Heat Transfer Theoretical Analysis Experimental Investigations and Industrial Systems Part 17 ppt

The Rate of Heat Flow through Non-Isothermal Vertical Flat Plate 629 The heat flux density is approximately 10 W/m 2 , and the heat transfer coefficients ~2 W/m 2 · K. 6. Conclusion Free convection along both sides of a vertical flat wall was considered within the framework of the laminar boundary-layer theory and for the case where only the temperatures of the fluid far away from the wall are known. It has been shown how to determine the average surface temperatures T 1 and T 2 together with the corresponding heat transfer coefficients in order for the equations (1) and (2) to yield the correct value for the total heat flow across the wall. In particular, if the small surface temperature variations θ L, R (x*) are neglected, the heat transfer from the wall to the fluid or vice versa is determined by the Pohlhausen solutions Θ L, R (ξ) only. The corresponding Nusselt number ()R Nu , for example, is obtained from (19a) by neglecting the J R term. This yields 1/4 1/4 3 () 00 2 0 0.471 R LR R R gH TT Nu T ν ⎛⎞ ⎛⎞ − = ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ . (34) It differs less that one percent as compared to the values given by eqs. (30a) and (31) which are valid for good thermal conductors. Consequently, the Pohlhausen solution can be therefore safely used in this case. For poor thermal conductors, like brick or concrete walls, the corrections may be more substantial. In particular, for a brick wall, the correction, obtained by comparing (32) and (34), is roughly 10 percent and it should be taken into account. In numerical calculations, the Newton method turned out to be sufficient in solving the equations for the temperature corrections to the Pohlhausen solution. The simple iteration procedure, however, was found to have a rather restricted range of validity (large thermal conductivity, large aspect ratio of the plate). 7. Mathematical note The system of equations (23a) and (23b) is defined for n = 1, 2,… only. For n = 0, the equations of both parts of the system simplify to a normalization conditions, 1 0 (*) * 1 L Fxdx = ∫ , 1 0 (*) * 1 R Fxdx = ∫ . (MN.1) It is natural to look for the functions F L and F R as elements of some linear subspace S m ⊂ r C ([0, 1]) where S m should become dense in r C ([0, 1]) as m → ∞. Let us denote the basis of S m as s i , i = 0,1,…, m. Then the unknown functions F L and F R could be written as Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 630 F L = , 1 m Li i i cs = ∑ , F R = , 1 m Ri i i cs = ∑ . We expect on physical grounds F L and F R to be rather smooth so r ∈ N could be assumed to be at least 2. This indicates that the Fourier coefficients of the unknown functions 1 0 (*)cos( *) * L Fx nxdx π ∫ , 1 0 (*)cos( *) * R Fx nxdx π ∫ , as well as of the other dependent quantities involved should decay at least as O(1/n 2 ) or faster. As a consequence, only a small part of the infinite system is expected to be significant for F L and F R . Thus, for a particular choice of m ∈ N only the equations n = 0, 1, , m are taken into account. This gives a system of 2(m + 1) nonlinear equations for the unknown coefficients c L := ,0 () m Li i c = , c R := ,0 () m Ri i c = , written in short as, f(c L ) = g 1 (c L , c R ), f(c R ) = g 2 (c L , c R ). (MN.2) The structure of the system (MN.2) follows from (23) but with (MN.1) added to each equations block. There are two important steps to be considered. The first is the choice of the subspace S m . We decided to try first perhaps the simplest approach, by choosing the subspace S m as the space of polynomials S m = P m of degree ≤ m. The numerical results turned out satisfactory. Alternatively, we could always switch to a proper spline space. The second step regards the