# Nonisothermal Free jets

Buoyant flows can be classified as55

• Buoyant jets when the buoyant force acts in the direction of the jet supply velocity at the origin, i. e., upward-projected heated air jet or downward-projected cooled air jet

• Negative buoyant jets when the buoyant force acts in the opposite direction, i. e., downward-projected heated air jet or upward-projected cooled air jet

• Nonbuoyant jets when the effect of buoyancy is negligible

• Plume when the buoyant force completely dominates the flow, as for flow generated with a heat source

In the general case, a buoyant jet has an initial momentum. In the region close to discharge, momentum forces dominate the flow, so it behaves like a nonbuoyant jet. There is an intermediate region where the influence of the ini­tial momentum forces becomes smaller and smaller. In the final region, the buoyancy forces completely dominate the flow and it behaves like a plume. When the jet is supplied at an angle to the vertical direction, it is turned up­ward by the buoyancy forces and behaves virtually like a vertical buoyant jet in a far field. A negative buoyant jet continuously loses momentum due the opposite direction of buoyancy forces to the supply air momentum and even­tually turns downward.

Between the nonbuoyant jet region and the plume region lies an interme­diate region in which the flow changes from the former to the latter.55 The ax­ial location of the beginning of this intermediate region depends primarily on the exit Froude number:

C _ _Zs_—— — (7.59)

GD0(T0-TJ’

The beginning of the transition zone in the linear jet is at approximately

^ = 0.5F2/3^j"1/3. (7.60)

Using the relation between the Froude number and the Archimedes number, Ar0 = 1/F, the length of the linear jet zone, x, where the buoyancy forces are negligibly small can be calculated as follows:

To characterize the relationship between the buoyancy forces and momen­tum flux in different cross-sections of a nonisothermal jet at some distance x, Grimitlyn56 proposed a local Archimedes number:

(7.62)

Where g is acceleration due to buoyancy. Equations for the local Archimedes

Number can be derived by substituting the expressions for axial velocity Vx and temperature differential AO* into Eq. (7,62):

(7.63)

For linear jets

(7.64)

Where the Archimedes number at the outlet is

(7.65)

Characterizing the ratio of buoyancy and inertial forces at the jet discharge from the outlet.

When calculating Ar0 for a linear jet, JA~0 is replaced by the width of the slot H0.

Introduction of the local Archimedes criterion helped to clarify noniso­

Thermal jet design procedure. Grimitlyn suggested critical local Archimedes

Number values, ArЈrit, below which a jet can be considered unaffected by buoyancy forces (moderate nonisothermal jet): ArT 0.1 for a compact jet, Ar^ < 0.15 for a linear jet.

A similar limitation for a linear jet from Eq. (7.64) at K1 = 2.5, h — = 2.0, T0 = 293°C, and = 313°C is Ar, < 0.14. ‘

Along with a constant velocity zone (Zone 1), there is a constant tempera­ture zone in a jet. Heat diffusion in a jet is more intense than momentum dif­fusion; therefore the core of constant temperatures fades away faster than that of constant velocities and the temperature profile is flatter than the velocity profile. Thus the length of the zone with constant temperature (Fig. 7.23) is shorter than the length of the constant velocity zone (Zone 1).5>51-58

From Tolmin’s theory and experimental data (e. g., Reichardt59) the rela­tionship between velocity profile and temperature profile in the jet cross-section can be expressed using an overall turbulent Prandtl number Pr = vt/ex,, where vt is a turbulent momentum exchange coefficient and at is a turbulent heat ex­change coefficient:

 (a)

Where 0 is the air temperature at the point of consideration, and 0,. is the max­imum air temperature in the cross-section of the jet,

A Prandtl number of 0.7 has been suggested for nonisothermal jets by Not — tage,60 Forstall and Shapiro,61 Corrsin and Uberoi,62 and Grimitlyn.^ Abramovich3 suggested a Prandtl number for a compact jet of 0.75, and for a linear jet, 0.5.

