Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٧٥ STUDYING HEAT TRANSFER AND FLUID FLOW DISTRIBUTION ON HEATED FIN INSIDE RECTANGULAR DUCT WITH USING TRIANGULAR PROTRUSION Dr. Ahmed Hashm Yousif Abas Naji Dr. Kareem Dakeel Jasem Mechanical Department Mechanical Department Mechanical Department AL-Diwania Technical AL-Diwania Technical AL-Diwania Technical Institute Institute Institute Abstract The structure of flow and heat transfer characteristics over a heated fin (at constant heat flux) in rectangular duct with using triangle protrusion has been studied numerically. An incompressible flow and finite volumes method (FVM) in general coordinates with collocated grid arrangement are used to solve two dimensions continuity, Navier – Stokes and energy equations. The Prandtl number and Reynolds number are takes as 0.71 and 2000 respectively. At present study, different lengths of base of triangle protrusion (d/H = 0, 0.26, 0.53, 0.8, 1.07, 1.33) at (l/H= 0.13) was used. The results were compared with the case of flow inside duct without using protrusion. The results indicate that heat transfer and pressure drop is affected by the length of protrusion base especially near the triangle protrusion. The results illustrate that the heat transfer will be increasing with decreases the length of protrusion base. This study identifies an enhancement in the average heat transfer with increasing in pressure drop. ����د ���ء ��� � ��� ز���� ����� دا� ��ى �درا�� ا��'�ل ا�%�ارة و��!�ن ا���ا +�+* ا�() آ�!. دا� ���.. د � ��5س ���* �3 ا2�1 ه�/. !��- . د 7 ��( � 7 8�. ا���( � 7 8�. ا���( � 8�. ا� ا�1!�ا� �– ا��:19 ا��'�* ا�1!�ا� � – ا��:19 ا��'�* !�ا� � ا�1–ا��:19 ا��'�* ا�<=>� �د���� ��� ���� ����اص ا������ل ا���� ارة وا� ����ن�1$��+0 (دا���* - �� ى -��,�(+* ا���(��) �'��& ز�#$��� �����#� أ! ��� درا���� ���ن C ا��8=�AB (FVM)@? أ��<�م = ���� ا�� ��م ا�����دة . ���ء -9'89 ا�67* �1!�د) 4 اري 1�2 � ، E�+@ @ ام�و@�? ا���< � ، ����ك – ��* -���دCت ����+ collocated grid ا�F �6G7+ ا���#�و�1��G�ا ��+K�#2 ��L�)��6'�� وا�ا���* . ا 1 ?�Lر Mأ�� ?�@ ��L0.71و . 2000ورL? ر�#���ز �= +2N@ ��#���ء @� درا�ة ا���L ل(d/H = 0, 0.26, 0.53, 0.8, 1.07, 1.33) ارة ���وا�����ل ا OAB���1 ط��GQ�ا &�'� �'& ا�T$+�� دا�* ا�� ى S+4 إن )(H ى أو��W . @? -��ر�� ا�#��VK -) �4�� ا� ��ن 1�ون ا��<�ام ����ء . ��9* ار@$�ع ا�� �'& ا��GQط �1�OAB وا�� +Gآ #��ء ا2�ا اMQ� أن VK��#�ا�Y�Tو� ��+$T�ا &'�1+# ه�M\ ا��را��� أن . L ب ا�#��ء" ��ل ا�� ارة OAB�ط ا�GQ1 #��ء -) ز��دة�ارة ���,^ �1!�د ا ��ل ا����ل ا��- . Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٧٦ Nomenclature Symbol Description Unit A Convection-diffusion discritization coefficient a1, a2 Transformation coefficients B Channel length m C p Static pressure coefficient d Length of protrusion base m D Diffusion term coefficient F Mass flux H Channel height m hij Local heat transfer coefficient W/(m 2 .K) J Jacobean of transformation m k Thermal conductivity W/(m.k) l Triangle protrusion height Nu Nusselt number P Source term in grid generation system of equation pressure N/m 2 P0 Total pressure N/m 2 Pij Local static pressure N/m 2 P r Prandtl number q Heat flux per unit area W/m 2 Q Source term in grid generation system of equation Re Reynolds number S Source term for the dependent variable Ф T Temperature K u Velocity in x-direction m/s U, V Contravariant velocities v Velocity in y- direction m/s x, y Cartesian coordinates Greek Symbols Symbol Description Unit ξ, η General coordinates Ф General variable α Thermal diffusivity µ Kinetic viscosity Kg/m3 ρ Density Kg/m3 Γ Coefficient of Diffusivity Abbreviation Symbol Description Unit 2-D Two dimensions FVM Finite volume method N-S Naveir-Stoke equation PDE Partial differential equation Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٧٧ Subscripts Symbol Description Unit b Bulk e,w,n,s Face of control volume E,W,N,S Neighbor nodes of node (P) f Fluid p Center of control volume ij Index control in ( x,y) or (ξ, η) direction in Inlet condition Ф Total source of Ф transport property w wall Introduction Compact heat exchangers are widely employed in engineering application, especially in automotive industry, air conditioning and refrigeration