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International Journal of Heat and Mass Transfer, 5, pp. 869-881, 1962-10-01
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Natural convection through rectangular openings in partitions. Pt. 2.
Horizontal partitions
Brown, W. G.
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Ser
THl
N21r 2 no. 170 c. 2 BLDGNATIONAL
RESEARCH
COUNCIL
CANADA
DIVISION O F BUILDING R E S E A R C H
NATURAL CONVECTION THROUGH RECTANGULAR
OPENINGS
IN PARTITIONS
PART I: VERTICAL PARTITIONS BY W. G. BROWN A N D K. R. SOLVASON PART 11: HORIZONTAL PARTITIONS BY W. G. BROWN
R E P R I N T E D FROM
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER VOL. 5. SEPTEMBER 1962. P. 8 5 9
-
8 6 8 AND 8 6 9-
881R E S E A R C H P A P E R N O . 170 O F T H E DIVISION O F BUILDING R E S E A R C H P R I C E 5 0 C E N T S OTTAWA O C T O B E R 1962 N R C 6 9 5 3 N R C 6 9 5 4
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1111. J . Heat Mass T r a ~ i ~ : f e r . Vol. 5, pp. 859-868. Pergamon P~.ess 1962. Printed i n Great Britain.
NATURAL CONVECTION THROUGH RECTANGULAR
OPENINGS IN PARTITIONS-1
VERTICAL PARTITIONS W. G . BROWN and K. R. SOLVASON Building Services Section, Division of Building Research,
National Research Cou~lcil, Ottawa, Canada
(Received 11 Decer~iber. 1961 arid in reoised fottrz 20 March 1962)
Abstract-The simple basic theory of natural convection across openings in vertical partitions has been generalized to include both heat and mass transfer. Experiments were carried out in a large heat-flow apparatus with opcnings from 3 to 12 in high and with air as the convecting fluid. Because of the high thermal resistance of the partition, the test results for the Nusselt number as a function of the Grashof number based on opening height in the range loG < Gr < loY would also be expected
to be directly applicable to mass transfer. NOMENCLATURE
dimensionless constants; exponent;
specific heat of fluid;
concentration; c,, c, concentrations in cavities I and 2 remote from the partition ;
expoilent ;
diffusion coeficient ofmass transfer; opciling area;
acceleration due to gravity;
opening height (vertical partition): thickness of horizontal partition; heat-transfer coeficient based on total opening area;
mass-transfer coefficient based on total opening area;
thermal conductivity of fluid ;
pressure loss;
width of opening in 1101-izontal partition ;
mass-transfer rate through opening; heat-transfer rate through opening; pressure ;
volumetric fluid-flow rate;
vertical distance between multiple openings ;
temperature; T I , T, temperatures in cavities 1 and 2 remote from the partition;
thickness of vertical partition; velocity;
width of opening in vertical parti- tion ;
distance in the direction of gravity;
= Apzlp = height of a fluid column of density p giving a pressure under gravity equal t o
A pgz ;
thermal diffusivity:
coefficient of conceiltratio~l and thermal expansion respectively; exponents;
dynamic viscosity; kinematic viscosity;
fluid density; p,, p,, density in cavities 1 and 2 remote from the partition; p, - p, = Ap, difference in density between the fluid in the two cavities; p = (p, f p,)/2, meail
density of fluid. Dimeilsionless groups
Gr, Grashof number based o n density difference, gApH"pv2;
Nu, Nusselt number, h ~ H / k ; Pr, Prandtl number, cPp//c;
Re, Reynolds number, VH/v; s c , Schmidt number, v / D ; Sh, Sherwood number, hnlH/ D.
860 W. G . B R O W N and K . R . S O L V A S O N
INTRODUCTION
A GREAT deal of attention has been given in
past studies of natural convection to problems of heat transfer involving vertical and horizontal plates and bodies of varying shapes. These studies and their application to practical situations, ranging from heating equipment to the cooling of turbine blades, have recently been reviewed by Schmidt [I]. In this review mention is also made of a type of natural convection that until now has received very little attention. This is the situation occurring at openings in partitions, for which Schmidt reports an optical investigation of the transient mixing of two fluids of different densities (carbon dioxide and air) separated by the opening in a vertical partition.
Apart from the transient case, the two basic aspects of natural convection through openings are those of steady conditions with vertical and horizontal partitions. In the present paper the theory and experimental results will be given for steady conditions with openings in vertical partitions. The case of horizontal partitions will be treated in a second paper (Part 2).
The theory for natural convection across openings in vertical partitions has proved to be quite simple, but until now it has not been fully developed. Emswiler [2] in 1926 treated the case of multiple openings in a wall and obtained an expression for the rate of flow of air in terms of temperature difference and Bernoulli's equa- tion for ideal flow. He did not consider the case of a single opening nor did he treat the heat- and mass-transfer aspects of the problem which, of course, can be generalized for all fluids.
As far as is known no direct measurements have been made to substantiate and extend the theory. Thismay be explained in part by measure- ment difficulties and by the fact that opening sizes of practical importance are rather large to be investigated in a laboratory. Recently, however, a large test unit, apparently the first of its kind, has been built at the National
THEORY
Single opening
The system under consideration is that of Fig. 1, in which two large sealed cavities are separated from one another by a vertical partition having a rectangular opening of height H and width W. The densities of the fluid remote from the wall are maintained at p, and
p,, the temperatures at TI and T, and the con-
centrations at c, and c,.
FIG. 1. Schematic representation of natural convection across an opening in a vertical partition.
Since the cavities are sealed, there is no net flow of fluid across the opening and the absolute pressure p o at the elevation of the opening centerline is everywhere equal. (This is the limiting case for density differences that are small compared with the mean density. The ensuing error in derivation of the heat- and mass-transfer equations will be negligible except for gases at very low temperatures.) In cavity no. 1 the pressure p at a level z below the center-
line will be
PI
= Po+
Plgz (1)and in cavity no. 2, at the same level it will be
Research council of Canada to allow direct being the acceleration due to gravity.
measurement of the heat transmittance of 8-ft The pressure difference between the two square walls. This apparatus, described pre- cavities at this level is thus
C O N V E C T I O N T H R O U G H R E C T A N G U L A R O P E N I N G S I N PARTITIONS-1 861
Similarly the pressure difference a t a level z above the centerline is just the negative of equation (3). The pressure difference p, - p2
can also be expressed as the height of a column of fluid
where p is the mean density
There is only limited information available for the relation between pressure head and velocity V for rectangular orifices at low flow rates. Consequently the flow will first be assumed to be ideal (frictionless), and then the influence of viscous forces will be considered.
