Mixing of two different temperature water flows as feedback controlled system mathematically modeled and simulated in MS Excel spreadsheet

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1 Mixing of wo differen emperare waer flows as feedback conrolled sysem mahemaically modeled and simlaed in MS Excel spreadshee S. Hbalovsky Absrac One of he mos imporan mehods in crren scienific and echnological research is process of modeling and simlaion of real experimen as well as modeling and simlaion of of feedback reglaed sysems. Sysem approach, modeling and simlaion are discipline wih is own heory and research mehodology. The paper focses o he heory of he process of modeling and simlaion, visalizaion of feedback conrolled sysem. Mlidisciplinary approach is poin o oo. Sep by sep here will be shown he process of creaion of saic, dynamic and feedback conrolled mahemaical model. Mahemaical model is spplemened by he simlaion model realized in MS Excel spreadshee. Visalizaion of he simlaion model is realized in MS Excel XY char. Keywords Modeling, comper simlaion, mahemaic model, simlaion model, saic characerisic, dynamic characerisic, feedback conrolled sysem. T I. INTRODUCTION HE erms sysem, model, simlaion, mlidisciplinary approach are imporan in crren approach o scienific, echnological and professional pracice. Many niversiies are realizing ha modeling and simlaion is becoming an imporan ool in solving and ndersanding nmeros and diverse problems. Modeling and simlaion, is becoming one of he academic programs of choice for sdens in all disciplines see e.g. in [1], [], [3], [4]. Modeling and simlaion is a discipline wih is own body of knowledge, heory, and research mehodology. To engage modeling and simlaion he mahemaical model of he real sysem has o be firs creaed. Models are approximaions for he real sysem. The model is hen followed by simlaion, which allows for he repeaed observaion of he model. Afer he simlaions model is verified, a hird sep akes place and ha is visalizaion of he model and he real sysem. The abiliy o define sysem, o bild p mahemaical model and o creae simlaion model develops logical hinking skills and imaginaion and is an inseparable par of a sden s sdy skills for hose sdying he specializaions Applied Informaics. In his paper we firs briefly inrodce o heory of modeling and simlaion as a mehod of mlidisciplinary invesigaion of feedback conrolled sysem. Similar inrodcion can be fond in [5], [6], [7]. Secondly we inrodce a case sdy illsraing sep by sep process modeling and simlaions of a feedback reglaed sysem sysem of mixing of cold and ho waer in flow-hrow waer ank. Firs he mahemaical model of he saic, dynamic and feedback reglaed sysem will be inrodced. Finally he comper simlaion model via MS Excel spreadshee will be shown. A. Modeling II. MODELING AND SIMULATION AS MULTIDISCIPLINARY EDUCATIONAL TOOL Modeling is a mehod ha is ofen sed in professional and scienific pracice in many fields of hman aciviy. The main goal of modeling is describe he conen, srcre and behavior of he real sysem represening a par of he realiy. The models are always only approaching of he realiy, becase he real sysems are sally more complex han he models are. The sysem homomorphism is applied in he process of modeling, which means ha each elemen and ineracion beween he elemens of he model corresponds o one elemen and ineracion of he modeled real sysem, b he reverse is no re. The model is always o be ndersood as simplificaion of he original. If he relaion of isomorphism is beween he model and real sysem he original model we cold no disingish beween he model and he original, which is discssed e.g. in [8]. The firs sep in he process of comper simlaion is creaion of mahemaical model of he sdied real sysem. The model can be obained eiher heoreically based on basic physical properies of he sysem, or nmerically by means of he measred vales. Deerminaion of parameers of heoreical model developed from empirical daa is called sysem idenificaion. The mahemaical model ms adeqaely describe he dependency sysem ops on is inps. Models of physical sysems are sally esablished as a sysem of mahemaical eqaions as will be shown in he following paragraphs of his paper. Isse 1, Volme 6, 1 46

