x→∞
= λ21 = 5.783 (4.143)
This would be considered the fully developed flow limit. You might recall that, for fully developed laminar flow in a pipe (with a parabolic velocity profile), the Nusselt number for isothermal wall conditions is the constant value of 3.66. We get a different value here because of the assumed plug flow velocity distribution.
4.6 Summary
A whole lot of details remain to be covered; such as superposition techniques in radial problems and spherical coordinate solutions. The latter topic will be addressed in a later chapter, and the former would extend directly from the examples in cartesian coordinates.
As a reference, a list of the general solution forms for problems in cartesian and cylindrical 2–D steady conditions is given below.
In cartesian coordinates, with homogeneous boundary conditions in the x direction, the solution will take the form
T = X∞ n=1
[Ancosh(λny) + Bnsinh(λny)] φn(x) (4.144) where φn is the eigenfunction for the x direction and λn is the eigenvalue. The eigenfunction will involve combinations of the trigonometric functions, i.e.,
φn= [cos(λnx), sin(λnx)] (4.145)
in which the square brackets denote a linear combination of the two functions.
In cylindrical r− z coordinates with r as the homogeneous direction, the solution will appear in the same form as before:
T = X∞ n=1
[Ancosh(λnz) + Bnsinh(λnz)] φn(r) (4.146) except the eigenfunctions φn(r) will now involve the ordinary Bessel functions of order 0:
φn(r) = [J0(λnr), Y0(λnr)] (4.147) If the homogeneous direction is z, the general solution will appear
T = X∞ n=1
[AnI0(λnr) + BnK0(λnr)] φn(z) (4.148) where the eigenfunctions φn(z) will involve the trigonometric functions as in Eq. (4.145).
And as always, the constants An and Bn are obtained from the BCs in the non–homogeneous direction and (when needed) the orthogonality relations for the eigenfunctions.
Exercises
1. A square, 2–D rod is exposed to identical convection conditions on the left and right faces.
The bottom surface is insulated, and the top surface receives a nonuniform heat flux given by q′′(x) = q0′′ exph
−a2(x − L/2)2i
in which q0′′ and a are constants. Formulate the problem for the temperature distribution in appropriate dimensionless form, and derive a solution using the SOV method. Note: the boundary conditions can be simplified by exploiting the symmetry of the problem.
2. A solid circular rod, of length L and radius R, has the ends at z = 0 and L maintained at T1
and T2. Electrical current flows through the rod which results in a uniform heat generation rate of q′′′ within the rod. The surface at r = R is cooled by convection to T∞.
(a) Using the superposition and SOV methods, determine the solution for the temperature distribution in the rod. Be sure to cast the problem in dimensionless variables.
(b) Re–derive the solution for the case of L → ∞. Recognize that in this limit there will no longer be an explicit BC stated at L, rather, the solution must asymptote to the correct behavior for z → ∞. Derive a formula for the total rate of heat transfer to/from the z = 0 surface of the wire, and plot the result (in appropriate dimensionless form) as a function of dimensionless generation rate q′′′ using BiR = hR/k = 1. Discuss the physical significance of your results.
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Figure 4.19: circular spreader, side view
3. A thermal spreader is a device used to transfer heat from a (typically small) integrated circuit to a cooling environment. Often they are designed to provide a larger heat transfer area to the convection environment than that occupied by the circuit, thereby ‘spreading’ out the heat much like a fin. A circular disk spreader is illustrated in Fig.4.19, which has a radius of R and a thickness of t. The circular IC, of radius Rc, is centered on the top of the spreader, and a uniform flux of q′′c enters the spreader from the IC. The bottom surface of the spreader is cooled by convection, and all other surfaces are adiabatic.
(a) Formulate the problem in appropriate dimensionless variables, and derive the analytical solution for the temperature distribution in the spreader. You will want to use R as the characteristic length and q′′cR/k as the characteristic temperature difference. Please note that this problem will admit a zero eigenvalue (λ0 = 0) with a non–zero zeroth eigenfunction (φ0 6= 0). It is critically important that you include the contribution of these terms.
(b) In the limit of Bi = hR/k → 0, and assuming t = t/R < 1, your solution should give the result of T → (Rc/R)2/Bi. Explain, using physical arguments, why this is the case.
(c) Make a plot of T (x = t, r = 0)· Bi (i.e., the dimensionless spreader temperature directly under the IC, multiplied by Bi) vs. dimensionless thickness t for Bi = 0.5, 1, and 5 and with Rc/R = 0.25. Using this plot, identify an optimum thickness of the spreader, i.e., that which minimizes the IC temperature for the fixed heat dissipation rate. Explain, using physical principles, why such an optimum occurs.
4. Consider the convective–diffusion problem examined in the last section, but now the boundary condition at the wall is a constant heat flux condition, i.e.,
k ∂T
Develop a solution for the dimensionless temperature profile in the tube as a function of r and x. Note that the zeroth eigenfunction will play a role in this situation. Also obtain
series solutions for the mean temperature Tm(x) and the Nusselt number N uD(x), and plot these quantities as a function of x using P e = 10 Note that the mean temperature will not
→ 0 for x → ∞, since heat is continuously being added to the fluid. Finally, obtain the fully–developed value for the Nusselt number.
General Multidimensional Conduction
5.1 Introduction
The problems examined in the previous chapters were restricted to two dimensions (space + time or space + space) and could be solved with the SOV/superposition methods. The first item of business in this chapter will be to extend SOV/superposition to problems involving three or more dimensions. In doing so we will retain, for time–dependent problems, the transient impulse model of the initial condition; this restriction will be lifted in the following chapter. We will also examine the variation of parameters method for solution of multidimensional conduction-type PDEs – which is somewhat more generalized than the SOV/superposition approach. Finally, we will introduce the concept of a semi–infinite domain and examine the steady flow of heat in such situations. A fundamentally different analytical method, known as the Fourier cosine transform, will be developed to describe the temperature field in the semi–infinite domain.