For ideal flow the Bernoulli equation can be assumed, i.e.
On integration from z = 0 to z = H/2, equation (5) gives for the total volumetric discharge through one half of the opening
The coefficient of discharge C has been inserted here as is customary for orifices. The value of
C ranges from about 0.6 for sharp-edged orifices to 0-8 for short tubes, and up to 0.98 for trumpet shapes. For submerged orifices, C tends to take on higher values.
With the flow Q is now associated: the heat-transfer rate
the mass-transfer rate
where c, is the specific heat.
Introducing now the heat-transfer coefficient IZT and the mass-transfer coefficient
Am,
defined asIZT =
4/
WH(Tl - Tz) andl1n2 = ri7/ WHp(cl - c ~ ) ,
equations (7) and (8) lead to the following equations in terms of dimensionless variables: for heat transfer
-
c
- -_.
3 d ( C r ) Pr, (9) for mass transfer
where the symbols are as defined in the Nomen- clature.
Equations (9) and (10) cannot be exact for all conditions owing to neglect of viscosity in equation (5) and neglect of thermal conductivity and diffusivity in equations (7) and (8). Con- sidering first viscosity," it will be remembered that for pure viscous flow velocity is lrectly proportional to the difference in pressure head, just as for laminar flow in a tube or viscous flow about a submerged object a limiting equation will occur of the form
where A is a dimeilsiollless constant.
Using equation (11) in place of equation (5) the Nusselt and Sherwood numbers become
If now the usual procedure is adopted and the Nusselt number is written
for a small range of the variables, then by equations (9) and (12) for negligible thermal
"
The following considerations are similar to those given in [4] for heat exchange in a vertical tube at low flow rates.t
A similar equation showing Q directly proportional to ( A p / p ) g ( H 3 / v ) can be derived from first principles for the case of a vertical slot.862 W. G . B R O W N and K . R . S O L V A S O N
conductance the exponent a on the Grashof number lies between $ and 1 and the exponent b on the Prandtl number is practically 1. The same reasoning holds for mass transfer.
For the other extreme, wherein the thermal conductivity or diffusion coefficient is very large, the heat- or mass-transfer phenomenon is similar to that of a solid, i.e.
Consequently, for the complete range of all variables (all fluids), the exponents on both
Gr and Pi in equation (13) lie between 0 and 1.
To be somewhat more specific about these exponents for various fluids some additional information can be obtained by comparing the estimated pure conduction or diffusion exchange obtained by analytical means with the convective transfer of equations (7) and (8). For air, for example, in the range of greatest interest, with say 50 degF temperature difference across a 1-ft square opening, the pure conduction heat transfer would be quite negligible compared with that of convection. Furthermore, the value of the Reynolds number Re, based on the mean flow velocity as determined from equation (6), lies close to the range covered in experiments on orifices which themselves validate equations (5) and (6). Hence, for air in this general range the final form of equation (13) is expected to be approximately
where E and [ are small compared with
4
and 1,respectively.
Il fultiple openings
Multiple openings in the vertical direction will increase the mass or heat transfer. In general, determination of the convection becomes com- plicated when openings of various sizes at irregular spacing are present. For equally spaced openings of the same size, however, it will be noted that the level of equal pressure in the two cavities falls on the center of gravity of the openings. Multiple openings will be discussed in greater detail in the following section on experimental results.
EXPERIMENTAL
Equipment
The test unit, already described in detail [3], consists of two large chambers 8 ft square, one of which is 4 f t deep and the other, since it contains an additional system to provide forced air circulation at will, 14 in deep. The first of these chambers can be maintained at approxi- mately 70°F and the other at variable tempera- ture down to -20°F. The walls of the unit are insulated and guarded to prevent heat losses. Fig. 2 shows the complete apparatus and its control equipment.
The partition in which the openings were cut was constructed of 2-in foamed polystyrene insulation on a ;-in plywood backing and was clamped and sealed between the two boxes of the test unit. Use of this wall of high thermal resistance has two experimental advantages :
(I) the heat transfer across the opening will be a large fraction of the total heat transfer, and (2) the large natural convection in the neighborhood of the opening will not be greatly influenced by coilvection over the wall proper. The convection conditions, then, approximate closely those which would occur when density differences are due to concentratio~l differences alone, and the test results for heat transfer would be expected t o apply for mass transfer as well.
Instrumentation
Thirty-gauge copper-constantan thermo- couples were installed at various locations to measure air and surface temperatures on both sides of the partition. The temperature control was sufficient t o maintain the air temperatures remote from the openings constant with time to within 0.2 degF. Total heat input to the warm side of the test unit was determined from con- tinuous recordings of d.c. voltage and current, the accuracy of the power input thus obtained being about 2 per cent.
Scope o f tests n17d procerlure
Tests were conducted for single rectangular openings of the following nominal sizes: 6 x 6 in, 6
x
12 in, 9x
9 in a n d 12x
12 in, with air temperature differences ranging from 15 to 85 degF. (Temperature on the warm sideFIG. 2. Test unit showing temperature control panel and measuring and recording instruments.
C O N V E C T I O N T H R O U G H R E C T A N G U L A R O P E N I N G S I N PARTITIONS-1 863
was maintained tl~roughout at about 72°F.) Tests were also made for 3
x
3-in openings but,since the heat flow for a single opening of this
2
;
size would be very small, seven openings were 2spaced horizontally on 104-in centers. The 5 "
ratio t/H of partition thickness t to opening
2
height H would be expected to affect the magni- - tude of results, and was varied by altering the
&
partition thickness in the neighborhood of the . o,,lopening, either by removing or adding insula- 2 0 3 0 4 o 5 o 6 0 70 M E A N TEMPERATURE OF
tion. Account was taken of the changed heat
transmission through this portion of the wall. LEGEND:
Several tests were also made with two equal NO FORCED AIR FLOW
3
x
3-in openings spaced vertically 15 in 0 AIR VELOCITY = l o o f ~ / ~ i ~apart on centers. A AIR VELOCITY = 2 0 0 f t / r n i n
In practical installations some forced air FIG. 3. Calibration o f the test partition.
movement in addition to natural convection might be present. Since the test unit already
contained a closed forced-air circulation system by assuming that both cavities behave as black on the cold side. it was used to obtain additional i.e.
test results with air flowing horizontally parallel
to the partition and opening with velocities of g r = W H ~ ( 7 : ~ - Tj) (15) 100 and 200 ft/min.
