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Research Papers: Forced Convection

# Unsteady Laminar Forced-Convective Tube Flow Under Dynamic Time-Dependent Heat Flux

[+] Author and Article Information
M. Fakoor-Pakdaman

Laboratory for Alternative Energy
Convection (LAEC),
School of Mechatronic Systems Engineering,
Simon Fraser University,
e-mail: mfakoorp@sfu.ca

Laboratory for Alternative Energy
Convection (LAEC),
School of Mechatronic Systems Engineering,
Simon Fraser University,
e-mail: mandishe@sfu.ca

Majid Bahrami

Laboratory for Alternative Energy
Convection (LAEC),
School of Mechatronic Systems Engineering,
Simon Fraser University,
e-mail: mbahrami@sfu.ca

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 3, 2013; final manuscript received November 20, 2013; published online February 12, 2014. Assoc. Editor: D. K. Tafti.

J. Heat Transfer 136(4), 041706 (Feb 12, 2014) (10 pages) Paper No: HT-13-1277; doi: 10.1115/1.4026119 History: Received June 03, 2013; Revised November 20, 2013

## Abstract

A new all-time model is developed to predict transient laminar forced convection heat transfer inside a circular tube under arbitrary time-dependent heat flux. Slug flow (SF) condition is assumed for the velocity profile inside the tube. The solution to the time-dependent energy equation for a step heat flux boundary condition is generalized for arbitrary time variations in surface heat flux using a Duhamel's integral technique. A cyclic time-dependent heat flux is considered and new compact closed-form relationships are proposed to predict (i) fluid temperature distribution inside the tube, (ii) fluid bulk temperature and (iii) the Nusselt number. A new definition, cyclic fully developed Nusselt number, is introduced and it is shown that in the thermally fully developed region the Nusselt number is not a function of axial location, but it varies with time and the angular frequency of the imposed heat flux. Optimum conditions are found which maximize the heat transfer rate of the unsteady laminar forced-convective tube flow. We also performed an independent numerical simulation using ansys fluent to validate the present analytical model. The comparison between the numerical and the present analytical model shows great agreement; a maximum relative difference less than 5.3%.

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## Figures

Fig. 1

Schematic of the two-region tube and the coordinate system

Fig. 2

The methodology adopted to find the transient thermal response of the tube flow under arbitrary time-dependent heat flux

Fig. 3

Variations of the dimensionless tube- wall temperature, Eqs. (13) and (14), versus the Fo number for a cyclic heat flux, q"=q"r[1+sin(8πFo)]

Fig. 4

Variations of the dimensionless fluid temperature, Eqs. (11) and (12), at an arbitrarily-chosen axial position of X = 0.4 and angular frequency of 8π at different radial positions across the tube against the Fo number for cyclic and step heat fluxes

Fig. 5

Variations of the dimensionless tube-wall temperature, Eqs. (13) and (14), at an arbitrarily chosen axial position of X = 0.4 against (a): the Fourier number for different angular frequencies of the heat flux, (b) the angular frequency at different Fourier numbers, and (c) the angular frequency and the Fourier number

Fig. 6

Variations of the local Nusselt number, Eqs. (21) and (22), at an arbitrarily chosen axial position of X = 0.4 against (a): The Fourier number for different angular frequencies and comparison with the “quasi-steady” model (b): the angular frequency for different Fourier numbers, and (c): the Fourier number and the angular frequency

Fig. 7

Variations of the applied heat flux, q" = q"r[1 + sin(8πFo)], and the corresponding Nusselt number for a few axial positions along the tube, Eqs. (21) and (22)

Fig. 8

Variations of the average Nusselt number, Eq. (24), with the angular frequency and comparison with the quasi-steady model

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