On the laminar forced convection with axial conduction in a circular tube with exponential wall heat flux

The values of the fully developed Nusselt number for laminar forced convection in a circular tube with axial conduction in the fluid and exponential wall heat flux are determined analytically. Moreover, the distinction between the concepts of bulk temperature and mixing-cup temperature, at low values of the Peclet number, is pointed out. Finally it is shown that, if the Nusselt number is defined with respect to the mixing-cup temperature, then the boundary condition of exponentially varying wall heat flux includes as particular cases the boundary conditions of uniform wall temperature and of convection with an external fluid.

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Über laminare Zwangskonvektion mit Längswärmeleitung in einem Kreisrohr mit exponentiell veränderlichem Wandwärmefluß

Zusammenfassung Es werden die Endwerte der Nusselt-Zahlen für vollausgebildete laminare Zwangskonvektion in einem Kreisrohr mit Längswärmeleitung und exponentiell veränderlichem Wandwärmefluß analytisch ermittelt. Besondere Betonung liegt auf dem Unterschied zwischen den Konzepten für die Mittel- und die Mischtemperatur bei niedrigen Peclet-Zahlen. Schließlich wird gezeigt, daß bei Definition der Nusselt-Zahl bezüglich der Mischtemperatur die Randbedingung exponentiell veränderlichen Randwärmeflusses die Spezialfälle konstanter Wandtemperatur und konvektiven Wärmeaustausches mit einem umgebenden Fluid einschließt.

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Nomenclature A _n dimensionless coefficients employed in the Appendix - Bi Biot numberBi=h _e r ₀/ - c _n dimensionless coefficients defined in Eq. (17) - c _p specific heat at constant pressure of the fluid within the tube, J kg^–1 K^–1] - f solution of Eq. (15) - h ₁,h ₂ specific enthalpies employed in Eqs. (2) and (4), J kg^–1] - h _e convection coefficient with a fluid outside the tube, W m^–2 K^–1] - rate of mass flow, kg s^–1] - Nu bulk Nusselt number,2r ₀ q _w /(T _w –T _b )] - Nu _H fully developed value of the bulk Nusselt number for the boundary condition of uniform wall heat flux - Nu _T fully developed value of the bulk Nusselt number for the boundary condition of uniform wall temperature - Nu ^* mixing Nusselt number,2r ₀ q _w /(T _w –T _m )] - Nu _C ^* fully developed value of the mixing Nusselt number for the boundary condition of convection with an external fluid - Nu _H ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall heat flux - Nu _T ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall temperature - Pe Peclet number, 2r ₀/ - q ₀ wall heat flux atx=0, W m^–2] - q _w wall heat flux, W m^–2] - r radial coordinate, m] - r ₀ radius of the tube, m] - s dimensionless radius,s=r/r ₀ - T temperature, K] - T ₀ temperature constant employed in Eq. (14), K] - T reference temperature of the fluid external to the tube, K] - T _b bulk temperature, K] - T _m mixing or mixing-cup temperature, K] - T _w wall temperature, K] - u velocity component in the axial direction, m s^–1] - mean value ofu, m s^–1] - x axial coordinate, m] Greek symbols thermal diffusivity of the fluid within the tube, m² s^–1] - exponent in wall heat flux variation, m^–1] - dimensionless parameter - dimensionless temperature =(T _w –T)/(T _w –T _b ) - ^* dimensionless temperature ^*=(T _w –T)/(T _w –T _m ) - thermal conductivity of the fluid within the tube, W m^–1 K^–1] - density of the fluid within the tube, kg m^–3]