,

R 90. ,Philips Res. Rep. 3, 338-360; 1948

DISSIPATION OF HEAT BY FREE CONVECTION *)

by' W. ELENBAAS 536,25

SummaryA body differing in température from the surroundings loses encrgynot only by radiation but also by conduction and convection. Thegeneral laws governing the latter dissipation of heat are investigatcd.The many quantities governing this transfer may be combined intoa few dimensionless numbers by means of similarity considerations.This leads to an important simplification and facilitates the surveyof experimentàl and theoretical results. First, the above-mentionedconsiderations -are applied to bodies that may be characterized byone linear dimension only (the vertical plate, the horizontal cylinder,the sphere), and further, to bodies characterized by two suchquantities (the vertical cylinder of finite height, cooling rihs, theinner surface of vertical tubes).

RésuméUn corps qui se'trouve à une température plus ëlevëeque celle de sonambiance perd de la chaleur par rayonnement et aussi par conductionet par convection. Le prësent article traite des.lois et principes gênë-raux qui régissent la perte de chaleûr par conduction et convection.Des considérations d'analogie permettent de combiner les diffërentes

, grandeurs qui apparaissent dans les équations fondamentales en tinpetit nombre d'indices numériques sans dimension. Il en rësulte dessimpIifications considérahles et cela nous petmet de donner un aperçusynthétique de données expérimcntales et de considérations thëori-ques, dans ce domaine.Avec cette façon de considérer les choses.Turtlcle traite successive-ment de corps caractérisés par une seule dimension linéaire (Ia plaqueverticale, Ie cylindre horizontal, et Ia sphère), et de corps caractériséspar deux dimensions Iinéaires (le cylindre vertical de longueur linie,les ailettes de refroidissement, et la .surface intêrieure d'un tuyauvertical).

Symbols and definitions

Suffix w means that the quantity, should be taken at the temperature'of the wall (Dw).

Suffix oo means' that the quantity should be taken at the temperatureofthe medium at a great distance from the body (Deo)'

Suffix m means that the mean should be taken, for instance:

Suffix s means that the quantity should be taken ~orthe hydrostatic state.

*) The paper is essentially .idcntical to that published in De Ingenieur 60, 0 21-0 34,1948, by the same author, under the same title.

~:..

NuGr

, RePrPéAcp(!-1]

{JgV

a

DISSJPATION OF .HEAT BY FREE CONVECTION

aldhT{}

tç[J

F[P. Re]b15SUT

rppsp

.= a ip; Nusselt's number.za g (!2 {J ewln2, Crashof''s number.(! V lln, Reynold's number.cp 1]1)., Prandtl's numbe~.

- (! cp V IIA, Péclet's number.heat conductivity.

= "specific heat at constant pressure.density.

339.

viscosity. _coefficient of cubic expansion; for' gaseous media, (J = -liT IX)'

- acceleration of gravity. ,- velocity; u, v, and 'UI are velocity. components in the X-,

y-, and z-directions. .' .'. ,- coefficient of heat transfer = the dissipation of heat per unit . i

of area per unit of time and per unit of temperature dif-ference. ' .

- mean value of a on the surface.linear dimeneion.

- diameter.height.temperature in °IC

__ temperature in oedifference of temperature as compared to remote sections ofthe medium, e = T - T IX) = {}~ {}co •

time.- quantity of heat dissipated per second.

heat-dissipating surface. I ,

- the so-called Poiseuille coefficient.spacing between the cooling ribs.

- plate thickness of the cooling ribs.surface of the cross-section of a cylindrical tube.circumference of the cross-section of a cylindrical tube.hydraulic radius of the cross-section = 2SIU.the angle that the plates or cylinders make with the vertical.

- pressure.hydrostatic pressure.

- p-ps·

1. Introduetion

In the anode of an X-ray tube or of a transmitting valve, in electrically.heated wires. (such as the filament of an incandescent lamp, or a heating-element of a~ electrical furnace), in the windings of a choke coil or a

./ .

". "

,

•",

340 W,. ELENBAAS

transformer, in thè plates of härrier-Iayer rectifiers, f and in many otherapparatus, energy is transformed into heat. As a consequence of theheat developed, the temperature of these bodies rises' above that of thesurroundings. As the temperature difference increases, the amount of heatdissipated rises until a state of equilibrium is reached, where the amountof hea t developed is equal to that dissipated." It is often desirable to calculate the temperature if the amount ofheat produced is known. Very often one aims at reducing the dimensions .of the apparatus, ox:parts thereof, as much as possible with a view tosaving space or lowering the weight, the load, and with it the temperature,being increased to thé highest permissible value. In a number of casesone wants the body, such as a furnace or a filament, to reach a certaintemperature. In order to ascertain the connection between the amountof heat ç[J dissipated per second; and the temperature differencewith respectto ,the surroundings, we shall first- investigate the mechanism of heattransfer. .

2. Radiation, conduction, convection

It is usual to divide the dissipation of energy into heat transferred byradiation, by conduction, and by convection. In this paper we shall notconsider radiation because the la,vs governing the radiation of heat aregenerally well known, an'd in ~ost cases it is possible to investigate thiscause of heat transfer separately, and studythe loss of heat by conductionand by convection apart from radiation. This is evident from the follow-ing ar~ment: if a body is cooled by a liquid, the difference in tempera-ture from the surroundings is usually so small that the loss of heat byradiation is negligible in' comparison to that by conduction and con-veetien. If the cooling is accomplished by a gas, the "amount of, ,heat

. . radiated may be considerable (e.g. in the case of the filament of a gasfilledincandescent lamp or of a heating element in an electric heating unit),but in this case the radiation penetrates the surrounding medium withoutappreciably heating it, and it is only absorbed when impinging upon the.wall of the tube or that of the :t:0om. Thus r~diation has practically noinfluence upon the conditions of the surrounding medium; hence the lossby conduction and by convection is practically unaffected hyvthe radia-tion loss, which therefore 'may be simply added to the former. To measurethe contribution of the radiation one may put the body in a vacuum, the.radiation losses then equalling the energy input when the equilibrium isreached, or one may measure th~ radiation by suitable means (thermo-pile, etc.).

