Jet Interaction

Diffuser jet interaction is a common case when air is supplied into ventilated rooms through multiple air diffusers (e. g., sidewall-mounted grills or ceiling-mounted air diffusers) or using speciality air distribution systems. Interacting jets can be supplied

• Parallel to each other

• In the opposite direction toward each other (e. g., from the outlets located on the opposite walls)

• Coaxially

• At an angle to each other

Studies of some common cases of jet interaction are discussed in this section.

Interaction of parallel air jets is the most common case. It was thor­oughly studied by Baturin,138 Koestel and Austin,139 Grimitlyn,56’140 Bashus and Kocheva,141 Posokhin,142 Nosovitsky,143 Kuzmina,144 Vasilyeva,14’ and Shepelev.42 Researchers developed equations describing velocities and tem­peratures in two or more interacting jets assuming that momentum and heat content of the flow through the elementary area in the cross-section of the resulting jet is equal to the sum of momentums and heat contents through the same area of the separate interacting jets. It was also assumed that sepa ­rate jets do not influence each other. Derivation of velocities based on these assumptions described by Shepelev42 is presented below.

Air velocity vy at any point (X, Y,Z) of the flow created by interaction of two parallel jets supplied from outlets located at a distance 2 a from each other (Fig. 7.50) can be described by

Fv = v + if2 , (7.140)

Where

V -, =

подпись: v -, =

X

подпись: x.141 )

(7.142)

подпись: (7.142)*1 voJK

V2 = ——e.

Air velocity in the resulting flow in the plane of the interacting jets’ axis (Z = 0) derived from Eqs. (7.140)-(7.142) is

KiVqJВ~q

X

Y-a Y | Y±a |2О

CX 1 cX

E v ’ + e

Jet Interaction

(7.143)

 

Air velocity on the axis of one of interacting jets (Y = a or Y = — a) is

CX

1 + e

K^JA, Vxl ~ X

Jet Interaction

7.144)

 

If X = (vxl — vx)/vx is the relative increase of axial air velocity in one of the in­teracting jets due to the influence of the other, then the length of the jet X"

Jet Interaction

Within which the influence of the interacting jet would be less than, e. g., 10% (X = 0.1) can be obtained from

X* = —…….. ■:… ;… -………….. — =….. ……. 26.4a, (7.145)

C 7-lnX(2 — X) J-X(i-X)

Which means that the axial velocity of the interacting jet is influenced by an­other jet only beginning with X* exceeding 13.2 times the distance between the outlets.

Velocity along the axis of symmetry between the interacting jets can be calculated assuming Y = 0 in Eq. (7.143):

= IJA46)

X

Air velocity along the jet supplied from the outlet with opening area equal to

2A0 is

_ KlV0j2A0

Vx = ~ • (7-l47)

If X = (vx — vx)/vx is the relative difference between the axial air veloc­ity in the jet with double opening area and the air velocity along the axis of symmetry of the interacting jets, then the length of the jet X*" within which this ratio would be less than, e. g., 10% (X = 0.1) can be obtained from

This approach can be used for interacting jets supplied from outlets with dif­ferent area of discharge with different initial air velocities. In this case the equation for air velocity in the flow of interacting jets will be

K!

X

_ !=JО r I Y + a

2 a I cX I ^ ~ cX

.^Ol^Ol6 + v02^2e

Jet Interaction

(7.149)

 

Comparison of the calculations using Eq. (7.143) with experimental data collected by Vasilyeva145 at v0l = 38.1 m/s, v02 = 36.6 m/s, D01 = 0.03 m, Dq2 = 0.04 m, a = 0.05 m is presented in Fig. 7.50 from Shepelev.42

Jet interaction should not be taken into account when the jets are closely adjacent to each other, are propagated in confined conditions, and entrainment of the ambient air is restricted. This may be the case for con­centrated air supply when air diffusers are uniformly positioned across the wall and the jets are replenished by the reverse flow, which decreases the jet velocity. This effect should be taken into consideration using the confine­ment coefficient Kc discussed in Section 7.4.5. For the same reason, jet inter­action should not be taken into consideration when air is supplied through the ceiling-mounted air diffusers and they are uniformly distributed across the ceiling.56

