasing role in computer-aided approaches for control systems design and optimization. Due to the present tremendous progress in computer tech- nology, analytical optimization techniques are being more and more replaced by systematic trial and error methods and evolutionary algorithms using digital sim- ulations of the processes. There is a clear trend at the moment towards such computer-assisted approaches. This implies that mathematieal modelling and simula- tion as a pre-condition will gain increasing importance. This is especially true for the field of automation and optimization in the chemical and process indus- tries, because here it is common for the plants and their models to be rather complex and non-linear, so that analytical design and optimization techniques fail or at least are extremely cumbersome. Maybe it is no exaggeration to anticipate that in the future the mathematical model will belong within the technical specification of any dynamic device used in a technical plant. The work of Professor Thomas is a highly important contribution to the attainment of these objectives in the field of process engineering. On the solid grounds of his long practical experience and expertise in the design of process control systems, he uses the systematic approach to modelling and simulation of dynamical systems in the process industries, rang- ing from the detailed understanding of the physical processes occurring on the plant to the codification of this understanding into a consistant and complete set of descriptive equations. With thoroughness and lucidity, the text explains how to simulate the dynamic behaviour of the major unit processes found in the chemical, oil, gas and power industries. Determined attempts have been made to derive the descriptive equations from the balance equations - the first princi- ples- in a clear, step by step, systematic manner, with every stage of the argument included. Thus, the book contributes to both the simulation of industrial plants by control engineers and a deep understand- ing of the quantitative relationships that govern the physical processes. Reflecting his exceptionally broad expertise in a wide variety of areas in applied con- trol theory, systems theory and engineering, Professor Thomas's treatment of modelling and simulation of industrial processes casts much light on the underlying theory and enables him to extend it in many important directions. The present volume is concerned, in the main, with the fundamental concepts of dynamic simulation- including thermodynamics and balance equations - and their application to the great variety of processes and their components in the process industries. This pro- vides indeed a good grounding for all those wishing to apply dynamic simulations for industrial process plant control. It serves for both undergraduate engineering students in electrical, mechanical and chemical engi- neering specializing in process control, starting from their second year, and for postgraduate control engi- neering students. However, it may also be considered as a very valuable reference book and practical help to control and chemical engineers already working in industry. The great variety of subsystems and technical devices occurring in plants of chemical and process industry are tackled in full detail and can be used directly to setup digital computer programms. There- fore, the book can be highly recommended to practical control engineers in this field. Professor Thomas's treatise is clearly a very impor- tant and comprehensive accomplishment. It deepens the understanding of the dynamic behaviour of techni- cal plants and their components and stimulates a more extensive application of modelling and simulation in the field of the process industries. XV Notation The wide range of subjects covered by the book causes occasional problems with duplication of symbols. Use has been made of generally recognized notation wher- ever possible, and normally the meaning of each sym- bol is clear enough in its context. However, a particular difficulty arises in any process engineering text from conflicting demands for the use of the letter v: both specific volume and velocity have strong claims. It has been decided in this book to use v to denote specific volume, and to assign to velocity the symbol, c, on the basis that c has an association with speed for most scientists and engineers, albeit the speed of light. SI units are assumed. Symbols Symbol d aq aj ~j A A Ai m ll Ftl AI AT {A} [A] a aopt A A,) b b Meaning Units stoichiometric coefficient stoichiometric coefficient of the ith component in the jth reaction jth nominally constant parameter value of jth nominally constant parameter expected in advance cross-sectional area m 2 constant used in Antoine equation for vapour pressure mass of chemical kmol constants used in Margules correlation for distillation throat area of nozzle; effective throat area of valve at a given valve opening m 2 effective throat area of valve