# Theoretical considerations

A full theoretical treatment yielding a design complete in every detail is not possible in this text. A theoretical approach takes the design only so far; beyond this, the design can be completed by assigning empirical values to the unsolved variables.

A heat balance must be struck between the water and the air:

^w^"w(^wl — ^w2) ^a(^a2— ^al) (11*1)

Where mw = mass flow rate of water in kg s-1

Cw = specific heat capacity of water in kJ kg-1 K-1

Rwl = inlet water temperature in °C

Fw2 = outlet water temperature in °C

/na = mass flow rate of air in kg of dry air per s

Ha = enthalpy of air at inlet in kJ kg"1 dry air

Ha2 = enthalpy of air at outlet in kJ kg-1 dry air

Although mw decreases by evaporation as it flows through the tower, it is regarded here as a constant for simplicity. It can he shown that the enthalpy difference between the water

and the air at any point in the tower is the force promoting heat and mass transfer, and Jackson (1951) shows that the following expression can be derived:

Z = 2-61”»(r»l ~ ?w2) (U 2)

KsaAhm

Where k = the coefficient of vapour diffusion in kg water per s m2 for a unit value of Ahm,

Ahm = the mean driving force, in kJ kg-1,

S = the wetted surface area per unit volume of packing, in nT1,

A = the cross-sectional area of the tower in m2,

Z = the height of the tower in m.

Using equation (11.1), a complementary form of equation (11.2) exists:

(1L3)

KsaAhm

The ‘volume transfer coefficient’, ks, is independent of atmospheric conditions and can be expressed by re-arranging equation (11.2).

In the vicinity of the water in the tower there is a thin film of saturated air, at a temperature rw, the same as the temperature of the water. This air has an enthalpy /iw which is greater than that of the ambient air, h. d. Thus, for any given value of /w, the water temperature, there exists an enthalpy difference, Ah, between the enthalpy of the film, hw, and the enthalpy of the ambient air:

Ah = hw — hA (11.4)

Water temperature, fw Fig. 11.5 An enthalpy-water temperature diagram showing the mean driving force for a Cooling tower (Ahm). |

The mean value of this is Ahm and is termed the mean driving force. Figure 11.5 shows this diagrammatically. The equilibrium line represents the variation in the value of hw with

Of tower Tower |

Respect to the water temperature, fw. The operating line shows the variation of the enthalpy of the air, with respect to fw.

The driving force, promoting heat transfer at any value of fw, is the difference given by equation (11.4). Since heat exchange occurs through the height of the tower, the value of rw reduces as it falls through the tower. Thus, Ah2 represents the driving force at the bottom of the tower and Ah{ at the top. The mean value occurs at some intermediate position.

The operating line is straight, being determined by equation (11.1). Hence, the slope of the line depends on the ratio of water flow to air flow, mw/ma. When mw/ma is zero, the driving force is zero and the tower is infinitely tall. Such a condition would arise if the operating line were AB’, tangential to the equilibrium curve, because at the point of tangential touching, Ah is zero. The converse case is AB" parallel to the abscissa. Here the tower is not tall but, to secure the wetted surface area, it must be of large cross-section. It is evident that increasing the value of ma with respect to mw, for a given tower, increases the mean driving force and produces a bigger tower capacity. Advantage is taken of this to secure optimal control of a tower. See section 11.6.

A typical value of mw/ma is 1.0.

EXAMPLE 11.1

Assuming a water-to-air mass flow ratio of 1.0 and an ambient wet-bulb of 20°C (sling), calculate the air quantity likely to be handled by a cooling tower used to cool water from

32° to 27°C, for a refrigeration plant having a coefficient of performance of 4.

Answer

Heat rejected at the condenser, per kW of refrigeration = 1.25 kW (because the COP is 4)

Water flow rate through the condenser = 1.25/(4.2 x 5)

= 0.0596 kg s"1

Hence the air flow rate through the cooling tower is also 0.0596 kg s_1.

Assuming an induced-draught tower, the fan handling saturated air at 20°C wet-bulb, then the humid volume is 0.8497 m3 kg-1 and the air flow rate is

0. 0596 x 0.8497 = 0.0506 m3s-1 for each kW of refrigeration

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