2 B. Simlaion The process of modeling is closely relaed o he simlaion. Simlaion can be ndersood as process of execing he model. Simlaion enables represenaion of he modeled real sysem and is behavior in real ime by means of comper. The simlaion enables also visalizaion and ediing of he model. A ypical simlaion model can be wrien boh hrogh specialized programming langages ha were designed specifically for he reqiremens of simlaions, or he simlaion model can be creaed in sandard programming langages and spreadshees (MS Excel). From he above consideraions, i is clear ha simlaion is a process ha rns on he comper. In some pblicaions, herefore, can be fond he erm comper simlaion. I generally is valid ha comper simlaion is a comperimplemened mehod sed for exploring, esing and analysis of properies of mahemaical models ha describe he behavior of he real sysems which canno be solved sing sandard analyical ools, se e.g. [9]. The simlaion models represened by execable comper program have o be isomorphic wih he mahemaical model ha is a represenaion. I means ha he mahemaical model and simlaion model have o represen he real sysem, is elemens, inernal ineracions and exernal ineracion wih he environmen in he same way. In or paper he real sysem is simlaed in MS Excel spreadshee and visalized in MS Excel XY dependency chars. C. Simlaion of feedback reglaed sysem Many real sysems are consrced so ha hey are able conrol hemselves by means of he feedback. Feedback is a mechanism or process ha is looped back o conrol a sysem wihin iself. Sch a loop is called a feedback loop. In he sysem conaining he feedback loop he inp characerisics or signals are inflenced by op characerisics or signals. There are wo ypes of he feedback: - posiive feedback, where increase of vale of ops increases vale of inps and conversely; - negaive feedback, where increase of vale of ops decreases vale of inps and conversely. Feedback of he reglaed sysem in engineering pracice akes place hrogh he conroller. The conroller is a device for inflencing he reglaed sysem, aomaed conrol and for achievemen and mainaining is desired sae. Typically is sed in negaive feedback of he sysem. The conroller's inp is sally no direcly moniored op vale of he enire sysem, b only deviaion from he desired vale. The conroller conrols he sysem o eiher he complee eliminaion of deviaion, or o keep deviaion wihin he prescribed limis. The conroller reads he saes of he sysem, eiher direcly or, if nobainable, is reconsrcing he saes from he model. The reading of he vales of sysem is in ime based on eiher coninos. Conrolling of he sysem can be eiher analog, digial or sep. The comper simlaion is essenial mehod no only for represenaion of he mahemaical model b can also be sed for invesigaion of he feedback conrolled sysems. Simlaion allows: - realizaion of siaion ha canno be invesigaed sing convenional real experimenal devices; - seing he parameer of he feedback and is opimizaion as before he real conrol sysem is made p; - visalizaion of he behavior of he feedback conrolled sysem. In or paper he real feedback conrolled sysem is simlaed in MS Excel spreadshee and visalized in MS Excel XY dependency chars. D. Significan fncion of he simlaion Simlaion has from he scienific poin of view several fncions see e.g. [9]. We will focs in his paper wo of hem and hey are: - replacing he real experimen; - developmen of edcaional process. 1) Replacemen of he real experimen This is an imporan and indispensable feare of simlaions and simlaion model becase i allows realize a siaion ha canno be invesigaed sing convenional real experimenal devices. The main advanage of simlaions is ha simlaions model allows changing of inp parameers, visalizaion and opimizaion he effecs of he real experimen. The simlaion is sally safey and cheaper. ) Developmen edcaional process The simlaion is very sefl from edcaional poin of view. Using he simlaion model and visalizaion of simlaion resls on he screen, sdens can beer ndersand he basic processes and sysems and develop heir iniion. I is also essenial ha he eaching by means of simlaion is mch cheaper and faser han he eaching carried by real experimen. In some cases providing he real experimen canno be feasible. Despie he fac ha experimenal edcaion in he laboraory canno be compleely replaced (becase sdens acqire manal dexeriy, hey learn o work wih real laboraory insrmens, hey learn o plan, implemen and evalae realisic experimen), he simlaions is a par and basic mehods of scienific knowledge. Sdens can easily learn heoreical fondaions of he laboraory asks. The simlaion model of he real laboraory ask can help hem o check some of he operaions performed in he laboraory. This can redce he direc lessons in he laboraory only o necessary ime for heir own experimenal measremens. Alernaively, lessons can be realized only by he simlaion models. In his case, i is imporan o noe ha sdens are deprived of conac wih he real device, so ha hey will no ge a fll picre of he implemenaion of he experimenal measremens. Isse 1, Volme 6, 1 47