at fully open m 2 6.023 x 1026 molecules of chemical A, = one kilogram-mole's worth concentration of chemical A kmol/m 3 vector of constant parameters vector of optimally chosen constant parameters n x n state matrix for a linear system matrix associated with distillation plate i constant stoichiometric coefficient bo Bu b.) B c C c Ca.j CB Cc Cij Cmax Cmin Cn Cp Cri Cro Cson Cv C(i) C C CI c; half of velocity deadband, half of backlash width adjustment coefficient used with pipeflow function constant used in Antoine equation for vapour pressure boiloff rate of component j from the liquid in plate i vector of boiloff rates on plate i n x I input matrix for a linear system signal produced by controller velocity stoichiometric coefficient gain of filter for white noise for parameter j average linear speed of turbine blade critical velocity - speed of sound at local conditions gain of transfer function, gij maximum value of controller output signal minimum value of controller output signal neutron speed specific heat at constant pressure velocity of incoming gas relative to turbine blade velocity of outgoing gas relative to turbine blade speed of sound in the fluid specific heat at constant volume vector associated with distillation plate i conductance constant used in Antoine equation for vapour pressure = C~/C,,, ratio of valve gas flow conductance to liquid flow conductance at a given valve opening = C*g/C,*,, ratio of gas sizing coefficient to liquid sizing coefficient, both at a given valve opening kmol/s m/s nv's m/s m/s J/(kg K) m/s n~s m/s J/(kg K) m 2 [(scf/US gall). (min/h)/ (psi)l/21 .~ XVll xviii Notation Cl, c;, Cd ~h = CJC~.,, ratio of valve gas flow conductance to liquid flow conductance for the valve as far as the throat only. Both conductances at a given valve opening = Cg/Cv,, ratio of gas sizing coefficient to liquid sizing coefficient, for the valve as far as the throat only, both at a given valve opening discharge coefficient valve friction coefficient for gas at high-pressure ratios CFcu valve friction coefficient for gas at high-pressure ratios at fully open C~ valve friction coefficient for liquid flow C~ = .vCc, gas flow conductance at valve opening, y C~ gas sizing coefficient at a given valve opening CG gas flow conductance for fully open valve C i concentration of precursor group i Ct value of Ct when the valve is fully open: Ct = Cc/Cv C~. line conductance C T total conductance of line plus valves and fittings C,, = yCv, liquid flow conduc- tance at valve opening, y C,*. liquid sizing coefficient at a given valve opening, equal to the valve capacity for water at 60*F C v liquid flow conductance for fully open valve C~, constant of proportionality for fully open valve, assuming that the differential pressure and specific volume are constant C~t ratio of measured velocity downstream of nozzle to the velocity that would have occurred if the expansion had been isentropic C,,, liquid flow conductance at a given valve opening for the valve as far as the throat only Cvr valve conductance to the valve throat at fully open [(scf/US gall). (min/h)/ (psi) I/2] m 2 scf/h/psi m 2 nuclei/ m 3 m 2 m 2 m 2 [US gall/ min/ (psi) j/2 ] m 2 kg/s m 2 m 2 c;, d dj D D D Di Dj Dr' dF dq dw e,n E E E~ emin f f A liquid sizing coefficient at a given valve opening, for the valve as far as the throat only constant defined by local text weighting fraction derivative term in controller output signal diameter constant used in Riedel equation for vapour pressure specific enthalpy drop across the ith stage of a turbine under isentropic conditions average partial heat of solution of component j valve size work done against friction in the small element by unit mass of the working fluid heat flux into the small element per unit mass flow = heat input per unit mass of the working fluid useful power abstracted from the small element per unit mass flow = useful work done by unit mass of the working fluid error, =difference between measured variable and setpoint error term after modification by limiting energy expression involved in estimating the pressure ratio across the valve that will lead to choked gas flow activation energy for reaction sum of the squared flow errors total vapour flow from distillation plate i to plate i + 1 vector of differences between model and plant measured transients vector of differences between model and plant measured transients with the optimal set of constant parameters Fanning friction factor function multiplying factor to account for the additional metal contained in the baffles, assumed to be at the same temperature as the heat exchanger shell [US gall/ min/ (psi)t/21 m J/kg J/kmol m J/kg J/kg J/kg J/kmol kg2/s 2 kmol/s fco,,,b combination function, combining f hpr and f lpr fe function derived from Fisher Universal Gas Sizing Equation f.aow generalized mass-flow function fhp~ high-pressure-ratio function ftp,, long-pipe approximation flow function flt,~ low-pressure-ratio function fLa liquid-gas