Shaft speed calculations

Near top speed, a fan may operate at a speed that is near or above the natural frequency of the wheel and shaft. Under such conditions, the fan can vibrate badly even when the wheel is clean and properly balanced. Whereas manufacturers often do not check the natural frequency of the wheel and shaft ia standard designs, many have suitable computer programs for such calculations. Frequency calculations should be made on large high speed fans. The first critical wheel and shaft speed of a fan that is subject to wheel deposits or out-of-balance wear should be about 25—50% above the normal operating speed. 

The stress in the crank shaft is calculated approximately from the power and speed as follows. Bear in mind that approximate calculations of this sort may be in error by up to a factor of 2 - but this makes no difference to the conclusions reached below. Referring to Fig. 16.9 ... 

To calculate the specific speed, N, it is necessary to select a reasonable shaft speed. First, calculate the approximate shaft power by assuming... 

The critical-speed calculation of a rotating shaft proceeds with equations to relate loads and deflections from station — 1 to station n. The shaft shear V can be computed using the following relationship ... 

Because the second section shares a common shaft with the first section, it is not necessary to look up a new impeller size. Apply the Section 1 impeller diameter, Equation 5.15, and the conversion constants of 12 in./ft and 60 sec/min. to calculate a shaft speed. 

With the shaft speed and the tip speed calculated in Step 9 for the Section 2 stages, calculate an impeller diameter using Equation 5.15. 

The male and female rotors act much like any bladed or gear unit. The number of lobes on the male rotor multiplied by the actual male shaft speed determines the rotor-passing frequency. In most cases, there are more lobes on the female than on the male. To ensure inclusion of all passing frequencies, the rotor-passing frequency of the female shaft also should be calculated. The passing frequency is equal to the number of lobes on the female rotor multiplied by the actual female shaft speed. 

Because this is a friction-driven motion, the cage turns much slower than the inner race of the bearing. Generally, the rate of rotation is slightly less than one-half of the shaft speed. FTF is calculated by the following equation ... 

Now the appropriate shaft speed for scaled-up production equipment can be calculated. The tank used for production batches has a capacity of 3780 L. It is equipped with a turbine-type agitator that has a shaft speed range of 20 58 rpm. The diameter of this tank is 167 cm. The diameter of the largest axial impeller is 87 cm. Given the diameter of the production tank, the cross-sectional area can be determined as... 

In the first type, the agitator shaft is connected to its drive shaft through a device which both resists and measures the rotation of the agitator shaft relative to the drive shaft. Separate static measurements yield the torque required to cause this amount of relative rotation of the driving and driven shafts, and the power being supplied to the agitator shaft is calculated as the product of the speed and the torque. A description of one such device is given by Stoops and Lovell (S8). 

A complete mechanical design should include a calculation of the critical speed. When the operating speed is close to or exceeds the critical shaft speed, vibrations can lead to mechanical instabilities and severe equipment damage. For more information, see the chapter by King [6] in Mixing in the Process Industries or the chapter by Dickey and Fasano in the Handbook of Industrial Mixing [1]. 

Equation (XI.28) cannot be used for practical calculations since the velocity Pn appearing in this equation characterizes the movement of particles at the surface, which is an indeterminate quantity. Hence, we will replace the velocity in this equation by the stirrer shaft speed , thus obtaining the following formula ... 

Among the models for calculation of the shaft speed needed for suspension, the equation of Zwietering [12] is based on numerous experimental results. The shaft speed necessary to reach a state where the particles are just suspended (njs) can be correlated as follows ... 

Once the process parameters and the rules for scale-up from the model dimensions have been determined by tests at various scales, the next step is to calculate the power rating for the industrial-scale system with the aid of the power characteristic. For Newtonian liquids this is no problem, in total contrast to non-Newtonian media, where the effective viscosity is always a function of agitator shaft speed [4]. 

Fig. 7 shows the variation of the maximum value of the dimensionless axial depth of gas invasion with shaft speed for the case of AP = 0.005 MPa. The larger the eccentricity ratio, the lower the shaft speed at which the gas invasion starts. However both the theory and the measurements show that the Interface reaches the oil end of the seal at the same shaft speed (Njj), regardless of z, though the Invasion starts at different speeds. Predictions are in good agreement with measurements. The calculation of the interface profile turned to be unstable with the increase in shaft speed, that is, with the... 

