Indian Journal of Engineering & Materials Sciences Yol. I, August 1994,pp

Indian Journal of Engineering & Materials Sciences Yol. I, August 1994,pp.181-188 Finite element analysis of laminar convection heat transfer and flow of the fluid bounded by V-corrugated vertical plates of different corrugation frequencies Department Mohammad Ali & M Nawsher Ali of Mechanical Engineering, Bangladesh Institute of Technology, Khulna 9203, Bangladesh Received 27 December 1993; accepted 5 May 1994 A study on steady-state natural convection heat transfer and flow characteristics of the fluid in a horizontal duct with V-corrugated vertical walls has been tarried out using control volume based Finite Element Method. The effects of corrugation frequency and Grashof numbers on local Nusselt number, total heat flux, vertical velocity and temperature distributions at the horizontal mid-plane have been investigated. Corrugation frequency from 1 to 3 and Grashof number from 103 to 105 have been considered in this study. The results are compared with the available information on natural convection and fluid flow. Natural convection is one of the important phenomena in everyday engineering problem. It is of particular importance in designing nuclear reactors, solar collectors and in many other design problems where heat transfer is occurring by natural convection. Heat transfer by natural convection depends on the convection currents developed by thermal expansion of the fluid particles. Further, the development of the flow is influenced by the shape of the heat transfer surfaces. Therefore, the investigation for different shapes of the heat transfer surfaces is necessary to ensure the efficient performance of the various heat transfer equipments.. There are several studies on convective heat transfer with corrugated walls. In these studies, however, the heat transfer has been considered only from a lower hot corrugated plate to an upper cold flat plate, both being horizontally placed. No research has, so far, been carried out on convective heat transfer with vertical hot and cold corrugated plates. Chinnapa 1 carried out an experimental investigation on natural convection heat transfer from a horizontal lower hot V-corrugated plate to an upper cold flat plate. He collected data for a range of Grashof numbers from 104 to 106 The author noticed a change in the flow pattern at Gr= 8 x 104, which he concluded as a transition point from laminar to turbulent flow. Randall et au studied local and average heat transfer coefficients for natural convection between a V-corrugated plate (600 V-angle) and a parallel flat plate to find the temperature distribution in the enclosed air space. From this temperature distribution they used the wall temperature gradient to estimate the local heat transfer coefficient. Local values of heat transfer coefficient were investigated over the entire V-corrugated surface area. The author recommended a correlation of the average heat flux which presents 10% higher value than that for parallel flat plates. Zhong et af.3 carried out a finite-difference study to determine the effects of variable properties on the temperature and velocity fields and the heat transfer rate in a differentia,lly heated, twodimensional square enclosure. Nayak and Cheny4 considered the problem of free and forced convection in a fully developed laminar steady flow through vertical ducts under the conditions of constant heat flux and uniform peripheral wall temperature. Chenoweth and Paoluccis" obtained steady-state, two-dimensional results from the transient Navier-Stokes equations given for laminar cunvective motion of a gas in an enclosed vertical slot with 'large horizontal temperature differences. System Description The problem schematic is shown in Fig. 1. The top and bottom walls of the enclosure are insulated and the left and right vertical walls are V-corrugated. The left and right walls are kept at constant temperature. The temperature of the left wall is Th and that of the right wall is ~, where Th > ~. The characteristic length of the square enclosure is L. The origin of the X-Y coordinate

182 \ INDIANJ. ENG. MATER. SCL AUGUST 1994 Insulat~ 'I I Insulated Insulated L Corrugation frequency = 1 ~ Insulated Corrugatio frequency = 2 Insulated V Corr~~ ~ Insulat~ l Fig. l-schematic system is located at the left bottom corner of the cavity. of the calculation domain u au ax + v au ay = _ ~ p ap ax + v (a2 ax~ + a2 ay~) + SU... (2) Basic Equations. y-momentum equation, The Navier-Stokes equations6 for two-dimensional, incompressible flow with constant properties in cartesian coordinates can be written as fol... (3) u av ax + v av ay = _ p~ ap ay + v (a2v ax2 + a2v) a/ + SV lows: Continuity equation, In the above equations, u and v represent the velocity components in the x and y directions respectively and p is the pressure. The source terms au av -+-=0... (1) SU and SV ax ijy consider the other body and surface forces in the x and y directions respectively and v is x-momentum equation, the kinematic viscosity.

