# Nonconfined and Nonstratified Environments

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich1 and Schmidt.2 Analytical equations to calculate velocities, tempera­tures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in

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FIGURE 7.59 Thermal plume above a horizontal surface.

Thermal plume cross-sections.3 These equations correspond to those determined experimentally by other researchers4,5 and are listed in Table 7.19. The equations in Table 7.19 were derived with the assumption that the heat source size was very small and did not account for the actual source dimensions.

The coefficients in the equations differ slightly in different references, de­pending on the entrainment coefficients used. The convective heat flux <t>, in W or W/m from the heat source, can be estimated from the energy consump­tion of the heat source <i>tot by

4> = k<Ptot (7.185)

The value of the coefficient k is 0.7-0.9 for pipes and ducts, 0.4-0.6 for smaller components, and 0.3-0.5 for larger machines and components.6 FIGURE 7.60 Plume from a point source.

Example 7.5.1

A point source has a convective heat output of 100 W (see Fig. 7.62). De­termine the airflow rate J m above the source.

Solution Table 7.19 gives the equation to be used:

Qv<. = 0.0054),/j2S/3. (7.186)

This gives

Q„. = 0.005- 1001/3 15/* — 0.023 m3/s.

7.5.2.2 Convection Flow along Vertical Surfaces

Convection flow along vertical surfaces (Fig. 7.63) is also of major inter­est in industrial ventilation, where large production units with a vertical ex­tension are often present. When the vertical extension of the surface is small, the convection flow is mainly laminar, but at larger extensions the flow is tur-   Rhe room. These surfaces are mostly treated as plumes from extended surfaces; see below.

In reality, heat sources are seldom a point, a line, or a plane vertical surface. The most common approach to account for the real source dimen­sions is to use a virtual source from which the airflow rates are calcu­lated;9’12 see Fig. 7.64. The virtual origin is located along the plume axis at a distance *t) on the other side of the real source surface. The adjustment of the point source model to the realistic sources using the virtual source method gives a reasonable estimate of the airflow rate in thermal plumes. The weakness of this method is in estimating the location of the virtual point source.

The method of determining a maximum case and a minimum case provides a tool for such estimation; see Fig. 7.65.12 In the maximum case, the real source is replaced by a virtual point source such that the border of the plume above the point source passes through the top edge of the real source (e. g., cylinder).

The minimum case is when the diameter of vena contracta of the plume is about 80% of the upper surface diameter and is located approxi­mately one-third of a diameter above the source. The spreading angle of the plume is set to 25". For low-temperature sources, Skistad12 recom­mends the maximum case, whereas the minimum case best fits the mea­surements for larger, high-temperature sources. —— 1…………………….. 1 i -— D Zf — 1.7-2.1D Virtual source

 FIGURE 7.66 Virtual source according to Morton.

For a flat heat source Morton et a!.1J suggested that the position of the vir­tual source be located at z0 = 1.7 -2.1 xD below the real source (Fig. 7.66). This corresponds well with the maximum and minimum method described above, which gives z0 = 1.47 -2.25 x D below a flat plate.

Mundt5 calculates the thickness of the boundary layer (see Table 7.20) at the top of a vertical extended heat source and adds this to the source radii, and then calculates the position of the virtual source as? o = 2.1 (D + 28) before us­ing the point source equation.

According to Bach et al.14 the volume flow from the vertical surfaces should be added to the volume flow calculated by the equations for point or line sources.

Example 7.5.3

Calculate the convection flow rate, qv at a height 2fi(xir = 2 m above the floor in the plume above a hot cylinder with a diameter of D = 0.66 in and a height of H — 0.66 m. The convective heat flux is <t> = 5 kW.

(a) The Maximum Case (Fig. 7.67) In the maximum case we get Zq = D/(2tan 12.5°) = 2.25D = 1.49 m,  