Tarnopolsky M. Room air distribution calculation. 2014

Tarnopolsky M. Room air distribution calculation. 2014 of 35

Room air distribution calculation

M. Tarnopolsky, Ph.D.

Abstract

The objectives of this task are to evaluate the performance of airflow simulation

techniques and to establish their viability as design tools. This paper outlines a semi-

empirical method for calculating the velocity and temperature fields in the room. The method

takes into consideration interaction between the various jets in the room, the walls of the

room and the counter flow between jet and enclosure.

Effect of the walls is calculated by the mirror method. To explore impact of initial

velocity and temperature difference profiles, inlet is divided to increments of elementary jets.

The velocity and temperature difference at point considered are determined by adding

momentum flux and excess heat, as supplied from all real, mirror, and elementary jets. Then

this solution has corrections of the velocity field for deviations from the equations of

continuity and momentum flux, taking into consideration the counter flow between jet and

enclosure. Similarly, the temperature field is calculated so that the heat balance equation is

preserved.

The results of the calculations are shown to be in good agreement with measurements

in scaled models. The reliable, practical, and efficient method can be used both in the

preliminary stages of the development of jet-supplying products and in room air distribution

design.

Introduction.

The ability to accurately predict air movement and temperature distribution in spaces

offers the potential for design engineers to evaluate and optimize room air distribution

systems at an early stage, leading to improved thermal comfort and ventilation effectiveness.

The purpose of this work is to create a reliable, practical, and efficient method, which

can be used both in the preliminary stages of the development of jet-supplying products and

in room air distribution design. --

-----------------------------------------------------------------------------------------------------------

Dr. Moshe Tarnopolsky is with Department of Civil and Environmental Engineering, Technion, Haifa, Israel

([email protected])


We expect that some expensive laboratory tests could be avoided if airflows in rooms

should become possible to predict using calculation with a certain degree of confidence.

Recent developments in measurement techniques open new possibilities to study these

phenomena, while calculation methods haven been produced in an attempt to describe this

flow.

The most fundamental research on jets has been performed using contoured nozzles

which produce a nearly uniform ‘‘top hat’’ velocity profile at the nozzle exit (Crow [1] and

Becker [2]).

There has been increasing evidence that the impact of initial conditions on the

downstream development of a flow is significant even in the far field (Wygnanski [3] and

George [4]).

Papers of Deo et al. [5], [6] report the statistical behavior of a plane jet discharging

into a large space by exploring the impacts of varying nozzle contraction profile. The nozzle-

exit profile varied by changing orifice-plates with different exit nozzle’s inner-wall

contraction radii. The results reveal consistent differences throughout the flow, extending

from the exit velocity profiles, through the near field, and into the far field. The exit velocity

profiles were found to depend systematically upon radii, with a gradual transition from being

saddle-backed for the case small radii to closely approximate a ‘‘top hat’’ for the cases of

large radii. In the self-similar far field, the rates of centerline velocity decay and jet spread

decreased in an asymptotic-like manner with an increase in radii.

Quinn et al. [7] compared sharp-edged and smoothly contoured round nozzles with

the same exit area. The experimental results showed the existence of distinct off-center peaks

in the mean stream wise velocity in the very near flow field of the jet from the sharp-edged

round nozzle. Stream wise velocity off-center peaks were absent in the smoothly contoured

nozzle jet flow which was found to spread faster than the sharp-edged slot jet flow in the

initial region.

The focus of the study by Quinn [8] was the relationship between speed of mixing in

the near flow field and slot aspect ratio in flows of turbulent free jets issuing from sharp-

edged rectangular jets. The results indicate that the speed of mixing in the very near flow

field increases as the slot aspect ratio increases.

The objective of the study by Cao et al. [9], [10], [11], and [12]) was to identify the

airflow patterns of attached plane non-isothermal jets after its impingement with the corner in

a room. After impingement in the corner, the jet flow reattached to the wall surface and the

flow directed parallel to the vertical wall, with the jet thickness growing downwards. Smith


[13] and Cao [9] found that the air velocity distribution in the ceiling-wall and wall-floor

corners is nearly symmetric and flow after impingement was as jet continuation. This means

that the reattached distance assumes approximately equal to separation distance in the

impinging zone and the maximum jet velocity at the separation point is approximately equal

to the jet velocity at the reattachment point after its impingement.

In experiments of Nielsen et al. [15], [16] and [16] attached plane isothermal jet was

discharged horizontally near the ceiling into a model space and removed from the model near

the floor in opposite wall. Corresponding results were obtained in an idealized wall-jet

configuration by Sehwarz and Cosart [17]. After impingement in the ceiling-wall and wall-

floor corners, the jet flow reattached to the floor surface and the flow directed parallel to the

floor backwards.

In experiments of Pelfrey and Liburdy [18], Pramanik and Das [19] plane isothermal

jet discharged from a nozzle adjacent to a plate parallel to the axis of the nozzle. Then jet

bended and attached to the solid boundary due to lower pressure region created between the

jet axis and the nearest solid boundary (Coanda effect). The downward curving jet, the nozzle

plate and the oblique wall formed a recirculation zone.

The air-jet discharged from a ceiling slot inlet in an isothermal ventilated enclosure in

experiments of Yu et al. [20]. Two geometrically similar scale models represented a model

and a prototype with ratio of 1:2. Velocity measurements provided by the same relative exit

velocity profile U/U0 but different Reynolds numbers. In experiments of Yu et al. [21] the

same models were used for non-isothermal conditions.

Shepelev [22] has performed the most fundamental research on jets, what we use in

our analysis.

Room flow analysis.

The objective of this task is to evaluate the performance of airflow calculation

techniques. Analysis assumes a jet supplying from a rectangular inlet to the room with the

rectangular cross-section (see Figure 1).

Initially, the effect of the sidewalls, ceiling and floor is taken into consideration by the

mirror method using virtual jets symmetrical to real jet (ignoring friction in the boundary

layer). The real jet is shown by solid lines and the virtual jets are shown by dotted lines in

cross-section 1 - 1 of Figure 1 (total number of jets are My and Mz along axes Y and Z,

http://fluidsengineering.asmedigitalcollection.asme.org/searchresults.aspx?q=J.%20R.%20R.%20Pelfrey&p=1&s=19&c=0&t=

http://fluidsengineering.asmedigitalcollection.asme.org/searchresults.aspx?q=J.%20A.%20Liburdy&p=1&s=19&c=0&t=


respectively). The velocity field obtained in this step will be denoted as current by the index c

(Uc).

