Merging of long rows of plumes: Crosswinds, multiple rows, and applications to cooling towers

The merging of a single row of plumes in a quiescent environment has been studied using irrotational flow theory [G. Rooney, J. Fluid Mech. 771, R1 (2015)]. The present study extends this theory by considering (i) two parallel rows of plumes in a quiescent environment, and (ii) a single row of plumes in a crosswind, with and without a pressure drag term. For plumes in two rows with and without offset, the effect of varying the spacing ratio on the plume dynamics is investigated. Two definitions of the contact height are suggested according to the shape of the velocity potential contours. For a single row of plumes in a crosswind, the governing equations are closed using an entrainment flux evaluated by the irrotational flow theory. This theory predicts the correct near- and far-field similarity solutions in both modest and strong crosswinds. A comparison of the theory in question to previous towing tank experiments yields satisfactory agreement in terms of plume trajectory. The present theory of single and dual rows of plumes is applied to long rows of cooling tower plumes.

Figure 1

(a) Schematic of a single row of equally spaced line sinks. The dashed rectangle indicates the field of view for the contours illustrated in panel (b). (b) Velocity potential contours for a range of $p$, i.e., $p=0.1$, 0.4, 0.7, 1, 1.2, 1.5, 2, and 5, where $p$ is the constant given in (3). The thick curve, which corresponds to $p=1$, represents the height of first contact, ${\widehat{z}}_{\text{fc}}$.

Figure 2

Evolution of $\widehat{w}$ [given by the solution of (21)] and $\widehat{Q}$ [given by the solution of (22)] as a function of $\widehat{z}$. The horizontal dashed lines denote the height of first contact, ${\widehat{z}}_{\text{fc}}=0.340$. The solid straight lines denote the near-field ($p<1$) and far-field ($p>1$) similarity scalings.

Figure 3

Surface plot illustrating plume merger in a long row of plumes. The plume boundary is shaded according to the height $0\le \widehat{z}\le 1$.

Figure 4

(a) Schematic of two nonoffset parallel rows of an infinite number of line sinks. The dashed rectangle indicates the field of view for the contours illustrated in panel (b). (b) Velocity potential contours for $b^{\prime }\equiv \pi b/a=\pi /2$. The contours start from (0, $\pi /2$) and expand outward with $p$ selected from the set {1, 2, 4, 8, 10, $cosh2b^{\prime }-1$, 11, $sinh2b^{\prime }$, 12, $cosh2b^{\prime }+1$, 15, 20, 30, 40, 50}. The thick half-solid and half-dashed contour corresponds to $p=cosh2b^{\prime }-1$, the thick dash-dotted contour corresponds to $p=sinh2b^{\prime }$, and the thick solid contour that extends into the corners corresponds to $p=cosh2b^{\prime }+1$. Within the dash-dotted contour ($p<sinh2b^{\prime }$), the solid and dashed parts of the contours correspond, respectively, to the positive and negative roots in (30).

Figure 5

Contour length, $l^{\prime }$, and cross-sectional area, $A^{\prime }$, as a function of $p$ for $b/a=0.5$. The horizontal line denotes a constant value of $\pi$.

Figure 6

Evolution of $\widehat{w}$ [given by the solution of (21)] and $\widehat{Q}$ [given by the solution of (22)] as a function of $\widehat{z}$ for spacing ratios of $b/a=0.25$, 0.5, and 1. Panels (a) and (b) denote the nonoffset case (Fig. 4), whereas panels (c) and (d) denote the offset case (Fig. 9). In all panels, ${\mathrm{\Gamma }}_0=1$.

Figure 7

Surface plot illustrating plume merger in the case of two parallel rows of nonoffset plumes with a spacing ratio of $b/a=0.5$.