ALl & ALl: NATURAL CONVECTION HEAT TRANSFER 183 By differentiating Eqs (2) and (3) with respect to y and x respectively and then subtracting the results of the former from the later, a single vorticity transport equation can be obtained, to yield, u-+ v-=-+-+ c3q c3q a2q a2q ae Grax ay ax2 ay2 ax. " (11) u-+v-=v -2+-2 + ---... (4) aw ax aw ay (a2w.ax ~w) ay (asv ax as") ay ". (12) where w is the vorticity, defined as, av _ au w= ax ijy Upon defining the stream function, 1jJ as a1jj = u ay _ a1jj = v ax ". (5)... (6)... (7) the Poisson equation relating w to 1jJ may be obtained by substituting Eqs (6) and (7) into Eq. (5)~ a21jj a21jj -2+~+W=O... (8) ax ay The details for implementing wand 1jJ conditions are available in the reference 7. Assuming the properties to be constant other than the density variation in the buoyant forces, the Boussinesq approximation8 which consists of retaining.only the variations of density in the buoyancy terms, may be used on Eq. (4) which results in where Gr and Pr are the Grashof and Prandtl numbers 10 respectively and defined as: Gr=gf3(Th- TJVlv2 Pr= via Here the parameters g, f3 and a represent the acceleration due to gravity, the coefficient of thermal expansion and the thermal diffusivity of the fluid respectively. The boundary follows:- conditions of the problem. are as (i) U= V= 0 at all walls (ii) lji= 0 at all walls (iii) e= 1 at left wall e= 0 at right wall C IV ) (ae) ay y=o - (ae) ay Y= I - 0 Method of Solution The calculation domain is first discretized9 and an example of discretization of the domain is shown in Fig. 2. Following the domain discretization, the integral formulation of the relevant transport equation 7 is impossed on each control u-+v-=v -2+-2 +gf3- aw ax aw ay (a2w ax a2w) ay at ax... (9) The energy transport equation for two-dimensional incompressible flow with constant properties6,9 can be written as, u at ax + v at ay = a (a2t ax2 + a2~ al)... (10) where a is the thermal diffusivity of the fluid. Eqs (4 )-( 10) can be normalised by introducing the following nondimensional quantities, x X=-, L y=l L' u= ul v ' v= vl v Q= wl2 V ' lji=!!! v' T- T. e=-_c Tt,-T:: Fig. 2-Domain discretization

184 INDIAN J. ENG. MATER. SCL, AUGUST 1994 volumell (Fig. 3) of the overall region. The solution of the governing equations is obtained iteratively. and once convergence of the equations has been achieved lated. the following quantities are calcu The local Nusselt number along the hot wall, Nuy, qnl = Nuy= k(th - '4) of) on... (13) The dimensionless total heat flux at the hot wall, Q=- Jy~ Y~ooN I of) -ds(y)... (14) Where S is the dimensionless distance measured along the corrugation of the wall and N is the dimensionless distance measured normal to same. In Eq. (13) qn is the heat flux rate per unit length, at the hot wall and kis the thermal conductivity. Results and Discussion In this investigation local Nusselt number along the hot wall, the total heat flux through the enclosure {ll1dvertical velocity and temperature distributions at the horizontal mid-plane have been examined and discussed for corrugation frequency (CF) 1, 2, 3, and Grashof number (Gr) 103, 10\ 105 The discussion of the results that follows are those obtained by using a 31 x 31 mesh. All the results are presented in dimensionless form. In this investigation the corrugation amplitude was fixed at 5% of the enclosure height for all runs, where the amplitude ''N.' is defined as half the horizontal distance measured from the left extremity of the left wall to its right extremity (see Fig. 1). Henceforth, the left and right extremitie~ of the hot wall will be referred to as the "trough" and "peak", respectively. A numerical value 10-5 was considered as the convergence criterion for receiving better accuracy of the results. It may be mentioned that the convergence checking was performed vorticities by calculating and coefficients the residue from the of the relevant nodal equations and comparing it with the convergence criterion. It was found that there was no appreciable change in results for lower value of convergence criterion than the above. Effects of corrugation frequency (CF) on totai heat flux-figs 4 and 5 show the streamline pattern of the fluid flow and isotherm plot into the enclosure for Grashof number 105, respectively and Table 1 shows the effects of CF on total heat flux (Q) with different Grashof numbers. This table shows that Q increases continuously with the increase in CF for Gr= 103 but for Gr= 104 and 1OS, Q decreases with increase in CF from 1 to 3. The increase of CF from 1 to 3 leads to a higher value of Q for Gr= lo3, which may be attributed to the enhancement of surface area, but the decrease in Q for higher Gr may be explained as the retardation of flow due to increased waviness of the corrugation. This behaviour may also be explained by asserting that at high Gr the fluid velocity increases near the peaks but drops near the troughs as the boundary layer tends to separate. Thus the fluid fails to maintain close contact near the troughs of the corrugation, resulting in decreased convection heat transfer, whereas for Table I-The variation of the total heat flux for corrugated walls with different Grashof numbers Grashof number 1.132 1.135 2.238 4.573 QforCF=2 2.271 4.753 2,295 4.837 1.126 3 1 Fig. 3-A typical control volume shown shaded Fig. 4-Streamline plot (Gr= 105 CF'= 3)