Then, this solution has corrections taking into consideration the counter flow between

jet and enclosure. The velocity field in the room (U) is calculated using a correction denoted

as a velocity shift by the index s (Us):

sc U - U = U , (1)

To explore impact of initial velocity and temperature difference profiles, inlet is

divided to Ny and Nz increments of elementary jets along axes Y and Z, respectively.

B0*H0

1

η

δ

Y

Z

Z

U

Xr

X 1

Cross-section 1-1

Hβ c

Be*He

U0

Ue

-2*Hr+2*H+Z

2*H+Z

B B Br-B BBr-BBr-B

-2*Br+2*B+Y

Hr

H

Hr-H

H

Hr-H

H

Hr-H

2*B+Y

B0*H0

1

η

δ

Y

Z

Z

U

Xr

X 1

Cross-section 1-1

Hβ c

Be*He

U0

Ue

-2*Hr+2*H+Z

2*H+Z

B B Br-B BBr-BBr-B

-2*Br+2*B+Y

Hr

H

Hr-H

H

Hr-H

H

Hr-H

2*B+Y

1

η

δ

Y

Z

Z

U

Xr

X 1

Cross-section 1-1

Hβ c

Be*He

U0

Ue

-2*Hr+2*H+Z

2*H+Z

B B Br-B BBr-BBr-B

-2*Br+2*B+Y

Hr

H

Hr-H

H

Hr-H

H

Hr-H

2*B+Y

Figure 1. Room jet interaction diagramme.


The current velocity (Uc) and temperature difference (Tc) at point considered are

determined by adding momentum flux and excess heat, as supplied from all real and mirror

jets jy and jz, and all inlet increments Ky and Kz, along axes Y and Z (see Tarnopolsky [23]):

d dU

HB g

=

gU

i

k

k

k

k

Nz

1=kz

Mz

1=jz

Ny

1=ky

My

1=jyoo

c 2112 1

, (2)

and

d dT U

HB

C = T U C ii

k

k

k

k

Nz

1=kz

Mz

1=jz

Ny

1=ky

My

1=jyoo

hcch 11

, (3)

where is the specific weight of the air at a given point:

T + T

T =

a

aa , (4)

Ui and Ti are air velocity and temperature difference induced by elementary jet i in

inlet with coordinates δ and η along axes Y and Z, respectively (according to Shepelev [22]):

]S C 2

) +(Z )+(Y[- U =U

22

2j

2j

mi

Exp (5)

and

)U

U(T =T

m

imi

, (6)

Yj and Zj are coordinates of a point in real and virtual jets jy and jz, m:

) - Y) + (-) - B() - M - (-j

= BY jyjy

yrj 11

2

1 (7)

and

Z- )(-1 + H(1 -2

)(-1 - (M - jH =Zj jz

jzz

r , (8)

The values of air velocity Um and temperature difference Tm on axis of i - elementary

jet have to be calculated by assuming conservation of momentum flux and excess heat of cell

made by increments ky and kz:

g

U =

g

dYdZU kkkkk

2i2H-Hr

H-

B-Br

B-

11 , (9)

and

kkkkkkhiih

H-Hr

H-

B-Br

B-

T U C = dYdZ T U C 11 , (10)

Taking into consideration Equations 4 through 6 and after further integration, we obtain:


kkkkk2

m2

a U SCU 1122 , (11)

and

kkkkkkhtmmah T U C = SCT U C 1122 , (12)

Empirical values of velocity profile constant C = 0.082 and turbulent Prandtl number

σ = 0.5 are used here. Ct is the temperature profile constant: 095.01

25.0

+ C = Ct , (13)

Equations 11 and 12 can be used to find air velocity Um and temperature difference Tm

on axis of i - elementary jet. After substitution of Equations 4, 5, 11, and 12 into Equations 2

and 3 and further integration, current velocity and temperature difference can be presented as:

zy0.5

ooa

cc U U )

H B

I g(=U (14)

and

zy0.5oocah

cc T T

)H B I (g C

Q = T , (15)

where Uy and Uz are the no dimensional components of the current velocity:

)]CS

YErf[-]CS

+Y(Erf[

U

U =U 0.5ijij

Ny

1=ky

My

1=jyy

o

k

1

2

5.0 , (16)

and

)]CS

-ZErf[-]

CS

+Z(Erf[

U

U=U 0.5ijij

Nz

1=kz

Mz

1=jzz

o

k 1

2

5.0

. (17)

Similarly Ty and Tz are given by:

])SC

-YErf[-]

SC

+Y(Erf[

UTU

TU=Tt

ij

t

ij

yoo

kkNy

1=ky

My

1=jyy

15.0

, (18)

and

])SC

-ZErf[-]

SC

+Z(Erf[

UTU

TU=Tt

ij

t

ij

zoo

kkNz

1=kz

Mz

1=jzz

15.0 , (19)

where Ic and Qc are the initial momentum flux and excess heat of the current calculated as the

product of sums of increments of momentum flux and excess heat in the inlet along the axes

Y and Z:


g

U =I kkkkk

2Nz

1=kz

Ny

1=kyc

11 , (20)

and

kkkkkkh

Nz

1=kz

Ny

1=kyc T U C=Q 11 . (21)

This solution have to be corrected taking into consideration the velocity “shift” (Us),

which is determined by the equation of continuity:

G = dZ dY U H-Hr

H-

B-Br

B-

, (22)

where G is the quantity of air in cross-section of the room. Accordingly, we obtain the

following from Equations 1, 14, and 22:

,U - H B

G = U e

rra

cs (23)

where Gc is the current quantity of air in considered cross-section of the room:

zy0.5

cooac L L )I H B (g = G ; (24)

where Ly and Lz are no dimensional discharge factors:

dY B

U = L

o

yB-Br

B-

y (25)

dZ H

U = L

o

zH-Hr

H-

z . (26)

Ue is the air-removal velocity in considered cross-section of the room:

rrae H B

G = U ; (27)

If distance from the inlet to the cross-section considered is less than the distance to the

outlet (X < Xe), the total quantity of the air in this cross-section of the room G equals to the

total quantity of the air in the inlet Go, and in another case (X = Xe or X > Xe) G equals to

zero.