efficient numerical solution of the system (MN.2). Inspection of the equations (23) reveals that the function f depends linearly on the unknowns. Since the functions g i are much more complicated, the direct iteration seems to be a cheap shortcut. So, with the starting choice incorporating the conditions (MN.1), (0) (0) ,, 1 2( 1) Li Ri cc m == + , i = 1, 2, …, m, (MN.3a) (0) (0) ,0 ,0 1 11 1 2( 1) m LR i cc mi = ==− + ∑ , (MN.3b) the direct iteration reads, f(c (1)k L + ) = g 1 (c ()k L , c ()k R ), f(c (1)k R + ) = g 2 (c ()k L , c ()k R ), k = 0, 1, … . The Rate of Heat Flow through Non-Isothermal Vertical Flat Plate 631 This approach was quite satisfactory for some parameter values, but failed to converge for others. Clearly, the map involved in this case ceases to be a contraction. However, the Newton method turned out to be the proper way to solve the system (MN.2). For any consistent data choice and particular m, only several Newton steps were needed. The initial values of the unknowns were again taken as in (MN.3). The Jacobian matrix J( c L , c R ), needed at each Newton step involving the solution of a system of linear equations, () () () () 1 () () () () () () 2 () (,) (,) () (,) kkkk LLLR kk LR kkkk RRLR J ⎛⎞⎛ ⎞ Δ− ⎜⎟⎜ ⎟ == ⎜⎟⎜ ⎟ Δ− ⎝⎠⎝ ⎠ cccc cc cccc fg fg , and a correction (1) () () (1) () () kkk LLL kkk RRR + + ⎛⎞⎛⎞⎛⎞ Δ ⎜⎟⎜⎟⎜⎟ =+ ⎜⎟⎜⎟⎜⎟ Δ ⎝⎠⎝⎠⎝⎠ ccc ccc , k = 0, 1, …, admits no close form and has to be computed numerically. It is a simple task to compute all the partial derivatives involved if the three basic terms that depend on the unknown coefficients are determined. A brief outline is as follows. For a given S m , let 0 () m ii s = be its basis, and F = 1 m ii i cs = ∑ , c := 0 () m ii c = , stands for F L and F R , and 0 0 0 () () () x m x ii i xFuducsudu ζ = == ∑ ∫∫ . Then () () j j Fx s x c ∂ = ∂ , 0 () () x j j xsudu c ζ ∂ = ∂ ∫ , j = 0, 1, …, m. Further, 11 00 ()cos( ) ()cos( ) j j Fx nxdx sx nxdx c ππ ∂ = ∂ ∫∫ , j = 0, 1, …, m, which yields all coefficients in both n = 0 equations as well as the parts of the elements in J that contribute by the partial derivatives of the function f. In order to compute ∂g i /∂c j , the following two terms have to be determined, Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 632 11 1 3/2 1/2 3/2 1/4 1/4 5/4 00 00 () 3 () 1 () cos( ) ( )cos( ) cos( ) ( ) 24 () () () x jj j Fx Fx Fx n x dx s x n x dx n x s u dudx c xx x πππ ζζ ζ ∂ =− ∂ ∫∫ ∫∫ and 1 3/4 1/2 0 11 3/4 3/2 1/2 1/4 000 () '() cos( ) () 2() '() '()() 3'() ( ) cos( ) cos( ) ( ) 4 2() () () j x jj j xFx nxdx c Fx Fxs x F xs x Fx xnxdx nxsududx Fx Fx x ζ π ζπ π ζ ∂ = ∂ − − ∫ ∫∫∫ where the prime indicates the ordinary derivative with respect to x. Since n is rather small, it turned out that the use of the Filon’s quadrature rules was not necessary. 8. Nomenclature a L,R (i) defined by eq. (24) A surface area of the wall F L, R (x*), ζ L, R (x*) defined by eq. (20c) g acceleration of gravity G = βg(T s – T 0R )H 3 /ν 2 Grashof number h L , R convection transfer coefficients to the left and right of the wall J L, R defined by eq. (15d) k thermal conductivity L, H thickness and height of the wall (,)LR Nu Nusselt numbers associated with the left- and right-hand surface of the wall P = ν/α Prandtl number Q  , q = /QA  heat flow, heat flow density T 0L , 0R air temperature far from the wall to the left and right of the wall T 1, 2 temperature of the left and right wall surface T s characteristic wall surface temperature U = k/L thermal transmittance u, v x- and y-component of the velocity field x* = x/H, y* = y/H dimensionless coordinates Greek symbols α thermal diffusivity β thermal-expansion coefficient of the air The Rate of Heat Flow through Non-Isothermal Vertical Flat Plate 633 γ defined by eq. (15c) ζ L,R (x*) defined by eq. (20c) η viscosity