As to the heat transfer by conduction and convection, it may he pointedout that the primary phenomenon is always conduction. For it can he

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, .

< .,..

DISSIPATION OF HEAT BY'FREE CONVECTION 341

. 'shown experimentally that at the surface of the body the medium ispractically at: rest with respect to that surface. Whether we have' a flowingmedium (in the case of convection) or whether the 'medium is stagnant(in the case of pure conduction), the heat at the' surface of the body mustalways be carried away by conduction only. Ifthe heat conductivity of themedium should he zero, heat transfer by convection would be im-possible too.

If n is the distance from a point in the medium to the surface of the ..body, measured along the normal to. the surface, and - d{}fdn the tem-perature gradient, the energy dlP transferred from a surface .element dF ,equals" , •

dw = - (Ad{}fdn)w dF, (1)

r '

""

where Ais the heat conductivity of the medium, while the suffix w means ....;.that for À. ~nd d{}fdn the values at the surface of the body should be takèn.

If the convection currents in the medium Increase, the value ~f d{}fdn.. "increases, and with it that of dlP. This may be understood by remembering

that the stronger the current, the faster the liquid heated by the bodyis carried away, the medium remaining cool to within a shorter distancefrom the surface of the body.' Hence the temperature gradient i~ larger.In fig. 1 the behaviour of the temperatpre as a function' of the distancefrom the surface, is represented in three different cases: (a) without any'flow (pure conduction), (b) in the case óf natural .or free convection; in

~, '

"

.~, I

I • .. '.

, I

-noFig.!. Temperature {}as a function of the distance n to the heat-dissipating surface. Thetemperature of the body is {}.. and that of the surroundings {}«l • ,

a. pure conduction without convection currents. 'b. natural cooling or free convection.c. forced convection.

3. The equation of heat dissipation

According to Newton's rule the heat transfer is proportional to the totalsurface area of the body and to the temperature difference between thebndy and thé surrounding.medium. If {}w represents the temperatur~ of thebody and {}co that of the medium at a great"distance f~om the surface, the

I •• flow of heatthrough an element dF ~f the surface is' given by

.,

, '

342 W. ELENBAAS

this case the motion of the liquid-is caused by gravitation in association.with, the de;"sity variatiob.s occurring as a result of the temperature varia-tions brought about by the heat-dissipating body, (c) in the case of forced'convection, where the motion is determined by external conditions. Inthis case natural convection occurs at the same time, but generally it maybe neglected in comparison, to the forced convection.o In the present paper we shall not deal with cooling by forced convection,restrioring ourselves further to stationary conditions, the energy transferif> not depending on time.

(2)

According to Newton's law a is a constant. However, in the case of naturalconvection this.simple rule with constant a does not hold. Nevertheiessone may retain equation (2), a then simply meaning the quántity required.If we introduce the quantity B = {}- f)°co , -the heat transfer (2) assumes

the form 'd4> --'a Bw dF, .

the total ~nergy transferred becoming

([J = J a e; dF .

If Bw is constant all over the surface, the body having a constant tem-- perature, then

cp = Bw f a dF . (3)

Introducing the average value a = (f a dF)jF, we obtain:

4>= e; a F. , (4)

o The above transformation, however, does not yet bring about thesolution of the problem,' all the difficulties being embodied in the quan-tity a. .

To find a relation from which a may be deduced, we shall submit the• state of flow and the temperature distribution ?f the medium to a closer..scrutiny.' ./

The velÓcities and the temperatures of the medium .will appear todepend on a number of quantities, such as the density (e), the viscosity ('17),

\

_----------------:---:---,---~-~---'~~ ....-. -.r- , '.

- ' 'DISSIPATION OF HEAT BY FREE CONVECTION 343

the heat conductivity (À), the coefficient of cubic expansion (fJ), thespecific heat at cons~ant pressure (cp) and ~he acceleration of gravity (g). ,Further, the conditions in the medium depend on the size and the shape'_of the ,heat-dissipating body_and on ew. In. addition; the differentialequations obtained appear to be of such a complicated form: that an'exact evaluation of a is in general not practicable. Calculations hav~ there-fore been carried out in the case of bodies of a very.simple shape only..whilst moreover use has been made of simplifying suppositions.· .Notwithstanding, ~rom consideration of the differential equations one

important conclusion may be drawn, namely, that the above-mentionedquantities and parameters only occur in special combinations. 'Thereforethe" theoretical and experimental resuÎts may be expressed as. relationsbetween these special combinations (characteristic numbers).

4. The velocity field, the pressure field and the temperature field

Let us consider a body cooled by natural convection. As observed above,the temperature differences give rise to convection currents in the sur-, , ,rounding medium. At a given point in the medium the temperature: and thevelocity have definite values not dependent on time. This means thatstationary conditions only are considered. Turbulence is not accounted for,it being of secondary importance, at least in the case of natural convectionand in the immediate neighbourhood of the heated body.Therefore we may say that a temporature field and a velocity field exist,

'to he described by the quantities {}(or e) and u, v, 10, the velocity compo-ncnts in the direction of the coordinate axes OX, 0 Y, ,oZ, which are func-tions of x, y and z, The Z-axis will be chosen pointing vertically upwards.

