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F4 Heat Transfer by Free Convection: Special Cases

Abstract

Heated vertical channels act as chimneys, i.e., buoyancy forces cause the surrounding fluid to flow toward the inlet and through the channel itself. Assume a channel with a constant wall temperature T w , a heated section of height h, and a longitudinal cross section of area f extending from the inlet to the outlet; and let fluid flow through the channel with a velocity distribution u. Then the heat transferred from the channel walls to the fluid is given by (1) $$\dot Q = \rho c_p \left[ {\int\limits_0^f u(T - T_{\rm{E}} )\;{\rm{d}}f} \right]_{\rm{h}} = A\alpha (T_{\rm{w}} - T_{\rm{E}} ),$$ where A is the area of the heated surface, T E is the temperature of the fluid entering the channel, and ϑ is the outlet temperature distribution. The average heat transfer coefficient for the entire channel is (2) $$\alpha = {{\dot Q} \over {A(T_{\rm{w}} - T_{\rm{E}} )}}.$$ It is described by the following relationship for vertical channels: (3) $${\rm{Nu}}_{\rm{S}} = {\rm{Nu}}_{\rm{S}} ({\rm{Gr}}_{\rm{S}}^ * \rm Pr),$$ where (4) $${\rm{Nu}}_{\rm{S}} = {{\alpha s} \over \lambda },$$ (5) $${\rm{Gr}}_{\rm{S}}^ * = {{g\beta (T_{\rm{w}} - T_{\rm{E}} )s^3 } \over {v^2 }}{s \over h},$$ (6) $$Pr = {v \over a}.$$ The coefficient of volume expansion β is determined as described in Chap. F1 . The reference temperature for the properties is $${\textstyle{1 \over 2}}(T_{\rm{w}} + T_{\rm{E}} )$$ . If the channel is planar and heated on one side, as illustrated in Fig. 1a, the characteristic length s from which Nu and Gr are calculated is the width of the channel, i.e., s = d. In plane channels heated on two sides, as illustrated in Fig. 1b, the characteristic length is half the channel width, i.e., s = d/2; and in heated tubes, as illustrated in Fig 1c, it is the channel radius s = r = d/2. The heat transfer area A and the channel cross section f can then be obtained from the channel length b. The following thus apply:

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Title
F4 Heat Transfer by Free Convection: Special Cases
Book Title
VDI Heat Atlas
Book DOI
10.1007/978-3-540-77877-6
Chapter DOI
10.1007/978-3-540-77877-6_122
Part of
VDI-Buch
Volume
Editors
Authors
  • Werner Kast (2_122)
  • Herbert Klan (2_122)
  • (Revised by André Thess) Send Email (1_122)
  • Author Affiliation
  • 2_122 Technische Universität Darmstadt, Darmstadt, Germany
  • 1_122 Technische Universität Ilmenau, Ilmenau, Germany
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