On the Selection of Design Parameters in the Design of Spray Ponds

P. A. Mukhachev

Associate Professor, Candidate of Technical Sciences

The principal parameters in the thermal calculation of spray ponds are:

1) the temperature of the cooled water—\(t_2^\circ\mathrm{C}\);

2) the cooling range—\(\Delta t = t_1 - t_2\), where \(t_1\) is the temperature of the water entering the installation;

3) the rainfall density—\(q\)—\(\mathrm{m^3}/\mathrm{m^2}\,\mathrm{h}\);

4) the head before the nozzles—\(H\) m water column.

The cooling of water drops falling through the air occurs mainly as a result of evaporation of the drop, as the latter tends to assume the wet-bulb temperature of the air.

The physical picture of the course of the cooling process of a falling drop may be represented as follows. A drop, having under a known approximation the shape of a sphere, falls in an atmosphere of air whose state is characterized by the dry-bulb temperature \(\tau_c\) and the relative humidity \(\varphi\) (the magnitude of the barometric pressure plays no essential role). The temperature at the surface of the drop tends to decrease to the temperature measured by a wetted thermometer. This circumstance entails the establishment of a temperature difference: a) between the center and the surface, \(t-\tau\); b) between the air and the surface \((\tau_c-\tau)\).

In the latter difference, the plus sign applies to the case when \(\tau_c > 0\) and \(\varphi < 100\%\), while the minus sign applies to the case when \(\varphi = 100\%\), or \(\tau_c < 0\). The heat lost by the drop is expended on the evaporation of water from the surface of the drop and partly participates in convective heat exchange between the air and the surface of the drop. The drop, owing to evaporation from its surface, is surrounded by a film through which water vapor diffuses into the air moving past the drop. In view of the presence of the film, the temperature at its surface is established somewhat above the wet-bulb temperature.

To determine the amount of heat lost by the cooling drop, one may use the theory of heat transfer in the cooling of bodies. For the heat loss by a sphere, Gröber gives the expression:

\[ Q = Q_0 \cdot \psi \left(\frac{aZ}{R^2};\, hR\right), \]

where \(Q_0\) is the excess heat in the sphere at the initial moment, in cal,

\[ Q_0 = \frac{4}{3}\pi R^3 c \gamma \Theta_c, \]

\(R\) — radius of the sphere, in m,
\(c\) — heat capacity, in cal/kg,

$\gamma$ — specific weight in kg/m³,

$\Theta_c$ — initial temperature of the body, in excess of the ambient temperature, in °C,

$\dfrac{aZ}{h^2}$ — Fourier homochronicity criterion,

$a$ — thermal diffusivity constant, $a=\dfrac{\lambda}{\gamma c}$,

$\lambda$ — coefficient of thermal conductivity in cal/m °C·h,

$Z$ — cooling time in h,

$hR$ — Nusselt criterion, $h=\dfrac{\alpha}{\lambda}$,

$\alpha$ — heat-transfer coefficient in cal/m² °C h. For water, at the temperature of atmospheric cooling: heat capacity $c=1$ cal/kg °C.

The temperature of the surrounding medium is the air temperature according to the wet-bulb thermometer; consequently, the initial temperature of the droplets, in excess of the ambient temperature, is $\Theta_c=t_1-\tau_m$,

the excess heat in a droplet is

\[ Q_0=\frac{4}{3}\pi R^3\gamma(t_1-\tau_m). \]

The quantity of heat lost by the droplet is

\[ Q_1=\frac{4}{3}\pi R^3\gamma(t_1-\tau_m)Y, \]

where

\[ Y=\psi\left(\frac{aZ}{R^2};hR\right). \]

The quantity of heat lost by 1 kg of cooled water will be

\[ \frac{Q_1}{\frac{4}{3}\pi R^3\gamma}=t_1-t_2=\Delta t, \]

therefore, the cooling zone is

\[ \Delta t=(t_1-\tau_m)Y. \tag{1} \]

Water falling on 1 m² of the basin surface will give up heat

\[ Q=q\Delta t\gamma=\varphi\gamma(t_1-\tau_m)Y \text{—cal/m²h}. \tag{2} \]

Equation (2) determines the possibility of maximum cooling of the water on the basis of the laws of heat transfer. On the other hand, the possibility of absorption of this heat by the air flowing through the spray plume is limited by the quantity of air and by the rate of diffusion of vapor into the air.

