THE EFFECT OF LONGITUDINAL VIBRATION ON LAMINAR FORCED CONVECTION HEAT TRANSFER IN A HORIZONTAL TUBE

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1 mber 3 Volume 3 September 26 Manal H. AL-Hafidh Ali Mohsen Rishem Ass. Prof. /Uniersity of Baghdad Mechanical Engineer ABSTRACT The effect of longitudinal ibration on the laminar forced conection heat transfer in a horizontal tube was studied in the present work. The flow inestigated is an internal, laminar, incompressible, deeloping, steady and oscillatory flow. The ibration ector was parallel to the fluid flow direction. The boundary layer was studied in the entrance region for Prandtl numbers greater than one and the hydrodynamic boundary layer grows faster than thermal boundary layer. The goerning equations which used were momentum equation in the axial direction, continuity and energy equation. The finite difference technique was introduced to transform the partial differential equations into algebraic equations and the later equations were soled using the upwind scheme. The ibrational Reynolds number used as indication to the ibration and ibrational elocity which is used as boundary condition. The effect of ibration on laminar forced conection heat transfer is to increase the local and aerage sselt number when the ibrational Reynolds number increases; therefore, heat transfer coefficient increases too. Correlations for local sselt number with ibration concluded for constant wall temperature and constant heat flux. (!"#$% &' 3(/!!2(!-.,+)*$ $ #9: **8Prandtl no.7 45% 6$$$ % )($54$!$;.<:($$$ 5)5$*'$ 45(8Upwind scheme7$4 '4?>$4 $ )!;=$.>)!;!"#($&@*!'@A48Re 7! :' 8sselt no.7)45 *!45 B>$$ "$! ;5)45)5,=C4(; () " 863

2 M H. AL-Hafidh and A M Rishem KEYWORD Forced Conection Heat Transfer, Oscillatory Flow, Flow in the Entrance Region, Tube Vibration, Vibration Reynolds mber. INTRODUCTION It is well known that the heat transfer rate is greater in a turbulent flow than in laminar flow. Vibration of the heat transfer surface gies rise to turbulence in the case of laminar boundary layers and increases the heat transfer rate []. Studies in this field can be broadly classified into two typesone in which the heat transfer surface is ibrated experimentally by two different methods. In the first method, the surface is held stationary and acoustic ibrations are established in the fluid medium surrounding the surface. In the second method, an oscillatory motion is impressed upon the surface itself. Theoretically the flow medium is subjected to pulsation, ibrations or agitation [2]. LITERATURE REVIEW Tessin and Jacob (953), [3] While studying the effects of starting length on heat transfer from a cylinder to a parallel gas stream, they found that the heat transfer coefficients were not appreciably altered when the cylinder was subjected to a ibrational elocity of a bout (2%) of the flow elocity. Anantanarayanan and Ramachandran (958), [4] obsered an increase of (3%) in the heat transfer from a ibrating horizontal nichrome wire, (.8 in) in diameter to an air stream flowing parallel to the wire. The wire was ibrated in a ertical plane with arying alues of ibrational elocity. The ratio of the ibrational elocity to the flow elocity used in their study aried from ( to 3%). Van der Hegge Zijnen (958), [5] reported that the heat transfer from a tungsten wire (.5 cm) in diameter to a normal air stream decreased when the wire was subjected to ibrations in the direction of the air stream. A maximum decrease of (4.3%) in the heat transfer coefficient was obsered when the ratio of the root-mean square elocity of ibration of the flow elocity was as high as (45%). Kenji Takahashi and Kazuo Endoh, (99) [] inestigated experimentally the effect of ibration on forced conection heat transfer from a ibrating sphere, a cylinder and a square-section tube to water. The obtained heat transfer data with the ibration effect is well correlated in terms of the energy dissipation calculated from the fluid drag acting on the ibrating bodies. Sreeniasan and Ramachandran (96), [2] inestigated the effect of ibration on heat transfer from a horizontal copper cylinder (.344) in diameter and (6 in) long. The cylinder was placed normal to an air stream and was sinusoidally ibrated in a direction perpendicular to the air stream. The flow elocity aried from (9 ft/s) to (92 ft/s), the double amplitude of ibration from (.75 cm) to (3.2 cm), and the frequency of ibration from (2) to (28 cycles/min). A transient technique was used to determine the heat transfer coefficients. Lemlich (96), [6] and [7] published two interesting studies of the effect of ibration on the forced conection heat transfer in a double pipe heat exchanger. In one study a clarinet mouthpiece placed on the center tube was used to obtain air ibration of about 6 Hz. He concluded that the aerage heat transfer was increased by (35%). In a similar study a specially designed pulsator was 864