Owing to the massiveness of the test apparatus, long periods of time were required to reach equilibrium conditions. In general, with natural convection alone the apparatus was allowed to run overnight before temperature and heat- flow readings were taken. On completing a natural convection test at a given temperaturc difference, forced air circulation was begun and equilibrium was again reached in a period of 3 or 4 h.
Before carrying out tests with the various openings it was first necessary to calibrate the blank partition. This was done by determining the total heat transfer at various temperature differences between the air on the two sides of the wall. Results are shown graphically i l l Fig. 3, where the total heat flow has been divided by the partition area (64 ftz) to obtain conductance. The slight irregularity in the test results with no forced air motion is due to the unavoidable variability of the film coefficients. This assertion is borne out by the tests with air velocities of 100 and 200 ft/min which show no irregular behaviour.
With an opening in the partition, a small portion of the total heat transfer occurs by radiation, the amount of which was calculated
where o = the Stefan-Boltzmann constant and subscripts denote cavity surface conditions. (Interchange with the edges of the opening was neglected.)
Test results
Natural convection (no ,for.ce~l-air. flonl). The air temperatures used in determining the Nusselt numbers were the averages from floor to ceiling on the two sides of the wall. Air properties were evaluated at the mean air temperature of both sides. Air temperatures were measured 15 in from the wall on the warm side and 8 in from the wall on the cold side. The temperature- measurement stations were 2 ft fiom the opening in a direction parallel to the wall. In the case of the 12 x 12-in opening the heat- transfer rate was sufficiently large to cause a considerable gradient in the remote air tempera- ture from floor to ceiling on both sides of the partition (Fig. 4). Since the two gradients did not differ greatly, however, it will be appreciated that only a small error is involved in basing all calculations on mean temperatures. Fig. 4 also shows the distortion of the air and surface temperatures in the plane of the opening center line caused by the double air flows.
W. G . B R O W N and K. R. S O L V A S O N
T E M P E R A T U R E O F
FIG. 4. Air and surface temperatures about a 12 x 12-in opening in a 2a-in insulatcd partition.
All experimeiltal results are given in Fig. 5, where the Nusselt number divided by the Praildtl number is ordinate and the Grashof number is abscissa in accordailce with equations (9), (12), (13) and (14). For comparison, the two theoretical extremes of equatioll (9) with
C = 0.6 and 1.0 are shown. In Fig. 5, the
influence of the additional variable, the ratio
t / H of the wall thickness t to the opening
height H, is apparent. It will be noted that for any given t / H ratio the slope of a line through the experimental results is always greater than 0.5 in accordance with equation (14). Further- more, the slope increases with decreasing Grashof number, i.e. tending more and more toward a value of 1.0 for very low values of GI. At the higller values of the Grashof number (between loi and 10" very little influence of
t/H is to be noted for the range t / H = 0.19-
0.38. This is further borne out by the test results for the rectangular openiilg 6 in high and 12 i n wide; they are apparently identical with those for the 6
x
6-in square. For Gr<
107 and for a greater range of t / H from 0.38 t o 0.75, the influence of t / H is marked.Since the Prandtl number for air in the range
of temperatures used in the tests was constant at 0.71 it was not possible to investigate its influence as a separate variable. A comparison of the N ~ u e l t numbers of the present work with those of natural convectioil over a vertical plate for the same range of Grashof numbers, however, shows very much Iugher values for the case of wall openings. For the vertical plate the thermal conductivity of the fluid plays a large role in the heat exchange, in that heat is transferred by pure conduction in a laminar layer of fluid on the plate, and it is found that the Nusselt number is proportional to P r 0 ' 2 h r
With the much hlgher values of NLI in the present work indicating a greater convectioil transfer, it would be expected that for most liquids and gases the exponent on Pr (or Sc) is close to 1-0 in accord with equations (9) and (12). (For a situation where turbulent flow occurs and in wluch the exponeilt on P r is even less than 0.25, see [4] for the case of heat transfer in a vertical tube.)
A series of tests was carried out with two 3
x
3-in openings spaced vertically 15 in between centers in the 2 5 i n partition. In this case, the Nusselt number based on the totalCONVECTION T H R O U G H RECTANGULAR O P E N I N G S I N PARTITIONS-1 865
FIG. 5. Natural convection across rectangular openings in a vertical partition.
area of the two openings and H = 3 in should be equal to one-half the difference between the Nusselt numbers for two single openings of height 18 and 12 in (see Appendix A). In Fig. 6 the measured values are plotted against theore- tical values as obtained by extrapolation from Fig. 5. Agreement is fairly good, but the measured values are somewhat lower than theoretical ones, presumably owing to the additional contracting effect of the narrow 3-in width. Since the ratio t/H used in these tests had the high value of 0.75 it can be expected that agreement between theory and measure- ment would be appreciably better for lower (more usual) t/H values.
Effect of a horizontal velocity parallel to the all. It was not the intention in tlis work to
study the effects of forced-air motion in detail but only to make approximate measurements. This restriction was demanded in part by the test apparatus itself, which allowed use of only one or two fans, each of 1000-ft3/min capacity.
The cross-sectional area for flow on the cold side of the wall being 10 fte, average horizontal velocities of 100 and 200 ft/min were available for the tests. Another consideration restricting the quantitative applicability of the test results is the geometry of the cold side box and the roughness of the wall; together these determine the velocity of approach profile at the wall opening.
The test results are given in Fig. 7, along with the superimposed mean curve for natural con- vection alone, as taken from Fig. 5, for t/H = 0.38. For an air velocity of 100 ft/min the Nusselt number is in every instance less than one-half of that for natural convection alone, but remains dependent on the Grashof number. With a n air velocity of 200 ft/min the Nusselt number becomes practically independent of
Gr but dependent on the opening size. Such a
result is to be expected for l i g h velocities, for then theory requires that the Nusselt number depend only on the Prandtl number and on the Reynolds number VHIv. It will be noted that
W. G . B R O W N and K. R . S O L V A S O N
T H E O R E T I C A L N U S S E L T N U M B E R
FIG. 6. Natural convection heat transfer across two 3 x 3 in square openings spaced vertically 15 in apart on centers in a 2)-in thick partition. (The Nusselt numbers are based on a height
of 3 in and a total area of 18 sq. in.)
3001
'
'
'
I I I / Ilo7 1 5 2 3 4 5 6 8 10' 15
G R A S H O : N U M B E R ( ~ r
= 9 ~ p ~ y ~
21')FIG. 7. Effect of a horizontal air flow parallel to the opening and partition. (The dashed curve is that for
natural convection with r/H = 0.38 from Fig. 5.)
the results for V = 200 ft/min cross the curve
for natural convection alone, i.e. if Gr is large the convection heat transfer with forced con- vection is less than that of natural convection alone, whereas for low Gr the heat transfer is greater if forced convection is present.