apparatus (Ahmed 2003). The improvement in design of these heat exchangers has attracted many researchers. The heat transfer coefficients on the flat surface are significantly law due to increase the boundary layer thickness and in order to increase the heat transfer between the gas and fin in plat fin heat exchanger, protrusions can be used on the flat surface (Isak,1998). Many researchers used different types of vortex generators to generate a longitudinal vortex to mixing the fluid near the wall with main fluid that lead to remove heat near the wall to the main flow. In plate – fin or tube heat exchangers the flow between the plates can be considered as a duct flow. To reduction the thermal resistance, the heat transfer coefficient need to be augmented (Tiggelbeek,1991). In plate – fin heat exchanger, the heat transfer coefficient is significantly law and in order to increase heat transfer between the fluid and plates, protrusions or vortex generators can be use and fluid flow and heat transfer on flat plates with using vortex generators was experimentally and numerically investigated. The researchers (Ahmed 2003), (Biswas, 1992) and (Fiebig, 1989) are investigated flow and heat transfer on flat – plate with using wing and winglet type vortex that the heat transfer enhanced by using vortex generators with increasing in pressure drop across the plates. The studies of (Tiggelbeek, 1991), (Isak,1998) and (Russell, 1982) investigated the flow and heat transfer from plates with using obstacles and vortex generators experimentally. The studies show that heat transfer enhanced with using obstacles and vortex generator. One relevant application using such flow configuration is the heat transfer between the flow and plates in the case of plate - fin heat exchangers,as shown in Figure (1) (Isak,1998). In the present paper, a numerical investigation is described to study forced convection heat transfer with using single triangle protrusion in a rectangular channel as a part of plate – fin heat exchanger with different ratio d/H = (0 , 0.26 ,0.53 ,0.8 ,1.07 ,1.33 ,) at (l/H =0.13) as showing in Figure (2) for Re=2000 and pr=0.71. The local heat transfer in front region of plate is relatively high and decreases toward the rear due to growth of boundary layer, therefore the strategy of this study is to place a triangle protrusion in the middle of channel to mixing the fluid near the wall with the main fluid and to increase the velocity of flow near the protrusion to sweep the hot area or to make a vortex in the flow. Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٧٨ Mathematical Model The mathematical analysis is presented for partial differential equation (PDE) which describe fully developing flow and heat transfer at Re=2000 in parallel plate channel. Incompressible flow and steady state is assembly in entrance region with neglecting body force, heat conduction and free convection. Grid Generation Figure (3) show the schematic of two dimensions body fitting grid used for present computation, this grid obtained by solving non homogeneous 2-D Laplace equations (Thompson, 1977). ξxx+ ξyy =P(ξ, η) (1) ηxx+ ηyy=Q(ξ, η) Where P and Q are used to cluster the grid near the wall to sense the velocity gradient because there is a friction between the wall and the fluid. Equations (1) are transformed to (ξ, η) coordinates by interchanging the roles of dependent variables. This yields the following system equations. g11xξξ –g21xξη +g22xηη =-J 2 (Pxξ+Qxη) (2) g11yξξ –g21yξη +g22yηη =-J 2 (Pyξ+Qyη) Where g11= 22 ηη yx + g21=2(xξxη+yξyη) g22= 22 ξξ yx + J=xξyη-xηyξ Because the geometry of the domain was complex the (2-D) N-S, continuity and energy equations must be transported to general coordinates (ξ, η). Physical and Computational Plane with collocated grid arrangement can be seen in Figure (4). Governing Equations The 2-D governing equations of mass, momentum and energy equations for steady incompressible flow can be written as follows:- Continuity equation 0= ∂ ∂ + ∂ ∂ y v x u (3) Momentum equation Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٧٩ 2 2 2 2 1 1 x v u y p y v v x v u y u v