Mathematical modeling for finding the thermal conductivity of solid materials

Transcription

1 Mathematcal modelng for fndng the thermal conductvty of sold materals Farhan Babu 1, Akhlesh Lodwal 1 PG Scholar, Assstant Professor Mechancal Engneerng Department Dev AhlyaVshwavdyalaya, Indore, Inda Abstract - Thermal conductvty refers to the ablty of materals to conduct heat. It plays crucal roles n the desgn of engneerng systems where temperature and thermal stress are major concerns. Typcal methods of thermal conductvty measurement can be categorzed as ether steady-state or non-steady state. The artcle deals wth transent heat conducton through the sold sphercal shape when t s mmersed nto the pool of hot water. The desgn of expermental set-up s purely based on natural convecton. So, varous dmensonless numbers Pr, Gr, Ra, Nu have been studed to fnd the natural heat transfer coeffcent (h). Durng transent heat conducton process through the materal some part of heat stores n the materal and temperatures-tme data s recorded by usng the data acquston system. Classcal lumped model s vald for lower Bot number. So, an effort has been done to mplement hgher Bot number. Based on analyss, modelng, temperature varaton wth tme and geometry of materal, soluton s obtaned to fnd the thermal conductvty of materals. Keywords: Transent heat conducton, Dmensonless numbers, Pr, Gr, Ra and Nu, Natural heat transfer coeffcent (h), Bot number (B), Fourer number (Fo) and thermal conductvty m) I. INTRODUCTION Thermal conductvty plays crucal roles n the desgn of engneerng systems where temperature and thermal stress are of concerns. It s an ntensve physcal property of a materal that relates the heat flow through the materal per unt area to temperature gradent across the materal. The thermal conductvty of a materal s bascally a measure of ts ablty to conduct heat. In a wde varety of applcatons rangng from buldng nsulaton to electroncs, t s mportant to determne the thermal conductvty of materals.extensve efforts have been made snce 1950s for the characterzaton of thermal conductvty. Typcal methods of thermal conductvty measurement can be categorzed as ether steady-state or non-steady state. In steady-state technques, equlbrum heat flux and temperature gradent are measured. Transent technques usually measure tme-dependent energy dsspaton process of a sample. Major drawbacks of ths technque nclude: (1) the testng sample should be relatvely large, n centmeter scale or even larger. () The test usually suffers from a long watng tme, up to a few hours, to reach steady state.the bggest challenge n the steady state technque s to accurately determne the heat flow through sample. However, f one has a standard materal whose thermal conductvty s known, the comparatve cut bar technque can be appled and the drect measurement of heat flow s unnecessary. The method s asked to nvestgate nvolves the transent heatng of sphercal shaped samples by the natural convecton means. The temperature of water s ncreased by supplyng heat to the water bath. The hot water rses up and cold water goes down as densty decreases wth ncrease n temperature for fluds and cycle begns to crculate the water. After mantanng the partcular temperature of the flud, a sold sphercal shape s ntally at ambent temperature, mmersed nto the hot water, untl the whole of the shape temperature rses up wth tme contnuously and t try to reaches the equlbrum/ steady state. Thus followng are the objectves of ths project work. To fnd the dmensonless numbers lke Prandtl number (Pr),) Raylegh number (Ra) and Grashof number (Gr) and natural heat transfer coeffcent (hc) by usng expermental data and correlaton of Nusselt number (Nu) for the sphercal shape. To fnd the Fourer number (Fo) and Bot number (B) by usng expermental data and Hesler chart for sphercal shape. To fnd the thermal conductvty and valdate the results of two dfferent materals (Km) by usng the analytcal technque and mathematcal modelng. To fnd the heat transfer due to conducton, convecton, radaton and total amount of heat utlze durng the varous processes. To plot thermal conductvty v/s temperature of two dfferent materals at varous temperatures. IJRTI Internatonal Journal for Research Trends and Innovaton ( 5

