Nusselt-fundamental Law Of Heat Transfer.pdf

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Fundamental Law of Heat Transfer [Natural and Forced Convection] Nusselt, W.

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Publisher’s version / Version de l'éditeur: https://doi.org/10.4224/20331615

Technical Translation (National Research Council of Canada), 1957

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PREFACE

The study of heat transfer through fluids to the enclosing walls of heated structures forms an important part of the work at the Division of Building Research,

In practice, temperatures, pressures and fluid velocities are not uniform over surfaces of heated structures and steady state conditions are seldom fulfilled, The effects of these variables are often neglected in calculations of heat flow, Simplified mean coefficients of fluid to surface and surface to fluid heat transfer have been developed which permit rapid estimates of heat flow, In more accurate work, however, reference to original theoretical work is necessary to obtain an indication of the permissible extent of simplifications, This paper by Dr. W, Nusselt presents the fundamentals of heat transfer by natural and forced convection. Both steady state and transient conditions are considered and the theory of dimensional analysis and models is clearly elucidated, This translation has therefore been prepared as a contribution from the Division, not only for its own studies of this matter, but also for the general information of others who are similarly interested. It is a pleasure to record that the translation has been prepared by two research officers of the Division's staff, Messrs. C, Wachmann and W. G. Brown. Their translation was m d e with the approval of the author. Mr. D.A. Sinclalr of the Translations Section of the National Research Council kindly checked the translation and rendered the translators valued assistance for which appreciation is here recorded.

Ottawa, J ~ Y , 1957,

R.F. Legget, Director.

NATIONAL RESEARCH COUNCIL OF CANADA Techniaal Translation TT-681

Title:

The fundamental law of heat transfer [~aturaland forced convection]

a as Grundgesetz des W&rme&erganges) Author :

W. Nusselt

Reference:

Geeundheits-Iwenieur, 38 (42): 477-482 and (43): 490496, 1915

Translators:

C. Wachmann and W.G. Brown, Division of Building Research, National Research Council

Translated with permission

T I E FUNDn_MENTAL LAW

OF HEAT TRANSFER

[NATURALAM) FORCED CONVECTION1 Heat exchange between a solid body and a surrounding liquid or gas has been determined by experiment for a nurnber of technically important cases. In a previous paper(' ), using similarity considerations of the differential equations, I established relationships which represent this process for the cooling of a gas stream flowing thpough a pipe and which the test results must satisfy. These theoretical results have been confirmed very well by my experiments.

In what follows I shall establish similar relationships for the case of a solid body at rest in st911 air and prove them by reference to available experiments. Suppose a large room is filled with a liquid or an elastic fluid. To begin with this has ever.ywhere zero speed and uniform temperature To0 absolute. Into this room is brought a hot solid , which is maintained by heat body having a uniform temperature T transfer from the centre. The heat emission from the body takes place in two different ways. One part is given off to the surroundings by radiation. If the surrounding fluid is a liquid then the layers next to the body surface absorb the entire radiated heat and are so warmed. But these quantities of heat are negligible compared with the amount of heat conducted away by the liquid. If the surrounding fluid is a gas, it is permeable to heat radiatiofi (except for a narrow absorption band deThe heat transfer to the enclosing pending on the kind of gas). walls by radiation then takes place without affecting the gas on the containing walls. It is assumed that these walls are large enough so that no noticeable change in their temperature To occurs. The laws of radiation will be discusspd in a later paper. The

-3following lines contain a formulation of heat flow by conduction. If h be the thermal conductivity of the surrounding medium, and T its temperature at a point with coordinates x, y and z, then the heat emitted by conduction from a surface element df of the body in time dt is

In this equation n is a coordinate of the normal to the surface element df,

The positive direction is towards the surroundings,

The total heat given off by the body in the time dt is obtained by integration of the right-hand side of equation (I) over the surface of the body, In the literature it is frequently asserted that the heat emission from a body has three causes( * ) radiation, conduction and convection, It is said that as a result of an upward movement of the heated air or an artificial air stream, cold particles are continuously being brought into contact with the surface of the body and removing heat from it, From this it would seem as though conduction and convection were two independent phenomena and that heat could be transferred by convection without the aid of conduction. This is not so, In actual fact the argument of equation (I) holds for both conduction and convection. If the thermal conductivity of the surrounding medium is zero, then no heat can be transferred either by conduction or by artificial convection. On the other hand, if the surrounding medium possesses a finite thermal conductivity then, if we disregard the radiation, all the heat is given off by conduction, But the quantity of heat removed is increased by flow of the medium, since through it the factor a T i.e., the temperature drop at the surface of the body, is increased If the flow of the surroundings is produced by buoyancy, this can be called natural flow or natural convection, If the movement of the air is produced by an external force one speaks of artificial convection, If there is no flow, then heat

x,

emission can be said to take place by pure conduction. Consequently, if we disregard radiation, the following four cases of heat transfer are possible: 1. 2.