function, used to approximate gas flow through a valve by analogy with the liquid flow case f,,o~ nozzle flow function fNV nozzle-valve function used to model gas flow through the valve by analogy with nozzle flow fNVA approximating function for fNV fpipe pipeflow function fPi function relating head to volume flow at design speed for a centrifugal pump fP2 function relating pump power demand to volume flow at design speed for a centrifugal pump fl,3 efficiency function, dependent on volume flow and speed for a centrifugal pump fsh,,~k shock correction factor for blade efficiency F frictional loss per unit mass of the working fluid along whole length of the pipe F force F mass flow in kilogram-mole units F fission rate of the reactor per m 3 of fuel FLi liquid feed flow to plate i f n-dimensional vector function of the state, x, and forcing variables, u g acceleration due to gravity g function gr neutron thermalization correction factor go (s) elemement of transfer function matrix, G(s) G mass velocity, = W/A G specific gravity with respect to water at 60~ G. specific gravity of gaswith respect to air at same temperature kg/s m W J/kg N kmol/s kmol/s m/s 2 kg/(m 2 s) Gj g G(s) h h H H Hp H, h i I IA ID J J J J, JB J~ k k k k k' k~ kd kd~ kp K K K Kb K~ K~ constant used in converting activity coefficient for component j to a different temperature range vector function dependent on the vector z transfer function matrix specific enthalpy sum of weighted squared deviations pump head Lagrange function polytropic head isentropic head vector function integral term in controller output signal adjusted value of integral term desaturated integral term general integer index moment of inertia Jacobian state matrix Jacobian matrix for parameter variations in companion model Jacobian input matrix Jacobian state matrix for companion model controller gain general constant, meaning dependent on local text forward velocity constant multiplication factor for the nuclear reactor backward velocity constant frequency factor for the reaction delayed neutron component of multiplication factor component of multiplication factor associated with delayed neutron group i prompt neutron component of multiplication factor = Aa (i)" j /(t~ja.~ + aja2) vapour pressure function energy loss in velocity heads heat transfer coefficient energy loss in velocity heads due to bend or fitting cavitation coefficient for a rotary valve at a given valve opening effective thermal conductivity of catalyst bed Notation xix K J/kg m J/kg J/kg kgm 2 Pa W/(m 2 K) W/m xx Notation Kc Kco. Kf Km KM KT KU Kvo I I L L,H Li mo moc roOD mb M MA Ms Mc Mt~" cavitation coefficient for a rotary valve at fully open energy loss in velocity heads due to contraction at the inlet energy loss in velocity heads due to pipe friction pressure recovery coefficient for liquid flow through valve at a given opening pressure recovery coefficient for liquid flow through valve at fully open total energy loss in velocity heads energy loss in velocity heads due to valve at a given opening energy loss in velocity heads at the fully open valve when the valve size matches the pipe diameter dimension of vector of forcing functions level average neutron lifetime length of component or pipe effective pipelength total liquid flow from distillation plate i to plate i-1 mass polytropic exponent for frictionally resisted adiabatic expansion polytropic exponent for frictionally resisted adiabatic expansion over the convergent part of the nozzle polytropic exponent for frictionally resisted adiabatic expansion over the convergent part of the nozzle when the flow is critical polytropic exponent for frictionally resisted adiabatic expansion over the convergent part of the nozzle for the design flow coefficient used in calculating pipe-flow coefficient, b0, at different pressure ratios mass in kilogram-moles mass of chemical A mass of chemical B mass of chemical C total liquid mass in distillation plate i m S m m kmol/s kg kmol kmol kmol kmoi kmol M Lij MR MH Mvij M n n n n nd ndi nG np N N NAK NI NF NRE p pep pij p, Ptca t, Ptwp Pl~p P P PD mass of component j in the liquid phase in distillation plate i kilogram-moles of reaction total vapour mass in distillation plate i mass of component j in the vapour phase in distillation plate i Mach number = ratio of velocity to the local sound velocity general index dimension of state vector polytropic index of gas expansion concentration of neutrons concentration of delayed neutrons concentration of delayed neutrons in group i number of neutrons in one of the M groups concentration of prompt neutrons number of cells rotational speed in revolutions per second number of molecules in a kilogram-mole, = 6.023 • 10 26 (Avogadro's number • 1000) concentration of fissile nuclei number of degrees of freedom for a gas Reynolds number pressure pressure at the critical point for the fluid (point of indefinite transition between liquid and vapour) partial pressure of component j in distillation plate