# 6.12: Specific Cutting Energy

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In the dredging industry, the specific cutting energy is described as:

The amount of energy, that has to be added to a volume unit of soil (e.g. sand) to excavate the soil.

The dimension of the specific cutting energy is: kN/m2 or kPa for sand and clay, while for rock often MN/m2 or MPa is used.

Adhesion, cohesion, gravity and the inertia forces will be neglected in the determination of the specific cutting energy. For the case as described above, cutting with a straight blade with the direction of the cutting velocity perpendicular to the blade (edge of the blade) and the specific cutting energy can be written:

$\ \mathrm{E}_{\mathrm{s p}}=\frac{\mathrm{F}_{\mathrm{h}} \cdot \mathrm{v}_{\mathrm{c}}}{\mathrm{h}_{\mathrm{i}} \cdot \mathrm{w} \cdot \mathrm{v}_{\mathrm{c}}}=\frac{\mathrm{F}_{\mathrm{h}}}{\mathrm{h}_{\mathrm{i}} \cdot \mathrm{w}}\tag{6-88}$

The method, with which the shear angle β is determined, is therefore equivalent with minimizing the specific cutting energy, for certain blade geometry and certain soil mechanical parameters. For the specific energy, for the non-cavitating cutting process, it can now be derived from equations (6-71) and (6-88), that:

$\ \mathrm{E}_{\mathrm{n} \mathrm{c}}=\mathrm{c}_{\mathrm{1}} \cdot \rho_{\mathrm{w}} \cdot \mathrm{g} \cdot \mathrm{v}_{\mathrm{c}} \cdot \mathrm{h}_{\mathrm{i}} \cdot \frac{\varepsilon}{\mathrm{k}_{\mathrm{m}}}\tag{6-89}$

For the specific energy, for the fully cavitating cutting process, can be written from equations (6-74) and (6-88):

$\ \mathrm{E}_{\mathrm{c a}}=\mathrm{d}_{1} \cdot \rho_{\mathrm{w}} \cdot \mathrm{g} \cdot(\mathrm{z}+\mathrm{1 0})\tag{6-90}$

From these equations can be derived that the specific cutting energy, for the non-cavitating cutting process is proportional to the cutting velocity, the layer-thickness and the volume strain and inversely proportional to the permeability. For the fully cavitating process the specific cutting energy is only dependent on the water depth.

Therefore it can be posed, that the specific cutting energy, for the fully cavitating cutting process is an upper limit, provided that the inertia forces, etc., can be neglected. At very high cutting velocities, however, the specific cutting energy, also for the cavitating process will increase as a result of the inertia forces and the water resistance.

# 6.12.1. Specific Energy and Production in Sand

As discussed previously, the cutting process in sand can be distinguished in a non-cavitating and a cavitating process, in which the cavitating process can be considered to be an upper limit to the cutting forces. Assuming that during an SPT test in water-saturated sand, the cavitating process will occur, because of the shock wise behavior during the SPT test, the SPT test will give information about the cavitating cutting process. Whether in practice, the cavitating cutting process will occur, depends on the soil mechanical parameters, the geometry of the cutting process and the operational parameters. The cavitating process gives an upper limit to the forces, power and thus the specific energy and a lower limit to the production and will therefore be used as a starting point for the calculations. For the specific energy of the cavitating cutting process, the following equation can be derived according to Miedema (1987 September):

$\ \mathrm{E}_{\mathrm{s p}}=\rho_{\mathrm{w}} \cdot \mathrm{g} \cdot(\mathrm{z}+\mathrm{1 0}) \cdot \mathrm{d}_{\mathrm{1}}\tag{6-91}$

The production, for an available power Pa, can be determined by:

$\ \mathrm{Q}=\frac{\mathrm{P}_{\mathrm{a}}}{\mathrm{E}_{\mathrm{s p}}}=\frac{\mathrm{P}_{\mathrm{a}}}{\rho_{\mathrm{w}} \cdot \mathrm{g} \cdot(\mathrm{z}+\mathrm{1 0}) \cdot \mathrm{d}_{\mathrm{1}}}\tag{6-92}$

