An inverse heat transfer problem for optimization of the thermal process in machining

Transcription

1 Sādhaā Vol. 36, Part 4, August 211, pp c Idia Academy of Scieces A iverse heat trasfer problem for optimizatio of the thermal process i machiig 1. Itroductio M GOSTIMIROVIC, P KOVAC ad M SEKULIC Departmet of Productio Egieerig, Faculty of Techical Sciece, Uiversity of Novi Sad, Trg D. Obradovica 6, 21 Novi Sad, Serbia marig@us.ac.rs; pkovac@us.ac.rs; milekos@us.ac.rs MS received 31 March 21; revised 11 February 211; accepted 27 May 211 Abstract. It is evidet that machiig process causes developmet of large quatities of thermal eergy withi a relatively arrow area of the cuttig zoe. The geerated thermal eergy ad the problems of its evacuatio from the cuttig zoe accout for high temperatures i machiig. These icreased temperatures exert a proouced egative effect o the tool ad workpiece. This paper takes a differet approach towards idetificatio of the thermal process i machiig, usig iverse heat trasfer problem. Iverse heat trasfer method allows the closest possible experimetal ad aalytical approximatio of thermal state for a machiig process. Based o a temperature measured at ay poit withi a workpiece, iverse method allows determiatio of a complete temperature field i the cuttig zoe as well as the heat flux distributio o the tool/workpiece iterface. By kowig the heat flux fuctio, oe defies criterium ad method of optimizatio, the iverse heat trasfer problem trasforms ito extreme case. Now, the task of optimizatio is to determie most favourable ratio betwee heat flux parameters i order to preserve exploitatio properties of the tool ad workpiece. Keywords. Machiig process; thermal eergy; temperature; heat flux; iverse problem; optimizatio. Moder maufacturig is facig complex demads o a daily basis. System flexibility, productivity of maufacturig ad high levels of machiig quality ad accuracy, are the most vital demads facig the market-orieted idustrial systems. Oly moder equipped idustrial systems shall be able to adjust their maufacturig process to these high market demads. There ca be little doubt that machiig techologies shall remai importat i moder maufacturig idustry. Especially importat are material machiig methods as the itegral part of the techological process of product maufacturig ad assembly. Basic advatages of machiig process are high productivity, precisio ad surface quality with the ability to cope with hard For correspodece 489

