National Research Tomsk Polytechnic University, Tomsk, Russia

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1 EPJ Web of Confeences 76, DOI: /epjconf/ C Owned by the authos, published by EDP Sciences, 014 Numeical evaluation of the measuement eo of tempeatue by suface themocouples in the conditions of incomplete themal contact with object of measuement Yuliana K. Atoshenko and Pavel A. Stizhak National Reseach Tomsk Polytechnic Univesity, Tomsk, Russia Abstact. One-dimensional and two-dimensional models of heat tansfe ae given in opeation fo eseach of the main chaacteistics of pocess of tempeatue measuement by means of suface themocouples. Results of numeical evaluations of the elative eo of the measuement aising owing to existence of ai gap between a sensitive element of the themocouple and object of measuement ae povided. It is evealed that the measuement eo in case of obsevance of necessay time of heating up set minimum equied time of measuement can be loweed to level of an allowed eo. 1. Intoduction Nowadays, tempeatue measuement is an integal pat of poductions of all industies and in all fields of activity of the peson. Tempeatue measuement is executed as with I aim monitoing of quality of couse of technological pocesses, and in systems of egulation of technological paametes. Themo tansfomes ae used fo emote tempeatue measuement, and also fo tansmission of measuing infomation fom management system esistance o themocouple. The senso type selection, mainly, is caied out depending on the ange of taken tempeatues, the admissible size of the sensitive element necessay fo time of establishment of indications and equiements to accuacy of measuements. Themocouples ae used in case of tempeatue measuement in monitoing systems o the egulations, measuements not equiing to high accuacy, and/o in cases when the fast esponse to change of the measued paamete tempeatue [1] is necessay. Nevetheless, time of establishment of indications in a themocouple usage time fo tempeatue measuement of a suface of any body, can be inceased significantly fo a numbe of easons. It has athe geat impact on the accuacy of measuements in case of non-compliance with necessay duation of heating of a sensitive element. Fo saving accuacy of detemination of tempeatue by themocouples in case of tempeatue measuement of a suface of object of measuement it is expedient to use pognostic models of nonstationay pocess of heat tansfe in system object of measuement the themocouple. A ow of opeations [ 4] is devoted to development of such models, nevetheless, the opeations devoted to pocess of heat tansfe in the themocouple fom the point of view of pediction of necessay duation of heating up pactically aen t pesent. This is an Open Access aticle distibuted unde the tems of the Ceative Commons Attibution License 4.0, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Aticle available at o

2 EPJ Web of Confeences Figue 1. Diagams of aea of the decision one-dimensional a and two-dimensional b of tasks of heat conduction: 1 themocouple junction; powde Al O ; potectivecove;4 aigap. Requiements to themocouples, and also equiements to the accuacy of measuements contain in the Intenational standads [5, 6]. In paticula, the standad [5] defines eight types of themocouples. In pactice fo measuement of tempeatues in the ange fom 00 Cto+1100 C in measuing systems of tempeatue boad application was found only by themocouples of types L, K, E. As efeence themocouples themocouples of types S and R ae applied. Allowed eos of all themocouples ae given in the standad [6]. It is necessay to mak that the measuement eo of tempeatue includes a systematic and accidental component. Thus the accidental eo in a cetain level is defined by duation of execution of measuements and can be educed by pediction of optimum duation of heating of the themocouple. The submesible themocouples which ae applying to tempeatue measuement of diffeent envionments, as a ule, possess athe small duation of establishment of indications. Time of esponse fo change of the taken tempeatue fo suface themocouples substantially is defined by value of ai gap between a suface of the themocouple and object of measuement. Detemination of necessay time of heating up of the themocouple can be executed by means of numeical modeling of nonstationay pocess of heat tansfe in the non-unifom system the themocouple ai gap object of measuements.. Physical model of heat tansfe In development pocess of pognostic model of heat tansfe one-dimensional and two-dimensional tasks of heat conduction wee consideed, diagams of which aea of the decision ae povided in Fig. 1. The following assumptions ae accepted in case of numeical modeling: 1 Heat-physical chaacteistics of the mateials enteing aea of the solution of the task don t depend on tempeatue; Tempeatue change within aea of the solution of the task Fig. 1a happens only in the diection of cylindical coodinate of fo one-dimensional setting. The end of pocess of heating up was defined at the time of achievement by a themocouple junction of tempeatue, othe than an allowed eo measued on value. Values of allowed eos ae given in Table 1. Fo aea solutions of the task Fig. 1 ae made the following sizes: H = 5mm;R= 5 mm. Thickness of ai gap between object of measuement and the themocouple along coodinates of z and vaied when caying out numeical modeling in the ange fom 1 mm to mm p.

