Plume Interaction
When a heat source is located close to a wall, the plume may attach to the wall; see Fig. 7.69. In this case the entrapment will be reduced compared to the entrainment in a free plume, and the attached plume can be regarded as half the plume from the source, with its mirror image on the other side of the wall; see Fig. 7.70.
When several heat sources are located close to each other, the plumes may merge into a single plume; see Fig. 7.72. In this case, the source should be regarded as one single source, with the heat emission equal to the sum of the
Heat emission from each of the sources:
Qt. = 0.005(E<t>)l/i*s/J (7.192)
Where ^<l> = the sum of the individual heat emissions (W), and z — the height above the virtual source (m). The total flow from N identical sources is then given by16
4v, n = <3v, inU%’ (7.193)
Where qLiJ is the volume flow in the plume from one of the sources.
7.5.1 Plumes in Confined Spaces
Figure 7.73 shows a plume in an open environment. The hot air from the source entrains ambient air into the convection currenr (the plume), thus making the air volume flow increase with height.
Imagine that we enclose the plume, as shown in Fig. 7.74. The plume still entrains air from the surroundings, but the available fresh air is limited. This means that fresh air will surround the plume only up to a certain level. Above this level, the entrained air has to be recirculated from the piume itself. This leads to a twozone flow model, with a layer of fresh air at the bottom, and warmer air from the plume at the top. The interface between the two layers is located at the height at which the entrained air in the plume equals the supplied air. This can be found from the volumeflow formulae of Section 7.5.2.
Example 7.5.4 Confined Plume
We now put the hot cylinder of Example 7.5.3 inside a room. The convective heat outpur is still <J> = 5 kW. The air is supplied at the floor, as shown in
Temperature distribution in a room enclosing a convection current. The air enters the room at low temperature, and is mixed slightly with the room air and heated by convection from the floor, thus making the air temperature at floor level higher than the supply temperature. The air temperature increases from the floor level up to the ceiling, more or less linearly, because the hottest air (in the core of the plume) rises to the ceiling due to its buoyancy, while the outer, cooler parts of the plume layer according to their temperature.
Plumes are influenced by the temperature stratification. The driving force of the plume is the temperature difference between the plume and the surroundings. When this difference diminishes, the plumes will disintegrate and spread horizontally in the room; see Fig. 7.77.
Batchelor17 noticed the influence of a temperature gradienr in the surroundings, and Morton et al.13 gave a method for calculating the maximum plume rise from a point source in surroundings with a temperature gradient. The volume flow rate in plumes in a room with a temperature stratification is slightly decreased compared to the volume flow rates calculated with the equations presented for nonstratified media.12 Jin18 studied the maximum plume rise height for plumes above welding arcs.
In the presence of the temperature gradient, the convective plume reaches its terminal height (zt), where the temperature difference between the plume and the ambient air disappears. Also, there is another point in the plume at which the air velocity equals zero. This is referred to as the maximum height of the plume The plume spreads horizontally between these two heights.
For a point source, Mundt3 gives the following plume rise formulae: Maximum height:
4(<^3/8
Zmax = 0.984) U4fzy (7.195)
Equilibrium height:
Td&Y3/* { dzt 
I jflU/n
(7.196)
Z, = 0.74<t>1/4
Where
DO/dz = vertical temperature gradient in the room air (°C/m)
<5 = convective heat from the source (W) zmax = maximum plume rise height (m) zt = equilibrium height of the plume (m)
Volume flow rate: The volume flow rate through a given height above the virtual point source in the plume can be found by the following calculation procedure, according to Mundt:3
1. Calculate the location of the virtual source and the corresponding z.
2. For the height z above the virtual source, calculate Zj according to the formula
Zi = 2.86z^,/!V1/4 (7.197)
If 2.125 < Z < 2.8, the density difference disappears and the calculations become uncertain; if Zj 2: 2.8, the plume has reached its maximum height below the actual level.
3. Calculate
= 0.004 + 0.039zj + 0.380zf — 0.062 • zf (7.198)
4. The volume flow rate in the plume through the height z can be found by
Qu = 0.00238<J>3/4[^j~5/8m1 , (7.199)
Where qv is the volume flow rate in m3/s.
For a line source, Mundt3 gives the following plume rise formulae:
Maximum height:
Z, MX = 0.51ct>i/3(jgj’/2 (7.200)
Equilibrium height:
Zt = 0.354>1/3f^)“l/2 (7.201)
Volume flow rate:
1. Calculate the location of the virtual source and the height z above the virtual source.
2. For the height z above the virtual source, calculate Z:
(7.2021 
Z = SJ%z(d%/dz)*,iQ~i/i
If 2.0 < z[ < 2 .95, the density difference disappears and the calculations become uncertain; if ^ ^ 2.95, the plume has reached its maximum height below the actual level.
.5. Calculate
(7.203) (7.204! 
Jв = 0.004 + 0.477j! + 0.029z? — 0.018zj
4. T he volume flow rate is given by
QvA = O. OO4824>/3(t/0/<i*)~1/2.y5
Where
Qr] is the volume flow rate in m7s in.
Example 7.5.5 Point Source in a Room with Thermal Stratification
We now put the cylinder of Example 7.5.4 inside a room with a vertical temperature gradient of 1.5 DC/m (see Fig. 7.78). In this case we assume that there are other heat sources in the room. We want to investigate how this temperature stratification influences the volume flow in the plume above the cylinder, and at what height the plume stops.
Following the formulae and calculation procedure of Section 7.5.5.J, we get
A. Location of the virtual source: We use the location calculated for the maximum case in Example 7.5.4, i. e.,
Z» = 0.83 nv
* i 
a 