i throat pressure for the nozzle or valve throat pressure at cavitation vapour pressure at the valve throat temperature vapour pressure power proportional term in controller output signal power demanded by the pump kmol kmol rxn kmol kmol neutrons/ m 3 neutrons/ m 3 neutrons/ m 3 neutrons/ m 3 neutrons/ m 3 r/s nuclei/ m 3 Pa Pa Pa Pa Pa Pa Pa W W PF PIMP P,. Pp Ps q Q Q Q, Q~ Qcrit Qs~ r re rj rlim p l)c rvap R R R= RB Ri R~ Rw s S S power expended against friction power expended by the impeller modified proportional term pumping power, i.e. useful power spent in raising the pressure of the fluid power supplied to the pump quality of steam heat volumetric flow rate volume flow in US gallons per min volume flow in cubic feet per hour critical or choked flow of gas through the valve equivalent volume flow in standard cubic feet per hour ratio of pressures at stations '1' and '2' critical pressure ratio for a gas reaction rate density, referred to the volume of the packed bed fraction of pressure ratio down to which an ordered expansion can occur ratio of the pressure at valve inlet to the pressure at the critical point for the fluid ratio of valve throat pressure at a given opening to the vapour pressure of the fluid at the valve-inlet temperature universal gas constant, value - 8314 remainder term, equal to the adjusted integral term less the integral term exponentially lagged version of the remainder term ratio of blade speed to incoming gas speed rate of radioactive decay of precursor group i valve rangeability, = ratio of maximum to minimum valve opening characteristic gas constant, =g/w specific entropy stiffness entropy W W W W J m3/s US gall/rain ft3/h standard ft3/h standard ft3/h kmol rxn/(m 3 s) J/(kmolK) nuclei/ (m 3 s) J/(kg K) J/(kg K) J/K Si total sidestream flow extracted from distillation plate i t time t j/2.i half-life of delayed neutron precursor group i T time constant T temperature Td derivative action time T i integral action time or reset time u specific internal energy U = QNo/N, ratio of flow to normalized speed Ui total internal energy of the contents of distillation plate i u /-dimensional vector of forcing variables specific volume volume volume at standard conditions (pressure = 14.7 psia, T = 520~ of an arbitrary mass of gas that has volume V at arbitrary conditions P s, Tt molecular weight polytropic specific work isentropic specific work mass flow critical flow for a gas cavitating flow for liquid through a rotary valve Wchoke choking flow of liquid through a valve (fractional) valve travel, fully shut = 0, fully open = 1 distance mole fraction of component j in the liquid phase in distillation plate i steam dryness fraction at the start of the expansion n-dimensional vector of system states n-dimensional vector of system states driven with variations in the nominally constant parameters y (fractional) valve opening, fully shut = 0, fully open = 1 Y0 mole fraction of component j in the vapour phase in distillation plate i expansion factor k-dimensional vector of model outputs t3 V Vscf W Wp Ws W w,. W cll t~ X XO Xo X Notation xxi kmol/s s s J/kg m3/s m3/kg m 3 standard cubic feet J/kg J/kg kg/s kg/s kg/s kg/s m xxii Notation Y k-dimensional vector of model outputs when the model is driven with variations in the nominally constant parameters z height relative to datum zuj mole fraction of component j in the liquid feed to distillation plate i Z compressibility factor, dependent on temperature and pressure. Z = l for an ideal gas z vector of unknowns defined by nonlinear, simultaneous equations g(z) 0 z vector of plant transient measurements at power to which concentration of chemical A is raised in forward reaction at' power to which concentration of chemical A is raised in backward reaction at a angle of turbine nozzle, measured relative to the direction of turbine wheel motion at~ composite term for net heat input to boiling vessel at2 angle of gas stream leaving turbine stage, measured relative to the direction of turbine wheel motion at2 composite term for total heat capacity of contents of boiling vessel at "~- combinations of variables for tl 9 att,,i~ distillation plate i, sometimes (n = 1 making particular reference to to 7) component j. atjd steady deviation from optimal value of nominally constant parameter j that causes the mean squared error to double atj~ steady deviation from optimal value of nominally constant parameter j = vector of variations to constants, a fl constant used in polynomial fl delayed neutron fraction fl power to which concentration of chemical B is raised in forward reaction fl' power to which concentration of chemical B is raised in backward reaction degrees W degrees J/K fll blade inlet angle f12 blade outlet angle fli delayed neutron fraction for group i ~in angle of approach to turbine blade of the incoming gas jet y ratio of the specific heats, Cp/C,., = index for isentropic expansion for a gas ~, power to which concentration of chemical C is raised in forward