THE EFFECTS OF DUCT SHAPE ON THE NUSSELT NUMBER

Transcription

1 Mathematical and Computational pplications, Vol, No, pp 79-88, 5 ssociation for Scientific Research THE EFFECTS OF DUCT SHPE ON THE NUSSELT NUMBER M Emin Erdoğan and C Erdem Imrak Faculty of Mechanical Engineering, Istanul Technical University, 339 Gumussuyu, Istanul, Turkey imrak@ituedutr stract- This article considers the effect of duct shape on the Nusselt numer In order to show the effects of duct shape on the Nusselt numer, four illustrative examples are given: the heat transfer in a duct of rectangular cross-section, heat transfer in a duct of semicircular cross-section, the heat transfer in a duct of circular crosssection and heat transfer etween two parallel plates The method used in this paper is general and it can e applied al flows It is not necessary to use thin plate analogy, first, the velocity distriution is otained and then the temperature distriution is found It is shown that the Nusselt numers calculated for the four examples cited depend not only on duct shape ut also on wall friction Key Words- Nusselt numer, duct shape, wall friction INTRODUCTION The effect of duct shape on heat transfer in a duct of uniform cross-section is investigated In order to show the effect of duct shape on heat transfer, heat transfer in a duct of rectangular cross-section, heat transfer in a duct of semicircular cross-section, heat transfer in a duct of circular cross-section, and heat transfer etween two parallel plates are considered The flow and the heat transfer in these ducts, apart from their theoretical interest, are of considerale practical importance and arise freuently in industrial processes Laminar flow forced convection in rectangular ducts has een given y Clark and Kays [] The heat transfer in ducts of rectangular, euilateral triangular, rightangled isoceles triangular, and semicircular cross-sections have een studied y Marco and Han [] They considered heat transfer in a duct of rectangular cross-section y making analogy to thin plate theory and they gave the conditions under which the solution is permissile By this analogy, first the temperature distriution is otained and then the velocity distriution is found In this paper, it is shown that it is not necessary to use the thin plate analogy; first, the velocity is otained and then the temperature distriution is found The method used in this paper is easier than that in [] and it can e extend to other ducts The values of the Nusselt numer in a duct of rectangular cross-section depend on the aspect ratio, / a The values of the Nusselt numer for heat transfer etween two parallel plates, / a, is / 7 or aout 8359 to eight decimal places and for heat transfer in a duct of suare crosssection, where / a, is aout 3897 to six decimal places The values of the Nusselt numer in [] differ from those in this paper Calculation of the Nusselt numer in a flow in which the velocity depends on two coordinates is different from that in a

2 8 M E Erdoğan and C E Imrak flow in which the velocity depends on one coordinate [3] The velocity distriution for flow in a duct of semicircular cross-section have een given y Eckert et al [] They presented the results of calculation for steady laminar flow Pressure drag and friction factor values for a duct of semicircular crosssection have een studied y Sparrow and Haji-Sheikh [5] The linearization method was applied to developing flow in these ducts [6] The steady flow in a curved semicircular duct has een investigated y Mashliyah [7] Calculation of the flow was made y solving the Navier-Stokes euation numerically The flow in a curved semicircular duct is composed of a main flow along the curved duct axis with a superimposed secondary flow comprehensive work on heat transfer in circular and non-circular ducts has een given in [8] In this paper, heat transfer in ducts of rectangular cross-section, semicircular cross-section, circular cross-section and the parallel plate duct are considered First the velocity distriution and then the temperature distriution are calculated Using the expressions of the velocity and the temperature for the considered ducts, values of the Nusselt numer are otained The calculation methods for velocity and temperature which depend on two coordinates are different than the methods for which they depend on only one coordinate The value of the Nusselt numer for a duct of rectangular crosssection depends on the aspect ratio, / a The value of the Nusselt numer is the smallest for a duct of suare cross-section for which the aspect ratio euals to unity The value of the Nusselt numer is the greatest for / a This value of the aspect ratio corresponds to the heat transfer etween two parallel plates The calculation in this paper is made y assuming that the surface temperature varies linearly along the duct In the case of a duct of semicircular cross-section the calculation of the Nusselt numer is straightforward and it is found that the value of the Nusselt numer is aout to nine decimal places It seems to e impossile to otain this value as a fraction In the case of a pipe of circular cross-section the value of the Nusselt numer is 8 /or aout to nine decimal places This value is smaller than that for a duct of semicircular cross-section In this paper, the values of the Nusselt numers otained are aout, 3577 for a duct of suare cross-section; 8 / (or ) for a circular pipe, aout 8887 for a duct of semicircular cross-section and / 7 (or 8359) for a parallel plate duct These values of the Nusselt numer show the effect of duct shape on heat transfer HET TRNSFER IN DUCT OF RECTNGULR CROSS-SECTION Consider the fluid in a duct of rectangular cross-section where sides are at x ±a and y ± as shown in Fig, it is assumed that a throughout the paper The governing euation for velocity is w w dp + () x y µ dz