/ z„= 0.83 m
<D = 5 kW/
Virtual source
B. Maximum plume rise: The maximum height above the virtual source is found from Eq. (7.195):
Zmax = 0.98 ■ 50001/4 • 1.5 5/8 = 7.08 m,
I. e., the maximum height is 7.08 m above the virtual source, which is
7.08m — 0.83m = 6.25 m above the flooi
Equilibrium height (Eq. 7.196):
Zt = 0.74 ■ 50001/4 ■ 1.5~3/8 = 5.34 m,
I. e., the equilibrium or terminal height is 5.34 m above the virtual source, which is
5.34 — 0.83 = 4.51 m above the floor.
C. Volume flow rate in the plume: To find the air volume flow rate through a level Zfloor = 3.5 m, we must first calculate z and then at this level with Eq. (7.197):
2! = 2.86 • (3.5 + 0.83) ■ 1.53/8 5000,/4 = 1,715
With Z we can calculate ml according to Eq. (7.198):
M{ = 0.004 + 0.039zj + Q.380zf — 0.062zf = 0.875 ,
Which gives the convection airflow rate from Eq. (7.199):
Q„ = 0.00238 • (50003/4) ■ 1.55/8 • 0.875 = 0.96 m3/s.
The volume flow calculated 3.5 m above the floor is slightly lower than that calculated for an isothermal atmosphere. The deviation between the volume flows calculated for isothermal room air and stratified room air can be seen from Fig. 7.79,
Example 7.5.6 Comparison with Numerical Modeling
Calculate maximum air velocity, airflow rate, and excessive temperature (relative to the ambient air temperature equal to 20 °C) in thermal plume above the heated cube (0.66 m x 0.66 m x 0.66 m) with convective heat production Wcony = 225 W, at heights of 2.0 m and 4.0 m above the floor level. Neglect temperature gradient along the room height. Compare the results with predictions made for the same case using CFD code19 (Fig. 7.80),
To calculate the thermal plume, the cube can be presented as a cylinder with a diameter equivalent to the hydraulic diameter of the top of the cube:
D — = 0.66 m.
4 x 0.66
In the minimum case,
Z0 = 0.8D/(2 tanl2.5°) = 1.19 m.
The virtual source is located below the floor level:
Zp = D/3 + Hza = 0.66/3 + 0.661.19 =0.31 m.
Thus the vertical distances to be used in the plume characteristics calculation are correspondingly equal to 2.31 m and 4.31 m.
Height above floor (m) FIGURE 7.79 Airflow rate in the plume above the cylinder of Example 7.S. S. 
The maximum velocity in the thermal plume, from Table 7.19, is v, , m = 0.128 • 2251/3 • 2.31 173 = 0.59 m/s
And
, 4 m 
= 0.128 ■ 2251/3 • 4.311/3 = 0.48 m/s.
The corresponding values of maximum velocities in the plume at heights 2.0 m and 4.0 m above the floor level, from the table in Fig. 7.80, are 0.54 m/s and 0.42 m/s.
The maximum excessive temperature in the thermal plume, from Table 7.19, is
A0 = 0.329 • 2252/3 ■ 2.31~5/3 = 3.0 °C
And
AO = 0.329 • 2252/3 • 4.31 s ; — 1.06 °C.
The corresponding values of maximum excessive temperatures in the plume at heights 2.0 m and 4.0 m above the floor level, from the table in Fig. 7.80, are
3.6 °C and 1.06 °C.
The maximum airflow in the thermal plume, from Table 7.19, is
QV’X = 0.005 • 2251/3 • 2.315/3 = 0.123 m3s
And
Qv<z = 0.005 • 2251/3 • 4.315/3 = 0.347 m3s .
The corresponding values of airflow rates in the plume at heights 2.0 m and 4.0 m above the floor level, from the table in Fig. 7.80, are 0.18 m3/s and 0.35 m3/s.
0.60 0.50 
I— 