2 49 M Gostimirovic et al materials ad complex surfaces (Khachfe & Jary 2). However, due to process of chip formatio durig machiig, it is evidet that machiig methods cause the developmet of large quatities of thermal eergy withi the cuttig zoe (Stepheso 1991). The geerated thermal eergy, located withi a relatively arrow area of the cuttig zoe, causes high temperatures (Milikic & Gostimirovic 1994). These icreased temperatures istataeously burst to a maximum, causig various physical ad chemical chages i the cuttig zoe ad exertig a proouced egative effect o the tool ad workpiece. Sice the mai task of machiig is to achieve satisfactory part quality with as large productivity as possible, special attetio is focused o the effect that the cuttig temperatures have o the chage of material properties i the workpiece surface layer ad machiig accuracy (Shaw 23). If the temperatures thus geerated are high eough to cause structural ad phase trasformatios of the workpiece material, the machied surface shall suffer from a umber of disadvatages. Should, i additio, dimesioal errors appear as well, the overall effect ca substatially dimiish exploitatio features of the fiished part. Obviously, efficiet machiig of parts, free of thermal defects i the workpiece, requires methods for optimizatio of thermal pheomea by regulatio of cuttig temperatures. Systematic research i machiig techologies has yielded a umber of various aalytical ad experimetal methods for determiatio of temperatures ot oly i the arrow ad wider cuttig zoe, but also i the machiig system (Gostimirovic & Kovac 28). Due to rudimetary measurig equipmet, first research o temperatures i machiig was mostly theoretical. Later advacemets i measurig equipmet allowed developmet of various experimetal methods for temperature measuremet i machiig, which have bee udergoig modificatio ad improvemet util today. However, although the temperature is a essetial parameter i machiig, its utilizatio for the purpose of optimizatio of machiig is fairly complex. The mai obstacle o the road to its utilizatio for optimizatio purposes lies i the difficult moitorig of cuttig temperatures. Therefore, efforts are aimed at improvig the existig ad developmet of ovel measuremet methods, while at the same time focusig o aalytical optimizatio models which ca successfully relate to cuttig temperatures (Li 1995). As the research so far has show, o-statioary ad o-liear processes that ivolve itesive heat coductio, such as machiig, ca be successfully solved usig iverse problems of heat trasfer (Özisik & Orlade 2; Shidfor & Tawakoli 22; Tikhe & Deshmukh 25). The iverse problems are today successfully applied i idetificatio, desig, cotrol ad optimizatio of thermal processes ad plats. I the case of machiig, the iverse problems so far has mostly bee used to idetify processes by approximatig heat flux or temperature field i the cuttig zoe (Kim et al 26). Whe the iverse heat trasfer problem is trasformed ito a extreme case it is practically the oly way to optimize heat loadig o the workpiece (Alifaov 1994). For a kow temperature measured at a poit withi the workpiece, umerical methods are used to approximate the total temperature field as well as the heat flux i the cuttig zoe. For the selected model of optimizatio, quality criteria ad boudaries, it is possible to arrive at optimal machiig coditios by cotrollig the heat flux. 2. Iverse heat trasfer problem of machiig 2.1 Geeral case of iverse problem The process of heat trasfer betwee solid bodies or betwee a system ad its eviromet is mostly cosidered from the stadpoit of mutual relatios betwee iput ad output process parameters. It is widely accepted that such process ca be schematized as i figure 1.

3 A iverse heat trasfer problem for optimizatio of the thermal process 491 Figure 1. Diagram of a thermal process. The first step i the research of ay thermal pheomeo is to model the real process. This meas that developmet of a model which is valid over a arrow domai limited by boudary coditios. The model, which describes a segmet of the real process, correlates iput u(t) ad output z(t) parameters which defie the state of the process at every momet i time t. If the iput parameters u(t) are kow ad output parameters z(t) defie process state i time, the the output parameters are a fuctio of iput parameters, i.e.; z = f (u, t). (1) The real thermal process is most ofte described aalytically. The goal is to set up a most adequate aalytical model, while, o the other side, keepig its form as simple as possible i order to facilitate solutio. Give the right mathematical method, the model thus defied, solves problems quickly ad efficietly. Aalytical model of thermal process most ofte takes the form of a system of differetial ad algebraic equatios. Sice such form is easily trasformed ito algorithm ad efficietly processed o computer, the differetial models are widespread today i the ivestigatio of thermal processes. If for the adopted thermal model there exist uique coditios (boudary ad iitial coditios, thermal ad physical properties ad geometry of a body or a system), the ay particular iput parameters of the thermal process shall result i that same or ay other thermal state defied by the temperature field of the aalysed object. Determiatio of the iput output relatioship is the direct heat trasfer problem. Coversely, the iverse heat trasfer problem is used to fid iput characteristics of the process for the kow temperature field (Beck et al 1985; Adreas 1989). If for every ukow parameter u there is a liear, smooth operator A which allows determiatio of output parameter z, the geeral case of iverse problem is formulated by the followig equatio: Au = z. (2) If we represet the ukow iput parameter of the thermal state with u(t),adifz(t) deotes the kow output parameter of the process, i area of D, the the iverse problem becomes: dz dt = f (z, u, t), (3)