x p y u v x u u ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ ρ ρ (4) Energy equation 2 2 y T y T v x T u ∂ ∂ = ∂ ∂ + ∂ ∂ α (5) The above equations can be written in a general transport equation as :- φφφ φφφρφρ S yyxxy v x u +      ∂ ∂ Γ ∂ ∂ +      ∂ ∂ Γ ∂ ∂ = ∂ ∂ + ∂ ∂ )()( )()( (6) This equation serves as a starting point for computational procedure in (FVM), by setting (Ф=1, ΓФ=0 and SФ=0) the continuity equation can be obtain, by setting (Ф=u, v and ΓФ=µ ), N-S equation can be obtain and by setting (Ф=T and ΓФ=µ /pr ) the energy equation can be obtain. The two terms in the left of the general transport equation (6) are convective terms, the first two terms in the right side of general transport equation (6) are diffusive terms and the last term is the source term. For the general curvilinear coordinates system (ξ, η), the general transport equation (6) can be transformed to the following form:- NEW S J a V J a U = ∂ Φ∂Γ −Φ ∂ ∂ + ∂ Φ∂Γ −Φ ∂ ∂ ΦΦ ) 2 () 1 ( η ρ ηξ ρ ξ (7) where (U, V) are the contravariant velocity components as shown in Figure (5). And SNEW =JS+SN where SN is the source term due to non orthogonal characteristic of grid system which disappear under orthogonal grid system. The transformation coefficients (a 1 , a 2 )are defined as:- 22 22 2 1 yx yx a a ηη ξξ += += (8) And the transformation metrics are calculated as follows J x J y J x J y y x y x ξ ξ η ξ η η ξ ξ = − = − = = (9) For collocated grid arrangement, all properties must be at the center of control volume (P) as shown in Figure (4). And the integration of equation (7) over control volume gives Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٠ NEW n s e w dvS J a V J a U =∆ ΦΓ −Φ+∆ ΦΓ −Φ ΦΦ ]) 2 ()[(]) 1 ()[( ξρηρ ηξ (10) Where dv=jξ jη To interpolate convection and diffusion terms, the power law difference scheme (Patenker, 1980)can be used, and the following algebraic equation can be obtained:- ApФp = AEФE+ AWФW+ ANФN+ ASФS+ dv SNEW (11) Where AE= De*A (│Pje│)+[[-Fe, 0]] AW=Dw*A (│Pjw│)+[[Fw, 0]] AN= Dn*A (│Pjn│)+[[-Fn, 0]] AS= Ds*A (│Pjs│)+[[Fs, 0]] Ap= AE+ AW+ AN+ AS when (F) is the mass flux and (D) is the diffusion term coefficient In order to avoid the checkerboard pressure field in collocated grid system, a momentum interpolation technique on cell face was successfully proposed by Rhie and Chow interpolation method that developed by ( Ferzigen, 1999). Boundary Conditions The (FVM) required that the boundary flux, either be known or expressed in term of known quantities on interior nodal value. the boundary of the present work include solid wall, inlet and outlet planes. 1 Inlet boundary Usually at inlet boundary, all quantities have to be prescribed as follows:- (u)inlet=uin (v)inlet=vin (p)inlet=pin (12) (T)inlet=Tin 2 Outlet boundary Always all values of dependent variables are unknown at exit boundary. It is often adequate to set outlet boundary values equal to intermediate up stream neighbor (e.g ФE= Фp). This is equivalent to saying the first derivative of property Ф in the direction normal to the boundary is equal to zero. 0 0 0 0 = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ x T x p x v x u (13) 3 Wall boundary The velocity on the wall equal to zero u=v=0 The first derivative of pressure in the direction normal to the wall equal to zero 0= ∂ ∂ y p (14) Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨١ For constant heat flux on bottom plane ijw T k y qT += * (15) And for adiabatic for top plate 0= ∂ ∂ y T (16) Heat transfer and pressure drop calculations The aim of present numerical study is to know logically the effect of triangle protrusion on local heat transfer and pressure drop by calculation Nusselt number (Nu) and pressure coefficient (C p ) as follows. 