Transient Heat Conduction, Lumped Element Method, Separation of Variables

1. Transient Heat
Conduction
PRESENTED BY: A.R. AMINIAN
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2. First things First….
 An Introduction to Metallurgical Engineering…
 Metallurgy…what does it mean?!
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3. 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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4. LEM Applicability
 Ex1: the Copper Ball
 Ex2: the Roast Beef
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5. 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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6. Lumped Element System:
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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7. Lumped Element System:
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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8. Lumped Element System:
the Definition
 Integrating from t = 0, at which T = Ti, to any time t, at which T = T(t),
gives
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9. Lumped Element System:
the Definition
 Taking the exponential of both sides and rearranging, we obtain
 where
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10. Lumped Element System:
the Definition
 The b is called the time constant
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11. Lumped Element System:
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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12. Lumped Element System:
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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13. Lumped Element System:
the Applications
 Metallurgical Analysis of Heat Transfer during
 Heat Treatment
 Casting
 Hot Forging
 Thermo-Forming
 Vacuum Thermo-Forming, and…
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14. Lumped Element Analysis:
other Apps. than Heat Transfer
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15. Transient Heat Conduction:
the Separation of Variables [1]
 an Intro to Nondimensionalization:
 Consider an original heat conduction problem:
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16. Transient Heat Conduction:
the Separation of Variables [1]
 Now, Nondimensionalizing the problem lead us to:
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17. Transient Heat Conduction:
the Separation of Variables [1]
 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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18. 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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19. 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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20. Separation of Variables:
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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21. Separation of Variables:
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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22. Separation of Variables:
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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23. Separation of Variables:
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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24. Separation of Variables:
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 𝑐 𝑛 are determined by the initial condition (2), namely
𝑐 𝑛 = 𝑢0, 𝑣 𝑛 𝑐, are the coefficients in the expansion of the function 𝑢0 𝑥
in the eigen functions 𝑣 𝑛 𝑥 𝑐(the Fourier coefficients).
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25. Separation of Variables:
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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26. Separation of Variables:
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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27. Separation of Variables:
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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28. Separation of Variables:
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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29. Separation of Variables:
the Applications [5]
 Mathematical Models of Heat Flow in Edge-Emitting Semiconductor
Lasers:
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30. Separation of Variables:
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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31. Separation of Variables:
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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32. 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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