Section A, Vol, 7

Appl. sci. Res.

IvrEASUREMENTS OF THE DISTRIBUTION OF HEAT AND MATTER IN A PLANE TURBULENT JET OF AIR by B. G. VAN DER HEGGE ZIJNEN Royal Dutch.Shell-Laborntory, Delft, of N.V. de Baxaafsche Petroleum Mautsch appij, The Hague, Holland

SUITlITlary

Measurements of the transfer of heat and matter across a plane turbulent jet of air, issuing with a velocity of 4000 cm S-l from a slit of 0.5 em x 10 em into still air are described. The initial temperature of the warm jet was from 15°C to 56°C above ambient air temperature. The experiments on the transfer of matter were carried out with 1% by volume of town gas added to the air taken in by the set-up. In both cases the Reynolds number was 13300. The paper describes the experimental equipment used and the experiments performed. The distributions of heat and matter prove to be identical; the transfers of momentum on the one hand, and of heat or matter on the other, are not in any way identical. The comparison of experimental data with theories shows a substantial lack of agreement. In the last section the coefficient of transfer of heat or matter is computed from experiment. Symbols = constan t = constant f = function F = function for U g = function h = height of a slit I = mixing length 1m = mixing length for momentum I r = mixing length for T U = local time-mean velocity in the jet in axial direction U 0 = discharge velocity of the jet U m = velocity in the axis of the jet V = local time-mean velocity in the jet in transverse direction X = axial distance from a cross-section of the jet to the apparent linesource

a c

-

277 -

278

B. G. VAX DER HEGGE ZIJNEN

X' = axial distance from a cross-section of the jet to the exit edge of the slit Y = transverse distance from a point in the jet to the axis Z = coordinate in the direction of the slit I' = common symbol for heat and matter r,n = common symbol for heat and matter in the axis of the jet So = coefficient of turbulent velocity exchange Er = coefficient of turbulent r exchange I) = distance ratio Y/X '/! = distance ratio for which U/U m or r;rm = 0.5 ()' = temperature excess at a point of the jet 80 = temperature excess at the exit of the slit (h = temperature excess upstream of the slit 11m = temperature excess in the axis of the jet p = density T = shear stress 'Pi = function ip? = fu fiction r1J = Gaussian error integral D = concentration at a point of the jet Q 1 = concentration upstream of the slit D m = concentration in the axis of the jet

§ 1. Introduction. The nature of the present research with a plane turbulent jet of air is essentially identical with that of the investigation of the distributions of momentum, heat and matter across an axially symmetrical turbulent jet of air, described in Appl, Sci. Res. Al (1949) 435. It has the same aim, namely to come to a better understanding of the transport mechanism of the quantities involved and to provide data for checking various theories proposed for the exchange across turbulent jets. The experimental results obtained with the momentum transfer in a plane turbulent jet were communicated in a previous paper 1); those obtained with the transfers of heat and matter are the subject of the present paper. Apparently there are only two sets of experiments on the heat. transfer across a plane jet: by Schmidt 2), who investigated an ascending plane stream of hot air, and by Reichardt 3). who worked with such small temperature excesses that external forces could be neglected. No measurements of the transfer of matter across a plane jet could be traced in literature. This being so, we began by measuring the temperature distribution across a horizontal jet of air issuing from a likewise horizontal slit of 0.5 em x 10 cm with a discharge velocity U 0 =