1 Heat transfer calculation To calculate Nusselt number (Nu) from heated wall at constant heat flux, first the temperature distribution can be calculated from the following equation. n T kq wall ∂ ∂ −= (17) Where n is the normal distance Then calculate local Nusselt number (Nuij) as follows:- qwall=qconv (18) qwall=h(ij)*(T(ij)-Tb) (19) Nu(ij)=h(ij)*H/kf (20) 2 Local static pressure coefficient The Local static pressure coefficient along the wall surface can be calculated as follows:- cp(ij)= 2 )( 5.0 )( in oij u pp ρ − (21) Results and discussion A number of computations have been performed at different (d/H) for protrusion. Figure (6) shows the stream lines for the flow in channel without using protrusion (free case). There is no change in direction of stream and no vortex appears. Figure (7) shows the stream lines for the flow in channel with using triangular protrusion with different (d/H). When (d/H) be a large value, the effect of protrusion is nearly equal to free case but when it be small, the procedure of velocity vectors will be changing when the flow reaches the protrusion and the direction of velocity will be changing toward the main flow, and due to the high momentum of the main flow in the middle of channel, the velocity will be change its direction behind the protrusion as in Figures (a, b). At small (d/H) the vortex will be appearing behind the protrusion because pressure different across the protrusions as in Figures (c, d, e and f). Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٢ Figure (8) shows the temperature contours for flow inside channel without using protrusion, the temperature gradient due to the thermal resistance (thermal boundary layer) clearly appeared, the thermal resistance at entrance of channel is equal to zero and it increases toward the rear due to the growth thermal boundary layer. Figure (9) shows the temperature contours for flow inside channel with using protrusion with different (d/H), the temperature gradient is different from the free case because of destroy the thermal boundary layer, also the figures shows that the heat transfer will be increasing in front of protrusion due to sweeping the heat toward the main flow but it be decreasing behind the protrusion. Figure (10) shows the pressure coefficient at different protrusion base along the channel. The figure shows that when the protrusion base be small [(d/H) be small], the pressure drops will be increasing near the position of protrusion since the different across the faces of protrusion and it will be decrease when increasing (d/H). Maximum local pressure drop occur was at d/H=0. Figure (11) shows the Nusselt number distribution along the plate surface for free case and with using triangle protrusion with different (d/H), the figure shows that the local heat transfer before the protrusion position for all cases are equal and its will be changing when the flow reaches the protrusion, heat transfer will be increasing at the front face of protrusion because of the flow will be sweeping the heat at this region and Nusselt number reaches maximum value at the tip of protrusion. At the rear face of protrusion, heat transfer will be increasing slightly because of the vortex of flow will be appearing in the rear of protrusion specially at small (d/H) and the effect of the triangle protrusion will be continuing to the rear of plate. Maximum local heat transfer occurs at d/H=0.26. Figure (12) shows the comparison of local Nusselt number for free case between the present study and reference (Ahmed, 2003) at the same conditions, the figure shows that the same procedure in two carves. Conclusions Local heat transfer and pressure drop of plate fins with using protrusions have been measured and compared with free case. The most important results can be summarized as follows:- • Protrusion accelerate the flow to sweep the heat from high heated region in plates, also it's mixing the hot fluid near the wall with the cold fluid at the main flow. • Average heat transfer enhanced (6% -21%) when using Protrusion with different ratio of (d/H). • Maximum average heat transfer enhanced when using Protrusion with (d/H= 0.26). • Pressure drop more than free case when using Protrusion and average pressure drop increased (2% -16%) with using Protrusions with different ratio of (d/H).. • Pressure drop increases when decrease d/H. Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٣ References 1- Ahmed, S. and Lars, D. "Numerical study of heat and flow in plate- fin heat exchanger with vortex generators " Turbulence heat and mass transfer4, pp 1155-1162, 2003. 