3 The Effects of Duct Shape on the Nusselt Numer 8 where w( x, y) is the axial velocity and dp / dz is the constant pressure gradient The oundary conditions are w ( ± a, y), w ( x, ±) Figure The coordinate system for a duct of rectangular cross-section The solution may e written in different forms, however, a convenient one is ( )( ) w p cos µ p x cosλ y () / µ dp / dz p where µ p ( p + ) π / a and λ ( + ) π / and p+ 3 ( ) p π ( p + )( + ) ( µ p + λ ) When / a goes to zero euation () reduces to w 3 ( ) cos y y lim 3 3 dp λ ( + ) a π µ dz This shows the velocity etween two parallel plates The governing euation for temperature is T T w T + (3) x y κ z where T is temperature andκ is thermal diffusivity The oundary conditions for T are T Tw To + ( T / z)z at the surface where T / z α is a constant and T T at z The solution suject to these oundary conditions is α dp T Tw Bp cos µ p x cosλ y, () κ µ dz p where p+ 3 ( ) Bp π ( p + )( + ) ( µ + λ ) When / a goes to zero euation () reduces to p

4 8 M E Erdoğan and C E Imrak T T w 8 ( ) cosλ y y y lim 5 5 α dp + π ( ) a κ µ dz This is the temperature distriution etween two parallel plates The Nusselt numer is the ratio of the convective conductance to the pure molecular conductance and is defined as Nu k( Tw TM ) S / D, where is the uantity of heat transferred per unit time from an immersed ody through an area S, Tw TM is a representative temperature difference, D is the hydraulic diameter or a representative length and T M is the mean temperature weighted with respect to the velocity given in the following form T wd T M wd where integrals are surface integrals and ( Tw T ) wd Tw TM wd and T ρ c w d z Hence inserting this expression, the Nusselt numer can e written as [] T D ρ c w d Nu z (5) S k w( T Tw ) d/ wd D / S is the ratio of the hydraulic diameter to the perimeter of the cross-section times unit length Hydraulic diameter is defined as four times the wetted area to the perimeter of the cross-section Using the surface temperature, euation (5) ecomes ( wd) D ρ cα Nu (6) S k w( T Tw ) d Inserting the expressions for w and ( T Tw ) and evaluating the surface integrals, one finds 6 / π p ( p + ) ( + ) ( p + ) ( / a ) + ( + ) Nu (7) a p ( p ) ( ) [( p + ) ( / a ) + ( + ) ] / a corresponds to the value of the Nusselt numer etween two parallel plates and since

5 The Effects of Duct Shape on the Nusselt Numer 83 π and π, p ( p + ) 8 ( + ) 96 one finds Nu / 7 This value is verified y calculating the Nusselt numer for a duct of two parallel plates which is aout to nine decimal places However, this value is given in [] as aout 8 / a corresponds to a duct of suare cross-section and the value of the Nusselt numer is aout to seven decimal places However, this value in [] is aout 36 The variation of the Nusselt numer with respect to / a is illustrated in Fig It is clearly seen that when / a tends to zero, the Nusselt numer reduces to that for the Nusselt numer etween two parallel plates and when / a is eual to unity, the Nusselt numer reduces to that for the Nusselt numer in a duct of suare cross-section The value of the Nusselt numer is the greatest for / a and the smallest for / a Figure The variation of the Nu numer with respect to the aspect ratio, / a 3 HET TRNSFER IN DUCT OF SEMICIRCULR CROSS-SECTION Consider the fluid in a duct of semicircular cross-section and the flow geometry is shown in Fig 3 The governing euation for velocity is w w w dp + +, (8) r r r r θ µ dz where w ( r, θ ) is the axial velocity and dp / dz is the constant pressure gradient The oundary conditions are w ( a, θ ), w ( r,), w ( r, π )

6 8 M E Erdoğan and C E Imrak Figure 3 Flow geometry for a duct of semicircular cross-section The solution of the governing euation suject to the oundary conditions is where w( n r, θ ) f θ, (9) n + ( r ) sin( n + ) n+ dp (/ π ) a r r f n ( r) n+ µ dz (n )(n + )(n + 3) a a The governing euation for temperature is + T T T w T + +, () r r r r θ κ z where T is temperature and κ is thermal diffusivity The oundary conditions for T are T Tw To + ( T / z)z at the surface where T / z α is a constant and T T at z The solution of euation () is where w f n+ ( r ) sin( n + ) n T T θ, () f n+ dp α (/ π ) a ( r) µ dz κ r ( )( )( 3) ( 3)( 5) n n + n + n n + a n+ n+ 3 r r 3 8( ) n+ n+ n + a a The Nusselt numer is defined y euation (5) For a duct of semicircular crosssection D / S π /( π + ) Inserting the expressions for w and T Tw into euation (6) and evaluating of the surface integrals, one finds ( π 8 ) Nu 56 ( π + ) where F n n r a n+ n+

