Heat transfer closed streamline flows

L. Heat Transfer at High Peclet Number Across Regions of Closed-Streamline Flow... 

L. HEAT TRANSFER AT HIGH PECLET NUMBER ACROSS REGIONS OF CLOSED-STREAMLINE FLOW... 

It is important to recognize that the analysis presented in this section is generally applicable to any high-Peclet-number heat transfer process that takes place across a region of closed-streamline flow. In particular, the limitation to small Reynolds number is not an intrinsic requirement for any of the development from Eq. (9-309) to Eq. (9-334). It is only in the specification of a particular form for the function V( ) that we require an analytic solution for f and thus restrict our attention to the creeping-flow limit. Indeed all of (9-309)-(9-334) apply for any closed-streamline flow at any Reynolds number, provided only... 

Poe, G. G., Closed streamline flows past rotating particles inertial effects, lateral migration, heat transfer, Ph. D. dissertation, Stanford Univ., 1975. 

Poe, G. G. and Acrivos, A., Closed streamline flows past small rotating particles heat transfer at high Peclet numbers, Int. J. Mult. Flow, Vol. 2, No. 4, pp. 365-377, 1976. 

When the flow in the boundary layer is turbulent, streamline flow persists in a thin region close to the surface called the laminar sub-layer. This region is of particular importance because, in heat or mass transfer, it is where the greater part of the resistance to transfer lies. High heat and mass transfer rates therefore depend on the laminar sublayer being thin. Separating the laminar sub-layer from the turbulent part of the boundary... 

Unfortunately, however, there are a large number of different types of flow conditions for which the boundary-layer form of the heat transfer correlation (9-255) is not applicable. This applies, basically, to any flow configuration in which the body is completely surrounded by a region of closed streamlines (or pathlines, if the flow is not 2D or axisymmetric). We will discuss high-Peclet-number heat transfer in such cases in Section L. Here, we consider... 

The correlation (9 230) was shown to be valid for heat transfer from solid bodies of arbitrary shape in a variety of arbitrary undisturbed flows, subject only to the condition that the body neither rotate nor be placed in an undisturbed flow that has closed streamlines at infinity. 

It was already indicated in the preceding sections that this thermal boundary-layer structure does not occur when a particle (or body) is entirely surrounded by closed streamlines (or closed stream surfaces). In this case, the convection process near the body can no longer transfer heat directly to the streaming flow where it is carried into the wake, but instead circulates it only in a closed path around the body. Thus the heat transfer process is fundamentally altered, because heat can escape from the body only by diffusing slowly across the region of closed streamlines (or stream surfaces). Because the size of this region is independent of Pe, the steady-state temperature gradients will be 0(1), and we expect that... 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

In the mass exchange problem for a circular cylinder freely suspended in linear shear flow, no diffusion boundary layer is formed as Pe - oo near the surface of the cylinder. The concentration distribution is sought in the form of a regular asymptotic expansion (4.8.12) in negative powers of the Peclet number. The mean Sherwood number remains finite as Pe - oo. This is due to the fact that mass and heat transfer to the cylinder is blocked by the region of closed circulation. As a result, mass and heat transfer to the surface is mainly determined by molecular diffusion in the direction orthogonal to the streamlines. In this case, the concentration is constant on each streamline (but is different on different streamlines). 

Duda and Vrentas [42] used this approach and found an infinite-series analytical solution for the closed-streamline axisymmetric flow in this cylinder. In a second paper [43], the corresponding developing heat transfer problem was solved using a formal Fourier series technique. The method allowed the calculation of time-dependent Nusselt numbers up to Is/Pedet numbers of up to 400. Extension to higher I/dh was prohibited as the eigenvalues of the solution became too close together as the aspect ratio was increased. 

---