Developing Thermal Flow

The thermal entry region for laminar flow can be calculated by [40] 

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Heat Transfer and Velocity Distribution in Hydrodynamically and Thermally Developed Laminar Flow in Conduits... 

The Circular Tube Thermal-Entry-Length, with Hydrodynamically Fully Developed Laminar Flow... 

Attention was then turned to developing duct flows. A numerical solution for thermally developing flow in a pipe was first considered. Attention was then turned to plane duct flow when both the velocity and temperature fields are simultaneously developing. An approximate solution based on the use of the boundary layer integral equations was discussed. 

Fig. 5-5 Local and average Nusselt numbers tor circular tube thermal entrance regions in fully developed laminar flow.

Table II also includes an estimate of the elevated temperatures reached in the contact areas. These estimates were developed on the basis of the TGA analysis as well as the IR data and appearance of the damage. Any evidence of thermal flow, charring, or chemical change in the impacted area of the film was taken as an indication that the T min of the TGA analysis was exceeded. As shown, these estimated local temperatures are quite high, and the question immediately arises as to their sources under impact conditions. There are three possible sources for this increase in temperature (a) internal damping, (b) differential surface displacement resulting from contact, or (c) fretting motion at the interface induced by the impact.

The developed resist patterns were heated at 200°C for 1 hour in a convection oven. Scanning electron microscopy shows no thermal deformation in the negative tone images (Figure 10). The positive tone images lost thickness consistent with thermolysis of the t-BOC side chain but also do not show evidence of thermal flow deformation. 

C Wilt a thermal boundary layer develop in flow over a surface even if both the fluid and the surface arc at the same temperature ... 

We start this chapter with a general physical description of internal flow, and the average velocity and average temperature. We continue with the discussion of the hydrodynamic, and thermal entry lengths, developing flow, and fully developed flow. We then obtain the velocity and temperature profiles for fully developed laminar flow, and develop relations for the friction factor and Nusselt nmnber. Hinally we present empirical relations for developing and full developed flows, and demonstrate their use. 

Aparecido, J., and Cotta, R. (1990) Thermally Developing Laminar Flow inside Rectangular Ducts, Int. J. Heat andMass Transfer. Vol. 33, pp. 341-347. 

Thermal modelling is a powerful way to predict the performance and the temperature response of MHP/ mHP. Unfortunately most of developed thermal models are 1-D models and empirical correlations are employed to determine fraction factor of vapour flow. 

Heat transfer coefficients in thermally fully developed, laminar flow... 

In the following we will show how heat transfer coefficients are calculated for thermally fully developed, laminar flow. In a corresponding manner the mass transfer coefficients with regard to fully developed concentration profile can be obtained. In order to show this fundamentally we will consider tubular flow. The explanations can easily be transferred to cover other types of channel flow. 

Table 3.2 Nusselt numbers Nu = ad JX in thermal, fully developed, laminar flow and resistance factors = Ap/(L/dh g/2w, ) in channels with different cross sections. Nut is the Nusselt number at constant wall temperature, Nuq that at constant heat flux at the wall and Re = wmdh/v is the Reynolds number.

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

S.P. Guidice, C. Nonino, and S. Savino, Effects of viscous dissipation and temperature dependent viscosity in thermally and simultaneously developing laminar flows in microchannels. International Journal of Heat and Fluid Flow 28, 15-27 (2007). 

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