Thermally and hydrodynamically developing flow

In flow that is neither hydrodynamically nor thermally fully developed the velocity and temperature profiles change along the flow path. Fig. 3.34 shows qualitatively some velocity and temperature profiles, under the assumption that the fluid flows into the tube at constant velocity and temperature. The wall temperature of the tube is lower than the inlet temperature of the fluid. 

The calculation of the velocities and temperatures requires the solution of the continuity, momentum and energy equations. An explicit solution of the system of equations is not possible. A numerical solution has been communicated by Stephan [3.24] and later by several other authors see [3.26], As a result the pressure drop Ap = pi — p between the pressure pi at the inlet and the pressure p at any tube cross section is obtained. It can be approximated by an empirical correlation [3.30] 

In the same way as shown in the previous section, the heat transfer coefficient and from that the mean Nusselt number Nume = amed/A (the index e stands for entry flow) can be obtained from the temperature profile. The Nusselt number can be calculated from an empirical equation of the form 

Pr = 0 and Pr — oo. Pr = 0 in this case means that the viscosity vanishes, but the thermal conductivity is finite. As no friction forces act in the fluid, the velocity at the inlet remains constant. This type of flow is known as plug flow. In the limiting case Pr — oo, because the viscosity is large in comparison to the thermal diffusivity, the flow is hydrodynamically but not thermally fully developed. At the limit Pr — oo equation (3.258) yields Nume/Num = 1. 

Vanishing Prandtl numbers Pr = 0 can also mean that the thermal diffusivity is approaching infinity, whilst the viscosity remains finite. Then the flow is already thermally fully developed at the inlet but not yet hydrodynamically fully developed. As the Peclet number disappears, Pe = w m d/a = 0, X 1 = L/(dPe) = oo. The Nusselt number is equal to that for thermal fully developed flow Num = 3.6568. 

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