To determine the trajectory of the jet, we have to calculate the effect of the

momentum flux and the buoyancy flux. The momentum flux in the cross-section of the room

is:

dZ dY gU = I

2H-Hr

H-

B-Br

B-

u

,


(28)

Taking into consideration Equations 1, 23 through 27 and after integration, we obtain:

ez2

y2

rr

oocu I+ ) L L

H B

H B - (1I = I , (29)

where Ie is momentum flux of air-removal in considered cross-section of room:

H Bg

U =I rr

e2

ae

. (30)

The buoyancy force, created by temperature-density difference between elementary

volume of current and ambient air, is:

dX dZ dY )-( = dI a

H-Hr

H-

B-Br

B-

t , (31)

Using Equations 4 and 15, and after approximate integration, Equation 31 becomes:

dX Q Q I g

H B

C T

Q=dI zy

0.5

c

ooa

ha

ct

, (32)

where Qy and Qz are no dimensional components of the excess heat:

dY B

T = Q

o

yB-Br

B-

y (33)

and

dZ H

T = Q

o

zH-Hr

H-

z . (34)

The components and the resultant momentum flux in a cross-section of room will be:

xycux I = I coscos , (35)

cxy I = I tan , (36)

t

X

0

cxz dI + I = I tan , (37)

and

)I + I + I( = I 0.5z

2y

2

x2 , (38)

The coordinates of the current trajectory can now be calculated:

dS I

I = X x

S

0 , (39)

and


dS I

I = B y

S

0 , (40)

and

dS I

I = H z

S

0 . (41)

Now ratios of the coordinates of current trajectory X

B and

X

H can be used as

tangents to find the angles of current trajectory αc and c along axes Y and Z.

The static pressure in cross-section of the room is given from the momentum flux

equation for closed space:

X

o rr

xos H B

dI P = P , (42)

Influence of the exit nozzle profile.

Deo et al. [5] and [6] studied influence of geometric profile of a long slot nozzle on

the properties of a jet. The nozzle facility, shown schematically in Figure 2, consists of an

open circuit wind tunnel, flow-straightening elements including a honeycomb and screens,

and a smooth contraction exit.

The jet was discharged horizontally at the midpoint into a laboratory space of

dimensions Xr =14 m (long) · Br=0.72 m (wide) · Hr =2.04 m (high). The nozzle-exit profile

of Bo = 0.72 m (width) and Ho = 0.01 m (height) was varied by changing orifice-plates with

different exit contraction radii over the range of 0 < rc / Ho < 3.6 (Ho = h and rc = r in Figure

2). The measurements were made at a slot-height based Reynolds number (Re) of 18000 and

a slot aspect ratio (span/height) of 72.


Figure 2. Schematic view of the wind tunnel and nozzle attachment in experiments of

Deo [5] and [6].

Calculated U using Equation 1 (solid lines) and measured (Deo [5], symbols) exit

vertical velocity profiles at relative distance of X / Ho =0.25 for rc / Ho= 0.45–3.60 and at X/

Ho = 1.25 for rc / Ho =0 compared in Figure 3, where Ux is the centerline mean velocity (rc /

Ho= r* in Figures 3 and 4).

Figure 3. Calculated (solid lines) and measured (Deo [5], symbols) exit velocity

profiles U/Ux at X/Ho = 0.25 for rc/Ho= 0.45–3.60 and at X/Ho = 1.25 for rc/Ho = 0.

Clearly, the exit velocity profiles depend on rc/Ho and specifically undergo a

substantial transition from being saddle-backed for rc/Ho= 0 to closely approximate a ‘‘top

hat’’ profile for rc/Ho = 1.80 and 3.60 (Figure 3).

One can see in Figure 4 satisfactory coincidence between measured axial jet velocity

(Deo [5], symbols) and calculated values Ux using Equation 1 by Y = 0 and Z = 0 (solid

Vertical velocity profiles.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Z / H0

U / U

x

0

0.45

0.9

1.8

3.6

Calculated rc / H0

X/Ho

Velocity of jet.

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20X / H0

Ux /

U0

0

0.45

0.9

1.8

3.6

Calculated rc / H0

X/Ho


lines), where Uo is the mean exit velocity.

Figure 4. Calculated (solid lines) and measured (Deo [5], symbols) evolution of the

axial jet velocity Ux/Uo for various cases of investigation.

The results of measurements and calculations obtained show that both the initial flow

and the downstream flow are dependent upon the ratio rc/Ho. As expected, there is a

consistent dependence of Ux/Uo on rc/Ho, in particular, the case of rc/Ho =0 being discernibly

different from that for other cases. A hump in Ux/Uo (at vena contracta about X/ Ho =2.0 for

the cases of rc/Ho <1.8) becomes obvious, although its magnitude tends to decrease as rc/Ho is

increased. For X/ Ho > 6, the decay rate of the axial velocity depends on rc/Ho, and for the

case of rc/Ho=0 decaying was at the highest rate. Overall, the effect of rc/Ho on the mean and

turbulence fields decreases as rc/Ho increases. In other words, even in the fully developed

state, a plane jet does not ‘forget’ its origin.

The focus of the study by Quinn et al. [7], [8] was the relationship between speed of

mixing in the near flow field and slot aspect ratio AR in flows of turbulent free jets issuing

from sharp-edged rectangular and round slots. Four slot aspect ratios (2, 5, 10, and 20), with

the same exit plane area of 0.00161 m2, were used. The exit plane equivalent diameter (De =

0.0453 m) of the slot was same as the diameter of a round slot with the same exit area as the

rectangular slot in all the flows considered. The exit plane equivalent diameter (De = 0.0453

m) of the rectangular slots was same as the diameter of a round slot with the same exit area as

the rectangular slot in all the flows considered. The mean stream wise velocity and stream

wise turbulence intensity at the center of the slot exit plane were at U0 = 60 m/s and 0.5%,

respectively, in all the jets.

The jets issue from flow facility (see Figure 5) into a cage, which extends Xr = 3.0 m

downstream and fitted with a Br=2.44 m (wide) · Hr =2.44 m (high). The downstream end of

the cage is open.


Figure 5. Plane view sketch of the flow facility in experiments of Quinn et al. [7] and

[8]

Calculated using Equation 1 by Z = 0 (solid lines) and measured (Quinn et al. [7],

symbols) mean radial velocity profiles for the sharp-edged and contoured round slots at

downstream location X/De = 2.66 compared in Figure 6.

