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3-D Numerical Modeling of Unsteady Heat Transfer in a Rectangular Duct

Kadir ISA1

Sakarya University, Adapazari, Sakarya, 54170

Ismail EKMEKCI2

Marmara University, Kadiköy, Istanbul, 34722

H. Riza GUVEN3, Nedim SOZBIR4

Sakarya University, Adapazari, Sakarya, 54170

In order to simulate heat producing elements on electronic circuits, heat input at the inlet of a rectangular duct is supposed to vary sinusoidally and transient forced convection for laminar flows are investigated numerically. This study focuses on a numerical and experimental analysis of unsteady forced convection in hydrodynamically developed and thermally developing laminar air flow in rectangular duct, subjected to periodic variation of the inlet temperature. The experiments are conducted on the range of Reynolds numbers of 1122, 1478, 1764 and 2225 for laminar flow and inlet frequencies (β) of 0.02, 0.04, 0.08, 0.16 and 0.24 Hz of the periodic heat input. Numerical results are obtained for the fully developed parabolic velocity profile under the developing temperature profile for the fifth kind boundary condition which is verified by computational fluid dynamics (CFD) code called STAR CCM+. Temperature variations along the centerline of the rectangular duct are observed to be thermal oscillations with the same frequency as the inlet periodic heat input and amplitudes those decayed exponentially with distance along the duct.

Nomenclaturea the width of the rectangular channel, mb half height of the rectangular channel, mm mass flow rate, kg/sPr Prandtl numberT temperature,Cx axial distance, mα dimensionless decay indexβ inlet frequency, HzRe Reynold numberΔT temperature amplitude, C, KDe hydraulic diameter of test section, mΔTi mean temperature at the inlet, C, K

I. Introduction This paper describes experimental and numerical studies of temperature distribution in the thermal entrance region of a rectangular channel. The main focus of the work is to present authentic experimental data on the nature of the temperature distribution along the centerline for a periodic inlet heat input at various frequencies and Reynolds numbers in the region.

In practical engineering applications, the temperature distribution for both solid and fluid varies with time and the heat transfer inside the duct, for example, a general passage between two boards in the computer, may be exposed to a number of planned or unplanned transients during normal operation. The unsteady state can produce undesirable

1 MSME, Ph.D. Student, Mechanical Engineering, Esentepe Kampusu, Adapazari, Sakarya, Turkey.2 Prof. Dr., Mechanical Engineering, Goztepe Kampusu, Kadikoy, Istanbul, Turkey.3 Prof. Dr., Mechanical Engineering, Esentepe Kampusu, Adapazari, Sakarya, Turkey.4 Assistant Prof. Dr., Mechanical Engineering, Esentepe Kampusu, Adapazari, Sakarya, Turkey.

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effects resulting in reduced thermal performance and increased thermal stresses, (with eventual mechanical failure). This phenomenon, when present, will affect the working condition of the computer chips placed on the circuit boards. Since these devices never attain steady state operation because of their nature to operate periodically in time, it is essential to know their transient response in order to provide an effective control system.

A search of the literature revealed a number of studies related to this investigation. Few simplified solutions to applications involving unsteady forced convection in ducts have been solved by different researchers1-4. These solutions just provide a basic understanding of unsteady problems. Kakaç et al5-9. have reported a series of experimental and theoretical studies of unsteady forced convection inside smooth ducts. Li et al10. made an experimental study of unsteady forced convection in a duct with and without arrays of block-like electronic components, simulating printed electronic circuit boards. Sözbir et al.11-13 made an analysis of unsteady forced flow in parallel plates channel with and without arrays of rectangular protruding surfaces. Cheroto et al14. studied ananalytical investigation of unsteady forced convection in parallel-plate channels. An alternative formulation isdeveloped, in which the problem is divided into real and imaginary formulations, and the analysis was performed by using these two coupled similar problems. Pimentel et al15. studied an analytical incompressible fully developed turbulent in ducts with rough walls. These studies are limited, based on the inlet heat input, block height and range of periodic inlet frequency.

In this paper, unsteady forced convection in a rectangular duct resulting from a time wise variation of the inlet temperature is numerically studied by means of Star CCM+ using 3-D implicit unsteady model. For the cases considered, the thermal responses of the system to the periodic variations have been evaluated after the initial transients have disappeared. Under such conditions, a general time dependent inlet condition can be expanded in terms of sine and cosine functions by use of Fourier series expansion. Based on this assumption, it is believed that results of sinusoidal variation of inlet temperature will provide a basic study with fundamental results for further research (theoretical) of unsteady forced convection under the general time dependent inlet condition.

The experiments are performed by Sözbir11 with a smooth parallel-plates channel and covered laminar and turbulent flow regimes. The presentation of results is structured to show the effects of inlet frequency (), the approach Reynolds number (Re), and the block dimensions on the temperature responses and the decay of the temperature amplitude along the duct.

II. Experiments

The experiments are conducted in a system originally designed and built by Ding16. An experimental study has been carried out to investigate the axial variation of inlet temperature and the impact of inlet frequency on decay indices in the thermal entrance region of a parallel plate channel. The investigation is conducted with both laminar and turbulent forced flows for Reynolds numbers ranging from 1,120 to 22,300 while the inlet heat input frequency varied from 0.02 Hz to 0.24 Hz. For this investigation, the basic geometry of the channel has a cross-sectional area

of 25425.4 mm2 (101 in.2).

A. Experimental Procedure

During the course of the experimental investigations the controlled parameters are the duct geometry, inlet temperature, and Reynolds number. A series of experiments are performed by changing the Reynolds number and inlet temperature frequency. For a specified inlet frequency and Reynolds number, the variation of the temperature amplitudes at different locations along the duct are obtained at thirteen measurement points along with, temperatures at the entrance and downstream of the orifice plate.

