Abstract:The thermal entry region in laminar forced convection of Herschel-Bulkley fluids is solved analytically through the integral transform technique, for both circular and parallel-plates ducts, which are maintained at a prescribed wall temperature or at a prescribed wall heat flux. The local Nusselt numbers are obtained with high accuracy in both developing and fully-developed thermal regions, and critical comparisons with previously reported numerical results are performed
“…However, the comparison between our results and those resulting from the analytical study carried out by Quaresma and Macêdo [29] (seen Table 5), has shown a deviation in our friction factor values not exceeding 0.4%. This deviation can be argued by the difference existing in the limit of weak shearing between the continuous viscoplastic equation (Eq.…”
Section: The Yield Shear Stress Influence On the Axial Velocity Profilessupporting
confidence: 58%
“…We thus note that the increase in Re causes Table 5 Friction factor (f Re) for various values of the Bingham number. (,) present work, (*) Quaresma and Macêdo [29]. a lengthening of the development length, an effect which is substantially reduced by the existence of a yield shear stress.…”
Section: The Influence Of the Inertia On The Axial Velocity Profilesmentioning
confidence: 50%
“…This deviation can be argued by the difference existing in the limit of weak shearing between the continuous viscoplastic equation (Eq. (5)) used in the present work and the ideal Bingham model (IBM) considered by [29]. Fig.…”
Section: The Yield Shear Stress Influence On the Axial Velocity Profilesmentioning
“…However, the comparison between our results and those resulting from the analytical study carried out by Quaresma and Macêdo [29] (seen Table 5), has shown a deviation in our friction factor values not exceeding 0.4%. This deviation can be argued by the difference existing in the limit of weak shearing between the continuous viscoplastic equation (Eq.…”
Section: The Yield Shear Stress Influence On the Axial Velocity Profilessupporting
confidence: 58%
“…We thus note that the increase in Re causes Table 5 Friction factor (f Re) for various values of the Bingham number. (,) present work, (*) Quaresma and Macêdo [29]. a lengthening of the development length, an effect which is substantially reduced by the existence of a yield shear stress.…”
Section: The Influence Of the Inertia On The Axial Velocity Profilesmentioning
confidence: 50%
“…This deviation can be argued by the difference existing in the limit of weak shearing between the continuous viscoplastic equation (Eq. (5)) used in the present work and the ideal Bingham model (IBM) considered by [29]. Fig.…”
Section: The Yield Shear Stress Influence On the Axial Velocity Profilesmentioning
“…It is the Generalized Integral Transform Technique -GITT [3], a method which has been used successfully to solve several diffusive problems such as those dealing with the flow in ducts with irregular geometries [4,5], with time varying coefficients and problems involving space dependence for the boundary conditions [6,7], problems with thermally and hydrodynamically developing flows [8,9], diffusive problems involving moving boundaries [10,11], problems of non-Newtonian fluid flows [12,13], among others.…”
“…However, the Graetz problem has been extended over problems that focus on turbulent flows [33,[41][42][43][44][45][46], slip flows [7,16,20,37], non-Newtonian flows [9][10][11][12]23,24,27,31,34,38], and forced convection in a porous medium [18,25,[27][28][29] and that include streamwise heat conduction [5,16,18,19,[23][24][25]28,29,39,[41][42][43][44][45][46], and viscous dissipation [1][2][3][4]9,10,16,18,[...…”
Forced convection heat transfer in a non-Newtonian fluid flow inside a pipe whose external surface is subjected to non-axisymmetric heat loads is investigated analytically. Fully developed laminar velocity distributions obtained by a power-law fluid rheology model are used, and viscous dissipation is taken into account. The effect of axial heat conduction is considered negligible. The physical properties are assumed to be constant. We consider that the smooth change in the velocity distribution inside the pipe is piecewise constant. The theoretical analysis of the heat transfer is performed by using an integral transform technique -Vodicka's method. An important feature of this approach is that it permits an arbitrary distribution of the surrounding medium temperature and an arbitrary velocity distribution of the fluid. This technique is verified by a comparison with the existing results. The effects of the Brinkman number and rheological properties on the distribution of the local Nusselt number are shown.
List of symbolsA ilm , B ilm constants in Eqs. (28) and (39) Br. Chiba et al. J m () Bessel function of the first kind of order m n number of partitions N u local Nusselt number = 2 h R/λ P m function defined by Eq. (24) Q heat generation term = Br[(1 + ν)/ν] ν+1 η (1+ν)/ν r radial coordinate R pipe radius (m) T fluid temperature (K) u axial fluid velocity (m/s), Eq. (2) u m mean axial fluid velocity (m/s) u max maximum fluid velocity = (1 + 3ν)u m /(1 + ν) U dimensionless velocity = u/u m W m function defined by Eq. (27) x axial coordinate X ilm Eigenfunctions corresponding to the lth eigenvalue for the ith region, Eqs. (28) and (39) Y m () Bessel function of the second kind of order m Greek symbols φ circumferential coordinate Φ lm function corresponding to the lth eigenvalue defined by Eqs. (37) and (39) γ lm lth eigenvalues η dimensionless radial coordinate = r/R κ power-law model parameter (Pa s ν ) λ thermal conductivity (W m −1 K −1 ) ν power-law model index θ dimensionless fluid temperature = (T − T s )/(T 0b − T s ) ρ fluid density (kg m −3 ) τ r x shear stress (Pa), Eq. (1) ξ dimensionless axial coordinate =λx/(u m R 2 ρc)
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