Иг 
— 
i 
1— 

0.30 0.20 0.10 0.00 
Jr 
♦…… 

Лч 
K 
4, :. 

“TP" 
‘X, . 

. .. X 
A 
■ 

0.1 
0.2 
0.5 
0.75 
1 
1.5 
2 
2.5 
3 
H 4 
5 

= 0 
0.06 
0.11 
0.25 
H (,_4 
0.33 
0.49 
0.54 
0.50 
0.45 
0.42І 
0.40 
ЯDt/H = 0.5 
0.07 
0.14 
0,25 
0.14 
0.33 
0.47 
0.50 
0.44 
0.35 
: (“О Ri ■o 
0.08 
AD?/H = 1.0 
0.06 
0.13 
0.26 
0.15 
0.32 
0.46 
0.48 
0.42 
0.32 
0.08 

*Dt/H= 2.0 
0.06 
0.12 
0.23 
0.14 
0.31 
R 0.41 
0.39 
0.27 
0.06 
Height above the floor level, m 
10.00 

ИЧ 
4n 

V* 
… 

Г* 
* 
;5Ar. 

. lb:: 
1 
♦— 
.. ♦ 

■ 

0.1 
0.2 
0.5 
0.75 
1 
1.5 
2 
2.5 
3 
4 
5 

Dt/H = 0 
2.76 
4.08 
5.56 
7.56 
5.96 
4.65 
3.60 
2.40 
1.66 
1.06 
0.84 
Dt/H = 0.5 
3.00 
4.36 
5.79 
7.51 
5.98 
4.46 
3.19 
1.94 
1.04 
0.16 
0.40 
A — Dt/H = 1.0 
2.83 
4.25 
6.38 
7.57 
5.96 
4.24 
2.81 
1.50 
0.52 
0.74 

•* Dt/H = 2.0 
2.24 
3.97 
5.80 
7.57 
5.66 
3.39 
1.42 
0.17 
1.05 
Height above the floor level, m 
Height above the floor level, m FIGURE 7.80 CDFpredicted values of maximum velocity V, temperature differential, 0^ (°С), and airflow, q (Us), in the horizontal crosssection of the buoyant plume above the heated cube (0.66 m x 0.66 m x 0.66 m, 225 W).19 
Airflow, L/s Max. temperature difference, °С Maximum velocity, V’, m/s 
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