4 492 M Gostimirovic et al with iitial, additioal ad boudary coditios: z t= = z z D=K = z K z D=S = u S. I equatio (3), u S is the solutio, i.e., the surface heat flux ad temperature o the boudary of body S, while fuctio z K represets the temperatures measured outside the body at a poit K. 2.2 Modellig of thermal state i machiig Oe of the essetial problems of machiig is the kowledge of fudametal laws which regulate geeratio ad distributio of thermal eergy as well as the character of the temperature fields i the cuttig zoe. The quatity of thermal eergy geerated i the machiig process is equivalet to mechaical work applied to machiig. Coversio of mechaical eergy ito thermal eergy takes place withi several characteristic zoes, with some zoes partially overlappig, figure 2 (left). O the flak ad face surface of a tool, mechaical work is trasformed ito heat due to itesive frictio betwee the tool, machied surface ad chip. Further trasformatio of mechaical work ito heat takes place durig chip formatio, through plastic deformatio of workpiece material. The thermal eergy thus geerated is evacuated by coductio, covectio ad radiatio. Heat siks, which take away the thermal eergy from the cuttig zoe, are tool, workpiece, chip, coolat ad eviromet. Characteristics of heat sources ad siks deped o a large umber of ifluetial factors which cotribute to a process of chip formatio durig machiig. Therefore, it is very difficult to defie the character of thermal eergy i the cuttig zoe. Similar to other thermal processes, correct evaluatio of the character of thermal eergy i machiig requires kowledge of the followig: type, dimesio, shape, distributio law, velocity, time ad stregth of thermal eergy. The role of mathematical theory behid thermal pheomea i machiig is to adopt the most adequate model of workpiece, tool ad their iter-relatioships. If the umerous variable parameters were take ito cosideratio, aalytical modellig of the machiig process would become a impossible task. Therefore, some simplificatios are ecessary where the fial solutio is verified by experimets. Despite simplificatio, such aalytical ad experimetal model yields reliable results. Oe ca assume that the total heat source o the machiig process q is the result of frictio betwee the tool, workpiece ad chip i the workpiece material shear plae. This total heat Figure 2. Model of a thermal state i machiig process.

5 A iverse heat trasfer problem for optimizatio of the thermal process 493 source, whose stregth varies withi a arrow rage, acts cotiuously, shiftig across the workpiece surface with costat velocity. The shape ad dimesios of the heat source deped o the tool/workpiece iterface. I machiig, cosiderig that the cuttig depth is may times smaller tha the legth ad width of tool/workpiece iterface, the heat source ca be treated as a strip of ifiite legth ad costat heat distributio, figure 2 (right). The assumptio of costat heat distributio across the iterface is a valid approximatio i case of the heatig of thi surface layers of tool ad workpiece material. 2.3 Iverse problem i machiig I the previously defied thermal model of machiig, heat trasfer alogside tool/workpiece iterface ca be disregarded, thus rederig the workpiece surface isolated. Furthermore, if we disregard the dissipatio of heat flow i the directio of heat source movemet, the the workpiece ca be approximated with a semi-ifiite plate, figure 3. Substitutio of the real workpiece with series of adiabatic thi plates is completely justified, bearig i mid that the heat source i machiig is geerated withi a small volume of workpiece material while the heat loadig of the surface workpiece layer is cosidered depth-wise. I that case, the followig is a more geeral case of differetial equatio of a oe-dimesioal heat coductio: T (x, t) C (T (x, t)) = ( ) T (x, t) x (, H) k (T (x, t)) t x x t (, t m ], (4) where T = T (x, t) - workpiece temperature at poit coordiate x at momet t; k -thermal coductivity; C = ρc - specific heat capacity (ρ - material desity, c - specific heat); H - thickess of the surface layer of workpiece material ad t m - largest time icremet. Now the aalytical form of iverse heat trasfer problem for machiig ca be described with differetial equatio (4) should be cosidered i cojuctio with the iitial temperature distributio, additioal ad boudary coditios. The iitial coditio refers to defiig a temperature profile i the workpiece at the iitial momet t = : T (x, t) t= = T (x) x [, H]. (5) Iitial coditio T (x) - iitial temperature Additioal coditio T K (t) - kow temperature Boudary coditio q (t) - ukow heat flux q * (t) - kow heat flux Figure 3. Schema of a ui-dimesioal iverse problem of the machiig process.