2 II. LITERATURE REVIEW The thermal conductvty of materals wdely used for ther applcatons n varous areas of Refrgeraton and Ar condtonng, Automobles, Industres and Engneerng etc.. In ths chapter, lteratures on thermal conductvty of sold materals and processes to fnd the thermal conductvty are dscussed n detals. The earlest work on Bot number were found. Su, G., Tan, Z., & Su, J. (007), An mproved lumped models for transent heat conducton n a slab wth temperature dependent thermal conductvty: where, they studed both coolng and heatng processes and presented solutons as a functon of the Bot number and a dmensonless parameter representng the heatng or coolng process. Prasad, D. (013). Analyss of transent heat conducton n dfferent geometres by polynomal approxmaton method: where, he shows heat transfer generally takes place by three modes such as conducton, convecton and radaton. Prakash, A., Mahmood, S., (013). Modfed lumped model for transent heat conducton sphercal shape: where, unsteady or transent heat conducton state mples a change wth tme, usually only of the temperature. In general, the flow of heat takes place n dfferent spatal coordnates. In some cases the analyss s done by consderng the varaton of temperature n one-dmenson. At a sphercal geometry to have one-dmensonal analyss a unform condton s appled to each concentrc surface whch bounds the regon. Wll, J. B., Kruyt, N. P., &Venner, C. H. (017). An expermental study of force convectve heat transfer from smooth, sold spheres: where, they use the rate of heat transfer due to natural convecton s estmated for the sphercal shape, based on a correlaton for the Nusselt number as a functon of the Raylegh number (Ra) and the Prandtl number (Pr) Where: 3 Gr {g (T -T )D }/ S 11 If, Ra 10 ; Pr 0.7, For the sphercal shape, Nusselt number s gven below, xRa Nusselt no. (Nu) = [1+{ } ] Pr and Ra = Gr Pr g s the gravtatonal acceleraton, β s the thermal expanson coeffcent at constant pressure of the water and ν s the knematc vscosty of the water and Pr s the prandtl number of water at gven temperature. III. METHODOLOGY When, the temperature of water ncreases, the hot water rses up and cold water goes down as densty decreases wth ncrease n temperature for fluds and cycle begns to crculate the water. A sold shape s ntally at ambent temperature, mmersed nto the hot water untl the whole of the shape temperature rses up wth tme contnuously and t try to reaches the equlbrum/ steady state. There s a convectve heat transfer at the surface of the shape.there s conductve heat transfer nsde of the shape due to the conducton of heat. In case of natural convecton, Nusselt number (Nu) s a functon of Grashof number (Gr) and Prandtl number (Pr). Natural convecton Nusselt number (Nu) = functon [Gr, Pr] By usng the densty, knematc, dynamc vscosty and thermal conductvty of water at partcular temperature, Prandtl number (Pr) can be found. It s a dmensonless number, named after the German physcst Ludwg Prandtl, s defned as the rato of momentum dffusvty to thermal dffusvty. Ths moton s caused by the buoyant force. The major force that ressts the moton s the vscous force. The Grashof number (Gr) s a way to quantfy the opposng forces. The Grashof number (Gr) s defned as: vscous dffuson rate c Pr andtl number (Pr) = = thermal dffuson rate k k c where : p momentum dffusvty nematc vscosty) k = thermal conductvty of water at temperature c = specfc heat of water p p IJRTI Internatonal Journal for Research Trends and Innovaton ( 6

3 Buoyancy force Grashof number = Vscous force 3 D gt Grashof number (Gr) = D = Dmeter of the shape = Densty of water g = Gravtatonal acceleraton T Temperature at the surface s T = Intal temperature = Coeffcnt of thermal expenson = Dynamc vscosty The Raylegh number (Ra) for a flud s a dmensonless number assocated wth buoyancy-drven flow, also known as free convecton or natural convecton. Ra = Gr Pr 11 If, Ra 10 ; Pr 0.7, For the sphercal shape, Nusselt number s gven, xRa Nusselt number (Nu) = [1+{ } ] Pr As, we know Nusselt number (Nu) s the rato of convectve heat transfer to conductve heat transfer across the surface boundary. Nusselt w Convectve heat transfer c c number (Nu) = Conductve heat transfer = k w kw k Natural heat transfer coeffcent (h c) = Nu w D Where : h D h D k s thermal conductvty of water at partcular temperature D s the dameter of the shape t The rate at whch heat s conducted across thelength of the body Fourer number (Fo) = = D The rate at whch heat s stored n the body s thermal dffusvty of materal D s dameter of the body = k c p k s conductvty of the body s densty of the body Intal temperature of the shape T, Temperature of 0 the hot water (T ) and Temperature of the shape at step change T Dmensonless temperature or T0 T Temperature profle 0 = T T Hesler chart for the sphercal shape IJRTI Internatonal Journal for Research Trends and Innovaton ( 7