If the quantity of air flowing toward the basin, referred to 1 m² of the latter, is denoted by $L$ kg/m²h, then the quantity of heat which this air can absorb when the humidity increases from the initial $\varphi_1$ to $\varphi_2=100\%$ will be

\[ Q_2=(I_2-I_1)L=(I_2-I_1)w\cdot\gamma_v\cdot3600, \]

where $I_1$ and $I_2$ are the heat contents of the air at entry into the spray plume and at $\varphi_2=100\%$,

$w$ — air velocity in m/s,

$\gamma_v$ — specific weight of air in kg/m³.

We assume that the process of saturation of the air with water vapor occurs at \(\tau_c=\mathrm{const}\).

For maximum cooling of the water it is necessary that the condition

\[ Q=Q_2, \]

i.e.,

\[ q\cdot \Delta t\cdot \gamma=3600\cdot w\cdot \gamma_v (I_2-I_1), \]

be fulfilled; hence the rain density is

\[ q=\frac{3600\cdot w\cdot \gamma_v\cdot (I_2-I_1)}{\Delta t\cdot \gamma}. \tag{3} \]

By special studies of the heat transfer and evaporation of drops, carried out in the steam-turbine laboratory of VTI, a formula was obtained for determining the evaporation coefficient \(\beta_p\) in the form:

\[ \beta_p=8.51\sqrt{\frac{w}{d}}\ \mathrm{kg}/\mathrm{m}^2\mathrm{hour}; \tag{4} \]

where

\(w\) is the air velocity, m/sec;
\(d\) is the drop diameter, m.

The derivation of the formula for the evaporation coefficient was carried out through the coefficient of diffusion of water vapor in air; consequently, \(\beta_p\) expresses the amount of vapor that can be taken up by the air passing by the drop, proceeding only from the phenomenon of diffusion. The amount of vapor obtained as a result of evaporation from the surface of the drop, and, consequently, also the evaporation coefficient, can be determined on the basis of thermal phenomena. For the amount of heat carried away by vapor from \(1\ \mathrm{m}^2\) of basin surface, one may write:

\[ Q=q\cdot \Delta t\cdot \gamma=n\cdot \beta\cdot f\cdot z\cdot r, \]

where \(n\) is the number of drops falling on \(1\ \mathrm{m}^2\) of basin surface per hour;
\(r\) is the latent heat of vaporization;
\(\beta\) is the evaporation coefficient;
\(f\) is the surface of one drop in \(\mathrm{m}^2\).

From the last expression,

\[ \beta=\frac{q\cdot \Delta t\cdot \gamma}{n\cdot f\cdot z\cdot r}. \]

Since

\[ n=\frac{q\cdot \gamma}{\frac{1}{6}\pi d^3\gamma},\qquad af=\pi d^2, \]

then

\[ \beta=\frac{d\cdot \Delta t\cdot \gamma}{6\cdot z\cdot r}. \tag{5} \]

The second condition for maximum cooling of the water is that \(\beta\leq \beta_p\).

The amount of water evaporating, as a percentage of the water being cooled, will be

\[ G_{\mathrm{ev}}=\frac{Q}{r\cdot q\cdot \gamma}\cdot 100 =\frac{q\cdot \Delta t\cdot \gamma}{r\cdot q\cdot \gamma}\cdot 100 =\frac{100\cdot \Delta t}{r}. \tag{6} \]

Equations 1, 3, 4, 5, and 6 make it fully possible to determine the basic parameters of spray ponds.

In practical calculations, certain simplifications may be allowed.

1. Assume: the coefficient of thermal conductivity \(\lambda = 0.52\) kcal/m·°C·h,
   the specific weight of water \(\gamma = 1000\) kg/m³,
   the specific weight of air \(\gamma_v = 1.2\) kg/m³,
   the air velocity \(w = 0.5\) m/sec, taking the wind velocity to be zero (complete calm),
   the latent heat of vaporization \(r = 580\) kcal/kg,
   the dimension of the droplet diameter in millimeters,
   the dimension of the cooling time in seconds.

2. Since, when a droplet is cooled in an air atmosphere with relative humidity \(\varphi < 100\%\), the heat flowing from the center of the droplet to its surface is practically instantaneously used for vaporization, then for this case the heat-transfer coefficient \(\alpha_0\) will be equal to infinity—\(\alpha_0 = \infty\). Consequently, the Nusselt criterion will also have the value \(hR = \infty\).

Fig. 1

Under the indicated assumptions, for the Fourier homochronicity criterion we obtain the expression

\[ \frac{az}{R^2} = 0.577 \frac{z}{d^2}. \]

The values of \(Y\) as a function of the quantities \(0.577 \dfrac{z}{d^2}\) are represented by the curve in Fig. 1.

