Transient heat conduction

Transient Heat

ConductionPRESENTED BY: A.R. AMINIAN

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First things First….

An Introduction to Metallurgical Engineering…

Metallurgy…what does it mean?!

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An intro to Lumped Element

Method (LEM) [1]

Lump means that the interior temperature 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)…

…Lumped system analysis, provides great simplification in certain

classes of heat transfer problems without much sacrifice from

accuracy.

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LEM Applicability

Ex1: the Copper Ball

Ex2: the Roast Beef

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Lumped Element System:

the Definition

Consider a body of arbitrary shape, 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.

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the Definition

assuming that 𝑇∞ > 𝑇𝑖, 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. An energy balance of the solid for

the time interval dt can be expressed as

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the Definition

Or:

Noting that m=ρV and dT=d(T-T∞) since T=constant, the Equation

above can be rearranged as

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the Definition

Integrating from t = 0, at which T = Ti, to any time t, at which T = T(t),

gives

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the Definition

Taking the exponential of both sides and rearranging, we obtain

where

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the Definition

The b is called the time constant

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the Definition

There are 2 point of views in the graph:

First:

The equation of b enables us to determine the temperature T(t) of a

body at time t, or alternatively, the time t required for the

temperature to reach a specified value T(t).

Second:

The temperature of a body approaches the ambient temperature T

exponentially. The temperature of the body changes rapidly at the beginning, but rather slowly later on.

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the Definition

A large value of b indicates that the body approaches the

environment temperature in a short time.

The larger the value of the exponent b, the higher the rate of decay

in temperature.

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the Applications

Metallurgical Analysis of Heat Transfer during

Heat Treatment

Casting

Hot Forging

Thermo-Forming

Vacuum Thermo-Forming, and…

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Lumped Element Analysis:

other Apps. than Heat Transfer

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Transient Heat Conduction:

the Separation of Variables [1]

an Intro to Nondimensionalization:

Consider an original heat conduction problem:

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Now, Nondimensionalizing the problem lead us to:

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Nondimensionalization reduces the number of independent

variables in one-dimensional transient conduction problems from 8

to 3, offering great convenience in the presentation of results.

The non-dimensionalized PDEs together with its boundary and initial

conditions can be solved using several analytical and numerical

techniques, including

the Laplace or other transform methods,

the method of separation of variables,

the finite difference method, and

the finite-element method.

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Separation of Variables [1 & 2]

The method developed by J. Fourier in 1820s and is based on expanding an arbitrary function (including a constant) in terms of Fourier series.

The method is applied by assuming the dependent variable to be a product of a number of functions, each being a function of a single independent variable.

This reduces the partial differential equation to a system of ordinary differential equations, each being a function of a single independent variable.

In the case of transient conduction in a plane wall, for example, the dependent variable is the solution function θ(X, τ), which is expressed as θ(X, τ) = F(X)G(τ), and the application of the method results in two ordinary differential equation, one in X and the other in τ.

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Separation of Variables:

Applicability

The method is applicable if:

(1) the geometry is simple and finite (such as a rectangular block, a

cylinder, or a sphere) so that the boundary surfaces can be described by simple mathematical functions, and

(2) the differential equation and the boundary and initial conditions

in their most simplified form are linear (no terms that involve products

of the dependent variable or its derivatives) and involve only one

nonhomogeneous term (a term without the dependent variable or

its derivatives).

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the Math Model [2]

The linear heat equation written in the form:

as the basic mathematical model.

Equation (1) describes heat transfer via conduction in a

nonhomogeneous isotropic medium and is supplemented by the

initial condition:

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the Math Model

and a homogeneous boundary condition, e.g. by the first-kind

condition

It is important for the method of separation of variables that the

boundary condition is homogeneous.

Therefore, if we deal with a problem with generic boundary

conditions, we should first pass to the problem with homogeneous conditions.

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the Math Model

The essence of the method of separation of variables (the Fourier

method) is the construction of particular solutions of (1) that can be

represented as a product:

where each factor depends on its own variable. Let us first consider

the case of a homogeneous equation (i.e. f (2, t) = 0 in (1)). We

substitute (4) into (1) and derive the equations for B(t) and v(x):

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the Math Model

According to (3), equation (5) is supplemented by the boundary

condition

The problem of (5) and (7) has nontrivial solutions only for some X

and is referred to as a spectral problem (the Sturm-Lioville problem).

The corresponding values of X are said to be eigenvalues and the

corresponding solutions v(x) are called eigen functions.

Let us number the eigenvalues of the problem of (5) and (7) in ascending order so that

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the Math Model

Given a solution of the spectral problem of (5) and (7), we can determine the general solution of (6) as

Let us now represent the solution of (1)-(3) with f(x,t) = 0 as a superposition of constructed particular solutions

The coefficients 𝑐𝑛 are determined by the initial condition (2), namely 𝑐𝑛 = 𝑢0, 𝑣𝑛 𝑐, are the coefficients in the expansion of the function 𝑢0 𝑥in the eigen functions 𝑣𝑛 𝑥 𝑐(the Fourier coefficients).

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the Math Model

Thus, we derived the solution with f(x, t) = 0 in the form:

In the case of nonhomogeneous equation (1), representation (9) in the method of separation of variables includes an additional term,

namely

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the Solution Form

We thus obtained the general solution (10) of the heat transfer

problem (1)-(3). The cases of the first- or second-kind boundary

conditions, mixed boundary conditions, etc. are proceeded with similarly.

Since the solution is represented as an infinite series, it is often

necessary to simplify the original problem to get a simpler solution.

The general solution is constructed given the solution of the spectral

problem in (5) and (7).

Note that the solution of this problem is known only in a few cases,

and most textbooks on heat transfer do not present these solutions.

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an Example

Let us find the solution of the simplest one-dimensional problem in

which x = (0,l). We consider the heat equation

with initial and boundary conditions in (2) and (3), respectively.

The corresponding eigenvalue problem (see (5) and (7)) becomes

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an Example

The eigenvalue problem in (12) and (13) has the solution

The solution of (11) is thus presented according to (lo), (14) and the

conditions in (2) and (3).

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the Applications [5]

Mathematical Models of Heat Flow in Edge-Emitting Semiconductor

Lasers:

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the Applications

Basic thermal behavior of an edge-emitting laser can be described

by the stationary heat conduction equation:

The heat power density is determined according to the crude

approximation:

Assuming no heat escape from the side walls:

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the Applications

Using the separation of variables approach, one obtains the solution

for T in two-fold form. In the layers above the active layer (n - even)

temperature is described by:

And for n - odd:

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References for Further Readings

1) Y.A. Çengel, Introduction to Thermodynamics & Heat Transfer, 2nd

ed., 2008, pp475-481.

2) A.A. Samarskii, Computational Heat Transfer, Vol. 1: Mathematical Modelling, 1995, pp62-65.

3) G.E. Myers, Analytical Methods in Conduction Heat Transfer, 1st ed.,

1966, pp74-85.

4) H.D. Bähr, K. Stephan, Heat and Mass Transfer, 3rd ed., 2011, pp144-

174.

5) V.S. Vikhrenko, Heat Transfer - Engineering Applications, 2011, pp7-24.

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