CALCULATION AND STUDY OF THE AERODYNAMIC DRAG OF HAMMERS

V. I. BABUROV, A. I. GROSHEV

(Presented by the Department of Mining Machines, Mine Transport, and Mining Mechanics)

A pneumatic hammer is a mechanical converter of the energy of compressed air into the kinetic energy of the striker, which, through impact, is transformed into useful work. Consequently, the smaller the losses of air pressure in the converter, the more fully the energy of the compressed air will be used, and the more perfect the machine as a whole will be \[4, 5\].

Compressed air, as a body and as a carrier of energy, is characterized by viscosity, density, flow rate, and velocity of motion.

Viscosity is inherent in all real gases and manifests itself during motion in the form of internal friction. The principal characteristics of compressed air under the conditions of the experiments are given in Table 1. The dynamic coefficient of viscosity at a temperature of \(20^\circ\text{C}\) is \(180.9 \cdot 10^{-6}\ \text{g}/\text{cm}\cdot\text{s}\) \[2\].

Table 1

Characteristics of compressed air at a temperature of \(20^\circ\text{C}\)

The regime of gas motion (laminar or turbulent) depends on the ratio of the forces of inertia and the forces of viscosity (internal friction) in the flow. This ratio is characterized by the dimensionless Reynolds number:

\[ Re=\frac{V_0D_0}{\nu}, \tag{1} \]

where

\[ V_0 \]

is the flow velocity, \(\text{m}/\text{s}\);

\[ D_0 \]

is the pipe diameter, \(\text{m}\).

For each installation there exists a certain range of “critical” values of the number \(Re\), at which the transition from one regime to another occurs.

The lower limit of the Reynolds number in the case of a circular cross section is 2300, while the upper limit depends on the conditions at the entrance to the pipe, the condition of the wall surface, and other causes.

The conversion of part of the flow energy into heat is caused by the molecular and turbulent viscosity of the moving medium. These losses of compressed-air energy are called friction losses \((\Delta H_{\text{fr}})\), or losses along the length of the pipeline. In addition, there are also local losses \((\Delta H_{\text{m}})\), which arise when the normal flow of the stream is locally disturbed, when it separates from the wall, when vortices are formed at places where the configuration of the pipeline changes, when obstacles are encountered, etc. Losses of velocity (dynamic) pressure at the outlet of the flow from the network into a large volume (the atmosphere) are also classed as local resistances (losses). In calculating the aerodynamic drag in hammers, the total losses were found by us by summation according to the superposition principle, i.e.,

\[ \Delta H_{\text{sum}}=\Delta H_{\text{fr}}+\Delta H_{\text{m}}. \tag{2} \]

Over the entire path from the compressor to the hammer, the compressed air loses part of its energy in overcoming local resistances and friction along the length. Estimation of these losses in hoses, the air-distribution device, and the hammer channels is of interest both in its formulation and in the possible conclusions.

Calculation of air-pressure losses in hoses

In most cases, the air supply to hammers is carried out through rigid and flexible pipelines of various sizes, both in length and in diameter. Let us calculate the pressure losses of compressed air in hoses (flexible pipelines) as a function of their length and of the air flow rate.

The hoses used in operation have an internal diameter of 16 mm. Their length from the rigid pipeline to the hammers is usually 10–20 m, and the air consumption, depending on the machine type size, is 0.5–2.5 m\(^3\)/min. Measurement and calculation of air-pressure losses in hoses was carried out using an experimental setup. The principal measuring elements of the setup were differential manometers. One of them was intended for accurate determination of the air flow rate, and the other for the pressure drop. As the throttling device of the differential manometer, a sharp-edged diaphragm 18 mm in diameter was selected. From the readings of this differential manometer, filled with tinted water, the air flow rate was calculated by the formula

\[ Q=\mu F_{0}\sqrt{\frac{2g}{\gamma_{1}}(P_{1}-P_{2})},\ \text{m}^{3}/\text{min}. \tag{3} \]

where

\[ \begin{aligned} \mu &\text{— coefficient of air flow through the diaphragm;}\\ F_{0} &\text{— area of the diaphragm opening, m}^{2};\\ \gamma_{1} &\text{— specific weight of compressed air, kg/m}^{3};\\ P_{1},\,P_{2} &\text{— readings of the differential manometer, kg/m}^{2}. \end{aligned} \]

For practical calculations this formula is modified as follows:

\[ Q=2.087\cdot 10^{-2}D_{0}^{2}\varepsilon K_{t}\alpha \sqrt{\frac{P_{1}-P_{2}}{\gamma_{1}}}, \tag{4} \]

where

$D_0$ is the diameter of the diaphragm orifice, m;
$\alpha$ is the flow coefficient;
$\varepsilon$ is a correction factor for gas expansion.