The calculation formulas 1, 3, 4, 5, 6 take the form:

\[ \Delta t = (t_1 - \tau_m)Y, \tag{1} \]

\[ q = \frac{2.16(I_2 - I_1)}{\Delta t}, \tag{3a} \]

\[ \beta_p = 268.8 \sqrt{\frac{w}{d}}, \tag{4a} \]

\[ \beta=\frac{d\cdot \Delta t}{z}, \tag{5a} \]

\[ G_{\text{us}}=0.172\,\Delta t. \tag{6a} \]

The data used in calculating spray installations are the temperature and humidity of the air, the quantity of water to be cooled, and the range of heating of the water in the condenser.

The range of heating of the water in the turbine condenser is determined from the heat balance of the condenser:

\[ \Delta t_{\text{h}}=\frac{D(i_2-t_{\text{k}})}{G}. \tag{7} \]

As the temperature of the cooling water changes, the vacuum of the turbine changes, and together with the latter the steam consumption \(D\), the heat content of the steam downstream of the turbine \(i_2\), and the condensate temperature \(t_{\text{k}}\) change.

Thus, the heating range, at a constant flow rate of cooling water \(G\), will depend on the temperature of the cooled water.

On the other hand, the cooling range is determined from equation 1, from which, for \(t_1=t_2+\Delta t\),

\[ \Delta t=\frac{(t_2-\tau_{\text{m}})Y}{1-Y}, \tag{7} \]

i.e., for a given sprayer design, head, and wet-bulb air temperature, it likewise depends only on the temperature of the cooled water, and this dependence is expressed by a straight line. Since the cooling range must be equal to the heating range, the common temperature of the cooled water for them can be found graphically, by plotting on the same scale the dependences superimposed on one another:

\[ \Delta t_{\text{h}}=f(t_2)\quad \text{and}\quad \Delta t=\varphi(t_2). \]

Temperature of the cooled water.

Fig. 2

The point of intersection of the straight line \(\Delta t=\varphi(t_2)\) with the curve \(\Delta t_{\text{h}}=f(t_2)\) gives the desired \(\Delta t\) and \(t_2\). Fig. 2 illustrates the construction of the indicated dependences; the straight line 2--2 is plotted for sprayers producing coarse spray, and the straight line 3--3 for sprayers producing finer spray.

For simplicity of solution, the difference \(i_2-t_{\text{k}}\) may be assumed constant and equal to that obtained when operating with normal vacuum. The excess steam consumption as a function of the increase in the temperature of the cooling water may be taken from the corresponding turbine characteristics.

Knowing \(\Delta t\), one can, by equation 3-a, determine the calculated rain density and check the condition \(\beta \leq \beta_{\text{r}}\). If the resulting calculated ...

but \(q\) proves to be greater than the rain density provided by the hydraulic operating regime of the spray nozzle, then several spray nozzles may be combined into a group with such a calculation that their total rain density does not exceed the design value. If, however, the design rain density is less than that provided by the hydraulic operating regime of the spray nozzle, or the requirement \(\beta \leqq \beta_p\) is not satisfied, then the design of the spray nozzle must be deemed unsatisfactory for the given conditions and another type must be selected.

The design area of the basin will be:

\[ F=\frac{1000G}{q}\ \text{m}^2, \]

The actual area of the basin will be determined by design after the spray nozzles and air corridors have been arranged in the basin.

The calculation procedure given above assumes as given: the type of spray nozzle, the head before the spray nozzle, and its hydraulic operating regime. The problem may be expanded by posing the question of selecting the optimum head before the spray nozzle. This selection can be made by means of an economic calculation for the particular installation.

With an increase in head, the time during which the droplets remain in the air increases and their diameter decreases, which entails a decrease in the temperature of the cooled water and a reduction in the turbine steam consumption. Knowing the cost of producing a ton of steam, we can construct the dependence:

\[ C_{\text{steam}}=f(H). \]

Furthermore, with a decrease in steam consumption, the heating zone will also decrease, and with it the cooling zone (provided \(G=\text{Const}\)), which will entail a reduction in the basin area and a reduction in allocations for depreciation and repair, i.e. \(C_{\text{dep}}=\varphi(H)\).

On the other hand, with an increase in head, the expenditure for electric power for pumping water will increase, which gives \(C_{\text{el}}=\psi(H)\).

The total curve \(C_{\text{steam}}+C_{\text{dep}}+C_{\text{el}}=F(H)\) will, at a certain point, give a minimum of expenditures and determine the optimum head.

The most correct calculation method will be one in which the calculation of the condensing installation is coordinated with the calculation of the basin.

Literature

1. D. N. Vyrubov. Heat transfer and evaporation of droplets. Journal of Technical Physics, 1939, vol. 9, issue 21.
2. Gröber. Die Erwärmung und Abkühlung einfacher geometrischer Körper. VDI No. 21, 1925.