Integral transform solution of natural convection in a cylinder cavity with uniform internal heat generation

Abstract

Purpose The purpose of this study is to propose the generalised integral transform technique to investigate the natural convection behaviour in a vertical cylinder under different boundary conditions, adiabatic and isothermal walls and various aspect ratios. Design/methodology/approach GITT was used to investigate the steady-state natural convection behaviour in a vertical cylinder with internal uniformed heat generation. The governing equations of natural convection were transferred to a set of ordinary differential equations by using the GITT methodology. The coefficients of the ODEs were determined by the integration of the eigenfunction of the auxiliary eigenvalue problems in the present natural convection problem. The ordinary differential equations were solved numerically by using the DBVPFD subroutine from the IMSL numerical library. The convergence was achieved reasonably by using low truncation orders. Findings GITT is a powerful computational tool to explain the convection phenomena in the cylindrical cavity. The convergence analysis shows that the hybrid analytical–numerical technique (GITT) has a good convergence performance in relatively low truncation orders in the stream-function and temperature fields. The effect of the Rayleigh number and aspect ratio on the natural convection behaviour under adiabatic and isothermal boundary conditions has been discussed in detail. Originality/value The present hybrid analytical–numerical methodology can be extended to solve various convection problems with more involved nonlinearities. It exhibits potential application to solve the convection problem in the nuclear, oil and gas industries.

Natural convection in a horizontal annular sector containing heat-generating porous medium

“…Natural convection in porous cavities is of great interest for industry and engineering applications (Lewis and Schrefler, 1998; Lewis et al , 2004), such as nuclear reactor safety and dry storage of nuclear spent fuel (Lisboa et al , 2018; Fu et al , 2018; Mohammadian and Zhang, 2019). Natural convection in cavities of different geometries with internal heat generation was investigated by using numerical, analytical and experimental methods, such as rectangular cavities (Lee and Goldstein, 1988; Joshi et al , 2006), vertical cylinders (Martin, 1967; Holzbecher and Steiff, 1995) and horizontal annulus (Shekar et al , 1984; Yuan et al , 2015).…”

“…As an alternative to purely numerical methods such as the finite element method (Lewis and Schrefler, 1998), an alternative hybrid numerical–analytical approach, the generalized integral transform technique (GITT), has been developed in recent decades as a powerful computational tool for the solution of non-transformable heat transfer and diffusion-convection problems (Cotta, 1993), with moving boundaries (Guerrero and Cotta, 1992; Guigon et al , 2007), nonlinear boundary conditions (Cotta et al , 2016), source terms (An et al , 2013; Fu et al , 2018) and cavities completely (Alves and Cotta, 2000) or partially (Lisboa et al , 2018) filled with porous media. Baohua and Cotta (1993) applied the GITT to study steady-state natural convection in a saturated porous vertical rectangular enclosure subjected to uniform internal heat generation.…”

Generalized integral transform solution for free vibration of orthotropic rectangular plates with free edges

“…The boundary conditions at free edges of the rectangular plate were treated exactly by carrying out integral transform of the boundary conditions along the free edge direction. Generalized integral transform technique is a hybrid analytical-numerical method that has been applied successfully in a wide range of flow and heat transfer problems [17][18][19][20], as well as in static and dynamic structural analyses [21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In this work, the free vibration of orthotropic thin rectangular plates with a pair of opposite edges clamped and one or two free edges (CSCF, CCCF, CFCF) is studied analytically by using generalized integral transform technique.…”

An exact GITT solution for static bending of clamped parallelogram plate resting on an elastic foundation

“…A double finite sine integral transform approach was employed to investigate the bending of the thin clamped orthotropic plates (Li et al, 2009). Recently, a hybrid analytical-numerical technology, known as generalised integral transform technique (GITT) (Cotta, 1993), has been proposed to solve high-order partial differential equations in the heat transfer and fluid flow behaviour (Cotta, 1998;Fu et al, 2018), structural vibration and deformation (An and Su, 2011;An et al, 2016) and dynamics fluidconveying pipe system (An and Su, 2015;An and Su, 2017), which illustrated excellent convergence and numerical stability behaviour and high accuracy. To the best of authors' knowledge, the GITT technique should be an excellent approach to solve the fourth-order partial differential governed equation of parallelogram plate bending problem.…”