Separate theoretical and numerical analyses have been conducted for the prediction of the mean bulkand wall-temperatures of hot fluids flowing inside horizontal tubes. Heat transmission between the internal forced flow and the external free flow of the surrounding fluid occurs through the solid wall of the tube. The mathematical formulation of this problem is expressed in terms of a parabolic, partial differential equation with a temperature-dependent, nonlinear boundary condition of third kind. The aim of the article is to critically examine the thermal response of this kind of in-tube flows utilizing two different mathematical models: a) a complete 2-D differential model and b) a largely simplified I-D lumped model. For the I-D lumped model, streamwise-mean values for the internal Nusselt numbers and the circumferential-mean values for the external Nusselt numbers have been taken from standard correlations that appear in basic textbooks. The combination of both mean Nusselt numbers leads to the calculation of a mean, equivalent Nusselt number, which serves to regulate the thermal interaction between the perpendicular fluid streams. For the two models tested, the computed results consistently demonstrate that the simplistic 1 -D lumped model provides accurate estimates of the mean bulk-and wall-temperatures, when compared with those computed with the rigorous 2-D differential model. The former is associated with hand calculations, whereas the latter inevitably necessitated a finite-difference methodology and a personal computer.
The influence of thermal radiation on laminar forced convection of a gray gas in a pipe flow is studied semi-analytically. The gas is considered as an absorbing-emitting medium and the 'thin gas model approximation is used to describe the radiative heat flux in the energy equation. Invoking the method of lines (MOL), the nonlinear boundary value problem is reduced to an initial value problem which is eventually solved by a standard Runge-Kutta integration scheme. Numerical results are presented graphically for a selected group of thermal parameters encompassing the thin gas behavior. Additionally, it is found that limiting cases namely: laminar plug flow and laminar parabolic flow, both in the absence of radiation define an envelope for the curves of bulk temperature and total Nusselt number describing the combined thermal process for a thin gas in laminar motion. In general, the comparisons reveal that these asymptotic solutions are valid when the entrance-to-wall temperature ratio is less than 2.
This report is concerned with a distinct instructional technique that facilitates the teaching of laminar forced convection flow through a horizontal tube exposed to external natural convection. Asimple one~dimensionallumped~basedformulation enabled the determination of the mean bulk and wall temperature distributions of the internal fluid flow as well as the total rate of heat transfer between two axial stations in the region of thermal development. Streamwise-mean values for the internal and circumferential-mean values for the external Nusselt numbers have been taken from standard correlations reported in undergraduate textbooks on heat transfer, leading to the immediate calculation of a mean, equivalent Nusselt number, which serves to regulate the thermal interaction between the two fluid streams. Representative results for a complete combination of internal and external fluids, expressed in terms of their respective thermal conductivity ratios, are discussed at length. The thermal quantities were computed with a pocket-size calculator and compared favorably with those based on a more general and rigorous two-dimensional differential formulation. The latter was utilized as a baseline solution, and inevitably required a finite-difference procedure on a personal computer.
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