6 494 M Gostimirovic et al A additioal coditio is the fact that at the poit of workpiece x = K ( < K H), thereis a kow temperature, measured durig a time iterval: T (x, t) x=k = T K (t) t [, t m ]. (6) Boudary coditios for the cosidered workpiece surface layer are defied by the boudary coditio of the secod order. The lower boudary coditio is defied by the kow heat flux: T (x, t) k (T (x, t)) x = q (t) t [, t m ], (7) x=h while the upper boudary coditio is defied by the ukow heat flux over the tool/workpiece iterface. T (x, t) k (T (x, t)) x = q (t) t [, t m ]. (8) x= The fial solutio of the iverse heat trasfer problem is the heat flux o tool/workpiece iterface q = q(t), ad the temperature field T = T (x, t) throughout the etire elemetary part of workpiece, D ={(x, t) : x [, H], t [, t m ]}. Furthermore, we ca trasform iverse heat trasfer problem ito a extreme case usig some method of optimizatio. Give the iput parameters, this would allow us to determie the optimal thermal state of the process for the aalytical model i had, so as to satisfy the state ad boudary fuctios for the give optimizatio criterio. For machiig, optimizatio of the thermal process is to determie the fuctio of the process state T = T (x, t), ad the cotrol fuctio q = q(t), so as to satisfy the aalytical form of the iverse problem for the machiig process, as defied by equatios (4) to (8). These fuctios are determied uder the coditio that the kow temperature at a particular fixed poit T K = T (K, t) is cosistet with the temperature T q = T(q, K, t) which is calculated based o cotrol fuctio q(t): J (q) = t m [ Tq (q, K, t) T K (K, t) ] 2 dt + rc (q), (9) where r c - weight coefficiet factor ad (q) - stabilizig fuctioal. It is kow with certaity that the additio of the stabilizig fuctioal cotributes to the quality of cotrol system. However, the ifluece of stabilizer (q) is quite low ad ca be disregarded, especially if the umber of icluded parameters is small. 3. Numerical solutio of iverse heat trasfer problem 3.1 Implicit method of fiite differeces Due to high complexity, differetial equatios of the secod order which describe the process of heat coductio i machiig are mostly solved usig umerical methods. These methods trasform exact differetial equatios ito approximate algebraic equatios. The first step with every umerical method is the discretizatio of space, i.e., approximatio of a thi, isolated plate of workpiece by a umber of elemetary pieces x, figure 4. Whe

7 A iverse heat trasfer problem for optimizatio of the thermal process 495 Figure 4. The shape of a ui-dimesioal mesh of iverse heat trasfer problem. dealig with a o-statioary heat coductio problem, the time of temperature chage ad heat flux are discretized by a t icremet. To solve the partial differetial equatio (4) a implicit form of the fiite differeces method was chose. The cocept of this method very much resembles the physical process, where the temperature or heat flux at each observed poit is calculated after a time icremet as the result of heat exchage with the eighbourig poits. Based o the five kow temperatures at the eighbourig poits, the temperature at the ext momet i time is calculated. The results obtaied by iverse problem of the heavily loaded thermal processes, are highly proe to errors due to chage of thermal ad physical material properties with temperature. Therefore, solvig of a o-liear iverse problem requires the thermal ad physical material properties to be expressed i the form of approximate fuctios of temperature, bearig i mid the coditios of smooth differetiatio. I that case, the aalytical model give i (4) should be substituted by aother, more appropriate form: 1 T α (T ) t ( ) = 2 T x 2 + k (T ) T 2 x (, H) k (T ) x t (, t m ], (1) where α = k/c - thermal diffusivity ad k dk/dt. Itroducig a eve-spaced triagular mesh {x h = h x, h =, H; t = t, =, m} ad usig Taylor approximatio, the first ad secod derivatives of the partial differetial equatio (1) ca be writte i the form of fiite differeces (Kreith & Black 198): 2 T x 2 = +1 h+1 2Th + h 1 ( h x 2 + ε A x 2 ) T x = T h+1 T h 1 ( + ε A x 2 ) (11) h 2 x T t = T h +1 Th + ε A ( t), t where ε A is a estimatio error. h