3 mber 3 Volume 3 September 26 used to obtain ibrations of about.5 Hz. In this study the oerall heat exchanger efficiency was increased by (8%). Deaer, Penny and Jefferson, (962), [8] studied the effect of low frequency and relatiely large amplitude ibration on the aerage heat transfer from a small wire to water. They also concluded that aboe a critical alue (af<.36 fps), increases in either the frequency or the amplitude increase the conectie coefficient of heat transfer. MATHAMATICAL MODEL The geometry of the tube and coordinates system are shown in figure () which represent two dimensional cylindrical coordinates (r, z), with a tube diameter of a (.7cm). The tube length introduced as thermal entrance length. To sole the flow equations (conseration of momentum, mass and energy equations) for water flows in a horizontal tube, some assumptions should be taken to fit the case, which is under study. These assumptions are: - - Steady state flow. 2- Two dimensional axisymmetric flow. 3- Incompressible flow. 4- Laminar flow. 5- No inlet swirl. 6- No heat generation. 7- Negligible body force, dissipation function and buoyancy effect. r Wall u in r o z C.L Fig (). Problem configuration and coordinates system. The goerning equations used in this problem were gien as: Momentum equation in axial direction:- 2 u u dp u u ρ + ρu = + µ ( r ) + 2 () r z dz r r r z Continuity equation u ( r) + r r z = (2) 865

4 M H. AL-Hafidh and A M Rishem The energy equation 2 T T µ T T ρ + ρu = ( r ) + 2 (3) r z Prr r r z Shear Stress From Newton's law of iscosity for simple shear flow: u τ w = µ (4) r r= R Local Friction Coefficient The local friction coefficient is defined by: C f τ w = (5) 2 ρuin 2 The aerage friction coefficient computed as follow: Cf = L L Cfdz (6) Bulk temperature The bulk (mixed-mean) temperature for the circular tube is: - T b R = R utrdr urdr (7) The friction coefficient and bulk temperature may be calculated by applying Simpson's rule. Local sselt mber The local sselt number is gien by: z = ( hd k) h Where d h =2 r o And so the local sselt numbers are gien in the following formula: For constant wall temperature: [ T r] r ro h ( Tw Tb ) = k = z [ T r] ] ( d ) /( T T ) = = (8) [ r ro h b w 866

5 mber 3 Volume 3 September 26 For constant heat flux: h( T T ) q So: w b = z [ q d ) ( T T )] ( h b w = (9) The mean sselt number is computed as = L L z dz Entrance Length For an internal flow we must be concerned with the existence of entrance and fully deeloped region. For laminar flow in a circular tube (Re 23), the hydrodynamic entry length may be obtained from an expression:- ( Z fd, h / D).5 Re () The thermal entry length may be expressed as: - ( Z fd, t / D).5Re Pr (2) (). Boundary Conditions Inlet Section Boundary Conditions In inlet section the axial elocity supposed to be uniform and its alue calculated from Reynolds number as follows:- u in Re µ = ρ d h The radial elocity in inlet section was equal to zero and the inlet temperature was ( T in = 2 C ) u = u in, = and T=T in Exit Section Boundary Conditions In the exit section the flow became fully deeloped, the axial and radial elocity gradient was zero and the temperature gradient was constant. So the boundary conditions were as follows: ( u z) =, = and ( T z) = Constant Boundary Conditions at the Wall The boundary conditions must be diided into two types, the forced conection heat transfer without ibration and the forced conection heat transfer with ibration. Wall Boundary conditions without Vibration 867