Further, the pressure in the medium may be described by a pressure field.However, the pressures and the velocities of the medium ar~ so closelyrelated that we shall consider them together under the heading "velocityfield"; .

A) Apart from boundary conditions~tli~ velocity field is detèrmined by{wo differential equations. The first equation is the fundamental hydro-dynamic equation of Navier and Stokes for a liquid- with internal friction.This equation is based on Newton's law: force = mass X acceleration,applied to an element of volume dxdydz, e dxdydz being the mass 'of theliquid in this element. The x-component of the acceleration is

du ou oudx oudy oudz ou - ou . ou ou-=-+--+--+--=-+u-+v-+w-.dt ot ox dt oydt oz dt ot ox oy ozAnalogous equations are valid for the y- and z-components dv/dt and.

dw/dt. :rhe first term of th:e right-hand member may be omitted becauseonly stationary conditions are considered; the ot~er terms are due, to the

·'

, .- "

.' ,

344·rw. ELENBAAS

fact that the volume element is displaced by the motion and then arrivesat places where the velocity componente have different values. The totalforce on the volume element, e.g. in the x-direction, consists of the follow-ing èomp~nents:

1) the force due to a pressure gradient

op-- dx dy dz·

ox '.2) the external force e Kx dXdydz where Kx is the total external force

for unit mass (for instance, in the case of gravity alone being active,Kx - Ky = 0, Kz.= -g) ;

3) the frictional force

1](02~ + 02:) dx dy dz,

oy . oz

due to the changes of the velocity 'gradients;4) a frictional force

1]~02U +'!~(OU + OV ., OW)2dxdY'dz, _? ox2 ox ox öy OZ S ,

being due to the deformation of the volume element considered 1). Thefinal form of Navier and Stoke's equation for the z-componerrt is:

, (OU ou OU) op.'e u-+v-+w- =--+eKx+, ox, oy oz , ox

s 0 (OU OV OW) 02u 02U 02U l+.1]?! o,x ox + oy + i);- +: ox2 + oy2 + OZ2 S' (Sa)

with análogous equations (Sb), and (Sc) for the y- and z-components .. 'In applying these equations to natural conyection a simplification arises

as tO,the terms

_ op + fJKx'ox ~ .

To see this we start from the initial state where the température of the, body is equal to that of the surroundings. In that case e is a function of z

.' only, es(z). The index s indicates hydrostatic equilibrium. The pressurein this case is the hydrostatic pressure Ps- The components eKx and eKyare zero, whereas eKz = - ges(z)., .,Then ps obeys the equations (Archimedes's law):

0ps 0ps 0ps''_ = 0, _ = 0, ---g es(z) = 0.ox _ oy oz'

(6)

." ~'. . '. -, I, 'DISSIPATION OF HEAT BY FREE CONVECTION , . 345'

. \. .The element of volume considered thus floats in the surroundirig medium.

'.' .if, howev~r, an element of volume ,~ith a higher temperature and .c~rre- "sponding lower density e is surrounded by a medium of a different density. e', the Archimedes force g e' and the force of gravity ge do not halance andthe element is subjected to a force in the' vertical direction, which sets themedium in motion, Thus a state of flow is established where the pressureis no more equal to the hydrostatic pressure, but differs-from it by a cer-tain amount, say P. Thus p = Ps +P. ', In equation (Sc) we have therefore: .

op , 0ps,,: oP '. Öps oP-- + eKz = - -, -~ - ge = - - - ges - - + g(es- e)·oz oz OZ OZ OZ

'. -

As, according to (6), - ops/äz - ges ' 0, we obtainJor the combined:.pressure forces and external forces: . -

"

oP- - + g(es - e) .oz ,

In vthe same way,the corresponding terms in equation (Sa, b) are re-duced to

.' ) "

- oP op-- and --,

ox oy

-,

respectively.Now' we' shall make the simplifying supposition that e does not appre- .

ciably depend on the pressure. In the case of a liquid this is evident, whereasin the case of a gas rt-implies that the differences in height occurring in ourproblem in connection with the dimensions of the body and the extent' of.the temperature field are s~ small that es has a con~tant value eco' ebeing a function of the temperature only.

Introducing the coefficient of ther~al expansion

: ,

\ I, '

we have:

For gases we have:

eco - e = e(T - T co )IT co = eelT 0:> or fJ = liTo:>.

Introducing these simplifications, eqs, (Sa, b, c) take on the form:

(OU ou OU)' oP [0 (OU ov. OW) 02u 02U 02U]e u-+v-+w- =~-+17 t- -+-+- +-+-+- (7a).ox oy OZ ox ox ox oy' OZ ox2 oy2 oz2

,.

1I

expressing the conservation of matter for a fixed element dxdydz in thestationary condition.!The boundary conditions for the solutions of (7) and (8) are that the

velocity at the surface of the body and at infinity shall be zero._. B) The temperature field is governed by a differential equation expressing

the equality of energy input' and output of a fixed, element of volumedxdydz.The heat developed by internal friction, being relatively small, is neglected.The energy transport through a surface element dydz being (!èpe'u dydz,

one obtains for the total energy input, raking into account eq. (8):

W.ELENBÀAS

, (OV OV·" OV) oPe u-+v-+w- =--+

. . ox oy. oz oy

[ 0 (OU OV' OW) 02V 02V 02vJ' . + 'YJ t oy ox+oy+Öz +ox2+oy2.+0;2

(OW OW OW) oP

e u-+v-, +iv- =---+gepe+ox oy OZ OZ /'

[0 (OU OV OW). 02w 02w 02W] •

+r] 1--+-+- +-+-+- (7c)oz ox oy OZ ox2 oy2 OZ2

(7b)

The second differential equation determining the velocity field is' theequation of continuity:

o((!u) o((!v) o((!w)-+-+--=0,ox . oy oz(8)

. (Oe oe 'oe) .-(!Cp u-+ v-+w- dxdydz.. ~ oy ~ .

