Feedback Control of Heat Transfer Systems by the Numerical Method of Lines

2009 ◽  
Author(s):  
David Brown ◽  
Chao Zhang ◽  
Jin Jiang
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Numerical Method ◽  
Direct Method ◽  
Method Of Lines ◽  
Finite Volume Methods ◽  
Current State ◽  
Heat Transfer System ◽  
Ordinary Time ◽  
Feedback Optimal Control

A direct method for feedback optimal control of fluid flow and heat transfer systems is investigated. The method consists of discretizing the spatial component of the governing partial differential equations using standard finite-volume methods (FVM) while leaving the transient term in its differential form. This partial discretization method is commonly known as the Numerical Method of Lines (NUMOL). The control is then applied to the partially discretized system of linear ordinary time differential equations. Of particular interest is the effectiveness of linearizing about the current state rather than the origin. The effectiveness of this method is investigated through numerical implementation on a simple one-dimensional heat transfer system.

A Numerical Study of Nonaxisymmetric Stokesian Flow in a Circular Tube

1981 ◽  
Vol 48 (3) ◽  
pp. 459-464
Author(s):  
J. Strigberger ◽  
A. Plotkin
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Circular Tube ◽  
Numerical Study ◽  
Mathematical Problem ◽  
Method Of Lines ◽  
Tube Axis ◽  
First Order ◽  
The Method Of Lines ◽  
Fixed Radius

A numerical study of the nonaxisymmetric Stokesian flow of a Newtonian fluid in a rigid circular tube of fixed radius has been performed. The analysis presented here is an integral part of the problem of modeling the flow of blood near the ostia of the intercostal arteries of rabbits in order to study a possible factor in the initiation of atherosclerosis. The method of lines is used to reduce the mathematical problem to one of solving a system of first-order ordinary differential equations along lines parallel to the tube axis. Solutions are obtained analytically using matrix eigenvalue techniques for the first two Fourier components of the flow and the accuracy of the numerical method is verified by suitable comparison with the results of independent computations.

Numerical Treatment of the Modified Burgers’ Equation via Backward Differentiation Formulas of Orders Two and Three

2018 ◽  
Vol 19 (7-8) ◽  
pp. 669-680 ◽  
Author(s):  
Vijitha Mukundan ◽  
Ashish Awasthi
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Burgers Equation ◽  
Method Of Lines ◽  
Step Method ◽  
Error Norm ◽  
Spatial Direction ◽  
Modified Burgers Equation

AbstractWe present an efficient numerical method for solving the nonlinear modified Burgers’ equation (MBE) using the multi-step method. The nonlinear MBE is first discretized along the spatial direction alone by using the method of lines technique, and this method converts the MBE to a nonlinear system of ordinary differential equations. Multistep methods are employed to solve the nonlinear system of ordinary differential equations. Applicability of the proposed numerical techniques is established through test examples. Discrete root mean square error norm $(L_{2})$ and maximum error norm $(L_{\infty})$ are computed and presented for demonstrating the accuracy of the present numerical method. Numerical experiments supported by figures shows that the proposed numerical scheme shows excellent agreement with exact solution and is superior to some existing numerical methods.

The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

2020 ◽  
pp. 030641902091042
Author(s):  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Method Of Lines ◽  
Adjoint System ◽  
One Dimensional ◽  
First Order ◽  
The Method Of Lines ◽  
Unsteady Heat Conduction

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

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