THE APPLICATION OF DISCRETE TRANSFER METHOD IN THE SIMULATION OF RADIATION HEATING OR COOLING

Transcription

1 , Volume, Number 1,.1-5, 2003 THE APPLICATION OF DISCRETE TRANSFER METHOD IN THE SIMULATION OF RADIATION HEATING OR COOLING Guangcai Gong, Kongqing Li and Gengxin Xie Civil Engineering College, Hunan University, China (Received 29 May 2002; Acceted 18 October 2002) ABSTRACT This aer resents a brief introduction of discrete transfer method (DTM) and the results comared ith other methods in the calculation of radiation heat transfer. It is concluded that DTM is very simle, reliable and valid in the calculation of radiation heat transfer. It as also shon that DTM should be advocated and alied in such fields as CFD, assessment of thermal comfort, design of radiant heating or cooling etc. 1. INTRODUCTION Radiation is a comlicate henomenon, articularly in an absorbing, emitting and scattering medium. The equations of radiant heat transfer are highly nonlinear. In ractice engineers adot various methods such as Monte-Carlo method (MCM), zonal method (ZM), flux method (FM), sherical harmonics method, discrete ordinate method to handle the radiation integral equations. MCM can be alied in the fields ith comlex boundary and handle various avelength radiation conveniently. Its result fluctuates around the true solution. By increasing the number of rays traced, it ill get the actuate result eventually. But it ill consume a lot of comutation time and may encounter strong challenges of converging difficulties. ZM is not suitable to solve the cases ith comlex boundary conditions. It also cannot deal ith nongray body and the cases in hich the radiant roerty is the function of temerature. FM is very simle, but ith oor recision, sometimes cannot accord ith the hysical henomena and cannot deal ith scattering, anisotroism medium. Those methods mentioned above consume a lot of comuter time and need high memory. Discrete transfer method (DTM) has the characteristic of all above methods. Its main idea is to consider the boundary surface as the radiant source and absortion. It divides energy emitted into the hemishere into finite number of rays and assumes that the radiation leaving the surface element in a certain range of solid angles can be aroximated by the single ray. Those rays are absorbed or reflected by the inner grid before they reach the boundary grids. The energy balances at each boundary surface [1-3]. The first descrition of DTM as resented by Lockood and Shah [1], and Doherty and Faireather [2]. Over the last several years scientists and technicians develo the method and have made raid rogress [3]. Coelho and Carvalho [] find that the DTM being used to calculate radiative heat transfer in combustion chambers is not conservative and ut forard the conservative formulation of the method by local correction of energy er unit time of the irradiation rays leaving the boundary. In order to consider the strong ray effects hich usually leads to a large error, Liu et al. [5] resented an imroved DTM in hich a total of nine discrete directions over a control angle instead of a single discrete direction as used in the traditional DTM as selected and the incident radiative intensity over this control angle is then relaced by a eighted-averaged intensity from these nine discrete directions. Talukdar and Mishra [6] investigated the effect of variable thermal conductivity on transient conduction and radiation heat transfer in a lanar medium. Versteeg et al. [7] discussed the truncation error due to the satial discretisation of the enclosure surface and medium conditions. Liu and Chen [8] investigated the accuracy and efficiency of the MDTM comared ith the solutions from the exact aroach, the DTM, and discrete ordinates method (DOM) ith three benchmark roblems covering different geometric and boundary conditions. Malalasekera [9] used a generalized eighted-sum-of-gases-discrete transfer deal ith a comlex geometry of a entroof sark-ignition engine and got the instantaneous radiative heat flux. Furthermore they integrated the WDT into the comutational fluid dynamics code KIVA-II. Versteeg et al. [10] examine aroximation errors in the heat flux integral of the discrete transfer method. Novo et al. [11] use the ray domain 1