This energy is carried away by h~at conduction ..- J..(oejox) dydz benig the conduction loss through the surface dydz,we obtain for the t~tal loss by conduction:

(02e 02e· (28)

-J.. --+-+- dxdydz.ox2, oy2 OZ2

• Equalling the two expressions one obtains the equation for the tempera-ture field in the form:

, . (Oe oe Oe) (02e' 02e' 02e)

, flcp' u-+v-+w- = J.. -+-+- .ox öy- oz , ox2 oy2 OZ2

The boundary conditions are: e = Bw at the surface of the heated bodyand e = 0 at infinity.

(9)

DISSIPATION OF HEAT BY FRE;E CONVECTION- 347

As stated above, no general s-olutions of eqs (7), (8) and' (9) have heen ',fomid. However, without writing do~vn the solution, important simpli-fications and a reduction of the number of variables and parameters mayboeobtained by comparing "corresponding", or "similar" states. Supposingthe heat transfer by natural convection to be known in one case, the trans-fer in the corresponding "similar" cases may be computed. In the caseof natural convection this principle was first applied by Nusselt 2).

((}UOW/OX)l (oP/ozh((}uow/àxh -. (OP/O~)2

(gg{Jeh(g(}{Je)2

('fJ02W/OY~)1('fJ02U}/oy2)2

, (10) I '

-5. Heat dissipation of similar hodies

Let us consider a number of similar bodies Ll' L2' ... , placed in the mediaMI' M2' ... , the outer boundaries of which are assumed either to be similar "or to lie at such distances as to have no influence. The shape of the bodiesbeing giv:en, they may .he characterized by one linear dimerÎsion (e.g.the diameter of a cylinder, the edge of a cube). These dimensions we callll' i2, •••• Now the velocity fields in ,MI and M2 will 'ne compared. Weshall call these fields similar if in every set of corresponding points(Xl/X2 = Yl/Y2 = Zl/Z2 = ll/l2) the corresponding velocities have the sameratio (ul/u2 ' vl/v2 = Wl/W2 = Vl/V2 ' a, where VI is the total velocity in,MI' etc.). Similar velocity fields are only possible if eqs (7) and (8) are thesame in either case and if the boundary conditions are equal. The lattercondition is always fulfilled, as V= 0 on the surface of the body and at theouter boundary' of the medium. The remaining necessary and- sufficientcondition is that the various corresponding terms in eqs (7) and (8) have'

. the same ratio. In this case the equationsmay be made identical by multi-plying them by an appropriate number and hence have identical so-lutions (the boundary conditions being equal). The similarity conditionsfor the ve~ocity fields of cases 1 and 2 are thus, for instance:

The equation (10) is valid for all terms, provided

((}PIlh,((}P/lh

(PIlh(Pjl)2

(g(}{Jeh ('fJV/l2);(ti! {Je)2 = ('fJV/l2)2 '

(lOa)

or, in symbols, A = B = C = D.Combining A and D, we obtain:

('(}Pil) ((}V21l) or ((1 Vl"n )1= (Q ~)2''fJ V/l2 1= 'fJVjl2 2 '/ '/

The quantity (}Vl/'fJ is called Reynold's number; it will he denoted by R~.It has zero dimension, being obtained by dividing a "force ofinertia" by a

(ll)

•

348 . W. ELENBAAS

. . \ .frictional force .. Equality of Re thus means that the ratio of the inertiaforce to the frictional force is the same in either case. It is immaterial forwhich pair of points the condition is written down, provided the points .

" are corresponding points. .Combination of C and D gives a second condition for similar velocity

~&: . -

,. I".

(gefJél2) = (ge(38l2), '

rJV 1 rJV 2·,.i.e., the frictional forces must be in the same ratio as the buoyancy forc~s.

Equalling A and B, we obtain as a third condition

(12) .

(PIe V2h = (PIe V2)2' (13)

"

PIe V2 is called Euler's nu~ber (Eu). ,Eqs (ll), (12) and (13) are equivalent to (10àf. Hence, the velocity

:fields are similar, provided the latter three equations are satis:fied. (Eq. (8)does not provide us with new conditions as the terms are all of the same-.kind.] ..

In the case of free convection eqs (ll), (12), (13) demand more than isnecessary; P and V not being given independently (as in the case of forced.convection], As a matter of fact, V is determined by the given quantitiesl, g, e- (3, 8, A and c. Therefore V may he eliminated from (11) and (12) ,in the case of natural convection, ..giving:

(14)...

The dimensionless quantity zag e2(3 81172 is called Grashof's number(9r). Eq. (14) being satisfied, (11) and (12) are fulfilled automatically.P .not being independently prescribed either, (13)'is alsofulfilled, Hencein the case of natural convection the numbers Re and Eu are of no im-portance, ..We thus arrive at the' conclusion that in the case of-natural cooling of

similar bodies'equality of Gr at eerreeponding .poinis is a sufficient conditionto ensure similarity 'of the eerreeponding velocity fields. Which pair of corre-sponding points is chosen is immaterial,. the temperature :fields beingsupposed similar at the same time. Usually 8~, the value of 8 at thesurface of the body, is inserted in (14). .

The next step is to investigate the conditions ensuring the similarityof the temperature :fields. ,, From (9) we obtain for the necessary condition:

., .