MEASUREMENTS IN A PLANE JET OF AIR

279

= 4000 em S-1. The Reynolds number, specified as: discharge velo-

city U 0 times the height of the slit h divided by the coefficient of kinematic viscosity v, amounted to 13300. The initial temperature excess ranged in the respective researches from about 15°C to 56°C. Measurements of the concentration across the unheated jet, to which about I % by volume of town gas had been added, followed. Both the initial temperature and the concentration of town gas were so low that the density throughout the jet was assumed to be constant. ' § 2. Suroey of theories on the heat transfer across plane jets. The theoretical background of the turbulent transfer of heat across plane jets is expounded in a report by Alexander, Baron and Comings 4), and also in the textbooks by Townsend 5), Pa i 6) and Goldstein 7). The problem of the transfers of heat and matter across a turbulent plane jet has been attacked theoretically along the same lines as the transfer of momentum by Pr an d t l s) (momentum-transfer theories). Taylor 9) (vorticity-transfer theory) and Reichardt 3) (inductive theory of the analogy between transfer by molecular motion and by turbulent agitation). These theories were discussed in 1). Prandtl assumes for the transfer of heat and matter a mixing length lover which a diffusing quantity remains constant when the fluid particles are transferred from one fluid layer to another. This definition allows for different magnitudes of the mixing lengths for momentum and for heat or matter. The magnitude of this mixing length must be found by experiment. When the mixing lengths are taken to be the same in both cases, To l lrn ie n's solution 10) of the velocity distribution from Pr a n d t l's momentum-transfer theory in its original form holds also for the distribution of heat and matter; both are similar. . Taylor's vorticity-transfer theory was worked out by Howarth 11), who like Tollmien assumed a mixing length over which the vorticity as well as the temperature of fluid particles instead of their momentum - remains constant before they are dissolved in their new environment. While according to Tollmien the temperature and velocity distributions are similar, according to Howarth the temperature

280

B. G. VAN DER HEGGE ZIJNEN

distribution is proportional to the square root of the velocity distribution when the same mixing length is used in both cases. Because Reichardt assumes perfect analogy between interchange by molecular agitation and by turbulent motion the molecular analogy theory yields similarity of temperature and velocity distributions. both to be represented by a Gaussian error function. § 3. Experimental equip-menlo In our equipment(fig. 1) a fan with a capacity of 1.5 m 3 min-1 supplied the air; the air passed an electrical heating element (only used for temperature measurements across a warm jet) and was led into a horizontal 4" tube of 1 m length at the end of which a likewise horizontal slit of 0.5 em X X 10 em was mounted. The entry edge of the slit .was rounded off with a radius of 0.9 cm. The static pressure in the tube served as a measure for the discharge velocity U o.

Fig. 1. Experimental equipment.

For reference purposes the initial temperature excess, i.e. the temperature in the 4" tube above ambient air temperature, 81 was measured by means of an alumel-chromel thermocouple, connected; to a sensitive galvanometer. The warm junction of this thermocouple has a diameter of 0.02 cm and was located just upstream of the slit. The cold junction of this thermocouple was at room temperature. A second thermocouple of alumel-chrornel, mounted on a standard with displacement in the direction of the jet and in transverse direction, was used to measure the local time-mean temperature excess (J in the jet. By applying the temperature ratio 8/81 variations in U 0, in the heat production and in the ambient air

MEASUREMENTS

I~

A PLANE JET OF AIR

281

temperature during the experiments were eliminated from the results. The question arises as to whether the thermocouple indicates the time-mean temperature correctly in turbulent flow. In general a thermocouple of finite size in a field of fluctuating temperature will not indicate the time-mean temperature correctly. It behaves in the same way as a hot-wire anemometer in a flow with a fluctuating velocity'. Actually the response to fluctuations of a thermocouple and of a hot-wire anemometer are analogous. A correction factor for the reading of a hot-wire anemometer with small fluctuations of its electrical resistance, i.e. of its temperature, is known. That correction depends on the intensity of the axial velocity fluctuations as well as on the intensity of the transverse fluctuations. It can be shown that the hot-wire anemometer reads slightly too high in the central part of a jet and too low in the outer part. The hot-wire correction is small and vanishes at that point in a plane jet where the local velocity is about ~ of the maximum velocity in the same cross-section. If we assume that this holds also for the thermocouple corrections, then a correction for errors in the time-mean temperature can be dispensed with. Of course, when the instantaneous temperature is to be measured, the thermal lag of the thermocouple must be compensated. Electrical techniques for compensating thermal lag of thermocouples and resistance thermometers are described by Shepard and Warshawsky 12). Since we worked with temperature ratios, it was sufficient to ascertain that the galvanometer readings varied linearly with the temperature excesses used. For the investigation of the transfer of matter across our jet about 1% by volume of town gas was added via a meter to the air taken in by the fan. For reference purposes the initial concentration of the air [h just upstream of the slit was determined ~y analyzing samples by means of a thermal-conductivity gas analyzer. The concentrations Q of the air across a section of the jet were measured by taking samples by means of a total-head tube as used in previous measurements of the velocity distribution 1). We worked 'with the concentration ratios QjQ 1 in order to eliminate variations in the gas supply during the measurements. The samples Q and Q 1 were taken in succession. The sensitivity of the setup was such that a concentration of town gas of 0.0 1~~ by volume could be detected.

