Experimental Investigation on Heat Transfer Coefficient From Horizontal Rectangular Ducts by Free Convection. Mohamed E. Ali Mechanical Engineering Department, King Saud University, P. O. Box 800, Riyadh 11421, Saudi Arabia. Phone : +966-1-467-6672, Fax : +966-1-467-6652, E:
[email protected] Abstract- experimental investigations have been reported on steady state natural convection from the outer surface of horizontal ducts in air. Five ducts have been used with aspect ratios of 2, 1 and 0.5. The ducts are heated using internal constant heat flux heating elements. The temperatures along the surface and peripheral directions of the duct wall were measured. Longitudinal (circumference averaged) heat transfer coefficients along the side of each duct are obtained for laminar and transition regime natural convection. Total overall averaged heat transfer coefficients are also obtained. Longitudinal (circumference averaged) Nusselt numbers are evaluated and correlated using the modified Rayleigh numbers for transition regime using the axial distance as a characteristic length. Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers, the aspect ratio and area ratio for the laminar and transition regimes. The longitudinal or total averaged heat transfer coefficients are observed to decrease in the laminar region and increase in the transition region. Laminar regimes are obtained only at very small heat fluxes otherwise transitions are observed. Keywords: Natural convection, rectangular ducts, experimental heat transfer, laminar and transition regimes.
1- INTRODUCTION Steady state natural convection from rectangular and square ducts has many engineering applications in cooling of electronic components, design of solar collectors and heat exchangers. Survey of the literature shows that correlations for natural convection from vertical plate, McAdams [1] and Churchill and Chu [2], horizontal surface, Goldstein et al. [3] and Lioyd [4], long horizontal cylinder, Morgan [5] and Churchill and Chu [6], spheres, Churchill [7], are reported for different thermal boundary conditions. Free convection simulation from an elliptic cylinder was studied by Badr and Shamsher [8] and by Mahfouz and Kocabiyik [9] and Correlations for natural convection from helical coils were reported by Ali [10-13] for different Prandtl numbers. On the other hand, there are limited correlations available in the literature for natural convection from the outer surface of such rectangular
ducts which motivates the current investigation. The approximation method suggested by Raithby and Hollands [14] to predict heat transfer from cylinders of various cross sections and for wide ranges of Prandtl and Rayleigh numbers was simplified by Hassani [15]. The free convection from a horizontal cylinder with cross section of arbitrary shape was theoretically analyzed for uniform surface temperature and uniform surface heat flux by Nakamura and Asako [16]. They also checked their theoretical results by experiments in water on cylinders of modified triangular and square cross section. However, their experimental values of the mean heat transfer coefficient were about 10 to 30 percent higher than the analytical values. Hassani [15] introduced a general correlation for predicting natural convection from isothermal two-dimensional bodies of arbitrary cross section. An experimental study was reported by Oosthuizen and Paul [17] for non-circular cylinders in air for single Rayleigh number using the transient method. Their results showed that, significant effects of the cross section orientation and shape on heat transfer rate were obtained. Furthermore, another experimental study using the same method by Oosthuizen and Bishop [18] was made on mixed convection heat transfer from square cylinders in two different orientations. Their results included some limited data for free convection in air, which correlated as:
Nu L = 0.424 Ra 0L.25 ,
(1)
3.5 x 10 5 ≤ Ra L ≤ 2.8 x 10 6
Recently, Zeitoun and Ali [19] have reported numerical simulations of natural convection heat transfer from isothermal horizontal rectangular cross section ducts in air. Their results show that at fixed Rayleigh number as the aspect ratio increases separation and circulation occurs on the top surface of the cross section duct and the corresponding behavior was observed through the isotherms. They have also obtained a general correlation using the aspect ratio as a parameter.
[
]
2
Nu L = 0.9 Γ −0.061 + 0.371 Γ 0.114 Ra 0L.1445 , 700 ≤ Ra L ≤10 8
(2)
This paper presents the results of an experimental investigation of laminar and transition to turbulence natural convection heat transfer from the outer surface of rectangular ducts with their axis horizontally. The 1
study focuses on the determination of axial (circumference averaged) and overall averaged heat transfer coefficient in non-dimensional form of Nusselt numbers. Furthermore, General correlation using Nusselt numbers as function of Rayleigh number, aspect ratio, and surface area ratio are obtained and the general trends of laminar and transition convection heat transfer are discussed.
