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This paper presents a statistical approach to analysis of gas pulsation in a positive displacement compressor in which excessive gas pulsation was identified as the major cause of fracture of the multi-ring valve plates. The pressure pulsation of the gas in the delivery pipe gave rise to large differential pressure at the point in time at which the valve opens or shuts. Consequently, the dynamic stresses within the plate, resulting from the hard impact between the plate and the seat on closure of the valve gave rise to the fracture. The study obtained a maximum likelihood estimate of peak pulsation levels in the operation of the compressor based on independent measurements. This was quantified analytically and used to predict the condition of the plant.
Key words: diagnostic evaluation, nitrogen compressor, valve plates, gas pulsation measurements, vibration switch, maximum likelihood estimate, parametric prediction, noise sequence
I N T R O D U C T I O N
All equipment failures exhibit three types of weakness (Kinchin, 1978). First, there is a technical fault associated with a failed component due to overstressing, either because the load was greater than the anticipated value or the part was not strong enough for the specified load. It is also possible that the part may not have been manufactured or fitted according to the design. Secondly, the failure may be due to organizational fault (Smart and Vertinsky, 1977). Nevertheless, engineering designers (Sievers and Flotow, 1988) have successfully reduced equipment failures, by isolating sections of equipment and treating them as quasi-closed systems,
but analysis of many industrial failures (Smart and Vertinsky, 1977) suggest that designers also need to be aware of conditions which challenge the assumption that component sub-assemblies can be treated independently.
Basically, there are two classes of compressors (McKenzie, 1980) namely, positive displacement and dynamic. Reciprocating piston compressors which are typically positive displacement type are widely used in the natural gas industry for a number of technical reasons, and as the name implies, a reciprocating compressor compresses gas by the action of a piston moving forwards and backwards in the cylinder. This principle is quite distinct from that of the dynamic compressor in that in the dynamic compressor air or gas is drawn into a rapidly rotating impeller wheel and accelerated to a high velocity. The stream is then discharged through a diffuser where its kinetic energy is converted into static pressure. The compression actions of the compressor give rise to pulses of the gas which, invariably, generate vibration in the piping system. The vibration levels would depend on the piping system design. Other factors which could lead to vibrations in the system include unbalance and misalignment (Cao et. al., 2000; Obinabo, 1981), mechanical looseness, bent shaft, bad anti-friction bearings, bad gears and electrical problems. There are several other factors that may be difficult to quantify (Cavailo et. al., 1999) but whatever the case, all these problems can be identified (Hsiao and Wang, 2000; Karimi et. al., 2004) while the machines are in their normal operations through on-condition vibration measurement (Odi-Owei and Okah-Avae, 1980) if it exists in the system.
The importance of vibration in machines stems from the fact that all machines vibrate (Yang et. al., 2001). Most of the factors contributing to this, such as the stack-up of manufacturing tolerances, frictional and centrifugal forces acting on the rotors of rotating machinery add up to some lower level of vibration present even in a newly installed machine (Shoureshi et. al., 1999; Obinabo, 1996). Any increase in vibration above this lower level (Tokhi and Veres, 2002) indicates that some changes have occurred in the mechanical or electrical condition of the machine. When the vibration level becomes excessive, then the severity of the problem has grown such that continued safe operation of the machine is in jeopardy.
This paper reports an account of the background to a nitrogen compressor failure in a steel-making plant. In particular, it analyses the vibration data obtained from measurements taken on the unit during commissioning and after repairs had been effected on the compressor unit
following the breakdown. The statistical method of maximum likelihood was employed to maximize the probability of obtaining the desired vibration level in the system.
The Nitrogen Compressor Unit: This unit is a reciprocating type and employs a two-stage compression process to attain the required level of the final gas pressure. The nitrogen gas is drawn into the first stage compression cylinder which is then closed off from the inlet. Its volume is forcibly reduced during which the gas is compressed. The capacity of the cylinder is fixed by a volume which, in turn, depends on the compression ratio. As the compression ratio increases, the volumetric efficiency of the compressor becomes less and mechanical stress limitations become more pronounced. This therefore accounts for the two-stage process involved in compressing the gas to the required pressure. On attainment of this pressure, the delivery valve opens to discharge the compressed nitrogen gas at a constant pressure.