6 M H. AL-Hafidh and A M Rishem At the wall, boundary conditions were taken from boundary layer theory that the axial and ertical elocity at the walls in the iscous flow must be equal zero, also the tube wall temperature was taken according to the case of constant wall temperature and of constant heat flux as following:- u = =, =, ( T = Tw ) (Constant wall temperature) u w And ( T r = q k ) (Constant heat flux) Wall Boundary conditions with Vibration The steady oscillatory flow was used in order to show the effect of ibration on the forced conection heat transfer and the effect of the ibration on the flow. Oscillatory flow [9] used ibrational elocity as boundary condition as follows: u = u w = 2πaf, =, T = Tw (Constant wall temperature) And ( T r = q k ) (Constant heat flux). Boundary Conditions in centerline The axial elocity gradient, radial elocity and temperature gradient occurred in the centerline was equal to zero so:- ( u r) =, = and ( T r) = SOLUTION METHODS Because of the complexity of the analytical solution of the set of the algebraic equations, they may be soled by finite difference technique. In this study the goerning equations are expressions of conseration of mass, momentum and energy equations. The Gauss-Siedal method was used to sole the continuity and momentum equations and Tri-diagonal method was used to sole the energy equation. A digital computer used to sole the finite difference equations using FORTRAN 9 program and Tec plot program used to plot the results. RESULTS AND CONCLUSIONS Reynolds number can be classified into two types; Reynolds number without ibration (Re) and Reynolds number with ibration (Re ) which is called ibrational Reynolds number. For each alue of Reynolds number (Re) three ibrational Reynolds number (Re ) were taken in order to show the effect of ibration on the laminar forced conection heat transfer. For Reynolds number (Re=), three ibrational Reynolds number were taken as (Re =75, and 25), for (Re=5), (Re =25, 5 and 75), and for (Re=2), Re (=75, 2 and 225) was taken. The inlet elocity calculated from the flow Reynolds number and ibrational elocity calculated from 3 ibrational Reynolds number. The properties of water were taken as follows ( ρ = 3kg / m, 4 o µ = 8.55 Pa.s, Pr=5.83 and k=.63w m. C Results without ibration / ). Temperature Variation for Constant Heat Flux: The temperature ariation in the axial and radial directions, for constant heat flux (q= W/m 2 ) and different Reynolds number (Re =) are shown in figure (2). The figure shows that the 868

7 mber 3 Volume 3 September 26 temperature increase in the axial and radial directions and a small increase in temperature contours can be obsered with the increases of flow Reynolds number. Temperature Variation for Constant Wall Temperature The temperature ariation in the axial and radial directions for constant wall temperature o ( Tw = C ) and different Reynolds number (Re =) are shown in figure (3). The figure shows that the temperature increase in the axial and radial directions and the change of flow Reynolds number causes a small change in temperature contours. Local sselt mber ( z ) The behaior of local sselt number with axial directions, for constant heat flux and for constant wall temperature is shown in figures (4a, b and c) respectiely. The local sselt number decreases with the axial directions in the entrance region. While in the fully deeloped region the local sselt number become constant with the axial directions. These figures show the effect of Reynolds number (Re=, 5 and 2) on the local sselt number. The alue of the local sselt number for constant heat flux is (4.364) and for constant wall temperature is (3.66) at the fully deeloped region (Z + =) because the length is introduce as thermal entrance length. Velocity Profile Deelopment of elocity profile along the axial directions, for different Reynolds number (Re=, 5 and 2) are shown in figures (5a, b and c) respectiely. The elocity profile + deeloped from the entrance region until ( Z. 55 ), after this axial distance the elocity profile becomes constant and the flow from this axial distance and on is said to be fully deeloped. The maximum elocity occurs at dimensionless radial distance ( R ) was equal to zero (centerline). Local Friction Coefficient: The ariations of the local friction coefficient along the axial directions is shown in figure (6) for different Reynolds number (Re=, 5 and 2), in which the elocity increase with the increasing of flow Reynolds number, while the local friction coefficient was decreased. The flow + became fully deeloped at the dimensionless axial distance ( Z. 55 ), therefore from this distance and on the local friction coefficient became constant. From these figures it will be obsered that the local friction coefficient decrease with the increasing of Reynolds number. The local friction coefficient alues decreased until (Z +.) but it was increased after this axial distance because of numerical solution accuracy and the assumptions which were taken in this study. a) Re=, q= W/m

