# Wet-bulb temperature

A distinction must be drawn between measured wet-bulb temperature and the temperature of adiabatic saturation, otherwise sometimes known as the thermodynamic wet-bulb temperature. The wet-bulb temperature is a value indicated on an ordinary thermometer, the bulb of which has been wrapped round with a wick, moistened in water. The initial temperature of the water used to wet the wick is of comparatively minor significance, but the cleanliness of the wick and the radiant heat exchange with surrounding surfaces are both important factors that influence the temperature indicated by a wet-bulb thermometer. On the other hand, the temperature of adiabatic saturation is that obtained purely from an equation representing an adiabatic heat exchange. It is somewhat unfortunate, in air and water-vapour mixtures, that the two are almost numerically identical at normal temperatures and pressures, which is not so in mixtures of other gases and vapours.

Wet-bulb temperature is not a property only of mixtures of dry air and water vapour. Any mixture of a non-condensable gas and a condensable vapour will have a wet-bulb temperature. It will also have a temperature of adiabatic saturation. Consider a droplet of water suspended in an environment of most air. Suppose that the temperature of the droplet is rw and that its corresponding vapour pressure is pw. The ambient moist air has a temperature t and a vapour pressure of ps.

Provided that pw exceeds ps, evaporation will take place and, to effect this, heat will flow from the environment into the droplet by convection and radiation. If the initial value of fw is greater than that of t, then, initially, some heat will flow from the drop itself to assist in the evaporation. Assuming that the original temperature of the water is less than the wet — bulb temperature of the ambient air, some of the heat gain to the drop from its surroundings will serve to raise the temperature of the drop itself, as well as providing for the evaporation. In due course, a state of equilibrium will be reached in which the sensible heat gain to the water exactly equals the latent heat loss from it, and the water itself will have taken up a steady temperature, t which is termed the wet-bulb temperature of the moist air surrounding the droplet.

The condition can be expressed by means of an equation:

(hc + hr)A(t — t’) = aAhfg(p’ss — Ps) (2.27)

Where

Hc = the coefficient of heat transfer through the gas film around the drop, by convection

Hr = the coefficient of heat transfer to the droplet by radiation from the surrounding surfaces,

A = the surface area of the droplet,

H{g = the latent heat of evaporation of water at the equilibrium temperature attained, a = the coefficient of diffusion for the molecules of steam as they travel from the parent body of liquid through the film of vapour and non-condensable gas surrounding it.

The heat transfer coefficients (hc + hr) are commonly written, in other contexts, simply as h.

Equation (2.27) can be re-arranged to give an expression for the vapour pressure of moist air in terms of measured values of the dry-bulb and wet-bulb temperatures:

A28)

The term (hc + hr)/(ahtg) is a function of barometric pressure and temperature.

Equation (2.28) usually appears in the following form, which is termed the psychrometric equation, due to Apjohn (1838):

Ps = PL — PmMt-t’) (2.29)

Where

Ps = vapour pressure

P’ss = saturated vapour pressure at a temperature t’ p. dl — atmospheric (barometric) pressure t = dry-bulb temperature, in °C t’ = wet-bulb temperature, in °C A = a constant having values as follows:

Wet-bulb > 0°C wet-bulb < 0°C

Screen 7.99 x lO’4 “C“1 7.20 x 10“4 “CT1

Sling or aspirated 6.66 x 10~4 °C_1 5.94 x 10“4 °C_1

Any units of pressure may be used in equation (2.29), provided they are consistent. EXAMPLE 2.14

Calculate the vapour pressure of moist air at a barometric pressure of 101.325 kPa if the measured dry-bulb temperature is 20°C and the measured sling wet-bulb is 15°C.

Answer

The saturated vapour pressure at the wet-bulb temperature is obtained from CIBSE tables for air at 15°C and 100 per cent saturation and is 1.704 kPa. Alternatively, the same value can be read from steam tables at a saturated temperature of 15°C. Then, using equation (2.29):

Ps = 1.704 — 101.325 x 6.6 x 10^(20 — 15)

= 1.370 kPa

CIBSE psychrometric tables quote 1.369 kPa for air at 20°C dry-bulb and 15°C sling wet — bulb.

The radiation component of the sensible heat gain to the droplet is really a complicating

Factor since it depends on the absolute temperatures of the surrounding surfaces and their emissivities, and also because the transfer of heat by radiation is independent of the amount of water vapour mixed with the air, in the case considered. The radiation is also independent of the velocity of airflow over the surface of the droplet. This fact can be taken advantage of to minimise the intrusive effect of hr. If the air velocity is made sufficiently large it has been found experimentally that (hc — hT)/hc can be made to approach unity. In fact, in an ambient air temperature of about 10°C its value decreases from about 1.04 to about 1.02 as the air velocity is increased from 4 m s_1 to 32 m s-1.

At this juncture it is worth observing that increasing the air velocity does not (contrary to what might be expected) lower the equilibrium temperature t’. As more mass of air flows over the droplet each second there is an increase in the transfer of sensible heat to the water, but this is offset by an increase in the latent heat loss from the droplet. Thus t’ will be unchanged. What will change, of course, is the time taken for the droplet to attain the equilibrium state. The evaporation rate is increased for an increase in air velocity and so the water more rapidly assumes the wet-bulb temperature of the surrounding air.

It is generally considered that an air velocity of 4.5 m s“1 is sufficient to give a stable wet-bulb reading.

In general,

C__MS ps Ma Pa

See equation (2.12) and so

Ms (Pss — Ps)

(gs’s — g) =

Ma Pa

It being assumed that pa, the partial pressure of the non-condensable dry air surrounding the droplet, is not much changed by variations in ps, for a given barometric pressure. The term g’s is the moisture content of saturated air at t’. Equation (2.28) can be re-arranged as

(ffss ~ j?) __ Ms (/ic + hr)

(t — t’) ~ Ma PaAfgOC

Multiplying above and below by c, the specific heat of humid air, the equation reaches the desired form

<2-30>

Where (Le) is a dimensionless quantity termed the Lewis number and equal to

Ms (hc + hr)

Ma p. dac

In moist air at a dry-bulb temperature of about 21°C and a barometric pressure of

101.325 kPa, the value of

Ms hc Ma paac

Is 0.945 for airflow velocities from 3.8 to 10.1 m s 1 and the value of

(hc + hr)

Hc

Is 1.058 at 5.1 m s’1. So for these sorts of values of velocity, temperature and pressure the Lewis number is unity.

As will be seen in the next section, this coincidence, that the Lewis number should equal unity for moist air at a normal condition of temperature and pressure, leads to the similarity between wet-bulb temperature and the temperature of adiabatic saturation.

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