DISCUSSION AND CONCLUSION
The test results for air in natural convection across rectangular openings in vertical partitions are in good agreement with theory. In particular the exponent on the Grashof number in the equation Nzc = BGrCLPr"s slightly greater than 0.5 as called for by consideration of the heat- transfer mechanism. Test results for a double opening are also in good agreement with theory. The ratio of wall thickness to opening size also
.
influences the Nusselt number.It was found that a horizontal forced-air flow parallel to the wall and opening could reduce the convection heat transfer under certain conditions.
The present tests encompass a range of the Grashof number from loG to lo8 and a range of the ratio of wall thickness to opening height of 0.19-0.75. For air, these results cover most cases of practical interest either for heat or mass transfer. F o r other fluids having Prandtl or Schmidt numbers greater than about 0.1 the results of the present tests would be expected
C O N V E C T I O N T H R O U G H R E C T A N G U L A R O P E N I N G S I N P A R T I T I O N S - 1 867
to apply approximately, provided that Gr and
t/H are witlin the ranges covered in the tests.
For fluids with very low Prandtl numbers the methods devised in Part 2 of this paper on openings in horizontal partitions may be used for approximate extrapolation of the test results.
ACKNOWLEDGEMENTS
This paper is a contribution of the Division of Building Research of the National Research Council of Canada and is published with the approval of the Director of the Division. Appreciation is due to Mr. A. G . Wilson, Head of the Building Services Section, National Research Council, for suggesting the project, and to Mr. J. Richardson for assistance with construction of the test partition and recording of results.
REFERENCES
1. E. SCHMIDT, Heat transfer by natural convection.
it~te~~r~ational Heat Transfer Cotzjkre~lce, 1961, Uni-
versity of Colorado. To be published by A S M E as "International Developments in Heat Transfer". 2. J. E. EMSWILER, The neutral zone in ventilation.
Trm~s. An~er. Soc. Heat. V e ~ l t . E ~ ~ g r s , 32, 59-74 (1 926). 3. K . R. SOLVASON, Large-scale wall heat-flow measur- ing apparatus. Trans. Amer. Soc. Hent. Refr. Air-Cord.
Engrs. 65, 541-550 (1959).
4. W. G. BROWN, Die ilberlagerung von erzwungener und natiirlicher Konvektion bei niedrigen Durch- siitzen in einem lotrechten Rohr. Forsck~~r~gsh. Ver. $sch. Ing. 480 (1960).
APPENDIX A A?Tziltiple Openings
When several openings of height Hl, Hz, H,,
etc. and width Wl, W2, W3, etc., are separated
vertically by distances S,,, S,,, etc., as in Fig. 8,
the elevation of the axis of equal pressure is determined from the condition that the total fluid flow across the openings above the axis is equal but opposite in direction to that below the axis. For the ideal case that equation (5) applies, this requires, after integration, that
i
1%
EQUAL- PRESSURE AXIS.
1-1
- - --S23 - x j p i r - W 7 1
n
FIG. 8. Designation for ~~lultiple openi~lgs in a vertical wall.
which equation can be solved for s.
The actual conditioils obtaining are more complicated than the above owing to non-ideal flow. It will be noted, however, that the terms in the above identity are each proportional to the product WNu, where Nu is the Nusselt number for single openings of height
( H I
+
SI2+
H,+
x) etc. Consequently, themost accurate calculation will obtain when the sums of WNu obtained from Fig. 5 are equated in place of equation (Al). Equation (Al) may be used to obtain a first estimate of s.
RCsum6-La thiorie classique de la convection B travers des orifices dans des parois verticales a ete genCralisee pour traduire B la fois les transports de chaleur et de rnasse. Les expiriences ont etC faites avec de I'air dans une grande installation dont les ouvertures avaient 7,5 a 30 c m de hauteur. Par suite de la grande resistance thermique de la paroi, les rCsultats des essais donnant le nombre de Nusselt en fonction du nombre de Grashof, pour des hauteurs d'ouverture correspondant a lo6 < Gr < lo8,
W . G . B R O W N and K. R . SOLVASON
Zusammenfassung-Die einfache Theorie der natiirlichen Konvektion durch 0ffnungen in senkrechten Trennwanden liess sich auf gleichzeitigen Warme- und Stoffubergang verallgerneinern. Die Versuche wurden in einer grossen W5rmeflussapparatur mit ~ffnungen von 76,2 mm bis 304,s mrn Hohe mit Luft als Konvektionsrnedium durchgefiihrt. Untersucht wurde die Nusselt-Zahl als Funktion der Grashof-Zahl rnit der ~ffnungshohe als kennzeichnende Lange im Bereich loB
<
Gr<
los. Wegen des grossen thermischen Widerstandes der Trennwand konnten die Ergebnisse direkt auf denStoffiibergang anwendbar sein.
h ~ ~ ~ ~ o ~ n q ~ i n - P a c c a r a ~ p ~ m a e ~ c n con\recTIIo ~el~no-11 ~1accoo6\1e11 IIa ocnone 06oGqe1111ofi npoc~eirruek Teopm e c ~ e c ~ n e 1 1 ~ 0 P I<oIIBeKqm .cepe3 oTnepcTLrn B B ~ ~ T P I I ~ ~ ~ ~ H L I X nepero-
p0nKaX. ~ K C ~ ~ ~ ~ I I \ I ~ I I T L ~ npOBOnE1nlICb Ha YC'raHOBkie C OTBepCTEIRaILI OT 3 A 0 12 A I O ~ ~ \ I O B
nb~co~ofi npII B O ~ ~ U I E I X TennoBLIx ~ a r p y a ~ a x ; B KasecTne TennoHocriTenn 11cnonb3onancn
noagyx. M O ~ K H O npeA11onoimTL, YTO Bnaronapn 6onb111oary Tep&itIsecI<oary conpoTEmneH5rIo
3kiCnepIIHeHTaJIbHbIe pe3yJIbTaTbI I I O 3aBEIC14>IOCTII IiPMTePIIR H Y C C ~ X ~ T ~ OT KpEITepIIR
Fpacrocja (aa x a p a ~ i ~ e p ~ ~ b ~ n p a a ~ e p nptIHaxaeTcn BbICOTa o ~ ~ e p c ~ m n ) , a a r i ~ ~ ~ o ~ ~ e ~ ~ ~ ~ o r o n npenenas lo6
<
Gr<
lo3, aroryr B L I T ~ npEr\reneIILI ,Inn \ ~ a c c o o B \ ~ e ~ ~ a .Inl. J . Ileal Mass Trarrsfer. Vol. 5, pp. 869-878. Perganion Press 1962. Printed in Great Britain.