reaction ?,' power to which concentration of chemical C is raised in backward reaction ~,q activity of component j on distillation plate i 8 small increment of quantity following (Sa~) 2 contributory variance of parameter j 8C~ production of nuclei of delayed neutron precursor group i due to absorption of neutrons in a fission event 8MR increase in the kilogram-moles of reaction A denoting incremental quantity of variable following a t standard deviation required by parameter j acting on its own to match measured variable i Ac. change of speed in the direction of turbine wheel motion Ahx actual change in specific enthalpy through the nozzle Ah,v, change in specific enthalpy through the nozzle for an isentropic expansion A p differential pressure At integration time-step AHj enthalpy of reaction j Ally internal energy of reaction j e a measure of the average height of the excrescences on the pipe surface e reactor elongation factor e/D relative roughness of the pipe surface r damping factor in transfer function gij(s), associated with output i and parameter j r/ efficiency r/s blade efficiency degrees degrees degrees nuclei kmol rxn m/s J/kg J/kg Pa S J/kmol rxn J/kmol rxn m OBa rIBN tic rltv rip rls 0 oi Om Op o, o~ K k # # P P a2a,j 2 tr$,i z" T r stroke $ rp(T) $ blade efficiency when there is no entry loss nozzle efficiency for the expansion taking place in the moving blades of a reaction stage distillation column efficiency nozzle efficiency pump efficiency stage efficiency angle height of the liquid on tray i measured value of plant variable, Of, plant variable setpoint for plant variable, Op height of the weir on distillation tray i bulk modulus of elasticity of the fluid eigenvalue molar fraction dynamic viscosity constant used in polynomial mass fraction nuclear power density averaged over the core constant used in pressure ratio polynomial degree of reaction in a turbine stage reactivity effective cross-sectional area for fission of each fissile nucleus variance of the nominally constant parameter, j variance to be associated with predicted variable, 0 variance of the companion model output, i time constant frictional shear stress standard deviation expected in advance for parameter j valve stroking time heat flux per unit length 'phi', = for(cp/T)dT, the temperature-dependent component of specific entropy heat flux white noise intensity state difference vector: X- x rad m m Pa -I s Pas W/m 3 dollars, niles m 2 s N/m 2 S W/m J/(kg K) W Notation xxiii X~ ve X O) O)a. j reactor flux averaged over the complete core vector of state deviations asso- ciated with state subvector x ") useful power extracted per unit length vector of outputs of companion model rotational speed in radians per second break frequency defining frequency content of the variation of parameter j undamped natural frequency of tranfer function gij relating the variance of output i to nomi- nally constant parameter, j Additional subscripts and superscripts 0 at time zero 0 at datum position 0 over convergent part of nozzle 0 at design conditions 0 model matching 0c over convergent part of nozzle in critical conditions 0D over convergent part of nozzle in design conditions 0,1,2, enumerative identifiers 1 at upstream station or inlet 2 at downstream station or outlet a station identifier a air at atmospheric ave average b due to bends and fittings b station identifier B 'boiloff' or evaporation; condensation when flow is negative B blade c critical or choked c of catalyst bed pellets c of the controller cc from fuel-pin cladding to coolant clad of the fuel-pin cladding con contraction cool of the coolant crit 9 critical cs critical and isentropic C relating to the distillate side of the distillation column condenser d demanded neutrons/ (m 2 s) W/m rad/s rad/s rad/s xxiv Notation d downcomer " d normally downstream di downcomer inlet do downcomer outlet D at design conditions e evaporator eft effective f friction f feed f of the fuel fc between fuel and cladding fuel fuel F friction g gas G denotes fully open valve, for gas flow G gas i general index i for isothermal expansion in inlet j general index k general index liq liquid L liquid L over the whole length m metalwork nuc nuclear N nozzle opt optimal out outlet over overall p at constant pressure p for a polytropic expansion P needed in practice P pump pins of the fuel pins prior prior r riser r due to the reaction recalc recalculated rev reversibly ri relative at the inlet ro relative at the outlet s setpoint s shell-side s under conditions of constant entropy sa under conditions of constant entropy from mid-stage to stage outlet si shell inside stroke associated with the stroke of the valve sw shell wall sws shell wall to shell-side fluid s ys system S stage t throat t total t tube-side ti tube. inside to tube, outside trans transferred tw tube wall twt from tube wall to tube-side fluid T total T in the stagnation state T theoretical tc throat, critical tot total tray associated with the distillation tray up normally upstream v associated with a valve; associated with liquid flow through valve v at constant volume yap vapour V vapour V denotes fully open valve, for liquid flow vc vena contracta w wall w water w in the direction of turbine wheel motion z due to height difference ^ specified per kilogram-mole 9 in US units [...]