Cooling Tower Performance

Cooling Tower Performance

Cooling Tower Performance

The tool rates a mechanical-draft cooling tower, counterflow or crossflow, by the Merkel method in the form used by Leeper in the U.S. Department of Energy report EGG-GTH-5775 (1981): the duty is turned into a required tower characteristic KaV/L, the fill is described by the power law KaV/L = C (L/G)⁻ⁿ, and the tower runs where the two are equal. The default case is the tower of that report's Example 2. Fill data, certified performance curves and the test code of the Cooling Technology Institute are not reproduced; C and n are entered or fitted to the design point.

Units

Water flow, air flow, altitude, fan pressure and pump head carry unit dropdowns and are converted when the unit system is switched. Temperatures are plain numbers in °C (SI) or °F (USCS) and are not converted by the switch; neither is the water price ($ per m³ or $ per 1,000 gal). Enthalpies are per unit mass of dry air; the USCS values use the datum of the inch-pound tables (dry air at 0 °F). KaV/L is dimensionless.

Design point

Range = hot water − cold water; approach = cold water − entering wet bulb. The water mass flow L is the volume flow times the density at the mean water temperature, and the heat load is Q = L c_w × range with c_w = 4.1868 kJ/kg·K (1 Btu/lb·°F). The dry-air mass flow G follows from the entered L/G, or from the fan air volume divided by the specific volume of the entering air, or from Leeper's rule for a first design that the air leaves saturated at the mean of the hot and cold water temperatures. The entering air enthalpy h₁ is that of saturated air at the wet bulb; the energy balance gives the leaving enthalpy h₂ = h₁ + (L/G) c_w × range, and the leaving air is taken as saturated at h₂. Moist-air properties are the perfect-gas relations with the Hyland and Wexler saturation pressure at the barometric pressure of the standard atmosphere for the entered altitude.

Merkel number

Merkel's equation combines sensible and latent transfer into one driving force, the difference between the enthalpy h_s of saturated air at the water temperature and the enthalpy h_a of the air stream:

KaV/L = ∫ c_w dT / (h_s − h_a), from the cold-water to the hot-water temperature, with h_a = h₁ + (L/G) c_w (T − T_cold) in counterflow.

The page evaluates it three ways. The four-point Chebyshev rule of cooling-tower practice takes the driving force at 0.1, 0.4, 0.6 and 0.9 of the range and gives c_w × range / 4 × Σ 1/(h_s − h_a). Those stations are the rounded abscissae of the equal-weight four-point Chebyshev quadrature (0.1027, 0.4062, 0.5938, 0.8973), which is shown as well. The reference value is composite Simpson integration over 200 intervals. The rounded rule is typically within one percent of the integral and falls further off when the approach is small. If the air line touches the saturation line inside the fill there is no driving force and no tower can carry the duty at that L/G; the page then reports the largest L/G that works.

The method assumes a Lewis factor of one, neglects the water lost by evaporation in the energy balance and does not fix the state of the leaving air (it is taken as saturated here). These are the assumptions of every Merkel-based rating.

Crossflow

In a crossflow fill the water falls while the air crosses, so the water of each column meets air of a different enthalpy and the single integral no longer applies. The same two transfer equations, c_w ∂T/∂y = −(KaV/L)(h_s − h_a) for the water and ∂h_a/∂x = (KaV/L)(L/G)(h_s − h_a) for the air (x and y running from 0 to 1 across and down the fill), are solved on a 40 × 40 grid of cells with the mean driving force of each cell, which conserves energy exactly. The required KaV/L is the value that brings the mixed water leaving the fill to the cold-water temperature. It is larger than the counterflow value for the same duty, commonly by 20 to 50 percent.

Characteristic and off-design point

The fill is described by KaV/L = C (L/G)⁻ⁿ. Leeper gives n ≈ 0.6 "for most conditions" (from the Cooling Tower Institute manual) and states the relation for L/G between one half and one and a half times the design value; the page flags an operating point outside that band. C is either passed through the design point, which makes the tower exactly adequate for the design duty, or entered for an existing fill, in which case the page also reports the cold-water temperature the tower delivers at the design duty.