3 The case sdy of his paper is ypical example of he simlaion of he real feedback conrolled sysem siable for edcaional prpose. The presened simlaion applicaion is creaed in MS Excel spreadshee and visalized in MS Excel XY dependency chars. E. Model verificaion and validaion Verificaion and validaion are imporan aspecs of he process modeling and simlaion. They are essenial prereqisies o he credible and reliable se of a model and is resls [1]. 1) Verificaion In modeling and simlaion, verificaion is ypically defined as he process of deermining if execable simlaion model is consisen wih is specificaion e.g. mahemaical model. Verificaion is also concerned wih wheher he model as designed will saisfy he reqiremens of he inended applicaion. Verificaion is concerned wih ransformaional accracy, i.e., i akes ino accon simplifying assmpions execable simlaion model. Typical qesions o be answered dring verificaion are: - Does he program code of he execable simlaion model correcly implemen he mahemaical model? - Does he simlaion model saisfy he inended ses of he model? - Does he execable model prodce resls when i is needed and in he reqired forma? ) Validaion In modeling and simlaion, validaion is he process of deermining he degree o which he model is an accrae represenaion of he real sysem. Validaion is concerned wih represenaional accracy, i.e., ha of represening he real sysem in he mahemaical model and he resls prodced by he execable simlaion model. The process of validaion assesses he accracy of he models. The accracy needed shold be considered wih respec o is inended ses, and differing degrees of reqired accracy may be refleced in he mehods sed for validaion. Typical qesions o be answered dring validaion are: - Is he mahemaical model a correc represenaion of he real sysem? - How close are he resls prodced by he simlaion execable model o he behavior of he real sysem? - Under wha range of inps are he model s resls credible and sefl? Validaion and verificaion are boh limaely aciviies ha compare one hing o anoher. Validaion compares real sysem and mahemaical model. Verificaion compares mahemaical model and execable simlaion model. Someimes validaion and verificaion are done simlaneosly in one process. Validaion of he mahemaical model as well as verificaion of he simlaion model of or real sysem seven sorey recificaion colmn are o be done simlaneosly by he comparison of he dependencies of he qaniies heoreically calclaed from he simlaion model wih he dependencies of qaniies experimenally measred on recificaion colmn. The whole process of ransformaion from a real sysem, he simlaion model and is visalizaion is shown in Fig. 1. Fig. 1 Process modeling and simlaion Here again le s smmarize ha he mahemaical model ha reflecs he real sysem has some limiaions and simplifying assmpions (he real sysem and mahemaical model are in homomorphic relaion). In conras, he simlaion model is only he comper expression of he mahemaical model (he mahemaical model and simlaion model are in isomorphic relaionship). F. Mlidisciplinary approach Anoher imporan benefi associaed wih he process of modeling and simlaion of real experimens is a mlidisciplinary approach, wiho which he process of idenifying he real sysem sing mahemaical and simlaion model and canno be realized. This is also emphasized in his paper. Mlidisciplinary approach generally means ha specialized disciplines are applied in a sdy of real sysem. These disciplines provide parial analysis of he real sysem. These mono-disciplinary analyses are inegraed o overall solion by inegraing he solver who has basic mli-disciplines knowledge. In or case sdy for disciplines are inegraed, namely, experimenal physics, hermodynamics, mahemaics and comper science. III. CASE STUDY MODELING AND SIMULATION OF REAL SYSTEM A. Problem definiion As i was already menioned in he previos secion, i is essenial o emphasize modeling and simlaion as one of he imporan edcaional ool, which helps sdens develop heir logical hinking, iniion and can sppor ndersanding of he real process and experimen. In he following pars of he paper case sdy sed in edcaion of sbjec Aomaion and Mecharonics in Universiy of Hradec Kralove will be described. Wihin he case sdy he process modeling and simlaion of feedback reglaed sysem will be shown. Similar examples can be Isse 1, Volme 6, 1 48