For the most common practical situation, when air is supplied by parallel jets from several diffusers placed in one plane and having the same outlet area A0 and discharge velocity v0, the resulting velocity on the axis of the coalesced flow V — can be found:56

• For compact and incomplete radial jets from

^#Kint (7.150)

• For linear jets from

Jet Interaction

The above relations can also be used as a first approximation to find the temperature drop in interacting jets by substituting A0T, A0O and AK2 for vy, v0, and K2, respectively. The values of interaction coefficient Kin( for even and odd numbers of outlets are given in Fig. 7.51, reproduced from Grimitlyn.56

7.4,6.2Interaction of Jets Supplied from Opposite Directions

For practical applications related to space ventilation and air-condit­ioning, interaction of similar jets with equal size and initial momentum sup­plied from the opposite walls (Fig. 7.52) were studied by Roeder,146 Conrad,147 Urbach,148 Regenscheit,149 and Smirnova.1

Jet Interaction

FIGURE 7.51 Coefficient Kintof interaction for the jets discharging from the opening located in a single row. Reproduced from Grimitlyn.56

Jet Interaction

FIGURE 7.52 Interaction of two jets supplied from the opposite walls.

Conrad147 compared impingement of two opposite linear jets attached to the ceiling with a linear jet changing its direction after impingement with a wall. For the attached jet, maximum air velocity along the jet can be described by

V ( bn "l0-375

VAm ’ (7J52)

Where m is a supply outlet characteristic that can range from 0.1 to 0.4,58 and X is the distance from the slot to the point of interest. After changing the jet direction, the velocity in the vertical jet can be obtained from

‘hзA 0.375

F L 1

4

/7

ML J

L-Y

V /

5

= K

(7.153)

Where Y = vertical distance from the ceiling to the point of interest, L = length of jet travel along the ceiling; K = 1, q = 0.2 when the jet travels vertically along the wall; K = 0.65, q = 1 in the case of two-jet interaction. In discussion of the data obtained by Conrad and Roeder, Regenscheit149 suggested that the values of K and q also de­pend upon the relative distances L/h0 and Y/fe0 and the characteristic m. Based on the data by Urbach,148 Regenscheit concluded that K and q parameters also depend upon ratios L/H and h0/H. Research data also show that air velocities in the com­bined vertical jet are lower than in the jet after its interaction with a wall.

The above data as well as studies of compact and radial jet interaction conducted by Smirnova, Avdeeva, and Gunes were summarized by Grimitlvn.1 Grimitlyn suggested that the air velocity in the jet resulting from impingement of two similar opposite jets is

V-l = O’* (7.154)

Where K°Јt can be evaluated from the graph in Fig. 7.53. The graph shows that smaller relative distance between jet supply outlets and the point of jet interaction, a/b0 (for linear jets) or a/ JA~0 (for radial and compact jets), re­sults in smaller air velocities in the combined jet.

7.4.6.3. Interaction of Coaxial Jets

During the last two decades, a new generation of HVAC systems with concen­trated air supply assisted by directing jets was introduced in several European coun­tries.150’151 In one common modification, the main streams of ventilating air (heated or chilled) are supplied through a small number of air openings (grills) at low initial velocities and distributed within the space by horizontal (coaxial with main streams)

Jet Interaction

FIGURE 7.53 Coefficient Kintof opposite jets interaction. Reproduced from Grimitlyn.1

And vertical (supplied perpendicular to the main streams), or only horizontal, direct­ing jets (Fig. 7.54). These jets are discharged at high velocities from nozzles having small outlet diameters. The air is delivered to these nozzles from a separate air han­dling unit. The same principle is utilized by the “air piston system,”151 in which hor­izontal directing jets are created by axial or radial fans located along the main streams. Analytical and experimental studies on the interaction of main streams of supplied air and horizontal directing jets in laboratories and in the field conducted by Zhivov25’152’152 laid the ground for the design method of such systems.

Interaction of the Free Isothermal Main Stream and Horizontal Directing Jets

The characteristic feature of the main stream and horizontal directing jet inter­action is that the directing jets are supplied through nozzles located at some distance from each other and from the outlet supplying the main stream (Fig. 7.55).