According to Abramovich,57 Regenscheit,64 and Shepelev,4 the relation be­tween the velocity distribution and the temperature distribution in the cross-sec­tion of nonisothermal compact, linear, or radial jets within Zone 3 can be expressed as

^SL = (0-e-)/(e*-ej= /J,

“Or a/ vx

And thus the Prandtl number is equal to 0.5. Table 7.14 lists some temperature equations as they are used for practical applications.

7.4.4.3 Centerline Temperature Differential in a Horizontally Supplied Jet

Compact Jet

The centerline temperature differential within the zone of fully established turbulent flow (Zone 3) of a nonisothermal jet can be derived using equations of momentum (Eq. (7.39)) and excessive heat conservation along the jet: 3,4’■*

V(6 — Qm)y dy, (7.68)

O

Where W0 = Cpp„i’o(0o — 0„)AO = excessive heat in supplied air, and Cp = spe­cific heat of air.

The equation for the centerline temperature differential in Zone 3 of the compact jet derived4 from Eq. (7.61) using the Gauss error-function tempera­ture profile (Table 7.14) is

TOC o "1-5" h z A0, = 1+ ct W0 1^1 _ (7 69)

* 2^cCpP„jM^x ‘

Equation (7.69) also can be presented as follows:

An (1 + a)OA0oyA^

A0X = ——f=———- *—, 17./0)

X 2Jttc<p x ’ ■ ■

where

A0s = To-T„ = Ј-^-. i././l)

The complex of coefficients having constant value (1 + cr)0/ 2 Jtt cc|> is called4 the thermal characteristic of the diffuser jet, K2, and characterizes the temperature decay along the air jet. Assuming perfect mixing in the room (i. e., 0OZ = 0„), 0.„ can be substituted for 0OZ, and Eq. (7.70) can be presented as follows:

!/ )~i ~71

L, J2)

As in the case of equations for the velocity decay computation, in the equations for the temperature decay computation, some researchers consider x to be a distance starting from some virtual source located at some distance x0 from the diffuser face; others for practical reasons neglect x0.

As in the case of the Kx characteristic, theoretical values of characteristic K2 depend upon supply conditions. According to Shepelev,4 in the case of air supply through a nozzle with a uniform outlet velocity profile, K-, x = (1 +Pr)/ (2tt0.0822). Thus, when = 6.88 and Pr = 0.7, K2 = 5.85. Grimitlyn1 suggests the following relation between K2 and Kt coefficients:

K2= . f7.73)

According to Helander’s unpublished data (Progress Report, Downward Pro­jection of Heated Air, January 6, 1951, referenced by Koestel58), K2 = 6.

E.-e-

 Author Round jet Linear jet Radial jet Shepelev4 ~4 (0.082*) E J |_SL. f 4 lO. l-rJ E 4 (in*) E Grimitlyn1 -0.7 (>/>• 0 5,)’ E . ao..s* >’0.5( “ * tan a0.5f ~ ^ VPr -0.7 (l’/yo s,)’ Tg “O. Sr >’0.5« “ — l ran a0 jj — x -0.7 ty E Vo. St = x tan “0.5f _ vfS “0.5r ‘ TPr Abramovich2’’ (1 — r/S3/V’4 (1 — y/S3/1) —

Linear Jet

Derivation of the equation for the centerline temperature differential in a linear jet is based on the same principles that are used in the case of a compact jet. For the linear diffuser jet, centerline temperature differential can be com­puted from the following equation:

~ 0QZ = K Ho (7 74)

0o-eo Z Vx‘ ■

The centerline temperature differential in Zone 3 of the diffuser jet is propor­tional to the value of the K2 coefficient, which, along with the coefficient,

Depends upon jet and diffuser types and supply conditions. The theoretical

Value of the K2 coefficient, according to Shepelev,4 is 2.49. Experimental data reported by Grimitlyn1 show K2 to be 2.0.

Equations for the centerline temperature differential in radial and in coni­cal jets (Fig. 7.21) is derived in the same way as for compact and linear jets4 and is similar to Eq. (7.70):

0*-0~ _ I 1 + Pr .9.1 fi~o {7J5)

00-0» 4 C sin а Ф x

The complex of parameters

 (7.76)

/ттсштф л/р

Is a thermal characteristic, K2. In the case of a radial jet (3 = 2ir and a = 90°. As­suming 0 = 1, <p = 1, с = 0.082, and Pr = 0.5, K7 = 1. For conical jet with a = 60° 0 = 2tt, 0 = 1, ф = 1, с = 0.082, and Pr ="0.5), K2 = 1.07. Table 7.15 lists some equations used for centerline temperature differential computation in horizontal jets.