282

B. G. VAN DER HEGGE ZIJNEN

---------------------_._--_ ..

§ 4. Temperature measurements in a direction parallel to the slit. Because the velocity measurements across the same jet executed in a direction parallel to the slit had shown that the velocity distribution is more or less saddle-shaped (1), fig. 2), temperature measurements were first made in the horizontal plane of symmetry, in a direction parallel to the slit (Z direction). The distance X' between the slit and the cross-section of the jet ranged from 0 to 20 em. This investigation was done to discover whether the temperature distribution in the plane of symmetry perpendicular to the slit is sufficiently two-dimensional for theoretical and experimental results to be comparable.

Fig. 2. Temperature ratio el8 0 across the jet in a direction parallel to the slit.

Fig. 2 shows that the temperature distribution of a jet heated to an initial temperature excess eo at the slit of 30°C above ambient air temperature and issuing with an initial velocity of 4000 ern s-1 from a slit of 0.5 em X 10 em is near enough two-dimensional across sections not farther away than twice the length of that slit, Some of the temperature curves show clearly the same saddleshape as the velocity curves; this is particularly so in the crosssections at X' = 6 cm = 12 times the height h of the slit. An explanation of this phenomenon is given in 1), § 5, namely large vortex rings surrounding the jet. Although the temperature field across our jet is not exactly twodimensional, it approaches the two-dimensional case sufficiently for a comparison of theory and experiment.

MEASUREMENTS IN .\ PLANE JET OF AIR

283

§ 5. Temperature measurements along the axis oj the [et, The temperature as measured along the axis of the jet is represented in fig. 3. Two runs were made: one with a discharge velocity Uo = 4000 ern S-1 and temperature excesses 80 = 15, 30 and 56°C, and another with U o = 1200 ern S-1 and 00 = 30 and 53°(. From the conservation of momentum and of heat in the plane jet it is to be expected that, if the density is constant throughout the jet. the axial temperature ratio Om/Do will satisfy a relation of the form Om/Oo = a/ y'X. (I) Here X is measured from the cross-section of the jet to the linesource from which the jet seems to originate, and a is a constant. Such a relation does not follow dearly from fig. 3; it is, however, approximated. There is a systematic difference between the results obtained with Ull = 1200 cm S-1 and with U 0 = 4000 ern S-I; it is possible that this is due to a difference in spread; the spread is somewhat greater at a lower velocity.

Fig. 3. Temperature ratio 8m /eo along the axis of the jet.

Take the mean of the temperature ratios measured with Uo = S-l. These experimental data satisfy approximately

= 4000 cm

8m

-

2.00

v

+

80 s: jh 0.6 The hypothetical line-source is situated at X = X' + 0.6h.

(2)

This is the same position as found for the apparent origin of the velocity field (d. 1)).

284

B. G. VAN DER HEGGE ZIJ:KE:-l

§ 6. Temperature measurements across the jet. Because the temperature ratio Om/OO at a location in the axis of the jet decreases systematically with increasing eo (U o = constant), as shown by fig. 3, a few measurements across the jet at the location X' = 5 cm were performed with U 0 = 4000 em S-l and 00 ranging from 11 "C to 56°e. Analogous measurements were made at the same -location with U o = 1200 em S-1 and eo ranging from 12.5 to 70°e. These measurements distinctly show that in both cases Om/Oo decreases continuously when 80 increases. This means that the halfwidth of the temperature curves measured increases slightly with increasing initial temperature of the jet. The temperature curves across the vertical line of symmetry of the horizontal jet are for all values of eo used very nearly symmetrical, so that buoyancy effects are negligible. • tII •

.

~

"

I

•

I

!

~

!

I

,,

.',

...k..-f!

v P.0 VII.. rill! I}

-'" 'V

~ r.:::-v:

•~ <.S

!

