Analytical solution and numerical simulation of the thermal entrance region problem for laminar flow through a circular pipe

In this paper, the assumptions implicited in Leveque’s approximation are re-examined, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth order (RK4) method. Finally, other important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail. A comparison with the previous study available in literature has been done and found an excellent agreement with the published data.

Analytical solution and numerical approaches of the generalized Levèque equation to predict the thermal boundary layer

In this paper, the assumptions implicit in Leveque's approximation are re-examined, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. By defining a similarity variable the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge-Kutta fourth order (RK4) method. Finally, others important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail.

Utilizing the Integral Technique to Determine the Similarity Variable in Classical Heat Transfer Problems: Thermal Boundary Layer Theory

In advanced heat transfer courses, a technique exists for reducing a partial differential equation, where the dependent variable is a function of two independent variables, to an ordinary differential equation where that same dependent variable becomes a function of only one. The key to this technique is finding out what the functional form of the similarity variable is to make such a transformation. The difficulty is that the form of the similarity variable is not intuitive, and many heat transfer textbooks do not reveal how this variable is found in classical problems such as viscous and thermal boundary layer theory. It turns out that one way to find this variable is by utilizing the integral technique. By employing the integral technique to boundary layer theory, it will be shown that when the approximate functional relationship for the dependent variable (temperature, velocity, etc) can be represented by an nth order polynomial, the similarity variable can be found very simply. This is seen to be a good tool especially in heat transfer education, but may have applications in research as well. The approach described here is a variation of a well-known technique used for isothermal momentum boundary layer consideration.

Similarity solution and Runge Kutta method to a thermal boundary layer model at the entrance region of a circular tube

Purpose The purpose of this paper is to re-examine the assumptions implicit in Leveque’s approximation, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth-order (RK4) method. Finally, other important thermal results obtained from this analysis, such as approximate Nusselt number in the thermal entrance region, was discussed in detail. After that, the analytical results of the present paper are validated with certain previous investigations which were found in the specialized literature. Design/methodology/approach By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This paper gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form and presents a similarity solution. The calculation methodology for numerical resolution is based on the RK4 technique. Findings The profiles of the solutions are provided from which the authors infer that the numerical and exact solutions agreed very well. Another result that the authors obtained from this paper is the number of Nusselt in the thermal entrance region for which a parametric study was carried out and discussed well for the impact of scientific contribution. Originality/value The novelty of this paper is the application of the RK4 with a step size control, as a sequential numerical method of a ODEs system compared with the exact similarity solution of the thermal boundary layer problem.

A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions

Fractal self-similarity of hadron interactions demonstrated by the [Formula: see text]-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable [Formula: see text] is a function of the momentum fractions [Formula: see text] and [Formula: see text] of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions [Formula: see text] and [Formula: see text] of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the [Formula: see text]-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

A similarity solution for reaction front propagation in a fracture–matrix system

We propose a new composite similarity variable, based on which a similarity solution is derived for reaction front propagation in fracture–matrix systems. The similarity solution neglects diffusion/dispersion within the fracture and assumes the existence of a sharp reaction front in the rock matrix. The reaction front location in the rock matrix is shown to follow a linear decrease with distance along the fracture. The reaction front propagation along the fracture is shown to scale like diffusion (i.e. as the square root of time). The similarity solution using the composite similarity variable appears to be applicable to a broad class of reactive transport problems involving mineral reactions in fracture–matrix systems. It also reproduces the solutions for non-reactive solute and heat transport when diffusion/dispersion/conduction are neglected in the fracture. We compared our similarity solution against numerical simulations for nonlinear reactive transport of an aqueous species with a mineral in the rock matrix. The similarity solutions agree very well with the numerical solutions, especially at later times when diffusion limitations are more pronounced. This article is part of the themed issue ‘Energy and the subsurface’.