The methods developed to examine steady–state heat transfer in a 2–D region can also be applied to convection–diffusion heat transfer in situations in which the velocity of the convective flow is fixed and known. Such a problem is illustrated in Fig. 4.17. Say flow enters a circular pipe with a uniform and constant velocity of u and an inlet temperature of T0. The walls of the pipe are maintained at a uniform temperature of Tw. We want to predict the temperature distribution of the fluid in the pipe and the rate of heat transfer to/from the wall.
This is not a fluid mechanics problem: we know the velocity distribution throughout the pipe.
This particular example, in which the velocity in the pipe is assumed uniform and constant, would be referred to as a plug–flow. A more realistic model, for laminar flow conditions, would be to use a parabolic velocity profile, yet the plug flow model will offer mathematical simplicity and would be more representative of certain turbulent flow conditions (in which the average velocity profile is mostly uniform over the cross sectional area of the pipe).
The steady form of the energy equation (in dimensional coordinates) will be u
with α = k/ρcp being the thermal diffusivity of the fluid. The term on the left, as you probably recognize, accounts for the axial convection of enthalpy in the flow. There is no corresponding r–directed term because there is no r component of velocity.
Take the radius of the pipe, R, to be the characteristic length. The problem can be made dimensionless by defining the variables as
T → T − Tw
T0− Tw, r → r
R, x → x R so that the DE now appears as
P e∂T
in which P e = uR/α is the Peclet number of the flow. The Peclet number is analogous to the Reynolds number; it is a ratio of the characteristic rates of axial convective and radial diffusive
heat transfer in the pipe. Often the Peclet number is defined using diameter D instead of radius R; we’ll use the radius definition to keep things more simple.
In pipe flow problems, the axial diffusion term (the second term on the right hand side) is often neglected – because the gradients in the axial direction (for developed flow conditions) will be relatively small compared to the radial gradients. If we removed this term, we would simply get a DE in the same form as the 1–D and transient problems in the previous chapter (i.e., a parabolic DE, with x taking the place of the time variable). However, for this developing flow problem we cannot, in general, neglect the axial diffusion term.
Boundary conditions to the dimensionless problem are
T (0, x) is finite (4.128)
T (1, x) = 0 (4.129)
T (r, 0) = 1 (4.130)
T (r, x → ∞) = 0 (4.131)
Note that the last boundary condition simply states that the fluid temperature will go to the wall temperature after sufficient distance in the pipe.
This problem can be solved with SOV methods, the solution of which will be outlined here.
The problem has a homogeneous direction (r), and we anticipate that the solution will be in the form
T (r, x) = X∞ n=1
Anφn(r) vn(x) (4.132)
Characteristic ODEs for φn(r) and vn(x) are obtained by substituting φn(r) · vn(x) into the PDE, Eq. (4.127);
1
r(rφ′n)′ = −λ2nφn (4.133)
P e v′n− vn′′= −λ2nvn (4.134) The eigenfunction ODE is in the same form as previous cylindrical problems, and we obtain
φn(r) = J0(λnr) (4.135)
J0(λn) = 0 (4.136)
Two independent solutions exist to Eq. (4.134), only one of which satisfies the zero condition at x → ∞. This solution is
vn(x) = exp x 2
hP e − 4λ2n+ P e21/2i
(4.137) The expansion coefficients An in the solution are obtained from the inhomogeneous condition at the inlet:
1 = X∞ n=1
Anφn(r) vn(0)
or, using vn(0) = 1 and the orthogonality of the eigenfunctions,
An= Z 1
0
φnr dr Z 1
0
φ2nr dr
(4.138)
This completes the solution for the temperature profile. In convective/diffusive problems such as this, it is often of interest to define the heat transfer coefficient h from the solution to the temperature profile. The heat transfer coefficient, for this pipe flow problem, is defined so that the heat flux to the pipe wall is given by
q′′(x) = h(x)(Tm(x) − Tw) (4.139) where Tm is the mean temperature of the fluid at position x (the quantities in the above equation are now dimensional ). The mean temperature is defined as
Tm= 2 R2
Z R 0
T (r, x) r dr (4.140)
and the heat flux is
q′′= −k ∂T
∂r
r=R
(4.141) If we now return to dimensionless coordinates and use the above three equations, we will obtain
hR k ≡ 1
2N uD = −
∂T
∂r
r=1
2 Z 1
0
T r dr
(4.142)
The dimensionless quantity N uD = hD/k is the Nusselt number based on pipe diameter. It has the same grouping of quantities as the Biot number, yet it has a fundamentally different interpretation.
The Nusselt number, as the above equation shows, is basically a dimensionless temperature gradient at the surface, scaled by a nondimensional mean temperature of the flow. Essentially, it is a nondimensional way of expressing the heat transfer coefficient h. The Biot number, on the other hand, relates conduction resistance within a solid to convection resistance from the solid.
A plot of N uD vs. dimensionless axial position x is given in Fig. 4.18, in which P e = 10. At the entrance to the pipe N uD → ∞, due to the instantaneous change in temperature of the wall.
As the flow progresses into the pipe both the heat flux and the mean temperature decrease to zero.
The ratio of the two, however, approaches a constant. In the large x limit it is easy to show that
1 2 3 4 5 x 10
20 30 40 50 Nu
Figure 4.18: N uD vs. length into pipe x for plug flow conditions
only one term is retained in the series solution for temperature. For this case the Nusselt number will become independent of both x and P e and equal to
N uD 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.