The coefficient d1 is the only unknown in the above equation. A relation between d1 and the SPT value of the sand and between the SPT value and the water depth has to be found. The dependence of d1 on the parameters αhi and hb can be estimated accurately. For normal sands there will be a relation between the angle of internal friction and the soil interface friction. Assume blade angles of 30, 45 and 60 degrees, a ratio of 3 for hb /hi and a soil/interface friction angle of 2/3 times the internal friction angle. For the coefficient d1 the following equations are found by regression:

$\ \mathrm{d}_{1}=\left(\mathrm{0.6 4}+\mathrm{0 .5 6} \cdot \mathrm{h}_{\mathrm{b}} / \mathrm{h}_{\mathrm{i}}\right)+\left(\mathrm{0 .0 1 6 4}+\mathrm{0 .0 0 8 5} \cdot \mathrm{h}_{\mathrm{b}} / \mathrm{h}_{\mathrm{i}}\right) \cdot \mathrm{S P T}_{\mathrm{1 0}}(\alpha=30 \text{ degrees})\tag{6-93}$

$\ \mathrm{d}_{1}=\left(\mathrm{0 .8 3}+\mathrm{0 .4 5} \cdot \mathrm{h}_{\mathrm{b}} / \mathrm{h}_{\mathrm{i}}\right)+\left(\mathrm{0 .0 2 6 8}+\mathrm{0 .0 0 8 5} \cdot \mathrm{h}_{\mathrm{b}} / \mathrm{h}_{\mathrm{i}}\right) \cdot \mathrm{S P T}_{\mathrm{1 0}}(\alpha=45\text{ degrees })\tag{6-94}$

$\ \mathrm{d}_{1}=\left(\mathrm{0 .9 9}+\mathrm{0 .3 9} \cdot \mathrm{h}_{\mathrm{b}} / \mathrm{h}_{\mathrm{i}}\right)+\left(\mathrm{0 .0 5 0 3}+\mathrm{0 .0 0 9 9} \cdot \mathrm{h}_{\mathrm{b}} / \mathrm{h}_{\mathrm{i}}\right) \cdot \mathrm{S P T}_{\mathrm{1 0}}(\alpha=60\text{ degrees })\tag{6-95}$

With: SPT10 = the SPT value normalized to 10 m water depth.

Lambe & Whitman (1979), page 78) and Miedema (1995) give the relation between the SPT value, the relative density RD (0-1) and the hydrostatic pressure in two graphs, see Figure 6-29. With some curve-fitting these graphs can be summarized with the following equation:

$\ \operatorname{SPT}=0.243 \cdot\left(82.5+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot(\mathrm{z}+10)\right) \cdot \mathrm{R D}^{2.52}\tag{6-96}$

And:

$\ \mathrm{RD}=\left(\frac{4.12 \cdot \mathrm{SPT}}{\left(82.5+\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot(\mathrm{z}+10)\right)}\right)^{0.397}\tag{6-97}$

Lambe & Whitman (1979), (page 148) and Miedema (1995) give the relation between the SPT value and the angle of internal friction, also in a graph, see Figure 6-28. This graph is valid up to 12 m in dry soil. With respect to the internal friction, the relation given in the graph has an accuracy of 3 degrees. A load of 12 m dry soil with a density of 1.67 ton/m3 equals a hydrostatic pressure of 20 m.w.c. An absolute hydrostatic pressure of 20 m.w.c. equals 10 m of water depth if cavitation is considered. Measured SPT values at any depth will have to be reduced to the value that would occur at 10 m water depth. This can be accomplished with the following equation (see Figure 6-30):

$\ \operatorname{SPT}_{10}=\frac{282.5}{\left(82.5+\rho_{\mathrm{l}} \cdot g \cdot(\mathrm{z}+10)\right)} \cdot \operatorname{SPT}_{\mathrm{z}}\tag{6-98}$

With the aim of curve-fitting, the relation between the SPT value reduced to 10 m water depth and the angle of internal friction can be summarized to:

$\ \varphi=51.5-25.9 \cdot \mathrm{e}^{-0.01753 \cdot \mathrm{SPT}_{10}}(+/- 3 \text{ degrees })\tag{6-99}$

For water depths of 0, 5, 10, 15, 20, 25 and 30 m and an available power of 100 kW the production is shown graphically for SPT values in the range of 0 to 100 SPT. Figure 6-31 shows the specific energy and Figure 6-32 the production for a 45 degree blade angle.