A)

W

I V t“

.Jjyj-‘ "

^………..

| — V-

2

_T — _____ _____________ 3

I ( "r— X7A V7k U

^o. xmax ^

va m m wa

Hr

^o. x

_ X, _

K

Jet Interaction

FIGURE 7.54 Concentrated air supply with directing jets: (a) with horizontal directing jets; (i>) with vertical directing jets; I—main stream; 2—main stream air diffuser; 3—horizontal directing jet; A—horizon­tal directing jet nozzle; 5—vertical directing jet; 6—vertical directing jet nozzle. Reproduced from Zhivov,131.

Jet Interaction

FIGURE 7.55 Schematic of free isothermal main stream and horizontal directing jet interaction: I— main stream (D0!, V0), /0I, d0l, Kn); 2—directing jet (D01, Vn, ln, d01 Kn), Reproduced from Zhivov, 131

Experimental studies with propane, as a tracer gas, introduced into the main stream showed that directing jets make the main stream narrower. Velocity profiles vr; in cross-sections of the resulting stream (created by the main stream and direct­ing jets) can be described by the formula derived, assuming the resulting stream mo­mentum is equal to the sum of interacting jet initial momenturns. For maximum velocity in the resulting airflow in the cross-section of the ( N — 1) nozzle, this equa-

Tion is

/ V. N ~ 1

(7.155)

подпись: (7.155)^02 foil Kill, V — i.

K2’

Jet Interaction

Where Kll and KJ2 are coefficients of velocity decay in the main stream and di­recting jets, respectively, Vq2 is the horizontal directing jet supply air velocity, Im and I02 are main stream and horizontal directing jet momentum, respec­tively, and J02 is Ae horizontal directing jet nozzle diameter.

To derive the equation for the jet boundary resulting from the interaction of coaxial main flow’ and a directing jet supplied at the distance /0 from the main outlet, this interaction was presented152 as the interaction of the main jet with a sink distributed along its axis (Fig. 7.55). Considering the influence of the directing jet on the main flow boundary as AY, the half width of the result­ing flow can be presented as

(7.156)

подпись: (7.156)Yh = AX — AY,

Where

Jet Interaction

(7.157)

A is the coefficient characterizing the angle y of the main flow divergence (Fig. 7.55) without the directing jets’ influence. The following relationship was de­rived for the resulting flow boundary:

Yb = AX l(MX)

Ft 12 *01

Where 8 is an experimental coefficient and X = X/lQ.

In the case of several directing jets interacting with a main stream, the above approach was used assuming that each following directing jet interacts with the resulting flow created by the main flow and the previous directing jets. The equation for the resulting flow boundary differs from Eq. (7.158) only by the expression for the 4>(X) function.

Interaction of the Confined Isothermal Main Stream

With Horizontal Directing Jets

The experimental studies152 show that the resulting jet length in the con­fined space can be divided into three zones (Fig. 7.56). In the first zone, there is an expansion of the resulting jet boundaries. The length of this zone (X’t) de­pends upon the relative momentum Uot/^oi) value and the relative distance (/,■/jAr) between the directing nozzles, where Ar is the room vertical cross­section area (br x hr). The distance (Xr) increases when the relative momentum (Whi) increases and the relative distance (/,/ jAr) decreases. In the second zone, the resulting jet width stays relatively constant. It expands up to the last nozzle, and its length is equal to (Xn — X,) and depends on the number of di­recting nozzles and the distance between them.

In the third zone, there is a significant decrease in resulting jet width. The cross-section in which the jet flow degrades is considered the end of the third zone. The length of the third zone (Xm — Xn) is practically equal to the length of the jet’s degradation zone in the confined space without directing jets, which is 2 jA~r. Within the studied range of parameters, the resulting jet throw (XUI) reached 10 jAr.