It is customary to express the air flow rate by its volume reduced to the free state. Therefore a correction for specific weight $\left(\dfrac{\gamma_1}{\gamma_0}\right)$ must be introduced into equation (4). In addition, the influence of temperature must be taken into account. In our case, the normal temperature is taken to be $20^\circ\mathrm{C}$. Then the temperature correction factor is $K_t=1$ [6].

The pressure drop in the manometer tubes is related to the difference in liquid levels in the differential manometer by the equation [6]:

\[ P_1-P_2=h\gamma_{\mathrm{l}}\cdot 10^4, \tag{5} \]

where

$h$ is the difference in liquid levels in the manometer tubes, m;
$\gamma_{\mathrm{l}}$ is the specific weight of the manometer liquid $(0.998\ \mathrm{g/cm^3})$, kg/m$^3$.

For a known air flow rate, the pressure losses in the hose are found in the following order:

1. The velocity of air motion is equal to

\[ V=\frac{Q}{F_{\mathrm{h}}}, \tag{6} \]

where

$\gamma_{\mathrm{h}}$ is the internal cross section of the hose, m$^2$.

1. The Reynolds number at this velocity will be

\[ Re=\frac{VD_{\mathrm{g}}}{\nu}, \qquad D_{\mathrm{g}}=\frac{4F}{\Pi}, \]

where

$D_{\mathrm{g}}$ is the hydraulic diameter of the adopted section, m;
$\Pi$ is the perimeter of the section, m.

1. Then the coefficient of friction along the length will be equal to

\[ \xi_{\mathrm{fr}}=\lambda\frac{l}{D_{\mathrm{g}}}, \]

where

$l$ is the length of the section (hose), m;
$\lambda$ is the coefficient of friction resistance per unit relative length of the section being calculated (determined from the tables and diagrams of I. E. Idelchik) [4].

1. Since the local resistance coefficient $\xi_{\mathrm{m}}=0$, therefore

\[ \xi_{\mathrm{sum}}=\xi_{\mathrm{fr}}+\xi_{\mathrm{m}}=\xi_{\mathrm{fr}}. \]

1. Then

\[ \xi_{\mathrm{ob}}=\xi_{\mathrm{sum}}\frac{\gamma_0}{\gamma_i}\left(\frac{F_{\mathrm{h}}}{F_{\mathrm{sh}}}\right)^2, \tag{7} \]

where

$\gamma_i$ is the specific weight of air at the corresponding pressure.

1. The pressure losses are determined from the data already found, namely:

\[ \Delta H=\xi_{\mathrm{ob}}\frac{\gamma_0 V^2}{2g}. \tag{8} \]

Table 2 gives the values of compressed-air pressure losses at 5 atm in the supply hose, 15 m long and 16 mm in diameter (GOST 8318-57), calculated in the order set forth. The calculations show

Table 2

Pressure losses of compressed air in a hose 15 m long

show that the resistances of hoses are proportional to their length and to the air flow rate. For practical engineering calculations we have proposed the empirical formula:

\[ \Delta H = 0.009 Q^{1.7} L,\ \text{kg/cm}^2, \tag{9} \]

where

$L$ is the hose length, m.

Determination of the Aerodynamic Resistances of Hammer Starting Devices

The determination of the resistances of starting devices was carried out theoretically and experimentally. If in hoses the principal resistances are resistances along the length, then in the structural elements of hammers these are local resistances. Therefore it is more convenient to calculate losses by summing the reduced resistance coefficients of the individual sections of the network.

The starting devices of the chipping and riveting hammers of the Tomsk Electromechanical Plant, MP and KE, are completely identical. Having divided the air-supply path of the starting device into 9 sections and having determined the losses at an air pressure in the network of 5 atm, we established that the modern designs of hammer starting devices are far from perfect.

The pressure losses calculated for the individual sections, for the entire starting device, were also checked by us experimentally. The difference amounts to thousandths, i.e., it is quite insignificant. The main point in an analytical calculation is the precise determination of the magnitude of the total resistance coefficient. It can also be found from the same formula (8), if the total air-pressure losses are measured experimentally.

