Isothermal Free Jet
For different types of free jets and air diffusers there are similarities in the resulting flows. Four major zones are recognized along a free jet. These zones, as described by Tuve,2 may be roughly defined in terms of the maximum or center core velocity that exists at the jet cross-section being considered (see Fig. 7.20a).
Zone 1 is a short zone, extending about two to six diffuser diameters (for compact and radial jets) or slot widths (for linear jets) from the diffuser face. In this zone, the centerline velocity of the jet remains nearly equal to the original supply velocity throughout its length.
Zone 2 is a transition zone, and its length depends upon the diffuser type. For a compact jet the transition zone typically extends to eight or ten diameters from the outlet. Within this zone, the maximum velocity may vary inversely with the square root of the distance from the outlet. Some researchers 3-5 suggest use of a simplified scheme of the jet (Fig. 7.206) with a transition cross-section for practical purposes.
Zone 3 is the zone of fully established turbulent flow. It has major engineering importance, since it is usually in this zone that the jet enters the occupied region. The length of this zone depends on the air jet shape, the type and size of supply air diffuser, the initial velocity of the air jet, and the turbulence characteristics of the ambient air.
Zone 4 is a terminal zone in which the residual velocity decays quickly into large-scale turbulence. Within a few diameters, the maximum velocity subsides to below 0.25 m/s. Though this zone has been studied by several researchers,6’7 its characteristics are still not well understood.
In some practical applications of air supply (e. g., multiple-jet ceiling diffusers, an annular jet collapsing into a compact jet, jets from rectangular outlets becoming round or elliptical, or multiple streams merging when air is supplied through perforated panels), measurements and accurate jet description in Zone 1 and Zone 2 may be difficult.
7.4.3.2 Velocity Distribution in Jet Cross-Section within Zone 3
Velocity distribution profiles in Zone 3 of the jet were found to be similar.8“10 They can be computed by applying momentum-transfer theory (Prandtl-Tollmein) and vorticity-transfer theory (Taylor-Goldstein). Modification of these theories with different assumptions has resulted in several equa-
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FIGURE 7.20 Turbulent |et (a) schematic with four zones; (b) simplified jet schematic. |
Tions for jet velocity profiles. These equation are presented in Table 7,10 and can be divided into two groups:
• Profiles with finite boundaries11’12 with a velocity of zero at the specified distance from the jet axis
• Profiles with indefinite boundaries13,14 with air velocity decreasing with distance from the axis asymptotically approaching zero
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TABLE 7.10 Equations for Velocity Profiles in a Free Jet.
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In the equations listed in Table 7.10, r(y) is the distance from the point of interest to the jet axis, and 8 is the distance to the jet boundary, which can be obtained from equations summarized in Table 7.11.
It has been demonstrated by Ruden, lfi Albertson et al.,10 Taylor et al.,i7 Keagy and Weller,18 Pai,19 Becher,20 Forthmann,9 Nottage et al.,21 Shepelev,4 Grimitlyn,1 and other researchers that the Gauss error-function profile by Rei — chardt is in agreement with data from studies of both nozzle jets and manufactured air diffusers supplying similar jets. This profile is utilized mostly by researchers using the analytical (semi-empirical) approach in air jet studies. Table 7.12 lists some modifications of the Gauss error-function velocity profile equations used for practical applications.
The Schlichting finite-boundaries profile is another one that is frequently used.23 Utilization of this profile is specifically fruitful for describing velocity distribution in complex flows, e. g., a jet in a cross-flow24 and jet interaction under the right angle.25 In such cases, distance from the jet axis, r(, to the point with an air velocity V,- is replaced by the parameter r, = (S,/7r)l/2, where S, is the area within a contour with a constant velocity Vj.
