Sub Module 2.6. Measurement of transient temperature

Transcription

1 Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure, pressure and flow rae vary wih ime. These are referred o as ransiens and he measuremen of hese ransiens is an imporan issue while designing or choosing he proper measuremen echnique and he probe. Here we look a he measuremen of emperaure ransiens. Temperaure sensor as a firs order sysem - Elecrical analogy Le us look a a ypical emperaure measuremen siuaion. We visualize he emperaure probe as a sysem ha is subjec o he emperaure ransien. The probe is exposed o he environmen whose emperaure changes wih ime and i is desired o follow he emperaure change as closely as possible. In Figure 44 we show he schemaic of he hermal model appropriae for his sudy. Probe a uniform emperaure T(), Characerisic lengh L ch. Fluid sream a T () C and h W/m 2 C Figure 44 Schemaic of a emperaure probe placed in a flowing medium The model assumes ha he probe is a a uniform emperaure wihin i a any ime. This means ha he probe is considered o be hermally lumped. The Indian Insiue of Technology Madras

2 Mechanical Measuremens Prof. S.P.Venkaeshan medium ha flows over he probe is a a emperaure ha may vary wih respec o ime. Iniially he probe is assumed o be a emperaure T. Le us assume ha he probe is characerized by he following physical parameers: Densiy of he probe maerial = ρ kg/m 3, Volume of he probe = V m 3, Surface area of he probe ha is exposed o he flowing fluid = S m 2, Specific hea of he probe maerial = C J/kg C, Hea ransfer coefficien for hea ransfer beween he probe and he surrounding medium = h W/m 2 C. By conservaion of energy, we have Rae of change of rae of hea ransfer beween = in ernal energy of probe he probe and he fluid (3) If we assume ha he probe is a a higher emperaure as compared o he fluid hea ransfer will be from he probe o he fluid and he inernal energy of he probe will reduce wih ime. Using he properies of he probe inroduced dt above, he lef hand side of Equaion 3 is given by ρvc. The righ hand d side of Equaion 3 is given by hs(t T ). Wih hese, afer some rearrangemen, Equaion 3 akes he form dt hs hs + T = T d ρvc ρvc (31) Noe ha his equaion holds even when he probe emperaure is lower han ρvc he fluid emperaure. The quaniy has he uni of ime and is referred o hs as he ime consan of he firs order sysem (firs order since he governing differenial equaion is a firs order ordinary differenial equaion). The firs order ime consan involves hermal and geomeric properies. The volume o surface area raio is a characerisic lengh dimension and is indicaed as L ch in Figure 44. Noing ha he produc of densiy and volume is he mass M of MC he probe, he firs order ime consan may also be wrien as =. The hs ime consan may be inerpreed in a differen way also, using elecrical Indian Insiue of Technology Madras

3 Mechanical Measuremens Prof. S.P.Venkaeshan analogy. The quaniy MC represens he hermal capaciy and he quaniy 1 represens he hermal resisance. Based on his inerpreaion an elecric hs analog may be made as shown in Figure 45. R V in C V ou Figure 45 Elecrical analog of a firs order hermal sysem In he elecric circui shown in Figure 45 he inpu volage represens he emperaure of he fluid, he oupu volage represens he emperaure of he probe, he resisance R represens he hermal resisance and he capaciance C represens he hermal mass (mass specific hea produc) of he probe. Equaion 31 may be rewrien as T dt T + = (32) d Noe ha Equaion 32 may be simplified using he inegraing facor e o wrie i as d d Te = T e This may be inegraed o ge Te (33) = T e d + A where A is a consan of inegraion. Using he iniial condiion T ( = ) = T we ge, afer minor simplificaion e T = Te + e T d (34) Indian Insiue of Technology Madras

4 Mechanical Measuremens Prof. S.P.Venkaeshan This is he general soluion o he problem. If he variaion of fluid emperaure wih ime is given, we may perform he indicaed inegraion o obain he response of he probe as a funcion of ime. Response o sep inpu If he fluid emperaure is consan bu differen from he iniial emperaure of he probe, he soluion is easily shown o be represened by T T T T = φ = e (35) The emperaure difference beween he probe and he fluid exponenially decreases wih ime. The variaion is indicaed in Figure 46. A he end of one ime consan he emperaure difference is some 37% of he iniial emperaure difference. Afer abou 5 ime consans he emperaure difference is quie negligible. φ / Figure 46 Response of a firs order sysem o a sep inpu A sep inpu may be experimenally realized by heaing he probe o an iniial emperaure in excess of he fluid emperaure and hen exposing i quickly o he fluid environmen. The probe emperaure is recorded as a funcion of ime. If i is ploed in he form ln(φ ) as a funcion of, he slope of he line is Indian Insiue of Technology Madras