According to Shepelev,42 the theoretical values of the K2/K^ ratio for air supply through nozzles with uniform velocity distribution at the outlet cross­section is 0.9 for compact jets and 0.95 for radial, conical, and linear jets. Prac­tically, for different types of air diffusers, this ratio can vary from 0.7 to 3.0.

Shepelev and Gelman 52 and Regenscheit53 computed air temperature along the first three zones of jets supplied from outlets with finite size using the method of superposition of the multiple-jet system. These jets originate from the points with supply air velocity equal to the average air velocity at the outlet.

Along with principles described in Section 7.4.3, this approach utilizes the following equations describing temperature distribution in a compact jet and a heat flux at a given point (x, y, z):

 E 2l°-082*J (7.77)

0-0^ = 1 + Pr JAо -%f

00-9«, 2×0.082 Jt: x

L‘(0 ~ 0-J _ 1 + Pr ^0 ~~2~fn.0^2xl (7 7«

1/0(00-0.) 2tt0.0822×2 ‘ ‘ ■’ ‘

TABLE 7.15 Centerline Temperature Differential in Horizontai Jets

 0-0^ «X-«- Author Round jet Linear jet Radial jet Shepelev4 Gnmitlyn22 »/A0 *2 X KJH, ~ x K, 1 — Ж,) a/3 sin a x Abramovich57 0.7 — + 0.29 *« Rf — 0.66—0.76 1.04 Йп + 0.41 A/B0 A = 0.09-0.1 2 ~ x Koestel, s 5.35- X — — Batumi14 9.24R,, C a/ 6() 3 „ I *n /0. 3.27 A/ * V 9« ^x~x4]+fy%

The heat flux through the finite element dA of the jet cross-section at the dis­tance x from the outlet can be calculated as

^(0-<U] = 1 + Pr dApMpl^f

I/o(0o-0«,) 2ir0.0822 x2 ‘ 1 ‘

The double integral of Eq. (7.79) across the outlet area 2A x 2B results in the following equation for a heat flux through a given point of a jet supplied through a rectangular outlet:

^yz(9,yj-9J = if f [T+Pr y + A f FT+Pr Y-A ) i’0(00-8J 4{ tf 2 0.082x a/ 2 0.082*J

X(“fiTEr^fe-CTf/i5^)- ,7’80)

Joint solution of Eqs. (7.68) and (7.80) results in the following equation for the temperature differential along the jet axis:

Erf fTTTr A erf /1ЈP? B…………………………

Ex«-e„_ V 2 0.082^ A/ 2 0.082s:

6O-e„ U-A— erf_JiZ ‘ ‘

A/ 0.082* 0.082a:

In the case when air is supplied through a slot with a width 2B (A = Eq. (7.81) can be converted into

Velocities and Temperatures in Vertical Nonisothermal Jets

Studies by Helander et al.,65-68 Knaak,69 Koestel,38 Shepelev,4 Regenscheit,70 and Grimitlyn5 resulted in equations for downward and upward projected diffuser

Jets.

For the circular jet, Regenscheit70 obtained empirical equations for the

Maximum velocity in the downward and upward vertical jets of heated and cooled air. For a compact (round) jet,

Where m = parameter characterizing the diffuser jet (m from 0.1 to 0.3). For a linear jet,

(7.84)

Based on theoretical analyses, Koestel,58 Shepelev,4 and Grimitlyn5 developed equations for velocities and temperatures in vertical heated and chilled air jets. The assumptions used by these authors are similar, and the method used is described in Koestel.58 The assumptions used in the analysis can be summarized as follows:

1. The jet of warmed or cooled air is projected into an unbounded atmosphere of still air of uniform temperature.

2. The only force opposing the downward flow of the heated air or upward flow of the cooled air is a buoyancy force. In their analysis, Helander and Jakowatz65 also suggested accounting for inertial forces due to the entrainment of room air. However, this suggestion is not in an agreement with a principle of momentum conservation used in most of the existing models for isothermal jets.

