Heat Transfer in Laminar Tube Flow

Consider the tube-flow system in Fig. 5-13. We wish to calculate the heat transfer under developed flow conditions when the flow remains laminar. The wall temperature is Tw, the radius of the tube is rc, and the velocity at the center of the tube is u0. It is assumed that the pressure is uniform at any cross section. The velocity distribution may be derived by considering the fluid element shown in Fig. 5-14. The pressure forces are balanced by the viscous- 

The heat flow conducted into the annular element is 

Net energy converted out = net heat conducted in or, neglecting second-order differentials, 

We assume that the heat flux at the wall is constant, so that the average fluid temperature must increase linearly with x, or 

This means that the temperature profiles will be similar at various x distances along the tube. The boundary conditions on Eq. 5-99 are 

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R.O.C. Guedes, R.M. Cotta, and N.C.L. Bram, Heat Transfer in Laminar Tube Flow with Wall Axial Conduction Effects, J. Thermophysics Heat Transfer, 5 (4), 508-513 (1991). 

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