Table 1. Physical dimensions for 1m length and 0.002 m thick of the ducts. Duct No. 1 2 3 4 5
Height (A) m 0.07 0.08 0.06 0.04 0.02
Width (B), m 0.07 0.04 0.03 0.04 0.04
Aspect ratio, Γ 1 2 2 1 0.5
Area ratio, As/Ac 57.143 75.0 100.0 100.0 150.0
2. EXPERIMENT SET-UP AND PROCEDURE Figure 1 shows a schematic cross section view of the duct (D) and the thermocouple locations in the longitudinal (axial) direction (TCW) on three sides of the duct. The ducts (D) were made from stainless steel (polished mild steel) with their dimensions listed in Table 1. An electrical heating element (H) (0.0066 m O.D.) was inserted into the center of the duct. Bakelite end plates (Bk, thermal conductivity = 0.15 W/mK [20] ) 0.0206 m thick were attached at both ends of the test duct (D). The surface temperature was measured at eleven points in the longitudinal direction of each duct at three surfaces (lower, upper and either side surface) as seen in Fig. 1. Thirty five calibrated iron- constantan (type J) self adhesive thermocouples (0.3 second time response with flattened bead) were stuck on the duct surfaces 0.1 m apart and two of them are stuck on the outer surface of the Bakelite end plates, one at each plate. Two thermocouples ( 0.01′′ or 0.25 mm dia.) are inserted into and flashed with the inside surface of the Bakelite end plates one at each plate as seen in Fig. 1, and one more thermocouple was mounted to measure the ambient air temperature. Ac
The duct was oriented horizontally using two vertical stands in a room away from the openings of the ventilation and air conditioning system to minimize any possible forced convection. Those thermocouples were connected to a forty channels Data Acquisition system, which in turn connected to a computer where the measured temperatures are stored for further analysis. The electrical power (Ac) to the heating element (H), is controlled by a voltage regulator (VR). The power consumed by the duct is measured by a Wattmeter (W) and assumed uniformly distributed along the duct length. The heat flux per unit surface area of the duct was calculated by dividing the consumed power (after deducting the heat loss through the Bakelite end plates) to the duct outer surface area. The input power to the duct increases for each duct from 10 to 350 W in a step of 4 W up to 26 W and after that increases by 10 W such that the maximum duct surface temperature does not exceed 160 oC. In other words, the experiment is repeated thirteen times corresponding to the various input power. Temperature measurements are taken after two hours of setting where the steady state should be reached. The procedure outlined above is used to generate natural convection heat transfer data in air (Prandtl number ≈ 0.69).
Power line W
VR
3. ANALYSIS OF EXPERIMENT Bk D H
TC W
DA
Longitudinal or axial (circumference averaged) and overall averaged Nusselt numbers as function of the modified and regular Rayleigh numbers were obtained from the measured temperature distributions. The heat generated inside the duct wall dissipates from the duct surface by convection and radiation in addition to the heat lost by conduction through the Bakelite end plates q Bk .
Input power = A s (q c + q r ) + A Bk q Bk
(3)
Where qc and qr, are the fraction of the heat flux dissipating from the duct surface by convection and radiation, respectively. The radiation heat flux is estimated using the total overall averaged surface Figure 1. Schematic of the experimental system showing the thermocouple locations in the longitudinal (Tcw) direction.
temperature
T at each run of the duct as following:
2
(
q r = ε σ T 4 − T∞4
)
(4)
11
h = ∑ h x / 11
(9)
x =1
and q Bk is determined using Fourier’s law through the Bakelite end plates. In Eq. (4), ε is the surface emissivity of the duct and was estimated as 0.27 for polished mild steel [21]. Measurements show that, the fraction of radiated heat transfer is 21.7 % of the total input power while the conduction heat lost through the bakelite end plates is 1.5 % at most.