3.

-

Heat emission by pure conduction; It 'I natural convection; l1 tt forced convection; II It

4.

18

tt

natural and forced convection.

In order to calculate the heat emission of the body by equation (A),the temperature drop at each point, on its surface must be known. This can be obtained by integration of the determining differential equations. Let x, y and z be the three axes of a right-angled coordinate system whose negative z axis is in line with the direction of gravity, Further let ,

q h P

c T

P t

u V W

be the viscosity, thermal conductivity, density (mass of unit volume), specific heat of unit mass, tt temperature, static pressure, l8 time, velocity components in the X-direction, tl 11 11 Y-direction, 11 11 tt tt 2-direction,

To begin with consider the cooling of a body through natural convection by its surrounding liquid or elastic fluid under a nuniber of simplifying assumptioiis. The thermal conductivity and viscosity of a fluid depend on the temperature; at first let them be assumed constant. Through the change of density p with temperature the hot particles acquire buoyancy. By approxi-

mation, this buoyancy which the fluid particles experience can be used instead of their weight as the inertia force in the direction of the z axis. In this theorem the potential energy is neglected so that pressures at large distances from the body all become equal, in other words independent of z. Let the pressure p be measured with this pressure as basis. Let the temperature of the surroundings be To snd the tenperature difference at a point be 8, so that T = To + O. At the surface of the body 43 = 0,. If the coefficient of expansion of the fluid. at a point is r, then the buoyancy thera mill be

where g is gravitational acceleration. Under these assumptions the equations of motion are as follows: f

The velocity components must also satisfy the continuity equation

When the first main theorem is applied to the energy contained in unit space we obtain the following equation:

=

a20

-;r (ax

+

a20

'7

ay

+

2;)

- 6 In this equation friction heat and the pressure gradient have been neglected, These five equations are sufficient to determine the five values, 0 , u, v, w and p, dependent on the coordinates, In addition the solution must satisfy the Following boundary conditions: At large distances from the body p nmst be zero.

a,

u, v, w and

On the surface of the body u, v, w = 0 and O = @

0

Although it is not possible to integrate these equations even for the simplest cases, it is still possible to obtain results of practical value on the basis of similarity considerations, In this way, conditions are obtained under which similar cooling takes place, If the emitted heat is known for one system it then becomes possible to calculate the heat emitted for any similar system. Consider two systems, I and 11, Let the values of the first system be denoted by the index I and those of the second by the index 2, The two cooling bodies must be geometrically similar, Let their linear dimensions bear the ratio of a of one to another, If dl and d2 are similarly located distances on them, then

To obtain similar flow, the coordinates of similarly located points must also bear the ratio a, i,e,, x2 = ax,, y2 = ay, , z, = az,, Let similar times be in the ratio p so that t, = St,. Let the ratio of the pressure at similarly located points be 6, i.e,,

The velocities at similarly located points are in the ratio

In addition let the constants of the defferential equations and the boundary conditions have the following fixed relationships*.

The question now is to decide whether it is possible for two systems I and I1 to be geometrically, mechanically and thermally similar, i,e,, to have constant values for the ratio values** a, y t 6, E , g, k t #, U, and x. If the answer to this question is affirmative, it then becomes necessary to decide whether the values can be chosen arbitrarily, The five differential equations (3), (4) and (5), of course, hold for both system, It is only necessary to add the indices I and 2, For instance, the third equation of (3) for system I becomes

and for system I1

Translators1 notes: + The relationship p2 = hp, has been apparently overlooked, ** p overlooked,

On the basis of the above relationship we now substitute the t b values with index 2 for those with index I in equation (3b), Then,

Correspondingly the four remaining differential equations of system I1 can be restated in terms of system I. Then equation (3c) oontaina only variables with suffix I. If the above similarity argument is possible, then equation (3c) r w t be identical to equation (3a). This will be so only if the following relationships are found for the ratio valuest

It is then possible to transform the remaining two equations of motion for system I1 into those for system I, and similarly with the continuity equation, In order to transform the heat movement ,equation, the following condition must be eatisfled:

Equations (6) and (7) show that among the 109 ratio values a,,.,~ the following relationships must hold:

It is seen from the above that similar flow is possible, but because of the five relationships 8 to 12 only five ratio values can be chosen arbitrarily. To ensure similarity all five equations

* ~ranslators'note: This should be I I.