2- Ahmed, H. Jallal, J and Hassan, K. "Enhancement heat transfer from heated cylinder inside duct by using vortex generators" PhD thesis, Mechanical Engineering Department University of Technology, Iraq, 2006. 3- Biswas, G. and Chattapdhyay, H "Heat transfer in channel with built - in wing type- vortex generators " Int. J. heat mass transfer vol.15, No.4, pp 803-814, 1992. 4- Fiebig, M. Brockmeier, U. Mitra, N. and Guntermann, T "Structure of velocity and temperature fields in laminar channel flow with longitudinal vortex generators " Numerical heat transfer, Part A, Vol.15, pp 281-302, 1989. 5- Ferzigen, J. and Peric, M. "Computational method for fluid dynamic" 2-th edition Springer, Berlin, 1999. 6- Isak, K. Teoman, A. Hayati, O. and Betiil, A. "Heat transfer and flow structure in rectangular channel with winglet type vortex generators " Tr. J. of Engineering and Environmental Science, vol.22, pp 185-195, 1998. 7- Patenker, S. V. "Numerical heat transfer and fluid flow. Hemispere Publishing Mc Graw- hill Book co. 1980. 8- Russell, C. Jones, T. and Lee, G. "heat transfer enhancement using vortex generators " Seventh int. heat transfer conf. Munich, Federal Republic of Germany, pp 283-288, 1982. 9- Tiggelbeek, S. Mitra, N. and Fiebig, M. "Comparison of wing type vortex generators for heat transfer enhancement in channel flows " ASME, vol. 116, pp 880-885, 1991. 10- Thompson, J. F. Thomas, F. C. and Mastin "TOMACA-A code for numerical generation of boundary - fitted curvilinear coordinate system on field containing any number or arbitrary 2 -D bodies " J of computational physics, vol/24, pp 274-303, 1977. 11- Yunliang, W. and Satoru, K. "Production of duct flows with a pressure- based procedure " Numerical heat transfer, Part A, vol/33, pp 723-748, 1998. (a) (b) (c) Figure (1) surface geometries for plat- fin heat exchanger (a) plain Rectangular fins, (b) plain triangular fins, (c) perforated fins Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٤ B flow H x l d Figure (2) proposed flow domain Figure (3) grid generated y Adiabatic wall Constant heat flux wall P P E E W W N N S S ( a ) (b) u v T Figure (4) (a) Physical Plane (x,y) (b)Computational Plane (ξ,η) Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٥ ξ x η V U y d/H=1.33 Figure (5) the contravariant velocity components Figure (6) Stream lines for free case Free d/H=0.53 d/H=1.07 d/H=0.8 (a) (b) (d) (c) Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٦ d/H=0d/H=0.26 1 1 2 3 4 5 5 7 8 9 11 11 57.95 10 43.11 9 38.78 8 37.33 7 37.03 6 37.01 5 37.00 4 37.00 3 37.00 2 37.00 1 37.00 T(c) Free case 10 8 6 5 4 3 2 1 1 10 65.34 9 52.89 8 43.21 7 39.18 6 37.31 5 37.10 4 37.02 3 37.00 2 37.00 1 37.00 T(c) d/H=1.33 Figure (8) Temperature contour for free case (f) (e) Figure (7) stream line distribution (a) d/H=1.33, (b) d/H=1.07 (c) d/H=0.8, (d) d/H=0.53, (e) d/H=0.26, (f) d/H=0 9 6 6 4 3 2 1 1 6 9 69.65 8 65.64 7 59.44 6 46.59 5 40.00 4 37.28 3 37.03 2 37.02 1 37.00 T(c) d/H=1.07 10 8 7 6 5 3 2 2 8 1 10 71.06 9 58.33 8 48.53 7 40.74 6 37.70 5 37.16 4 37.03 3 37.01 2 37.00 1 37.00 T(c) d/H=0.8 8 8 6 3 3 2 1 1 7 1 10 73.49 9 63.88 8 53.93 7 49.51 6 47.11 5 44.87 4 40.64 3 37.31 2 37.02 1 37.00 T(c) d/H=0.53 (a) (d) (c) (b) (e) (f) 10 8 6 5 3 2 4 5 10 79.22 9 53.62 8 50.76 7 46.88 6 44.32 5 42.20 4 38.29 3 37.20 2 37.04 1 36.11 T(c) d/H=0.26 8 7 6 5 2 1 8 4 2 2 10 84.82 9 79.68 8 70.72 7 51.03 6 44.29 5 40.39 4 38.21 3 37.59 2 37.04 1 36.11 T(c) d/H=0 Figure (9) Temperature counters (a) d/H=1.33, (b) d/H=1.07 (c) d/H=0.8, (d) d/H=0.53, (e) d/H=0.26, (f) d/H=0 Al-Qadisiya Journal For Engineering Sciences Vol. 2 No. ٢ Year 2009 ٢٨٧ Figure (10) pressure coefficient distribution Figure (11) Nusselt number distribution Figure (12) Comparison of Nusselt number with reference [1] 0 1 2 3 4 5 6 7 8 9 10 X/H -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 C p d/H=0 d/H=0.26 d/H=0.53 d/H=0.8 d/H=1.07 d/H=1.33 Free 0 2 4 6 8 10 X/H 5 10 15 20 25 30 35 40 N u d/H=0 d/H=0.26 d/H=0.53 d/H=0.8 d/H=1.07 d/H=1.33 Free 0 2 4 6 8 10 x/H 5 10 15 20 25 30 35 40 45 50 N u present study Refrence[1]