·

· ··

··, •

i I

\

,

I

I I

.\ f!.,11

~~ :::---

://~v.

'(Ill 'Ill

I

!

'.

!

··· ·•

·

i X'.lc:ra _ \SCM

-, "" _l.5C11t _l ~

_ 4 c"'

... 'i .. ,

C"I

I

t 1

f--

elrl

.. 7 em .. ,

~

eft!

f--

... 9 ...... ..oj!>

~r-

... 1: 0:""

f--

_I' cJ'I'I

.1. _11

;1'1'1 I;fI'l

f--

-10 tm

~~~ ~t>.. ~\'~\\;t~ ~ ~ ~ ,\\~ ~;.t\.~ ~ ~::.: ~ ~ ~

zs

Fig, 4. Temperature ratio 8/8 0 across the jet as a function of Y,

After this preliminary investigation a series of traverses was made with U o = 4000 em S-l, 80 = 30°C and X' ranging from 1 to 20 em (2h to 40k). Fig. 4 shows the ratios of local temperature excess 8 and initial temperature excess 80 as a function of the vertical distance Y from the axis of the jet. In the central part of the first two, or perhaps three, curves a core of uniform temperature can be observed; such a core follows also from fig. 3. It indicates that the turbulent flow across the jet is not yet fully developed at distances of less than 6h from. the slit.

}rEASUREME:-iTS

I~

285

A PLANE JET OF AIR

In fig. 5 all the temperature curves for the sections X' = 5 to 20 em have been reduced to the same length scale by plotting the ratio fJj8 rn as a function of the distance ratio YJ= Y/X = Y/(X' +0.6h).

Although the experimental data scatter, they appear to arrange themselves along a single curve. Hence the temperature distributions are similar. The half-width of the dimensionless temperature curves appears to be '171 = 0.1415. In the left-hand part of fig. 5 the mean curve through Reichard t's data was drawn with the assumption of coincidence with our data at 1']1 = 0.1415. In the region YJ = 0 to 0.20 Reichardt's results compare favourably with ours. 11/9,..

I'\!n'" UI\J,,"-

~or--r--,--,---r-.-~Ml'l~-r--;--:-r--r-:-,

I

i

I

! I

,020

010

Fig. 5. Temperature ratio ()j()m, concentration ratio QjDm and velocity ratio UjU m as a function of the distance ratio n-

§ 7. Concentration measurements. Previous investigations with a round jet of air (App!. Sci. Res. A 1 (1949) 435) had shown that 'in turbulent shear flow diffusion is independent of the nature of the quantity diffusing. A single test run showed that this is also the case in a plane turbulent jet. Such a test run was made in the plane of symmetry perpendicular to the slit at a distance X' = 10 em, with a discharge velocity U 0 = 4000 em 5-1 and an initial concentration of town gas of about 1% by volume. The ratio of the local concentration Q and the concentration at the centre of the same Appl, sci. Res. A 7

286

B. G. VAN DER HEGGE ZIJNEN

cross-section Q m is plotted in fig. 6 as a function of i:'. This figure shows also the temperature ratio G/O m (XI = IOcm; U o = 4000 em 5-1; 1J0 = 30°C) and the velocity ratio across a jet of pure, unheated air (XI = 10 em; U o = 4000 em S-I). It appears that the concentration ratio and the temperature ratio actually coincide within the limits of experimental accuracy. Hence heat and matter will be designated by the common symbol r. Fig. 5 shows that the spread of r is considerably greater than that of velocity. The mean half-widths 7}t of the velocity and T fields amount to 0.096 and 0.1415 respectively, hence the halfwidth for is 1.47 times the half-width for velocity. Across an axially symmetrical jet TJi for was 1. 17 times 'It for velocity.

r

r

~

V/U",

~

fl8 M lS ... 1O·CI

•

tlJl"ll'Ilfl, .. 1"'4)

I

.. I---+--'!--A----+-----;.-~-+_-__I

."I-----Jf--.f--+--......I----l--\--"t----1

] Yc'"

Fig. 6. Concentration ratio {)/Dm , temperature ratio 810 m and velocity ratio U/U m across the jet at XI = 10 ern as a function of Y.

§ 8. Comparison between theory and experiment. In the right-hand