ALl & AU: NATURAL CONVECTION HEAT TRANSFER 185 Gr= 103, the low vertical velocities thus generated enable the fluid to maintain better contact with the corrugated wall. Thus with increasing CF the corresponding enhancement of heat transfer surface leads to an increase in total heat flux at low Gr, but for the case of high Gr, the lower velocities and consequent decrease in convective heat transfer at the troughs offsets the effect of increased surface area. Effects of corrugation frequency on local Nu~ selt number-the decreasing nature in convection heat transfer is evident from Figs 6-8 where it may be observed that the local Nusse\t number which is identical to the dimensionless local heat flux attains minimum values at the troughs of the corrugation. It can be noted from these figures that there is a significant increase in local Nusselt number at the peaks of the corrugation. The Shope of the vertical wall OJ z~.. -.J.0E t) ;Jc OJZ 0 ua 10 12 Grid size =(31x 31) >- -A--A- Gr = 104 C F=2 -<>-<>- ~ Gr r = 10 103 5~ Fig. 5-Isotherm plot (Gr= 105 CF= 3) 02 0'" 06 0'8 1 0 Oimt'nsionless vt'rtical distance ( Y) 14 Grid size = (31 x 31 ) Fig. 7- Local Nusselt n.umber distribution (CF=2) on the hot wall 12 >:' :J Z ~ 10 L."D E :J C 8 II III III ~ 6 Iii u.3 " 2 Gr ': trf C.F = 3 Gr = to"} t 03 12 >- 10 OJ Z ~ 8.0 E ;J c OJ Z 4 t:l u a -.J GrId size =(31)(31) Gr = 104 C Gr G r =!OS} T03 o 0'0 0 2 0'4 0'6 0 8 1 0 Dimens ion less vertic~1 dlstanc e (V) Fig. 6- Local Nusselt number distribution on the hot wall (CF=3) 0_ 00 0 2 0 " tl6 (J ll 1 0 Oimt'nsionless vt'rtlcal dlslonc e (Y) Fig. 8-Local Nusselt number distribution on the hot wall (CF=l)

186 JNDIAN J. ENG. MATER. sel, AUGUST 1994 reason for this is that the peaks cause the fluid to come in contact more intimately with the surface, resulting in large convection heat transfer and consequently the local Nusselt number increases. These figures also show that the peak values of Nuy decrease with increasing vertical distance along the corrugated wall. This may be explained by the fact that the colder fluid collects at the bottom-left corner of the enclosure creating a large temperature gradient with thc hot wall, which is the main driving force for heat transfer and as it moves up and receives heat, the temperature gradient decreases, causing the decrease in local Nusselt number. Effects of corrugation frequency (CF) on vertical velocity and temperature distribution-figs 9 20- and 10 exhibit the effects of CF on vertical velocity distribution at the horizontal mid-plane for 10 Gr= 105 and 103, respectively. Fig. 9 indicates >. 0, that the peak value of 0the vertical velocity dec reases with increase in_i. CF. 0 I-3This o~ trend can be ex,,,,.. 0 2 ' 0 040'6 0 8, plained by examining Fig. 11 which indicates that -10 -,al. the temperature gradient is lower for higher CF, causing a lower buoyant force and hence a lower vertical velocity. Because of this lower velocity, the strength of convection heat transfer decreases with increasing CF. But in Fig. 10 the vertical velocity increases with increase in CF, which leads to an increase in overall heat transfer. Effects of Gr on total heat flux-fig. 12 shows tj:le variation of Q as a function of Gr for different CF and portrays the comparison between the results of the present study and Hussain7 Here it can be pointed out that for higher values of Gr the rate of increase of Q increases with decreasing CF and the different curves "cross" at around Gr= 103 indicating a trend reversal which have been discussed earlier..0 I I Gr = 103 t, I ~ I Grief size = (31 x31) 30., Present lor CF = 1.,. Present lor C F = 2 l] Present tor C F = 3 1, 80-0 > - 20-6 60 _40 20-800 Gr = 105 Grid size = (31 x31) -e-<>- Present lor CF = 1 ~ Presenl tor C F = 2 ~ Present tor CF = 3 Fig. 10-Vertical velocity distribution at the horizontal midplane (Gr= 103) 1-0 0 9. ạ o 0 8 CD 0 7 ~ 0 6 o \.. 0 5 <1> \.. ;:J a 0'" \.. C!J 0. 0 3 E <1> l- 0 2 0,1. o ạ x G r = 105 Grid size =(31 x 31) Present tor CF = ~,. Present tor CF = 2 : D Present tor CF = 3 o~xijbbi8i!~!i:!i!fiio..6 ė 02 0'4 x Fig. 9-Vertical velocity distribution at the horizontal mid 'plane for different corrugation frequencies (Gr= 105) 06 \"0 o I L I I I d o 0 2 0'" 0,6 0,8 1 0 Fig. 11- Temperature distribution at the horizontal mid-plane (Gr= 105) X