NATURAL CONVECTION THROUGH RECTANGULAR
OPENINGS
IN PARTITIONS-2
HORIZONTAL PARTITIONS" W. C . BROWN
Building Services Section, Division of Building Research, National Research Council, Ottawa, Canada
(Received 11 December 1961 and in revised forrr~ 20 March 1962)
Abstract-Natural convection through square openings in a horizontal partition for the case of heavier fluid above the partition is investigated using air as the fluid niediuni. The test results relating the Nusselt number to the Grashof number and to the ratio of opening thickness to opening width are found to agree generally with the requirements of theory. Although the Prandtl number for air remained constant in all tests, it was nevertheless possible, with the help of the theory, to show the approximate influence to be expected for any value of the Prandtl number. Because of the high thermal resistance of the partition material the test results may also be expected t o apply directly t o mass transfer. The following range of variables was covered in the tests: Grashof number G r ~ r based on partition thickness and air temperature difference across the opening, 3 :< lo4 < G r l ~ < 4 x lo7; ratio of partition thickness to the side of the square opening H/L, 0.0825 < H / L < 0.66, with openings
of 6 X 6 in, 9 x 9 in and 12 x 12 in.
IN A previous paper (Part I), the theory and
experimental results for natural convection across openings in vertical partitions were given. In this situation buoyancy forces resulting from temperature or concentration differences between the fluids on either side of a vertical partition cause a fluid interchange across a partition opening with resultant heat or mass transport.
To complete the iilvestigation of natural convection across openiilgs in partitions the theory and experimental results for horizontal partitions will now be presented. As far as is known no previous work of this kind has been carried out. The situation of greatest importance is that where the fluid above the opening in a partition has the greater density; an unstable condition then arises, and an interchange of lighter and heavier fluid takes place. A rather surprising result of this interchange, which would not be immediately recognized but which is predicted by the theory and was verified
-
For Nomenclature, see Part I.
experimentally, is that heat- or mass-transfer rates increase with increasing partition thickness. The following theoretical considerations are carried out in considerable detail in order t o introduce a method of analysis suited to con- vection problems generally. The need for a systematic approach lo such problems arises, in particular, when several din~ensionless variables are to be related. For this condition it is often difficult, if not impossible, to obtain a relationship among the variables by experi- mental means alone that is not partly or entirely empirical. In experimental work an attempt is often made to relate the dimensionless variables of the problem as products of powers, but since the range of variation of one or more of the variables is usually limited, n o complete equation can be written for the phenomenon. Moreover, the exponents on the dimensionless variables themselves are often inter-dependent ;
consequently even small experimental errors may cause some of the exponents to have apparent values that a theoretical investigation could have shown to be impossible.
870 W. G . B R O W N
THEORY and for the heavier fluid flowing downward In the situation shown in Fig. 1, two sealed
cavities containing fluid at densities p, and V;
p, (pl
>
p,), with temperatures T, and T2 and Pr - PI = - P I T - 11+
P ~ S H (2) concentrations c, and c,, are separated by apartition of thickness H having an opening of or, combining equations (1) and ( 2 ) , characteristic width
L
(length of a side for asquare opening). The partition is assumed to be Vl V.?
impermeable to heat or mass transfer. (PI - P,) g H = PI
+
~e2
f (11 f 12). ( 3 )Here g is the acceleration due t o gravity and
.
I, and I, are pressure losses due t o entrance into.
the opening and fluid friction.The condition of no net flow across tlle opening requires that
where El and E, are the cross-sectional areas over which the flow occurs.
If p, and p, do not differ greatly it is reasonable to assume that both V, and V , and consequently El and E, are approximately equal. With the further assumption that I,
-.
12, equation (3) then becomes :4 P 21. ( 5 ) *
(p%fi)gb
= 4 g H = V 2 + -P
From knowledge of the general behaviour of
FIG. 1. Schematic representation o f natural convection fluid flow in pipes and conduits it can be through an opening in a horizolltal partition. assumed that for small ranges of all variables
Since the condition of the fluid at the opening is inherently unstable, no steady distribution of flow can be assumed. For determining the general relationship between variables, however, any flow distribution in wluch tht: lighter fluid flows upward with velocity V, and the heavier fluid flows dowilward with velocity V, may be considered. The pressures p, and p, a t the level of the top and bottom of the partition are presumed everywhere constant in the horizontal plane.
Neglecting any interchange of fluid in the horizontal direction, Bernoulli's equation yields, for the lighter fluid flowing upward
where C is a constant and v is the kinematic ' viscosity. The exponent u must lie between ..
0 and 1 and exponent b must lie between 0 and -1. (Provided only that the flow is not transitional, i.e. as occurs in a pipe at the critical /' value of Re.)
::: In a study of natural co~ivection in a11 insulated vertical tube with the higher temperature a t the lower end, Grassmann [I] takes a similar approach in deriving the flow equations. Owing to the large ratio of height t o opening size (pipe diariieter), however, his assumptions for the pressure losses d o not apply to the present prob- lem. Sinlilarly, the assumption that all heat transfer takes place in the lateral direction between the two streanis of fluid is not valid in the present case.
CONVECTION T H R O U G H RECTANGULAR OPENINGS I N PARTITIONS-2 871
Insertion of equation (6) into equation (5) for heat transfer
gives IITH
,-= Nu11
Again for a small range of C (H/L)a (VL/v)b, = C3GrIf1i ( z i b d )
(;)
(ucZ-bd)i(~+b(Z)equation (7) can be approximated by P r (13)
A P a V L b d and for mass transfer
- gH = V2Cl
[(;)
I-;)
]
('1 b,HP --- = S/ZH
D
where d must now lie between 0 and 1. Cl is
(;I
((r(i-bfZ)lC2-kb~Z) another constant. From equation (8) = ~ ~ ~ ~ . ~ 1 1 ( 2 + b a )at7 L bd
V 2 + b ( Z = $ ~ I ~ l ( z )
(;)
(9) = c3GrHll(z+b[1) SC. (14)P
where the exponent brl lies now between 0 and
- 1 and at1 lies between 0 and 1, with C, as a
new constant.
Having obtained an expression for velocity, the hcat- and mass-transfer rates across the opening become respectively
and
with c, as the specific heat.
