Forced convection of thermally developing laminar flow in circular sector ducts

Forced convection of thermally developing flow in circular sector ducts

laminar

Q. M. LEI and A. C. TRUPP Department of Mechanical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received 24 May 1989 and in final form 10 Oclober 1989)

Abstract-Numerical solutions are presented for laminar, forced convection heat transfer in the thermal entrance region of circular sector ducts using the HI and H2 thermal boundary conditions. With fully developed hydrodynamics, results are given for duct apex angles of 20’. 45”. 90”. 130”. 180”, 270”. and 360”. For the HI condition. decreasing apex angle increases the duct comer effects, hence enhancing local and fully developed Nusselt numbers and shortening thermal entrance sections. For the H2 condition, the apex angle influences the thermal quantities differently. The data presented are compared with the limited available solutions and good agreement is found.

INTRODUCTION ANALYSES OF forced convection heat transfer in circular sector ducts are often encountered in engineering applications, e.g. for flow through a multi-passage tube. Also, such analysis provides a lower bound for the problem of combined free and forced convection which has received much attention in recent years. Primarily, the present investigation was motivated by our experimental study of buoyancy effects on forced convection in a horizontal semicircular duct [l, 21. Since one practical heating means is heating by electric resistance wires (like our experimental set-up), this analysis considers the boundary condition of axially uniform heat flux. The peripheral conditions employed are uniform wall temperature and uniform wall heat flux, denoted by HI and H2, respectively. Previous theoretical investigations for circular sector ducts have been mainly limited to laminar fluid tlow and forced convection in the fully developed region. Some of these characteristics have been compiled by Shah and London [3]. Lei and Trupp [4-6] have recently extended this work and provided the fully developed pressure drop data and Nusselt numbers over the entire apex angle range for a number of thermal boundary conditions. For the problem of hydrodynamically fully developed and thermally developing flow, Manglik and Bergles [7] numerically predicted Nusselt numbers for an isothermal semicircular tube. Regarding the HI condition, only one investigation for a semicircular tube was reported by Hong and Bergles [S]. When studying internally fined tubes, Prakash and Liu [9] reported local Nusselt numbers for three circular sector ducts, but for the flow of simultaneous development of velocities and temperature. This theoretical study involves the developing temperature with the fully developed hydrodynamics which corresponds to the behavior of fluids with high Prandtl numbers. Results presented in this paper for the HI and HZ conditions over the

whole apex angle range of circular sector ducts are also useful for designing compact heat exchangers. ANALYTICAL FORMULATION

Figure 1 shows the duct cross-section under consideration. The analysis is limited to the steady, laminar flow of incompressible Newtonian fluids with constant properties. The fluid axial heat conduction is considered to be negligible. For circular tubes at least, this idealization is valid [3], except for the immediate neighborhood of the duct inlet, providing (Re Pr) > 50. Like most laminar flow analyses, this study also neglects viscous dissipation within the fluid. Using dimensionless variables and parameters defined in the Nomenclature, the governing differential equations can be written as follows : momentum equation I?‘)( 2+$+f$=c

(1)

energy equation Z’T 1 ZT 1 d’T dr’+;-&+-?~=H’g+A.

(

>

(2)

The boundary conditions for equation (1) are w(r, 0) = K( 1,O) = 0,

and

de r, C$)=O a’“(

(3)

where the last condition corresponds to the symmetry line defined by 0 = 4. Equation (2) is supplemented by the following boundary conditions : at x = 0 T=O

at I > 0

167.5

for the Hi condition

forallrand0

(4)

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Q. M. LEI and A. C.

TKUPP

NOMENCLATURE

A

T, t

C

Fw, r:,

geometry parameter, 2(4 + I)/+ dimensionless pressure drop parameter, (R$~FV)(~P/C?X) specific heat CP hydraulic diameter, 24R,,/(4 + 1) 4 HI axially uniform wall heat flux with uniform peripheral wall temperature HZ axially and peripherally uniform heat flux ti peripherally averaged heat transfer coefficient, equation (9) k fluid thermal conductivity thermal entrance length, Li-, L: L, = X,/(0, Re, Pr) and L: = X,/(Ro Reo Pr) Nu, Nu, peripherally averaged Nusselt number, equations (10) and (12) II normal coordinate P pressure Prandtl number, pc,/k Pr peripherally averaged wall heat flux at q” each section radial coordinate, r = R/R0 R. r radius of the duct R, Re,, Reo Reynolds number, Reh = p D,, w/p and Re, = pR,r/p

