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71 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications Parametric Studies of Top Heat Loss Coefficient of Double Glazed Flat Plate Solar Collectors Anand Bisen P.P. Dass Department of Mechanical Engineering, Govt. Kalaniketan Polytechnic College, Jabalpur (M.P.), India Department of Mechanical Engineering, Jabalpur Engineering College, Jabalpur (M.P.), India Rajeev Jain Department of Mechanical Engineering, Govt. Kalaniketan Polytechnic College, Jabalpur (M.P.), India Abstract: The performance of any solar collector is largely affected by various parameters. Estimation of various heat losses in the flat plate solar collector is important for their thermal performance evaluation and upward heat losses have a major contribution in the total heat losses in flat plate collectors. The top heat loss coefficient is a function of various parameters which includes the temperature of the absorber plate Tp, ambient temperature Ta and glass cover temperatures Tg1 and T g2, wind heat transfer coefficient hw, emissivity of absorber plate Îp and glass cover Îg, spacing between the absorber plate and glass cover L1, L2, tilt angle of the collector b, number of glass covers N and thickness of glass covers Lg1, Lg2.The present work aims at the studies of effect of estimation of air properties by two sets of correlation on the estimation of top heat loss coefficient of double glazed flat plate solar collector by estimating glass cover temperatures by using numerical solution technique. This comparison of air properties effects on top heat loss coefficient (Ut) has been studied for the given range of absorber plate temperature (Tp), Ambient temperature (T a) and wind heat transfer coefficient (hw) using MATLAB. Keywords: flat plate collector, top heat loss coefficient, absorber plate temperature, ambient temperature and wind heat transfer coefficient. (Fig. 1) is a special kind of heat exchanger which converts solar radiation energy to heat energy by using usually air, water or oil. Flat-plate collectors can be designed for applications requiring moderate temperature (T < 100 0C). They are relatively inexpensive, easily constructed, require little maintenance and use both beam and diffuse radiation of the sun. They are well suited to the drying of agricultural crops. The major applications of these units are in solar water heating, building heating, air conditioning, and industrial process heat. A review of correlations quantifying heat transfer has been carried out by Holland et.al (1976). They described equation to calculate the Nusselt number for natural convection between parallel plates. Correlations suggested to calculate Nusselt number (Nu) are Nu If È 1 1.44 É1 È Ra Ø É Ù 1, Ê 5830 Ú Ê then, N u 1708 Ø R a ÙÚ ËÈ R a Ø Û ÌÉ Ù 1Ü Ê Ú Í 5830 Ý È 1708 Ø Ù Ê Ra Ú 1 1.44 É Buchberg et.al (1976) have given correlation to calculate the Nusselt number for natural convection between parallel Solar energy is the most clean energy source in the world. plates. Buchberg et al., (4) has suggested correlations to The sun emits approximately 1353 (W/m2) on to a surface of calculate Nusselt number (Nu) are, earth if there is no atmospheric layer.(Thekaekara and If R < 1708, then, N = 1 a u Drummond 1971).The world receives 170 trillion (kW) solar If (Ra < 1708) & (R a1 < 5900), then, energy and 30% of this energy is reflected back to the space, 1708 Ø È 47% is transformed to low temperature heat energy, 23% is N u 1 1.44 É1 R a ÙÚ used for evaporation/rainfall cycle in the Biosphere and less Ê than 0.5% is used in the kinetic energy of the wind, waves If (R < 5900) & (R < 9.23 × 10 4), then, a a and photosynthesis of plants (Duffie and Beckman 1991, Kalogirou 2004).Thus Solar Energy option has been identified N u 0.229 R 0.252 a as one of the promising alternative energy source for If (R < 9.23 × 10 4) & (R < 10 6), then, a a future. Various devices (i.e. Solar Collector) are used for N u K R a0.285 thermal collection and storage of this energy. A solar