5 Mechanical Measuremens Prof. S.P.Venkaeshan negaive reciprocal of he ime consan. In fac, his is one mehod of measuring he ime consan. Example 15 shows how his is done. Example 15 A emperaure probe was heaed by immersing i in boiling waer and is hen quickly ransferred in o a fluid medium a a emperaure of T amb = 25 C. The emperaure difference beween he probe and he medium in which i is immersed is recorded as given below: (s) T-T amb Wha is he ime consan of he probe in his siuaion? o The daa is ploed on a semi-log graph as shown here. I is seen ha i is well represened by a sraigh line whose equaion is given as an inse in he plo. EXCEL was used o obain he bes fi. Semi-log plo ln(t - Tamb) Linear (ln(t - Tamb)) ln(t-t amb ) ln(t-t amb ) = R 2 = ,s o The slope of he line is and hence he ime consan is Indian Insiue of Technology Madras

6 Mechanical Measuremens Prof. S.P.Venkaeshan 1 1 = = = s slope.665 o The correlaion coefficien of he linear fi is This shows ha he daa has been colleced carefully. Indian Insiue of Technology Madras

7 Mechanical Measuremens Prof. S.P.Venkaeshan A noe on ime consan I is clear from our discussion above ha he ime consan of a sysem (in his case he emperaure probe) is no s propery of he sysem. I depends on parameers ha relae o he sysem as well as he parameers ha define he ineracion beween he sysem and he surrounding medium (whose emperaure we are rying o measure, as i changes wih ime). The ime consan is he raio of hermal mass of he sysem o he conducance (reciprocal of he hermal resisance) beween he sysem and he medium. I is also clear now how we can manipulae he ime consan. Thermal mass reducion is one possibiliy. The oher possibiliy is he reducion of he hermal resisance. This may be achieved by increasing he inerface area beween he sysem and he medium. In general his means a reducion in he characerisic dimension L ch of he sysem. A hermocouple aached o a hin foil will accomplish his. The characerisic dimension is equal o half he foil hickness, if hea ransfer akes place from boh sides of he foil. Anoher way of accomplishing his is o use very hin hermocouple wires so ha he bead a he juncion has very small volume and hence he hermal mass. Indeed hese are he mehods used in pracice and hin film sensors are commercially available. Response o a ramp inpu In applicaions involving maerial characerizaion heaing rae is conrolled o follow a predeermined program heaing. The measuremen of he corresponding emperaure is o be made so ha he emperaure sensor follows he emperaure very closely. Consider he case of linear heaing and possibly linear emperaure rise of a medium. Imagine an oven being urned on wih a consan amoun of elecrical hea inpu. We would like o measure he emperaure of he oven given by ( ) = T R (36) T + The general soluion o he problem is given by (using Equaion 34) Indian Insiue of Technology Madras

8 Mechanical Measuremens Prof. S.P.Venkaeshan T e T R = A + e d e d + (37) where A is a consan of inegraion. The firs inegral on he righ hand side is easily obained as ( e 1). Second inegral on he righ hand side is obained by inegraion by pars, as follows. ( e 1) 2 e d = e e d = e (38) If he iniial emperaure of he firs order sysem is T i, hen A = Ti, since boh he inegrals vanish for = (he lower and upper limi will be he same). On rearrangemen, he soluion is () = ( Ti T + R ) e + ( T + R R ) (39) T We noice ha as he ransien par ends o zero (ransien par is he exponenial decaying par) and he seady par (his par survives for >> ) yields T o + R T() = R. (4) The seady sae response has a lag equal o R wih respec o he inpu. Indian Insiue of Technology Madras

9 Mechanical Measuremens Prof. S.P.Venkaeshan Temp eraure T, o C Inpu Response Time, s Figure 47 Typical response of a firs order sysem o ramp inpu Figure 47 shows he response of firs order sysem o a ramp inpu. The case shown corresponds o Ti= 2 C, T = 35 C, R =.15 C/s and = 1 s. For ( > 5 = 5 s) he probe follows he linear emperaure rise wih a lag of R =.15 1 = 1.5 C. In his case i is advisable o rea his as a sysemaic error and add i o he indicaed emperaure o ge he correc oven emperaure. Response o a periodic inpu There are many applicaions ha involve periodic variaions in emperaure. For example, he walls of an inernal combusion engine cylinder are exposed o periodic heaing and hence will show periodic emperaure variaion. Of course, he waveform represening he periodic emperaure variaion may be of a complex shape (non sinusoidal). In ha case he waveform may be spli up in o is Fourier componens. The response of he probe can also be sudied as ha due o a ypical Fourier componen and combine such responses o ge he acual response. Hence we look a a periodic sinusoidal inpu given by = T cos( ) (41) T a ω In he above expression T a is he ampliude of he inpu wave and is he circular frequency. We may use he general soluion given by Equaion 34 Indian Insiue of Technology Madras