T=T,

atr=

I,

and at 0 = 0,

O
(note that T, is a function dT -=0 a

atO=4,

(54

of X)

O
for the H2 condition dT -=I & ST -= i;o LST -=0 al

-r

atr=I,

0<8<4

atU=O,

O
atO=$,

I

O
FIG. I. Cross-section of the circular

T w, )t‘ w x, x

temperature, T = (t- f,)/(q”R,,/k) average wall temperature at any section, Tw = (7, - t,)/(q”R,/k) constant wall temperature both peripherally and axially axial velocity, )z1= W/W cross-sectional mean axial velocity axial coordinate, .Y= X/( R, Re, Pr).

Greek symbols fluid mean conduction path length angular coordinate fluid dynamic viscosity fluid density half apex angle of the circular sector duct.

Subscripts fd HI H2 T” W .Y

fully developed conditions for the H 1 boundary condition for the H2 boundary condition bulk mean thermal entrance length or the T boundary condition at duct wall axially local.

It is noted that the wall temperatures (T,) at each cross-section are not known in advance for both the cases. To solve for T, a solution method is first to use equation (4) at the duct inlet. Then the solution procedure is progressed along the axial direction. At each axial station, wall temperatures must be initially guessed while equation (2), together with its boundary conditions, is solved for the T distribution. Wall temperatures are then corrected by determining the dimensionless bulk mean temperature. After several iterations, the correct velocity and local temperature fields must ensure that

(6b) (64

+ T,” = 2 4” si

1 0

Twr dr d0 = 0.

(7)

As the solution step is marched, the temperature profile gradually develops. In a far downstream region, dTjdx approaches zero and the temperature field becomes independent of the axial coordinate. Consequently, in such a fully developed region, equation (2) reduces to

sector duct.

Complete series solutions for this equation with both the HI and H2 boundary conditions have been provided previously. The accurate fully developed Nusselt numbers reported in refs. [5. 61 were used in this

Forced convecuon heat transfer in circular sec:or ducts

with grid size selection as discussed later. The cross-sectional-average heat transfer coefficient, 6, is defined in customary manner as

study in connection

q” = /;cL -1,).

(9)

Using the dimensionless peripheral average wall temperature TWand the hydraulic diameter D,, the Nusselt number is given by (‘0) where FW= T, for the HI condition,

but for the HZ

condition, T,,,can be evaluated by 7, = &(l

T(r,O)dr+j;

T(l,O)dB).

(11)

Since L),,is a function of the duct apex angle, to better isolate the 4 effect and to facilitate comparison to a circular tube, the Nusselt number may be defined, on the basis of duct radius &,, as

Nu_liR, ’ o=--=--. k

Tw

(‘2)

Information on the thermal entrance length is also of practical importance. Thus, in this analysis. values of thermal entrance lengths were taken as the dimensionless distances where the local Nusselt numbers first drop to within 1.05 and 1.01 times the fully developed Nusselt numbers, denoted by L$j and ,!I$,, respectively. Similarly, the conventional thermal entrance length, LT, was correspondingly defined with the axial distance normalized by (D, Re,, f’r). The relationship between LT and G for circular sector ducts is given byt LT =

(

2

2L;. >

(‘3)

COMPUTATIONAL PROCEDURE The finite difference equation (2) was formulated using control volume integration. For the velocity field, the exact solution to equation (1) with equation (3) was computed from the series expression in ref. [4]. At a given axial station, equation (2) with equation (5) or equation (6) was numerically solved by a band storage linear equation solver. Iterations were needed for obtaining the correct Tdistribution. Thus, a number of corrections (2-S for most stations) were made on the wall temperature for the HI condition and on an arbitrary nodal temperature for the H2 condition, while ensuring that the bulk mean temperature was less than 10e6 in magnitude. For each case, the solution procedure was marched along x until the local

TInasmuch as one of the objectives in this study is to examine the q5effect, the use of these two pairs of definitions for IVUand & is useful.