I

8 to 12 rnust be satisfied.

eel.

The ratio The boundary conditions require only Q0, = value p of the velocities can be chosen arbitrarily since the velocity does not appear in the boundary conditions. To begin with we only want to observe the state of inertia and the emitted heat during a long period of time, hence the vibration period of periodic fluctuations of flow, which may ariae through vortex detachment, appears negligibly small, Consequently the heat emitted is independent of time, p., g, and 6 can therefore be chosen to correspond to equations ( 8 ) , (9) and (10). The other ratio values a, y, E, tj, 5, v,$, and'%depend on the material constants and boundary conditione. Among them the conditional equations (11) and (12) must hold, If we introduce into both equations the atate values of syetema I and I1 w e obtain

and

If these equations are satisfied, then heat emission takes place in a similar fashion. It is of course also possible to have several systems for comparison which must all have the same values for ratios definable by equations (lla) and (12a). In this way the most general conditions are obtained which give

and

Hence for comparable systems equations (1 lb) and (1 2b) must have constant values. The numbers B and C can of course take any

desirable positive value, For each corresponding pair of values a nuniber of similar cases are obtained. If, for instance, for one system the heat Q, emitted by a body in unit time is known, e,g, from an experiment, then it is possible to calculate from Qi the heat emitted for a similar system in the following way, According to (1 )

a@,

dQ, = -I1 r

el,

(13)

is the heat emitted by a surface element of body I in the unit time, For the same time, the heat emitted by a similarly located part of body II l a

aoz

dQ2 = hZdf2

an, '

('4)

If in this equation we put

J

w e obtain

dQ, =

-E

a@1

a # h1df,3q

a

The relationship of equations (16) and (13) gives

Now

and hence

By integration over the entire surface of the body the heat emission from the second body is

To make this equation valid equations (lia) and (12a) M U S ~ be satisfied, The terms with common indices in equation (20) may now be brought to the same aide

If

w e consider a nuniber of systems it follows that provided

they are chosen so that B and 0 have the same values in equatione (lib) and (l~b), then

must also remain constant, In addition, if the magnitudes of d, p, g, r, Ow, q, h , and c are altered in such a way that B and C always yield the same constant values, then D also remains constant, and the emitted heat becomes

Thus, if the values of B, C and D from one experiment are known it is possible to calculate the heat emission Q for an infinite nurriber of systems. Suppose, for example, that the heat emission of a body in the atmosphere under given conditions has been measured as Q, = 120 kg,cal. per hour, Let the temperature difference be 1O0C, and the size of the body, d = 0.2 m, What does the above relationship tell with respect to cooling under other conditions? If we consider cooling in tb same atmosphere, the valuesh, c, q, p, g a n d r inequations (ilb), (12b) and (20~) are constant, Equation (la) is thereby immediately satisfied. For all possible systems equation (llb) must also be fulfilled,

The term GW d3 must be constant and equal to 10 Therefore by equation (20)

If for example d, is chosen 0 . 1 then, ,@

x

0.2 = 0.08.

becomes 80 and henoe

Further, B and C can be chosen arbitrarily, For each matched pair of values B and C a value D is obtained, Consequently D is a function of B and C

Therefore, the h e ~ temission of a body is

This is the most general expression which can be set up to express the heat emission of a body immersed in a fluid under the foregoing a~sumptions. It shows that the heat loss per h o w depends on: size of the body, temperature excess above the surroundings, thermal conductivity of the fluid, viscosity of the fluid, density of the fluid, specific heat of the fluid, coefficient of expansion of the fluld, acceleration due to gravity, This dependence is not an arbitrary one for all independent variables, by means of equation (21 ) the nuniber of independents is reduced to two, Experimental determination of function is therefore substantially simplified, It is only necessary to carry out two series of tests in which the factors contained in fractions B and C are varied one at a time, for example, h and eW

or c and d, At the same time we immediately obtain the dependence of the heat Q on those factors, which had not been altered during the tests, In practice it has been found appropriate to introduce into heat emission calculations the so-called coefficient of heat tranemission, By this is understood a quantity of heat a which is given up by the unit surface per h o w and per l otemperature difference, In this way the heat emitted per unit time is

S i n ~ eF is proportional to d2, by equation (22),

Let it be assumed for the sake of argument that the cooling fluid is a gas, e,g, atmospheric air, then r can be calculated from the gas law. If po is the remote density at temperatur@ To, then

From this the buoyancy is obtained,

and the coef'fici'ent of expansion is 1

P

= To +

@

.