2 decomosition and the satial domain decomosition in the alication of DTM and achieve good accurate result. Docherty and Faireather in their aer [2] get the rediction of radiative transfer from non-homogeneous combustion roducts using the DTM. In this study, e use the discrete transfer method to solve the radiation heat transfer in the design and assessment of radiation heating esecially the net heat gain of the targets (such as ersons) hich is very imortant for the engineers but is still far from solution. 2. THE PHYSICAL IDEA OF DIS- CRETE TRANSFER METHOD A detailed descrition of the discrete transfer method can be found in Lockood and Shah [1]. We ill only consider asects of the algorithm of relevance to the issue addressed in the resent article. The grids used for both DTM method and flo field simulation are identical. We call the boundary element surface that emits energy as radiating surface, surface that receives energy is termed as receiving surface (See Fig. 1). In Fig. 1, P i and P j are the boundary grids hose center oints are P i and P j. At the boundary grid P i, e can dra N = N θ Nφ rays hich divide the hemishere of P i into many differential solid angles. The direction of each differential solid angle (dω r ) is Ω r. The rays are called characteristics line. The N rays ill intersect other boundary faces at N oints. At each radiating surface, rays are fired at discrete values of the zenith and azimuth angles (θ = 0 ~ 90, φ = 0 ~ 2π, covering the radiating hemishere). Each ray is then traced to determine the control volumes it intercets as ell as its length ithin each control volume. The transfer equation for thermal radiation along a ray, neglecting scattering can be ritten as the folloing formulation. di aσt = ai ds π (1) The intensity entering and leaving the control volume (or grid) is then comuted by integrating the radiation transfer equation (1) along a series of rays emanating from the boundary faces in each discrete control volume as: I σtn as n 1 (1 e ) as ne = I (2) π here T g is the temerature of the medium in the control volume (K); a is absortion coefficient; s is the ray s length ithin the grid (or control volume), m; σ is Stefan Boltzmann constant (5.672E-8 Wm -2 K - ); I n and I n1 are the intensity of the ray on entry and exit at the nth control volume. The radiation intensity aroaching the oint P i is integrated to yield the incident radiation heat flux, q, as: N q = IdΩ I sin θcosθdφ (3) r r= 1 here Ω is the hemisherical solid angle; and I is the intensity of the incoming ray. The net radiation heat flux from the element P i is then comuted as a sum of the reflected ortion of q and the emissive oer of the element: q = (1 ε )q ε σt () here ε is the emissivity of all; and T is the temerature off all. Hence, for grey alled enclosures, the element boundary condition for the radiation intensity I op of the ray emanating from the oint P i can be calculated aroximately using the folloing equation. IoP = q P / π (5) here I op is the radiation intensity of the element boundary at the oint P i. 3. THE FLOW CHART OF DISCRETE TRANSFER METHOD Fig. 1: Delineation of grid, receiving surface and radiating surface Since the initial value of the radiation intensity emanating form the boundary surface is not knon exactly, the incident flux and emitted intensity make the discrete transfer method a guess and correct rocedure in hich an estimate of the incident flux distribution is iteratively imroved. So e can dra the flochart of DTM in Fig. 2. 2

3 For the comromise beteen comutation time and recision, the convergence criterion is set to in the rogram. Of course the smaller the criterion value, the more comutation time is needed, and the more accurate the rediction of radiative heat distribution is obtained. The calculation results are shon in Figs. to 8 and Tables 2 to. Because of symmetry of the calculated room, it is easy to verify the results qualitatively. Furthermore, the total heat gain is in very good agreement ith the results comuted by other methods. The net heat gain of the erson is also aroximately equal to the results using the vie factor. So e can say the results are reliable and believable. Fig. : Net heat contours of surface 0 Fig. 2: The flochart of DTM. APPLICATIONS The calculation examle is shon in Fig. 3. The alls (or ceiling and floor) are labeled by the number from 0 to 5. The arameters of both the enclosures and erson are ritten in Table 1. Room size OA=5m OC=m OG=3m Fig. 5: Net heat contours of the surface 1 Person Height 1.7m Weight 70kg Label ABCO surface 0 GDEF surface 1 GDAO surface 2 FEBC surface 3 GFCO surface ADEB surface 5 Fig. 3: The calculation examle delineation Fig. 6: Net heat contours of the surface 2 3

