Figure 6.4 shows an idealized form of the flow patterns and the variation of the surface and liquid temperatures in the regions designated by A, B, and C for the case of a uniform wall heat flux. Under steady state one-dimensional conditions the tube surface temperature in region A (convective heat transfer to single-phase liquid), is given by:

and where *q*;*SPMquot*; is the heat flux, *Per* is the heated perimeter, *G* is the mass velocity, *A* is the flow area and is the liquid specific heat. Also is the temperature difference between the wall surface and the mean bulk liquid temperature at a given length *z* from the tube inlet, h is the heat transfer coefficient to single-phase liquid under forced convection. The liquid in the channel may be in laminar or turbulent flow, in either case the laws governing the heat transfer are well established; for example, heat transfer in turbulent flow in a circular tube can be estimated by the well-known Dittus-Boelter equation.

This relation is valid for heating in fully developed vertical upflow in *z*/*D* > 50 and *Re* > 10,000.

where is the hydraulic diameter, is the liquid viscosity and is the liquid thermal conductivity.

For the case of a given constant wall temperature, the temperature difference will decrease, as well as the heat flux. From an energy balance this is represented by a logarithmic decrease in the temperature difference.

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