Introducing the heat-transfer coefficient h~
and the mass-transfer coefficient /I,,, defined as /IT = q/L2(To - TI)
and
equations (10-12) lead t o the following equations in dimensionless form:
Here k is the thermal conductivity of the fluid and D is the diffusion coefficient.
NuH = Nusselt number based on partition thickness,
SIIH = Sherwood number based on partition
thickness,
G ~ H = Grashof number based o n partition thickness,
P r = Prandtl number, Sc = Schmidt number.
By summing the exponents on H in equation (13) or (14) it is readily found that either h~ or
h,,& is proportional t o H(l-(LIZ)/ (2+b(Z) and because
of the limits on nd and bd then (1 - ad)/(2 f bd) lies between 0 and 1. Thus the heat or mass transfer will either remain constant or increase with increasing partition thickness. Equations (13) and (14) can also be used as a partial check on experimental results because the pernlissible range for the expo~lent on one dimensionless group is conditioned by the exponent on the other. If, for example, bcl is found t o be -0.2 then the exponent on (LIH) in equations (13) and (14) must lie between (0
+
0-2)/1-8 =4
and (1
+
0.2)/1.8 =g.
In employing equations (11) and (12) it was tacitly assumed that essentially no heat or mass transfer takes place by thermal conduction or mass diffusion in the fluid. For fluids with high thermal conductivity and diffusion coefficients additional consideration must be made.
872 W . G . B R O W N
Assuming negligible heat or mass transfer in the horizontal direction, the equations for the heat and mass conservation in the opening are
and
where a(= 1c/pcp) is the thermal diffusivity and
s is the distance in the vertical direction. Inte- gration of these equations gives the expressions
exp (Vz/a) - 1
T - TI = (T2 - TI)
[exp (VHIal - 1
(for temperature), and
(for concentration). It will be noted that equations (17) and (18) reduce to the pure conduction and diffusion forms
for zero velocity or large a and D ; and to the pure convection forms T = T1 (constant) and c = c, (constant) for high velocities or small
a and D.
The heat and mass being transported across area El E L2/2 are respectively
and
where (i, -I- c,T) is the enthalpy of the fluid.
After substituting for dT/ds and dc/dz, obtained from equations (17) and (18), equations (19) and (20) become
TI exp ( VH/a) - T, exp (VHIu) - 1 and
The net heat and mass transfer across the parti- tion due to fluid flow in both directions is thus respectively,
4
= vpc, - TI exp ( VHla) - T2 exp(VH/a)-1-1
TI exp-
-
(--
VH/a) - T2 exp (- VHIU) - 1 L2 exp (VH/a)+
1 = VpcP2
(TI - T2) exp ( VHIa)-
1 andIt will be recognized that these equations reduce to the pure convection forms:
and
for high velocity or low a o r D. Similarly, for
low velocity or lugh a and D the equations reduce to the pure conduction and diffusion forms :
and
(c1 - c2)
l i l = pDL2
-
H . (28) Equation (10) can now be inserted into equations (23) and (24) and the Nus~elt and Sherwood 'numbers evaluated. It will be noted, however,
.
that the density difference A p, which was initially assumed to be equal to p, - p,, must now betaken as an average value since both tempera- 3 s
ture and concentration, o n wluch density depends, now vary througl~out the height H.
Writing
- = BYAT
+
/ ~ , , A cP (29)
where
/ 3 ~
and/3?n
are the coefficients of thermal and mass expansion for the fluid, then the average of AT and Ac over the height H can be inserted to obtain the average Ap.CONVECTION T H R O U G H R E C T A N G U L A R O P E N I N G S I N PARTITIONS-2 873
From equation (17) the average temperature difference over the height H between upward- and downward-flowing fluid is
ATavg =
H exp (VHIa) - 1
1
exp (- Vz/a) - 1 exp (- VH/a) - 1I?
Similarly, from equation (18) the average con- centration difference is
exp(VH/D)+ 1 2 0 A cavg =:(cl - ~ 2 ) -
-1.
(31)exp ( VHID) - 1 VH Equations (30) and (31) can now be inserted into equation (29), and this with equation (10) can be inserted into equations (23) and (24) t o obtain the Nusselt and Sherwood numbers. Consequently, for the general case of all fluids the relationship between the Nusselt or Sher- wood numbers and the remaining variables can be expressed as
Nu = f [GI.H, Pi', H/L, ~ T ( T ~ - TI), SC] (32) and
where f signifies the same function in both cases. Equations (32) and (33) are an interesting example, obtained directly from theory, in which heat and mass transfer are interrelated. The same kind of relationship would be obtained, of course, in the case of two-component mass transfer in place of heat transfer plus mass transfer. For the special case P r = Sc, equations
(32) and (33) reduce to
Nu = fl(Gi.~, HIL, Pr) (34) and
Sh = , f , ( G r ~ , H/L, SC). (35) Application of the theory
The foregoing equations are approximate to a considerable degree. Nevertheless they can be used in conjunction with existing data from other flow problems to estimate the general magnitude of Nu and Sh, which would be
expected in a practical situation. In conjunction with limited experimental results obtained with a given fluid, the equations may also be used t o extrapolate data for other fluids. T o illustrate these procedures, the conditions to be expected for air, the fluid used in the tests reported in the following section on experimental results, will be considered. Returning to equation (6), it is kilowll from hydraulic experiments that the head loss 1 at a square entrance into a pipe can be expressed as 1 = 0.5p(V2/2). Similarly, for a
re-entrant pipe 1 = 1.Op(V2/2).
Assuming that these values apply approxi- mately for the situation in Fig. 1, equation (5) becomes
With heat transfer alone, and with practical conditions of T2 = 70°F, TI = 30°F, H = 1 in and L = 6 in, for which the mean temperature
is 50°F with a = 0.78 ft2/h and v = 0.56 ft2/h,
equation (23) can now be investigated.
With the given data and equation (37) the term VH/a has a value between 100 and 125; thus equation (23) reduces t o the form of equation (1 I). Equation (30) becomes
It is now necessary to determine whether equation (36) can be expected t o have the approximately correct form in the range of given conditions. Strictly speaking this cannot be done without knowing the distribution of flow in the opening. It seems reasonable to assume, however, that flow will occur somewhat as indicated in Fig.. 1, in which case the Reynolds number, which indicates the range of validity of equation (36) for hydraulic flow in orifices, can be evaluated using about one-half of the opening width L as characteristic length. The Reynolds number VLI2v so obtained has the value of about 400, which is sufficiently close to the range covered in orifice experiments t o indicate the validity of equation (36).
8 74 W. G. BROWN
Forming the Nusselt number using equation (37) yields the theoretical relation for air
Experimental data for air would be expected to take the form of equation (1 3).
