Майнор / Книги / Chapter-3---Conduction--One-dimensional-transient-and-_2021_Heat-Transfer-En

Substituting in Eq. (3.37) we get the following

Please note that Eq. (3.44) is an ordinary differential equation (ODE), as opposed to Eq. (3.37), which is a partial differential equation (PDE). Eq. (3.44) is a secondorder differential equation that supports two conditions. These are

η= 0; θ = 0

η→ ∞; θ =θi

The second condition is quite interesting. This actually means that, x → ∞ as well at t = 0, θ = θi. Hence, our proposal to introduce a new variable η, frequently referred to as a similarity variable has helped us to maintain the integrity of the original governing equation, as now we have only two conditions for θ, as opposed to three in the original PDE.

In order to obtain the constant A, We need to integrate Eq. (3.50) from 0 to ∞ and the corresponding limits for θ are 0 and θi.

76 CHAPTER 3  Conduction

Combining Eqs. (3.51) and (3.52), we have

At this point in time, we can invoke the use of error function, erf(η) as follows,

Consequent upon the invoking of erf(η), Eq. (3.54) may be rewritten as,

Eq. (3.56) uses the information that erf(∞) = 1.

The error function can be tabulated in a form that is easy to use. Values of erf(η) against η are given in Table 3.1.

Table 3.1  Error function table.

Eq. (3.57) can be used to determine the temperature at any point “x” at any time “t” in the slab. Please note that nowhere does the length scale L of the problem appear in the final solution.

3.4  The method of separation of variables

Let us now move on to the more general and less restrictive case where the dimension-

less temperature φ = θ = f (ξ, Bi, Fo). Consider a one-dimensional slab of thickness

θi

2L. Let ρ,cp and k be the density, specific heat capacity, and thermal conductivity of the body respectively. The body is initially at a temperature of Ti throughout. At time t = 0, the two ends at x = ±L are suddenly brought to T∞ with T∞ < Ti. The problem of transient conduction begins with the challenge being our ability to get T(x,t) in the domain for −L ≤ x ≤ L and for all t ≥ 0.

Let us now try to sketch qualitatively the variation of φ across the slab at various times, as shown in Fig. 3.5

From the figure it is seen that at “early” times a substantial part of the slab is still at Ti, and it is in this regime that the semi-infinite approximation is valid. Additionally,

FIGURE 3.5

Qualitative variation of temperature time history for various time instances in an infinitely long slab of thickness 2L.

†An incorrect choice of λ2 instead of –λ2 would have resulted in a solution for temperature that is exponential in x and periodic in t, which completely goes against the physics of the problem.

80 CHAPTER 3  Conduction

In Eq. (3.75), An is as yet unknown. To obtain An we invoke the initial condition. At t = 0 s; φ = φi = 1

n=1

We need to use the property of a set of functions called orthogonal functions to

obtain An.

Orthogonal functions

An infinite set of functions g1 (x), g2 (x), g3 (x)…..gn (x) is said to be orthogonal in

the closed interval a ≤ x ≤ b, if ∫ab gm (x)gn (x) = 0; for m ≠ n.

Many functions like sinx and cosx exhibit orthogonality. Hence if we multiply Eq. (3.77) by cos(λnx) on both sides and integrate from 0 to 1, we can exploit the property of orthogonal functions to evaluate An.

Please note that the right hand side of the equation 3.78 was originally a summa­ tion. However, in the view of the orthogonality property of cosλnx, only the term with cos2λnx will remain in the series, as all other terms will be zero.

84 CHAPTER 3  Conduction

FIGURE 3.7

Temperature distribution chart for a plane wall with thickness 2L initially at a uniform temperature of Ti subjected to convection on both sides with a heat transfer coefficient of h and an ambient temperature of T∞.

FIGURE 3.8

Enthalpy chart for a plane wall with thickness 2L initially at a uniform temperature of Ti subjected to convection on both sides with a heat transfer coefficient of h and an ambient temperature of T∞.