8 496 M Gostimirovic et al Substitutig equatios (11) ito equatio (1) oe derives equatio of o-statioary uidimesioal heat coductio i the form of fiite differeces: rh h 1 + ( 1 + 2rh ) T +1 h rh h+1 = T h + r h k h ( T 4kh h+1 Th 1 ) 2, (12) where r h = α h t/ x2. Discrete represetatio of heat flux q(t) ad q (t) requires itroductio of fiite differeces of the secod order, bearig i mid the accumulatio of heat withi a particular elemet of x thickess (Alifaov 1994): q +1 = k q +1 = k H 1 + C x x 2 TH +1 H 1 + CH x The discrete form of the iitial temperature distributio is: T t x 2 H T H t. (13) T h = T (h x) h =, H =. (14) Expressios (12) to (14) result i a system of liear algebraic equatios i implicit form of fiite differeces. The equatios are used to calculate the ukow heat flux q +1 ad the temperature field i the cuttig zoe h (h =, 1,..., K 1, K + 1,..., H) as follows: [ ] R {T} = {B}. (15) Solvig the matrix system (15) requires the iitial task to be divided ito two parts. First, a stadaloe system is calculated: [ ] R2 {T2 } = {B 2 }, (16) that is: (1 + 2r K +1 ) r K +1 r K +2 (1 + 2r K +2 ) r K r H 1 (1 + 2r H 1 ) r H 1 2r H1 (1 + 2r H ) k+1 k+2. H 1 TH +1 = b k+1 b k+2. b H 1 b H The matrix equatio (16) is solved as the direct task of heat coductio withi the area D 2 = {(x, t) : x [K, H], t [, t m ]}. The solutio yields the ukow temperatures h (h = K + 1,...,H). Oce vector T 2 is determied, oe ca tackle the problem of iverse heat trasfer i the area of D 1 ={(x, t) : x [, K ], t [, t m ]} usig the system: [ ] R1 {T1 } = {B 1 }, (17).

9 that is: A iverse heat trasfer problem for optimizatio of the thermal process 497 2r r1 ( ) 1 + 2r 1 r 1... rk ( ) r K 2 rk 2 rk ( ) r K 1 rk 1 ( 1 + 2r ) b1 =., b K 1 b K 2 ˆb K i equatios (16) ad (17) where: u q. K 2 K 3 K 1 u q = 2 α t k x q+1 + T b i i=1;k 2 i=k+2;h 1 b i i=k 1 i=k+1 = b i b i = T i = b i + r i k b H = TH + 2 a H t k q +1 ˆb H x + r i k i 4k i ( T i+1 Ti 1 ) 2 K = b K ( 1 + 2rK ) T +1 K +1 + r K K +1. From the system (17), startig from the first ukow temperature K 1, vector T 1 is calculated. This vector represets the ukow heat flux q +1 ad the ukow temperature h (h =,...,K 1). 3.2 Iterative optimizatio method I order to establish optimal cotrol over the previously defied extreme case of iverse problem, the total search iterative method was selected as the method of optimizatio. It allows optimizatio of liear ad oliear goal fuctios, with or without costraits. It is efficiet ad allows relatively simple solutio of eve the most complex optimizatio problem. Usig iterative algorithms, through gradual approach to the maximum over a umber of successive steps, the smooth fuctio q(t) is give i the form of a fiite vector: q = { q 1, q 2,..., q m} T, (18)