Figure 6. Calculated (solid lines) and measured (Quinn et al. [7], symbols) mean

velocity profiles for the sharp-edged and contoured round slots in the central XY plane at X/De

= 2.66.

The four rectangular jets calculated U using Equation 1 by Y = 0 or Z = 0 (solid lines)

and measured (Quinn [8], symbols) mean velocity profiles in the central XY and XZ planes at

downstream location X/De= 10, are presented in Figures 7 and 8, respectively. Y0.5 and Z0.5

are the half-velocity width in the central XY and XZ planes, correspondingly.

Velocity profiles.

0.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1 2 3Y / Y0.5

U / U

x 1s

1c

Calculated AR


Figure 7. Calculated (solid lines) and measured (Quinn [8], symbols) mean stream

wise velocity profiles for the four rectangular jets in the central XY plane at X/De = 10.


wise velocity profiles for the four rectangular jets in the central XZ plane at X/De = 10.

Calculated using Equation 1 by Y = 0 and Z = 0 (solid lines) and measured (Quinn [8],

symbols) mean stream wise velocity along the jet centerline Ux is shown in Fig. 9 for round

jets issuing from a sharp-edged slot and from a contoured nozzle and for the four rectangular

jets.


0.0

0.4

0.8

1.2

-2.5 -1.5 -0.5 0.5 1.5 2.5

Z/Z0.5

U /

Ux 2

5

10

20

Calculated AR

Horizontal velocity profiles.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-2.5 -1.5 -0.5 0.5 1.5 2.5

Y/Y0.5

U / U

x

2

5

10

20

Calculated AR

Velocity of jet.

0.0

0.4

0.8

1.2

1.6

0 2 4 6X / De

Ux /

U0

1s

1c

2

5

10

20

Calculated AR



wise velocity along the jet centerline

Attached plane jet after its impingement with the corner.

The objective of the study by Cao et al. [9], [10], [11], and [12] was the indoor

airflow of an attached plane jet after its impingement with the corner in a room (see Figure

10). To avoid a high air velocity in the occupied zone, the maximum air velocity, V1, of the

attached plane jet should be predictable and designed accurately before the air enters the

occupied zone in room conditions.

The jet was discharged horizontally close to ceiling into a laboratory space of

dimensions Br=2.0 m (wide) · Hr =4.3 m (high) through the slot of Bo = 2.0 m (width) and Ho

= 0.03 m (height).

The experiment used initial jet velocities (U0) 1.0, and 2.0 m/s by three initial

temperature differences (T0) of 0, -3, and -6 0C.

The distance between the slot and impinging wall (Xr) was 1.5 m, and 2.0 m. Before

the impingement of an attached plane jet in the ceiling-wall corner, the jet was bound by the

ceiling surface on its upper side in room conditions.

Figure 10. Application of an attached plane jet in a room (Cao [9])

The above analysis can also be used to calculate the air distribution for the case of a

plane jet by Bo = Br. The stream wise mean velocity distributions are presented in Figure 11,

where the calculated U using Equation 1 (solid lines) and measured (Cao [9], symbols)

velocity profiles are normalized by the local maximum velocity Ux, and by the jet half-width

distance Z0.5.


Figure 11. Calculated (solid lines) and measured (Cao [9], symbols) mean stream

wise velocity profiles for attached plane jet in the central XZ plane by U0=1.0 m/s.

The profiles express similarity to some extent in each case by the mean exit velocity

Uo =1.0 m/s. Most of the data could fit the calculating curves using Equation 18 quite well,

except very close to the jet slot.

Figure 12 shows comparison of calculated values Ux using Equation 1 by Y = 0 and Z = 0

(solid lines) and measured ([9], filled symbols) maximum jet velocity by U0=1.0 m/s and

T0=0 0C.

Figure 12. Comparison of calculated (solid lines) and measured (Cao [9], symbols)

maximum jet velocity by U0=1.0 m/s and T0=00C.

After impingement in the corner, the jet flow reattached to the wall surface and the

flow directed parallel to the vertical wall, with the jet thickness growing downwards.

Earlier studies of Cao [9] revealed that the air velocity distribution in the ceiling-wall

and wall-floor corners is nearly symmetric and flow can be considered as jet continuation.

This means that the reattached distance assumes approximately equal to separation distance

in the impinging zone and the maximum jet velocity at the separation point is approximately

equal to the jet velocity at the reattachment point after its impingement.


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Z/Z0.5

U/U

x2

4

6

10

14

22

30

Calculated X/H0

Velocity of jet.

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6X, m

Ux, m

/s

1.0 0

Calculated U0 T0


Analyzing earlier studies of Smith [13] and Cao [9], the separation distance in the

corner can be expressed as jet thickness at the distance from the jet slot to the corner.

Then the distance from the slot to the separation point in the ceiling-wall corner (Scw),

and to the measured sections of jet near vertical wall (Xw) can be calculated as:

,ZXS cwrcw (43)

and

ZZSX cwcww (44)

where Zcw is the jet thickness at the distance from the jet slot to the ceiling-wall corner (Xa).

At the distance to the wall Xr = 2.0 m the jet thickness was calculated as Zcw = 0.44 m, and

the distance to the separation point Scw = 1.56 m.

The distance from the slot to the wall-floor corner (Xwf), and to the separation point in

the wall-floor corner (Swf), can be calculated as:

ZHXX cwrcwwf , (45)

and

ZXS wfwfwf , (46)

where Zwf is the jet thickness at the distance from the jet slot to the wall-floor corner (Xwf). At

the distance to the the wall Xr = 2.0 m the distance from the jet slot to the wall-floor corner

floor Xwf = 5.42 m the jet thickness was calculated as Zcw = 0.95 m, and the distance to the

separation point Scw = 4.47 m.

Comparison of calculated using Equation 1 by Y = 0 and Z = 0 (solid lines) and

measured (Cao [10], symbols) maximum jet velocity after impingement in the ceiling-wall

corner (by X = Xw, initial jet velocity U0 =1.0 m/s, and three initial temperature differences T0

of 0, -3, and -6 0C) are presented in Figures 13, 14, and15, correspondingly.



maximum jet velocity after impingement in the ceiling-wall corner by U0=1.0 m/s and T0=0 0C.


maximum jet velocity after impingement in the ceiling-wall corner by U0=1.0 m/s and T0=-3 0C.


maximum jet velocity after impingement in the ceiling-wall corner by U0=1.0 m/s and T0=-6 0C.