DISSIPATION OF HEAT BY FREE CONVECTIO,N ,

This leads tor '(e cp V l/A)l = (£I cp V l/Ah·

This quantity is called Péclet's number (Pé). On th~ same grounds asgiven above, V may be eliminated (dividing by Re), giving:

The quantity cp 17/Aiscalled Prandtl's number (Pr). In t~e case ofnaturalconvection and the velocity fields being similar, ,the fulfilment of (16) .suffices to ensure similàr temperatuf fields., 'Therefore, when considering similar bodies cooled by natural convection,the velocity and the te'!"perature fields are similar provided Gr and Pr are equal

,at cortesponding points.Supposing these conditions to be satisfied, one may determine the ratio

of the amounts of heat dissipated in either case. In similar temperature .fields the energies transferred in unit time by equal surface elements areaccording to (1) in the proportion (A Bw/lh to (A Bw/lh, and hence the..energies dissipated by t~e whole surface per second have the ratio:

, (17)

This follows also .from (9), bearing in mind that corresponding volume I

elements are in the ratio ll: l23•According to (4),

tP1 : tP2 = (a Bwl2h : (a ewl2)2'Combining (17) and (18), we have

(ä uA.h = (a l/A)2 .. ,

Fro~ this the ratio al: lÏ2 and, by (4), that of tP1: tP2 may be calculated.The quantity a l/A is called Nusselt's number (Nu). 'We thus arrive at the conclusion: If, with natural cooling of two similar

- bodies the numbers Gr and Pr at corresponding points are the same, the velocityand temperosure fields are similar and hence the Nu-numbers are likewise thesame; thus Nu is a function of Gr and Pr only: Therefore the solution ofour problem takes the form: ,

". or

, This.important simplification makes the survey of results much easier.A further simplification is obtained in the case where corresponding

3-:t9

(15)

(16)

. (18)-,

(19)

(20)

"(20a)

· ,350 W. ELENBAAS

media are gases having the same number of atoms in the molecule, for Pr, is a quantity containing only properties of the medium and is moreover .independent of the temperature, being dependent only on the n~mber ofatoms in the molecule.

For molecules' with n degrees of freedom, we have:

Pr = (n + 2)/(n + t).When restricting ourselves to gases having the same number of atoms

per molecule, eq. (20a) may thus be reduced to.-

Nu = j~(Gr).: - (21)

For bodies of a given shape the heat transfer is then represented by onesingle curve. ,

When comparing .the heat transfer in arbitrary media eq. (20) should beused. This equationmay be represented by a number of curves, Nu =f(Gr)~having Pr as a parameter. In the first approximation a much simplerrepresentation of facts mayalso be obtained in this case. Rememberingthat with natural cooling the velocities are generally small, the inertia-forces may generally be neglected in comparison to the frictional forcesànd 'the buoy,ancy forces. Therefore, in order to ensure similarity, only(12) has to, be satisfied. V not being independently given, the heat transfer

, is a function of a' combination of (12) and (15) that does not contain V.S~ch a combination being

we have:

g e fJ el2 • e cp Vl __Gr .PI'TJV À. '

Nu = fa(Gr . Pr) . (22)

Thus neglecting inertia. forces in the case of natural convection, Nu is afunction of one variable only, viz. Gr' Pr. "-

It is necessary now to co'nsider the following complications. Until nowthe quantities fJ, 'Y}, À. and cp were treated as constants. In realitythey dependmore or less on the temperature and hence assume different values at

, different points Of the température field. Iri. addition e was consideredconstant, whereas in the' problem 'of natural convection the dependence ofe on the temperature is essential. If fJ does not depend on the temperature,the density'fields are similar, and in this case it is' immaterial ~vhich densityis suhstituted in (ll), provided the values arc taken at correspondingpoints. In the case of gases fJ is independent of the temperature e and evenequal for different gases (fJ . liT o:J)' whereas the temperature depend-ence of 'Y}, À. arid cp is almost' the same for different gases. However, when

/

/

DISSIPATION OF HEAT BY FREE CONYECTION .351

... ~ '" '. ' .comparing a gas with a liquid, this no longer holds, 'Y} and A increasing withincreasing temperature in the case of a gas,_whereas in the case of liquidsthe reverse is true. '

r:I.'hisis the.reason why (22) must not be used when comparing gases andliquids. Therefore it is .not permissible to apply the data with a liquidmodel to the' problem of cooling by a gaseous medium.

Nusselt proposed to use mean values em, 'Y}m, etc., for the tempenature-dependent quantities occurring in the 'characteristic numbers ör, if thissliould result in too large a scattering of measuring points, to in,trod~ce a"new parameter (ew/T o::J) (cf. Hermann 3)). Eq. (22) is then to be replaced by

Nu = f;(Gr . Pr, ew/T ee ) , (23)" ,

where some arrangement still has to he made as to the temperature (be itTw or T ec or an intermediate temperture ) at which the values of (!> 'Y}, •

etc., occurring in Gr, Pr and Nu shall be taken. Although (23) is moreaccurate than (22), it is generally permissible to use (22), provided e, 'Y},

etc., are taken at adequately chosen tempertures.

6. ':fhe dissipation of heat hy hodies characterized hy one linear dimeneiononly _

We shall begin by considering theform of-the function jj (eq. (22)) for anumber of bodies having simple shapes that may be characterized by. one linear dimension only. These considerations will be based partlyonresults of measurements, partlyon theoretical foundations as laid down 'in the formulae (4) to (9). Calculations are only possible in the simplestcases 'and with the aid of simplifying assumptions. Tt 'is always supposedthat the outer boundaries of the medium have no influence on the problem.

The quantities Nu, Gr, and Pr will be p~ovided with two suffixes, the- first indicating the characteristic d.imension chosen (h = height, d =

diameter, etc.) and the second whether À, e, etc., are taken at the surfacetemperat~e (suffix w) or whether' an intermediate value (suffix m) isused: For instance, NuJ,m means a dJAm' where

(Jw

Àm=: f À(fJ) dO,.CIte

(J",

(24}

'In Gr we always substitute fJw for e.Nu, G,rand Pr being dimensionless,it is immaterial which units are used," provided they are consistent *).