part of fig. 5 a comparison is made between our results and the theoretical T distributions according to Prandtl's momentum-. transfer theory in its original form (computation by Toll m ie n 10) with the same mixing length for momentum and T), Taylor's vorticity transfer theory (computation by Howarth 11)) and according to Reichardt's molecular-analogy theory 3) (Gaussian error function). All the theoretical lines in fig. 5 are made to match with the mean curve through our points of observation at the location 'YJI: = 0.1415. Tollm(en's theory with equal mixing lengths does not agree

~!EASURE:MENTS

IN A PLANE JET OF AIR

287

with experiment: at the centre of the jet the theoretical line is too pointed and beyond the half-width r is predicted too high. The theory allows, however, for different mixing lengths: lm for momentum transfer and lr for r transfer. The ratios between them follow from the half-widths llim and ll!r respectively as measured and from Tollmien's assumption that l is constant across the jet and proportional to X: l

From T'o llrn i e n's table

10)

= eX.

(3)

for the plane jet:

'1]1 =

.3;-

0.96v 2c 2 .

(4)

Since the ratio of the measured half-widths is 1.47, the ratio between the mixing lengths should be (1.47)3/ 2 = 1.78. This theory does not hold. Vorticity-transfer theory (Howarth) with the same mixing lengths for r and for vorticity yields in the central part of the jet almost the same r distribution as momentum-transfer theory; here both theories are equally unsatisfactory. Beyond the half-width, however, vorticity-transfer theory is in better agreement with observation than momentum-transfer theory, but is still unsatisfactory. Molecular-analogy theory (the assumption of a Gaussian r distribution) holds better in the central part of the plane jet when the theoretical and experimental curves are made to match at the half-width. Beyond the half-width molecular-analogy theory yields too high values of This is also the case when according to Reichardt the constant a in the error function for the T distribution

r.

(5)

is taken to be half the constant in the error function for the velocity distribution. If a for the velocity distribution is now taken to be 75 (d. 1)). according to Reichardt the r curve should satisfy (6)

yielding for the half-width 1)1 = 0.136 instead of 0.1415 as measured, i.e. 4 % less. The curve according to (6) is seen in the right-hand part of fig. 5.

288

B. G. VAN DER HEGGE

ZIJNE~

It appears that all theories considered here are unsatisfactory; the transport of momentum and of r are of a different character. § 9. Ratio between the coefficients of turbulent momentum exchange and 01 r transfer. The theories considered above assume a constant ratio of the coefficient of turbulent velocity exchange Ell specified by T

dU = pev dY

(7)

-.

(T = shear stress; p = density: dU/dY = velocity gradient) and the analogous coefficient Sr for the transfer of any transferrable quantity r. In order to find that ratio, we start from the fundamental equation of motion and of r transfer respectively, which reduce with the usual simplifying assumptions for the plane jet to

au

U eX U

ar ax

eo

+v --l-

(; (

BY = aY

v ar

ev

I

=

~_ BY

&U)

cv aY

(8)

>

(e r av Or).

(9)

Moreover the equation of continuity

au

av

ax

BY

-+--=0 and the equations for the conservation of momentum and of axial direction:

(10)

r

in

+00

J U2 d Y = const.,

(1 I)

-00

+00

fur dY =

canst.

(12)

must be satisfied. The boundary conditions are Y = 0: U = c.; Y = co: U = 0; r

au/oy = = o.

0;

r= r.; ario"l[ = o,lf

(13)

\Ve introduce the dimensionless distance parametera, put U = (fl(·\lf(I]),

( 14)

r

(15)

= !Pz(X)g(1])

MEASUREMENTS IN A PLANE JET OF AIR