collector I. INTRODUCTION 72 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications A Malhotra et.al. (1981) have suggested that Nusselt number correlations of a single gap are extended to a two cover system. It is found by using two covers there is an overall saving of more then 50% in convective losses yielding better result. N. Akhtar (1999) reported that for evaluating the top heat loss factor it is adequate to calculate the individual heat transfer coefficients based Figure 1: Cross section of basic flat plate solar collector on mean temperature of glass cover rather than two separate relations for inner and outer surfaces. He has suggested correlations for calculating top heat loss coefficient of corresponding enclosed space, three correlations are, Upward heat loss from the absorber plate to the first glass cover is given by, Qt = (hcpg1+ hrpg1) (TP Tg1) From the first glass cover to the second glass cover by, Qt = (hcg1g2 + hrg1g2) (Tg1 Tg2) and from the second glass cover to the atmosphere by , Qt = (hw + hrg2a) (Tg2 Ta) T.P. Woodman (1977) has provided the basic equations for calculating collector performance as a function of design and operating parameters. This equation is used to investigate the sensitivity of collector output to ambient air temperature, absorber plate temperature and heat transfer coefficient of wind. He suggested that use of two cover plates instead of one is shown to be nearly always beneficial. heat balance equations.An additional advantage of the proposed technique over the semi-empirical equations is that results can be obtained for different values of sky temperature, using any given correlation for convective heat transfer in the air gap spacing, and for any given values of fluid (air in the present case) properties. Sekhar et. al. (2009) have found that the emissivity of the absorber plate has a significant impact on the top loss coefficient and consequently on the efficiency of the Flat plate collector. The efficiency of FPC is found to increase with increasing ambient temperature. There is no significant impact of tilt angle on the top loss coefficient. Need of the present work: To find effect of estimation of air properties by two sets of correlations for the estimation of top heat loss coefficient of double glazed flat plate solar collector by estimating glass cover temperatures. Nusselt number (Nu) can be calculated by the correlations suggested by Holland et al. [6] and Buchhberg et al. [4] to estimate the convective heat transfer coefficient This is required to show the variations in results obtained by the above said correlations. Objective of Present study: The present work aims at the studies of effect of estimation of air properties by two set on the estimation of top heat loss coefficient of double glazed flat plate solar collector by estimating glass cover temperatures by using numerical solution technique. This comparison of thermo-physical air properties effects on top heat loss coefficient (U t) has been studied for the given range of absorber plate temperature (Tp), Ambient temperature (T a) and wind heat transfer coefficient (h w) using MATLAB. The range of variables for which the proposed equations have been evaluated in given Table 1. Table 1: Range of Variables Variables Range Ambient Temperature (Ta) 288K to 318K Absorber Plate temperature (Tp) Heat Transfer coefficient of wind (hw) 373K to 423K 5w/m2-K to 25 w/m2-K Emissivity of Absorber Plate (T p) 0.9 Kutscher, et al., (1993) have showed that the radiation heat loss is determined by considering losses to both the sky and 0.88 Emissivity of glass (Tg) the ground and Convective heat losses are obtained by Thickness of first glass cover (L g1) 0.004 m integrating the product of the temperature and velocity profiles ) 0.004 m Thickness of second glass cover (L g2 in the boundary layer at the downwind edge of the collector. Space between Absorber plate & This convective heat loss is then expressed in terms of the First glass cover (L1) 0.098 m thermal equivalent length of