*)' In the figures showing the coefficienta, this quantity is always expressed iIll'watts/~2, per degree. The heat transfer in kcal/m'' per degree per hour is obtained from this bymultiplication by 0'86. '

"

• I

Î

, .

352 W. ELENBAAS

a) The flat. vertical plate

The heat transfer by Hat vé'rtical plates in air was measured hy Schmidtand. Beckmann 4) and King 6), that in oil by Lorentz 6): Plotting NUh,~ ~sa function of (Gr· Pr )h,w (h = height of the plate) on a doubly logarith-mic scale, the measuring points are- approximately on a straight line withan inclination of t;. The proportionality factor lies between 0'52 and0'65. The experimental results are thus' approximately represented by

NUh,w = 0'6 (Gr· Pr)~:w' (25)

This formula is valid for a limited range only, namely for 104 < Gr' Pr< 109• This means that in air of 1 atm and ew = 10 and 100 oe, the corre-sponding limits for h are It < h < 70 cm and! < h < 35 cm, respec-. tively. When using a height equal to twice the upper Iimit, the error madein using (25) is small, so that formula (25) may be said' to cover animportant range of practical applications. .

A formula of the form (25) mayalso be obtained by calculation. For avertical plate in ,air this calculation was carried out bySchmidt, Pohlhausenand Beckmann 4) making use of the following simplifying suppositions:(1) "the How is laminar;(2) where the température difference and' the velocity are appreciable,the thickness of the layer is small with respect to the height;

\ (3) the dimension of the plate in horizontal direction (y-direction infig. 2) is so large that edge effects may be neglected, all derivatives with, respect to' y being zero. .

Under these suppositions it follows from (7-), (8) and (9) that e/ew is afunction Of ' .

x (ge2wfJew)"'.'YJw2Z .-. . '

,-x ~gi~f!8w)~4'1:1 z; w

Fig. 2. 'I'emperature field in front of a warm vertical plate. as calculated with formulae(7) to (9). At given circumstances ejew is only a function of x/i'" Equi-temperature .planes thus bend away from the plate with increasing height. (Taken from Schmidt andBeckmann ').) ,

,,

' ....- ,.

DISSIPATION OF HEAT BY FREE CONVECTION 35~

In fig. 2 this function is represented. It appears that at a fixed distance, x from the surface of the plate, the temperature increases with Increasing z.According to (1) , .

, • ' , _ (g(J2wpew)'"d{lj= - Aw(öejöx)w'dF = C Aw ew 'YJw

2Z. dz, (26)

where dqi is the energy;,tr.a,:r,:!~fe:.;r,~dper ,èm width between zand z + 'dz,and C is a constant. The 'heat loskes 'per unit of surface area thus appear

_- .... ,",,'"'l,_ "

to decrease with increasing z. This is a consequence ofthe fact that theair,when moving upwards, is 'warmed already by lower zones of the plate;which results in a decrease' of - (öejöx)w with increasing z. Accordingto eq. (26) the energy transfer is infinite for z = O.This, however, is 'of noconsequence, the total energy transfer being formed by integration, withthe result: \

(h3g(J2 pe ),/. '/

{Ij= Cl Aw e~. 'YJ:: w = Cl Aw e; Grh:tv -:-- (27)

As, by definition, qi = a hew, it follows fróm (27) that , iI.:» -t.NUh,w =- = Cl Grh,w •

. . Aw

For air Schmidt and Beckmann found the value 0'48 for Cl' .Extcnding,eq. (28) to other media in accordance with eq. (22), one obtains'

'/ . "NUh,w = 0'525 (Gr· Pr)h;w, (29)

, (28)

ew=fOO°

(Jw=30°

8w=.1rr% 20 40 60 80

-h(em)

_'

Fig. 3,Heat dissipation in watts per m2 and per degree of temperature difference of verticalplates in air of 1 atm and room temperature for different excess temperatures calculatedwith (25). The dimension of the plate in a horizontal direction is supposed to be largecompared .to the height.

354 W. ELENBAAS ,The factor 0'525 arises from division of 0'48 by (0'73)"', 0'73 being the

value of Pr for air. 'The calculations of Schmidt, Pohlhausen and Beck-mann are thus in fair agreement with the experimental result (eq. (25)).

I. The limited validity of (29) is caused by the fact that outside .the regionof 'values indicated above the fust and the second suppositions are no

I longer fulfill~d. At the upper limit the flow ceases to he laminar and he-,comes turbulent, whereas at the lower limit the second supposition does' nolonger hold. If the third supposition is not fulfilled, the çooling is increasedby the edge effects. In the experiments this may he avoided by applyingtransverse plates to the edges or by making the dimensions in the y-direc-tion very large.. In fig. 3 the heat transfer a by vertical plates in air at 1 atm and 20 oeis plotted for a number of ew-values as a function of h; according to (25).

b ) Th~ horizontal cylinder

The heat dissipation of horizontal cylinders in diatomic gases has been. "investigated experimentally several times. In these experiments the lengthof the cylinder is so chosen that edge effects may be neglected, the transferbeing proportional to the length. Nusselt was the fust to arrange theexperimental results by plotting NUd,m vs Gra,m' Afterwards Hermann 7)classified the whole material taking into account the dependence onew/T ee (cf. eq. (23)).

