Plug flow, also known as piston flow and slug flow, is a flow with a homogeneous velocity profile across the entire flow section: u = w = const (w = ^{1}/^{∞}A∫_{A}udA is the section average velocity). Examples of plug flow heat transfer are: the motion of the long bars of various profiles, wires, threads pulling through a heat-treatment zone (for instance, in hardening and annealing devices); the motion of granular bodies (the difference in velocity near the wall and in the core of the flow is in this case only 10-12 percent). A plug flow model can also be used to study heat transfer in a laminar flow of molten metals within the entrance region of a pipe (i.e., when 0 ≤ x l_{eh}, where l_{eh} is the bydrodynamic entrance region length and x the distance from the tube entrance). This is possible as a result of the low values of Prandtl number for molten metals (Pr 0.01− 0.03). The length to diameter ratio l_{eh}/d_{h} 0.03 Re (here d_{h} is the hydraulic diameter, Re is the Reynolds number) considerably exceeds the length to diameter ratio of the entrance region l_{et}/d_{n}. Note that thermal l_{et}/d_{n} 10^{−1} Pe (Pe = ud_{h}/κ = Re Pr is the Pecl_{et} number), i.e., l_{et}/l_{eh} 3Pr 0.03 - 0.1. Therefore, the temperature field has a chance to stabilize as long as the velocity profile is at the initial stage of development and remains close to that for plug flow. The accepted model of plug flow allows us to present the flow/energy equation in the following form (it is assumed that die physical properties of the liquid, the density ρ, heat capacity c, and thermal conductivity λ are constant, and that turbulence is absent):in the Cartesian system of coordinates

and in a cylindrical system of coordinates

where k = κ/u = d_{n}/Pe is the constant (k = λ/(ρc) is the thermal diffusivity), q_{v} is the volumetric heat generation rate. The equations are related to those which are considered in the heat conduction theory and can be solved by the analytical and numerical methods developed for heat transfer equations.

#### REFERENCES

Carlslaw, G., Jaeger, D. (1964) *Heat Transfer in Solid Bodies*, Moscow, Nauka.

Petukhov, B. S. (1967) *Heat Transfer and Resistance in Laminar Fluid Flow Through Pipe*, Moscow, Energy.