Alternative derivation of equipotential-plane dip in .NET Build 3 of 9 barcode in .NET Alternative derivation of equipotential-plane dip

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Alternative derivation of equipotential-plane dip using barcode development for .net control to generate, create code 39 image in .net applications. .NET Framework 4.0 Consider the situation in Fig barcode 3/9 for .NET ure 8.7.

There is a conduit along the bed between points 1 and 2. We wish to determine under what conditions the upglacier slope of the hill will be parallel to an equipotential plane so that water in the conduit will not ow. The ice pressure at (1) is Pi1 = i g(h1 + h2 + h), and that at (2) is Pi2 = i gh2 .

In the absence of water ow and conduit closure, the pressure in the water at (1), Pw , would be the sum of Pi2 plus the hydrostatic head in the conduit, w gh1 . If Pi1 > Pw , the conduit will begin to close and water will be forced out over the hill. Thus, the condition we seek is Pi1 = Pw , or:.

i g(h 1 + h 2 + h) = i gh 2 + w gh 1 (8.8). Solving this for h1 , dividin .net framework Code 3 of 9 g by x, noting that = h/ x and tan = h1 / x, and inserting numerical values for the densities leads directly to Equation (8.7) Q.


Melt rates in conduits h2 h1 (1). Figure 8.7. Sketch illustrati ng alternative derivation of dip of equipotential planes in a glacier.

. Melt rates in conduits Let us now consider the rate of melting of conduit walls, following Shreve (1972). The total amount of energy available per unit length of conduit, s, per unit time is:. Q m3 s s N/m2 m s m= N m J USS Code 39 for .NET = s s (8.9).

Some of this energy must be u sed to warm the water to keep it at the pressure melting point as ice thins in the downglacier direction. The rest is available to melt ice, thus:. m s (2 r i L) + w Cw m kg bar code 39 for .NET J m3 kg (H z) i g s Q = Q s s s m kg J K kg m m3 m (8.10) 2 3 kgK 3 s2 m m m s N/m C.

m m s Here, r is the radius of the .net framework 3 of 9 conduit, L is the latent heat of fusion, Cw is the heat capacity of water, and C is the change in the melting point per unit of pressure (see Equation (2.2)).

As you will see from inspection of the terms and the dimensions of the various quantities in them, the rst term on the left is the energy used to melt tunnel walls, and the second is the energy needed to warm the water to keep it at the pressure melting point. Here, we have implicitly taken the positive s-direction to be upglacier, in the direction opposite to that of the water ow. Thus, both / s and (H z)/ s are positive.

It is common to de ne k = w Cw C. Inserting numerical values ( w = 1000 kg m 3 , Cw = 4180 J kg 1 K 1 , and C = 0.074 10 6 K Pa 1 ) we nd that k = 0.

309 and that it is dimensionless. If we assume that the water is saturated with air, and adjust C accordingly, k = 0.410.

. Water ow in and under glaciers Then, using Equation (8.5) and dividing by m(2 r i L) + k z w g s s s yields:. s (8.11). or solving for m:. Q (1 k) m= z + k w g s s 2 r i L (8.12). It is interesting to insert s ome numbers into this equation to get a sense of the magnitude of m. Consider a horizontal tunnel so z/ s = 0. Suppose the tunnel has a diameter of 0.

5 m and that it is under a glacier with a surface slope of 0.01. We now need a relation between Q and the tunnel roughness.

The Gaukler Manning Strickler equation is one of two that are commonly used for such calculations. It is:. v= Q R 2/3 S 1/2 = 2 r n (8.1 Code 3 of 9 for .NET 3).

Here, v is the mean velocity over the tunnel cross section, R is the hydraulic radius of the tunnel, or the cross-sectional area divided by the perimeter (so R = r/2 in circular tunnels), S is the nondimensional headloss:. S= 1 w g s (8.14). which is approximately equal to the glacier surface slope, and n is known as the Manning roughness coef cient. For smooth channels, n may be as low as 0.005 m 1/3 s, but studies of oods, called j kulhlaups, resulting o from drainage of ice-dammed lakes through subglacial conduits yield values ranging from 0.

08 to 0.12 m 1/3 s (Bj rnsson, 1992). A still higher o value was obtained from dye-trace experiments on Storglaci ren; ow a 1/3 velocities there suggested n 0.

2 m s (Seaberg et al., 1988; Hock and Hooke, 1993). Where roughness elements on the tunnel walls and oor are large in comparison with the tunnel size, n will be higher; this is probably responsible for the relatively high value from Storglaci ren.

a 1/3 Choosing an intermediate value of 0.1 m s, Equation (8.13) gives a mean velocity of about 0.

25 m s 1 , or Q 0.05 m3 s 1 , and Equation (8.14) gives / s 98 N m 3 .

Whence m 0.22 m a 1 . This may not seem like a lot but, volumetrically, the amount of ice melted in a year is 2.

6 times the size of the original conduit. A consequence of this melting and the resulting inward ow of ice towards the conduit is that structures such as foliation in the ice are also bent inward. A beautiful example of this is shown in Figure 8.

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