289

and get from (I I), with the function F defined by '7

F = ff(r))dr;,

( 16)

o

I

U =

I

yJX f(rj) =

v / X P.

(17)

From (10) n

V

I = y'X

t J' f(1))d1)]

[Yif(YJ) -

I (IJF' - -}F) viX

=

(18)

o (the prime denotes differentiation with respect to 1)), and from (12) : I

F=--g

( 19)

viX .

Inserting (17) and (18) we get finally from (8)

(GVF")

o (7 -t-(FP) =- - or) o'Yj V'X

(20)

or (21) Because of the boundary conditions the integration constant is zero. Correspondingly we get from (9) erg' Fcr=-2--

viX'

l:>

again with an integration constant =

F'

g

Inserting U m and

Ev

F"

= ; ; g;- =

U

r

=

(22)

o. From (21) and (22) ev oU/8r; ErOF/(1) .

(23)

r m , the ratio of the transfer coefficients becomes a a;;ln(U/U m) 8

(24)

-~-ln(riTm) 01)

For numerical values UjU m and Firm must be known functions of

rj.

290

B. G. VAN DER HEGGE ZIJNEN

We have found that the experimental velocity data satisfy (25)

Hence

o

-In(U/Um ) = 0"1

1501].

(26)

From the mean of the left-hand and right-hand part of fig. 5 we derived by trial and error

rlrm

= (1

+ 301]2 + 22001]4 -

3000Qn6)e-757J~.

(27)

Fig. 7. Temperature ratio 818 m as a function of TJ from interpolation formula and from experiment.

Fig. 7 shows that this relation holds satisfactorily throughout the entire region investigated. From (27) (28) Finally we get from (24), (26) and (28)

+ 88001]2 - 18000Qn4 + 307]2 + 22007]4 - 300001]6)

60 150(1

(29)

Fig. 8 gives the reciprocal value of (29), namely Briev, as a function oft].

From the formulae given above it follows that the ratio

£r/£v

is

291

MEASUREMENTS IN A PLANE JET OF AIR

constant throughout the entire jet and equals unity if the velocity distribution and the r distribution are the same functions ofY] (To l lm i e n's case for equal mixing lengths). If, however, both distributions satisfy a Gaussian error function with different constants, the ratio Erlev is constant across the jet and equal to the ratio of those constants (Reichardt's case). In all other cases Erie/) is a function of the velocity ratio and of the distance ratio.

/

/

.~

\

,,

e..

~

~

".

."

.oo

r.

<,

r-.

0"

111'11

Fig. 8. Ratio between the coefficients of r transfer and of turbulent momentum exchange as a function of the distance ratio 7)-

The magnitude of definition (7): T

el)

at any point in the jet follows from its

dU pcv dU = pev-- = --

X

dY

(30)

d'i]

together with (25), which we put in the general form U/U11I = e- a/q 3

(31) .

and the equation for the shear stress (1), eq. (28)) -r

y'"l'C 2y'a 2 I

- - = - - - - - ea'l) pU2

2


.(32)

where

fr

as a function of n-

292

:>fEASDRE~fENTS

IN A PLANE JET OF AIR

REFEl{ENCES I} Hec:ge Zijnen, 13. G. van d e r , Appl. Sci. Res. A 7 (1958) 256. 2) Schmidt, \,V., Z. angcw, ;"Iath. "'Iech. :H (1941) 265. 3) Reichardt, H., Forsch. a.d. Geb, Ing. wesens I:. B (1942); V.D.!. Forschungsheft 414, 2nd ed. 1951. 41 Alexander, L. G .. T. Baron and E. W. Comings, Univ. of Illinois Eng. Ex p, Station 413, 1953. Sl Townsend, A. A., The structure of turbulent shear flow, Cambridge 1956, ell. 8, p. 172. 6) Pai, S. t., Viscous flow theory, Princeton N.J. 1957, If cu. 7, p. 115; also FhiiJ dynamics of jets, );ew York 1954. 7) Goldstein, S., Modern developments in fluid dynamics, Oxford 1943. 8) Prand t l, L., Z. angew. ",ratli. "'lech.;; {192S) 136; Verh. 2 Int. Kongress f. tech. \kch. ZUrich 1926; Z. angew.';"lath. i\feeh.;!~ (1942) 241. 9) Taylor, G. 1., Proc. Roy. Soc. Lond'on .\ 1:1;; (1932) 685. to) Tollmien, W., Z. anzew. ;"Iath. Mech. t) (!926) 468. 11) ~Ll\varth, L~, Proc, Cambridge Phil. Soe.:v. (1938) 185. (2) Shepard, C. E. and 1. Warshawsky, N.A.C.A. Techn. Note 2703, 1952.