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1、2.2.6 Pump priming If the impeller speed, the radius of the impeller, and the velocity of the fluid leaving the impeller are constant, the developed head is the same for fluids of all densities and is the same for liquids and gases The increase in pressure, however, is the product of the developed h
2、ead and the fluid density. If a pump develops, say, a head of 30.5m and is full of water, the increase in pressure is 2.95 atm. If the pump is full of air at ordinary density, the pressure increase is about 0.0035atm. It is called air bound and pump can accomplish nothing until the air has been repl
3、aced by a liquid. Positive-displacement pumps are not subject to air binding. 2.2.7 pump selection Before a centrifugal pump is selected, its application must be clearly understood. It is necessary to know the liquid to be handled, the total head, capacity and, in most cases, the temperature, viscos
4、ity, vapor pressure, and specific gravity. In chemical industry, the task of pumps election is frequently further complicated by the presence of solids in the liquid and liquid corrosion characteristics requiring special materials of construction. The performance curve of centrifugal pump shows the
5、pumps capacity and total head. It also shows required power, and NPSH requirement over a range of flow rates. It also shows the pumps best efficiency point (BEP). The pump operates most cost effectively when the operating point is close to the BEP.Pumps can generally be ordered with a variety of imp
6、eller sizes. Each impeller has a separate performance curve (see Figure ). To select a midrange impeller that can be trimmed or replaced to meet higher or lower flow rate requirements. To minimize pumping system energy consumption, select a pump so the system curve intersects the pump curve within 2
7、0% of its BEP. Select a pump with high efficiency contours over your range of expected operating points. A few points of efficiency improvement can save significant energy over the life of the pump.2.3 Gas-Moving Machinery Gas-moving machinery comprises mechanical devices used for compressing and mo
8、ving gases. They are often classified from the standpoint of the pressure heads producedClassification:fans for low pressures, discharge heads are low, from about 0.1 m to 1.5 m H2O. blowers for intermediate pressures, compressors for high pressures 2.3.1 Fans The commonest method for moving small v
9、olumes of gas at low pressures is by means of a fan. Large fans are usually centrifugal and their operating principle is similar to that of centrifugal pumps. In a centrifugal fan, the centrifugal force produced by the rotor causes a compression of the gas, called the static pressure head. Also, sin
10、ce the velocity of the gas is increased, a velocity head is produced. Writing mechanical energy balance equation between the inlet and outletFriction loss is accounted for mechanical efficiency, and potential energy difference between two stations is negligible, so setting z2-z1=0Customarily u1=0 fo
11、r the fans, rearranging the equation aboveCustomarily, individual term of the equation above is imparted by head, as follow2.3.2 Blowers and compressors For handling gas volumes at higher pressure rises than fans, several distinct types of equipment are used. Turbo-blowers or centrifugal compressors
12、Rotary blowers and compressorReciprocating compressors Turbo-blowers or centrifugal compressors are widely used to move large volumes of gas for pressure rises from about 5 kPa to several thousand kPa. The principles of operation for a turbo-blower are the same as for a centrifugal pump. Rotary blow
13、ers are machines of the positive-displacement type and are essentially constant-volume flow-rate machines with variable discharge pressure. Changing the speed will change the volume flow rate. The pressures up to about 1000 kPa can be obtained, depending on the type. Reciprocating compressors which
14、are of the positive displacement type using pistons are available for higher pressures. Multistage machines are also available for pressures up to 10 000 kPa or more Equations for Compression of GasesWhen the pressure on a compressible fluid is increased adiabatically, the temperature of the fluid a
15、lso increases. the work required for compressing a kilogram of fluid is larger than if the compression were isothermal. The temperature rise has a number of disadvantages. Because the specific volume of the fluid increases with temperature,Excessive temperatures lead to problems with lubricants, stu
16、ffing boxes, and materials of construction. The fluid may be one that cannot tolerate high temperatures without decomposing For the isentropic (adiabatic and frictionless) pressure change of an ideal gas, the variables relations are, For a given gas, the temperature ratio increases with an increase