irradiated absorber, and analysis shows that this loss can be very low for large collectors even Space between first and second under windy conditions. Mullick and Samdarshi (1994) have 0.012 m glass cover (L2) presented a generalized analytical equation for the top heat loss factor of a flat-plate collector with one or more glass covers. The maximum computational errors resulting from II. METHODOLOGY the use of the analytical equation with several simplifications Estimation of air properties by two sets of correlation on are ±5 percent compared to numerical solution of the set of the estimation of top heat loss coefficient of double glazed 73 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications flat-plate solar collector by estimating glass cover temperatures Q t (h cpg1 h rpg1 ) (Tp Tg1) (1) by using numerical solution technique. This comparison of air properties effects on top heat loss coefficient (U t) has been and from the first glass cover to the second glass cover by studied for the given range of absorber plate temperature (T p), Q t (h cg1g2 h rg1g2 ) (Tg1 Tg2) (2) Ambient temperature (Ta) and wind heat transfer coefficient and from the second glass cover to the atmosphere by (hw) using MATLAB. Q t (h w h rg2a ) (Tg2 Ta ) (3) Energy balance equations of flat-plate collectors In steady state, the performance of a flat-plate solar collector can be described by the useful gain from the collector, Qu, which is defined as, Qu=Ac[S-UL(Tp-Ta)]+ where Ac is gross area of the collector, the first term is the absorbed solar energy and second term represents the heat loss from the collector, TP and Ta or Plate temperature and ambient temperature. Heat loss from the collector In solar collectors, the solar energy absorbed by the absorber plate is distributed to useful gain and to thermal losses through the top, bottom, and edges. Top heat loss through the cover system To evaluate the heat loss through the cover systems, all of the convection and radiation heat transfer mechanisms between parallel plates and between the plate and the sky must be considered as shown in Figures 1, 2 and 3. In this study the net radiation method is applied to obtain the expression for the heat loss for the general cover system of flat-plate solar collectors. Figure 3: Thermal network for a two cover flat plate collector (a) in terms of conduction, convection and radiation (b) in terms of resistances between plates. Where, h cpg1 h rpg1 h cg1g2 Figure 2: Heat transfer mechanisms through a cover system with two glass cover Calculation of top heat loss coefficient The top loss co-efficient is evaluated by considering convection and re-radiation losses from the absorber plate in the upward direction. Upward heat loss from the absorber plate to the first glass cover h cg1g2 h rg2a2 K1 N u1 L1 (4) V Î Þ 2 2 Ñ 1 1 Ñ (Tp Tg1 ) (Tp Tg1 ) Ï 1ß ÑÐ °p °g Ñà K 2 Nu 2 L2 V È 2 1Ø ÉÊ °g ÙÚ V · °g (5) (6) 2 (Tg1 2 ) (Tg1 Tg2 ) Tg2 (Tg2 4 Tsky 4 ) Tg2 Ta Measurable input variables in a solar collector. TP = Plate temperature Ta = Ambient temperature (7) (8) 74 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications Îg = Emissivity of the glass Îp = Emissivity of absorber plate Buchberg (4) equation to calculate Nusselt number (N u1) (19) If Ra1 < 1708, then N u1 = 1 L2 = Spacing between absorber plate and glazing L1 = Spacing between first and second glazing If (Ra1 > 1708) & (R a1 < 5900), then È N u1 Lg = Thickness of glass covers k = Thermal conductivity of air 1 1.446 É1 Ê 1708 Ø R a1 ÙÚ (20) If (Ra1 > 5900) & (R a1 < 9.23 × 10 4), then, Using numerical solution method programming in MATLAB N u1 0.229 R 0.252 has been done. The properties of air with mean temperature as a1 independent variable is evaluated. 