10 498 M Gostimirovic et al where q = q(t ) for = 1,m are compoets derived by discretizatio of the chose time mesh. For computer applicatio, the itegral quality criterio, equatio (9), is replaced by the fiite sum: J (q) m [ = t Tq (q, K, t ) T K (K, t ) ] 2, (19) =1 where t - time icremet ad t - time poit of a eve discretizatio of smooth fuctios T q (q, K, t ) ad T K (K, t ). I order to miimize the differeces betwee the kow ad the calculated temperatures, a iterative method of optimizatio is used, which gives exact solutio usig the followig form: q i+1 (t) = q i (t) + q i (t) i =, 1, 2,..., (2) where q (t) - iitial approximatio ad q i (t) - iteratio step. J(q i ) is calculated by applyig the iterative optimizatio algorithm for particular values of the fuctioal q i (t). The procedure is repeated, decreasig the value of the fuctioal J(q i+1 )< J(q i ), util its miimum is reached. The iterative gradiet method procedure is cosidered fiished for a sufficietly small fuctioal, which meas that the calculated T q (q, K, t ) ad the measured temepartures T K (K, t ) are very close or almost idetical. Our cosideratio of the problem of optimizatio usig iterative method supposes that it is the ukow solutio of the iverse problem that is miimizig the goal fuctio. However, if the iput data cotai certai error ad discretizatio itervals are small eough to preclude self-regulatio, the fial solutio shall be approached i a oscillatig maer. It is therefore advisable to stop the iterative process at some iteratio i order to avoid substatial oscillatios of the solutio. Accordigly, for the allowed level of optimizatio (equatio (2)), the followig stoppig criterio is valid: J (q) = ε 2. (21) Here ε represets estimatio of iput data error, ad is calculated as: t m ε 2 = σ 2 (t) dt = t m =1 [T K (K, t ) T m ], (22) where σ 2 (t) - dispersio of fuctio T K (K, t ) ad T m = T K (K, t )/m - average value of measured temperatures. 4. Verificatio of iverse problem i the machiig 4.1 Experimetatio As the proposed system uses experimet ad aalytical model to optimize thermal process i machiig, it requires distributio of temperatures to be determied experimetally at a poit withi the workpiece, figure 5(bottom). Temperatures i the workpiece were measured at various distaces from the measurig poit to the cotact surface of the workpiece ad the tool. The experimetal work was carried out o a surface gridig machie ( Majevica type CF 412 CNC). The workpiece material was HSS - high speed steel (B.S. BM 42) at 66 HRc hardess. The tool was alumium oxide wheel ( Witerthur type 53 A8 F15V PMF, diameter D s =

11 A iverse heat trasfer problem for optimizatio of the thermal process 499 Figure 5. Experimetally obtaied temperature distributio i time withi the workpiece surface layer. 4 mm). The depth of cut was a =,5 mm, the workpiece speed was v w = 5 mm/s ad the wheel speed was v s = 3 m/s. A water-based coolat (emulsio 6%) was used durig the machiig test. For measuremet, processig ad cotrol of cuttig temperatures, a moder iformatio system was used. The temperature was measured i the workpiece surface layer usig a thermocouple (type K, φ,2 mm) built ito the workpiece at a specified clearace from the tool/ workpiece iterface, figure 5(top). Applicatio of thermocouple is simple, reliable ad costefficiet, ad does ot iterfere with the real machiig coditios. 4.2 The results of iverse method I this case of verificatio, to ivestigate the machiig thermal process by iverse heat trasfer problem, followig iput parameters were take. Thermo-physical properties of workpiece material (high speed steel B.S. BM 42): thermal coductivity k = 21,378 +,275 T W/m C ad thermal diffusivity α = m 2 /s. Spatial discretizatio: umber of the iterior poits h = 7 with space step x =,5 mm. Temporal discretizatio: umber of the time icremet