In this way the values of the resistance coefficients of the starting devices of hammers of various designs were found. For the hammers of the Gorky Automobile Plant PM-D8 it is 1.62; for those of the Izhevsk Mechanical Plant R-1, 2, 3—2.0; for those of the Nikolaev Plant named after Nosenko RM-35, 55, 75—1.7, respectively, and the air-pressure losses in them are: 0.31 kg/cm², 0.38 kg/cm², and 0.32 kg/cm², i.e., considerably higher in comparison with hammers of the MP and KE type (see Table 3). The magnitude of the aerodynamic resistances of hammer starting devices can also be evaluated as a percentage by the index $\eta$, which is equal to the ratio:

\[ \eta = \frac{P_0 - \Delta H}{P_0} \cdot 100\%, \tag{10} \]

where

\(P_0\)—air pressure at the inlet to the starting device, atm;
\(\eta\)—index of the perfection of the design of the starting device (handle); the values of \(\eta\) are given in Table 3.

Table 3

Air-pressure losses in hammer starting devices at 5 atm

Analysis of the magnitudes of the resistances of individual sections of the air-conducting path shows that the largest of them are the frontal resistances of the ball shutoff device. Thus, for the case considered, the resistance coefficient of the handle path without the spring, ball, and pusher is 0.44, and the losses in this case are equal to 0.16 kg/cm², which amounts to 57% of the sum of all losses.

An important aspect of these studies is also that they made it possible to compare and evaluate the magnitudes of local resistances and lengthwise resistances.

In hammers of the MP and KE types, about 5–10% of the pressure losses of the total magnitude are losses caused by lengthwise resistance, while the remaining 80–95% of the losses are due to the presence of local resistances. In other hammer designs this ratio differs still more. The data presented show how important it is, in the design and manufacture of hammers, to provide for the maximum reduction of resistances to the motion of air.

The magnitude of the resistance coefficient depends not only on the design of the hammer elements, but also on the air flow rate and its pressure in the network. For this reason, under the same conditions the losses for KE-type hammers are greater than for MP-type hammers, which have a lower compressed-air consumption.

Calculation and Study of the Aerodynamic Resistances of Hammer Air-Distribution Devices

Significant resistances to air motion are present in air-distribution systems. These resistances occur both in the distribution mechanism and in the channels through which air enters the rear and front cavities of the hammer. Depending on the design features of the air distributor and the size type of the machine, the hydraulic resistances will naturally be different.

Using the calculation method set forth above, we calculated the resistances of the air-distribution device of the MP-5 chipping hammer.

In general, when air moves into the front chamber of the hammer, the total coefficient of aerodynamic drag is 3.6, i.e., almost 2 times higher than the drag coefficient of the starting device.

The supply of air to the rear chamber of the hammer is carried out along a shorter path; therefore the aerodynamic drags of this duct are much smaller. Calculations show that they are equal to 1.96. The maximum values of the compressed-air pressures in the hammer chambers are in approximately the same ratio. The losses in the distributing device of the BS3-135V hammer are somewhat smaller. They are, respectively, \(0.47\ \mathrm{kg/cm^2}\) and \(0.30\ \mathrm{kg/cm^2}\).

Table 4 gives the values of the coefficients of aerodynamic drag of the hammers as a whole. From these data it follows that, for

Table 4

Coefficients of hydraulic resistance

hammers with valve air distribution, large aerodynamic drags are characteristic in comparison with hammers having spool-type distribution.

Conclusions

1. The investigations we carried out and the analysis of hammer designs from the standpoint of the magnitude of aerodynamic drags showed that, in creating these machines, measures to reduce the resistance of the air duct are not provided at all. Nevertheless, fairly simple design changes make it possible to noticeably reduce the resistance of hammers and thereby increase their energy parameters.

2. As a result of reducing aerodynamic drags, it is possible to increase the impact power of the machine with only a slight increase in air consumption.

Literature

1. V. I. Baburov. Study of pneumatic chipping-and-riveting hammers. Candidate dissertation, Tomsk, 1964.

2. N. B. Vargafik. Handbook of the Thermophysical Properties of Gases and Liquids. Gosfizmatizdat, 1963.

3. V. F. Gorbunov, V. I. Baburov, et al. Hand-Held Pneumatic Hammers. Mashgiz, 1967.

4. I. V. Idelchik. Hydraulic Resistances. Gosenergoizdat, 1961.

5. A. S. Ilyichev. Mining Pneumatic Installations. Gosgortekhizdat, 1954.