Velocity of jet.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 0.5 1.0 1.5 2.0Uz, m/s

Hr

- Z

, m

1 -6

Calculated U0 T0

Velocity of jet.-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.0 0.5 1.0 1.5Uz, m/s

Z, m

1 0

Calculated U0 T0

Velocity of jet.

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0 0.5 1.0 1.5 2.0Uz, m/s

Hr -

Z, m

1 -3

Calculated U0 T0


The flow directed parallel to the vertical wall influenced by buoyancy force and the

corresponding air velocity is higher than in the isothermal ones by T0 = -3 0C and is the

highest by T0 = -6 0.

The results show the similar maximum velocity decay trend in three cases and

indicate that the calculated maximum velocity can obtain a certain agreement compared with

all measured cases except the ceiling-wall corner and the wall-floor corner region

Attached isothermal jet in a room.

In experiments of Nielsen et al. [15], [16], and [16] the jet was discharged

horizontally near the ceiling into a model space of dimensions Xr = 9 m (long) · Br = 3.0 m

(wide) · Hr = 3.0 m (high) with inlet velocity U0 = 0.455 m/s and inlet temperature difference

To = 0°C (see Figure 16). The inlet Reynolds number is 5,000 and the turbulence intensity 4

%. The room is specified by ratios of H0/Hr = 0.056, He/Hr = 0.16, where H0 is the inlet slot

height, He the exhaust height. (Xr = L, Hr = H, Br = W, X = x, Z = y, Ho = h, and He = t in

Figures 15 - 21)

Figure 16. Model of the room in experiments of Nielsen [16].

The aim is to compare the calculated results with this data (Nielsen [15], [16], and

[16]). In particular the inlet profiles and profiles on two vertical locations X = Hr and X = 2

Hr.

Two values of the width of the inlet slot (B0/Br = 0.5 and 1.0) were used to assist the

determination of the extent of influence of a three dimensional inlet flow on the velocity

characteristics of the room. The inlet was preceded by a smooth, plane contraction of area

ratio 2 and resulted in the initial profiles shown on Figures 17 and 18. As can be seen, the

mean profiles are symmetric about axis. Transverse profiles along the centerline of the slot

indicate, over the central 80 percent of both slots, differences of less than two percent of the


bulk velocity. Measurements show that, within experimental uncertainty, no dimensional

profiles are independent of Reynolds number.

Calculated U using Equation 1 (solid lines) and measured (Nielsen [15], symbols)

mean velocity profiles in the central XY plane and two XZ planes for B0/Br = 1.0, Y / Br = 0

and 0.4 at downstream location X/ Hr= 0.05, are presented in Figures 17 and 18, respectively.

Figure 17. Calculated (solid lines) and measured (Nielsen [15], symbols) inlet

velocity profile U/U0 in the central XY plane for B0/Br = 1.0, at downstream section X/Hr=

0.05.

Figure 18. Calculated (solid lines) and measured (Nielsen [15], symbols) inlet

velocity profiles U/U0 in two XZ planes for B0/Br = 1.0, Y/Br = 0 and 0.4 at downstream

section X/Hr= 0.05.

The profiles were unfortunately influenced by the inlet geometry, which had a

contraction in Y – direction and constant height.


-0.50

-0.25

0.00

0.25

0.50

0.0 0.5 1.0

U/U0

Z/H

0

0.05 0

0.05 0.4

Calculated X/Hr Y/Br

Horizontal velocity profile.

-0.50

-0.25

0.00

0.25

0.50

0.0 0.5 1.0 1.5

U/U0

Y/B

0

0.05 0

Calculated X/Hr Z/Hr


The flow near the floor can be considered as jet continuation. After impingement in

the wall-floor corner, the jet flow reattached to the floor surface and the flow directed parallel

to the floor backwards.

Then the distance from the slot to the measured sections of the back flow near floor

(Xb) can be calculated as:

XZXSX wfrwfb , (47)

Flow in the room can be calculated taking into consideration the superposition of the

jet and the backward flow near floor, by adding the velocity components at each point and

later correcting the velocity field for deviations from the equations of continuity and

momentum flux. To correct the velocity field for deviation from the equation of continuity the air-

removal velocity in considered cross-section of the flow near floor (Uef) should be equal to

zero. To correct the velocity field for deviation from the equation of momentum flux the

momentum flux of the flow near floor (Ib) at distance from the slot to considered cross-

section (Xb) should be:

rrsab H )BP(PI = I , (48)

and the initial momentum flux of the jet (I by X = 0) should be recalculated as follows:

bc I I = I . (49)

Figures 19 and 20 show comparison between calculated using Equations 1, 43

through 49 (solid lines), and measured (Nielsen [15], symbols) mean velocity profiles at

sections X=Hr and X=2Hr in XZ planes for B0/Br = 1.0, Y/Br = 0 and Y/Br = 0.4,

correspondingly.


0.0

0.2

0.4

0.6

0.8

1.0

-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7

U/U0

Z/H

r

0.0

0.2

0.4

0.6

0.8

1.0

-1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7

U/U0

Z/H

r

1 02 0



Figure 19. Comparison between calculated (solid lines) and measured (Nielsen

[15], symbols) mean velocity profiles at sections X= Hr and X = 2 Hr in XZ plane for B0/Br =

1.0, Y/Br equal to zero.

Figure 20. Comparison between calculated (solid lines) and measured (Nielsen

[15], symbols) mean velocity profiles at sections X= Hr and X = 2 Hr in XZ plane for B0/Br =

1.0, Y/Br = 0.4.

The general trends of velocity are represented reasonably well by all simulation

approaches but some discrepancies exist in certain areas. In general, the calculations, do not

predict recirculation in the corners, and under-predict particularly near the floor at section

X=Hr.

Figure 21 present comparison between calculated using Equations 1, 43 through 49

(solid lines), and measured (Nielsen [14], symbols) mean velocity profiles at sections X=Hr

and X=2Hr in XZ planes for two inlet arrangements (B0/Br = 0.5 and 1.0), at two values of Y

(Y/Br = 0 and 0.4). Both sets of measurements indicate a major region of recirculation.