10 Mechanical Measuremens Prof. S.P.Venkaeshan and perform he indicaed inegraion o ge he response of he probe. The seps are lef as exercise o he suden. Finally he response is given by 1 T e Ta cos( ω an ( ω) ) + (42) T = Toe a 2 2 ( 1 + ω ) Transien response ω Seady sae response Again for large, he ransien erms drop off and he seady sae response survives. There is a reducion in he ampliude of he response and also a ime lag wih respec o he inpu wave. Ampliude reducion and he ime lag (or phase lag) depend on he produc of he circular frequency and he ime consan. The variaions are as shown in Figure Ampliude reducion facor Phase lag/ (9 degree) Circular frequency ime consan produc, ω Figure 48 Response of a firs order sysem o periodic inpu In order o bring ou he feaures of he response of he probe, we make a plo (Figure 49) ha shows boh he inpu and oupu responses, for a ypical case. Indian Insiue of Technology Madras

11 Mechanical Measuremens Prof. S.P.Venkaeshan 1.8 Temperaure raio T/T Oupu response Time lag -.2 Inpu Time, s Figure 49 Response of a firs order sysem o periodic inpu T The case shown in Figure 33 corresponds o T a o =.25; ω = 1rad s and = 1 s. The oupu response has an iniial ransien ha adjuss he iniial mismach beween he probe emperaure and he imposed emperaure. By abou 4 o 5 ime consans (4 o 5 seconds since he ime consan has been aken as 1 second) he probe response has seled down o a response ha follows he inpu bu wih a ime lag and an ampliude reducion as is clear from Figure 33. Indian Insiue of Technology Madras

12 Mechanical Measuremens Prof. S.P.Venkaeshan Example 16 The ime consan of a firs order hermal sysem is given as.55 min. The uncerainy in he value of he ime consan is given o be ±.1 min. The iniial emperaure excess of he sysem over and above he ambien emperaure is 45 o C. I is desired o deermine he sysem emperaure excess and is uncerainy a he end of 5 s from he sar. Hin: I is known ha he emperaure excess follows he formula () ( ) T T = e where T() is he emperaure excess a any ime, T() is he emperaure excess a = and is he ime consan. o We shall conver all imes given o s so ha hings are consisen. The ime consan is =.55 min =.55 6 s = 33 s o We need he emperaure excess a ( ) ( ) = T 5 = T e = 45e = 9.89 C s from he sar. Hence o We would like o calculae he uncerainy in his value. We shall assume ha his is due o he error in he ime consan alone. Δ=±.1 min =±.1 6 =±.6 s o The influence coefficien I is given by = T 5 I = = 45 e =.454 C / s 2 33 Hence he uncerainy in he esimaed emperaure excess is: ( ) Δ T 5 =± I Δ=± =.272 C Indian Insiue of Technology Madras

13 Mechanical Measuremens Prof. S.P.Venkaeshan Example 17 o A cerain firs order sysem has he following specificaions: Maerial: copper shell of wall hickness 1 mm, ouer radius 6 mm Fluid: Air a 3 C Iniial emperaure of shell: 5 C o How long should one wai for he emperaure of he shell o reach 4 C? Assume ha hea ransfer is by free convecion. Use suiable correlaion (from a hea ransfer ex) o solve he problem. o Hea ransfer coefficien calculaion: Hea ransfer beween he shell and he air is by naural convecion. The 1 / 4 appropriae correlaion for he Nussel number is given by Nu = Ra where Ra is he Rayleigh number. The characerisic lengh scale is he sphere diameer. The air properies are calculaed a he mean emperaure a =. From he given daa, we have D = 12 mm =.12 m, T m = (5 + 3) / 2 = 4 C The air properies required are read off a able of properies: 6 2 ν= m / s, Pr =.71, k =.27 W / m C The isobaric compressibiliy of air is calculaed based on ideal gas assumpion. Thus 1 1 β= = = K T amb 3 1 The emperaure difference for calculaing he Rayleigh number is aken as he mean shell emperaure during he cooling process minus he ambien emperaure. We are ineresed in deermining he ime o cool from 5 o C. Hence he mean shell emperaure is T Shell = = 45 C. The 2 emperaure difference is Δ T= TShell Tamb = 45 3= 15 C. The value of he Indian Insiue of Technology Madras

14 Mechanical Measuremens Prof. S.P.Venkaeshan acceleraion due o graviy is aken as 2 g = 9.8 m/s. The Rayleigh number is hen calculaed as gβδ TD Ra = Pr =.71 = ν 6 ( ) o The Nussel number is hen calculaed as 1/4 1/4 Nu = Ra = = 4.9 The hea ransfer coefficien is hen calculaed as o Time consan calculaion: Nu k h = = = 11.7 W / m K D.12 Copper shell properies are Copper shell hickness is 3 ρ= 8954 kg / m, C = J / kg C δ =.1 m Mass of he copper shell is calculaed as M =ρπd δ= 8954 π.12.1 = kg Surface area of shell exposed o he fluid is S =π D =π.12 = m The ime consan is hen esimaed as 3 MC = = = 31 s hs o Cooling follows an exponenial process. Hence we have, he ime 4 a which he shell emperaure is 4 C, = 31 ln = s 5 3 Indian Insiue of Technology Madras