EXPERIMENTAL
Tests were carried out in the large wall-panel test unit used in previous tests on natural convection through openings in vertical parti- tions [2]. The apparatus [3] consists of two boxes 8 ft square and 4 ft deep (Fig. 2). One box (the warm side) is maintained at approximately 72°F by means of water circulated through the tubing of a panel on its inside wall. A separate tubing arrangement in the wall is separated from the panel by insulation and maintained automatically at the same temperature to prevent heat flow to the environment from the inner wall panel. The cold side can be maintained at any temperature down to about -20°F by means of a low temperature water-glycol-alcohol liquid from a separate cooling system that flows in the tubing of a wall panel and in a secondary finned tubing arrangement.
Since the test apparatus was designed for use with vertical walls it was necessary to build a special wall section (Fig. 2) in order to obtain a horizontal partition in which openings of various sizes could be cut for the tests. The test section was built in the form of a cubical box, 3 ft on a side, protruding from the wall.
By constructing all parts of the wall and test section of insulating material, a twofold advan- tage was afforded, (1) a large portion of the total %
heat transfer would occur across an opening, and (2) the convection conditions approximate . closely those that would occur with density differences due to concentrations alone, the result being that the heat transfer test results would be expected to apply for mass transfer as well. The wall, with the exception of the top and bottom of the test section, was constructed of 2-in foamed polystyrene insulation on a :-in plywood backing. The partitions forming the top and bottom of the test section consisted solely of layers of foamed polystyrene.
Because of space limitations and for ready access to the test section the two boxes of the apparatus were separated by a distance of 2 ft. An insulated wall was then built around this region to assure a minimum load on the cooling system of the cold-side box.
-. INSULATION INSULATION GUARD PANEL HEATING 2 IN INSULATION WARM SIDE
PUMP AND ACCESS WATER
RESERVOIR
TUBING
SIDE P A N E L
COOLING
CONVECTION T H R O U G H R E C T A N G U L A R O P E N I N G S I N PARTITIONS-2 875
Thirty-gauge copper-constantan thermo- couples were arranged to measure the air temperature at five locations 10 in above the test partition and opening and also at five locations at the elevation of the centre of the box-like test section. Each set of thermocouples was arranged in the form of a square, 18 in on a side, with one thermocouple in the center directly above or below the center of a square opening in the test partition. Additional thermo- couples were instalied to measure the air temperature in the warm- and cold-side boxes at locations remote from the test section.
The temperature control for both warm and cold sides was sufficient to maintain the air temperature at any given location constant to withill 0.2 degF. The total heat input to the warm side was obtained directly from continuous d.c. watt-meter recordings, the accuracy of the power input thus obtained being about 2 per cent.
Scope of tests and procedure
Tests were carried out with single square openings of nominal size: 6 x 6 in, 9 :< 9 in and 12 :< 12 in, with air temperature difference across the opening ranging from about 20 to 90 degF. The thickness of the partition of foamed polystyrene insulation was varied from 1 to 8 in. One set of tests was also made with a 12
s
12-in opening in an 8-in thick partition bevelled at a45" angle to a thickness of 2 in. A few tests were also made with an opening in the lower partition of the test section. For this case a stable situatioil with no convection was to be expected.
Before carrying out tests with various openings it was necessary to calibrate the entire wall and test section with a blank partition of given thickness in place. This was done by deter- mining the total heat transfer at various tem- perature differences between the air in the warm- and cold-side boxes. The results are given graphically in Fig. 3 where the heat flow in Btu/degF is plotted against air temperature difference. (The warm-side air temperature was maintained throughout at 72°F.)
With an opening in the partition, a small portion of the total heat transfer takes place by radiation, the amount of which was calculated by assuming that both tlle warm- and cold- side boxes behaved as black bodies, i.e.
where o is the Stefan-Boltzmann coilstant and
subscripts denote surface conditions. (Inter- change with the edges of the opening was negiected.)
Test reszllts
In accordailce with equations (39) and (13) a relationship is t o be expected between NuII/Pr and Grn, with the ratio of partition height to
AIR TEMPERATURE DIFFERENCE ( O F )
4lNG I N 81) PARTITION: H/L = 0 . 6 6
I
.
12'; 12" OPEA o 6'k 6 " OPENING IN 4" PARTITION: H/L = 0 . 6 5 A 12" 12" OPENING IN 2" PARTITION: H/L ' 0. 163 A 6 " x 6 " OPENING IN I " PARTITION: H/L = 0 . 1 6 4 12"x 12" OPENING I N 4 " PARTITION: H/L = 0 . 3 3 0 3 oI
1
1 1 1 I 1 I1
1 1 1 1J
2x101 4 6 8 10' 2 4 6 8 IOr 2 4 6 8 loT 2 4 5x10' ~r~ = ~ ~ A T H ~ v 'FIG. 4. E x p e r i m e n t a l r e s u l t s for t h e N u s s e l t n u m b e r d i v i d e d by t h e Prandtl n u m b e r a s a f u n c t i o n o f t h c G r a s h o f n u m b e r
CONVECTION T H R O U G H R E C T A N G U L A R O P E N I N G S I N PARTITIONS-2 877
opening width H/L as a secondary variable. These dimensionless groups were evaluated for all tests using the average of TI and T, to obtain air properties; the results are given in Fig. 4. (The air temperatures TI and T, employed in calculations are the averages of readings of the four outer thermocouples above and below the opening.) From Fig. 4 it will be noted that agreement with the ideal situatio~l as represented by equation (39) is fairly good, but that the experimental results are all somewhat lower than called for by this equation. This effect, as might be expected, is most pronounced at h g h values of H/L because fluid friction was neglected in equation (39).
As may be seen from Fig. 4 the experimental accuracy of the data is not sufficient to deter- mine the effect of the term H/L with great accuracy. The data were rearranged, however, in accordance with equation (1 3), assuming several values of the exponeilt on H/L in the term N L I H ( H / L ) ~ " ~ / P ~ and plotting the results against the Grashof number. A value of exp =
4
appeared to correlate the data with fair accuracy (Fig. 5). A mean curve through the points in this figure gives the equation :
From the discussion following equation (14) it will be appreciated that the exponents o n GTII and (LIH) fall within the range expected from theoretical considerations. The range of validity of the equation is for
3 x 10" GI'H
<
4 :i10'and
0.0825
<
H/L<
0.66,and should be correct for ally value of the Prandtl number greater than that of air (Pr = 0.71).
