The thermal entrance length Leh is used to outline the boundary between the fully developed heat flow and the non-fully developed heat flow in a heated/cooled pipe flow of a fluid.
Contents
Overview
A fully developed heat flow in a pipe can be considered in the following situation. If the tube wall of the pipe is constantly heated or cooled so the heat flux from the wall to the fluid via convection is a fixed value, then the bulk temperature of the fluid increases steadily at a fixed rate along the flow direction. An example can be a pipe covered entirely by an electrical heating pad, and the flow is introduced after a uniform heat flux from the pad is achieved. At a distance away from the entrance of the fluid, the fully developed heat flow is achieved when the heat transfer coefficient of the fluid becomes constant, and the temperature profile has the same shape along the flow. This distance is defined as the thermal entrance length, which is important for engineers to design efficient heat transfer processes.
Quantitative measurement
Quantitatively, If x is chosen to be the axis parallel to the pipe and x = 0 is chosen as the commencing point of the pipe flow, the thermal entrance length is defined as the distance (x >0) required for the Nusselt number Nu associated with the pipe flow to decrease to within 5% of its value for a fully developed heat flow [1].
Depends on different flow conditions (laminar, turbulent, shapes of entrance, etc.), the Nusselt number has different dependence on Reynolds number, Prandtl number and the friction factor of the flow.
Simple scenario
A simple example is a laminar flow which is already hydrodynamically fully developed at x=0, and a constant and uniform pipe wall temperature is maintained. In this case, the thermal entrance length can be calculated by a simplified equation written as:
(Leh (5%))/D=0.033 ReDPr [1]where D is the pipe diameter, ReD the Reynolds number and Pr is the Prandtl number.
Given Reynolds number is a constant for a hydrodynamically fully developed flow (where the velocity of the flow remains unchanged), the equation above indicates that the thermal entrance length is proportional to the Prandtl number [1], which is defined as the ratio of momentum diffusion rate to the thermal diffusion rate of a fluid. That is, a low Pr material, with its thermal diffusion rate closer to its momentum diffusion rate, can achieve a fully developed heat flow at a shorter distance compared to a high Pr material in this situation.
Complex scenario
In a more complicate scenario (turbulent flow, rectangular entrance, open end entrance, etc.), there is seldom an easy method to calculate the thermal entrance length. If the flow is laminar, i.e. Reynolds number is equal or less than 2100/2300, the thermal entrance length can be within 5 diameters for high Pr and low Pr materials given there is no large perturbation or eddies [1]. For gases and water at higher temperatures, the Prandtl number is close to 1 and the thermal entrance length can vary between 15 and 40 diameters [1]. Overall, the determination of the thermal entrance length can be difficult and requires an understanding of the heat and fluid transfer phenomenon, and a list of different calculations and charts can be found in various reading sources.