κ L, R defined in eq. (15e) ν = η/ρ kinematic viscosity ξ G R 1/4 y*/(4x*) 1/4 ρ mass density Ф L, R (ξ), Pohlhausen solution Θ L, R ( ξ ) temperature function associated with Ф L, R (ξ) (*,) R x φ ξ , (*,) R x θ ξ corrections to the Pohlhausen solution introduced in eqs. (5), (6) ψ stream function introduced in eq. (5) Subscripts, superscripts a air L, R left, right s surface w wall * dimensionless coordinate based on H 9. References Grimson, J. (1971). Advanced Fluid Dynamics and Heat Transfer, McGraw-Hill, Maidenhead, pp. 215-219 Kao, T.T.; G.A. Domoto, G.A. & Elrod, H.G. (1977). Free convection along a nonisothermal vertical flat plate, Transactions of the ASME, February 1977, 72-78, ISSN: 0021-9223 Landau, L.D.; Lifshitz, E.M. (1987). Fluid Mechanics, ISBN 0-08-033933-6, Pergamon Press, Oxford, pp. 219-220 Miyamoto, M.; Sumikawa, J.; Akiyoshi, T. & Nakamura, T. (1980). Effects of axial heat conduction in a vertical flat plate on free convection heat transfer, International Journal of Heat and Mass Transfer, 23, 1545-53, ISSN: 0017-9310 Ostrach, S. (1953). An analysis of laminar free convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, NACA Report 1111, 63-79 Pohlhausen, H. (1921). Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit Kleiner Wärmeleitung, ZAMM 1, 115-21, ISSN Pop, I. & Ingham, D.B. (2001). Convective Heat Transfer, ISBN 0 08 043878 4, Pergamon Press, Oxford, pp. 181-198 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 634 Pozzi, A. & Lupo, M. (1988). The coupling of conduction with laminar natural convection along a flat plate, International Journal of Heat and Mass Transfer, 31, 1807-14, ISSN: 0017-9310 Vynnycky, M. & Kimura, S. (1996). Conjugate free convection due to a heated vertical plate, International Journal of Heat and Mass Transfer, 39, 1067-80, ISSN: 0017-9310 25 Conjugate Flow and Heat Transfer of Turbine Cascades Jun Zeng and Xiongjie Qing China Gas Turbine Establishment P. R. China 1. Introduction Heat transfer design of HPT airfoils is a challenging work. HPT usually requires much cooling air to guarantee its life and durability, but that will affect thermal efficiency of turbine and fuel consumption of engine [1] . The amount of blade cooling air depends on the prediction accuracy of temperature field around turbine airfoil surface, which is related to the prediction accuracy of temperature field in laminar-turbulent transition region. The laminar-turbulent transition is very important in modern turbine design. On suction side of turbine airfoil, the flow is relaminarized under the significant negative pressure gradient. In succession, when the relaminarized flow meets enough large positive pressure gradient, laminar-turbulent transition appears. In transition region, the mechanisms of flow and heat transfer are complicated, so it is hard to simulate the region. Methods of conjugate flow and heat transfer analysis have been discussed extensively. In 1995, Bohn [2] simulated the heat transfer along Mark II [3] cascade using a two-dimensional (2D) conjugate method at the transonic condition (the exit isentropic Mach number is 1.04). In his simulation, the turbulence model used was Baldwin-Lomax model. The result was that the max difference between the predicted temperature along the cascade and the test data was not larger than 15K. The method was also used to simulate C3X [3] turbine cascade at 2D boundary conditions by Bohn [4] , and a good agreement between the prediction results and the test data was gotten. Bohn also published a paper [5] in which Mark II turbine cascade with thermal barrier coatings was calculated in 2D cases. The ZrO 2 coatings with a thickness of 0.125mm bonded a 0.06mm MCrA1Y layer were applied. There were two configurations of the coatings. The problems and the influence of coatings on