7 The Effects of Duct Shape on the Nusselt Numer 85 F n ( n + )(n 3)(n )(n + ) (n + 3) (n + 5) + 8(n 3)(n )(n + ) (n + 3) (n + 5)(n + 7) 3( n + ) (n )(n + ) (n + 3) (n + 5) + 3 6( n + )(n )(n + ) (n + 3) (n + 7) The value of the Nusselt numer is aout 8887 to nine decimal places This value of the Nusselt numer is comparale to that for a pipe of circular crosssection HET TRNSFER IN PIPE OF CIRCULR CROSS-SECTION Consider the fluid in a pipe of circular cross-section as shown in Fig The governing euation for velocity is w w dp +, () r r r µ dz where w(r) is the axial velocity and dp / dz is the constant pressure gradient The oundary condition is w ( a), where a is the radius of the pipe The solution of the governing euation satisfying the oundary condition and the uniformity at r is r w w a where w is the velocity at r Figure Flow geometry for flow in a circular pipe The governing euation for temperature is T T w T + (3) r r r κ z

8 86 M E Erdoğan and C E Imrak The oundary conditions for T are T T T + ( T / z)z at the surface where T / z α is a constant and T T at z The solution of euation (3) is α w r r T Tw a 3 () 6κ a a The maximum temperature is at the center of the pipe The Nusselt numer is defined y euation (5) For a pipe of circular cross-section D / S / π Since T / z α is a constant, euation (5) can e written as ( wd) D ρ cα Nu S k w( T Tw ) d Inserting the expressions for w and T Tw into this expression and performing the surface integrals, one finds Nu 8/ or aout to nine decimal places It is clearly seen that the value of the Nusselt numer for a duct of semicircular crosssection is larger than that for a pipe of circular cross-section 5 HET TRNSFER BETWEEN TWO PRLLEL PLTES Consider the fluid etween two parallel plates as shown in Fig 5 The governing euation for velocity is d u dp, dy µ dx where u(y) is the velocity and dp / dx is the constant pressure gradient The oundary condition is u ( ±) The solution of the governing euation is u u [ ( y / )], where u is the velocity at y w o Figure 5 Flow geometry for flow etween two parallel plates The governing euation for temperature is d T u dt dy κ dx

9 The Effects of Duct Shape on the Nusselt Numer 87 The oundary conditions for T are T Tu T + ( T / x)x at the surface where dt / dx α is a constant, and T T at x The solution suject to the condition at the surface is α u y y T T w κ 6 The Nusselt numer is defined y euation (5) For a parallel plate duct Since dt / dx α is a constant, euation (5) can e written as ( u d) D / S is D ρ cα Nu S k u( T Tw ) d Inserting the expressions for u and T Tw into this expression and evaluating the surface integrals, one finds Nu / 7 or aout to nine decimal places This value of the Nusselt numer is the highest for the examples considered 6 CONCLUSIONS The effect of duct shape on the Nusselt numer is investigated In order to show the effect of duct shape, four illustrative examples are given One of them is heat transfer in a duct of rectangular cross-section, the second is heat transfer in a duct of semicircular cross-section, the third is heat transfer in a pipe of circular cross-section and the fourth is heat transfer etween two parallel plates The methods used in this paper can e applied to all flows It is necessary to use the thin plate analogy First, the velocity distriution is otained and then the temperature distriution is found The values of the Nusselt numers are: aout 3597 for a duct of suare cross-section, aout for a circular pipe, aout 8887 for a duct of semicircular cross-section and aout for a flat duct Values of the Nusselt numers otained in the four examples considered show clearly that the Nusselt numer depends on the duct shape cknowledgment-the authors thank Professor Mehmet Pakdemirli and the refree 7 REFERENCES SH Clark and WM Kays, Laminar flow forced convection in rectangular tues, Trans SME 75, , 953 SM Marco and LS Han, note on limiting laminar Nusselt numer in ducts with constant temperature gradient y analogy to thin plate theory, Trans SME 77, 65-63, S Goldstein (Ed), Modern Developments in Fluid Dynamics,Vol, second edn, Roder Pu, New York, 965 ERG Eckert, TPJr Irwin and JT Yen, Local laminar heat transfer in wedgeshaped passages, Trans SME 8, 33-38, ER Sparrow and Haji-Sheikh, Laminar heat transfer and pressure drag in isoceles

10 88 M E Erdoğan and C E Imrak triangular, right triangular and circular sector ducts, Trans SME 87, 6-7, HM Soliman, BB Munis and BC Trap, Laminar flow in the entrance region of circular sector ducts, Trans SME, J ppl Mech 9, 6-6, 98 7 JH Mashliyah, On laminar flow in curved semicircular ducts, J Fluid Mech 99, 69-79, 98 8 S Kakaç, RK Shah and W nng (Eds), Handook of single-phase convective heat transfer, John Wiley & Sons, New York, 987