## General Equation

The freezing or thawing of a sludge layer can be described by Equation 9.2:

where

m = Proportionality coefficient (cm (°C-d)-1/2) = 2.04 cm (°C-d)-1/2 = 0.60 in. (°F-d)1/2.

AT = Temperature difference between 0°C (32° F) and the average ambient air temperature during the period of interest (°C; °F). t = Time period of concern (d).

Equation 9.2 has been in general use for many years to predict the depth of ice formation on ponds and streams. The proportionality coefficient m is related to the thermal conductivity, density, and latent heat of fusion for the material being frozen or thawed. A median value of 2.04 was experimentally determined for wastewater sludges in the range of 0 to 7% solids (Reed et al., 1984). The same value is applicable to water treatment and industrial sludges in the same concentration range.

The freezing or thawing index in Equation 9.2 is an environmental characteristic for a particular location. It can be calculated from weather records and can also be found directly in other sources (Whiting, 1975). The factor AT in Equation 9.2 is the difference between the average air temperature during the period of concern and 32°F (0°C). Example 9.1 illustrates the basic calculation procedure.

Example 9.1. Determination of Freezing Index

The average daily air temperatures for a 5-d period are listed below. Calculate the freezing index for that period.

Mean Temperature Day (°C)

Solution

1. The average air temperature during the period is -4°C.

2. The freezing index for the period is AT d = [0 - (-4)](5) = 20°C-d.

The rate of freezing decreases with time under steady-state temperatures, because the frozen material acts as an insulating barrier between the cold ambient air and the remaining unfrozen sludge. As a result, it is possible to freeze a greater total depth of sludge in a given time if the sludge is applied in thin layers.

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