3. The air entrained by the jet has room air temperature.

4. A velocity profile and a temperature difference profile have shapes that can be approximated by an error-function type curve.

For practical use the influence of buoyancy forces on temperature and ve­locity decay in vertical nonisothermal jets, as proposed by Grimitlyn,56 can be accounted for by the coefficient Kn of nonisothermality. For compact jets,

85)

(7.86!

Where K„ for a compact jet can be computed as

For linear jets,

(7.89)

Where

(7.90)

By applying the Ar* criterion, Eqs. (7.87) and (7.90) can be transformed into

 (7.91)

Kr, = V1 ±aArr >

Where a = 2.5 for axially symmetric and incomplete radial jets and a — 1.8 for linear jets. The plus sign in Eq. (7.91) corresponds to the situation when the directions of buoyancy and inertia forces coincide, whereas the minus sign corresponds to their counteraction. This equation can be used for vertical nonisothermal jets at Ar^ ^ 0.25.

Nonisothermal Jet Throw

The throw of downward-projected heated jets or upward-projected chilled jets can be derived from Eqs. (7.85) and (7.88) for Kn equal to some value, e. g., 0.1. Helander and Jakowatz,65 in their work on heated jets projected down­ward, have called attention to some of the differences between the actual con­ditions and those assumed for analysis. One of these is the radial escape of warm air in the terminal zone of a hot stream projected downward. This escap­ing warm air then rises and causes a change in ambient conditions for the up­per part of the jet. The terminal zone and the edges of the jet are zones of marked instability, with definite surges and fluctuations, so that the jet enve­lope is very difficult to define or to determine experimentally. In the closure to the paper presented by Knaak,69 Dr. Helander suggested that from the point of view of practical application, the distance to the beginning of the unstable, ter­minal zone of the jet is about 80% of the jet throw.

The data from Baturin and Shepelev,71 Helander et al.,66 Regenscheit/’4 Turner,72 Shepelev,4 Seban et al.,73 Sato et al.,74 Grimitlyn,56 Mizucbina,75 and Weidemann and Han el76 show that maximum downward/upward travel of heated/cooled compact jets can be evaluated using the equation

 (7.92)

^■max = ______

JВO JВTq

Where the coefficient a varies between 1.59 and 2.57 with a mean value of

A = 1.8 ± 0.3.

Considering that most of the data were obtained using round nozzles with jet characteristics close to ideal discharge conditions (K^ = 6.88, K2 = 5.85), Eq. (7.92) can be presented as56

 Z,

 Max

 (7.93)

 D JKTKFq’

Max_ 0.67KP D (K2Ar0)2/3 ‘

Trajectory of Horizontal and Inclined Jets

Buoyancy forces influence the trajectory of horizontally projected air jets or air jets supplied at some angle to the horizontal plane (Fig. 7.24). Most nonisothermal air jet studies were devoted to horizontally projected compact air jets. Based on the analytical studies,3,42’77-80 the trajectory axis of inclined jets can be described by a polynomial function

(7.951

Similar equations were suggested by experimental studies.81-85 Some au­thors determine the trajectory axis from equations of another kind—parabola, for example.84’85

In some studies on inclined air jets, the equation for the trajectory’ differs from Eq. (7.95) by the additional term as follows:

 Tg «(
 JAo
 A

 (7.96)

 JAo

T’aliev86 and Schneider87 derived the equations for the trajectories by nu­merical methods. Experimental data for the trajectory of the inclined jet (a0 & 0) were obtained only by Fleishbacker and Schneider.80

As was shown by Zhivov,88 the main difference in most of the equa­tions for the jet trajectory is the value of the coefficient i)j. The differences in experimental data obtained by different authors are mainly due to the diffi­culties in the measurements of nonisothermal air jets supplied with low ini­tial velocities (2-10 m/s). There is also a different understanding of the term “air jet trajectory.” Some authors mean the points with maximum ve­locity values, while others mean the centers of gravity of the cross-sections of the jet.