4 Fgure 3.1: Hesler chart for the sphercal shape The heat flow experences two resstances, the frst wthn the sold metal (whch s nfluenced by both the sze and composton of the sphere), and the second at the surface of the sphere. The conductve component s measured under the same condtons as the heat convecton. Fgure 3.: Schematc dagram of constant temperature water bath wth sold sphercal shape Conductve heat transfer resstance wthn the body Bot number (B ) = Convectve heat transfer resstance across the surface of the body = Bot number (B ) = Where: c c Lc km hl c c 1 km h c hl c c km h s heat transfer coeffcent So, Thermal conductvty of the materal k h L s the characterstc length of the shape k s the thermal conductvty of the shape of the materal m B m c Lc IV. EXPERIMENTAL SET-UP DEVELOPMENT The objectves of the present study mentoned n the prevous chapter, are manly attaned by specfcally desgned expermental set-up to perform temperature measurements durng transent heat conducton. The current chapter provdes the detals of expermental set-up, technque used for the temperature measurements n the present study. Ths chapter s organzed as per the followng subsectons: () Expermental set-up () Technque for the temperature measurements. Expermental set up used for ths work s shown fgure (Fgure 3.1). The expermental set up comprses of a constant temperature water bath wth nbult heater, a temperature measurng unt, data loggng software and a Computer. IJRTI Internatonal Journal for Research Trends and Innovaton ( 8

5 Fgure 4.1: Expermental set-up Sold sphercal shapes Two dentcal sold sphercal shapes of dfferent materals (Stanless steel and Brass) have been used n ths experment for ther analyss. Each shape has been ftted wth a thermocouple (electrcal temperature sensor) that s located at the precse centre of the sphere. Fgure 4.: Sphercal shape of Steel and Brass materals EXPERIMENTAL PROCEDURE After takng measurement of ntal teperature of the shape, the sold shape s mmersed n a water bath and measure the temperatures durng transent heat conducton and t wll also be rocorded by usng data acquston system. Before, mmersng the sold shape nto the water bath, ts temperature has to be establshed at desred temperature. After, the bath emperature s stablzed. The sold shape s mmersed nto the water bath. Frst, look the sold to fsh-eye on the frame. Insure, about data recordng. Then, lower the sold nto the water bath untl completely submerged. Don t drop the sold n the bath, but try to get t mmersed quckely. Tme s noted down at the nstant, the sold s mmersed. When, the sold has reached the temperature (step change) lower than the bath temperature and t tres to acheve steady state. Remove the sold and hang t back over the plastc tub so that t can be cool to room temperature. Make sure that the data are saved on the hard dsk for the backup. Repeat the same procedre to the other materal. V. EXPERIMENTAL DATA IMPLEMENTATION AND ANALYSIS Table 5.1 Propertes of water at 60, 70 and 80 C IJRTI Internatonal Journal for Research Trends and Innovaton ( 9

6 Table 5. Expermental Data Steel materal Analyss : STEEL MATERIAL (Dameter 45 mm) " Case 1" Materal Steel Shape - Sphercal Intal shape temperature (T ) = 7C Water bath temperature (T )= 6C Measured temperature (T )= 60C at tme (t) = 180 seconds Pr opertes of water at 60C Densty ( ) = kg/m Knematc vs cos ty ( ) = 0.478x10 m / s Dynamc vscosty ( ) = 471x10 f -6 m / Thermal conductvty ) = W/mC Prandtl number (Pr) = 3.0 Volumetrc expenson of the materal 1 1 = T ( T T ) / Usng correlaton for free convecton 3 D gt Grashof number (Gr) = Ra = Gr.Pr 11 If, Ra 10 ; Pr 0.7 By calculaton; Gr =.97x10 Pr 3.0 So, Ra = 8.97x xRa Nusselt no. (Nu) = [1+{ } ] Pr and Nusselt no. (Nu) = 135 s Natural heat transfer coeffcent (h) = Nu k w = 135 * W...(1) -3 D 45x10 m C Bath Temp.6 C Steel T = 7C IJRTI Internatonal Journal for Research Trends and Innovaton ( 10

7 T T 0 Dmensonless temperature ( 0) = T T Fourer number (F o)= t 3.3 D From Hesler chart Bot number (B )= 0.7 hxl Bot number (B) = k m c Thermal conductvty of materal L c m) = B h From equaton...(1) x10 m) = h 0.077xh...() 0.7 W m) = m C Fgure 5.1: Transent heat conducton behavor of steel materal at 60 C Table 5.3 Propertes of water at 60, 70 and 80 C IJRTI Internatonal Journal for Research Trends and Innovaton ( 11

8 Table 5.4 Expermental data Brass materal Analyss : BRASS (Dameter 45 mm) Case 1 Materal Brass Shape - Sphercal Intal shape temperature (T ) = 7C Water bath temperature (T ) = 6C Measured temperature ( T ) = 60C 0 at tme (t) = 130 seconds Bath Temp. 6C Brass T = 7C Pr opertes of water at 60C Densty ( ) = kg/m Knematc vs Dynamc f 3-6 cos ty ( ) = 0.478x10 m / s -6 vscosty ( ) = 471x10 m / Thermal conductvty ) = W/mC Prandtl number (Pr) = 3.0 Volumetrc expenson of the materal 1 1 = T ( T T ) / s Usng correlaton for free convecton 3 D gt Grashof number (Gr) = Ra = Gr.Pr If, Ra ; Pr 0.7 By calculaton: Gr =.97x10 Pr 3.0 So, Ra = 8.97x xRa Nusselt no. (Nu) = [1+{ } ] Pr and Nusselt no. (Nu) = 135 Nu 135 W Heat transfer coeffcent (h) = k w = * (1) -3 D 45x10 m C IJRTI Internatonal Journal for Research Trends and Innovaton ( 1