3.a Axial (circumference averaged) heat transfer coefficient In this case the circumference averaged surface temperature at any x station in the longitudinal direction for each constant heat flux (run) is determined as: 3
Tx = ∑ Txj / 3 ,
(5)
j=1
where j is the number of thermocouples in the circumference direction at any x station along the surface of the duct . The arithmetic mean surface temperature is calculated along the axial direction for each run as:
θ x = 0.5 (Tx + T∞ ) , x = 1,2,…..,11
(6)
Therefore for each heat flux (run) there are eleven T x longitudinal temperature measurements. Therefore, once the electrical input power to the duct is measured, q Bk from Fourier’s law, qr from eq. (4) and qc from Eq. (3) then the axial (circumference averaged) heat transfer coefficient hx can be calculated from:
hx =
qc , Tx − T∞
x = 1,2,3,…,11
(7)
The non-dimensional Nusselt and the modified Rayleigh numbers are defined as
Nu x =
hx x k
, Ra *x =
gβqc x 4 νkα
(8)
All physical properties are evaluated at the axial circumference averaged mean temperature θx for each qc. 3.b Total overall averaged heat transfer coefficient In this case the circumference averaged heat transfer coefficient hx is first evaluated at each x station as in Eq. (7) and then the overall longitudinal average
h is obtained as:
Therefore, each heat flux qc is presented by only one overall averaged heat transfer coefficient on contrary to case (3.a) where qc was presented by eleven hx’s along the longitudinal direction given by Eq. (7). All circumference averaged physical properties are first obtained at θx than the overall averaged properties are obtained the same way following Eq. (9). The nondimensional overall quantities Nusselt and the modified Rayleigh numbers are defined using the characteristic length L = A + B as:
Nu L =
g β q c L4 hL , Ra *L = k νkα
(10)
In order to compare the present results with similar previous results; another way of averaged results using the overall averaged temperature is also used. In this way the temperature is first circumference averaged following Eq. (5) then the overall average temperature is obtained as: 11
T = ∑ Tx / 11
(11)
x =1
All physical properties are obtained in this case at T at each heat flux. The overall averaged heat transfer coefficient for constant heat flux is determined from:
h=
qc (T − T∞ )
(12)
and Nusselt and Rayleigh numbers using L = A + B as a characteristic length are defined as:
Nu L =
g β (T − T∞ ) L3 hL , Ra L = k να
(13)
3.c Experimental uncertainty The error in measuring the temperature is ± 0.2 C and the error in measuring the electrical input power is in the range ± 0.25 to ± 4.9 W, in addition to 1.02 % uncertainty in measuring the surface area. These errors give uncertainties of 2.2 %, 3.4 %, 3.03 % and 2.24 % in the calculated convection heat flux, heat transfer coefficient, circumference averaged Nusselt numbers o
*
Nux and the modified Rayleigh numbers Ra x respectively, at most. Furthermore, the uncertainty in the overall averaged Nusselt numbers
Nu L and
Rayleigh numbers Ra L were estimated as 3.43 % and 3.31 % at most, respectively, using the method outlined in Holman [22].
3
4. RESULTS AND DISCUSSIONS
6
qc= 874.21 W/m2
5
Comparison between the axial circumference averaged heat transfer coefficient (▲) and the overall averaged heat transfer coefficient (solid lines) is presented in Fig. 3 for duct number 2. Figure 3 (a) shows the transition regime where heat transfer coefficient increases as the heat flux increases. However, in Fig.3 (b) heat transfer coefficient decreases as the heat flux increases which confirms that laminar regime is achieved. It worth mentioning that other symbols are used in Fig 3(b) only for clarifying the figure but all the data are for duct number 2 and corresponding to the temperature distributions given in Fig. 2(b). It should be mentioned that other ducts in Table 2 give similar effects. 10
qc = 874.21 W/m2
9
8
hx (W/m2 K )
Experimental data points were obtained with rectangular ducts oriented horizontally in air. Figure 2 shows the axial circumference averaged surface temperature normalized by the ambient temperature T ∞ vs. the non-dimensional axial (longitudinal) duct length for three selected values of q c using duct number 2 (Table 1). As seen in Fig. 2(a), the temperature distribution at low heat flux is almost unaffected by the end effects where the heat lost through the Bakelite end plates is minimum. As the heat flux inside the duct increases the surface temperature increases and the end effects become more distinguishable. The distance between the dashed lines in Fig. 2(a) shows that the temperature is almost uniform and least affected by the end effects. Therefore, in order to avoid the end effects, the test section of the duct is chosen to be between these two dashed lines where the study is focused. Figure 2(b) shows the axial temperature distributions as in Fig. 2(a) but for lower values of heat flux corresponding to the laminar regime as will discussed in Fig. 3(b).