ALl & ALl: NATURAL CONVECTION HEAT TRANSFER 187 5 5 --'-- C F: 2 ( Pr~s~nt ---- -- CF: 3}- 1 ----- CF: O(R~f. 7) study) 4 4 a 3 d J Slope = 0'132 C = 2'OOh 105 I 8 12 16 20 24 28 36 lc r]0'311 It rf'019 2 Fig. 13-Graphical representation of the correlation of V-corrugation 1 103 104 Gr Fig. 12- Total heat flux vs Grashof number Table 2 - Total heat flux for grids 31 x 31 and 49 x 49 Grashof number = 105 Grid size 31 x 31 49 x 49 QforCF= 4.8371 4.8372 I QforCF=2 4.7529 4.7531 105 Qfor CF= 3 4.5732 4.5734 Effects of grid refinement on total heat flux-to observe the effect of mesh size a grid refinement study has been made using 49 x 49 mesh for Grashof number 105. The results of the runs that, follows the Table 2, show that the total heat flux for the mesh 31 x 31 agrees quite well with that of 49 x 49 mesh. Correlation-On the basis of the above results a correlation equation can be developed in terms of Q, Gr and CF which is found to be capable of correlating the computed data within ± 1.4%. The correlation is: Q= 0.132 [Gr]0311 [CF]-00l9 where, 1000 ~ Gr~ 100000 1 ~CF~3 The graphical representation of the above correlating equation is shown in Fig. 13. Conclusions The results obtained in this study indicate the followings: 1 The overall heat transfer decreases with the increase of corrugation frequency for higher Grashof number but the trend is reversed for lower Grashof number. 2 At low Grashof number the conduction mode prevails and as corrugation frequency increases, total heat flux is enhanced for the enhancement of surface area whereas for high Grashof number the fluid fails to collect heat by transport due to increasing corrugation. 3 The peak values of local Nusselt number decrease with the increase of vertical distance along the corrugated surface and local Nusselt number increases significantly at the peaks and attains minimum values at the troughs of the corrugation. 4 The vertical velocity of the fluid decreases with the increase of corrugation frequency for high Grashof number because of the lower temperature gradient as the boundary layer tends to separat~ but the trend is reversed for low Grashof number. 5 The decreasing trend of total heat flux may find application in practical situations where heat transfer reduction is desired across large temperature diffcrences by increasing the corrugation of the vertical wall. Acknowledgement The authors are highly grateful to Computer Centre of Bangladesh University of Engineering and Technology, Bangladesh for providing facilities of the numerical computation. Nomenclature u = horizontal velocity, m/s v = vertical velocity, m/s U - dimensionless horizontal velocity

188 INDIAN J. ENG. MATER. SCI., AUGUST 1994 v = dimensionless vertical velocity x = horizontal cartesian coordinate, m y = vertical cartesian coordinate, m X = dimensionless horizontal cartesian coordinate Y = dimensionless vertical cartesian coordinate J1. = absolute viscosity, kg/ms v = kinematic viscosity, m2/s p = density, kg/m3 1/1 = stream function, m2/s lji = dimensionless stream function w.= vorticity, S-I Q = dimensionless vorticity a = thermal diffusivity, m2/s fj = coefficient of thermal expansion, 11K g = acceleration due to gravity, mls2 h = convective heat transfer coefficient, W/m2 K k = thermal conductivity, W/m K Gr= Grashof number Pr = Prandtl number References 1 Chinnappa J C V, Int J Heat-Mass Transfer, 13 (1970) 117-123. 2 Randall R K, Mitchell J W & El-Wakil M M, J Heat Transfer, Trans ASME, 101 (1979) 120~125. 3 Zhong Z Y, Yang K T & Lloyd J R, J Heat Transfer, 18 (1985) 113-146. 4 Nayak A L & Cheny P, Int J Heat Mass Transfer, 18 (1975) 227-236. 5 Chenoweth D R & Paolucci S, J Fluid Mech, 169 (1986) 173-210. 6 Yuan S W, Foundation of fluid mechanics (Prentice-Hall ofindia, New Delhi), 1969, 103-139. 7 Husain S R, Extensions of the control volume based finite element method for fluid flow problems, Ph D Thesis, Texas A & M University, 1987. 8 Turner J S, Buoyancy effects in fluids (Cambridge University Press, Cambridge), 1973,6-24. 9 Patankar S V, Numerical heat transfer and fluid flow (D C Hornisphere, Washington), 1980. 10 Kreith 'F, Principles of heat transfer, 3rd ed (Harper & Row), 1976,309-407. 11 Baliga B R & Patankar S V, Num Heat Transfer, 3 (1980) 393-409.