# 6.12.2. The Transition Cavitating/Non-Cavitating

Although the SPT value only applies to the cavitating cutting process, it is necessary to have a good understanding of the transition between the non-cavitating and the cavitating cutting process. Based on the theory in Miedema (1987 September), an equation has been derived for this transition. If this equation is valid, the cavitating cutting process will occur.

$\ \mathrm{v}_{\mathrm{c}}>\frac{\mathrm{d}_{1} \cdot(\mathrm{z}+\mathrm{1 0}) \cdot \mathrm{k}_{\mathrm{m}}}{\mathrm{c}_{\mathrm{1}} \cdot \mathrm{h}_{\mathrm{i}} \cdot \varepsilon}\tag{6-100}$

The ratio d1/c1 appears to have an almost constant value for a given blade angle, independent of the soil mechanical properties. For a blade angle of 30 degrees this ratio equals 11.9. For a blade angle of 45 degrees this ratio equals 7.72 and for a blade angle of 60 degrees this ratio equals 6.14. The ratio ε/khas a value in the range of 1000 to 10000 for medium to hard packed sands. At a given layer thickness and water depth, the transition cutting velocity can be determined using the above equation. At a given cutting velocity and water depth, the transition layer thickness can be determined.

# 6.12.3. Conclusions Specific Energy

To check the validity of the above derived theory, research has been carried out in the laboratory of the chair of Dredging Technology of the Delft University of Technology. The tests are carried out in hard packed water saturated sand, with a blade of 0.3 m by 0.2 m. The blade had cutting angles of 30, 45 and 60 degrees and deviation angles of 0, 15, 30 and 45 degrees. The layer thicknesses were 2.5, 5 and 10 cm and the drag velocities 0.25, 0.5 and 1 m/s. Figure 6-57 shows the results with a deviation angle of 0 degrees, while Figure 6-58 shows the results with a deviation angle of 45 degrees. The lines in this figure show the theoretical forces. As can be seen, the measured forces match the theoretical forces well.

Based on two graphs from Lambe & Whitman (1979) and an equation for the specific energy from Miedema (1987 September) and (1995), relations are derived for the SPT value as a function of the hydrostatic pressure and of the angle of internal friction as a function of the SPT value. With these equations also the influence of water depth on the production can be determined. The specific energies as measured from the tests are shown in Figure 6-57 and Figure 6-58. It can be seen that the deviated blade results in a lower specific energy. These figures also show the upper limit for the cavitating cutting process. For small velocities and/or layer thicknesses, the specific energy ranges from 0 to the cavitating value. The tests are carried out in sand with an angle of internal friction of 40 degrees. According to Figure 6-28 this should give an SPT value of 33. An SPT value of 33 at a water depth of about 0 m, gives according to Figure 6-31, a specific energy of about 450-500 kPa. This matches the specific energy as shown in Figure 6-57.

All derivations are based on a cavitating cutting process. For small SPT values it is however not sure whether cavitation will occur. A non-cavitating cutting process will give smaller forces and power and thus a higher production. At small SPT values however the production will be limited by the bull-dozer effect or by the possible range of the operational parameters such as the cutting velocity.

The calculation method used remains a lower limit approach with respect to the production and can thus be considered conservative. For an exact prediction of the production all of the required soil mechanical properties will have to be known. As stated, limitations following from the hydraulic system are not taken into consideration.

# 6.12.4. Wear and Side Effects

In the previous chapters the blades are assumed to have a reasonable sharp blade tip and a positive clearance angle. A two dimensional cutting process has also been assumed. In dredging practice these circumstances are hardly encountered. It is however difficult to introduce a concept like wear in the theoretical model, because for every wear stage the water pressures have to be determined numerically again.