4 6 If (Ra1 < 9.23 × 10 ) & (Ra1 < 10 ), then, Air properties (21) 0.285 K1 R a1 (22) For accurate prediction of collector performance it is Arithmetic mean temperature of space between first glass necessary to evaluate properties of the working fluid (air) to cover and second glass cover, calculate the convective heat transfer coefficient. Properties Tg1 Tg2 of air evaluated at the arithmetic mean of the corresponding Tm (23) surface temperatures 2 N u1 Tp Tg1 Tm (9) 2 Following correlations for air properties are used to calculate Raleigh number (Ra) Air properties correlation-I [7] K1 3 108 Tm2 104 Tm 4 105 (10) V1 [9 105 Tm 2 0.040 Tm 4.17] 106 (11) Pr1 1.057 0.06 log Tm (12) Air properties correlation- II [10,11] K1 = 0.0002067 Tm0.85 V1 = 9(10 ) Tm Pr1 = 1.0602 0.602 log Tm 10 (13) (14) (15) 1.72 Raleigh number (Ra) of enclosed space can be calculated by using air properties of corresponding enclosed space R a1 9.8 (TP Tg1 ) (L1 )3 (Pr ) V12 È TP Tg Ø ÉÊ Ù 2 Ú (16) N u1 1 if È R a1 Ø 3 É Ù 1, Ê 5830 Ú then N u1 È 1708 Ø Ù Ê R a1 Ú 1 1.44 É 9.8 (Tg1 Tg2 ) (L 2 )3 (Pr 2 ) (24) (Tg1 Tg2 ) 2 V2 2 The natural convection heat transfer coefficient for the enclosed space between the first glass cover and second glass cover is calculated by correlations suggested by Holland et.al. (6) and Buchberg et.al.(4) as discussed earlier (Eqn. 17 to 22).The convective heat transfer coefficients (h cpg1, hcg1g2,) and Radiative heat transfer coefficients (h rpg1, hrg1g2 and hrg2a) can be evaluated by equations discussed earlier. (Eqn. 4 to 8). Ra2 Calculation of top heat loss coefficient UPg1, Ug1g2, and Ug2a are given by correlations [5] Between Plate and First Glass Cover (U pg1) U pg1 (17) (18) (h cpg1 h rpg1) (25) Between first glass cover and second glass cover (U g1g2) U g1g2 A review of correlations quantifying heat transfer has been carried out by Holland. et.al. (6) and Buchberg et.al.(4). They recommended use of the following correlations to calculate Nusselt number (Nu) For (Nu1) Space between the absorber plate and first glass cover is calculated by Holland et.al.(6) Equation to calculate Nusselt number (Nu1) 1 Ë Û 1708 Ø ÌÈ R a1 Ø 3 È Ü É 1 1.44 É1 Ù 1Ü R a1 ÙÚ ÌÍÊ 5830 Ú Ê Ý Raleigh number (Ra2) of enclosed space can be calculated by using air properties of corresponding enclosed space (source: Duffie and Beckman) (h cg1g2 h rg1g2 ) (26) Between second glass cover & ambient air (U g2a) U g2a (h w h rg2a ) (27) and Top heat loss coefficients may be given by, Ut (h cpg1 h rpg1 ) 1 (h cg1g2 h rg1g2 ) 1 (h w h rg2a )1 2L g / k g (28) where, Lg = Thickness of the glass cover kg = Thermal conductivity of glass Overall heat transfer loss coefficient (U L) can be calculated by following correlation Ul U t U b Us (29) 75 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications K L1 by Holland and Buchberg. The comparison of air properties effects on top heat loss coefficient (U t) has been studied for the given range of absorber plate temperature (T p), Ambient If side heat loss co-efficient (U s) is assumed to be constant. Temperature (Ta) and wind heat transfer coefficient (h w) UL Ut Ub (31) using MATLAB. The present study will help in calculating upward heat losses for a flat plate solar collector with double cover system. III. RESULTS AND DISCUSSION where, Ub (30) A way to describe the top heat loss coefficient of double glazed flat plate solar collector using numerical computing environment MATLAB the results have been presented in a graphical manner. In the present study effects of three important parameters Ambient temperature (Ta) Absorber plate Temperature (Tp) and heat transfer coefficient of wind (hw), on over all heat transfer coefficient has been discussed. In the present study, top heat loss coefficient (U t) is calculated from equation 28. Convective heat transfer coefficients (hcpg1, hcg1g2) are calculated from equations 4 to 6. For natural convection, Nusselt number (Nu) is calculated from correlations suggested by Holland et al., (6) and Buchberg et al., (4) from equations 17 to 22. The comparison of top heat loss coefficient (U t) evaluated by Holland et al., (6) and Buchberg et al., (4), result as shown in Tables 2 and 3. Percentage change or variation in Ut is 0.35% to 0.33% [by Holland et al., (6)] and percentage change or variation in Ut is 0.42 % to 0.40% [by Buchberg et