9 0 Dmensonless temperature ( 0) = T T Fourer number (F )= o From Hesler chart Bot number (B ) = hxl Bot number (B) = k m c T T t 4.0, tme t = 0 sec. D Lc Thermal conductvty of materal m) = h B x10 m) = h 0.04 xh... () W m) = m C Fgure 5.: Transent heat conducton behavor of steel materal at 60 C VI. EXPERIMENAL RESULTS AND DISCUSSION Table 6.1 Heat transfer due to conducton, convecton and radaton of Steel and Brass materal Steel materal Brass materal IJRTI Internatonal Journal for Research Trends and Innovaton ( 13

10 Table 6. Natural heat transfer coeffcent, thermal conductvty m) of Steel and Brass materals at 60, 70 and 80 C Plots transent heat conducton behavor of Steel and Brass materals at 60 C, 70 C and 80 C Case-1( at 60 C) Case-1( at 70 C) Case-1( at 80 C) Fgure 6.1: Transent heat conducton behavor of Steel and Brass materals at 60 C, 70 C and 80 C Plots heat transfer v/s temperature for Steel and Brass materal Steel materal Brass materal Steel and Brass materal Fgure 6.: Heat transfer due to conducton, convecton and radaton n Steel and Brass materals at 60 C, 70 C and 80 C IJRTI Internatonal Journal for Research Trends and Innovaton ( 14

11 Plots thermal conductvty v/s temperature for Steel and Brass materals Steel materal Brass materal Steel and Brass materal VII. RESULTS &CONCLUSION Fgure 6.3: Thermal conductvty of Steel and Brass materals at 60 C, 70 C and 80 C A sphercal shape s used whch thermal conductvty s to be measured. A K type of thermocouple s used due to envronmental consderaton, low prce, easly avalablty and accurate measurement of temperature. The shape s mmersed nto the constant temperature water bath, mantaned at partcular temperature. Bead of the thermocouple s located at the centre of the sphere for ts temperature measurement durng transent heat conducton. Data s recorded by usng data acquston system. Subsequently, few dmensonless numbers lke Prandtl (Pr), Grashof (Gr), Raylegh (Ra), Nusselt (Nu) have been found by usng the propertes of water at partcular temperature. Based on these dmensonless numbers, natural heat transfer coeffcent (h) has W been found n the range of 1700 to 64. Due to transent heat conducton through the shape, some part of heat s also m C absorbed. So, ths has been found that Fourer number (Fo) ncreases wth the bath temperature. It s n the range of 3.30 to 4.00 for steel materal and 7.10 to 8.49 for brass materal. The values of Bot numbers (B) are as 0.7 to 0.35 for steel and to for brass materal. Thermal conductvty of both Steel and Brass materals have almost been found constant durng temperature range 55 C to 90 C W and ther values are 5 to 54 and 110 to 113 for Steel and Brass respectvely. REFERENCES m C [1] G. Su, Z. Tan, & J. Su, Improved lumped models for transent heat conducton n a slab wth temperature dependent thermal conductvty, Appled Mathematcal Modelng 33 (009) [] A. Prakash, S. Mohammad, Modfed lumped model for transent heat conducton n sphercal shape, Amercan Internatonal Journal of Research n Scence, Technology, Engneerng and Mathematcs [3] D. Prasad, Analyss of transent heat conducton n dfferent geometres by polynomal approxmaton method, Internatonal Journal of Mechancal Engneerng and Robotcs Research [4] N. Haque, A. Paul, Study of mproved lumped parameter n transent heat conducton, Internatonal Journal of Engneerng Scence and Management ISSN: [5] A. Wllam, J. Edward, Equatons of propertes as a functon of temperature for seven fluds, Computer Methods n Appled Mechancs and Engneerng 61 (1987) North-Holland. [6] B. Xu, P. L, C. Chan, Extendng the valdty of lumped capactance method for large Bot number n thermal storage applcaton, Tucson, AZ 8571, USA. [7] Keshavraz P. and Taher M. (007), An Improved lumped analyss for transent heat conducton by usng the polynomal approxmaton method, Heat and MassTransfer, Vol. 43, pp IJRTI Internatonal Journal for Research Trends and Innovaton ( 15