8 M H. AL-Hafidh and A M Rishem Fig (2): Contour of temperature for constant heat flux constant heat flux constant wall tem perature 3 2 a) Re= Z + Fig (3): Contour of temperature for constant wall temperature (T w = o C ). 6 5 constant heat flux constant wall temperature b) Re= Z + 87

9 mber 3 Volume 3 September R * Z + =. Z + =. Z + =.2 Z + =.3 Z + =.4 Z + =.5 Z + =.5 5 Z + =.6 Z + = u / u in a) Re= Fig (4). axial distribution for local sselt number without ibration. 87

10 M H. AL-Hafidh and A M Rishem.8 R * Z + =. Z + =. Z + =.2 Z + =.3 Z + =.4 Z + =.5 Z + =.5 5 Z + =.6 Z + = u /u b) Re=5 in Fig (5).deeloping profile of axial elocity without ibration..8 R * Z + =. Z + =. Z + =.2 Z + =.3 Z + =.4 Z + =.5 Z + =.55 Z + =.6 Z + = c) Re=2 u/u in 872

11 mber 3 Volume 3 September Re= Re=5 Re=2.3 C f.2. Fig (6).: axial distribution of local friction coefficient without ibration. Results with ibration: The parameters effects on the problem are frequency (f), amplitude (a) and the product of them which is represents by ibrational Reynolds number (Re ). The results as follows: Temperature Variation for Constant Heat Flux and Constant Wall Temperature: - The effect of ibration on the temperature for constant heat flux and constant wall temperature are as follows: the case of Reynolds number (Re=) and ibrational Reynolds number (Re = ) are shown in figures (7) and (8). For this case the temperature increased along the axial distance + ( Z ), and the temperature increased along the radial distance ( R ). The other cases had the same behaior. Generally when ibrational Reynolds number increased the lines of temperature contour were shift with small distance to the left. Local sselt mber ( ) and Aerage sselt mber ( ): z The effect of ibration on local and aerage sselt numbers are shown in figures (9a and b), () and () where Reynolds number (Re=) and ibrational Reynolds number ( Re =), for constant heat flux and constant wall temperature respectiely. A ibrational Reynolds number ( Re =) represents the behaior of local sselt number without ibration. Generally local and aerage sselt numbers increases when a ibrational Reynolds number increases because of the modifications of the flow in laminar boundary layer; earlier transition from laminar to turbulent boundary layer flow. For all cases fully deeloped occurs at the axial distance (Z + =) because the length is introduce as thermal entrance length. Velocity Profile: Z + For the case of Reynolds number (Re=) and ibrational Reynolds number ( Re =75, and 25), the elocity profile is shown in figures (2 a) at (Z + =.4) and (2 b) at (Z + =.) respectiely. For the first figure the cure of (Re>Re ), the elocity profile from the axial distance + ( Z. 25 ) was constant to the exit section, therefore, the flow was fully deeloped from 873

12 M H. AL-Hafidh and A M Rishem + ( Z. 25 ) to the exit section. In this figures the elocity profile was for turbulent flow because of the ibration influence, in the second cure of (Re=Re ), the elocity profile is uniform from inlet section to exit section of the circular tube (flat elocity) and for the third cure of (Re<Re ), the flow + from axial distance ( Z. 25 ) to the exit section so the elocity profile is constant and the flow is fully deeloped. Local Friction Coefficient ( C ) and Aerage Friction Coefficient ( C ): - f The local and aerage friction coefficient are affected by ibration and this is illustrated in figures (3) and (4) respectiely for Reynolds number (Re=) and ibrational Reynolds number ( Re =75, and 25). A ibrational Reynolds number ( Re =) represents the behaior of local friction coefficient without ibration. Generally the local and aerage friction coefficients decreases for a ibrational Reynolds number increases, since the axial elocity gradient decreases in case of the ibration influence. For (Re> Re ) local friction coefficient decreases at entrance region until axial + distance ( Z. 25 ), then from this distance to the exit section the local friction coefficient behaior became constant (fully deeloped flow). For (Re= Re ), local friction coefficient alues were zero because the axial elocity gradient is zero whereas the flow is uniform from the inlet section to the exit section. For (Re< Re ), local friction coefficient alues were negatie because the alues of the axial elocity gradient is negatie. The local friction coefficient alues decreases until (Z +.) but its increases after this axial distance because of numerical solution accuracy and the assumptions which taken in this study. f R * Re = Z + Fig (7). Contour of temperature for constant heat flux. 874