1677

Nusselt number was at least within 0.1% of the corresponding fully developed value as numerically generated from equation (8). Decisions on the adequacy of the cross-sectional mesh size adopted for each duct geometry were mainly guided by comparing a numerical value of (iv& to the exact value obtained from the series solution [5,61. Table 1 gives these comparisons and the mesh sizes used in this investigation. For both the cases, the chosen mesh sizes were taken to be fine enough to make a grid value of (Nu)rd within 0.5% of the exact solution value. Regarding the axial step size, the marching step size As for the study was a!so determined by numerical experimentation. For example, for 24 = 180”. with the cross-sectional mesh size (22 x 25) fixed and the first step size Ax = lo-“, two numericai tests were conducted near the entrance. One let AX be increased by 5% and another by 25% for each consecutive station. Local Nusselt numbers from these two tests showed no noticeable difference. This insensitivity to the axial step size is mainly attributed to the use of the accurate direct solver for simultaneous linear equations at each cross-section. For all cases, it was finally decided to use the following pattern. The first step size was taken as Ar = 10b6. Then Ax was increased by 20% for each subsequent step size. Upon reaching 8 x 10e4, AX remained unchanged for the rest of the entrance section. Using these three-dimensional meshes, specific rests were run which positively confirmed that the three-dimensional temperature profile for each case was indeed converging to the two-dimensional fully developed temperature profile for the same cross-sectional grid. RESULTS AND DlSCUSSlON The following section provides information on the thermal entrance region of seven circular sector ducts. For the largest apex angle (24 = 360’): the results correspond to a circular tube with one internal full fin. Fully developed results

Nusselt numbers for hydrodynamically and thermally fully developed flow were generated by solving equation (8) numerically. These results, together with values determined from series solutions 15. 61, are given in Table 1. For both the H 1 and HZ conditions, all present Nusselt numbers fell within 0.506 of the exact solution values with an average difference of about O.l%, This excellent agreement can be seen (for NuO only) in Fig. 2, which shows variations of fully developed Nusselt numbers with duct apx angle. Both Nusselt numbers (Aiu based on D,, and -Vu, based on Ro) have been piotted, but as can be seen, (in& tends to mask the 4 effect. For instance. (NuH& decreases with decreasing $I which is accompanied by increasing corner effects. In fact, for the H 1 condition, this increase of corner effects should enhance the duct capability of thermal energy transfer. The reason for

1678

Q. M. LEI and Table

A. C.

I. Mesh size and fully developed results (&I,),

rX0

20 45 90 130 180 270

22x 15 22X I7 22x20 22 x 22 22 X 25 30x35 22 x 30 50x40 22x32

360

I

-

Lei and Trupp

Present

Exact [5]

Present

Exact [6]

2.7683 3.2820 3.7460 3.9485 4.0597 4.2113

2.7633 3.2792 3.7440 3.9466 4.0880 4.2178

0.3692 1.7592 2.9871 3.0282 2.92 IO

0.3710 I .7630 2.9870 3.0280 2.9200

2.7704

2.7690

4.2653

4.2852 2.6876

2.6870

/

r

[5,6]

20 (Deg) FIG. 2. Fully developed

Nusselt numbers for circular ducts.

(‘~~H,)k!

_

this is as follows: since I&,;,,= -k(c’t/dn)/(t,, --tr,), where t,. is constant, introducing the fluid mean effective conduction path length 6, (as in ref. [3]) yields h-,, = k(tW - t,)/&/(fW- t,) = k/6,. Accordingly, a circular sector duct with smaller 4 should have a shorter overall 6, value, and this means a higher average heat transfer coefficient. Concerning the H2 condition, a parallel analysis suggests that wall temperatures at corners are higher than temperatures for surfaces away from the corners. Hence since /;H2 involves the evaluation of average wall temperature, the Cpeffect on &? should not be as marked as on I&,,. On the other hand, as is shown in Fig. 2, Nzco follows the expected trends of heat transfer coefficients versus apex angle for both cases. Note that a maximum for (Nu,,&~ occurs at 24 z 60”. This particular circular sector duct has an (Nu,,&,, value which falls between the Nusselt values for equilateral triangular ducts with two rounded corners and no rounded corners [3].