Figure 1 Basic Arrangement of the Nitrogen Compressor unit
showing Sensing Points for Vibration Measurement
In the compressor, nitrogen enters for the first stage compression at a pressure of 1.5 bars and a temperature of 400 C. It finally leaves the system at a desired pressure and temperature of 30 kilo bars and 42°C respectively. The temperature of the nitrogen gas leaving the cylinder after each stage of compression depends on the energy transferred to or from the gas by the compressor
unit. The maximum amount of energy that may be transferred by the unit is represented by an adiabatic reversible (isentropic) process and this is a standard against which all compressors are compared (McKenzie 1980). Other components integral with the compressor unit include the inter stage-piping, aftercoolers and separators as well as the pulsation dampers all of which are incorporated to take care of the high temperatures and pulsations associated with the compression of the gas. There are also other control systems installed on the unit to shut down the compressor should any of the following malfunctions occur.
(1) Over speed,
(3) Excess vibration,
(3) Low lubrication oil pressure.
(4) High jacket water temperature.
(5) Abnormal gas discharge temperature,
(6) Abnormal suction and discharge pressures,
(7) High bearing temperature,
(8) Low cooling water inlet,
Connected with the vibration level on the compressor unit and mounted on the outside of the first stage compression cylinder is the excessive vibration trip called the vibration switch which is designed to shut down the compressor should the vibration level exceed a predetermined value. The switch includes a vibration detecting mechanism which also functions as a mechanical amplifier to operate a snap acting switch to energize, through relays, the solenoid servo spill valve and shut down the engine when the selected level of vibration is exceeded. This detector mechanism consists of an armature suspended on flexure pivot and restrained from motion by a permanent magnet acting through a small air gap. It also consists of a compression spring which provides an adjustable opposing force to the magnet. Whenever the peak vibration inertia plus the adjustable spring force exceeds the holding force of the magnet, the armature is released and it snaps against a second latch magnet, simultaneously actuating the snap-acting switch. The switch has no moving parts except under trip conditions and is designed to last many years without maintenance. A normal practice is to replace the switch should any trouble be experienced and the defective piece be returned to the manufacturer for rectification.
Pulsation of the Nitrogen Gas in the Compressor Unit: Pulsation of the nitrogen gas in the piping network caused decreased operational efficiency of the compressor and produced vibrations which, in turn became excessive and dangerous especially when the frequency of the pulsation coincided with the resonant frequency of the piping. In the piping network, the coupling points were the piping bends, closed ends of vessels and headers and restrictions such as orifices, valves and reducers. In providing pulsation dampers to the compressor system it is necessary to avoid, any piping spans or piping component that are mechanically resonant at any of the excitation frequencies. To this end, the connection between the pulsation damper and the compressor cylinder is normally designed to be as short as possible and straight, and where its length exceeds twice the inside diameter of the cylinder connection, it is enlarged to minimize flow restriction. Once this is accomplished, then the vibration level in the piping system should be within acceptable criteria assuming the basic supporting structure of the compressor system is properly designed and monitored against excessive vibration.
parameter estimation of peak vibration levels
The vibrating model of the compressor unit was described mathematically as a square, linear, open-loop system S(A, B, C), and represented generally by the following state space model (Obinabo and Anyasi, 2007)
where , and are the , and dimensional state-, input-, and output-vectors and A, B and C are constant matrices of appropriate dimensions. The time series model of (1) was obtained as
The partial fractions required for obtaining a time solution for distinct roots was obtained by the method of residues. If are the poles of the function with multiplicities , the partial fraction expansion
was obtained, and the inversion integral has the form
For distinct roots, the following was obtained
where , are the parameters of the system, and n the order of the system whose output-input relation can be put in the form (6)
x(j) + aix(j - i) = biu (j - i) 7
using the shift operator defined by in equation (7) to yield
where is the observed output signal, is the applied input signal, N is the number of samples and is the variation or noise sequence. The polynomial operators A, B, and C are defined as follows
With the continuous model discretized the following was obtained
where the parameter set a, b is related to the parameter set , via the relation
and T is the sampling interval in seconds. The infinite variations or noise of the discrete measurements due to aliasing that may result from the discretization process is negligible. This is
justified if and are independent of all k and s. This is a reasonable assumption as long as the identification is performed for data acquired from experiments, where is a priori known sequence. In some practical situations, this assumption is often violated when operating records are used because in such a case the input may depend on the output through feedback. The canonical model (Chang and Wang, 1984; Chen and Hsiao, 1965) can be made equivalent to the discrete-time model if the following conditions are satisfied
Maximum Likelihood Estimates of the Vibration Level: The statistical method of maximum likelihood is employed to maximize the probability of obtaining the desired vibration level in the system. Consequently, a loss function is defined as follows
which will be minimized with respect to the vibration parameter set . The residual numbers are defined by
so that the vibration parameter set a' and b' that makes in (17) minimum will be the estimates of the vibration level of the system.