Off design, the water flow, air flow and heat load are entered as percentages of design and the wet bulb as a temperature. Then L/G = design × water % / air %, range = design range × load % / water %, and the cold-water temperature is found by trial so that the KaV/L required for (cold water + range, cold water, wet bulb, L/G) equals C (L/G)⁻ⁿ. This is the trial-and-error procedure of Leeper's Examples 2 and 3. Figure 1 repeats the solution over a span of wet bulbs; Figure 2 shows the demand curves of the design and off-design duties against L/G with the characteristic line, whose crossings are the operating points.

Effectiveness model

If the saturation enthalpy is replaced by a straight line over the water range, with slope f′ = (h_s,hot − h_s,cold) / range, the Merkel equations become those of a heat exchanger between an air stream of capacity rate G and a water stream of equivalent capacity rate L c_w / f′. With NTU = KaV divided by the smaller capacity rate and C* the ratio of smaller to larger, the effectiveness is that of a counterflow exchanger, ε = [1 − e^{−NTU(1 − C*)}] / [1 − C* e^{−NTU(1 − C*)}], or of a crossflow exchanger with both streams unmixed, and the heat rejected per unit of the smaller capacity rate is ε (h_s,hot − λ − h₁). The term λ = (h_s,hot + h_s,cold − 2 h_s,mean) / 4 allows for the curvature of the saturation line. This is the construction published by Jaber and Webb (1989); it is written here from the mathematics and checked numerically (for a straight saturation line it reproduces the Merkel integral exactly), not against the paper. It is shown beside the Merkel result and usually agrees within a few tenths of a degree.

Water balance

Evaporation is taken from the humidity balance of the air, E = G (W₂ − W₁), W₁ from the entered dry bulb and wet bulb and W₂ saturated at h₂; Leeper calls this the more accurate of his two methods. Beside it the page prints the rule of thumb E = factor × range in °F × water flow and the upper bound Q / h_fg that would apply if all the heat left as vapour. Leeper's factor is 0.001 (one percent of the flow per 10 °F of range), which is close to that upper bound; the common figure 0.00085 corresponds to about 85 percent of the heat leaving as latent heat. Drift is a percentage of the circulating flow; 0.02 percent is the emission factor of EPA AP-42 for induced-draft towers. Following the Federal Energy Management Program fact sheet, blowdown is whatever must leave besides evaporation to hold the cycles of concentration: blowdown + drift = E / (cycles − 1), and make-up = evaporation + drift + blowdown = E × cycles / (cycles − 1). The table of cycles shows the diminishing return of higher cycles; going from three to six cycles cuts make-up by 20 percent and blowdown by 60 percent.

Fan and pump

Fan input power = air volume × fan total pressure / efficiency, with the leaving air volume for an induced-draft fan and the entering volume for a forced-draft fan; Leeper's rule of 1 hp per 8,000 actual cubic feet per minute is printed for comparison. Pump power = ρ g × flow × head / efficiency for the head the tower adds to the circuit (static lift to the distribution deck, nozzle pressure and pipe friction, entered as one figure; Leeper suggests the fill height plus 10 ft).

Checks and limits

The test script reproduces the three examples of the Leeper report to hand accuracy: its enthalpies are hand-table values and its integrals are hand sums, so the tool gives KaV/L = 1.437 where Example 1 prints 1.408, 1.523 where Example 2 prints 1.49, and cold water at 79.6 °F and 89.9 °F where Examples 2 and 3 print 79.5 and 89.6 °F. The Merkel integral, the off-design points, the crossflow solution and the effectiveness relations are checked against an independent numpy / scipy implementation. Not covered: natural-draft towers, fill pressure drop and fan curves, the Poppe method and the state of the leaving air, plume visibility and fogging, freezing, and supplier ratings, which govern for an actual tower. The wet bulb must be above −5 °C and the hot water below 80 °C.