7
6
391.0
5
4
4 3
Tx/T
3
2 0.0
391.0
31.34
0.2
0.4
0.6
0.8
1.0
X/l 8
2
qc = 12.04 W/m2
31.34 1
0.2
0.4
0.6
0.8
1.0
x/l 1.4
qc= 31.34 W/m2
hx (W/m2 K )
0 0.0
7
6
5
20.74
1.3
4
31.34
Tx/T 1.2
3 0.0
20.74
0.2
0.4
0.6
0.8
1.0
X/l 11.4
1.1
1.0 0.0
0.2
0.4
0.6
0.8
1.0
Figure 3. Axial circumference averaged heat transfer coefficient along the duct surface for some selected heat fluxes for duct number 2; (a) associated with transition regime and (b) corresponding to laminar regime
x/l
Figure 2. Circumference averaged dimensionless axial temperature distributions along the duct surface for some selected heat fluxes for duct number 2; (a) associated with transition regime and (b) corresponding to laminar regime.
The axial circumference averaged Nusselt numbers vs. the modified Rayleigh numbers are shown in Fig.4. corresponding to the test section defined by the dashed lines in Fig. 2. using all ducts for all heat fluxes. Since the modified Rayleigh number is function of qc and x4 then the following observations can be drawn 4
from this figure (i) at any fixed x station along the duct length as the heat flux increases the Nusselt number decreases up to a minimum critical value then increases as qc increases, (ii) the decrease in Nux at fixed x and at different heat flux corresponds to an increase in Ra*x as indicated by the dawn ward inclined arrow, (iii) at fixed heat flux as x increases along the duct surface Nux increases which corresponds to an increase in Ra*x as indicated by the upward inclined arrow, (iv) below the solid line, all the data are less sensitive to be distinguished either for qc or x and collapse on each other and the general trend is Nu x increases as Ra*x increases therefore, this region can be characterized as a transition region as will be seen in Fig. 5., (v) the data above the solid line, as mentioned earlier, can be identified either by qc or x and not collapsing on each other and the general trend is Nux decreases as Ra*x increases at fixed location on the duct surface for different heat flux, therefore, this region is defined as a laminar region and will be treated on the overall averaged basis as will be seen in Figs. 6 and 7. 4
4
15% 2
Nux 100 8 6 4
Duct No. 1 2 3 4 5
2
10 1E+7
1E+8
1E+9
1E+10
1E+11
1E+12
1E+13
Ra*x Figure 5. Local circumference averaged Nusselt numbers vs. the modified Rayleigh numbers for the transition regime. Solid line present the data fit given by Eq. (14).
Eq. (10) are shown in Fig 6. where the laminar and transition regions are characterized by a decrease or increase in the heat transfer coefficient respectively, since h ∝ Nu L for each duct. The dashed line presents the fitting locus of the critical points segregating the laminar and transition regimes, where the exact critical points are circled for each duct. The correlation of the fitting locus is given by
2
Nux 100 8 6 4
Nu L = 0.168
Duct No. 1 2 3 4 5
2
(15)
with a correlation coefficient of 95.93%. 5
1E+9
1E+10
1E+11
1E+12
1E+13
Figure 5 is constructed to obtain a correlation in the transition region for the data below the solid line in Fig. 4. A least square power law fit through the data set yields the following correlation
(Ra )
* 0.237 x
Nu x = 0.355
1.9 x10 ≤ Ra ≤ 7.0 x10 * x
,
11
(14)
with a correlation coefficient (R-squared) of 96.5% with an error band of ± 15% where 94.8% of the data fall within this band. The overall averaged results using the definitions of NuL and
Ra *L given by
4
3
NuL
Tr an sit ion
1E+8
Ra*x Figure 4. Local circumference averaged Nusselt numbers vs. the modified Rayleigh numbers; solid line separating the laminar data (above the line) and the transition data (below the line). The inclined upward arrow shows the transition direction while the down ward arrow presents the laminar direction.
8
* 0.287 x
La m in ar
10 1E+7
(Ra )
2
10 1E+6
1E+7
Ra*L
1E+8
Duct No. 1 2 3 4 5 1E+9
Figure 6. The overall averaged Nusselt number profiles. Dashed line presents the data fit (Eq. (15)) through the critical points marked by circles. Figure 7 shows the overall averaged Nusselt numbers vs. the modified Rayleigh numbers where the laminar data is correlated using the aspect ratio and the area ratio. Dashed lines present the fitting correlation for duct numbers 1 and 2 which given by
5
(
Nu L = 12.414 Ra A for s Ac
)
* − 0.237 L
As Ac
1.259
Γ
−1.226
(
Nu L = 0.256 Ra
, (16)