Figure 8

The contact heights ${\widehat{z}}_{c,1}$ and ${\widehat{z}}_{c,2}$ plotted as a function of the spacing ratio $b/a$. ${\widehat{z}}_{c,1}$ is defined as the elevation where $p$ is closest to $cosh2b^{\prime }-1$, and ${\widehat{z}}_{c,2}$ is defined as the elevation where $p$ is closest to $cosh2b^{\prime }+1$. In all cases, ${\mathrm{\Gamma }}_0=1$.

Figure 9

(a) Schematic of two parallel rows of an infinite number of line sinks with an offset. The dashed rectangle indicates the field of view for the contours illustrated in (b). (b) Velocity potential contours for $b^{\prime }\equiv \pi b/a=\pi /4$. The contours start from (0, $\pi /4$) and expand outward with $p$ selected from the set {0.5, 1.5, $sinh2b^{\prime }, cosh2b^{\prime }$, 3, 4, 5, 8, 10}. The thick half-solid and half-dashed contour corresponds to $p=sinh2b^{\prime }$, and the dash-dotted contour corresponds to $p=cosh2b^{\prime }$. The solid and dashed curves correspond, respectively, to the positive and negative roots in (55).

Figure 10

Contour length, $l^{\prime }$, and cross-sectional area, $A^{\prime }$, as a function of $p$ for $b/a=0.5$. The horizontal line denotes a constant value of $\pi$.

Figure 11

Surface plot illustrating plume merger in the case of two offset parallel rows of plumes with a spacing ratio of $b/a=0.5$.

Figure 12

As in Fig. 8 but for the case of offset plumes with an offset distance of $a/2$. ${\widehat{z}}_{c,1}$ is defined as the elevation where $p$ is closest to $sinh2b^{\prime }$, and ${\widehat{z}}_{c,2}$ is defined as the elevation where $p$ is closest to $cosh2b^{\prime }$. In all cases, ${\mathrm{\Gamma }}_0=1$.

Figure 13

Nondimensional effective entrainment perimeter as a function of $p$. For two rows of plumes, the spacing ratio is fixed as $b/a=0.2$.

Figure 14

Evolution of plume trajectory (a), volume flux (b), and horizontal (c) and vertical (d) velocities. The horizontal dashed (solid) line denotes the height of first contact for $\overline{a}\equiv a/L_B=0.2$ ($\overline{a}=1$). In both cases, ${\mathrm{\Gamma }}_0=1$.

Figure 15

Contact height, ${\overline{z}}_c$, as a function of $\overline{a}\equiv a/L_B$. ${\overline{z}}_c$ is defined as the elevation where $p$ is closest to 1. In all cases, ${\mathrm{\Gamma }}_0=1$.

Figure 16

Effects of varying the source Froude number, ${\mathrm{Fr}}_0$ (a), ambient to plume source velocity ratio, $R$ (b) and port spacing ratio, $a/D$ (c)–(f) on the plume trajectory. The thin curves represent the theory without a drag term (Secs. 4a and 4b) whereas the counterpart thick curves represent the theory of Sec. 4d with $C_D=1$. The experimental data are taken from Appendix A of Kannberg and Davis [24]. Note that the uppermost diamond in (c) was measured using a different method compared to the other experimental data [24].

Figure 17

Top view of single and dual rows of cooling tower cells. The black circles denote cells at the center, and the gray half-circles denote the half-cells at the two ends.

Figure 18

Relative humidity profiles for single (a),(b) and dual (c),(d) rows of plumes. The model input parameters are specified in Table 1. Note, in particular, that $\frac{{\dot{m}}_d}{{\dot{m}}_w}$ represents the ratio of dry to wet air mass flow rate.

Figure 19

Comparison between the present irrotational flow theory and the theory of Wu and Koh [19]. The horizontal solid and dashed lines denote the respective heights of first contact for the above theories. In all cases, ${\mathrm{\Gamma }}_0=1$.

Figure 20

Comparison between theory and experiment for cases 1–6 of Ref. [34]. The solid and dashed curves nearly overlap except for large $x/D$.

Figure 21

As in Fig. 20 but for cases 7–12.