In the derivation of equation (24) constant r was assumed, With gases, this occurs only for small temperature differences;

here we get

For c with gases the specific heat at constant pressure cP is to be substituted. By further substituting for density P the specific weight y , the coefficient of heat transmission for the cooling of a body in a stationary gas with a small temperature difference becomes

Now in accordance with experiments on the kinetic theory of gases

A =

E

CV T),

(29)

where e is constant for a particular gas. In a previous paper(1) I have demonstrated that c depends only on the atomic nunher as follows: Atomic n w e r I 2 3 4 5 6 t3 2.50 1.74 1.57 1.23 1.28 1.24 e,;

In expression (29) it is now possible to substitute cP for

according to an expression of heat theory

where x also depends only on the atomic number as follows: I 2 3 4 5 6 Atomic number x 1.66 1.40 1.27 1.28 1.28 1.25 We now substitute the value from equation (2%) in (29) and get

A =

& - 1

x

P 'T),

C

In other words the value C depends only on the atomic nuniber of the gas and varies between 1,5 and 1, If we extend uonsiderations only to gases of similar atomic nunibere, e,g, diatomio gases, hyebogen, oxygen, nitrogen, air, stc, then C remains constant and the coefficient of heat transmission depends only on B. The formula is then simplified to,

(30) If factor

$ is transferred to the left

side and fraction

ad is designated as A, h then In other words the problem of the cooling of a solid body of given shape in a stationary volume of diatomic gagl for small temperature differenaes can b e reduced to the determination of a function of a single variable, It is therefore only necessary to know the dependence of the heat transfer on one of the five factors of which B is made up, The dependence on the other factors followe. If, for instance, the linear dimensions of a body are altered and the heat transfer process is measured in air at the same temperature difference, and if, for instance, a is independent of the dimension^, lee,, of d, it follows that:

Herein, however, C is a constant to be determined by experiment and yo is the speuific weight of the gas at the normal oon-

dition of 1 5° and I atmosphere, Or if it is found on examining a body at different temperatures that a is proportional to @&, it follows from (30) = (.' 1

'12 To

)"' .

,

.,

4 '0; .y02-. pa. 2882 CIA ~ ~ ' g . , l z . ~TOY . "

~

. .

.

. -

' .

It is unfortunate that there are no series of experiments available in which temperature differences had been chosen small enough that expression (30) would be satisfactory. In actual fact the temperature difference is always appreciable, Under this condition the density, the conductivity, the viscoeity and the uoefficient of expansion depend on the temperature. These are quantities which had been assumed constant in the derivation of expression (30). These restricting assumptions will now be dropped, On the basis of the following consideration it I s possible to render expression (30) approximately valid for this general case as well, It is certain that the temperature difference O at small distances from the surface of the body has zero value, where coneguently the temperature of the surroundings is constant, The temperature drop is then restricted to a thin layer adjacentto tkebody, and it would appear approximately correct to substitute in the differential equation the mean values of temperature dependent variables, obtained by integration in th temperature range Ow. We obtain

If To is the temperature of the surroundings and T, the wall temperature, the according to our definition we get

furthermore

and

where yo 16 the specific weight of the gas at surrounding temperature To. It is now possible to obtain for this case an accurate expression from the differential equation. Because of the temperature dependence of p, h , q, cp equations ( 3 ) , (4) and (5) now expand to

bT bw bybs

bT bw bzby

b T b u- +.-h y b z

A t great distance from the body let 1

At the surface let

Now from the gas law, *o

P = Po*T9 and

if the small variation of pressure is neglected. The variation of q, h and c with temperature shall now be represented by an P exponential relationship which is possible within broad limits, as will be shown later by example. Thus

If we again consider two similar system, it follows:

i . e , similarity can exist only if the ratio of absolute surface and space temperatures remains constant, From the differential equation itself follows the constancy of the relationships

and

Hence A becomes a functior, of the three variables B,C, thus

a (B,c,D)

A =

and D,

(42)

Thus equation (43) gives the relationship for the cooling of a hot body in a gas at any temperature difference. It is immaterial whether the values of h, p, q and cp are substituted at T, or To, since each of these values can be transferred from the value at one of two positions to the other position by multiplication by a power of Since A depends on the ratio, D = 'n &o Tw then equation (43) is satisfied for all these cases. , To

5.