3 Table 1. Limits of allowed eos of themocouple [6]. Themophysical Basis of Enegy Technologies Themocouple s type Pemissible deviation limit fom ated diect cuent chaacteistic, C L toleance class ±, 5 in the ange of tempeatues fom 40 to 00 C; ±0, 0075 t in the ange of tempeatues fom 00 to 800 C S toleance class ±1, 5 in the ange of tempeatues fom 0 to 600 C K 1 toleance class ±1, 5 in the ange of tempeatues fom 40 to 75 C ±0, 004 t in the ange of tempeatues fom 75 to 1000 C. Mathematical model and decision methods The system of diffeential equations descibing pocess of heattansfe within one-dimensional model, has the following appeaance: c 1 ρ 1 1 c ρ c ρ c 4 ρ 4 4 = λ 1 = λ = λ = λ 4 The two-dimensional model is descibed by system of equations: 1, t>0, 0 << 1, 1, t>0, 1 <<,, t>0, ><, 4, t>0, << 4. 4 c 1 ρ 1 1 = λ 1 T T 1, t>0, 0 << z 1, z <z<h; 5 c ρ c ρ = λ T = λ T T, t>0, 0 << z, z <z<z ; 1 <<, z <z<h; + T, t>0, 0 << z, z 1 <z<z ; <<, z <z<h; 6 7 c 4 ρ 4 4 = λ 4 T T 4, t>0, 0 <<L,0<z<z z 1 ; <<, z <z<h; Whee adial coodinate, m; z axial coodinate, m; ρ density, kg/ m; c specific heat capacity, J/kg C; λ coefficient of heat conduction, W / m C; indexes: 1 themocouple s junction, powde of an oxide of aluminum, a potective cove, 4 ai p. 8

4 EPJ Web of Confeences Tempeatue distibution in an initial time-point in the system making aea of the solution of the task ae defined by the following initial conditions: t = 0; T = T 0,0<<R,0<z<H, whee T 0 tempeatue coesponding to efeence conditions: T 0 = 0 C. The bounday conditions which have been set on an axis of symmety = 0: = 0, = 0. 9 Bounday conditions on bounday = R: = R; T = T. 10 T tempeatue of a heating element. Bounday conditions on boundaies on an axis z: z = H, = 0; z = 0, T = T. 11 On boundaies an Al O themocouple powde seal, the powde Al O -potective a cove, a potective cove ai Fig. 1 conditions of the IV kind wee accepted: T 1 1, z = T 1, z ; T 1, z = T, z ; λ 1 1 = λ =1 ; λ 1 z = λ =1 z=z z ; z=z T, z = T, z ; T, z = T, z ; λ = λ = ; 1 λ = z = λ 1 z=z z ; z=z T, z = T 4, z ; T, z 1 = T 4, z 1 ; = λ 4 4 ; = = = λ 4 4. z=z1 z=z1 λ λ z Conditions 1 ae valid fo one-dimensional model, conditions wee applied to two-dimensional model 1, 1. The aea of the solution of the task Fig. 1 is boken into the unifom gid consisting of 40 nodes. The slot pitch on adial and axial coodinates is equal.5 10 mm.thesteponatempoal gid changed in the ange fom 10 4 to 10 s fo eduction of volume of computation and incease of accuacy of the decision. Systems of Eqs. 1 4 and 5 8 with the appopiate initial and bounday conditions decided using a method of finite diffeences [7]. The solution of the diffeence analogs of the diffeential equations epesenting linea algebaic equations was caied out by a local and one-dimensional method [7]. The po-ace method was applied to the decision of system of the diffeence equations on the basis of the implicit fou-point diagam [7]. The consevatism veification of applied diffeence schemes was conducted to estimate the confidence of numeical simulation esults simila to [8 10] and the compaison with expeiment esults was accomplished. z 4. Results and discussion Heat-physical chaacteistics of basic elements of consideed systems ae povided in Table. The descibed models wee used fo eseach of necessay duation of heating up of diffeent types of themocouples K, L, S. Results of numeical modeling ae given in Table p.4