The analytical method of jet trajectory study developed by Shepelev4 al­lows the derivation of several other useful features and is worth describing. On the schematic of a nonisothermal jet supplied at some angle a0 to the horizon (Fig. 7.25), S is the jet’s axis, X is the horizontal axis, and Z is the vertical axis. The ordinate of the trajectory of this jet can be described as z = xtga + Az, where Az is the jet’s rise due to buoyancy forces. To evaluate Az, the elementary volume dW with a mass equal to dm = psdV on the jet’s trajectory was considered. The buoyancy force influencing this volume can be described as dP = g(p„ — ps). Vertical acceleration of the volume under the consideration is / = dP/dm — g(pO0 — pj/ps ~g(Ts — T„)/Ts. Vertical acceleration can be presented with the help of the vertical velocity compo­nent j = dVz/dr, where the time interval dr can be described as dS/V5. Based on these equations, the vertical component of air velocity can be pre­sented as

S J. _ J

 0- l. A-U. O- • — 2, A-12,0- e- 3, A -13, ■p — 9- 4, A -14,0­0- 5,0-15,^- Tf — 6, a-i6, v- 9- 7,a-17,T — d — 8, Q—18, ‘7­4-9, 0-19, V- • -10, «-20, V- + — -21, 22 _ / 23, 24, / Y 25, 26, I Y / A 2/, 28, 79 . A >30, 31 Ј N. 4 V 1 © 0 V & -*2 H r * •“V L x* ■V O 0< 9 A I V
 X tgg0 — VВ0

 Vz J’

 0 2 4 6 8 10 12 14 16 18 20 22 23 24 25

‘ X y «2 Ar0

.Va0/ Kr Cos «„

The ratio of local temperature difference and velocity on the jet’s axis in Eq. (7.97) can be substituted by

= |2Xj. Q,Z, ls> (7.98>

I’s Kr~ l’o ‘

Resulting in

K2gT0-T,

 V, =

O °° <^ /7 99j

Considering that = dz/dr, vz/ us = dz/ dS and vs = K1v0jA~0 /S, the equation for calculating Az can be rewritten as

 K2 gT0-Tt K, TK

Or

‘ — (7.100)

0 Vs ‘

A<: = . (7.101)

3KiT00VqJA()

Substituting ac cos a0 for S, and Ar0 for the complex of parameters, the re­sulting equation for the trajectory is

Z = ^a0±±f|Ar0(^) (7.102)

When a chilled air jet is supplied at the angle a0 upward, it will cross the level of the supply outlet at the distance x0. This distance can be calculated by substituting z = 0 in Eq. (7.102):

— V^cosa. Vsma.

The abscissa and ordinate of the jet vortex in the case of upward inclined cold air jet supply or downward inclined warm air supply were derived from Eq. (7.103):

K i cosan /sina0

«, — —……… ~ <7-K)4>

JK2 Ar0

= 2Ki(sina0p2

" 3 JK7KF0 ‘ 1 >

The ratio xv to zv depends only on tga0: zv/xv =2/3 tga0, and the ratio of xv/x0 has a constant value equal to 0.578. To clarify the trajectory equation of inclined jets for the cases of air supply through different types of nozzles and grills, a series of experiments were conducted.88 The trajectory coordi­nates were defined as the points where the mean values of the temperatures and velocities reached their maximum in the vertical cross-sections of the jet. It is important to mention that, in such experiments, one meets with a number of problems, such as deformation of temperature and velocity profiles and fluctuation of the air jet trajectory, which reduce the accuracy in the results. The mean value of the coefficient ‘P’ obtained from experimental data (Fig. 7.25) is 0.47 ± 0.06. Thus the trajectory of the nonisothermal jet supplied through different types of outlets can be calculated from

The accuracy of this value is sufficient (Fig. 7.26) to be used in designing the trajectory of inclined ventilation jets at an angle a0 s ± 45°. Using the exper­imental value of the coefficient, the equation for the vortex abscissa xb can be presented as follows:

Cosa0A/ sina0

 A 07

X„ =

Jlx0.47(K1/K})Ar0

 (b)

However, this clarification does not affect the ordinate-to-abscissa ratio, which remains equal to 0.578.