Beyond the third jet zone, there is a stagnant zone in which the velocity values are relatively uniform and have an unstable direction. There is reverse flow in zones I through III which is located between the jet boundaries and the cylinder walls. The maximum value of the velocity in the reverse flow is in the cross-section at the end of zone I at the distance Xv The following equation was derived to calculate the length of the first zone Xt:

W 0.755,, 0 ^021Av

",60’

The average experimental value of the coefficient 0 is 1.7 with a standard de­viation (ct9) of 0.05. Equation (7.160) allows one to calculate the momentum ratio (Iq2/hi) required to extend the length of zone I to the value equal to

Xj, given that the distance between the directing nozzles is equal to The

Graph presented in Fig. 7.56 is plotted according to Eq. (7.160) for Kvl and Kj2 equal to 6.2. The maximum value of reverse flow velocity (vKV) was found to be in the cross-section at X equal to Xj:

Jet Interaction

Verse flow and confining surfaces increases the value of fx to 0.2. Based on Eq. (7.162) and experimental |jl values, the maximum initial air temperature sup­plied by the main stream is limited by

O 0 — — Kh17fitdoij; ,-7 — tss*

©o r ~ Kj. /2 ‘ ’ ( . .166)

Where a is a coefficient equal to 2.65 for a free jet and 7.07 for a confined jet.

7.4.6.4 Interaction of jets Supplied at an Angle to Each Other

There are only a few studies of air jets supplied at some angle a (0° < a < 90°) to each other. To predict characteristics (trajectory, velocity decay, etc.) of the flow resulting from interaction of two jets supplied at some angle toward each other, Hudenko153 proposed to sum momentums of interact ing jets as in the case w’ith parallel jets. He has estimated that the error of predic tion will be smaller at a smaller

• Interaction angle

• Distance between the supply nozzles

• Difference in the nozzle sizes

• Supply air velocity values

Meshalin154 conducted experimental studies of twro equal jets supplied at angles of 15°, 30°, and 45° to each other. Based on the results of his studies, the author concluded that

• The turbulent mass transfer in the flow resulting from the interaction of the two jets is more extensive than in a single jet under the same supply conditions.

• The intensity of mass transfer in the flow’ in the plane of the interacting jets’ axis is lower than in the plane of symmetry.

• The intensity of mass transfer in the resulting flow increases with the angle of interaction.

Numerous studies of jets supplied into a uniform or nonuniform cross-flowr were conducted in application to such areas as air pollution control, burning processes, etc. Detailed discussion of these studies is beyond the current review. However, some results of these studies will be mentioned as needed in the following section.

Interaction of a Free Isothermal Main Stream with Directing Jets

Supplied at a Right Angle to the Main Stream

As in the case of the interaction of coaxial directing jets, the interaction of main streams w’ith directing jets supplied at a right angle was studied25’131 to develop a design method for air distribution with horizontal and vertical di­recting jets.

The discussion of the interaction of air jets supplied at some angle to each other shows that application of the method of superposition of the interacting jets’ momentums and surplus heat to predict velocity and temperatures in the combined flow results in inaccuracy when two unequal jets are supplied at a right angle. A different approach was undertaken in the studies of interaction of the main stream with vertical directing jets.25’131

Visualization studies of the resulting flow showed that the directing jet changes its initial direction due to its interaction with a main stream. The interaction results in two separate flows; the first is a continuation of the main stream and the second is a continuation of the directing jet. The spe­cific feature of this interaction is that vertical directing nozzles are located within a main stream (Fig. 7.57). The median diameter of the directing jet is significantly (several times) smaller than the main stream diameter within a zone of their interaction. Thus, the interaction of the main stream and the vertical directing jet can be seen as an interaction of the axisym — metric (directing) jet with an infinite cross-draft with a nonuniform veloc­ity profile.

An analytical solution of the interaction in the case of isothermal main and directing jets, assume that the main stream (Fig. 7.57), supplied with ini­tial velocity (v01) through a nozzle that has internal diameter is develop­ing within a zone ( — l0, 0) as a free jet. The momentum (Ij) of the jet within the zone ( — l0 + lc, 0) remains equal to the initial momentum (/0|)> and the ve­locity distribution in the cross-section of interaction in the plane X V remains the same within the zone (0, X^). The axisymmetric main stream within the zone (0, XA) is substituted by the linear flow with velocity profile that can be described by the formula

I>i = vme ‘ du ‘, Xee [0, XA], c = 0.082 (7.167)

The directing jet is supplied at a right angle to the main stream axis with an initial velocity of v03 from the nozzle with an inner diameter (ti0i) located at the distance (/0) from the plane of main stream supply and at the distance Y0 from its geometrical axis. The momentum vector compo­nent along the Y axis remains constant and equal to the initial momentum (Fig. 7.57): ‘