0.0

0.2

0.4

0.6

0.8

1.0

-0.5 0.0 0.5 1.0 1.5U/U0

Z/H

r

0.0

0.2

0.4

0.6

0.8

1.0

-1.5 -1.0 -0.5 0.0 0.5

U/U0

Z/H

r

1 0.4

2 0.4


U / U0


Figure 21, Comparison between calculated (solid lines), and measured (Nielsen [14],

two inlet for planes XZin r2H=Xand rH=X sat section svelocity profilemean s) symbol

.)0.4and 0= rBY/( Yat two values of ,)0.5 and 1.0= rB/0B( arrangements

, 43 using Equation 1 xUalculated cbetween show comparison 22 Figure

through 49 by Z = 0 (solid lines) and measured (Nielsen [15], symbols) axial velocity in XZ

.)0.4and 0= rBY/( Yat two values of ,)0.5 and 1.0= rB/0B( two inlet arrangements forplanes

These measurements are compared with the corresponding results obtained in an idealized

wall-jet configuration by Sehwarz and Cosart [17]

Figure 22. Comparison between calculated Ux (solid lines) and measured

(Nielsen [15], symbols) axial velocity in XZ planes for two inlet arrangements (B0/Br = 0.5

and 1.0), at two values of Y (Y/Br = 0 and 0.4).

The calculation of velocity decay corresponded quite well with measurements except

that the measured recirculation in the corner was not predicted.

0.0

0.2

0.4

0.6

0.8

1.0

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

0.0

0.2

0.4

0.6

0.8

1.0

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Z/H

r

U/U0

Z/H

r

U/U0


0.5 1 0

0.5 1 - 0.4

0.5 2 0

0.5 2 - 0.4

1 2 0

1 2 - 0.4

Calculated Bo/Br X/Hr Y/Br

0.01

0.02

0.04

0.08

0.16

0.32

0.64

2 4 8 16 32 64 128

Ux/

U0

X/H0

Velocity of jet.

0.5 0

0.5 -0.4

1 0

1 -0.4

Calculated Bo/Br Y/Br


Offset plane jet.

It should be noted that plane jets are greatly affected by the ceiling or floor, as a result

of the Coanda effect. The static pressure difference between the lower and upper regions of

the current (relative to the current axis) creates a vertical pressure force:

PBrdX = IX

0

p , (50)

where ΔP is elementary static pressure difference between the lower and upper regions of the

current and calculated as momentum flux difference between these current regions:

B

dY )

H

dZU

g -

H - H

dZ U

g( = P

r

20

H-r

2H-Hr

0

B-Br

B-

. (51)

The jet trajectory can be calculated using Equations 41 and 49:

dX I

I + I = H pzX

0 . (52)

If jet attaches to the plate, after impingement, it creates forward and backward flows

near plate, which can be considered as jet continuation. The momentum fluxes of forward If

and backward Ib flows can be determined from preservation of momentum flux before

impingement I and its projection on the plate:

bf I I = I , (53)

bfc I - I = I cos , (54)

accordingly, we obtain the following from Equations 52 and 53:

2/)cos1( cf I I , (55)

2/)cos1( cb I I . (56)

In experiments of Pelfrey and Liburdy [18] and Pramanik and Das [19] a plane jet is

discharged from a nozzle adjacent to a plate parallel to the axis of the nozzle, it bends and

attaches to the plate due to low pressure region created between the jet axis and the solid

boundary (Coanda effect) as shown in Figure 23. The nozzle width was fixed at 12.5 mm and

the span at 150 mm (aspect ratio of 12). The flow exiting the nozzle discharged above a

horizontal plate by an offset ratio 7.0 and was contained by two Plexiglas sidewalls.




Figure 23. Offset jet flow geometry in experiments of Pelfrey and Liburdy [18].

All data presented were for an offset ratio (H/H0) of seven and nozzle Reynolds

number (Re) of 15000 and inlet temperature difference T0 = 0°C. At the nozzle exit the

velocity profile was essentially flat (U0=17.5 m/s). However, a slight skewness was found

(see Figure 23).

Figure 23. Comparison between calculated (solid lines) and measured (Pelfrey

and Liburdy [18], symbols) nozzle exit velocity profile.

Figure 24 shows comparison between calculated using Equation 1, 43 through 48 by Y

= 0 (solid lines) and measured (Pelfrey and Liburdy [18], symbols) the jet development from

the nozzle exit to seventeen nozzle widths (H0) downstream. It is obvious that the flow was

immediately affected by the secondary recirculation in the lower left corner. The attachment

distance along the plate was found to be 13 H0.

Exit velocity profile

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.92 0.94 0.96 0.98 1.00 1.02U/U0

Z/H

0

Z/Ho

Calculated U/U0


0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

X/H0

Z/H

0

0.1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

Calculated U m/s, X/H0








Figure 24. Comparison between calculated (solid lines) and measured (Pelfrey

and Liburdy [18], symbols) vertical mean velocity profiles.

Figure 25 shows comparison between calculated jet trajectory using Equation 44

(solid red line) and back flow (dashed blue line) and measured mean velocity vectors (Pelfrey

and Liburdy [18], arrows).

Figure 25. Comparison between calculated jet trajectory (solid line) and measured

(Pelfrey and Liburdy [18], arrows) mean velocity vectors. One can see satisfactory

coincidence between these measurements and calculated using Equation 1 by Y = 0 velocity

contours in Figure 26.

Figure 26. Calculated jet velocity contours.

Reasonable agreement between the experimental results and calculations has been

observed. For further validation of the calculation result, a comparison between calculated

Velocity vectors and trajectory of jet.

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

X/H0

Z/H

0 Z/Ho

Zb/Ho

Calculated Z/H0

0 2 4 6 8 10 12 14 160

2

4

6

8 Calculated U/U0

X/H0

Z/H0

Velocity contours

0.9-1.0

0.8-0.9

0.7-0.8

0.6-0.7

0.5-0.6

0.4-0.5

0.3-0.4

0.2-0.3

0.1-0.2

0.0-0.1

-0.1-0.0

-0.2--0.1

-0.3--0.2








using Equation 1 by Y = 0 and Z = 0 (solid lines) and measured (Pramanik and Das [19] and

Pelfrey and Liburdy [18], symbols) jet axial velocity is presented in Figure 27. The

calculated centerline velocity decay matches well in the potential core region barring a small

region near the impingement distance about 15 H0.where the experimental results show a

sudden dip in the pattern.