 La)

 X.

 D0 = 30 / /— / A R—< W 1 Hi ♦/ / SS
 12 16 20 24 28 32 36 38

 Y_ ‘I An

 12

 16

 20

 FIGURE 7.26 Trajectory of a nonisothermal jet supplied at an angle: (a) — a0 = 30°; (7.106). Reproduced from Zhivov. BB

 J c c R r

 FIGURE 7.28 Jet attachment with air supply through outlets located at some distance from the surface: (o)multiple jets; (b)long slot |et (c)rectangular jet, (d) rectangular corner jet: (e)general sche­matic. Reproduced from Nielsen90.

Fig. 7.29. These data indicate that the parameter Kwal1 = K1wall/Kl reflecting the influence of the wall on the velocity decay along the jet increases from 1 to

1 .4 with distance from the outlet. For a compact jet Kwa11 = 1 when x < 5d0 for a linear jet #Cwal1 reaches its maximum value of 1.4 only at x < 20Ј>0, where b() is an outlet width.

Studies of wall jets54-55 show that they have two layers: a turbulent boundary layer close to the wall and an outer shear layer. The thickness of the boundary wall layer can be neglected for practical purpose. Accordingly, to compute the maximum velocity in the wall jet, researchers4-’*’4,89 apply the method of images by treating the wall jet as one-half of a free jet. Appli­cation of this method gives a relationship between the characteristics of a wall jet and a free jet, which results in a correction factor equal to J2. This approach has some inaccuracy even with linear and radial jets, in a For a three-dimensional wall jet, the procedure is even more approximate. Discus­sion by Etheridge and Sandberg89 of some previous studies of attached jets indicates some loss of momentum in an attached jet due to the friction against the surface. The authors compiled information from previous stud*

 J. O • I* M O O’ 1! C ‘ • • I O -1 • -2

TOC o "1-5" h z -4— 1—, m I.. ……………………………………. t,.— ——— ——— 1…… — I……………. .J—

0 10 20 30 40 50 60 70

, , x/d0

 I X A X • 1 • I* M X T ■ X X • -1
 0 20 40 60 80 100 120 140 Ih xlh<>

 V V V ,w p 1
 0 4 8 12 16 20 X/d0

(c)

FIGURE 7.29 Correction parameter KW3B reflecting the influence of the wall: (o) compact jets (experimental data from V. Mitkalinny, A. Abdushev, V. Baharev and L. Fedorov); (b) linear jets (experi­mental data from W. Kerka and Z. Sakipov); (c) radial jets (experimental data from N. Gelman). Repro­duced from Grimitlyn.56

Ies, which is summarized in Table 7.16. According to the equations pre­sented in Table 7.16, the maximum velocity in a wall jet is inversely proportional to the distance from the outlet in a different power compared to a free jet.

It is not uncommon to supply air into the room with jets attached both to the ceiling and to the wall surfaces.1’90 Air jets can be parallel to both surfaces or be directed at some angle to one or both surfaces (Fig. 7.28). Studies of compact wall jets supplied parallel to both surfaces reported by Grimitlyn1 show that the correction factor value is in the range from 1.6 to 1.7, which means that restriction of entrainment from two sides reduces velocity decay by 20% to 30% compared to the case of a wall jet.

When a jet is supplied at some distance from the surface, the attachment oc­curs when the distance between the outlet and the surface is below a critical dis­tance; otherwise the jet will propagate as a free jet.97 If the jet attaches to the surface, the flow downstream of the attached point is similar to that of a wall jet. For a compact isothermal jet, the critical distance for jet attachment to the surface is Lcm = 6/4g/2.98 For Ltrit < 6Aq’2 the velocity decay coefficient K]

TABLE 7,16 Maximum Velocity Decay in Wall Jets

 Type of jet Decay of the maximum velocity Reference Linear wall jet -0.555 — X -0. ’75 ~ X Schwarz and Cosart^ Regenscheir*4 Radial wall jet -1.12 -1.15 ~ .V Bakke*3 Wasciike-‘4 Rectangular wall jet Hq/L = 0.025 Hq/L = 0.05 Hq/L = 0.01 H0/L = 1.0 -1.15 — X -1.09 ~ X -1.15 ~ X -1.14 — X Sforza and Herbst95 Compact wall jet -1 ~ X Nielsen-^

Becomes greater than it would be in the case of free jet, and should be corrected using correction factor F (see Fig. 7.30} to compensate for surface proximity.