-

It viill now be proved that the above equation of approximation (28a) satisfies the conditions of equation (43). To begin with it is necessary to use the relationships given by equation (40) in order to calculate the mean values according to equation (33). This becomes

A,,,

= lo( t t t

+ I)

=Ao

Tw

To- -

c,

nri-1-

= c g 0 ( r + 1) --

T,=

and I*,,, =

1 T o . I)--In B -

,,-n,

ib-1"

I) - 1

1.

( t n f-

1)

nm+1-.

2

1--

If these values are now substituted into equation (28a), this becomes

This equation indicates that A is dependent on B, C and D, which was to be proved. Of course equation (28b) represents only one special case of the much more general relationship (43). When equation (28b) is satisfied by experiments, it loses as a result of this proof, the approximation character inherent i=1 its derivation. Equation (28a) shall now be tested in an example, A nmiber of experiments are avail~iblefor a circular cylinder suspended horizontally in stationary air, Confining ourselves first of all to the establishment of an equation for heat transfer to a diatomic gas, equation (28a) is then simplified because of the constant value of C, Then,

whereby and

For the determination of the function i2 a series of

experiments is sufficient in which B is varied within as broad limits as possible. Now B can be varied in many different ways by jointly or independently altering the factors of which it is composed. If me consider cooling in air it is then possible to alter diameter d of the pipe or the pressure of the air, then the air temperature To and the wall temperature Tw. The greatest change in B is obtained when the diameter d is varied, since it is as the third power, If a series of expe~imentsis now , and hence also carried out in which d is varied while p, To, T a, remain constant, the function is obtained and therewith a solution to our problem. Thus even without performing a series of tests with variable temperature difference, we know the dependence of heat transfer on temperature difference. Experiments are available in which d, p and Tw were varied, From these, a proof of equation (44) and a determination of function @ is possible. Equation (44)presupposes a knowledge of the thermal conductivity and viscosity of air, According to the kinetic gas theory TI is independent of pressure an8 depends only on temperature. This dependence on temperature is influenced by the law of molecular attraction which is introduced, In the ternperature range of -180 to 1200°C. ~utherland's formula agrees well with the experiments, It is:

C and qo are constant for any specific case.

and ?I0

= 1,69 x

lo-'

kg. sec, m.-2

For air

The thermal conductivity can then be calculated from equation (30). It is

where cv is the true specific heat for unit mass at constant pressure, According to Pier the man specific heat of air between 0 and t for I kg, weight is

From this the true specific heat is obtained as

or, if the absolute temperatme T is sribstituted,

substituting the values of equations (49b) and (47) in equation (30) and taking into account the fact that according to the latest experiments( 3 ) the thermal conductivity of air

then the thermal conductivity of air is given by the expression

The values of q and h , as dependent on temperature, are given in Table I and Figures 1 and 2.

In place of equation (47) an exponenti~lformula has often been used with the exponent for air, rn = 0.765. The expression

is then obtained,

Between 200 and 800' absolute, the difference between the two expressions is less than 3%. Table I also contains the integrals of h and 7-l between 0 and To absolute for calculation of the mean values h, and qm.

Four useful series of experiments by Kennelly, Wright and ~~levelt'~);~amsler(~); ~an.gmulr(~); and Bylevelt I'[ are available, The experimental results and the values of A and B calculated from them are given in Tables 11, 111, IV and V. Unfortunately none of the results contain all the values needed for the calculation of A and B. Kennelly gives an a i r ternperature T, = 291" absolute for only one series of experiments. This value was used as a basis for the other series of experiments, Wamsler did not measure the air pressure, Since the experiments were carried out at IXunich the mean prevailing barometric pressure of 715 mm, of mercury at O°C, was presupposed as basis. L w u i r does not give the air pressure and only the temperature of the surroundings, To = 300' absolut,e for one series of experiments. An air temperature of 300° absolute and an air pressure of' 750 mm. are therefore applied to his experiment. The experiments of Bylevelt were carried out in Dresden. For them the mean prevailing a i r pressure of 750 mm, was used as a b a s i s . The material of the wires was as follows: Kennelly Langmuir Bylevelt Vlamsler

-

Copper Platinum Nickel and tantalum Wrought iron gas pipe.