/3cosp = /03, (7.168)

Jet Interaction

FIGURE 7.57 Schematic of free isothermal main stream and vertical directing jet interaction: I — main stream (D0,, V0,, /ol, dol, K,,); 2—directing jet (D03, VM, l03, dm, Kl3). Reproduced from Zhivov. 131

Erfl Xz_Xq) + erff — j-

Clo

CL

Cl

0

R

/

].5~

1 —

Rfr]

V

V-, _ _

Y^COs(Ј/,-3,w) =

.71)

Where r, = JSj/it, Rfr = b/2, and 5, is a cross-sectional area limited by the constant velocity line. The joint solution of Eqs. (7.168), (7.170), and (7.171) results in the following expression for the maximum velocity along the directing jet:

Where p is the angle between the Y axis and the tangent to the directing jet tra­jectory

. ■, TT<in; , ,,

103 — Po.^03-4^• i/.ibJi

The aerodynamic force (P) of the main stream and the momentum of the

Injected air (iinj) change the X component of the directing jet. The X compo­

Nent of the directing jet can be calculated from

/.jsinfl = P +1,nj. (7.170)

Experimental studies have shown that velocity distribution in the cross­section of the directing jet can be described by the same equation as those in the axisymmetric jet in a cross-draft,

 

Jet Interaction

= oi Vf„ Y ’

 

.172)

 

Where

 

M2 1 ! + a_~Jcosp ‘

Qk

 

.173)

 

When p < 25°,7cosP = 1.

The differential equation for the directing jet trajectory was obtained by the joint solution of Eqs. (7.168) and (7.170) assuming,/cosp = 1:

 

Tga-dX + ±A

R&P 7y f———- 1 — T v °r———————- ’

Tt I Jf|,? ^ Xv l03

 

(7.174)

 

Where

 

Y-Y,

 

Jet Interaction

A = e

 

— e

 

(7.175)

 

The equation for the directing jet trajectory was obtained by the integra­tion of Eq. (7.174) at X = 0, Y = 0:

7.176)

подпись: 7.176)_ ck0mll01

X — T „ J— D, 3 Xu *03

Y-^Hk

Erf

Y-Y,

Where B = (Y-Y0) + Yrr

Jet Interaction
Jet Interaction

‘ 2

 

Based on Eqs. (7.175) and (7.177) one can conclude that beyond the bound­ary of the main stream the directing jet has a straight trajectory.

Visualization studies of the directing jet showed that after interacting with a main stream the directing jet has a straight trajectory when (3 is less than 50°. At a greater value of p (?gp > 1.2) the directing jet trajectory is significantly curved.

In the case of a nonisothermal directing jet, the above assumptions are true, except that the momentum vector component along the Y axis changes due to the buoyancy force:

(7.178)

подпись: (7.178)Dlv3 = ±dG

The amount of heat W3 along the directing jets remains constant.

Jet InteractionExperimental studies have shown131 that the temperature distribution in the cross-section of the directing jets can be described as follows:

(7.179)

Based on the above assumptions, the following equations were derived to

Jet InteractionCalculate the velocity and temperature decay along the nonisothermal direct ing jet:

Jet Interaction(7.180)

(7.181)

A 1 gdnt 0m-er ( y I2

подпись: a 1 gdnt 0m-er ( y i2
 
Where

(7.182)

And

Jet Interaction

(7.183)

In the case when there is no main stream (v()l = 0, q = and 0 = 0), Eqs.

(7.180) To (7.183) reduce to those for free jets.

Joint solution of Eqs. (7.170) and (7.178) allows one to calculate the maximum amount of heat supplied by a directing jet with the assumption that the jet reaches the occupied zone (vm3 > 0.1 m/s) and (tg$n-tg$)/tg$ is less than 0.2 at the point where it enters the occupied zone. The maxi­mum initial temperature difference of the air supplied by vertical directing jet is

Jet Interaction

Based on the experimental results, the following values of coefficients were found: A = 9.5, k = 0.25, k0 = 3.3 at 0° < (3 < 25° and = 2.4 at 25° < p < 50°.

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