,8

TRUPP

sector

Therndiy developing flow heat tramfer Results for Nrr,,,,, and Nr!,,,, in the thermal entrance region for the seven circular sector ducts are presented in Table 2. For convenience. the dimensionless axial distance x is normalized by the corresponding thermal entrance length L$, as previously defined (values for I$, will be tabulated later). It should be noted that Table 2 can also be used to estimate the flow length average Nusselt number which is often required in a heat exchanger design. Local Nusselt numbers for the HI condition. Figure 3 shows the variations of the local- HI Nusselt numbers for four of the circular sector ducts. As indicated before, the hydraulic diameter Dh is intentionally removed from the plot to better disclose the apex angle erect. For each circular sector duct, Nu,,,,, exhibits the expected trend of a monotonic decrease with x. approaching the limiting value of (Nuo,,,,)rd in the fully developed region. Except for very near the duct inlet (_r < 8 x IO-“), local values for NL(~.~.~, demonstrate the same 4 variation as the (Nu”.~& values. namely an increase with decreasing apex angle. For comparison purposes. Fig. 3 also includes the results of other investigators for other ducts. As can be seen in Fig. 3, all non-circular ducts including the one internal full-fin tube (24 = 360’) possess higher Nusselt numbers than a circular tube. except for very close to the duct inlet. For a semicircular tube (24 = 180”). the present Nu~,<,~, values compare well in Fig. 3 with the results of Hong and Bergles [3, 81. Differences between the local Nusselt numbers averaged 2.7%, but were always within 7%. Figure 3 also shows a comparison with two non-circular ducts whose Nusselt numbers fall between the curves of circular sector ducts with 24 = 45” and 90’. These two ducts are the equilateral triangular duct (comparable with the circular sector duct of 24 = 60”) and the right-angled isosceles triangular duct (comparable with the circular sector duct of 24 = 90’). Using the same basis of R0 (see Fig. 3 for RO definition), the N~t~.~,n, values for these two ducts appear to have correct trends with s as well as with 4. For instance. for the right-angled isosceles higher values of N+u, triangular duct are definitely expected when compared to the values for the circular sector duct of 24 = 90’.

0.0001 0.0005 0.0010 0.0030 0.0050 0.0070 O.OIOn u.02uu 0.0300 O.tMQO 0.0500 0.0700 0.1000 o.t5@4I 0.2000 0.2500 0.3000 0.4000 OSOOLl 0.6000 0.7000 0.8000 0.9000 1.0000 1.2000 1.5000

37.852 22.653 17.880 12.349 10.461 9.386 8.385 6.733 5.916 5.404 5.040 4.547 4.099 3.680 3.436 3.281 3.169 3.024 2.937 2.883 2.848 2.824 2.807 2.796 2.182 2.774

2+=20"

9.268 4.812 3.583 2.236 1.796 I.559 1.343 I.013 0.864 0.775 0.713 0.633 0.563 0.499 0.463 0.441 0.425 0.405 0.393 0.385 0.380 0.377 0.375 0.373 0.371 0.370

40.809 27.270 22.203 15.888 13.542 12.133 IO.763 8.472 7.379 6.709 6.233 5.614 5.050 4.509 4.189 3.975 3.824 3.627 3.508 3.433 3.384 3.352 3.330 3.315 3.298 3.287

28.988 17.214 13.308 8.706 7.124 6.243 5.441 4.210 3.641 3.306 3.069 2.754 2.475 2.223 2.084 1.998 1.940 I.871 1.834 I.812 1.798 1.789 1.782 I.777 I.770 I.765

2$J=45"

44.494 31.850 26.497 18.756 15.682 13.915 12.251 9.629 8.405 7.651 7.129 6.418 5.764 5.143 4.778 4.537 4.365 4.141 4.005 3.920 3.863 3.826 3.801 3.783 3.764 3.752

29 =w

49.603 32.231 25.198 16.392 13.363 II.732 10.242 7.Y4Y 6.891 6.246 5.800 5.189 4.639 4.120 3.818 3.619 3.479 3.297 3.190 3.122 3.079 3.050 3.030 3.017 3.002 2.992