4 fond e.g. in [1], [4], [5], [6], [11]. B. Descripion of he real experimenal sysem Real experimenal sysem ha is simlaed in his paper is schemaically shown on Fig.. Fig. 3 XY graph of he saic characerisic Fig. Real experimenal sysem The sysem consiss of he waer conainer wih volme V, wo waer inflows he firs of cold waer wih emperare T 1 and flow rae Q 1 and he second of warm waer wih emperare T and flow rae Q. Overflow from he ank discharges waer a he emperare T and flow rae Q. The waer is perfecly mixed. The sysem enables changes he emperares T 1 and T as well as he flow raes Q 1 and Q. This simple sysem enables represen sep by sep process of he creaion of a mahemaical model and simlaion model for visalizaion of: - saic characerisics; - ransien characerisics; - process of feedback. The visalizaion of he characerisic is via he ables of he appropriae vale characerisic dependencies and via he XY dependency chars. Deailed descripions of he dependencies are a scope of he following pars. IV. MATHEMATICAL MODEL In his secion he ransformaion of above described real sysem (seven-sorey recificaion colmn) will be described. From mlidisciplinary poin of view he disciplines like physics and mahemaics will be involved in. The derivaion of he common mahemaical model is e.g. based on [1]. A. Saic model To measre he saic properies of he sysems i is necessary o realize he nmber of measremens so ha he given vale of he inp variable (inp signal) is applied o he sysem inp and afer he sysem sabilizaion, i.e., afer erminaion he ransien phenomenon, we measre he vale of he op variables y (op signal). Measremen is repeaed for differen vales of he inp signal. This provides he se of XY poins - he dependency of he op variables on inp variables y(). By charing his dependency in XY graph he saic characerisics of he sysem can be reached - see Fig. 3. The ransiion fncion which describes he saic properies of he sysem can be obained by approximaion of he measred daa via siable mahemaical mehod. Usally he ransfer fncion is approximaed by polynomial fncion of he given order. This fncion is ndersood as saic mahemaical model in he analyical form. This mahemaical model allow for any vale of he inp signal (from measred range) calclae he corresponding op vale. Saic properies of or discssed sysem can be described by following eqaions: Q Q 1 Q, (1) TQ T Q, () T Q 1 1 where he inp variables are T 1, T, Q 1 and Q, and op variables T and Q for he descripion see par. III. A very imporan parameer, which describes he saic behavior of he reglaed sysem, is acion gain Z of he sysem. I is defined as he raio of op vale y o he vale of he inp variable in seady sae. If here is a reglaed sysem, a he poin P (see Figre ), he acion gain of he sysem he poin P can be approximaely calclaed as he raio of change of he op signal - y, and he change of he inp signal - : y Z. (3) If he dependence y() is described analyically, he gain of he sysem can be calclaed as he vale of he firs derivaive of his fncion a given operaing poin P. dy Z. (4) d If we know he vale of Z for given sysem and reqired vale of op signal, he vale of inp signal, which has o be applied o inp of he sysem, can be calclaed o achieve he reqired op vales. In or discssed sysem he gain of op variable T can be Isse 1, Volme 6, 1 49

5 sdied depending on one of he op variables T 1, T, Q 1 and Q. B. Dynamic model ransiion characerisic The measremens of dynamic characerisics of he sysem mean he invesigaion of he ransiion sae of he sysem. The inp signal as an appropriae fncion of ime is applied o he sysem inp and he response of he sysem is measred, i.e. ime dependency of he op signal is measred. The ime dependence of he inp signal is sally seleced in he form of a sep fncion when he inp signal is change from he iniial vale () o he vale S (). This sep vale change can be expressed by following formlas: ( ) ( ). (5) ( ) ( ) S Time dependence of he op signal y() = f(()) afer he sep change of he inp signal () shown on he Fig. 4 is called as ransiion characerisic. The op signal is sabilizes o he vale of y S (). The solion of his differenial eqaion is: or T T e T T, (8) T where T e T 1 e, (9) V, (1) Q 1 Q is ime consan of he waer ank. T Q T Q 1 1 T,, (11) Q Q 1 is seady sae emperare of waer a he beginning of measremen (before sep change of he inp vales T 1, T, Q 1 and Q ) and T Q T Q 1 1 T,, (11) Q Q 1 is seady sae emperare of waer afer he measremen of he ransien characerisic (for he acal inp vales T 1, T, Q 1 and Q ). The differenial eqaion (9) can be convered o nmerical recrren eqiaion: Fig. 4 XY graph of he ransiion characerisic The dynamic characerisics of sysems are mahemaically described by differenial eqaions, sally linear differenial eqaions of n h order wih consan coefficiens and consan iniial condiions. A dynamic model of he sysem can herefore be represened eiher graphically by he ransien characerisic or analyically by he differenial eqaion. The resl of he differenial eqaion is he ime dependence of he op vale o he sep change of he inp vale. The dynamic properies of or discssed sysem can be described by he following eqaion: d VT TQ T1Q1 TQ. (6) d For a consan vale of he ank volme V he eqaion can be rewrien: T d TQ V T1Q1 TQ. (7) d T s s 1, (1) k T 1 e Tk e where