A Heat Transfer Textbook Lienhard Solution Manual
Rae Puleo
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This introduction to heat and mass transfer, oriented toward engineering students, may be downloaded without charge. The eBook is fully illustrated, typeset in searchable pdf format, with internal and external links.
The present new similarity analysis method is applied for similarity transformation of governing partial differential equations for laminar forced convection without consideration of variable physical properties and viscous thermal dissipation, and then the complete similarity mathematical model is obtained. In the transformed dimensionless equations, dimensionless similarity velocity components W x (Ƞ) and W y (Ƞ) describe clearly the forced convection momentum field. The rigorous numerical solutions on velocity and temperature fields, as well as heat transfer equations, are obtained, which are very well coincident to Blasius and Pohlhausen solutions. All these prove that the present new similarity analysis method is valid.
This paper deals with the numerical solution of the random Cauchyone-dimensional heat model. We propose a random finite difference numericalscheme to construct numerical approximations to the solution stochasticprocess. We establish sufficient conditions in order to guarantee theconsistency and stability of the proposed random numerical scheme. Thetheoretical results are illustrated by means of an example where reliableapproximations of the mean and standard deviation to the solution stochasticprocess are given.
The heat is the energy which flows from the higher to the lowertemperature and the transport coefficient depends on the specific modetransfer. The transfer modes are the diffusive transport of thermal energy(the conduction mode), the exchange of heat between a moving fluid and anadjoining wall (the convection mode), and the radiation mode where all bodiescan emit thermal radiation [1, 2]. In a metal rod with nonuniformtemperature, heat is transferred from regions of higher temperature toregions of lower temperature. Usually, the physical principles for heattransfer are heat energy of a body with uniform properties, Fourier'slaw of heat transfer, and conservation of energy [2-4].
When dealing with a partial differential equation together withthe initial and boundary conditions, it is crucial to obtain a well-posedproblem. The extent of the spatial domain is another division for the partialdifferential equation that makes one method of solution preferable overanother. Spatial domain may be a finite interval or an infinite interval,such as the whole real line. If the spatial domain is unbounded, the boundaryconditions are not an important issue and in that case the problem is calledinitial value problem (IVP). In mathematics, a pure IVP is usually referredto as a Cauchy problem [3, 5]. This paper is concerned with the study ofrandom finite difference schemes (one of the most widely used methods forengineering models) to the Cauchy problem for the one-dimensional random heatequation with unbounded spatial domain
In this IVP (1)-(2), t is the time variable, x is the spacecoordinate, [u.sub.t] and [u.sub.xx] denote the first and the secondderivatives with respect to t and x, respectively, and [beta] is a randomvariable defined in a probability space ([OMEGA], F, P). In addition,[u.sub.0](x) is an initial deterministic data function. Expression (1) is arandom parabolic partial differential equation for temperature u(x, t) in aheat conducting insulated impurity rod along the x-axis since theconductivity coefficient, [beta], is assumed to be a random variable. Thephysical significance of thermal diffusion coefficient is associated with thespeed of the flux of heat into the material when changes of temperature takeplace over the time. The heating propagation rate is proportional to thethermal diffusivity [6]. As it is stated in the thermodynamics' laws,[beta] should be a function of two independent and intensive dynamicproperties (usually, temperature and pressure) [7]. From the secondthermodynamics law, it is required that [beta] be positive. In this paper, wetake [beta] as a random variable since the randomness of heat transferdepends on the randomness of the conductivity coefficient. The randomness of[beta] may be from the impurity material properties used to make the rod.
In this paper we have studied the randomized Cauchy heat model byassuming that the diffusion coefficient is a random variable and consideringa deterministic initial condition over an unbounded domain. Thus, boundaryconditions have not been required. We have proposed a random finitedifference scheme for solving this model. The mean square consistency of therandom finite difference scheme has been studied. Sufficient conditions forthe mean square stability of the random finite difference scheme have beenprovided. The numerical experiments show that the proposed random finitedifference scheme gives reliable approximations for the mean and the standarddeviation of the solution stochastic process.
[13] M. N. Koleva, "Numerical solution of the heat equationin unbounded domains using quasi-uniform grids," in Large-ScaleScientific Computing: 5th International Conference, LSSC 2005, Sozopol,Bulgaria, June 6-10, 2005. Revised Papers, vol. 3743 of Lecture Notes inComputer Science, pp. 509-517, Springer, Berlin, Germany, 2006.
[17] M.-C. Casaban, J.-C. Cortes, B. Garcia-Mora, and L. Jodar,"Analytic-numerical solution of random boundary value heat problems in asemi-infinite bar," Abstract and Applied Analysis, vol. 2013, Article ID676372, 9 pages, 2013.
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