12 5 M Gostimirovic et al m = 25 with time step t =,25 s. Iitial temperature distributio: experimetally determied temperatures (figure 5) for the t = s. Boudary coditio: heat flux over the lower boudary of the cosidered workpiece surface layer q (t) = W/m 2. Additioal coditio: the kow temperature distributio measured outside the workpiece at depth z = 1 mm (figure 5), i.e., at poit K = 2. Based o the previously preseted, the total temperature field i workpiece first was obtaied by computatio. Calculated temperature chage over time i the workpiece surface layer, as well as the model the temperature field i the cuttig zoe, are show i figure 6. The temperature distributio i the workpiece is computed by iverse heat trasfer problem. It defies the thermal process i the machiig. The computed time ad depth-related chage of temperature i the iterface zoe of the workpiece surface layer (figure 6), shows a high degree of coformity with the experimetally obtaied results (figure 5). Oce temperature field i the workpiece is determied, the ukow heat flux i the tool/workpiece iterface is calculated. I this case of verificatio, the cotact temperature was ot allowed to exceed the critical temperig temperature, which was experimetally established at 55 C for the selected high speed steel. The computed temperature ad heat flux, show i figure 7, defie the heat loadig i the workpiece surface layer. The computed distributio of heat flux over the tool/workpiece iterface clearly shows the direct relatioship betwee heat flux parameters, i.e., the power of heat ad its total active time. The greater the power of heat flux, the shorter the active time, ad vice versa. 4.3 Optimizatio of heat flux parameters Usig extreme case of iverse heat trasfer problem for optimizatio of the machiig thermal process requires selectio of machiig coditios such that the surface heat loadig does ot exceed the limits allowed. Thus, optimizatio of heat loadig of workpiece surface layer is required, which relies o heat flux parameters. More specifically, the process of optimizatio Figure 6. Computatio of temperature field i the workpiece surface layer.

13 A iverse heat trasfer problem for optimizatio of the thermal process 51 meas determiatio of most favourable ratio betwee heat power ad its active time, for the previously established distributio of heat flux over the tool/workpiece iterface, figure 7. Uder the coditio of maximum machiig productivity (Q w = v w a = max), the process of optimizatio is coducted by completely searchig the bouded solutio space (v w, a). Thus optimal workpiece speed v w ad cuttig depth a are determied which ca be iside or o the very boudary of the search space. Based o the previously proposed model of optimizatio, the defied criterio of optimizatio ad the state fuctios ad boudaries, the optimal ratio of heat flux parameters is calculated ad subsequetly used to derive optimal machiig coditios, figure 8. Parameters thus derived, Figure 7. Heat flux ad temperature distributio over the tool/workpiece iterface. Figure 8. Optimal machiig coditios is calculated of most favourable ratio betwee heat power q ad its active time t.

14 52 M Gostimirovic et al yield maximum machiig productivity with the temperature beig kept below the dagerous level which could compromise fuctioal properties of fiished parts. The compariso betwee the computed optimal machiig coditios with the maximum allowed oes, which were derived experimetally, shows very little differeces. The differeces ca be explaied by a large umber of parameters which were either omitted from aalysis or had to be estimated i the course of aalytical modellig of the machiig process. 5. Coclusios Based o this ivestigatio, followig coclusios ca be made: The iverse heat trasfer problems are applied i idetificatio, desig, cotrol ad optimizatio of machiig thermal processes based o aalytical models ad experimetal results is gaiig popularity. I the optimizatio over thermal state i machiig, the extreme case of iverse heat trasfer problem is practically the oly way to reliably approximate the allowed heat loadig o workpiece ad tool. Aalytical iverse heat trasfer problem allows approximatio of a complete temperature field ad heat flux distributio i the cuttig zoe. The iverse heat trasfer problem was solved usig method of fiite differeces i implicit form, where the cocept of this method very much resembles the physical process. As this is a experimetal ad aalytical system for computatio of heat loadig of machiig process, it requires a exact, experimetally obtaied temperature distributio at a sigle poit withi the cuttig zoe. The stability of the iverse problem umerical solutio largely depeds o the iitial ad boudary coditios, thermal ad physical properties of the machiig process ad the choice of temporal ad spatial discretizatio. The aalytically obtaied temperature field i the cuttig zoe largely agrees with the experimetal results. Optimizatio of thermal state of cuttig zoe requires determiatio of the fuctio of the process state ad the cotrol fuctio, so as to satisfy the aalytical form of the iverse heat trasfer problem. Total search iterative method was selected as the method of optimizatio. Optimizatio of machiig coditios allows the parameters of heat flux to be kept withi limits which guaratee fuctioal properties of the tool ad workpiece. List of symbols A liear smooth operator a mm cuttig depth C J/m 3 C specific heat capacity c J/kg C specific heat D aalysed area D s mm diameter of gridig wheel h mm coordiate of approximatio H mm thickess of workpiece boudary layer J optimizatio criterio (fuctioal)