7.4.3.3 Centerline Velocity in Zone 3
Compact Jet
Centerline velocity in Zone 3 of the supply jet can be calculated from the equations based on the principle of momentum conservation along the jet: 26 27
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Linear jet |
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Radial jet |
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Author Round jet |
M0 = 2trp I’’ V2 y dy (7.39)
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TABLE 7.11 |
Equations for Jet Boundaries |
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5 |
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Author |
Round |
Linear |
Radial |
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Tollmem11 |
0.151* |
0.272.x |
— |
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Reichardt13 |
0.085* |
— |
— |
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Heskestad15 |
— |
— |
0.288* |
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Tuve2 |
— |
— |
0.185* |
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TABLE 7.12 Practical Modifications of the Gauss Error-Function Velocity Profile Equations |
Where M0 = pvqAq = initial jet momentum and y* = distance from the axis to the jet boundary.
Application of the Gauss error-function equation for velocity profile in the form proposed by Shepelev (Table 7.12) in Eq. (7.39) results in the following formula for the centerline velocity in Zone 3 of the compact jet:
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(7.40) !7.4i) |
JьcJ p„x — Equation (7.40) can be also presented as
_ 0<Ј VoJВQ
, zC~ hr x
Where
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(7.42) |
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9 = |
And
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R l |
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A T-<P = |
 |
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||
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||
Are the coefficients of the velocity distribution at the diffuser outlet.
When outlet velocity distribution is uniform, <f> — 1, and vQ = average air velocity at the diffuser outlet (vQ = q0/A0), The complex of coefficients ®if>/Jпic reflecting the conditions of the air supply has a constant value for a given situation and is called the dynamic characteristic of the diffuser jet.4 Dynamic characteristic describes the intensity of velocity decay along the air jet
Axis:
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‘.44) |
Qtp
Jvc
The above approach for the centerline velocity computation was utilized by different researchers using other velocity profiles and resulted in
‘.45)
Theoretical values of characteristic depend upon the type of velocity profile equation and supply conditions assumed. According to Shepelev, Kl = 6.88; another estimate is = 6.7.28 The Schlichting profile results in K: = 7.4, and with the Tollmein profile Kt = 7.76.29 According to experimental studies reported by Tuve,2 the range of Kt characteristic for compact jets discharged from round outlets varies between 5.7 and 7 depending upon supply air velocity and type of outlet. Analysis of experimental data from different researchers by Rodi30 indicates that Kt is close to 7.
Some researchers (e. g., Abramovich,26 Baturin,31 Rajaratnam,32 and Nielsen and Moller33) consider x to be the distance from a point located at some distance xQ upstream from the diffuser face. Equations for the jet boundaries and velocity profile used in the centerline velocity derivation assume that the jet is supplied from the point source. Addition of the distance x0 to the distance from the outlet corrects for the influence of the outlet size on the jet geometry. For practical reasons some researchers neglect x{).
Linear Jet
The equations for centerline velocities in a linear diffuser jet can be derived using the same principles as in the case of a compact jet. For a linear jet:
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(7.46)
Where Hy = height of the slot.
Similar to the case with compact air jet supply, theoretical values of characteristic Kj depend upon the type of velocity profile equation and supply conditions. According to Shepelev, K] = 2.62. The Gortler profile results in iCj = 2.43 and the Tollmein profile in K1 = 2.51.29 Becher34 reported the Kl characteristic for a linear jet to be equal to 2.55. Experimental results by Heskestad,35 Miller and Comings,36 van der Hegge Zijnen,37>38 Gutmark and Wygnanski,39 and Kotsovinos and List40 appear to satisfy K1 = 2.43.
Radial Jet
The application of the principle of momentum conservation to the radial jet by Koestel41 resulted in the following equation for the centerline velocity
(Fig. 7.21):
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(7.47) |
JK(H0/R0)cosQ[K(H0/R0)cos6 + 1]
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JR(R — R0) R0
The value of the numerator of the right-hand side of Eq. (7.47) depends on the geometric configuration of the outlet (R0, H0, cos0 ). The denominator repre sents the dimensionless distance from the outlet. For a given diffuser or a plaque Eq. (7.47) becomes
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(7.48) |
Vr _ CRp
VO JR(R-Ro)
Experimental C values for a radial slot and a radial nozzle tested by Koestel are 1.13 and 1.19, respectively. Eq. (7.48) is similar to Eq. (7.45) if the distance from the outlet is large enough so that (R — R0) is approximately equal to R. The theoretical value of the characteristic K1 in Eq. (7.45) applied to a radial jet is equal to 1.05, according to Shepelev.42
Similar derivations by Regenscheit43 resulted in the following equation for the radial jet centerline velocity:
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(7.49)
Where
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(7.50)
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D„
| H.