In fig. 4, being valid for diatomic gases, the most probable course of the

1.5·r-----r-----.-----,------.-----r~--_,.'T~f.q-----T-----r---_,-----+-- __~----~

-O~T---~~i-----~Ok-----~2~----47-----~e~--~8---- log 6rd,w

Fig. 4-. Log NUd,IDas a function of log Grd,w for two values of ew/T", for horizontalcylinders in air, after Hermann 7); -- experimental, - - - calculated with (31). .

"

DISSIPATION OF"HEAT BY FREE CONVECTION\

355

function is shown for two values of 8~/T 00 in terms of Grd,w and NUd,w.It is seen that for Grd,w> 104 the values of NUd,w are practically inde-pendent of 8w/T 00 ~ In this region the results may be summarized in aformula that does not contain ew/T 00' viz"

(30)

The suffix to mayalso be replaced by m,'

Also in this case a calculation of the heat transfer has proved to bepossible. For an infinite1y long horizontal cylinder Hermann 7) computed,in the case of diatomic gases (in a ma~lller analogous to that of Schmidt,Pohlhausen and Beckmann in the case of a vertical plate):

NUdw = 0'37 GrJ'w..' , (31)

In deriving (31), the same simplifying assumptions were made as wereused in the derivations of (26) to (29). Therefore eq. (31) is also valid in alimited region only (namely for 104 < Grd,w < 5'108), which forms partof the region where, according to fig. 4;NUd,wdoes not depend on 8w/T 00

and where the empirical formula (30) holds.Whereas in the case of the vertical plate the existence of a lower limit

to the validity of (29) does not imply a serious restrietion on most of thepractical applications, it does in the case of a cylinder, the cooling-of acylinder with a diameter <1 cm in air of atmospheric pressure not beingadequately described by (30) and (31). Horizontal 'wires are quite beyondthe region of validity of these equations.

Langmuir 8), who carried out measurements on wires having diametersfrom 40 (1- to 510 (1-, was able to interpret his results by assuming the wireto be surrounded by 'a thin layer of finite thickness through which theheat is solely transferred by conduction, whereas 'outside this layer thegas has the temperature Too' The thickness of the layer was assumed to be 'determined hy the condition that the temperature gradient at the boundary(the angle 15 in fig. 5) has a value independent of the diameter of the wire.Combining .Langmuirs results with those of Nusselt, one obtains a for-

mula with a much larger range of applicability 9), namely:

I'

"

Nu3 d,m e-6/Nud,m = Grd,m/500. ' (32)

The factor 1/500 was found by experiment; it is Y~li~ in the case ofdiatomic gases. ,

For large values of Grd,m, NUd,m is large 'too; hence the exponential

function approaches 1,NUd,mb~ing proportional to Gr;J:m in this region. "-The diameter d occurring in the first power in NUd,m (= ad/Àm) and in thethird power in Grd,m (= d3g rfm{3 8w/'YJ2m), we find that for large valuesof Grd,m, a is independent of the diameter. .

., .

I .r:

'.

. ,.

\..'

356 'Wo "EL!>NBÁAS

• A ...

F~r small values of Grd,m, owing to the large i~fluence ofthe exponentialfunction, NUd,mdecreases slowly with decreasing Grd,m' In this region, 'therefore, a dl À~ almost has a constant value. As a means the heat transferper cm2 and as the circumference of the cylinder is proportional to d, theto~al heat transfer per cm of length is proportional to ad. For small valuesof Grd,m the dissipation of heat per cm of length, as it occurs in the case

Fig. 5. Radial température course i~ the medium surrounding a wire. According to thesupposition of Langmuir the temperature gradient at the boundary of the film in whichonly pure conduction plays a part (the angle ij) doesnot depend on the diameter ofthe wire.

of thin wires,.is thus almost independent of the diameter. Langmuir'sinvention of coiled filaments in gasfilled incandescent lamps is an applic-ation of this property. The effective length of a coil being much smallerthan that of a straight filament, the energy loss by convection is much less

, in the former case. Although (32) fairly checks the experimental results,there are deviations to an amount of about 20 per cent. Moreover (32) onlyh~lds for diatomic gases. A slightly more complicated formula, which is atthe same time valid for different media, runs 10):

. : N 3 -6/Nud m' (Gr· Pr)d,wu dw e • =

, . 235i(Gr· Pr)d,w/"

(33)

The function i (Gr· Pr)d,w occurring in (33) has, been determinedexperimentally. Fig. 6 is.a graphical representation of it. For (Gr·Pr)d,w< 104, i.is a constant (_:__1). With inçreasing value of (Gr· Pr)d,w, iincreases. ThIs makes the increase of Nu less than it would be ifiwereconstant. We may interpret the occurrence of the function iby imaginingthat in the case of a relatively thin layer (large value of d) the medium in.' \the layer is appreciably heated on flowing past the cylinder. As a result,r

"

DISSIPATION OF HEAT BY FREE CONVECTION 357 ,

the value of a iri the higher parts of the cylinder is smaller th~n that in the'lower parts, ,resulting in a smaller .value of a (this effect is to he comparedwith that occurring in the case of the vertical plate (eq. (26)). As, withathin layer, Hermann's theory is applicable, one should expect the influenceoff to be discernable likewise for Gr> 104; this is confirmed by experiment, .as may be seen from fig. 6. .

f{Gr.Pr)

+I

3.0d,w

, ,V..,~

V-. /2.

./V

'tOJ

V/'

~VV i.-----t05 104 - 2 5 K? 2. 5 ta5 2 5

"

- (Br.Pr )d,w 'Fig. 6. The function! (Gr· Pr)d,IOoccurring in the formulae (33) and -(34),.as determinedexperimenta~y. For (Gr' Pr)d,w < 104, t:= 1.

