Saturated or onephase flow

When one immiscible fluid, such as humid air or water, fills the pore space, flow is saturated relative to that phase. Except through very coarse snow, such as large-grained depth hoar, the flow is slow enough to be described by Darcy's law. In this case, the flow velocity of the fluid vk is proportional to the combined pressure and gravitational forces. Thus,

pressure gravitational gradient force

Here, K is the saturated or intrinsic permeability, x is a spatial coordinate along the direction of flow, ^ is the angle between the flow direction and a downward vertical, g is the acceleration due to gravity, and pk, nk, and Pk are the pressure, dynamic viscosity, and density of fluid k (a = air, t = water) respectively. Since snow is rarely water-saturated (except when water ponds above ice layers or frozen soil), Equation (2.19) is primarily used to model air flow.

Saturated or air permeability Saturated or intrinsic permeability is a property of the pore structure and determines the ease and rate at which snow transmits fluids. K is typically measured by forcing air through a snow sample while measuring the flow rate and pressure drop (Chacho and Johnson, 1987; Albert et al., 2000). It is therefore often referred to as "air" permeability. Air permeability varies widely with snow type and changes over time as a result of metamorphism. In cold sites experiencing little or no melt conditions or wind drifting, the permeability of snow usually increases over time as its texture coarsens. Below the top several meters in firn, however, permeability can decrease as the pressure of overlying layers causes compaction and crystal sintering (Albert et al., 2000). In seasonal snow, the permeability of the surface layers can experience dramatic changes within the span of a day as aresult of insolation and surface heating (Albert and Perron, 2000).

Figure 2.12 (after Jordan etal., 1999b) compiles measurements of Kfrom several investigators and compares them with the ice fraction (1 — 0). Measurements range over two orders of magnitude from 3 x 10—10 m—2 for fine-grained, wind-packed snow (low porosity and small pore size) to 600 x 10—10 m—2 for large-grained depth hoar (medium porosity and very large pore size). The dashed lines indicate general ranges of permeability for different snow categories observed by Bader etal. (1939). Most measurements fall within the outer boundaries of their classification scheme. Those reported by Sturm (1991) and Jordan et al. (1999b) for depth hoar in Alaska and Northern Canada are notably higher and most likely reflect extremely large pores and possibly the presence of vertical mesopores (Arons and Colbeck, 1995). Albert and Perron (2000) also showed that ice layers in seasonal snowpacks are permeable, although their permeability is substantially lower than the surrounding snow.

The scatter in Fig. 2.12 demonstrates that porosity alone is not a useful indicator of K. If the pore structure is idealized as bundles of tubes, K becomes proportional to the square of the tube or pore diameter as well as to porosity (Dullien, 1992). Other factors, such as pore shape, pore interconnectivity, size distribution, and tortuosity also influence K. While the tubular model works well for soils and denser snow, it is unrealistic for light snow, when particle shape and specific area are dominant factors. Because pore size is difficult to measure directly, it is usually estimated from a combination of porosity (or snow density) and grain diameter. By binning his permeability measurements into equi-density groups, Shimizu (1970)

Snow Particle Size And Snow Density

Figure 2.12. Measured permeability K versus ice fraction (1 - 0) based on the laboratory observations of Bender (1957), Ishida and Shimizu (1958), Shimizu (1960), Keeler (1969b), Shimizu (1970), Conway and Abrahamson (1984), Buser and Good (1986), Sommerfeld and Rocchio (1989), Sturm (1991), and Jordan et al. (1999b). Dotted lines indicate Bader et al. (1939) classification scheme (after Jordan et al., 1999b, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission).

Figure 2.12. Measured permeability K versus ice fraction (1 - 0) based on the laboratory observations of Bender (1957), Ishida and Shimizu (1958), Shimizu (1960), Keeler (1969b), Shimizu (1970), Conway and Abrahamson (1984), Buser and Good (1986), Sommerfeld and Rocchio (1989), Sturm (1991), and Jordan et al. (1999b). Dotted lines indicate Bader et al. (1939) classification scheme (after Jordan et al., 1999b, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission).

decoupled the relationships with density and grain size and derived the widely used formula:

It is interesting to note that his empirically derived function has the correct theoretically derived units of length squared. If we normalize K to d2, as shown in Fig. 2.13 (after Jordan et al., 1999b), the resulting relationship shows a clear decrease in K with ice fraction. While Shimizu's data set was limited to fine-grained, wind-packed snow, the trends in Fig. 2.13 suggest that his function is also applicable to other snow types. More recent measurements of Luciano and Albert (2002), however, found that his formula yielded permeability estimates that vary from observations by more than an order of magnitude.