al., (4)]. It shows as plate temperature increases percentage change in Ut decreases.The effect of ambient temperature (Ta) on top heat loss coefficient (U t) with variable absorber plate temperature (Tp), results shown in (Graphs 4, 5 and 6), shows that loss factor increases as the ambient temperature also increases. This trend implies that collector losses will be minimum in the early hours in the morning and maximum at higher absorber plate temperatures. The effects of absorber plate temperature (Tp) on top heat loss coefficient (U t) with variable heat transfer coefficient of wind (hw) (Graphs 1, 2 and 3) indicates that the heat transfer coefficient of wind can contribute to losses in the collector. The loss coefficient increases as the heat transfer coefficient of wind also increases and this is attributed to the increased convective and radiative losses from the glazing cover to the surrounding. IV. CONCLUSION REFERENCES [1] Akhtar, N., (1999). Upward Heat Losses in Solar Thermal Collectors, Ph.D. Thesis, CES, I.I.T. Delhi. [2] ASHRAE, (1986). Standard 93-86, Methods of Testing to determine the thermal performance of solar collectors, American society of Heating, Refrigeration and Air Conditioning Engineers. [3] Malhotra, H.P. Garg and Usha Rani (1980). Minimizing convective heat losses in flat plate solar collector. Solar energy, volume 25, Issue 6, pp. 521-526. [4] Buchberg, H., I. Catton and D.K. Edwards (1976). Natural convection in enclosed spaces, A review of application to solar energy collector, J. Heat Transfer, Trans. ASME, 98 (2): 182-188. [5] Duffie, J.A. and W.A. Beckman (1991). Solar Engineering of Thermal Processes, 2nd edition, Wiley Interscience, New York. [6] Hollands, K.G.T., T.E. Unny, G.D. Raithby and L. Konicek, (1976). Free convective heat transfer across inclined air layers. J. Heat Transfer, Trans. ASME, Ser. C 98 (2), 189193. [7] Holman, J.P., Heat transfer Tata Mcgraw Hill, Delhi, 9th edition 2004. [8] Kalogirou, S.A. (2004). Solar Thermal Collectors and Applications, Progress in Energy and Combustion Science, No. 30 pp. 231-295. [9] Malhotra, A., H.P. Garg and A. Patil (1981). Heat loss calculation of flat plate solar collector. J .Thermal Engg. Vol. 2: 59-62. [10] Mullick, S.C. and Samdarshi, S.K. (1988). An Improved Technique for Computing the Top Heat Loss Factor of Flat Collector with Single Glazing. ASME Journal of Solar Energy Engineering, Vol. 110 : 262-267. [11] Mullick, S.C. and Samdarshi, S.K. (1991). An analytical equation for Top Heat Loss Factor of Flat Plate Collector with Double Glazing.ASME. Journal of Solar Energy Engineering, Vol.113:117-122. An analytical study has been conducted to evaluate the top heat loss coefficient of a flat plate solar collector. In the [12] Sekhar Y. Raja, Sharma, K.V. and Rao M. Basaveswara (2009) evaluation of heat loss from the solar collector, two sets of Eevaluation of heat loss coefficients in solar flat plate correlations have been used for estimating air properties. collectorsARPN journal of engineering and applied sciences, For natural convection heat transfer between configuration Vol. 4, no. 5. correlations of Holland nd Buchberg was used. The effects [13] Woodman, T.P. (1977). The effect of design and operating of absorber plate temperature (Tp), ambient temperature (Ta) parameters on the performance of flat plate solar collectorand wind heat transfer coefficient (hw) on top heat loss Calculation method and detailed appraisal. Solar Energy, Vol. transfer coefficient are evaluated by correlations suggested 19(3):263-270 76 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications V. NOMENCLATURE A : Collector area,m Tp Ta : Plate temperature,0C : Air temperature, 0C Tg1 Tg2 : Temperature of first glass cover, 0C : Temperature of second glass cover, 0C Îg Îp : Emissivity of the glass : Emissivity of the absorber plate L2 L1 : Spacing between absorber plate and first glazing : Spacing between first and second glazing Lg Tm : Thickness of the glass cover : Mean temperature between two heat transfer