46.674 34.084 28.141 19.409 16.138 14.302 12.609 Y.Y46 8.700 7.922 7.384 6.648 5.981 5.350 4.981 4.737 4.565 4.341 4.207 4.123 4.067 4.030 4.005 3.988 3.968 3.955

51.663 32.330 24.606 15.655 12.791 II.264 9.882 7.731 6.721 6.096 5.662 5.085 4.565 4.080 3.799 3.616 3.487 3.321 3.222 3.160 3.118 3.091 3.072 3.058 3.043 3.033

48.602 35.454 28.907 19.545 16.216 14.379 12.700 IO.046 8.802 8.024 7.480 6.746 6.081 5.457 5.093 4.854 4.686 4.470 4.342 4.261 4.208 4.172 4.148 4.131 4.110 4.097 51.923 30.605 22.881 14.585 12.024 10.626 9.344 7.320 6.368 5.781 5.377 4.839 4.356 3.906 3.646 3.475 3.356 3.200 3.107 3.047 3.007 2.981 2.963 2.950 2.935 2.926

24 = 180"

regionofcircular sector ducts

29 = 130"

Table 2.Variationof Nusseltnumbersintheentrance

58.871 36.330 28.161 19.004 15.909 14.185 12.579 10.011 8.714 8.011 7.474 6.759 6.115 5.511 5.162 4.934 4.775 4.572 4.452 4.376 4.326 4.293 4.270 4.253 4.234 4.220

49.247 26.664 19.850 12.930 IO.715 9.506 8.382 6.577 5.738 5.227 4.873 4.405 3.986 3.599 3.377 3.233 3.133 3.004 2.927 2.879 2.846 2.824 2.809 2.798 2.785 2.776

60.794 34.856 27.429 18.758 15.747 14.061 12.487 9.960 8.739 7.984 7.456 6.752 6.121 5.529 5.189 4.968 4.814 4.618 4.502 4.428 4.380 4.347 4.323 4.308 4.288 4.275

45.641 23.739 17.870 II.795 9.807 8.697 7.663 6.043 5.292 4.835 4.520 4.105 3.737 3.402 3.212 3.090 3.006 2.898 2.832 2.789 2.760 2.740 2.725 2.715 2.702 2.693

1680

Q. M. LEI and A. C. TRUPP

Circular .~cror

ducts

go’,.

180”,

Froln

Shah and London [3]

X/(RofieoP+)

z =

FIG. 3. Comparison of Nu~,~.~,, with other published results.

As demonstrated before, this is because the fluid mean effective conduction path length for the right-angled isosceles triangular duct is relatively shorter. Local Nusselt numbers for the H2 condition. Variations of Nu~.~,~?for the thermally developing flow for the seven circuiar sector ducts plus a square duct are graphically shown in Fig. 4. Like NuOr,n,, for each duct, Nu ,,r.H2monotonically drops as .Yincreases. In the far downstream region, each curve approaches its correct fully developed value of (Nu,&,. It should also be pointed out that, near the entrance, Fig. 4 appears to show flatter slopes for NuO,.,,? than for

I-’

“““‘I

’ “““‘I

’ “““‘I + -

further downstream. This misleading appearance is due to the logarithmic scale plotting. The actual slopes of NIL,, versus axial distance for all cases of this study were always steepest near the entrance. Concerning the Cp effect, for x 2 6 x lo-‘, local values of Nu,,~,,, for all circular sector ducts in Fig. 4 follow the same pattern as (Nu&~~ versus 4, as displayed in Fig. 2, i.e. increasing the apex angle up to 60’ gives rise to an increase in Nz(~.~,~~ which then decreases with further increases in 4. For the middle region of the flow development, the 4 effect on NI.I~~.“~ is very small for 24 > 45’. Figure 4 also provides one

! “““‘I Square duct,

’ “““‘I

RO

cl1 Sastri [lo]

Chandrupatla and

z = X/(R,,Re,Pr) FIG. 4.

’ ’ “““I

Local Nusselt numbers for the H2 condition.