the thermal efficiency were solved by the same solver and evaluated. On the basis, the 3D numerical investigation of conjugate flow and heat transfer about Mark II with thermal barrier coating was done [6] . The uncoated vane was also used to validate the 3D method. The influence of the reduced cooling fluid mass flow on the thermal stresses was discussed in detail. York [7] used 3D conjugate method to simulate C3X turbine cascade. Because no transition model was used, the simulated external HTC (EHTC) at the leading edge stagnation point and laminar region had low precision. Facchini [8] made another 3D conjugate heat transfer simulation of C3X, however there was an obvious difference of HTC between simulation results and experimental data. Sheng [9] researched 3D conjugate flow and heat transfer Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 636 method of turbine. In some reference, the effect of transition on conjugate flow and heat transfer was not mentioned. Because of laminar-turbulent transition on suction side of turbine airfoil and the transition consequentially effects conjugate flow and heat transfer, high precision transition models must be researched. The best method for the simulation of conjugate flow and heat transfer case is Large Eddy Simulation (LES) or Direct Numerical Simulations (DNS). The two methods have high accuracy of predicting flow and heat transfer in transition region, but they are not suitable for engineering application nowadays, because they are too costly. At present, the better method for conjugate simulation is still using two-equation turbulence model with transition model. In this paper, the numerical method considering transition was used to predict 2D and 3D conjugate flow and heat transfer. T3A flat plate, VKI HPT stator, VKI HPT rotor and MARK II stator were calculated. T3A flat plate was used to validate the accuracy of aerodynamic simulation, and the conjugate flow and heat transfer cases of other blades were calculated to validate the method. In the conjugate simulations, the effect of various turbulence models and inlet turbulence intensities on heat transfer were investigated. 2. Numerical method 2.1 Governing equations The governing equations were: Continuity equations: 0 i i U tx ρ ρ ∂ ∂ + = ∂∂ Momentum equations (N-S equations): 2 3 i jjj ii eff eff j i jj ii j UU U U p UU tx xx xx xx ρ ρ μμ ⎛⎞ ⎛⎞⎛⎞ ∂∂∂ ∂ ∂∂ ∂∂ ⎜⎟ ⎜⎟⎜⎟ +=−+ +− ⎜⎟⎜⎟ ⎜⎟ ∂∂ ∂∂ ∂∂ ∂ ∂ ⎝⎠⎝⎠ ⎝⎠ Energy equations: * * 2 3 jj ii eff i eff i iii jj ii j UU p Uh U hT UU txtxxx xx x x ρ ρ λμ μ ⎛⎞ ⎛⎞⎛⎞ ∂∂ ⎛⎞ ∂ ∂∂ ∂∂∂∂ ∂ ⎜⎟ ⎜⎟⎜⎟ +=+ + +− ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ∂∂∂∂∂∂ ∂∂ ∂ ∂ ⎝⎠ ⎝⎠⎝⎠ ⎝⎠ For thermal conduction of solid, when there is no thermal source, the governing equations were: ii cT T txx ρ λ ⎛⎞ ∂∂∂ = ⎜⎟ ∂∂∂ ⎝⎠ On the interface of fluid and solid, heat flux is equivalent. The governing equations were discretized with finite volume method. By means of solving continuity equations and Momentum equations simultaneously, the uncoupling of pressure and temperature was resolved. Convection term has second-order precision. Conjugate Flow and Heat Transfer of Turbine Cascades 637 2.2 Turbulence model Advanced turbulence model with high accuracy of describing turbulence nature must be used to get exact flow field, especially for engineering problems. In order to improve accuracy of flow and heat transfer analysis, Menter [10] developed SST turbulence model. The model assimilated the advantages of k-ω model and k-ε model. It used k-ω model near wall and k-ε model far from wall, having high accuracy of predicting flow field near wall and avoiding strong sensitivity to free stream conditions. A number of test cases were predicted by means of the model, proving that the model has high accuracy of conjugate flow and heat transfer problem especially for large adverse pressure gradient [11] . 