Figure 27. Comparison between calculated (solid lines) and measured (Pramanik

and Das [19], symbols) jet axial velocity.

Non-isothermal attached plane jet in a room.

The air-jet discharged from a ceiling slot inlet in a non-isothermal ventilated enclosure in

experiments of Yu et al. [20]. Two geometrically similar scale models representing a model

and a prototype with ratio of 1:2 (see Figure 28). Both models had an exhaust plenum

between the test room and exhaust duct to reduce the effect of exhaust airflow from the slot

outlet (Xr = L, Hr = H, Br = W, Ho = h, and He = d in Figure 28). In experiments of Yu et al.

[21] the same models were used for isothermal conditions.

Axial velocity of jet.

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25

X/H0

Ux/U

0

Calculated Ux/U0




Figure 28. The scheme and dimensions (prototype/model) of test chambers in

experiments of Yu et al. [20].

Typical vertical velocity profiles of a plane wall isothermal jet (T0 = 0°C) at values for

X of 0.25 Xr, 0.5 Xr, and 0.75 Xr with inlet airspeed of U0=3.41 and U0=3.56 m/s for the

model and prototype respectively in experiments of Yu et al. [21] are compared with

calculated values using Equation 1 by Y = 0 (solid lines) in Figure 29. In calculations, relative

exit velocity profile U/U0 for the model and prototype was the same.

Figure 29. Comparison between calculated (solid lines) and measured (Yu, et al.

[21], symbols o for prototype, and x for model) vertical velocity profiles with inlet airspeed

of U0=3.41 and U0=3.56 m/s for the model and prototype respectively and inlet temperature

difference T0 = 0°C, (U=u, air-jet velocity).

The measured peak velocities of the isothermal jet (Yu et al. [21], empty symbols for

prototype, and filled symbols for model) at specific distances from the inlet are compared

with calculated values using Equation 1 by Y = 0 and Z = 0 (solid lines for prototype, and

dashed lines for model) as a function of Reynolds number Re (based on U0 and H0) in Figure

30.


0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

0.00 0.25 0.50 0.75 1.00

X/Xr

Z/H

r

0.25 0.5

0.50 0.5

0.75 0.5

0.25 1.0

0.50 1.0

0.75 1.0

Calculated X/Xr Hr, m


Figure 30. Comparison between calculated (solid lines for prototype, and dashed

lines for model) and measured (Yu et al. [21], empty symbols for prototype, and filled

symbols for model) peak velocities by T0 = 0°C.

One can see satisfactory coincidence between these measurements and calculated

velocities of the isothermal jets by the same relative exit velocity profile U/U0 but different

Reynolds numbers Re for model and prototype.

In experiments of Yu [20] the floor of the prototype was fitted with insulated heat

panels to simulate occupied region surface temperature the airflow patterns were observed

from the contours of velocity and temperature in the chamber.

Figures 31 and 32 shows comparison between calculated trajectory of jet using

Equation 44 by Y = 0 (solid red line) and back flow (dashed blue line) with measured velocity

and temperature contours (Yu [20], contours) by initial jet velocities U0= 0.78 m/s and

temperature difference T0 = -20°C.

Axial velocity of jet.

0

1

2

3

4

5

6

0 1000 2000 3000 4000 5000

Re

Ux,

m/s 0.25 0.5

0.50 0.5

0.75 0.5

0.25 1.0

0.50 1.0

0.75 1.0

Calculated X/Xr Hr, m


Figure 31. Comparison between calculated trajectory of jet (solid red line) and

back flow (dashed blue line) with measured velocity contours (Yu [20], contours) by initial

jet velocities U0= 0.78 m/s and temperature difference T0 = -20°C,

Figure 32. Comparison between calculated trajectory of jet (solid red line) and

back flow (dashed blue line) with measured temperature contours (Yu [20], contours) by

initial jet velocity U0= 0.78 m/s and temperature difference T0 = -20°C.

One can see satisfactory coincidence between these measurements (see Figures 31

and 32) and calculated velocity using Equation 1 by Y = 0 and temperature using Equation 15

by Y = 0 contours (see Figures 33 and 34), correspondingly.

Figure 33. Calculated velocity contours by initial jet velocity U0= 0.78 m/s and

temperature difference T0 = -20°C.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.0

0.2

0.4

0.6

0.8

1.0

Z, m

X, m

Calculated Ux/U0

Velocity contours.

1.0-1.1

0.9-1.0

0.8-0.9

0.7-0.8

0.6-0.7

0.5-0.6

0.4-0.5

0.3-0.4

0.2-0.3

0.1-0.2

0.0-0.1

-0.1-0.0

-0.2--0.1

-0.3--0.2

-0.4--0.3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

H, m

X, m

Temperature contours and trajectory of jet.

H

Hb

Calculated H, m


Figure 34. Calculated temperature contours by initial jet velocity U0= 0.78 m/s

and temperature difference T0 = -20°C.

The trajectory, velocity, and temperature of jet predicted by calculation matches well

barring a small region near the floor at impingement distance about 1.0, m.

Conclusions.

This study presents basic concepts of a semi-empirical method for calculating the

velocity and temperature fields in rooms. The method takes into consideration the interaction

between the various jets in the room, the boundaries of the room and the counter flow

between jet and enclosure by adding the momentum flux of all jets at each point.

To explore impact of initial velocity and temperature difference profiles, inlet divided

to increments of elementary jets. The effect of the walls is calculated by the mirror method.

The velocity and temperature difference at point considered are determined by adding

momentum flux and excess heat, as supplied from all real, mirror, and elementary jets. Then

this solution has corrections of the velocity field for deviations from the equations of

continuity and momentum flux, taking into consideration the counter flow between jet and

enclosure. Similarly, the temperature field is calculated so that the heat balance equation is

preserved.

Two- and three-dimensional calculations compared with measurements performed

under isothermal and buoyant conditions. Almost all the calculations can predict the

isothermal airflow patterns with an acceptable degree of realism. In many cases, the predicted

occupied zone air velocities are within a band indicating general compliance with

expectation. Small recirculation areas in the corners of the room were usually not predicted

though their impact is believed small.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.0

0.2

0.4

0.6

0.8

1.0

Z, m

X, m

Calculated T, deg

Temperature contours.