IRq-Dq ———— *-j
(fc)
[ FIGURE 7.21 Schematic of a radial jet: (a) general case 0 a 0 (b) air supplied through a diffuser with a plaque: 0 = 0.41
Referring to the data from Baturin,44 Regenscheit evaluated aF to be
0. 377. At a significant distance from the supply outlet (R0/R —->0), Eq. (7.49) can be transformed into Eq. (7.45).
Equations (7.45) and (7.46) show that air velocity along compact, linear, and radial jets is proportional to the value of K1. This parameter depends upon
1. Jet type
2. Diffuser type
3. Initial air velocity45 or Reynolds number46-48
A fruitful approach for velocity computation in the first three zones of jets supplied from outlets with finite size was developed based on the hypothesis that momentum diffuses with distance from the source in the same manner as heat energy.49’50 This approach, developed by Elrod,51 Shepelev and Gelman,52 and Regenscheit,53 utilizes the method of superposition of jet momentum from the multiple-jet system. These jets originate from the points with supply air veloc
Ity equal to the average air velocity at the outlet. This approach utilizes the following principles:
1. Momentum conservation along the jet
2. Air velocity in each jet cross-section, described using the Reichardt Gauss error-function profile
3. Constant angle of divergence along the jet
This method was applied by Shepelev and Gelman 52 to air supply through a rectangular outlet with dimensions 2L x2B (Fig. 7.22), resulting in the following equation for air velocities in the jet cross-section located at the distance x from the outlet:
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L^l + er(L±l |
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Erf |
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Cx |
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Cx |
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Cx |
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Erf^ + erfB + 2 |
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Cx |
 |
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From Eq. (7.52) the centerline velocity can be calculated by substituting y = 0 and z = 0:
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(7.53) |
Vr = v0 |erf— erf— .
Equations (7.52) and (7.53) describe air velocity in cross-sections of the jet located in Zone 1 through Zone 3. The shape of the outlet can be from square (2B x IB) to infinite slot with a width IB (L = In the case of a linear jet supplied through the slot, Eqs. (7.52) and (7.53) become
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‘■ + erf |
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Cx |
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B + z Cx |
 |
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Ind
"* = y<Werf^- (7’55)
The solution for a circular jet as presented by Elrod51 and Regensche. it:’3 cannot be evaluated in closed form. The exact solution can be determined only for the centerline velocity:
Diffuser jet throw, L, is a parameter commonly used in air diffuser sizing defined as the distance from the diffuser face to the jet cross-section where the centerline velocity equals a terminal velocity vx (vx is often assumed to be 0.25 m/s).
Therefore, the throw (L) can be determined by velocity decay equations with vx
Equal to the terminal velocity:
L = K, JX№. [7.57)
Entrainment ratio is another jet characteristic commonly used in air distribution design practice. Specifically, it is used in analytical multizone models (see Chapter 8) when one needs to evaluate the total airflow rate transported by the jet to some distance from a diffuser face. Airflow rate in the jet, Qx, can be derived by integrating the air velocity profile within the jet boundaries:
Qx = 2tt(‘ Vydy. (7.58)
Equations for airflow rate computation in compact, linear, and radial jets are presented in Table 7.13.
For a given area of diffuser opening A0, the entrainment ratio is proportional to the distance x (for compact, radial, and conical diffuser jets)
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TABLE 7.13 Airflow Rate through a Jet Cross-sectional Area
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Or proportional to the square root of the distance x (for linear jets). For the same type of jet, the entrainment ratio is less with a large Kj than with a small Kt. Radial and conical diffuser jets have a smaller entrainment ratio than compact or incomplete radial jets with the same K:l value. Linear diffuser jets have a smaller entrainment ratio than radial and conical jets.
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