... response of liquid level to a range of forcing functions imposed on inlet valve demanded travels and on the setpoint for liquid level We would then adjust the gain of the level controller to give good control over the range of liquid levels expected in plant operation We shall now use the mathematical model just derived to illustrate some general features of dynamic simulation 2.3 The general form of the simulation. .. Conditions for emergence from saturation for P + I controllers with integral desaturation, Pages 374-377 Index, Pages 379-390 1 Introduction Much of control engineering literature has concentrated on the problem of controlling a plant when a mathematical model of that plant is at hand, at which time a large number of effective techniques become available to help design the control system Unfortunately for. .. dimension or order of the plant as we modelled it would go up from 4 to 5 If we had needed to make an allowance for the temperature of the metal in the tank, 10 Simulation of Industrial Processes for Control Engineers then the additional state variable would have pushed the order up to 6 Of course, the plant itself would not have changed, merely our perception of how it worked The question of when the model... to the conservation laws for mass and momentum, and we can expect every dynamic simulation of an industrial process to need to invoke one or more of these laws The interpretation of these laws as they apply to different types of processes leads to different forms for the describing equations This chapter will begin by reviewing the thermodynamic relations needed for process simulation, and it will go... - - t 0 , u is an ldimensional vector of forcing variables, f is a 8 Simulation of Industrial Processes for Control Engineers vector function that depends on the states, x, on the forcing variables, u, and (sometimes) directly on time itself, t (The direct dependence on time can allow for the change in parameters over time in a known manner, such as the ageing of catalyst in a catalyst bed It would... It aims also to be of practical help to control and chemical engineers already working in industry The level is suitable for control engineering simulations for industrial process plant and simulations aimed at evaluating different plant operational strategies, as well as the programming of real-time plant analysers and operatortraining simulators 2 Fundamental concepts of dynamic simulation 2.1 Introduction... Jsfi (2.47) dt where: is an n x 1 vector of state deviations from an operating point defined by the state vector, x, and the input vector, u, fi is an l x I vector of input deviations from the input vector, u, J is the n x n Jacobian matrix, defined as: Ofl Oxl Of Ox Of 2 Of 2 Oxl Ox2 Ox Of, Oxl J~ Ofl Ox2 Of n Ox2 c'~2 c~,, -2Cvs~ W3 = Cv3Y3 9 (2.48) Ofl 0/gl 0U2 Of 2 Ou= 0f2 Ou2 igUl 0 0 0 1 r2 J 0 0... We find the eigenvalues by setting the determinant to zero: , Of, , Ou2 0 0 Of 2 Of n Out (2.2) u l"1 Oft " Out 9 to the form: Ox Ofl Of 1 Of 2 , while the n x ! input-Jacobian matrix, Jn, is defined as: ~-~ - xx~~ CAp 1 Jn-" 1 -~Cvs (2.51) This may be simplified using equation (2.2), repeated below: Ofl Ox,, ,, At3 x xsvr~ Of, , 9 kv ~'3 0 Of, , Out ~ We evaluate the Jacobian matrices at a particular... nonlinear simultaneous equations, the principles of which will now be explained We may describe a system of n nonlinear, 16 S i m u l a t i o n of Industrial Processes for Control Engineers simultaneous equations in the n unknowns of the vector z by the vector equation: g(z) = 0 (2.71 ) Applying a truncated Taylor's formula in the vicinity of the jth estimate of the roots, z (j), gives g(ztJ+l)) = g(ztJ))... after the mathematical model has been derived The major task facing the control engineer working in the process industries is the detailed understanding of the physical processes occurring on the plant and the 2 Simulation of Industrial Processes for Control Engineers codification of this understanding into a consistent and complete set of descriptive equations This is the background against which the book . of the physical processes occurring on the plant and the 2 Simulation of Industrial Processes for Control Engineers codification of this understanding into a consistent and complete set of. needed for the analysis of stabil- ity and robustness of the control system, on the other hand it is used for the (nowadays exclusively digital) computer simulation of the plant in order to perform. Simulation of Industrial Processes for Control Engineers by Philip J. Thomas • ISBN: 0750641614 • Publisher: Elsevier Science & Technology Books • Pub. Date: July 1999 Foreword