T(k) is seady sae emperare of waer in previos inerval of measremen, T(k + 1) seady sae emperare in he following ime inerval and S is sampling ime inerval. The raio of vales of y S and s eqals o he gain of he sysem, which is idenical wih he hickness calclaed from he saic characerisics of he sysem. The raio Z y S ( ) (13) S eqals he acion gain of saic characerisic given by eqaion (3). C. Mahemaical model of he feedback conrolled sysem The reglaor or conroller is inegraed par of conrol circi, which incldes a converer, a smmaion elemen, a cenral conrol elemen ha conrols he aciviy of he conrol circi and power amplifier. Hereafer, nder he erm conroller will be ndersood he cenral elemen. Isse 1, Volme 6, 1 5

6 Conrollers can be divided based on energy income o eiher direc, which ake power spply direcly from conrol circi or indirec, which has he exernal power spply. According o he characer of he media, which carry he conrol signal he conrollers can be divided ino mechanic, pnemaic, hydralic or elecric. According he characer of he conrol signal we disingish analog conrollers and digial conrollers. Analog conrollers can be from he ransmission dynamics poin of view divided o proporional, inegral and derivaive. In he conrol echnology are widely sed are proporional inegral derivaive conrollers (PID conrollers). In he following par he mahemaical descripion of he conrollers dynamics will be discssed. Le s assme ideal conrollers working wiho any delay. The mos imporan characerisic sed for mahemaical descripion of he conrol sysems are: 1. Error difference beween reqired vale (se-poin) y R () and crren measred op vale y(): e( ) y ( ) y( ) (14) R. Conrolled inp acion signal (). I is also imporan o appreciae ha he conrol acion coneracs he change in he conrolled variable y, i.e. ha y is fncion of y(). 1) Proporional (P) conroller: Response of he P conroller is described by following eqiaion: ( ) r e( ), (15) where consan r is proporional gain of P conroller. Regarding he fac, ha he acion signal is mliple of he error difference e, i is clear ha he proporional conroller is always working wih he permanen conrol deviaion. ) Inegral (I) conroller: Response of he I conroller is described by following eqiaion: r e d 1, (16) where consan r 1 is inegral gain of I conroller. I conroller conrols wih zero conrol deviaion. 3) Derivaive (D) conroller: Response of he D conroller is described by following eqiaion: de r, (17) d 1 where consan r 1 is derivaive gain of D conroller. D conroller conrols wih zero conrol deviaion. Iself D conroller canno be se for conrol. Usally combinaions of PD, respecively PID are sed. Proporional and inegral erms affecs he qaliy of reglaion in seadysae, derivaive erm affecs process of he conrol especially in he ransiion sae. 4) PID analog conroller: Response of he PID conroller is described by following eqiaion: or r e r eτ 1 de dτ r, (18) 1 d r e eτ 1 de dτ K d K d i, (19) where K i is inegral consan and K d is derivaive consan. PID conroller is sed o conrol complex dynamic sysems. 5) PID digial conroller: Despie he fac he reglaed sysems are analog sysem, heir conrol can be realized by digial conroller sally implemened by digial comper. Digial conroller a cerain ime inervals he sampling inerval, scans he vales of conrolled variables y and ransmis he corresponding vales of conrol variable. Mahemaical descripion of he response of digial PID conroller can be derived from (19), where inegraion erm is approximaed by nmeric smmaion and derivaive erm is approximaed by he firs nmeric differences. The PID digial conroller can be in k h conrol sep described by following eqiaion: k ek k 1 e j e j 1 r K, () K i j ek ek 1 K d K where K is sampling inerval. The eqiaion () can be rewrien o he incremenal form sed in comper simlaion: k q ek q ek 1 1 q ek k 1, (1) where (k) vale of conrol acion signal in k h sep, and e(k) = y R (k) y(k) is error difference beween desired vale and reqired vale in k h sep. The parameers q, q 1, q has he following form: Isse 1, Volme 6, 1 51