2.3 Transition model Transition has significant effect on heat transfer. In order to predict conjugate flow and heat transfer of turbine cascades, turbulence model must be coupled with transition model. Experience modified transition models include zero-equation model, one-equation model and two-equation model. Intermittency is given in zero-equation model. In one-equation model, user-defined transition Reynolds number is used to solve intermittency, avoiding solving another equation so that reduce computation time, but the model does not consider effect of turbulence intensity and pressure gradient on transition. Two-equation model connects free stream turbulence intensity with transition momentum thickness Reynolds number at the onset of transition and solves two transport equations. One is used to calculate intermittency and the other is used to calculate momentum thickness Reynolds number. The two equations couple with production terms in SST turbulence model. The two-equation model can solve the transition caused by shock wave or separation. In order to predict complex cascade flow field with high accuracy and improve the solving precision of temperature and external heat transfer coefficient along airfoils, the modified two-equation transition model developed by Menter [12] was used. The transport equation of intermittency γ in the two-equation transition model is: 1122 () () () t j t jjj U PE PE xxx γγγγ ργ ρ γγ μμ ⎛⎞ ∂ ∂∂∂ ⎜⎟ +=−+−++ ⎜⎟ ∂∂ ∂ ∂ ⎝⎠ where: 0.5 1 2() length onset PFSF γ ργ = ; 11 EP γγ γ = ; 2 0.06 turb PF γ ρ Ωγ = ; 22 50EP γγ γ = ; ( ) 23 max ,0 onset onset onset FFF=−; () 4 0.25 T R turb Fe − = ; 1 Re 2.193Re v onset c F θ = ; ( ) ( ) 4 211 min max , ,2.0 onset onset onset FFF= ; 3 3 max 1 ,0 2.5 T onset R F ⎛⎞ ⎛⎞ ⎜⎟ =− ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ; 2 Re v yS ρ μ = ; T k R ρ μ ω = ; 2 i j i j SSS= ; 2 i j i j Ω ΩΩ = ; 1 2 j i ij j i U U S xx ⎛⎞ ∂ ∂ ⎜⎟ =+ ⎜⎟ ∂∂ ⎝⎠ ; 1 2 j i ij j i U U xx Ω ⎛⎞ ∂ ∂ ⎜⎟ =− ⎜⎟ ∂∂ ⎝⎠ Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 638 For the transition caused by separation, the correction is: ( ) max , e ff se p γγγ = where: 4 20 Re min 2 max 1,0 ,2 ; 3.235Re T R v sep reattach t reattach c FFFe θ θ γ ⎛⎞ − ⎜⎟ ⎝⎠ ⎛⎞ ⎛⎞ ⎛⎞ =• − =⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ; The transport equation of transition momentum thickness Reynolds number Re t θ  in the two-equation transition model is: (Re) (Re) Re 2( ) t jt tt tt jj j U P xx x θ θθ θ ρ ρ μμ ⎛⎞ ∂ ∂∂ ∂ ⎜⎟ +=+•+ ⎜⎟ ∂∂ ∂ ∂ ⎝⎠   where: ( ) () 1 2 1 500 0.03 Re Re 1 ; tttt PFt t U θθθθ ρ μ ρ =−−=  ; 4 2 0.02 min max ,1 ,1 10.02 y twake FFe δ θ γ ⎛⎞ − ⎜⎟ ⎝⎠ ⎛⎞ ⎛⎞ − ⎜⎟ ⎛⎞ ⎜⎟ =•− ⎜⎟ ⎜⎟ ⎜⎟ − ⎝⎠ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ 50 Re ; 7.5 ; t BL BL BL BL y UU θ Ω μ δδδθθ ρ =• = =  ; 5 Re 2 2 110 ; Re wake y Fe ω ω ρω μ ⎛⎞ − ⎜⎟ × ⎝⎠ == ; Re t θ is the transition momentum thickness Reynolds number in experiment. 3. Validation 3.1 T3A flat plate The geometric configuration and the boundary conditions of ERCOFTAC T3A [13] flat plate were shown in figure 1. The case was used to validate SST turbulence model with transition. Inlet Outlet Plate Sym U=5.4m/s Tu=0.03 P=1bar Fig. 1. Geometric configuration and boundary conditions of T3A [...]