- 25

All the observations are included in Figure 3 with log B plotted on the abscissa and log A on the ordinate. The points can be smoothed out satisfactorily to give the curve shown, The coordinates of this curve are given in Table VI. In this way A is determined as a function of B, lee.,

From the figure or the table it is possible to obtain corresponding values of A from values of B from 1OW5 to In the region of B = lo4 to lo7 it 18 possible to represent this dependence by a power expression. For this we obtain

For values of B less than 1 Oms it appears that A approaches a constant value hence,

In other words the coefficient of heat transmission is inversely proportional to the pipe diameter and proportional to the mean thermal conductivity, For values of B greater than Io7 it appears that quite soon the point is reached where the coefficient of heat transmission becomes independent of the pipe diameter, From this it follows that

The relationship obtained in Table VL may now be illustrated by a few examples, Since the density of the air depends on the

pressure and temperature, the coefficient of heat transmission is dependent on four factors: air pressure, air temperature, surface temperature, pipe d i m e ter.

In the first example let the pipe diameter be varied, Let the air pressure be one atmosphere, The air temperature will be 288' absolute and the temperature difference 10°C,, i. e, the surface temperature is 298' absolute. The calculated values of the coefficient of heat transnlission are given in Table VII, from which the particularly great influence of small diameters can be understood. If equation (51c) is true for very small diameters then the heat emission by a wire is independent of the diameter and air pressure and depends only on the mean thermal conductivity of the air.

,

Let a 1" gas pipe be suspended horizontally in air at 288O and heated to 298" absolute. For this case Table VIII and Figure 5 indicate the influence of air pressure on the coefficient of heat transmission. For pressures over I atmosphere the coefficient of heat transmission is approximtely proportional to the square root of ; in a vacuum the influence of pressure the pressure (a = c is sn~ller.

If the air temperature is varied while keeping the air pressure, diameter, and temperature difference constant, the variation of the coefficient of heat transmission is small, as shown in the example in Table IX and Figure 6. This is calculated for d = 0.025 m, p = l atnq and Ow = l O°C. By comparison, the influence of the temperature is important, The expression shows next that at given temperature difference the

coefficient of heat transmission is the same irrespective of whether the heat flows from the pipe into the air or vice versa, It is possible thus to interchange the wall and air temperatures without altering the heat transmission. Table X and Figure 7 contain the coefficient of heat transmission for different temperature differences for a pipe of 0,025 m, diameter and an air pressure of I atmosphere and air temperature of 288' absolute, An attempt was made to use the general equation (43) infitead of equation (28a) for the evaluation of experiments. According to this equation, if we are satisfied with setting up a formula for diatomic gases, the value A depends on two independent variables B and D. If the edqerinental values were recorded on a rectangular coordinate system they would have to lie in one plane. For the clear determination of this plane the experi~aenta1 data was found to be insufficient and too inaccurate, The relationship already obt~ined

expressed in Figure 3 and Table VI is valid only for a long pipe of circular cross section susper~dedhorizontally in a stationary diatomic gas, e,g, air, The relationship will now be extended to any chosen gas or liquid for which expression (24) is valid vihich states that A is a function of the two independents B and C, Equation (51e) is only one particular case of equztion (24) which results from s'~.stitutionof the value of C for air, This value is calculated from the values of h , q and cP (it should be noted here that cD is the specific heat of unit mas obtained in the technical sys%ern of *its from the specific heat of the unit weight multiplied by the acceleration due to gravity equal to 9.81 ) giving

Equation (24) can be simplified considerably by carrying out a few simplifications in ths differential equation (3). If it is assu;z?ed that thc friction has little effect on the flow process, then the viscosity term on the right side of the flow equation can be neglechcd, carrying through the similarity consideration with these sirqlif'ied equations ve then obtain

in other words a function with one independent variable. This equation is a special case of equation (24), since the independent variable is B This can also be written as

.