- ..In (33) NUd,w'as' well as NUd,m occurs. This makes the parameter

ew/T r::IJ superfluous, In order to determine a, Grd,w being known, onemay write NUd,w as NUd,m (A.m/Àw). Bringing the factor (io.m/Àw)3 onthe right side of (33), NU3d,mexp (-;-:-6/Nud,m) is known, hence NUd,mand a may be calculated. The value of Àm may be computed from (24).In the. table 'at the end of the paper Àm is given for air and water. In' fig. 7NUd,w is plo~ted against (Gr'Pr)d,w, according to the measurements (seefig. 4) and according to eq. (33). The agreement is' good. The theoretical

, curves meet: at large values' of (Gr· Pr)d,w, the influence of the function-exp (- 6/Nud,m) being negligible for large values of NUd,m' The dependenceon ew/T eo is thus rightly described by (33). The agreement for (Gr· Pr)d,w.> 104 is trivial, because the functionfin fig. 6 is chosen in such a way that(33.) describes the experiments.,Fig. 8 shows the dissipation o~ heat of horizontal cylinders in air (a~

"..

, -

358 W. ELENBAAS

~;r-----.------r--~-.~.-----r-----.----~logNud.w

t '

. ---tog(Gr.Pr)d,w

Fig, 7. -- log NUd,.oas a function of log (Gr· Pr )d,IV for horizontal cylinders according .to (33). - - - The experimental curves of fig. 4 now plotted as a function of (Gr· Pr)d,1O"

00)'..

t'~ 1-.1'-

~l')'. ~l'

~ ~?};.,=20"c

l'. tatm,~

<,ew 1000 -

50<," <, 500

~ <, V 300200

~ ~ V tOO,

~ ~~ 30

2 ~ " ,.---tOI "~

~~ ~ s::: ::::r-to :::::-:

"" - --5

......

,, ,

,

, .

d.005 001 002 005 2 5-deem)

toat 02 as

Fig. 8. The heat dissipation in watts per ma and per degree of temperature difference for .horizontal cylinders in air of 1 atm and room temperature as a function of the diameter,for different excess temperatures, calculated with (33). The length of the cylinders is large.as compared to the diameter. .

DISSIPATION OF HEAT BY FREE 'CONVECTION 359. '

- 20° C) as a function of d, calculated with the aid of (33), use having beenmade of the curve of fig. 6.

c) The sphere

In a way analogous to the derivation of (33) one finds for the heat dissi-pation of a sphere 10):, - .

(Gr· Pr)d,w 'NU3d,w(1-2JNud,m)6 = 235f(Gr·Pr)d,w· .

For' small values of (Gr· Pr)d,w, NUd,m according to (34) apprcacheethe value 2. This means that the situation approaóhes that of pure conduc-tion, the thickness .of the Langmuir layer becoming large as comparedto the' radius. In this case ','le may compute the heat dissipation (/J asfollows: Through each concentrio sphere with a radi~s R > id the heat (/Jflows through the spherical section 4n R2. Thus we have:

,(34)

- <P = -4 n R2 A(#) dWdR (35)

with the boundary conditions e . ew for R = id ~nd e = 0 for 11= 00.

From (35), we obtain:<P dR

A(#) d# = - --,, 4n R2, ..and integrating:

(36)

As, according to (24)~

we obtain from (36):<P = 2;"; d e; Am.

As by definition of a we have <P.= nd2aew, we obtain adJÀm= 2. 'It is well .known that in the case of the heat-dissipating cylinder this

reasoning leads to a constant temperature field all around the cylinderwith zero dissipation.

A comparison of (33) and (34) shows that, for small values of (Gr' Pr)d,w 'the heat transfer per cm2 of a sphere exceeds that of a horizontal cylinderof the same diameter, if the conditions are the same, whereas the twovalues 'meet at large values of (Gr· Pr)d,w.,

Actually in the case of a small sphere the' experiment gives ~ heattransfer superior to that of a cylinder of the same diameter at the sameexcess temperature ew. In fig, 9.we show the heat transfer of spheres in ,air (1 atm, 20°C) according to (34), making use of fig. 6.

, ,

. ,

I

360 W. ÈLENBAÁS

30

~

1''0~

............. r---~

:----...r--,~t-I-t-

~ F=::::: ;:::: I'-- I'-- ew=10~ t:-t-10 t-.... r---..:: t-.. 51<, ......... t-... t-... t-... 3...... t.... r--... t-..<, t-.... t-.....

~............. i"'- r-,

" t-. .5 r--- 1'--..

- 7l,=20°C1atm

2 -,

,1 '

000000

20000

3010.'

. ,

1 2 105-deem)

IFig. 9. The heat dissipation in watts per m2 and per degree of temperature difference for .spheres in air of 1 atm and room temperature as a function of the diameter, for differentexcess temperatures, calculated with (34.).

"1,_

(to be continued}

Eindhoven, April 1948

REFERENCES'

1) L. Prandtl & O. Tiet jens, Hydro und Aeromechanik, Bd Ir, p. 60; Handbuchexpo Phys. lVI p. 47-71; H. Lamb. Hydrodynamics, Cambridge.

2). W. Nusselt, Gesundheits Ing. 38, 477, 1915.r 3) R. Her manu, Phys. Z. 33, 4.25, 1932. ,4) E. Schmidt & W. Beckmann, Tech. Mech. Thcrmodyn. 1, 34.1, 391, 1930 .

• ó) W. J. King, Mech. 'Etng. 54, 347, 1932.6) H. A.' Lorentz, Z. tech. Phys. 15,.362, 1934.7) R. Hermann, Forschungsheft 379; Beilage Forseh. au( dem Geb.,des Ingw. 1936.S) I. Langmuir, Phys. Rev. 34; 40, 1912.9) W. Elenbaa~, Physica 6, 380, 1939.10) W. Elenbaas," Physica 9, 665, 194.2.