’ ‘f

Forced convection heat transfer in circular sector ducts

1681

Present result8 Mqlik

ad Bergles [7]

z = X/(RoReoPr) FIG. 5. Local Nusselt numbers for a semicircular duct with different boundary conditions.

to the Nu,,~,~* results of Chandrupatla and Sastri [lo], who studied non-Newtonian fluids in a square duct. Their Newtonian results of Nu,,.~,~~ appear between the curves of the circular sector ducts for 24 = 90” and 130”. This would also be the case if their Newtonian results of NhX.HIwere plotted in Fig. 3 (not plotted because of crowding). Effect of thermal boundary conditions on NuOx.The differences between Nusselt numbers for the different thermal boundary conditions are considered next for each geometry. All results obtained in this investigation indicated that, except for the region very near the duct inlet, Nu,,~,“, was always higher than Nu,,,,~. This expected outcome is shown graphically in Fig. 5 using the semicircular tube as a typical example. It can be seen that, near the duct inlet, NuOx,,,,starts to As the flow continues to develop, the exceed NuOX.HP difference between them increases and reaches up to about 40% (i.e. (Nu~,~,)~(Nu~,“~)~~ x 1.4) in the fully developed region. Figure 5 also shows the developing Nusselt numbers for constant wall temperature in both the axial and peripheral directions (T) which were calculated from the correlation equation of the axial length mean Nusselt number (after conversion to Nu,,~.~) reported by Manglik and Bergles [7]. As illustrated, the Nu,,~.~values fall between the curves of Nuo~,~, and Nu~.~.~~as the flow approaches the fully developed region. Thermal entrance length. Table 3 lists results of the thermal entrance lengths for the Hl and H2 conditions. For the semicircular tube (24 = 180’), the present values of 0.05395 and 0.09747 for b,,,, and &,.“, compare reasonably well with 0.0525 and 0.0893 respectively (obtained by interpolating the thermal entrance results of Hong and Bergles [3, 81 comparison

using the (Nu&, value of 4.108 [8]). The differences are 3% for J&, and 8% for &,,“,. For each geometry and boundary condition, Table 3 presents significantly higher values for Lr, than for Lr,,. AS a result, the LTJ.“, entrance lengths average 56% of the LT,,“, values while results for kIH2 average 54% of the Lr,,H2 values. These indicate (as expected) that the flow develops very gradually near the fully developed region. Furthermore, each &,HZ value listed in Table 3 is higher than its LT.H,counterpart. On average, the thermal entrance length for the H2 condition is 2.8 times longer than the section for the Hl condition. This indicates that the Hl condition, when imposed on a circular sector duct, is a much stronger thermal boundary condition (compared to H2) such that the temperature field requires a much shorter axial length for its full development. The thermal entrance results of Chandrupatla and Sastri IlO] for a square duct also show that the H2 condition results in longer thermal entrance lengths than the H 1 condition for Newronian as well as non-Newtonian fluids. For the Hl condition, a study of the data in Table 3 was made to determine the effect of apex angle on thermal entrance length. For LT.“,, for large 4, thermal entrance lengths appear almost constant; however, for 24 6 IOO”, &,“, (which is norma!ized by Q,) increases with decreasing 4. In fact, physical variations of thermal entrance length are better viewed in terms of g.,, (normalized by &), which show a monotonically increasing trend with 4. For small 4, duct corner effects assist early temperature development so that the trend is expected, Note that increasing the apex angle up to 360” (which forms a circular tube with one internal full fin) results in the largest thermal entrance length for each base, e.g. L&,, E

1682

Q. M. LEI and

A. C.

Table 3. Results of thermal

TKUPP

entrance

length

20

45

90

%J 130

180

270

360

&.“I &.,,I &HZ L&HZ

0.01004 0.01822 0.07607 0.13466

0.02345 0.04044 0.04524 0.09878

0.04501 0.07749 0.05393 0.09377

0.06203 0.10919 0.09367 0.16624

0.08057 0.14556 0.14896 0.26542

0.10657 0.19935 0.26030 0.48814

0.12583 0.23870 0.39179 0.75257

L TS.tlI

0.11372 0.20631 0.861’4 1.52464

0.07373 0.12716 0.14225 0.31060

0.05814 0.10011 0.06967 0.12114

0.05489 0.09663 0.08290 0.14712

0.05395 0.09747 0.09975 0.17773

0.05406 0.10112 0.13203 0.24760

0.05167 0.10371 0.17023 0.32698

0.12583. This value is still shorter than 0.17221 for a circular tube [3] which has no ‘corner effects’ at all. For the H2 condition, again, L:,,,, rather than L-,,,,: provides a better picture of the 4 effect. Beyond 24 2 45’, corner effects diminish as the L:.,: values increase gradually with $. It is noted that minima occur at 24 z 4.5. for L&: and at 24 2 90 for may be attributed to the G.kIZ. Such behavior versus 4 pattern which has a maximum (ok& value at 24 z 60’, as shown in Fig. 2.