7 K K d q r 1, () K K i K K d q r 1, (3) 1 K K i q r K d. (4) K Above menioned general eqiaions of PID conrol sysem can be applied o or discssed real sysem - of mixing of cold and ho waer in flow-hrow waer ank. In his case he conrol acion signal is given by one of he inp vales T 1, T, Q 1 or Q ; reqired vale y R is given by reqired emperare T R and crren measred vale y is given by crren emperare T. In he case he conrolled acion signal is given by Q he eqiaion (1) has a form: k q ek q ek q ek Q k 1 Q, (5) 1 1 where e(k) is given by: e( k) T ( k) T( k) (6) R and q, q 1, q are calclaed from formlas (), (3), (4). Similar eqaions are valid if conrolled acion signal is given by Q 1, T 1 or T. 6) Properies of he closed conrol reglaed sysem For opimal realizaion of he reglaion of he feedback conrolled sysem i is necessary o explore no only he saic and dynamic properies of he reglaed sysem b also he behavior of a closed conrol circi. The ineres is aken in: - sabiliy of he conrolled feedback sysem; - response of he conrolled inp signal and measred op signal y dring he ransiion from he iniial o he new operaing poin (dring he sep change of he reqired vale y R ); - eliminaing he inflence of random disrbances ha affec he feedback. There are a nmber of crieria for assessing he qaliy of he conrol process. In pracice, however, a very simple assessmen of conrolled reglaion process of sep ransiion is sed, see Fig. 5. Qaliy of he conrol reglaed process is characerized by: - over-deviaion of he measred op signal y; - deviaion period T m ; - period of he reglaed conrol sep T %. All hese hree vales are minimized in opimal conrol reglaed process. In he synhesis of reglaory process i is necessary o: - selec he appropriae conroller (P, D, I, PID); - opimally he parameers of he conrolled sysem. Fig. 5 Qaliy qanyies of he reglaion process V. COMPUTER SIMULATION MODEL In his secion he above menioned mahemaic models (1. saic;. ransiion dynamic; 3. feedback reglaed) of he real sysem will be ransformed o he comper simlaion model. The comper model is o be realized in MS Excel spreadshee. Visalizaion of he saic and dynamic behavior of he characerisic as well as he behavior of he characerisic dring he reglaion conrol process is o be done by drawing he appropriae characerisic dependency in MS Excel XY-char. The basic ime ni of he simlaion models is aken as 1 mine. The op vales of he simlaion model are calclaed from inp vales based on nmerical calclaions direcly in MS Excel cells. Simlaion models are consrced so ha he given row of he MS able conains he inp vales and calclaed op vales corresponding o he given ime from he beginning of he simlaion. A. Saic simlaion model The inp variables of he saic simlaion model are he flow raes Q 1 and Q and he corresponding flow waer emperares T 1 and T. The inp vales are enered by heir iniial vales a he beginning of he simlaion and sep incremen of his vales dring one mine. The op variables are Q, T and Z. Q is calclaed according o eqaion (1), T is calclaed according o he eqaion (). As an example, he gain Z of emperare T depending on he Q 1 and Q is deermined. The able wih he inp and op vales is shown on he Fig. 6. The inp vales are enered in he rows 5 and 6. The calclaions of op vales needed for simlaion model are carried o in he cells down from row 1. In or pariclar case, we have limied he calclaion of he saic simlaion model in he ime inerval, min. Isse 1, Volme 6, 1 5

8 Fig. 9 show he gain Z of emperare T depending on flow raes Q 1 and Q. Fig. 6 Simlaion model of saic characerisics in MS Excel able The visalizaion of he simlaion model via he XY char - he ime dependence of he op vales T and inp vales T 1, T is shown on he Fig. 7, ime dependence Q, Q 1 and Q is shown on he Fig. 8. Fig. 9 Simlaion model of saic characerisic ime dependence of Z TQ1 and Z TQ B. Dynamic simlaion model ransiion characerisic The inp variables of he dynamic simlaion model are he flow raes Q 1 and Q, he corresponding flow waer emperares T 1 and T and sep change of his vales Q 1, Q, T 1 and T. The inp vales are enered by heir iniial vale a he beginning of he measremen of he ransiion characerisic and sep change of his vales. To calclae ime consan of he waer ank he volme V is oher vale needed o be enered. The op is emperare of he waer in he ank T calclaed according boh analyical model given by (9) and nmeric recrren calclaion model given by (1). Fig. 1 shows he MS Excel able o which cells he inp vales are enered (rows 5 and 6). Fig. 7 Simlaion model of saic characerisic ime dependence of flow raes Fig. 8 Simlaion model of saic characerisic ime dependence of flow raes Fig. 1 Simlaion model of dynamic ransiion characerisics in MS Excel able The calclaion of ransiion characerisic of emperare T is carried down from he row 1. In or pariclar case, we have limied he calclaion of he ransiion characerisic in he ime inerval, 3 min see Fig. 11. Isse 1, Volme 6, 1 53

