Estude fácil! Tem muito documento disponível na Docsity
Ganhe pontos ajudando outros esrudantes ou compre um plano Premium
Prepare-se para as provas
Estude fácil! Tem muito documento disponível na Docsity
Prepare-se para as provas com trabalhos de outros alunos como você, aqui na Docsity
Os melhores documentos à venda: Trabalhos de alunos formados
Prepare-se com as videoaulas e exercícios resolvidos criados a partir da grade da sua Universidade
Responda perguntas de provas passadas e avalie sua preparação.
Ganhe pontos para baixar
Ganhe pontos ajudando outros esrudantes ou compre um plano Premium
Comunidade
Peça ajuda à comunidade e tire suas dúvidas relacionadas ao estudo
Descubra as melhores universidades em seu país de acordo com os usuários da Docsity
Guias grátis
Baixe gratuitamente nossos guias de estudo, métodos para diminuir a ansiedade, dicas de TCC preparadas pelos professores da Docsity
The analysis of transient heat conduction in solids, focusing on one and two-dimensional conduction, lumped capacitance method, and validity of the method. It also discusses transient conduction in semi-infinite solids and multidimensional systems. Examples and applications are included for better understanding.
O que você vai aprender
Tipologia: Exercícios
1 / 76
Esta página não é visível na pré-visualização
Não perca as partes importantes!
1/21/2018 Heat Transfer 1
Prepared and Presented by: Tariku Negash Sustainable Energy Engineering (MSc) E-mail: thismuch2015@gmail.com Lecturer at Mechanical Engineering Department Institute of Technology, Debre Markos University, Debre Markos, Ethiopia Jan, 2018
2
4
5
In heat transfer analysis, some bodies are observed to behave like a “lump” A compact mass Interior temperature of some bodies remains essentially uniform at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of time only, T ( t ). Heat transfer analysis that utilizes this idealization is known as lumped system analysis. which provides great simplification in certain classes of heat transfer problems without much sacrifice from accuracy.
7
Consider a body of arbitrary shape shown fig. At time 𝑡 = 0 , the body is placed into a medium at temperature 𝑇∞, and heat transfer takes place between the body and its environment, with a heat transfer coefficient h. For the sake of discussion, we will assume that 𝑇∞ > 𝑇𝑖, but the analysis is equally valid for the opposite case. We assume lumped system analysis to be applicable, so that the temperature remains uniform within the body at all times and changes with time only, T = T(t). During a differential time interval dt , the temperature of the body rises by a differential amount dT.
8 An energy balance of the solid for the time interval dt can be expressed as and
𝑠𝑖𝑛𝑐𝑒 𝑇∞is constant Can be rearranged as (2) Integrating with T = Ti at t = 0 T = T ( t ) at t = t Taking the exponential of both sides and rearranging, we obtain ( 3 ) (^) Where, is + ve quantity
10 i. The temperature of a body approaches the ambient temperature 𝑻∞ exponentially. ii. The temperature of the body changes rapidly at the beginning, but rather slowly later on. iii. A large value of b indicates that the body will approach the environment temperature in a short time iv. The larger the value of the exponent b, the higher the rate of decay in temperature. Note that b is proportional to the surface area, but inversely proportional to the mass and the specific heat of the body. Which means that it takes longer to heat or cool a larger mass, especially when it has a large specific heat
11 The rate of convection heat transfer between the body and its environment at time t The total amount of heat transfer between the body and the surrounding medium over the time interval t = 0 to t
Once the temperature T(t) at time t is available from Eq. (3) , the rate of convection heat transfer between the body and its environment at that time can be determined from Newton’s law of cooling as (7) The maximum heat transfer between the body and its surroundings when body reaches the surrounding Tem (𝑻∞).
13 When a solid body is being heated by the hotter fluid surrounding it (such as a potato being baked in an oven), heat is first convected to the body and subsequently conducted within the body. The Biot number is the ratio of the internal resistance of a body to heat conduction to its external resistance to heat convection. Therefore, a small Biot number represents small resistance to heat conduction, and thus small temperature gradients within the body. Lumped system analysis assumes a uniform temperature distribution throughout the body, which will be the case only when the thermal resistance of the body to heat conduction (the conduction resistance) is zero.
14 Then the question we must answer is, How much accuracy are we willing to sacrifice for the convenience of the lumped system analysis?
When this criterion is satisfied, the temperatures within the body relative to the surroundings (i.e., 𝑻 − 𝑻∞) remain within 5 percent of each other even for well-rounded geometries such as a spherical ball. Thus, when Bi < 0.1 , the variation of temperature with location within the body will be slight and can reasonably be approximated as being uniform.
16 4.5 Transient Heat Conduction In Large Plane Walls, Long Cylinders, and Spheres With Spatial Effects Consider the variation of temperature with time a nd position in one- dimensional problems such as those associated with a large plane wall, a long cylinder, and a sphere. ( c) A sphere
17 At time t = 0 , each geometry is placed in a large medium that is at a constant temperature 𝑻∞ and kept in that medium for t > 0. Heat transfer takes place between these bodies and their environments by convection with a uniform and constant heat transfer coefficient h. By neglecting radiation or incorporate the radiation effect into the convection heat transfer coefficient h The variation of the temperature profile with time in the plane wall is illustrated in Fig. When the wall is first exposed to the surrounding medium at 𝑻∞ < 𝑻𝒊 at t= 0 , the entire wall is at its initial temperature 𝑻𝒊. But the wall temperature at and near the surfaces starts to drop as a result of heat transfer from the wall to the surrounding medium. This creates a t emperature gradient in the wall and initiates heat conduction from the inner parts of the wall toward its outer surfaces.
19
There is clear motivation to present the solution in tabular or graphical form. However, the solution involves the parameters 𝒙, 𝑳, 𝒕, 𝒌, 𝜶, 𝒉, 𝑻𝒊 𝒂𝒏𝒅 𝑻∞ , which are too many to make any graphical presentation of the results practical. In order to reduce the number of parameters, we non-dimensionalize the problem by defining the following dimensionless quantities: ( 10 ) (11) ( 12 ) ( 13 )
20 Non-dimensionalization reduces the number of independent variables in one-dimensional transient conduction problems from 8 to 3 , offering great convenience in the presentation of results. Dimensionless differential equation Dimensionless BC’s Dimensionless initial condition