15 A iverse heat trasfer problem for optimizatio of the thermal process 53 K poit outside of workpiece k W/m C thermal coductivity Q w mm 3 /mm s specific material removal rate q W/m 2 heat flux q W/m 2 kow heat flux time of approximatio r c weight coefficiet factor T C temperature T c C critical temperig temperature T C iitial temperature T K C kow temperature at poit K T q C calculated temperature t s time t m s largest time icremet v c m/s cuttig speed v w mm/s workpiece speed u iput parameter x mm coordiate z output parameter z mm distace of from machied surface α m 2 /s thermal diffusivity t s time icremet x mm elemetary piece ε error ρ kg/m 3 material desity stabilizig fuctioal Refereces Alifaov O M 1994 Iverse heat trasfer problems. (Berli: Spriger-Verlag) Adreas N A 1989 A iverse fiite elemet method for directly formulated free boudary problems. It. J. Numerical Methods i Eg., 28: Beck J V, Blackwell B, Clair C R 1985 Iverse Heat Coductio: Ill-posed Problems. (New York: Wiley- Iterciece Publicatio) Gostimirovic M, Kovac P 28 The thermal state of the workpiece surface layer durig productivity gridig. Maufacturig Techology Joural for Sciece, Research ad Productio, Prague, 8: Kim H J, Kim N K, Kwak J S 26 Heat flux distributio model by sequetial algorithm of iverse heat trasfer determiig workpiece temperature i creep feed gridig. Iteratioal Joural of Machie Tools ad Maufacture, No 46: Khachfe R A, Jary Y 2 Numerical solutio of 2-D o liear iverse heat coductio problems usig fiite elemet techiques. Numerical Heat trasfer, A Iteratioal Joural of Computatio ad Methodology, Part B, 37: Kreith F, Black W 198 Basic heat trasfer. (New York: Harper ad Row) Li J 1995 Iverse estimatio of the tool-work iterface temperature i ed millig. Iteratioal Joural of Machie Tools ad Maufacture, 35(5): Milikic D, Gostimirovic M 1994 Ifluece of tribological ad geometrical parameters of the gridig process o the magitude of cuttig temperature. Tribologija u idustriji, 16(3): Özisik N M, Orlade R B 2 Iverse heat trasfer: fudametal ad applicatios. (New York: Taylor & Fracis)

16 54 M Gostimirovic et al Shaw M C 23 The size effect i metal cuttig. Sadhaa, 28(Part 5): Shidfor A, Tawakoli K 22 A iverse heat coductio problem. Southeast Asio Bulletio of Mathematics, 26: Stepheso D A 1991 A iverse method for ivestigatig deformatio zoe temperatures i metal cuttig. Joural of Egieerig for Idustry, 113: Tikhe A K, Deshmukh K C 25 Iverse trasiet thermoelastic deformatios i thi circular plates. Sadhaa, 3(Part 5):