9 Iniial vale of he acion signal (e.g. Q ) and oher inp vales (Q 1, T 1, T ) are enered ino he cells B6:E6. Vale of Q (or oher acion signal Q 1 or T 1 or T ) is calclaed in he range B15:E65 of he shee based on (4). The calclaion formla enered e.g. in cells D18 is as follows: =IF($A$1=;$J$6*K17+$K$6*K16+$L$6*K15+D17;$D$6). The condiion $A$1= checks if Radio Bon Q is checked. If yes, Q is acion signal and is vale is calclaed based on (5), oherwise he vale of he Q is consan and eqals vales of he cell D6 - see Fig. 1. Fig. 11 Simlaion model of ime dependency of ransiion characerisic T C. Simlaion model of he feedback conrolled sysem Simlaion model of he conrolled reglaed sysem is represened by ime dependencies of inp and op variables. The model is creaed in he MS Excel spreadshee and dependencies are visalized by means of MS Excel XY char. In or pariclar case, we have limied he simlaion of he feedback conrol in he ime inerval, 5 min. The main principles of he simlaion model will be described in he following ex. 1) Seing he acion signal The conrolled inp acion signal can be one of he inp qaniies Q 1, Q, T 1 or T. If e.g. conrolled inp acion signal is Q only is vale is changing dring he simlaion. Oher inps Q 1, T 1, T are se in he beginning and are dring simlaion of he conrol process consan. Seing he inp acion signal is given by radio bons see Fig. 1. Fig. 1 Simlaion model of PID conroller in MS Excel able ) Seing he reqired emperare TR Oher inp qaniy is reqired emperare T R of he mixed waer in he conainer. Vale of T R is changed sepwise a given ime of he conrol feedback process. Sep change of vale T R inflences he change of he vale of acion signal (e.g. Q ). In or pariclar case a he beginning of he simlaion in ime inerval 8 min we assme ha T R = T S, where T S in emperare of seady sae given by (11). Res ime inerval of he simlaion 95 min is spli o wo pars - 99 min and 35 min. Two sep changes of he vale of T R are arisen a he beginning of hose ime inervals. These sep changes inflence vale of he inp acion signal (e.g. Q ) by feedback mechanism and ime dependence of his signal is visalized on he XY char. Sep changes of he vale T R are in or presened simlaion model realized by specific way ha enables animaion of he simlaion char. The principle of his animaion is ha he vale of he sep change T R can sep by sep aler from iniial o final vale and each sep by sep change is visalized as a new ime dependence of inp and op variables in MS Excel XY simlaion char. The principle of an animaion is recording of a seqence of images which slighly differ. Principle of he animaion in MS Excel char is ha he char has o gradally change by changing of he char sorce daa, namely T R. Iniial seing of he vales based on he T R is sep by sep changed is in or pariclar simlaion model realized in he range of cells A8:E1 see Fig. 1. The gradally change of he T R is realized by ieraive recalclaion of he vale of appropriae cell in MS Excel. In or pariclar case he circlar reference formla =IF(E1>=D1;;E1+1) is enered ino he cell E1. This formla expresses ha he vale of he cell E1 afer one ieraive recalclaion sep increases by 1. Ieraive recalclaion sars by pressing of F9 key. By repeaed pressings, evenally by holding down he key F9, he vale of cell E1 will sep by sep increase by 1 p o vale enered in cell D1, which deermine nmber of he animaion seps. The vale of he cell E1 is sep animaion parameer based on acal vale of T R cells B1, C1 resp. is calclaed by he formla Isse 1, Volme 6, 1 54

10 =B1+(B11-B1)/$D$1*$E$1 for he cell B1. Acal vale of T R from he cell B1 is copied o he cells J4:J44 (ime inerval 99 min) and acal vale of T R from he cell C1 is copied o he cells J45:J65 (ime inerval 35 min). 3) Oher inp parameers The oher consan ha has o be iniialized in he beginning of he simlaion are V, K, K i, K d and r. The vale of hese consans deermines he ype of he conrol process (P, PI, PD or PID conroller), becase regarding (19) and () hey enable negleced some of he erms of his eqiaion. Table I shows possible vales of hese consan for given ype of he conrol process. Table I Vales of consan r, K i, K d for given ype of conrol process r K i K d P conroller,5 1 PI conroller,5 1 PD conroller,5 1 PID conroller,5 1 Inp vales V, K, K i, K d and r are enered o he cells in he row 6. 4) Op variable The op variable is he acal emperare T of he mixre waer. The vale of he emperare T is calclaed in he colmn I of he Excel simlaion model based on (1). 5) Examples of simlaion models The firs example of he simlaion model is shown on he Fig. 13 and Fig. 14. Fig. 14 Simlaion model of P conroller ime dependencies of acion signal Q (), Q 1 () and Q() I is he model of P ype of conroller he vales K i = 1 and vale K d =. The acion inp signal is Q = 1 l/min. The oher inp vales are Q 1 = 5 l/min, T 1 = 15 C and T 1 = 6 C. The reqired emperare T R = {45, 4, 5} C. Fig. 13 shows ime dependencies of he emperares T 1 (), T (), T R () and T(); Fig. 14 shows he ime dependencies of acion signal Q (), seady flow rae Q 1 () and flow rae Q(). The second example of he simlaion model is shown on he Fig. 15 and Fig. 16. I is he model of PID ype of conroller for he same iniial vales as in previos example. Only he vales K i = 1 and vale K d =. Fig. 13 Simlaion model of P conroller ime dependencies of T 1 (), T (), T R () and T() Fig. 15 Simlaion model of PID conroller ime dependencies of T 1 (), T (), T R () and T() Isse 1, Volme 6, 1 55