nar convection in a horizontal semicircular duct, the present analytical results will serve as lower boundaries for the data. ~~kno,~(~,c!clsnren(--This research was supported Natural Canada.

and

Engineering

Research

bq the

Council

of

REFERENCES

I Q. M. Lci and A. C. Trupp. 2.

CONCLUDING

Sciences

REMARKS

Correlation of laminar mixed convection for a horizontal semicircular duct. /.?rh C
for constant property, laminar fluid flow in the thermal entrance region of circular sector ducts. With the fully developed velocity field, the employed thermal boundary conditions HI and H2 simulate non-circular ducts with high and low thermal conductivities in the peripheral direction, respectively. Results of forced convection heat transfer in the fully developed region agreed excellently with the existing data. Local Nusselt numbers for the semicircular tube with the HI condition compared well with results in refs. [3, 81. The data presented, which cover the entire apex angle range, were used to study the apex angle effects on heat transfer quantities of interest. It was observed that the apex angle influences these quantities differently for these two boundary conditions. Compared to H2, the HI condition is thermally stronger. hence resulting in higher heat transfer coefficients and shorter thermal entrance lengths. However, in the absence of buoyancy effects, real situations would generally lie between the two, e.g. for a duct with an electric resistance wiring heating. On the other hand, for our experimental investigation of combined lamiIn this study,

numerical

solutions

were obtained

3. 4.

5

6. 7.

8.

9.

IO.

Forced convection heat transfer in circular sector ducts CONVECTION FORCEE DUN ECOULEMENT LAMINAIRE THERMIQUEMENT ETABLI DANS DES CONDUITES EN SECTEUR CIRCULAIRE R&m&-Des solutions numeriques sont present& pour la convection thermique laminaire dans la r&on dent& des conduites en secteur circulaire avec les conditions aux limites thermiques Hl et H2. Pour des regimes hydrodynamiques etablis, les rbuitats sont don& pour des angles au centre de 20”. 45”, 90”, 130’. I 80",270’ et 360”. Pour la condition HI, une decroissance de I’angle augmente les effets de coin, provoquant l’accroissement des nombres de Nusselt locaux et raccourcissant les longueurs d’etablissement thermique. Pour la condition H2, l’angle au centre influence differemment les caracttristiques thermiques. Les resultats present& sont compares aux solutions limit&es disponibles et on trouve un bon accord.

ERZWUNGENE KONVEKTION IM THERMISCHEN EINLAUFGEBIET EINER LAMINAREN STRGMUNG IN KREISSEKTOR-FGRMIGEN KANALEN Zusammenfassung-Die laminare erzwungene Konvektion im thermischen Einlaufgebiet von kreissektorforrnigen Kanllen wird numerisch untersucht. Dabei werden als Randbedingungen konstante Wandtemperatur und konstante Wlrmestromdichte verwendet. Fur hydrodynamisch vollstlndig ausgebildete Striimung werden Ergebnisse fiir die folgenden Sektorwinkel angegeben : 20’; 45 ; 90’ ; 130’ : 180’ ; 270’ und 360”. Bei konstanter Wandtemperatur steigt die Bedeutung der Winkeieffekte mit abnehmendem Sektorwinkel an, was zu einer Erhcihung der Brtlichen und vollstlndig entwickelten Nusselt-Zahl fiihrt. Das therm&he Einlaufgebiet wird dadurch kleiner. Fiir aufgeprlgte Warmestromdichte ist der EinfluO des Sektorwinkels uneinheitlich. Die Ergebnisse werden mit den begrenzt vorhandenen Liisungen verglichen, die Obereinstimmung ist gut.

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