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

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

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Abstract

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.

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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Abbreviations

A n :

dimensionless coefficients employed in the Appendix

Bi :

Biot numberBi=h e r 0/λ

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]

\(\dot m\) :

rate of mass flow, [kg s−1]

Nu :

bulk Nusselt number,2r 0 q w /[λ(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 /[λ(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, 2ūr 0/α

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]

ū :

mean value ofu, [m s−1]

x :

axial coordinate, [m]

α:

thermal diffusivity of the fluid within the tube, [m2 s−1]

β:

exponent in wall heat flux variation, [m−1]

\(\hat \beta \) :

dimensionless parameter\(\hat \beta = Pe r_0 \beta \)

ϑ:

dimensionless temperature ϑ=(T w −T)/(T w −T b )

ϑ* :

dimensionless temperature ϑ*=(T w −T)/(T w −T m )

λ:

thermal conductivity of the fluid within the tube, [W m−1 K−1]

ϱ:

density of the fluid within the tube, [kg m−3]

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  1. Istituto di Fisica Tecnica Facoltà di Ingegneria, Università di Bologna, Viale Risorgimento 2, I-40136, Bologna, Italy

    A. Barletta &  E. Zanchini

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  1. A. Barletta
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  2. E. Zanchini
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Barletta, A., Zanchini, E. On the laminar forced convection with axial conduction in a circular tube with exponential wall heat flux. Heat and Mass Transfer 30, 283–290 (1995). https://doi.org/10.1007/BF01463917

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