Differential Transformation Method

Differential Transformation Method

DOI: 10.4018/978-1-5225-2713-8.ch006
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Abstract

In this chapter, a new linearization procedure based on Differential Transformation Method (DTM) will be presented. The procedure begins with solving nonlinear differential equation by DTM. The effectiveness of the procedure is verified using a heat transfer nonlinear equation. The simulation result shows the significance of the proposed technique.
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Background

Porous media has an extensive use of applications like catalytic and inert packed bed reactors, enhancing drying efficiency, filtering, insulation, lubrication (Kaviany, 1995), stabilization of non-uniform flow (Hasanpour et al., 2011) and enhancing oil and natural gas production. A high thermal conductivity porous substrates are used to enhance forced convection heat transfer in many engineering applications such as reactor cooling, heat exchangers, and solar collectors (Alkam & Al-Nimr, 1999). The concept of using fins made of porous materials is firstly introduced by Kiwan and Al-Nimr (2001) and they introduced Darcy model for analyzing the porous fins for the first time in Refs. (Kiwan, 2007; Kiwan & Zeitoun, 2008). Heat exchanger industries are looking for more compact in and more cost-effective heat exchanger manufacturing techniques which lead the method to use porous fins to enhance heat transfer (Hamdan & Al-Nimr, 2010). The heat-transfer enhancement between two parallel-plate channels was investigated by adding porous fin through the channel (Hamdan, Al-Nimr & Alkam, 2000) and by adding porous insert to one side of the duct walls (Alkam, Al—Nimr & Hamdan, 2002). An analytical prediction for performance of porous fins was presented by Kundu and Bhanja (2011). In their work, the influence of some dependent parameters on the performances and optimization conditions was studied for the selection of a design criterion of porous fins. Also it was concluded that a clear difference in results for heat transfer rate at the optimum point is noticed for the different models of predictions and consequently it can be highlighted that the selection of the actual model is necessary for the realistic implementation of the design in concern. Kiwan (2007) investigated the thermal analysis of natural convection porous fins. He used a method based on energy balance and Darcy’s model to formulate the heat transfer equations and the thermal performance of porous fins was studied for three types of fins. It was found that the heat transfer rate from porous fin can exceed that of a solid fin. Gorla and Bakier (2011) performed a study on natural convection and radiation in rectangular profile fin. Their results showed that the radiation transfers more heat than a similar model without radiation. Domairry and Fazeli (2009) solved the nonlinear straight fin differential equation to evaluate the temperature distribution and fin efficiency. Also temperature distribution for annual fins with temperature-dependent thermal conductivity was studied by Ganji et al. (2011). The analytical approaches like Differential Transformation Method (DTM) is used for solving the scientific and engineering cases.

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