Figure 2.13 also shows theoretical curves for assemblages of thin discs and spheres. For a given porosity and grain radius, beds consisting of thin discs with an aspect ratio of 25 have approximately eight times the specific area of those consisting

1,000

"Q

0.01

0.001

0.0001

0.00001

* Shimizu Function

• Sommerfeld and Rocchio O Hardy and Albert O Jordan et al.

• Sommerfeld and Rocchio O Hardy and Albert O Jordan et al.

Figure 2.13. Reduced permeability K d 2 versus ice fraction (1 - 0) based on laboratory observations of Ishida and Shimizu (1958), Shimizu (1960), Shimizu (1970), Sommerfeld and Rocchio (1989), Hardy and Albert (1993), and Jordan et al. (1999b). Lines show the reduced Shimizu function (Eq. 2.20) and theoretical solutions (Jordan et al., 1999b) for beds of thin discs (a = diameter, b = thickness, a/b = 25) and spheres. Discs and spheres have equal radii (after Jordan et al., 1999b, copyright 1999; copyright John Wiley & Sons Limited. Reproduced with permission).

of spheres and therefore exert a higher drag on fluids. This suggests that, in addition to grain radius and porosity, the surface-to-volume ratio - typically higher for new snow - is an important parameter in characterizing snow permeability.

The natural stratification of the snow cover usually causes it to be more resistant to flow in the vertical than in the horizontal direction (Ishida and Shimizu, 1958; Dullien, 1992). Luciano and Albert (2002) reported measurements of anisotropy or directionality of permeability in snow and firn field measurements, and concluded that differences in permeability between layers accounts for greater variation than directional differences within a single layer. Because vertical resistances in layered media are computed in series, permeability is controlled by the most resistant (or least permeable) strata.

Forced and natural convection of air Air movement in porous media can result from two mechanisms: (1) pressure differences that force the flow (windpumping or forced convection) or (2) a temperature gradient that induces buoyancy-driven thermal convection (also termed natural

Table 2.4 Temperature difference per meter for the onset of natural convection.

Table 2.4 Temperature difference per meter for the onset of natural convection.

Snow type

x 10-10 m2

Wm-1K-1

ATcrit

Depth hoar

200

200

0.05

10

15

New snow

80

200

0.10

50

77

Old snow

100

300

0.20

80

123

Firn

40

300

0.20

200

308

Wind-pack

10

300

0.20

800

1234

convection). There are occasions when forced and natural convection of air can occur within natural snow.

When a large temperature gradient exists between a warm substrate and a cold surface, air density changes and causes buoyancy-driven air circulation within porous media; this is known as natural convection. Powers et al. (1985) detail the theory of natural convection in snow. The non-dimensional Rayleigh number governs both the onset of convection and its intensity, and is given by

In Equation (2.21), p0a is the reference density of air, ft is the coefficient of thermal expansion, AT is the driving temperature difference, HS is the depth of the snow, and A = W(PaCp,a).

The critical Rayleigh number Racrit for the onset of natural convection is approximately 27 when the surface is permeable with constant temperature and the bottom boundary is impermeable with constant temperature. When the bottom boundary is instead impermeable with constant heat flux, the critical Rayleigh number is approximately 18. For common values of permeability and thermal conductivity of various snow types, Table 2.4 lists the temperature difference that would be required for the onset of natural convection in a 1 m deep snowpack under the two bottom boundary conditions. Because natural temperature gradients of 50 °C per meter are very rare, it is highly unlikely that natural convection will occur in natural snow except in the case of packs composed of depth hoar with no intervening layers. Sturm and Johnson (1992) did measure temperatures in highly porous, thin subarctic snowpacks (composed primarily of large depth hoar crystals) that support the existence of natural convection in that case.

Forced convection in snow is caused by natural pressure changes. It has been shown that turbulent winds over a flat surface, barometric surface pressure changes, and winds over surface relief can all cause pressure perturbations that propagate vertically into the snow (Colbeck, 1989a; Clarke and Waddington, 1991). Barometric pressure changes may cause slow, low-velocity air movement in near-surface snow. Turbulence in the winds causes high-frequency pressure fluctuations that propagate millimeters or centimeters into the snow. The "form drag" pressure differences, caused by air flow over surface roughness (such as across sastrugi), can cause stronger and more sustained air flow deeper within the snow.

Snow layering will affect ventilation through differences in the permeability of the strata. Albert (1996) showed that, even under the assumption that snow within a given layer is isotropic, differences between the layers affect subsurface air flow fields. For example, buried high-permeability (e.g. hoar) layers can serve as channels for increased lateral flow through the snowpack.

The forced convection of air in snow can affect other processes in snow, such as sublimation (Albert, 2002), chemical changes (Waddington etal., 1996; McConnell et al., 1998; Albert et al., 2002), or heat transfer (Albert and Hardy, 1995; Jordan, et al., 2003; Andreas, et al., 2004).

Renewable Energy 101

Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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