13 mber 3 Volume 3 September 26 R * Re = Z * Figure (8): Contour of temperature for constant wall temperature. 5 4 = = 7 5 = = Z + a) q=const = = 7 5 = = Z + b) T w =const. Figure (9): The effect of ibration on deeloping No. 875

14 M H. AL-Hafidh and A M Rishem Figure (): effect of ibration on for (q= W/m 2 ) at Re= Figure (): effect of Re on at Re= for (T w = C o ). 876

15 mber 3 Volume 3 September = = 7 5 = = 2 5 R *.4.2 a) Z + = U.8 Re = Re =75 Re = Re =25..6 R *.4.2 b) Z + = U Figure (2): deeloping profile of axial elocity with ibration. 877

16 M H. AL-Hafidh and A M Rishem CONCLUSIONS WITH VIBRATION: - The ibration influence is to conert the flow from laminar to turbulent flow. 2- The growth of hydrodynamic boundary layer with ibration is much faster than that without ibration. 3- Local sselt number with ibration is greater than that without ibration and that means increases in the heat transfer coefficient. 4- New correlation predicated for local sselt number due to ibration effect as follows: Vibrational local sselt number for constant wall temperature as follows: - D (Re Pr ) = % L D.3 +.Pr(Re ) L Where Percentage of increasing in = % = Vibrational local sselt number for constant heat flux is: - D.86(Re Pr ) = % L D +.Pr(Re ) L This correlation is Valid for Pr=5.83 or if RePrD/L< REFRENCES [] Kenji Takahashi and Kazuo Endoh. (99), Anew correlation method for the effect of ibration on forced conection heat transfer Journal of chemical engineering of Japan. Vol.23, No., pp [2] K.Sreeniasan and A.Ramachandran, (96) Effect of ibration on heat transfer from a horizontal cylinder to a normal air stream Int.heat mass transfer, Vol.3, pp [3] W. Tessin and M.Jakob, (953), Influence of unheated starting section on heat transfer from a cylinder to gas streams parallel to the axis Trans.Amer.Soc.Mech.Engrs, Vol.75, pp [4] R.Anantanarayanan and A.Ramachandran, (958), Effect of ibration on heat transfer from a wire to air in parallel flow Trans.Amer.Soc.Mech.Engrs, Vol.8, pp [5] B.G.Van Der Hegge Zijnen, (958), Heat transfer from horizontal cylinders to a turbulent air flow Appl.Sci.Res.A7, pp [6] R., Lemlich, (96), A musical heat exchanger Journal of heat transfer, Trans.ASME.Vol.83, pp [7] R., Lemlich, (96), Vibration and pulsation boost heat transfer Chemical engineering, Vol.68, pp [8] F.K., Deaer, W.R., Penny, and T.B., Jefferson, (962), Heat transfer from an oscillating horizontal wire to water Journal of heat transfer, Trans.ASME.Vol.84, series C, pp [9] Takahiko Tanahashi, Tsuneyo Ando, Hideki Kawashima and Teruyuki Kawashima, (985), Flow in the entrance region of a porous pipe Bulletin of JSME, Vol.28, No.237, pp

17 mber 3 Volume 3 September 26 NOMENCLATURE a = amplitude [m] a o = projected area [m 2 ] Cf = local friction coefficient Cd = drag coefficient d h = hydraulic diameter (2r o ) [m] f = frequency of ibration [Hz] = local sselt number Pr = Prandtl number q = heat flux [W/m 2 ] r =tube radius [m] r o = outer radius [m] R * = dimensionless radius (r/r o ) Re = Flow Reynolds number T = temperature [ C ] u = axial elocity [m/s] U = dimensionless axial elocity (u/u in ) = radial elocity [m/s] Z + =axial distance (z/l) GREEK SYMBOLS ρ = density [kg/m 3 ] µ = iscosity [Pa.s] k = thermal conductiity [W/m. C ] ω =angular frequency of ibration [rad/s] τ = wall shear stress [N/m 2 ] w SUPERSCRIPT ( ) = aerage quantity SUBSCRIPT = ibration w = wall in = inlet b = bulk 879

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