11 Fig. 16 Simlaion model of PID conroller ime dependencies of acion signal Q (), Q 1 () and Q() Fig. 18 Simlaion model of PID conroller ime dependencies of acion signal Q (), Q 1 () and Q() of non-conrolable sysem The las example is he simlaion model of nonconrollable sysem - see Fig. 17 and Fig. 18. I is he model of PID ype of conroller he vales K i = 1 and vale K d =. The acion inp signal is Q = 1 l/min. The oher inp vales are Q 1 = 5 l/min, T 1 = C and T 1 = 6 C. The reqired emperare T R = {4, 55, } C. In his case he opimal conrol reglaed process is no reached, he vale over-deviaion of he measred op signal y, deviaion period T m and period of he reglaed conrol sep T % has no minimm. VI. CONCLUSION Modeling and comper simlaion provides new mehodology of experimenal research in crren science and high school edcaion see e.g. [13] In he paper here is offered he case sdy of process modeling and simlaion of feedback reglaed sysem mixing of cold and ho waer in flow-hrow waer ank. Sep by sep here is shown he process of mahemaical modeling of saic, dynamic and feedback reglaion of he sysem and process of creaion of he simlaion model in MS Excel spreadshee and visalizaion in MS Excel XY char. ACKNOWLEDGMENT This research was sppored by he Research projec of he Facly of Science of Universiy of Hradec Kralove No. 13. Fig. 17 Simlaion model of PID conroller ime dependencies of T 1 (), T (), T R () and T() of non-conrollable sysem REFERENCES [1] S. Hbalovsky, E. Milkova, Modelling of a real siaion as a mehod of he algorihmic hinking developmen, in Proc. WSEAS/IASME Inernaional Conference on Edcaional Technologies (EDUTE 1), WSEAS Press, Kanoi, Sosse, Tnisia, 1, pp [] V. Jehlicka, Inerdisciplinary relaions in eaching of programming, in Proc. WSEAS/IASME Applied comping conference 1 (ACC'1), WSEAS Press, Timisoara, Romania, 1. pp [3] S. Hbalovsky, The sysem approach o eaching of algorihm developmen, in Proc. WSEAS/IASME Applied comping conference 1 (ACC'1), WSEAS Press, Timisoara, Romania, 1. pp -7. [4] S. Hbalovsky, M. Msilek, Crypoanalysis as a mehod of he sysem approach in he algorihm developmen, in Proc. WSEAS/IASME Applied comping conference 1 (ACC'1), WSEAS Press, Timisoara, Romania, 1. pp [5] S. Hbalovsky, E. Milkova, P. Prazak, Modeling of a Real Siaion as a Mehod of he Algorihmic Thinking Developmen and Recrsively Given Seqences, WSEAS Trans. on Informaion Science & Applicaions, Isse 8, vol. 7, Ag. 1, pp [6] S. Hbalovsky, Modelling of real kinemaics siaion as a mehod of he sysem approach o he algorihm developmen hinking, Isse 1, Volme 6, 1 56

12 Inernaional jornal of applied mahemaics and informaics, Vol. 4, No. 4, 1, pp [7] S. Hbalovsky, M. Msílek, Aomaic crypoanalysis of he monoalphabeical sbsiion as a mehod of he sysem approach in he algorihm developmen hinking, Inernaional jornal of applied mahemaics and informaics, Vol.4, No.4, 1, pp [8] J. Bailer, M. Daniela, Tracing he Developmen of Models in he Philosophy of Science, Magnani, Nersessian and Thagard, 1999, pp [9] S. Harmann, The World as a Process: Simlaions in he Naral and Social Sciences, In R. Hegselmann, e al., Modelling and Simlaion in he Social Sciences from he Philosophy of Science Poin of View, Theory and Decision Library. Dordrech: Klwer, 1996, pp [1] J. A. Sokolowski, C. M. Banks, Principles of Modeling and Simlaion A Mlidisciplinary Approach, Wiley Pblicaion, New Jersey, 9, pp [11] V. Jehlicka, S. Hbalovsky, Simlaion models sed in disance edcaion, In Proc. Use of e-learning in he raining of professionals in he knowledge sociey. Kaowice : Sdio Noa Irenesz Olsza, 1. [1] M. Araki, PID Conrol. Conrol sysems, roboics and aomaion, Vol.,. [13] J. Benacka, Solion o projecile moion wih qadraic drag and graphing he rajecory in spreadshees, Inernaional Jornal of Mahemaical Edcaion in Science and Technology, Vol. 41, No. 3, 1, pp Sepan Hbalovsky was born in Trnov, Czech Repblic in 197, he obained maser degree in edcaion of mahemaics, physics and comper science in 1995 and docor degree in heory of edcaion in physics in 1998 boh in Facly of Mahemaics and Physics, Charles niversiy in Prage, Czech Repblic. He worked 5 years as maser of mahemaics, physics and comper science on several secondary schools. He works as assisan professor on Universiy of Hradec Kralove from 6. He ineresed in algorihm developmen, programming and comper modelling. RNDr. Sepan Hbalovsky, Ph.D. is member of Union of Czech Mahemaicians and Physicis. Isse 1, Volme 6, 1 57