Fig. 4 shows a very good conformance between measured and calculated capacitances, which gives confidence in the method and, more specifically, in the lubricant film calculations used. The Lubcheck measurements clearly indicated full film separation in all the contacts, which was confirmed by the calculated film thicknesses. The composite roughness of the ball/ring contacts was 0.05 pm (Table 1), which means that even at a shaft speed of 500 rpm there is full film separation with a calculated A >4 (A is the ratio of lubricant film thickness to composite RMS surface roughness). It is interesting to note that to the total bearing capacitance of 600-1100 pF, the stray capacitance contributes 230 pF and the unloaded zone 200 pF. 

Fig. 5 shows the measured and the calculated capacitances of a 22220 CC spherical roller bearing as a function of shaft speed at two test loads 50 kN and 140 kN. The test bearings were lubricated with Shell Turbo T68 oil, with an inlet temperature of 49 C and a flow rate of 2.2 1/min. The test bearing temperature was first stabilised at an outer ring temperature of 58 C at a shaft speed of 1500 rpm. 

If motor speed matches the input shaft speed, a simple mechanical coupling can be used. But if it turns at a speed different from that recommended or calculated for the equipment, a speed-conversion drive is needed. It includes pulley and belt, gear, or chain and sprocket. 

The problem is how do we know the exact size sprocket that we need to reach the desired speed Use the same formula that was used to calculate shaft speed, only switch the location of the driven shaft speed and the driven sprocket size ... 

Let s say that we have a problem similar to the ones that we just did, but we want to change the shaft speed of the driven unit. If we know the speed we are looking for, we can use the formula above to calculate the sprocket size required. 

Calculate the number of teeth on the large sproeket by multiplying the number of teeth on the small sprocket by the desired drive ratio (faster shaft speed/slower shaft speed). Round the result to the nearest integral number of teeth. 

If the shaft speed is near the natural frequency, increase the solid shaft size to the next in. larger, or metric equivalent. Then redo the critical speed calculation. Again, if the operating/critical speed ratio meets the natural... 

The minimum shaft diameter will be the greater of the two values calculated in eqs. (21-5) and (21-6). For practical purposes, most mixer shafts are made from bar stock, so standard sizes are nsually available in or 1 in. increments or certain multiples of millimeters. For critical speed calculations, the next larger standard shaft diameter shonld be nsed. 

A hollow shaft, made from pipe, can increase the stiffness and reduce the weight (mass) of a mixer shaft in critical speed calculations. Such changes will increase the natnral freqnency and extend the allowable shaft length or operating speed. When determining the appropriate shaft size for the strength of a hollow shaft, begin with the dimensions for standard available pipe or tnbe. Then compute the shear and tensile stress values and compare them with the allowable values. The equations for combined shear and tensile limits in hollow shafts are, respectively. 

The method for determining the deflection is very similar to calculating the critical speed. However, the hydraulic forces on the shaft are now taken into account. Also, the frequency is a known value. Hydraulic forces are determined based on the impeller torque. Because hydraulic forces used by mixer manufacturers already include the effect of dynamics, speed (frequency) is already included in the magnitnde. Consequently, a forced response at the shaft speed would not be appropriate because the results would reflect the effect of frequency twice. Therefore, determining the forced response for the static condition is necessary (frequency = 0). From the bending moment for each position along with the torque from the impeller(s), the tensile and shear stresses can be calculated for each position. The static condition can only be calcnlated where the forcing freqnency is effectively zero compared with the natnral frequency, and such an analysis requires a 4 x 5 matrix. 

For transients that operate far from the design condition, it was necessary to extrapolate the maps for the transient calculations. If JIMO were pursued, performance maps would be required for a wider range of mass flow rates and shaft speeds. 

In RELAP5-3D, the shaft speed is calculated by balancing the torques of the compressor, turbine, and alternator plus any additional losses. 

Among the transients analyzed are an overspeed transient and startup. These transients result in mass flow rates and shaft speeds outside the ranges in the performance maps provided. To model transients that extend beyond the range covered by the performance maps, extrapolations to both lower and higher mass flow rates were performed. In addition, extrapolations to lower and higher shaft speeds were also included in the input model. Simple linear extrapolation was used. Use of these extrapolations demonstrates the system behavior trends and the ability of the computer code to provide reasonable calculations. If the JIMO project would have continued, more explicit performance maps for flow rates and shaft speeds beyond those currently available would have been required. 

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