surfaces 2 Tp Ut(h) Ut(b) % change 383 3.8912 3.9070 0.404402 384 3.9019 3.9176 0.400756 385 3.9127 3.9283 0.397118 386 3.9236 3.9391 0.393491 387 3.9346 3.9500 0.389873 388 3.9456 3.9609 0.386276 389 3.9566 3.9719 0.385206 390 3.9678 3.9829 0.379121 391 3.979 3.9940 0.375563 392 3.9902 4.0052 0.374513 393 4.0015 4.0158 0.356093 Upg1 : Over all heat transfer co-eff. between absorber plate and first glass cover (U1 ) Ug1g2 : Overall heat transfer co-eff. between first and second glass cover (U 2 ) 394 4.0129 4.0272 0.355085 395 4.0243 4.0386 0.354083 396 4.0358 4.0500 0.350617 Ug2a : Overall heat transfer co-eff. between second glass cover and Ambient air (U3 ) hrpg1 : Radiative heat transfer co-eff. between plate and first glass cover 397 4.0474 4.0615 0.347162 398 4.0581 4.0731 0.36827 399 4.0698 4.0847 0.364776 400 4.0815 4.0964 0.363734 hcpg1 : Convective heat transfer co-eff. between plate and first glass cover hcg1g2 : Convective heat transfer co-eff. between first and second glass cover 401 4.0933 4.1081 0.360264 402 4.1052 4.1199 0.356805 403 4.1171 4.1318 0.355777 hrg2a : Radiative heat transfer co-eff. between second glass cover and ambient air hrg1g2 : Radiative heat transfer co-eff. between first and second glass cover 404 4.129 4.1437 0.354755 405 4.141 4.1556 0.351333 406 4.153 4.1676 0.350322 407 4.1651 4.1797 0.349307 408 4.1773 4.1918 0.345913 409 4.1894 4.2039 0.344918 410 4.2017 4.2161 0.341548 Table 2: Comparison of Top Heat Loss Co-efficient by Holland and Buchberg Equation with Air Property Correlation - I L1 = 0.098 m; L2 = 0.012 m; Tp = 373:1:423K; Ta = 303K; hw =5 w/m2-K 411 4.214 4.2284 0.340554 412 4.226 4.2406 0.344291 Tp Ut(h) Ut(b) % change 413 4.2384 4.2530 0.343287 373 3.7915 3.8050 0.354796 414 4.2508 4.2654 0.342289 374 375 3.8012 3.8111 3.8148 3.8247 0.356506 0.355583 415 4.2633 4.2779 0.341289 416 4.2758 4.2904 0.340295 376 377 3.8211 3.8312 3.8347 3.8448 0.354656 0.353725 417 4.2884 4.3029 0.336982 418 4.301 4.3155 0.335998 4.3136 4.3282 0.337323 378 379 3.8413 3.8516 3.8550 3.8652 0.355383 0.351858 419 420 4.3263 4.3409 0.336336 380 381 3.8619 3.8723 3.8755 3.8860 0.350922 0.352548 421 4.3391 4.3537 0.335347 422 4.3519 4.3665 0.334364 382 3.8805 3.8964 0.408069 423 4.3647 4.3793 0.333387 77 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications Table 3: Comparison of Top Heat Loss Co-efficient by Holland and Buchberg Equation with Air Property Correlation - II L1 = 0.098 m; L2 = 0.012 m; Tp = 373:1:423K; Ta = 303K; hw =5 w/m2-K Tp Ut(h) Ut(b) % change 408 4.1301 4.1471 0.409925 409 4.1423 4.1593 0.408723 410 4.1545 4.1715 0.407527 411 4.1667 4.1837 0.406339 Tp Ut(h) Ut(b) % change 373 3.7446 3.7605 0.422816 412 4.1790 4.1960 0.405148 374 3.7544 3.7703 0.421717 375 3.7643 3.7802 0.420613 413 4.1914 4.2084 0.403954 376 3.7743 3.7902 0.419503 414 4.2038 4.2208 0.402767 377 3.7844 3.8003 0.418388 415 4.2162 4.2332 0.401587 378 3.7945 3.8105 0.419892 416 4.2287 4.2457 0.400405 379 3.8048 3.8207 0.416154 417 4.2412 4.2583 0.401569 380 3.8151 3.8310 0.415035 418 4.2538 4.2709 0.400384 381 3.8255 3.8414 0.413912 419 4.2662 4.2835 0.403875 382 3.8360 3.8519 0.412783 420 4.2789 4.2963 0.405000 383 3.8465 3.8625 0.414239 421 4.2916 4.3090 0.403806 384 3.8572 3.8731 0.410524 422 4.3044 4.3218 0.402610 385 3.8679 3.8838 0.409393 423 4.3172 4.3347 0.403719 386 3.8787 3.8946 0.408258 387 3.8895 3.9054 0.407129 388 3.9004 3.9163 0.405995 389 3.9098 3.9273 0.445599 390 3.9209 3.9383 0.441815 391 3.9321 3.9494 0.438041 392 3.9433 3.9606 0.436803 393 3.9546 3.9718 0.433053 394 3.9660 3.9831 0.429314 395 3.9774 3.9945 0.428089 396 3.9889 4.0059 0.424374 397 4.0004 4.0173 0.420681 398 4.0120 4.0289 0.419469 399 4.0236 4.0401 0.408406 400 4.0353 4.0518 0.407226 401 4.047 4.0635 0.406054 402 4.0588 4.0753 0.404878 403 4.0706 4.0871 0.403709 404 4.0825 4.0990 0.402537 405 4.0939 4.1110 0.415957 406 4.1059 4.1230 0.414747 407 4.1180 4.1350 0.411125 MIT International Journal of Mechanical Engineering Vol. 1 No. 2 Aug 2011, pp 71-78 ISSN No. 2230 – 7699 © MIT Publications 78