INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6340 (Print) ISSN 0976 6359 (Online) Volume 6, Issue 3, March (2015), pp. 14-20 IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2015): 8.8293 (Calculated by GISI) www.jifactor.com IJMET I A E M E EFFECT ON HEAT TRANSFER AND THERMAL DEVELOPMENT OF A RADIATIVELY PARTICIPATING FLUID IN A CHANNEL FLOW NDZANA Benoît Senior Lecturer, National Advanced School of Engineering, University of Yaounde I, Cameroon BIYA MOTTO Frederic, Senior Lecturer, Faculty of Sciences, University of Yaounde I, Cameroon LEKINI NKODO Claude Bernard P.H.D. Student; National Advanced School of Engineering, University of Yaounde I, Cameroon ABSTRACT The paper deals with Simultaneous heat transfer by convection and radiation in a channel flow between two infinite black parallel plates is investigated. The effect of radiation on the heat transfer and the full thermal development of the flow is studied. The effect of scattering albedo, conductionradiation parameter and the optical thickness are examined. The radiation is shown to substantially alter the heat transfer downstream before the thermally fully developed conditions. The full thermal development is shown to exist for the constant wall temperature case, while it is pushed further downstream and could not be seen for the constant wall heat flux case. While the radiation greatly affects the heat transfer when the fluid is heated, for the cooling case radiation effect decreases along the stream wise direction and vanishes at the fully developed conditions. Keyword: Heat Transfer; Thermal Development; Channel Flow; Albedo; Radiation. 14
NOMENCLATURE = hh = = = h h = h h = =, = h =h = h = = = ℵ= = ℵ = h = = = = Ω= INTRODUCTION 4 The analysis of combined modes of heat transfer has been the subject of many investigators. Different numerical techniques, iterative in nature and differing from each other in the way the radiation part is handled, have been employed to solve the combined radiation convection problems. The present work focuses on the effect of radiation on the heat transfer and the full thermal development of an absorbing, emitting and anisotropically scattering fluid between two infinite black parallel plates, as it combines with convection. The problem is to solve the energy equation, a parabolic partial differential equation, which couples with the equation of radiative transfer, an integro-differential equation. A multilayer technique is employed to obtain the radiation contribution. Scaling laws derived in [1] and [2] have been utilized to treat the general case of an absorbing, emitting and anisotropically scattering fluid by modelling it as an equivalent non-scattering medium. The effect of some parameters such as the scattering albedo, the optical thickness and the conduction radiation parameter are considered. GOVERNING EQUATIONS The governing equation to be solved, with the assumptions of constant properties, no internal heat generation and no heat production due to viscous dissipation, is written as: = (1) Where the second term in the right hand side is the radiation contribution 15
The boundary and inlet conditions are: = at =0 =0 at = (2) = or = at =0 In a dimensionless form, the energy equation is written as: = ξ ℵ ℵ (3) The conduction radiation parameter represents the relative importance of conduction to radiation in the transverse direction of the flow. The divergence of the radiative heat flux is written from [3] as: ℵ = 1 [4 ℵ ] (4) Where ℵ us the average radiative intensity given by: ℵ =2 ℵ (5) The radiation intensity ℵ, is governed by the equation of radiative transfer:,ω +,Ω = 1 +,Ω Ω,Ω Ω (6) A solution of the equation of transfer is needed in the analysis of this problem. However, the knowledge of the temperature is necessary. We see clearly that the equation of radiative transfer couples with the energy equation. For an anisotropically scattering medium, even with the use of approximate methods, the difficulty in solving the equation of transfer still remains. Thus, we seek a solution for the anisotropically scattering medium by solving the equation of transfer for a simpler problem. This is achieved by using the scaling laws which model an anisotropically scattering layer by a non-scattering one. DISCRETIZATION The energy equation is discretized using the control volume approach [4] and a fully implicit scheme. The generation term or the radiation contribution term is linearized using the tangent method [4]. The discretized energy equation for a node (i,j) is written as:, +, +, = (7) Where = ξ ℵ (8) 16
=1+2 ℵ +4, (9) = ℵ (10) =, + 3, +, (11) The system of equations obtained is solved by a tridiagonal matrix algorithm. A the first step, the, s the inlet temperature. COMPUTATION CONSIDERATION The full radiation solution coupled with the convection problem is so computationnaly intensive. Although the scaling greatly simplifies the radiation solution and reduces the calculation time required to obtain the radiation contribution, the sudden change in the boundary conditions causes the problem to be very stiff and presents numerical difficulties regardless of the computation technique used for the radiation portion of the solution. Because of the stiffness of the problem at the entrance region, very small axial steps are necessary to ensure acceptable accuracy. The present work focuses also on reducing the computation time by generating a non-uniform grid in the axial direction, which is keyed to the pure convection Nusselt number development. This allows for very small steps in the beginning where all the large changes are taking place. Then, the larger steps downstream still yield accurate results and significantly