III. Numerical Study

It is assumed that the flow through the duct has negligible viscous dissipation and axial diffusion compared to the convection in the axial direction. Incompressible flow and constant fluid thermophysical properties are assumed since the velocity of the air flow in the experiment is low and the fluid temperature difference is small in the duct.

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A. The Fifth Kind of Boundary Condition

Because thickness and heat capacitance of the wall are not negligible according to applied numerical model, boundary condition which accounts for both external convection and wall heat capacitance is derived.

byt

tzyxTLc

y

tzyxTkTtzyxTh w ,0

),,,()(

),,,(),,,(*

(1)

where h* =(1/h+kw/L)-1 is equivalent heat transfer coefficient, kw,

wand c

ware the conductivity, density and

specific heat of the wall material, respectively.

IV. Results and Discussions

A. Experimental and Numerical Results of Smooth Duct

Experiments are conducted in the smooth rectangular duct to compliment the main objectives of the studies. The temperature variation with time at predetermined locations are measured and recorded. The thermal response is shown to be periodic in nature, fluctuating with a frequency approximately equivalent to that impose at the inlet and propagating with small but increasing phase lags between successive points as the downstream distance from the inlet increased. Amplitudes of minimum values of the thermocouple outputs, as

T T T ( ) / .max min 2 (2)

Fig. 1 shows comparison of the experimental and numerical results for laminar flow (Re 2225 and β=0.24 Hz) . The experimental data follow a straight line on these semilog plots, wherein regression lines of the experimentalresult are shown solid. Linear appearance of these plots is due to the exponential decay of temperature amplitude along the duct. The slopes directly gives the decay index, , defined as;

amp

i

x DT

Te e

/ (3)

Figure 1. Centerline temperature amplitudes decays of experimental and numerical studies along the smooth duct in laminar flow, =0.24 Hz, Re2225.

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Figure 3. Centerline temperature amplitudes decays of experimental and numerical studies along the smooth duct in laminar flow, =0.02 Hz, Re1479.

Figure 2. Centerline temperature amplitudes decays of experimental and numerical studies along the smooth duct in laminar flow, =0.04 Hz, Re1764.

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V. Conclusions

Temperatures in ducts without blocks oscillate at approximately the same frequency as the inlet temperature oscillations. The temperature amplitude decays along the duct exponentially. Away from the thermal entrance, the decay of the amplitude is dominant by one term, since the experimental and numerical results are almost linear in the semilog plots. The transient heat transfer problem is solved by means of CFD code called Star CCM+ using finite volume method in the thermal entrance region of the duct. Numerical results are compared with experimental results by Sözbir11. Results are submitted as the form of figures below.

1. Temperature variations along the centerline of the rectangular duct are unsteady and sinusoidal.2. The amplitudes of temperature variation along the centerline of the rectangular duct are decayed exponentially

with distance along the duct. 3. The slope of amplitudes of temperature variation along the centerline of the rectangular duct is increasing with

frequency of heat inlet at the same Re number, but it is decreasing with Re number at the same frequency of heat inlet.

4. The present model is acceptable to predict the temperature distribution inside the duct and the decay of the temperature along the duct for timewise varying inlet temperature in the laminar thermal entrance region. The experimental results and numerical analyses are in an acceptable agreement.

AcknowledgmentsThe authors acknowledge the support by the Council of Scientific Research Projects (BAPK) of Sakarya

University – TR under project number 2006-50-02-051.

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Walls”, Int. J. Heat Transfer, Vol.11, pp.837-853, 1968.1Kakaç, S., “A General Analytical Solution to the Equation of Transient Forced Convection with Fully Developed Flow”, Int.

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J. Heat Transfer, Vol.29, No.10, pp.1449-1501, 1986.5Kakaç, S., Li, W. .and Cotta, R.M., “Theoretical and Experimental Study of Transient Laminar Forced Convection in a Duct

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of Block-Like Electronic Component”, Report, University of Miami, Coral Gables, Florida, 1995.12Sözbir, N., Brown, D. M., Kakaç, S., Arik, M. and Santos, C.A.C., “Unsteady forced flow in parallel plates channel with

and without arrays of rectangular protruding surfaces”, 9th Int. Conference on Thermal Engineering and Thermogrammetry, May 31-June 2, 1995, Budapest, Hungary.

13Sözbir, N., Brown, D.M., Santos, C.A.C., Kakaç, S. and Güven, H.R., “Experimental Investigation of Unsteady Forced Convection in a Channel with and without Arrays of Rectangular Protruding Surfaces", Journal of Thermal Sciences and Technology, Vol: 17, No:4, 1995.

14Cheroto, S., Santos, C.A.C., Kakaç, S., “Hybrid-Analytical Investigation of Unsteady Forced Convection in parallel-Plate Channels for Thermally Developing Flow”, Heat and Mass Transfer, Warme und Stoffubertragung, vol. 32, pp 317-324, 1997.

15Pimentel, L.C.G., Cotta, R.M., Kakaç, S., “Fully developed turbulent Flow in ducts with symmetric and asymmetric rough walls”, Chemical Engineering Journal 74, 147-153, 1999.

16Ding, Y.,“Experimental Investigation of Transient Forced Convection in Ducts for a Timewise Varying Inlet Temperature”,M.S. Thesis, University of Miami, Florida, 1987.

17ASME Standards, “Measurement of fluid flow in pipe using orifice, nozzle and venture”, MCF-3M-1984.