An analytical solution to non-axisymmetric heat transfer with viscous dissipation for non-Newtonian fluids in laminar forced flow

Abstract

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)

Review and Analysis of Electro-Magnetohydrodynamic Flow and Heat Transport in Microchannels

“…Their results highlight that the effects of the viscous dissipation parameter vary the temperature distribution. Many researchers [136][137][138][139][140][141] introduced the viscous energy dissipation on the fully developed laminar flow in circular ducts. They showed the effect of viscous dissipation on the axial distribution of wall heat flux for power-law fluids in a thermally developed region.…”

Thermodynamically Consistent Limiting Nusselt Number in the Viscous Dissipative Non-Newtonian Couette Flows

“…Most of the earlier works on forced convective heat transfer of non-Newtonian fluids, including the effects of viscous dissipation, have focused on the generalized Newtonian fluid model. The studies reported by several researchers, such as Tso et al, 2 Koliatwong et al, 3 Etemad et al, 4 Barletta, 5 and Chiba et al, 6 have included the power-law fluid model, and that of Min et al 7 has included the Bingham fluid model in delineating the consequential effects of viscous dissipation on the heat transfer characteristics.…”