5 Table. Heat-physical chaacteistics of mateials [11]. Themophysical Basis of Enegy Technologies No Specific heat Heat conduction mateial Name of the mateial capacity c, coefficient λ, Density Fig. 1 J/kg C W/m C ρ, kg/m 1 Themocouple s junction type S Themocouple s junction type L Themocouple s junction type K Powde AlO Stainless steel sheath steel Ai Table. Duation of heating up of the TEP sensitive element, s. Two- One- Two- One- Two- Onedimensional dimensional dimensional dimensional dimensional dimensional T, K model model model model model model L type K type S type Fo the themocouple of L type at a tempeatue ove 570 K and fo the themocouple of K type at a tempeatue ove 650 K necessay duation of heating up inceases slightly because the allowed eo in the specified ange of tempeatues has not constant chaacte but depends on the taken tempeatue. The esults eceived by means of one-dimensional model consideably diffe fom the esults defined by two-dimensional model. It is caused by that the two-dimensional model consides pocess of heating not only fom vetical boundaies but also fom lowe bound. Howeve the one-dimensional model can be used fo pediction of time of heating up of the themocouples placed in the funaces o heating cameas which length consideably exceeds diamete of the themocouple. Dependences of the elative eo of measuements on duation of heating up of the themocouple of L type ae given in Fig. up to the tempeatue of 550 K in case of diffeent values of value of ai gap. The Fig. testifies that duation of heating up of the themocouple can pomote measuing accuacy incease. Value of ai gap has essential impact on a measuement eo in case of non-compliance with minimum necessay time of heating up of the themocouple. Dependences of the elative eo of measuement of diffeent tempeatues ae given in Fig. in case of value of ai gap of 1 mm between the themocouple and object of measuement. The dependences povided in Fig. testify that in the pesence of ai gap between the themocouple and object of measuement the eo on condition of satisfaction of the themocouple to all technical equiements can be educed by means of obsevance of minimum necessay duation of heating up of the themocouple. Tempeatue measuement eo depending on assignment of measuement can have a negative impact on a ow of factos. In paticula, in a usage time of a measuing signal of tempeatue in 0104-p.5

6 EPJ Web of Confeences δt, % 1 4 t, c Figue. Dependence of the elative eo of tempeatue measuement on duation of heating up of the themocouple of L type up to the tempeatue of 550 K: 1 value of ai gap of mm, value of ai gap of mm, valueofaigapof1mm,4 anallowedeo. δt, % 1 t,c Figue. Dependence of the elative eo of measuement on duation of heating up of the themocouple of L type in the conditions of ai gap of 1 mm at tempeatues: K, 550 K, 50 K. management systems as signals on which egulation is caied out can lead to eatic establishment of adjustable paamete that can cause an excessive consumption of fuel, damage of technology equipment and to othe negative consequences. 5. Conclusion By esults of numeical eseach of pocess of heat tansfe in case of themocouple heating up, it is possible to daw the following outputs: 1 the one-dimensional model can t be used fo pediction of necessay duation of heating up of the themocouple used fo tempeatue measuement of a suface of object in the conditions of heating up fom the lowe and vetical boundaies; 0104-p.6

7 Themophysical Basis of Enegy Technologies value of ai gap has essential impact on a measuement eo, thus, eo eduction in this case can achieve, obseving minimum necessay time of heating up of the themocouple; numeical modeling of pocess of heat tansfe can be used fo detemination of duation of heating up of the themocouple up to the given tempeatue in case of what the necessay duation of heating slightly changes in the field of tempeatues in which the allowed eo has no constant value, but is defined by the taken tempeatue. This wok was suppoted by the Russian Foundation fo Basic Reseach No Refeences [1] J. Sulcine, Contol Engineeing, 46, [] T. V. Boovkova, V. N. Yeliseyev, and I.I. Lopukhov, Physics of Paticles and Nuclei Lette, 5, [] G. V. Kuznetsov and K. M. Mukhammadeev, Jounal of Engineeing Themophysics, 19, [4] A. E. Segall, Intenational Jounal of Heat and Mass Tansfe, 44, [5] IEC Intenational standad. Themocouples. Pat 1: Refeence tables, 1995 [6] IEC Intenational standad. Themocouples. Pat : Toleances, 1989 [7] A.A. Samaskii, The Theoy of Diffeence Schemes Macel Dekke, Inc., USA, 001 [8] G.V. Kuznetsov G.V. and P.A. Stizhak, Intenational Jounal of Heat and Mass Tansfe, 5, [9] D.O. Glushkov, G.V. Kuznetsov, P.A. Stizhak, Russian Jounal of Physical Chemisty B., [10] D.O. Glushkov, P.A. Stizhak, Jounal of Engineeing Themophysics, [11] N.B. Vagaftik, L.P. Filipov, A.A. Tazimanov, E. E. Totskii, Handbook of Themal Conductivity of Liquids and Gases, CRC Pess, Inc., Boca Raton, p.7