Critical depth

Based on the relative amount of iron in the critical depth, f The critical depth is taken as that beyond which only 1% of the quanta in the emergent beam are produced. 

Inspection of the table shows that the quotient a/Wj e is in fact nearly constant that I changes much less rapidly than W e] and that the critical depth has doubled when the highest oxide is reached. All three conditions are reflections of the (positive) absorption effect that occurs in this binary system when iron is replaced by oxygen, which has a lower mass absorption coefficient. 

In the earlier discussion of critical depth (6.5), it was pointed out that this depth is determined by the density, and by the mass absorption coefficients of the sample for the incident beam and for the analytical line. With light elements, the coefficient for the analytical line can be so large that the coefficient of the incident beam may be neglected in... 

This value of D corresponds to the critical depth for flow in a channel. 

Thus a hydraulic jump can occur provided that the depth of the liquid is less than the critical depth. After the jump the depth will be greater than the critical depth, the flow having changed from rapid to tranquil. 

It is an unfortunate characteristic of propellants that they invariably bum to detonation if there is more than a critical depth of powder above the point of ignition. This depth depends greatly on the composition and on the grain size. It may vary from about 10 cm to several metres. In processing, the critical depth for the product being made is not exceeded unless full precautions for handling a detonating explosive are taken. 

Recent Developments in the Optimization and Control of Nitration in the Continuous Manufacture of Trinitrotoluene , ibid (Feb 1977) 40) WM. Stirrat RM. Rindner, Critical Depth Tests of Bulk TNT Flake Explosive , ARLCD-TR-78003, USA Arm Res Dev Command, Dover, NJ (1978) 41) J.R. 

The velocity and rate of discharge occurring at the critical depth are termed Vc and qc = qmax, the critical velocity and flow, respectively. On account of the greater area, the velocity of upper-stage flow is slower than the critical and is called subcritical velocity likewise, supercritical velocity occurs at lower-stage conditions. Combining Eqs. (10.110) and (10.113) yields a simple expression for critical velocity,... 

For any value of E there exists a critical depth, given by Eq. (10.113), for which the flow is a maximum. 

When flow occurs at critical depth, both Eqs. (10.113) and (10.115) are satisfied and the velocity head is one-half the depth. 

For simplicity of explanation, the treatment of critical depth, is confined to wide rectangular channels. We shall now consider an irregular section (no measurable channel walls or smooth channel surfaces) of area A carrying a flow Q. Thus, Eq. (10.111) becomes... 

This may now be set equal to zero and solved for the value of the critical depth for the given flow. As A may or may not be a reasonable function of y, it is helpful to observe that dA = B dy and thus dAJdy = B, the width of the water surface. Substituting this in the preceding expression results in... 

It has already been pointed out that the cross section most commonly encountered in open-channel hydraulics is not rectangular but trapezoidal. As repeated trial-and-error solutions of Eq. (10.119) become very tedious, practicing hydraulic engineers avail themselves of numerous tables and curves which have been prepared for finding the critical depth in trapezoidal channels of any bottom width and side slopes. 

The M3 scenario. This occurs because of an upstream control, as by the sluice gate. The bed slope is not sufficient to sustain lower-stage flow, and, at a certain point determined by energy and momentum relations, the water surface will pass through a hydraulic jump to upper-stage flow unless this is made unnecessary by the existence of a free overfall before the M3 crest reaches critical depth. 

The S scenarios. These are steep slope cases. They may be analyzed in much the same fashion as the M scenarios, having due regard for downstream control in the case of upper-stage flow and upstream control for lower-stage flow. Thus a dam or an obstruction on a steep slope produces an Si scenario, which approaches the horizontal asymptotically but cannot so approach the uniform depth line, which lies below the critical depth. Therefore this curve must be preceded by a hydraulic jump. The S2 scenario shows accelerated lower-stage flow, smoothly... 

When water from a reservoir enters a canal in which the depth is greater than critical, there will be a drop in the surface owing to the velocity head and also to any friction loss at entrance. Water flowing over a dam in a channel is such a case. But if the uniform depth is less than the critical value, the water level at entrance will drop to the critical depth and no farther, no matter how low the water level may be in the stream below. The maximum flow in the canal is then limited by this factor. When the channel dam immediately slopes downstream, the surface curve will have a point of inflection at about the entrance section and although a steep slope curve downstream may be changed with different channel conditions, the portion of this curve upstream from this point of inflection remains unaltered. 

---