Also not clear is, if the assumption that the sand shears along a straight line will also lead to a good correlation with the model tests with worn blades. Only for the case with a sharp blade and a clearance angle of -1o a model test is performed.

It is however possible to introduce the wear effects and the side effects simply in the theory with empirical parameters. To do this the theoretical model is slightly modified. No longer are the horizontal and the vertical forces used, but the total cutting force and its angle with the direction of the velocity component perpendicular to the blade edge are used. Figure 6-33 shows the dimensionless forces c1c2, and cfor the non-cavitating cutting process and the dimensionless forces d1d2 and dt for the cavitating process.

For the total dimensionless cutting forces it can be written:

$\ \begin{array}{left}\quad\quad\quad\text{non-cavitating}\quad\quad\quad\quad\quad\text{cavitating}\\ \mathrm{c}_{\mathrm{t}}=\sqrt{\left(\mathrm{c}_{\mathrm{1}} \cdot \mathrm{c}_{\mathrm{1}}+\mathrm{c}_{\mathrm{2}} \cdot \mathrm{c}_{\mathrm{2}}\right)} \quad \quad\mathrm{d}_{\mathrm{t}}=\sqrt{\left(\mathrm{d}_{\mathrm{1}} \cdot \mathrm{d}_{\mathrm{1}}+\mathrm{d}_{\mathrm{2}} \cdot \mathrm{d}_{\mathrm{2}}\right)}\end{array}\tag{6-101}$

For the angle the force makes with the direction of the velocity component perpendicular to the blade edge:

$\ \theta_{\mathrm{t}}=\operatorname{atn}\left(\frac{\mathrm{c}_{2}}{\mathrm{c}_{1}}\right) \quad \quad\quad\quad\quad\Theta_{\mathrm{t}}=\operatorname{atn}\left(\frac{\mathrm{d}_{2}}{\mathrm{d}_{1}}\right)\tag{6-102}$

It is proposed to introduce the wear and side effects, introducing a wear factor cs (ds) and a wear angle θs (Θs) according to:

$\ \mathrm{c}_{\mathrm{t s}}=\mathrm{c}_{\mathrm{t}} \cdot \mathrm{c}_{\mathrm{s}} \quad\quad\quad\quad\quad\quad\quad \mathrm{d}_{\mathrm{t} \mathrm{s}}=\mathrm{d}_{\mathrm{t}} \cdot \mathrm{d}_{\mathrm{s}}\tag{6-103}$

And

$\ \theta_{\mathrm{ts}}=\theta_{\mathrm{t}}+\theta_{\mathrm{s}} \quad\quad\quad\quad\quad\quad\quad \Theta_{\mathrm{ts}}=\Theta_{\mathrm{t}}+\Theta_{\mathrm{s}}\tag{6-104}$

For the side effects, introducing a factor cr (dr) and an angle θ(Θr), we can now write:

$\ \mathrm{c}_{\mathrm{tr}}=\mathrm{c}_{\mathrm{t}} \cdot \mathrm{c}_{\mathrm{r}}\quad\quad\quad\quad\quad\quad\quad\mathrm{d_{tr}=d_t\cdot d_r} \tag{6-105}$

And

$\ \theta_{\mathrm{tr}}=\theta_{\mathrm{t}}+\theta_{\mathrm{r}} \quad \quad \Theta_{\mathrm{tr}}=\Theta_{\mathrm{t}}+\Theta_{\mathrm{r}}\tag{6-106}$

In particular the angle of rotation of the total cutting force as a result of wear, has a large influence on the force needed for the haul motion of cutter-suction and cutter-wheel dredgers. Figure 6-34 and Figure 6-35 give an impression of the expected effects of the wear and the side effects.

The angle the forces make with the velocity direction θtΘt, where this angle is positive when directed downward.

The influence of wear on the magnitude and the direction of the dimensionless cutting forces ct or dt for the non-cavitating cutting process.

The influence of side effects on the magnitude and the direction of the dimensionless cutting forces ct or dt for the non-cavitating cutting process.

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