The few tests carried out with an opening in the lower partition of the test section indicated pure conduction heat flow througll stratified air. The measured heat flow in this case was less than the experimental accuracy of measurement. Extrapolation of the test results for low Prnndtl
or Schmidt numbers
Experimental data may be used to estimate the value of the Nusselt or Sherwood ilumbers for low values of the Prandtl or Schmidt numbers.
Making use of equation (30) the "effective" Grashof number is :
where G ~ H is the Grashof number based on the temperature difference (TI - T,). Forming the dimensionless group VH/a with the help of equations (lo), (13) and (42) leads t o
where ( N ~ ~ f ~ / P r ) a i r is the value of the Nusselt number divided by the Prandtl number obtained with air.
Inserting equation (10) into equation (23) yields for the Nusselt number
NufI = C3 [exp (VH1a)
+
I] exp ( VHIa) - IL (ad-bd)l(e+bd)
(Gl.c)l/(2 4-6,)
(,)
Pr,-
-
ex^
exp ( ( Vl$/a) VH/u)+
-
I] 1 Y(H/L)(ad-b["I(2-kbcI) P ra i r
(44) By inserting various values of (Nll~/Pr)air corresponding to the range of tests into equation (43), valucs of VH/a are obtained which can be inserted into equation (42) to obtain Gi.~{and into equation (44) to obtain NUH(H/L)(""~~)I("~~)/P~ for any given value of Pr. An example is given in Fig. 6 for P r = 0.01 as would be obtained with liquid sodium. (The solid portion of the curves corresponds to the range covered in the tests with air.) It will be noted that for P r = 0.01 and for a value of GYH
<
lo5 practi- cally all heat transfer is due t o conduction even though fluid mixing and circulation still occurs.DISCUSSION A h 4 CONCLUSION
Equation (41) and Fig. 5, representing the relation obtained in heat-transfer tests with air as the fluid medium, should also be directly
880 W . G. B R O W N
applicable within the range of Gra and HIL
covered in the tests for any value of Pr greater than about 0.1. The test results obtained may have been influenced to some extent by the configuration of the test section itself: of necessity it was relatively small compared with the openings due to space limitations within the test apparatus. So far as is known, however, no previous tests of this kind have been carried out and it is to be expected that the results are sufficiently accurate for practical purposes. The method devised for extending the data for fluids having low Prandtl or Schmidt numbers is not to be considered exact, owing principally to the inherent approximations involved in equations (15-31). For example, it was assumcd that the temperatures on both sides of the partitions were everywhere constant, when in reality there is always a temperature gradient extending beyond the opening. Also, a gradient in temperature through the opening would be expected to have an additional influence on the velocity not included by defining the effective Grashof number as in equation (42). For
practical purposes, however, the methods given here should be useful in estimating either heat or mass transfer for a wide range of conditions.
ACKNOWLEDGEMENTS
The author is indebted to Mr. K. R. Solvason, Research Officer at the National Research Council, for use of the test apparatus and for discussions during progress of the work; and t o Mr. J. J. M. Lavoie, Research Technician, for carrying out the construction and calculations involved in the tests. This paper is a contribution of t h e ' Division of Building Research, National Research Council of Canada, and is p~~blished with the approval of the Director of the Division.
REFERENCES
1. P. GRASSMANN, Theory given in Freie Kormektioti irlz
set~krechtetz Rolrr (von STEFAN B ~ ~ H L E R ) . Diplomar- beit E.T.H., Zurich (1957).
2. W. G . BROWN and K. R. SOLVASON, Natural con- vection through rectangular openings in partitions-1. Vertical partitions. Irlt. J. Heat Mass Tratlsfer, 5, 859-868 (1962).
3. K . R . SOLVASON, Large-scale wall heat-flow nleasuring apparatus. Trans. A/,ler. Soc. Heat. Refi. Air-Corzcl.
Et~grs, 65, 541-550 (1959).
RCsumC-Cet article etudie la convection naturelle de I'air ;I travers des ouvertures carrees dans une
paroi horizontale pour le cas du fluide le plus lourd au-dessus de la paroi. Les resultats d'essais donnant le nombre de Nusselt en fonction du nornbre de Grashof et du rapport profondeur sur largeur des ouvertures concordent generalement bien avec la theorie. Bien que le nombre de Prandtl de l'air reste bien constant dans tous les essais, il a Cte nearnoins possible, i l'aide de la thiorie, de montrer approxi- mativement l'influence de chaque nombre de Prandtl. Par suite de la grande risistance thernlique de la paroi les resultats des mesures peuvent &tre directenlent appliques a u transport de masse. Les domnines suivants des variables ont ete explores au cours des essais: nombre de Grashof base sur l'epaisseur de la paroi et la dirirence de temperature de l'air travers l'ouverture 3 : 101 < GrH < 4 x 10';
rapport de l'epaisseur de la paroi la section droite de l'ouverture 0,0825 < H/L < 0,66 avec des ouvertures de 15 cm2, 23 cm3, 30 cm2.
Zusammenfassung-Die naturliche Konvektion durch quadratische 0ffnungen in einer waagerechten Trennwand mit dem schwereren Mcdiurn oben wurde fiir Luft als Konvektionsn~ediun~ untersucht. Die Ergebnisse liefern die Abhangigkeit der Nusselt-Zahl von der Grashof-Zahl und dem Verhaltnis Offnungsdicke zu ~ffnungsweite und stirnmen in1 allgen~einen rnit der Theorie iiberein. Obwohl die Prandtl-Zahl der Luft fur alle Versuche konstant blieb, wares doch moglich. mit Hilfe der Theorie, angenlhert den zu erwartenden Einfluss anderer Prandtl-Zahlen zu bestin~n~en. Wegen des grossen therrnischen Widerstands des Tre~~nwandmaterials konnten die Ergebnisse auch dirckt auf den Stoffiibergang anzuwenden sein. Die Versuche umfassten folgenden Bereich von Variablen: Grashof- Zahl GrII auf die Trennwanddicke und die Differenz der Lufttemperaturcn beiderseits der Offilung bezogen 3 x lo4 < Gr11 < 4 x lo7; Verhiltnis der Trcnnwanddicke zu Breite der quadratischen ORnung HIL, 0,0825 < H / L < 0,66 bei Offnungen von 152 x 152 mm; 228 :-: 228 rnm und 305 x
CONVECTION T H R O U G H R E C T A N G U L A R OPENINGS I N PARTITIONS-2 881 rpacrocpa 11 o~rroure~rrre\r T O J I ~ L Z I T L I OTBepcTIIn I i ero ruIrpkIae, B o6uear corxacy1oTcn c
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