4 Fig. 7: Net heat contours of the surface 3 Fig. 8: Net heat contours of the surface or 5 Table 1: The arameter of the alls and erson Label or name Person * Temerature ( o C) Emissivity * outer surface arameter of clothing Table 2: Net radiation heat gain of the erson under different temerature of radiator Temerature ( o C) Net radiation heat (W) Table 3: Convergence time and mean relative error under difference Nθ, Nφ (comared ith the results calculated using zonal method) Nθ = 50 Nθ = 100 Nθ = 150 Nθ = 100 Nθ = 100 Nφ = 50 Nφ = 50 Nφ = 50 Nφ = 75 Nφ = 100 Convergence time (s) * Relative error (%) * Datum obtained by the comuter ith CPU: PENTIUM-MMX166 MEM: 32M Table : Relative error comared ith the results by other methods (Each surface as divided into 100 zones) Label of enclosure Maximum Comared ith zonal method -1.22% 0.06% 0.0% -0.5% -1.79% -1.19% -1.22% Comared ith Monte-Carlo method 1.67% -0.87% 0.67% -0.3% 1.3% -3.2% 2.10% For the same roblem, it took about 8minutes to converge by Monte Carlo method. Meantime vie factor and both direction and total exchange area of the erson can be obtained using the DTM too (omitted). 5. CONCLUSIONS From the calculation results, e can get the distribution of radiant energy. If the radiant energy as used as the heat source and handled as the second boundary condition, it is ready for the

5 simulation of flo field. It can also be used in the study of thermal comfort of eole. Furthermore, it is very exedient to estimate the total radiant heat and to redict the effect of the roject of heating or cooling by radiator during the design stage. 11. P.J. Novo, P.J. Coelho and M.G. Carvalho, Parallelization of the discrete transfer method, Heat Transfer: Part B - Fundamentals, Vol. 35, No. 2, (1999). REFERENCES 1. F.C. Lockood and N.G. Shah, A ne radiation solution method for incororation in general combustion rediction rocedures, 18th Symosium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, (1981). 2. P. Doherty and M. Faireather, Prediction of radiative transfer from nonhomoqeneous combustion roducts using the discrete transfer method, Combustion and Flame, Vol. 71, (1988). 3. P.S. Cumber, Imrovements to the discrete transfer method radiation heat transfer, International Journal of Heat and Mass Transfer, Vol. 38, No. 12, (1995).. P.J Coelho and M.G. Carvalho, A conservative formulation of the discrete transfer method, Journal of Heat Transfer, Vol. 119, No. 1, (1997). 5. J. Liu, S.J. Zhang and Y.S. Chen, An imroved discrete transfer method for 3-D surface radiation calculation, Numerical Heat Transfer: Part B - Fundamentals, Vol. 2, No. 3, (2002). 6. P. Talukdar and S.C. Mishra, Transient conduction and radiation heat transfer ith variable thermal conductivity, Numerical Heat Transfer: Part A -Alications, Vol. 1, No. 8, (2002). 7. H.K. Versteeg, J.C. Henson and W. Malalasekera, Satial discretization errors in the heat flux integral of the discrete transfer method, Numerical Heat Transfer: Part B - Fundamentals, Vol. 38 No., (2000). 8. J. Liu and Y.S. Chen, Prediction of surface radiative heat transfer using the modified discrete transfer method, Numerical Heat Transfer: Part B - Fundamentals, Vol. 38, No., (2000). 9. W.C. Malalasekera, Full-cycle firing simulation of a ent-roof sark-ignition engine ith visualization of the flo structure, flame roagation and relative heat flux, Proceedings of the Institution of Mechanical Engineers - Part D - Journal of Automobile Engineering, Vol. 21, No. 8, (2000). 10. H.K. Versteeg, J. C. Henson and W. Malalasekera, Aroximation errors in the heat flux integral of the discrete transfer method, Part 1: Transarent Media, Numerical Heat Transfer: Part B - Fundamentals, Vol. 36, No., (1999). 5