The length of the recirculation zone, xa, for a linear jet (the distance to the point of jet attachment to the surface) was studied by Sawyer," Miller and Comings,36 and Bourque and Newman.100 The results of these studies, sum­marized in Awbi,97 show that the length of the recirculation zone (Fig. 7.31) is proportional to the distance from the outlet to the surface and can be de­scribed as

Xa/H0 = 0.73(D/H0) — 2.3, (7.109)

Where HQ is the width of the outlet.

Sandberg et al. tot conducted similar tests with a heated linear jet so that the buoyancy forces opposed the forces due to the lower pressure in the circu­lation zone (bubble). Based on the results of these tests, it was concluded that

 Distance of jet centerline from surface,

V^o

 D/h FIGURE 7.31 Effect of supply distance from surface on the attachment distance for a linear jet. Reproduced from Awbi.97

Heating the jet does not change the location of the attachment point. For 5 < D/H0 < 13 the length of the circulation zone followed the relation

Xa/H0 = 1.175(D/H0) + 6.25 . (7.110)

Calculation of xa/H0 using Eqs. (7.109) and (7.110) results in significantly different results. For D/H0 = 8, xa/HQ =3.54 according to Eq. (7.109), and xa/H0 = 15.65 according to Eq. (7.110).

The length of the circulation zone (bubble), Lb, created when the linear jet is supplied at an angle a to the surface was studied experimentally by Bourque and Newman100 and theoretically by Sawyer." The effect of the angle between the jet axis at the outlet and the surface on the length of the circulation bubble is shown in Fig. 7.32, reproduced from Awbi.97 The data presented in Fig. 7.32 show that at sufficiently high Reynolds number the length of the circula­tion zone is independent of the Reynolds number.

Linear jet attachment to a plane not parallel to the supply direction was studied by Katz.103 The critical angle, 0C, of the plane to the jet supply direction, as indicated in Fig. 7.33, was found to be dependent on the supply velocity (Reynolds number). It also depends on the distance of the plane edge from the supply oudet (see Fig. 7.34).

Baturin54 studied air jets supplied from rectangular nozzles at some angle to the plane with an edge of the nozzle coincident with the plane. The results of his studies indicate that the critical value of the angle of the jet supply direction to the plane is 45°. It was also shown that the jet supplied through a rectangular outlet with a noz­zle located at some distance from the plane does not attach to the surface.

7.4.4.6 Jet Separation

When the temperature of an air jet attached to the ceiling is lower than the temperature of the ambient air, the jet will remain attached to the ceiling until the downward buoyant force becomes greater than the upward static pressure (Coanda force). At this point, the jet separates from the ceiling and

 A° FIGURE 7.32 Effect of supply jet angle on recirculation bubble length. Experimental data and theo­retical curve from Sawyer.42 Reproduced from Awtn.97

 FIGURE 7.34 Critical angle. Wall is located at different distances from the air supply outlet. Repro­duced from Katz.103

Begins a downward-curving trajectory.104 Studies of nonisothermal jets con­ducted by Grimitlyn,5 Schwenke,105 Nielsen and Moller,106 Miller,104 Ander­son et a!.,107 and Kirkpatrick et al.108 showed that the distance to the point of the jet’s separation can be computed using the following equation:

 .111)

_ a

Ь ■

 For linear diffuser jets109 10.5

(Ar0)

A = 2.5H0 and b = 2/3. For compact diffuser jets a = 1.6Aq3 and b — O.5.107’108 According to theoretical analysis and ex­

Perimental data collected by Grimitlyn,5 the separation distance of jets could be expressed by equations similar to Eq. (7.111) considering diffuser charac­teristics Kj and K2. For compact and incomplete radial jets,

 (7.112)

 For linear jets,

0.55KiJД~o

OAKi^Ho (K2Ar0)2/* ■