... test one reduced gradually 647 Conjugate Flow and Heat Transfer of Turbine Cascades Fig 17 Temperature distribution (condition 1 and 2) Fig 18 EHTC distribution (condition 1 and 2) a) Condition 1 Fig 19 Pressure distribution of different turbulence models b) Condition 2 648 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 4.1.2 The effect of turbulence models... different inlet turbulence intensities 650 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems a) b) Fig 23 Computational grid(3D) of MARKⅡcascade Fig 24 Pressure distribution (2D and 3D) Conjugate Flow and Heat Transfer of Turbine Cascades 651 a） 2D simulation b） 3D simulation Fig 25 Contours of Mach Number and Temperature (2D and 3D) 4.2 3D simulation The 3D simulation... wave appeared resulting in laminar-turbulent transition Strong heat transfer made temperature curve becoming abrupt in the region of transition, and the maximum 646 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems a） condition 1 b） condition 2 Fig 16 Contours of Mach Number and Temperature (condition 1 and 2) appeared at the end of the transition region The numerical... 654 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 7 References [1] Mahmoud L Mansour; Khosro Molla Hosseini & Jong S Liu ”Assessment of the impact of laminar-turbulent transition on the accuracy of heat transfer coefficient prediction in high pressure turbines”.ASME GT2006-90273 [2] Bohn D.; Bonhoff B & Schonenborn H ”Combined aerodynamic and thermal analysis. .. agreed with the test one The onset of transition was downstream predicted and the transition was predicted more sharply than test In the low turbulence intensity, the method has high accuracy of predicting EHTC with various Reynolds number 642 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems Comparing figure 8 with figure 9, it can be seen that EHTC increased... transfer simulations, which is applied to validating codes or the accuracy of methods 644 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems Geometric data Chord/mm Pitch/mm Height/mm Stagger angle/◦ Throat/mm Leading edge thickness/mm 136.22 129.74 76.2 63.69 39.83 12.80 Fig 13 Sketch and geometric data of MARKⅡcascade 4.1 2D simulation The 2D simulation used unstructured... upstream The reason may be that the endwall had some influence on the flow The difference between Mach contours were shown in figure 25 652 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems The temperature distribution of 2D and 3D was shown in figure 26 (also can be seen in figure 25) All the results agreed well with the test data, but there were some dissimilarities... Conjugate Flow and Heat Transfer of Turbine Cascades of T3A flat plate showed that the method had high accuracy on predicting flow All the other results indicated that the method had high accuracy on simulating conjugate flow and heat transfer For the conjugate flow and heat transfer problems, two steps were applied The first step was that no internal cooling cascades, VKI HPT stator and rotor, were... figure 4 Total number of nodes was 123328 and total number of elements was 91233 Fig 4 Hexahedral mesh of T3A Figure 5 showed the simulation results by means of hexahedral mesh The result with transition model was also in good agreement with the test data The trend of the predicted 640 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems skin friction with transition... M.S.; Turner E.R.; Nealy D.A & York R.E ”Analytical and experimental evaluation of the heat distribution over the surfaces of turbine vanes” NASA CR168015, 1983 [4] Bohn D & Heuer T “Conjugate flow and heat transfer calculation of a high pressure turbine nozzle guide vane” AIAA 2001-3304 [5] Bohn D.; Heuer T & Kortmann “Numerical conjugate flow and heat transfer investigation of a transonic convection-cooled . simulation results and experimental data. Sheng [9] researched 3D conjugate flow and heat transfer Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 636 method. Strong heat transfer made temperature curve becoming abrupt in the region of transition, and the maximum Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems. (2001). Convective Heat Transfer, ISBN 0 08 043878 4, Pergamon Press, Oxford, pp. 181-198 Heat Transfer - Theoretical Analysis, Experimental Investigations and Industrial Systems 634 Pozzi,