reduce the computation time. Using uniform axial steps, up to 243 seconds of CPU time on a single processor Cray 2 is necessary for stable solutions. With a non-uniform axial grid and taking advantage of the problem symmetry, the computation time is reduced to 80 to 100 seconds with comparable accuracy. Some numbers of study case for the grid generation is shown in table 1. TABLE 1: =1.0 =0.01 ℵ =2.0 Uniform grid Non-uniform grid Entrance step size 0.0001 0.0001 0.0001 Largest step 0.0001 0.0288 0.0288 Downstream position 1.0000 0.9877 0.9877 Number of steps 10000 711 711 Nusselt number 7.5405 7.5401 7.5500 CPU time (cray 2) 243 sec 56 sec 21 sec RESULTS Non-uniform ½ channel The effect of radiation on the heat transfer is shown through a study of the effect of various parameters. Figure 1 shows the development of the bulk temperature which increases slowly in the entrance region and then drastically goes to a maximum constant value downstream where the fully developed condition is reached. The bulk temperature is seen to be higher in the entrance region for lower albedo values since the medium is then more absorbing. The albedo is seen to have no effect in the fully developed flow. A plot of the total (convection + radiation) Nusselt number along the axial direction figure 2 shows first the usual decrease at the entrance region, but then unlike the pure 17
convection case which reaches a constant limiting value, it passes by a minimum and then sharply increases to come finally to a limiting value. These results are in agreement with the results presented by Chawla and Chan [5] for the entry region. The radiation effect dominates past the minimum point. The minimum occurs earlier and is higher for a less scattering medium. Figure 3 shows the effect of radiation being stronger for smaller conduction-radiation parameter ie conduction dominating radiation in the transverse direction, and also as the medium is optically thicker. FIGURE 1: Effect of albedo on bulk temperature FIGURE 2: Effect of albedo on total Nusselt number 18
FIGURE 3: Effect of cond-rad parameter on total Nu If a hot fluid enters the channel and is cooled, the scattering is shown to affect the heat transfer in the entrance region. The effect of radiation vanishes downstream as the fluid is cooled down to finally reach the fully developed condition. The same limiting solution as the pure convection case is then reached. This is illustrated in figure 4. FIGURE 4: Effect of albedo on total Nu (Cooling case) 19
The same behaviour, that is the Nusselt number drastically increases beyond a certain axial location, and the same effects of the albedo, optical thickness and the conduction-radiation parameter, is seen in the heat transfer when the walls are a source of heat flux to the fluid which enters cold. But the thermally fully developed conditions seem to be pushed further downstream. CONCLUSION The effect of radiation as it combines with convection is determined for this channel flow. Unlike a pure convection problem, with a radiatively participating fluide the heat transfer coefficient is substancially altered downstream before the full thermal development. The effect of radiation is stronger for lower values of and for higher optical thicknesses. If the walls are cold and the fluid enters with a uniform hot temperature, the radiation effect vanishes along the axial direction as the fluid is cooled down. The same trend of the Nusselt number is seen for constant heat flux boundaries, going by a minimum then dramatically increasing. However the thermally fully developed conditions are pushed futher downstream and could not be seen. REFERENCES 1. H. Lee, and R. O. Buckius, Scaling anisotropic scattering in radiation heat transfer for a planar medium, Journal of Heat Transfer, vol. 104, 1982, pp 68-75 2. H. Lee, and R. O. Buckius, Reducing scattering to non-scattering problems in radiation heat transfer, Int. J. Heat Mass Transfer, vol. 26, 1983, pp 1055-1062 3. M. NecatiOzisik, Radiative transfer and interactions with conduction and convection, A Wiley-interscience publication, 1973 4. S.V. Patankar, Numerical heat transfer and fluid flow, Mc Graw-Hill, New York, 1980 5. T. C. Chawla, and S. H. Chan, Spline collocation solution of combined radiation convection in thermally developing flows with scattering, Numerical Heat Transfer, vol. 3, 1980, pp 47-65 6. S. Tiwari, D. Chakraborty, G. Biswas, P.K. Panigrahi Numerical prediction of flow and heat transfer in a channel in the presence of a built-in circular tube with and without an integral splitter, International journal of heat and mass transfer, volume 48 issue 2, January 2005. 7. Chandra A., Chhabra R.P., Flow over and forced convection heat transfer in Newtonian fluids from asemi-circular cylinder, Int. J. of H.and M. T. 54 (2011) 225-241. 8. Ashish Kumar, Dr. Ajeet Kumar Rai and Vivek Sachan, An Experimental Study of Heat Transfer In A Corrugated Plate Heat Exchanger International Journal of Mechanical Engineering & Technology (IJMET), Volume 5, Issue 9, 2014, pp. 286-292, ISSN Print: 0976 6340, ISSN Online: 0976 6359. 9. Sunil Jamra, Pravin Kumar Singh and Pankaj Dubey, Experimental Analysis of Heat Transfer Enhancementin Circular Double Tube Heat Exchanger Using Inserts International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 3, 2012, pp. 306-314, ISSN Print: 0976 6340, ISSN Online: 0976 6359. 10. Doddeshi B C, Vilas Watve, Manu S and Dr. Sanjeevamurthy, Experimental Investigation of Flow Condensation Heat Transfer In Rectangular Minichannel International Journal of Mechanical Engineering & Technology (IJMET), Volume 5, Issue 9, 2014, pp. 251-258, ISSN Print: 0976 6340, ISSN Online: 0976 6359. 20