The well-known Boussinesq formula('1 is based on this simplification; it is a special case of equation (53a) since it can be

written

Instead of this it also possible to make another simplification in the dynamic equation by neglecting the acceleration term, Then the similarity co~zsidepa'cionyields the formula

which is also a f'unction with one independent variable. Also this equation (55) i s a special case of equation (24) since it can be written

The vrell-known formula by Lorenz(9) is a s p e c i a l case o f this equation, since i t reads

NOIT for heat emission the state of the fluid layer at the

surface of the emitt21:g body i s of fundamental importance and the friction there 113s strollig influence on the fluid flo~v,,€30 that it seems to me the approxifi:zttion forming the basis of equation (55a), v~hicl~ neglects the inertia and viscosity terms in the basic equations gTves a flo'vy t-~hichdiffers but little from actual conditions, In m y opinion, therefore, eqpation (55) i s an approximtion tha* can be very useful in practice. This equation has becn ar~ivedat on the assumptLon that h , T, cp and r are independent of the temperature. The same considerations that lead to equation (&) show that equation (55a) is valid also for real liq~ddswhen mean values obtainea by integration betvieen TJ7 and To are u s e b For air, was fomd,

for diatcmic gases, the equation

(51e)

From equation (55a) it fo1lo:i.s then that for every gas and every liquid,

.

I In this, accordip~to equation (52), Go is 2. 0%

Therefore the coefficient of heat transmission for the coolii7,a or heating of a horizontally situated plpe In a gas or liquid is

In this equation: W = a function shown by Figure 3 and Table VI, QW = temperature difference (surface temperature minus surrounding temperature), d = the diameter of a horizontally positioned pipe, pm = the mean density of the fluid cooling the pipe, % = the mean coefficient of viscosity, h, = the mean thermal conductivity, c = the mean specific heat of unit mass, Pm I;n = the mean coefficient of expansion, g = acceleration due to gravity, As

an example, consider the cooling of a pipe of d = 0,025

diameter in water. Let the water temperature be 15OC. and the pipe snrf ace temperature 2 5 ' ~ . Then,

Since the specific weight of water at 15OC. is y = 999.1 2 kg. and at 25'C. is Y = 997.06 kg. nJ, then the mean value of density is,

The mean specific heat of unit c = 0,9987 P

4

mas

is

9,81 = 9.80 kg. cal, a kg,"sec.

The mean viscosity of water is

-1

QC. - 1

m,

Of eourae, the mean thermal conductivity m a t also be expressed in heat units per second, This is (40)

= 143

10"

kg. cal. aec.

.I

~4-I

c.-'

.

From the change in Uensity of the water we obtain the mean coefficient of expansion

These values give,

Therewith according to Table VI,

or the required coefficient of heat transmission is,

= 305 kg. ca1.

ill.a

e

m

O C . ~ ~

Experiments for this case unfortunately are not available, But this value corresponds very well with practical experience, It is to be expected that for other bodies the nature of function W remains the same. The forrmlare developed up to now referred to the cooling sf a solid body in a stationary fluid, This body shall now be cooled by an air stream with velocity w, in a given direction, Hence the hydro- and thermodynamic equations remain the aame, Only the boundary conditions are partly different. A t great distance

the air velocity is no longer zero, but is equal to the vector If again two geometrically and thermally airnilas systems w2 * are considered, then the velocity ratio becomes

i,e., equal to the ratio of the velocities of the air streams, In order to have eirnilarfty fn both system, not only m e t B and C be constant according t o equations ( l l b ) and (1 Zb), but also according to equation (8) the fractfon

must remain constant,

Consequently

lee,, a function of three independent variables, or,

If cooling in a gas is considered, then in addition D appears as an independent variable, so the coefficient of heat transmission is

If the velocity of the air stream is very large then the influence of gravity disappears and

Again introducing the mean values for the variables which vary with temperature, and restricting ourselves to diatomic gases, then

which is a formula I have checked and found correct by experiments on heat transmission in a pipe. This case will be taken up later. All cases so far considered have referred to the equilibrium condition. In addition to this, let us consider a further variable condition in which the surface temperature of the body has initially the temperature of the swroundlngs To and then is , at which it is maintained heated suddenly to the temperatme T permanently. In the steady-state process equation (60) is valid. In order to obtain the value of the coefficient of heat transmission during the variable period, however, it is necessary to repeat the considerations observed earlier. As a new boundary condition, there is now the fact that at time t = 0 the air temperature at all points is To. Now, if two aimilar systems are again considered then equation (9) requires that

is constant.