The approach considered in the foregoing is formulated as one of finding the coefficients of the prediction model
so that the mean square prediction error
is as small as possible. Thus the assumption of Gaussian distribution of the variation sequence may be relaxed, but on the other hand, the stochastic properties of the estimates are lost. In the model given in (4), the residues are obtained as
For system order n greater than 1, as given above becomes computationally difficult and highly nonlinear. Whereas in the model of (14), the residues given by (18) are easier and quicker to compute for any given system order. Since the two models are equivalent via the transformations defined in (16), we are free to use any of them
We now consider an example which should help to illustrate the points made in the foregoing sections; and show that a maximum likelihood estimate of peak vibration levels k in the operation of the nitrogen compressor based on N independent measurements of peak vibration levels T (namely ) can be quantified analytically. Here, we assume the time between successive data acquisition on the plant to be distributed with a probability density function of the form
where k is any constant
if we have independent measurements of peak vibration levels (), then
Taking log likelihood of the function gives
Maximizing with respect to and gives
Letting gives the maximum likelihood of the estimates as follows
From the Cramer-Rao bound
Thus as and the estimate is constant.
RESULTS AND DISCUSSION
As part of the commissioning of the Nitrogen compressor, vibration measurements were taken using magnetic vibration pick-ups mounted on the identified sensing points on the compressor unit (fig. 1). These readings were taken on the bearing housing and major structures of the compressor unit in a direction perpendicular to the rotating centre line of the parts of the compressor. Similar readings were also taken on the unit after a major repair had bean effected on it following the breakdown of the compressor unit. The data so obtained are shown in tables 1 and 2 respectively and analyzed using the General Machinery Vibration Severity Charts.
Vibration Data Obtained During the
Commissioning of the Compressor Unit
Vibration Velocity (in/sec)
* 1 mil = One thousandth of an inch.
Vibration Data Obtained After Repairs
Vibration Velocity (sec)
* I mil = One thousandth of an inch.
Deductions made using the General Machinery Vibration Severity Charts indicate that the vibration level on the compressor unit was within the acceptable limit using the data in tables 1 and 2. The breakdown itself was attributed to excessive vibration with the attendant fracture of the multi-ring valve plates. The pressure pulsation of the nitrogen gas in the delivery pipe gave rise to large differential pressure at the point in time at which the valve opens or shuts and the associated dynamic stresses within the plate, resulting from the hard impact between the plate and the seat on closure of the valve gave rise to the fracture.
the valve plates: The failure of the valve plates was due to either one or a combination or the following:
Pressure pulsation of the nitrogen gas in the delivery pipe which gave rise to large differential pressure at the point in time at which the valve opens or shuts,
Excessive cylinder lubrication or inadequate removal of water by suction separators. Both of these gave rise to the liquid droplets observed on the valve plates which gave rise to intense local loading. Pitting on some of the examined plates could be due to excessive oil lubrication or high temperature as some of the plates were oily and blackened when collected from the dump.
The quality of the material of manufacture of the valve plates was considered and our deduction is that fully machined plates give a much longer life as they do not incorporate points of stress concentration or sharp edges.
The valves operate under different conditions and, concerning the motion of the valve plate, it is important whether they operate under low or high pressure. In low pressure valves, the motion of the plate is affected considerably by the springs, because the force exerted by the springs is governed by the effects due to the nitrogen gas flowing through the valve. In high pressure valves the plate is governed completely by the dynamic effects of the gas. It lifts and bears on the stop plate very quickly. It is only in the final phase of the opening of the valve that oscillations sometimes occur, and this is due to gas pulsations.
The dynamic stresses within the plates, resulting from the hard impact between the plate and the seat on closure of the valve are responsible for the failure of the valve plates. A significant
improvement in the prevention of this failure is to reduce the valve lift. Excessive lift results in valve instability which causes the valve to slam many times during a single discharge or suction event.
The Vibration Switch: The malfunctioning of the vibration switch was traced to:
(1) Abnormal discharge pressure since most of the tripping occurred at pressures much below the desired discharge pressure of 30.4 bars.
(2) Malfunctioning of the cooling water inlet valve and the associated light indicator on the control panel.
(3) Faulty time delay relay for lubrication oil pump.
A general check and bridging of the control systems on the unit revealed, shockingly, low cooling water inlet into the heat exchanger. Appropriate corrective measures would require that the faulty light indicator for this control unit be replaced.
Having considered these issues, it is worth stressing the importance of vibration monitoring on the compressor unit and, of course, on continuous basis so that any vibration problem can be quickly pinpointed and got rid of at the early stages thereby prolonging the life span of the unit.
The authors gratefully acknowledge the assistance received from the technical staff of the Iron and Steel industry plant for this research.