40-41

39-40

38-39

37-38

36-37

35-36

34-35

33-34

32-33

31-32

30-31

29-30

28-29

27-28

26-27

25-26

24-25

23-24

22-23

21-22


In buoyant flow, it has been shown that calculations can predict flow pattern, velocity

and temperature distribution, although with a reliability reduced from that demonstrated for

isothermal flow.

After impingement in the corner, the jet flow reattached to the wall surface and the

flow directed parallel to the wall, with the jet thickness growing downwards. Experiments and

calculations revealed that the air velocity distribution in the ceiling-wall and wall-floor

corners is nearly symmetric and flow after impingement can be considered as jet

continuation.

Calculation of the supply jet characteristics from number of models (approaches) by

dividing inlet to increments of elementary jets reveal consistent differences throughout the

near field. In the self-similar far field, the rates of centerline velocity and temperature decay

decreased in an asymptotic-like manner. It would be helpful if the manufacturers of air

diffusers would publish some near field data (e.g. profiles in front of the device) with their

technical specifications.

NOMENCLATURE.

AR slot aspect ratio

B, H coordinates of current trajectory along axes Y and Z (given by Equations 40, 41, and

52), m

Bo, Ho width and height of inlet, respectively, along axes Y and Z, m

Br, Hr, Xr width, height, and length of room, respectively, along axes Y, Z, and X, m

C empirical constant (a value of C = 0.082 is used here)

Ch specific heat of air (at constant pressure), kJ//kg/ oK

Ct, temperature profile constant (given by Equation 13)

De equivalent diameter of the rectangular slot, m

dIt buoyancy force, created by temperature-density difference between elementary volume

of current and ambient air (given by Equation 32), N

Do, diameter of round inlet, m

G quantity of air moving trough cross-section considered of room (given by Equation 22),

N s-1

g acceleration due to gravity, m/s2

Gc quantity of air in current cross-section considered (given by Equation 24),

N s-1


I, Ix, Iy, Iz components and resultant momentum flux in cross-section of room (given by

Equations 35-38, 49, and 53), N

Ib the momentum flux of the backward flow near floor (given by Equation 48, 5655), N

Ic initial momentum flux of the current (given by Equation 20), N

Ie momentum flux of air removal in ross-section considered (given by Equation 30), N

If the momentum flux of the forward flow near floor (given by Equation 55), N

Ip vertical pressure force created by static pressure difference between the lower and

upper regions of the current (relative to the axis) and (given by Equation 42, 50), N

Iu momentum flux in cross-section of room (given by Equation 29), N

i ordinal number of elementary jet

Jy, My ordinal and total numbers of real and virtual jets interacted, respectively, along axis Y

Jz, Mz ordinal and total numbers of real and virtual jets interacted, respectively, along axis Z

Ky, Ny ordinal and total numbers of increments, respectively, in inlet along axis Y

Kz, Nz ordinal and total numbers of increments, respectively, in inlet along axis Z

Ly, Lz no dimensional discharge factors (given by Equations 25, 26)

Pa static pressure in the cross-section of outlet (by X=Xa), Pa

Po static pressure in the cross-section of inlet (by X=0), Pa

Ps static pressure in current cross-section considered of room (given by Equation 42), Pa

Qc inlet excess heat (given by Equation 21), W

Qy, Qz no dimensional factors of excess heat (given by Equations 33, 34)

r orifice radius, m

S distance from inlet to cross section considered of current along jet axis, m

Scw distance from the slot to the separation point in the ceiling-wall corner (given by

Equation 43), m

Swf distance from the slot to the to the separation point in the wall-floor corner (given by

Equation 46), m

T temperature difference at point considered, oC

Ta absolute temperature of ambient air, oK

Tc current temperature difference (given by Equation 15), oC

Ti temperature difference induced by i - elementary jet, (given by Equation 6), oC

Tk inlet temperature difference of cell made by increments Ky and Kz, oC

Tm temperature difference on axis of i - elementary jet (selected from Equations

11 and 12), oC

To inlet axial temperature difference, oC


Ty, Tz no dimensional factors of current temperature difference (given Equations 18, 19)

U velocity at point considered (given by Equation 1), m/s

Uc current velocity at point considered (given by Equation 14), m/s

Ue air-removal velocity in cross-section considered of room (given by Equations 27), m/S

Ui air velocity induced by i - elementary jet (given by Equation 5), m/s

Uk inlet velocity of cell made by increments Ky and Kz, m/s

Um air velocity on axis of i - elementary jet (selected from Equations 11), m/s

Uo inlet axial velocity, m/s

Us velocity “shift” (given by Equation 23), m/s

Ux axial velocity (given by Equation 1, Y = 0, and Z = 0), m/s

Uy, Uz no dimensional factors of current velocity (given by Equations 16, 17)

X, Y, Z coordinates of point considered, m

Xa distance from inlet to outlet position, m

Xb distance from inlet to the measured sections of the back flow near floor, m

Xw distance from slot to the measured sections of jet near vertical wall (given by

Equation 44), m

Xwf distance from the jet slot to the wall-floor corner (given by Equation 45), m

Y0.5, Z0.5 no dimensional half-velocity width in the central XY and XZ planes,

correspondingly

Yj, Zj coordinates of point considered in jet j (given by Equations 7, 8), m

Zcw jet thickness at the distance from the jet slot to the ceiling-wall corner (Xr), m

Zwf jet thickness at the distance from the jet slot to the wall-floor corner (Xwf), m

αc angle between current trajectory projection in XY plane and X axis, degrees

c angle between current trajectory projection in XZ plane and X axis, degrees

xy angle between current trajectory and XY plane [ xy = atan(cosαc tan c)], degrees

δ, η coordinates of elementary jet i in inlet along axes Y and Z, m

δk, δk+1 start and end of increment Ky, respectively, in inlet along axis Y, m

ΔP elementary static pressure difference between the lower and upper current parts (given

by Equation 51), Pa

specific weight of the air at point considered (given by Equation 4), N m

a specific weight of ambient air, N m

ηk ,ηk+1 start and end of increment Kz, respectively, in inlet along axis Z, m

σ turbulent Prandtl number (a value of σ = 0.5 is used here)


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Tarnopolsky M. Room air distribution calculation. 2014

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