Ice will commence to form on the surface of the FWS wetland when the bulk water temperature reaches 37.4°F (3°C) and will continue as long as the temperature remains at or below 32°F (0°C). In northern climates where extremely low air temperatures can persist for very long periods, the FWS wetland may not be a feasible year-round treatment because extensive ice formation can result in physical failure of the system. The thickness or depth of ice that will form over a 1-d period can be estimated with Equation 6.19:

y = [(0(T)/(5)(Q)][(Tm - Tir)/( + yi/k +1/Us) - U( - Tm)] (6.19)

where y = Thickness of ice formation per day (ft/d; m/d). t = Time period of concern (d).

TABLE 6.17

Thermal Conductivities for Wetland Components

TABLE 6.17

Thermal Conductivities for Wetland Components

Material |
k (Btu/ft2hr°F) |
K (W/m2-°C) |

Air (no convection) |
0.014 |
0.024 |

Snow (new, loose) |
0.046 |
0.08 |

Snow (long-term) |
0.133 |
0.23 |

Ice (at 32°F) |
1.277 |
2.21 |

Water (at 32°F) |
0.335 |
0.58 |

Wetland litter layer (dry) |
0.029 |
0.05 |

Dry gravel (25% moisture) |
0.867 |
1.5 |

Saturated gravel |
1.156 |
2.0 |

Dry soila |
0.462 |
0.8 |

a This is native soil underlying the wetland bed. Heat transfer is into the wetland bed during the winter and from the wetland bed during the summer. Assume a 3-ft (1-m) depth for this soil layer.

a This is native soil underlying the wetland bed. Heat transfer is into the wetland bed during the winter and from the wetland bed during the summer. Assume a 3-ft (1-m) depth for this soil layer.

Source: Reed, S.C. et al., Natural Systems for Waste Management and Treatment, 2nd ed., McGraw-Hill, New York, 1995. With permission.

T = Time conversion factor (24 hr/d; 86,400 s/d). 8 = Density of ice (57.2 lb/ft3; 917 kg/m3). Q = Latent heat (144 Btu/lb; 334,944 J/kg). Tm = Melting point of ice (32°F; 0°C).

Tair = Average air temperature during time period of concern (°F; °C).

ks = Conductivity of snow (from Table 6.17).

ki = Conductivity of ice (from Table 6.17).

Us = Heat transfer coefficient at the wetland surface, W/m2-°C (Btu/ft2-hr-°F).

= 1.5 W/m2-°C (0.264 Btu/ft2-hr-°F) for dense marsh vegetation. = 10-25 W/m2-°C (1.761-4.403 Btu/ft2-hr-°F) for open water; high value used for windy conditions with no snow cover. Ui = Heat transfer coefficient water to ice (from Equation 6.18). Tw = Average water temperature during period of concern (from Equation 6.17).

It is necessary to repeat the calculation for each day of interest with appropriate adjustments in the depth of ice and snow in Equation 6.19. The time period of concern for the previous FWS thermal models is equal to the design HRT for the wetland; in this case, the time period of concern may be the entire winter season if significant periods of subfreezing temperatures persist. A reasonable first approximation of potential ice formation can be achieved by using the average monthly air temperatures (in the coldest winter of record) during the period of concern. This model was also derived from Ashton (1986) with the assistance of Darryl Calkins (USA CRREL; Hanover, NH).

The rate of ice formation will be the highest on the first day of freezing, when neither an ice cover nor a snow layer is present to retard heat losses. In addition, the final term in Equation 6.19 is usually small and can be neglected for estimation purposes. As a result, Equation 6.19 reduces to the Stefan formulation (Stefan, 1891):

where y = Depth of ice that will form over time period t (ft; m). m = Proportionality coefficient (ft/°F0S-d0.s; m/°C0S-d0.s).

= 0.066 ft^P^d05 (0.027 mTC^-d05) for open-water zones with no snow.

= 0.044 ft/°F5-d05 (0.018 m^C05^05) for open-water zones with snow. = 0.024 ftTF^-d05 (0.010 m/X^-d05) for wetland with dense vegetation and litter.

Tair = Average air temperature during time period t (°F; °C).

t = Number of days in the period of interest (d).

Equation 6.20 can be used to estimate total ice formation on FWS wetlands over the entire winter season or for shorter time periods if desired. This equation can be used to determine the feasibility of winter operations for a wetland in locations with very low winter temperatures. For example, a site with persistent air temperatures at -13°F (-25°C) would result in a wetland that is 1.5 ft (0.45 m) deep freezing to the bottom in about 84 days.

The term (Tm - Tair)(t) is the freezing index and is an environmental characteristic for a particular location (values can be found in published references). Equation 6.20 is also used in Chapter 9 of this book to determine the depth of sludge that can be frozen for dewatering purposes.

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