What are the correct equations for calculating HLIs and AHLUs?

Calculation of the Heat Load Index (HLI) requires Temperature (T) in °C, Relative Humidity (RH) expressed as a percentage, Wind Speed (WS) in m/s and Black Globe Temperature (BGT) in °C. Of these, T, RH and WS are routinely measured by the great majority of weather stations. Although sensors for measuring BGT exist, these are not normally included as part of the standard weather station and must be ordered from a suitable supplier. In the absence of a BGT sensor, the BGT can be inferred from measurements of T and Solar Radiation (SR).

The equation for calculating BGT from T and SR is:

BGT = 1.33 x T – 2.65 x sqrt(T) + 3.21 x log(SR + 1) + 3.5
where:
• log is the logarithm (base 10) function
• sqrt is the square root function

In the past, the HLI was calculated using one of two equations, depending on whether the BGT is above or below 25°C as follows:

if ( BGT is below 25 ) then
HLILO = 1.3 x BGT + 0.28 x RH – WS + 10.66
else
HLIHI= 1.55 x BGT + 0.38 x RH – 0.5 x WS + exp (2.4 – WS) + 8.62
where:
• exp is the exponentiation function
and the HLI value was taken as either HLIHI or HLILO depending on the BGT value.

One issue that has been identified is that large jumps in HLI occur under some circumstances when the BGT passes through 25°C – for example from 24.9°C to 25.1°C. To overcome this, a blending function – S(BGT) – is used to produce a smooth transition in HLI values calculated using the two different equations. The blending function is:

S(BGT) = 1 / (1 + exp(-(BGT – 25) / 2.25))

Using this blending function, a value of the HLI is calculated as follows:

HLI = S(BGT) * HLIHI + (1 – S(BGT)) * HLILO
where HLIHI and HLILO are defined above.

Finally, the HLI value is not allowed to decrease below 50. HLI values smaller than 50 are set to 50.

if ( HLI < 50 ) HLI = 50

The Accumulated Heat Load Unit (AHLU) represents the amount of heat accumulated in cattle over a period of time. The rate of accumulation depends on the current HLI value. Large HLI values result in a more rapid increase in AHLU, conversely, low HLI values result in a decrease of the AHLU (ie the cattle cool down and recover). Whether cattle recover or become stressed depends on the value of certain thresholds. The first threshold occurs at a HLI value of 77. For HLI values below 77, the cattle cool down and recover. The second or UPPER_THRESHOLD depends on the type and condition of the cattle and their pen environment. Its value ranges from about 80 for unacclimatised (and possibly compromised) black angus cattle to about 95 for acclimatised Brahman cattle. The range of HLI values between 77 and the upper threshold is called the thermo-neutral zone. For this zone, cattle neither recover nor become stressed.

It is important that the correct UPPER_THRESHOLD is used otherwise the AHLU values will erroneously indicate the state of the cattle.  The UPPER_THRESHOLD can be calculated using the Katestone on-line calculator at: http://chlt.katestone.com.au/toolbox/rap-calculator/

The equation for calculating AHLU is as follows:

AHLUCURRENT = AHLUPREVIOUS + INCREMENT

For example, the AHLU at 2pm is calculated by adding an INCREMENT to the AHLU at 1pm. The INCREMENT is calculated as follows:

If (HLI is between 77 and the upper threshold) then INCREMENT = ZERO

If ( HLI < 77) then INCREMENT = INTERVAL * (HLI – 77) / 2

If(HLI > UPPER_THRESHOLD) then INCREMENT = INTERVAL * (HLI – UPPER_THRESHOLD)

where
INTERVAL is the time interval between successive HLI measurements. Its value is 1.0 for an interval of one hour, 0.5 for an interval of 30 minutes, 0.25 for an interval of 15 minutes and so on.

Some points worth noting:

• The INCREMENT can be positive or negative.
• If HLI is below 77, then INCREMENT is halved (ie the rate of recovery or heat loss is half of the rate of heat accumulation.
• AHLU values do not go below zero. If any calculation results in an AHLU value below zero, it is set to zero.

The following table contains sample data that you can use to test your implementation of the HLI equations above:

 Relative Humidity = 33% Black Globe Temperature Wind Speed 20o 25o 30o 2 m/s 50.0 55.4 67.1 4 m/s 50.0 53.3 64.8 6 m/s 50.0 51.7 63.5 Relative Humidity = 66% Black Globe Temperature Wind Speed 20o 25o 30o 2 m/s 54.3 66.3 79.3 4 m/s 52.3 64.2 77.0 6 m/s 50.4 62.6 75.8 Relative Humidity = 99% Black Globe Temperature Wind Speed 20o 25o 30o 2 m/s 63.9 77.2 91.5 4 m/s 61.9 75.0 89.2 6 m/s 59.9 73.5 88.0

The following table contains data that you can use to test your implementation of the AHLU equations:

 HLI AHLU-80 AHLU-86 AHLU-91 70 0 0 0 72 0 0 0 74 0 0 0 76 0 0 0 78 0 0 0 80 0 0 0 82 2 0 0 84 6 0 0 86 12 0 0 88 20 2 0 90 30 6 0 92 42 12 1 94 56 20 4 96 72 30 9 98 90 42 16 100 110 56 25 98 128 68 32 96 144 78 37 94 158 86 40 92 170 92 41 90 180 96 41 88 188 98 41 86 194 98 41 84 198 98 41 82 200 98 41 80 200 98 41 78 200 98 41 76 199.5 97.5 40.5 74 198 96 39 72 195.5 93.5 36.5 70 192 90 33 68 187.5 85.5 28.5 66 182 80 23 64 175.5 73.5 16.5 62 168 66 9 60 159.5 57.5 0.5 58 150 48 0 56 139.5 37.5 0 54 128 26 0 52 115.5 13.5 0 50 102 0 0 50 88.5 0 0 50 75 0 0 50 61.5 0 0 50 48 0 0 50 34.5 0 0 50 21 0 0 50 7.5 0 0 50 0 0 0 50 0 0 0 50 0 0 0