Concequently equation (60) emands to

This equation indicates that the coefficient of heat transmission depends on the time t that has elapsed since cooling of the body began. I discovered this phenomenon some time ago in connection with another case A quantity o f gas w a s en-

.

closed in a container and suddenly raised to high temperatures while the walls remained cold. The rate of cooling was measured by an indicator, In a diagram, the actual coefficient of heat transmission calculated from the pressure dres per unit time was plotted, against gas,temperature. The laver the temperature, the smaller the coefficient of heat transmission. A nuniber of experiments were now carried out with different heating of the gas and for each experiment the a T curve was plotted. For the different experiments at the same gas temperature, the wall temperature and the density were also the same, hence the same values could have been expected, i.e,, one curve for all experiments. If w e disregard the first part of the cooling process for each experiment, this view is correct. The values in the first part of the cooling process are highest in all experiments. This phenomenon, in which the coefficient of heat transmission at the beginning of the cooling process was greatly magnified, could only be attributed to the influence of time. At that time I proved this observation neglecting the force of gravity, a simplification which is now eliminated in the above derivation. In the cooling of the above body the coefficient of heat transmission will accordingly be appreciably higher at the beginning of the cooling process than the actual value according to equation (28a). Eventually, however, the influence of the time ceases and the coefficient of heat transmission for the steadystate enters the picture.

-

Beginning with the differential equations of' flow and heat aonduction a relationship for the cooliw of a body in a diatomic gas is set up, which has been checked and conf'irmed by experimental data available in the literature for the cooling in air of a horizontal cylinder, The relationship shows that the coefficient of heat transmission depends on the surface temperature, the gas

temperature, the temperature difference, the diameter of the pipe and the air pressure, From the relationship an approximation equation is derived for other gases and liquids,

References

7.

1,

Der WHrmeiiberga in Rohrleitungen. e eat transfer in pipe lines Z.VDI , 1909, p. 1750; Mitteilmgen Qber Forschungsarbeiten, no, 89,

2 .

Chwolson, Z.B, Lehrbuch der Physik, v. 111, p, 306, or Winkelmann, Handbuck der Physilr;, v, 111, p, 436.

3.

Moser,

Ober die WHrmeleitfghigkeit von Gasen und Dbpfen bei h6heren Tenrperaturen (on the thermal conductivits of gases and vapours at higher temperatures). Diss, 19.13 .

/

4.

The convection of heat from small copper wires, Am. Inst. Elec, Engrs, 28 (I): 363, 1909.

5.

Die WHrmeabgabe geheizter Kl)rper an Luft e eat transfer from heated bodies to air), Mitteillungen iber Forschungsarbeiten, no, 98 and 99, 1911.

6,

Convection and conduction of heat ingases.

7.

Die ktinstliche Konvection am elecktrischen Hitzdrahte (~rtificialconvection at electrically heated Diss. Dresden, 1915. wires).

8,

Boussinesq. Miae en Qquation des ph&non&nes de convection calorifique et apervu SUP le pouvoir refroidissant des fluides he setting up of equations for the phenomena of thermal convection and the estimation Journal de of the cooling capacity of fluids). Physique, 1902, p. 65,

9.

Lorenz, Lo Uber das Leitverm6gen der Metalle fYlr WUrme und ElektrizitHt he thermal and electrical conductivity of metals). Wied, Ann. 13: 582, 1881,

34:'401, 1912.

Trans.

Phys. Rev.

10,

Nusselt.

Die OberflHchenkendensation des Wasserdampfes surface condensation of steam) to appear shortly in Z,VDI.

11, Nusselt.

he

Der WHrmellbergang in der Gasmachlne; die Abhangigkeit der WHrmetibergangszahl von der Zeit eat transfer in the gas engine, the relation between the ooefficient of heat transmission and time). Z.VDI, 1914, p e 361.

T A B U VII

TABLE V Xxperinents of Eylevelt

TABLE: VIII

TABLE VI -

B

I

I

log A

logB

I

-

1 I

A

air

FreSSUr= Atm. abq.

,

1

COQ{/U

*

AW

W

cr\L

L LL

'*-L.~-'

~

cowu tr."?

ais

w. T. ' abs.

abs.

200 288

300 400 MH)

600

2.94 2,74 2,72 2,63

t W r n d d & C -

kf

700 800

2.62 2,07 2.70 2,76

000

1000

2,m 2.69

3,38 I

TABLE X

LT

e a t . ketni:* "c:'

Fig.

1

Temperature dependence of thermal conductivity of air

temp.O

absolute

Fig. 2 Temperature dependence of v i s c o s i t y of a i r

Dependence of the coefficient of heat transmission on pipe diameter

"0

10 ---

20

30 40 50 60 70 80 pressure (atmospheres).

90 IOOat

Dependence of coefficient of heat transmission on pressure

-

.

temp ( Obb 8.)

Fig. 6

Dependence of coefficient of heat transmission on temperature

Fig. 7

Degendenoe of coefficient of heat transmission on temperature difference