17、in the compression ratio p2/p1. In blowers with a compression ratio below about 3 or 4, the adiabatic temperature rise is not large, and no special provision is made to reduce it. In compressors, however, where the compression ratio may be as high as 10 or more, the isentropic temperature becomes ex
18、cessive. Also, since friction, the heat from friction is absorbed by the gas, and temperatures well above the isentropic temperature are attained. Compressors, therefore, are cooled by jackets through which cold water or refrigerant is circulated. In small cooled compressors, the exit gas temperatur
19、e may approach that at the inlet, and isothermal compression is achieved. In very small ones, air cooling by external fins cast integrally with the cylinder is sufficient. In larger units, where cooling capacity is limited, a path different from isothermal or adiabatic compression, called polytropic
20、 compression, is followed.A positive-displacement blower is shown in Fig Positive-displacement blowerA single-stage blower can discharge gas at 0.4 to 1 atm gauge, a two-stage blower at 2 atm. The blower shown in Fig. has two lobes. Three-lobe machines are also common. Centrifugal blowers A single-s
21、tage centrifugal blower is shown in Fig. The operating speed is high-3600 r/min or more. High speeds and large impeller diameters are required because very high heads are needed to generate modest pressure ratios. Positive-displacement compressors Rotary positive-displacement compressors can be used
22、 for discharge pressures up to about 6 atm. For high to very high discharge pressures and modest flow rates, reciprocating compressors are the most common type. the temperature rise is important. The cylinder walls and cylinder heads are cored for cooling jackets using water or refrigerant. When the
23、 required compression ratio is greater than can be achieved in one cylinder, multistage compressors are used. Between each stage are coolers, which are tubular heat exchangers cooled by water or refrigerant. Centrifugal compressors also known as turbo-compressors belong to the roto-dynamic type of c
24、ompressors. In these compressors the required pressure rise takes place due to the continuous conversion of angular momentum imparted to the gas by a high-speed impeller into static pressure. Low-pressure gas enters the compressor through the eye of the impeller (1). The impeller (2) consists of a n
25、umber of blades, which form flow passages (3). From the eye, the gas enters the flow passages formed by the impeller blades, which rotate at very high speed. As the gas flows through the blade passages towards the tip of the impeller, it gains momentum and its static pressure also increases. From th
26、e tip of the impeller, the gas flows into a stationary diffuser (4). In the diffuser, the gas is decelerated and as a result the dynamic pressure drop is converted into static pressure rise, thus increasing the static pressure further. The gas from the diffuser enters the volute casing (5) where fur
27、ther conversion of velocity into static pressure takes place due to the divergent shape of the volute. Finally, the pressurized gas leaves the compressor from the volute casing (6). The blades of the compressor, either forward curved or backward curved or radial. Backward curved blades were used in
28、the older compressors, whereas the modern centrifugal compressors use mostly radial blades. diffuserThe stationary diffuser can be vaned or vaneless. As the name implies, in vaned diffuser vanes are used in the diffuser to form flow passages. The vanes can be fixed or adjustable. Vaned diffusersVane
29、d diffusers are compact compared to the vaneless diffusers and are commonly used for high discharge pressure applications. However, the presence of vanes in the diffusers can give rise to shocks, as the gas velocities at the tip of the impeller blade could reach sonic velocities in large, high-speed
30、 centrifugal compressors. vaneless diffusersIn vaneless diffusers the velocity of gas in the diffuser decreases and static pressure increases as the radius increases. As a result, for a required pressure rise, the required size of the vaneless diffuser could be large compared to vaned diffuser. Howe
31、ver, the problem of shock due to supersonic velocities at the tip does not arise with vaneless diffusers as the velocity can be diffused smoothly. guide vaneGenerally adjustable guide vanes or pre-rotation vanes are added at the inlet (eye) of the impeller for capacity control.Unlike reciprocating c
32、ompressors, centrifugal compressors are steady-flow devices, hence they are subjected to less vibration and noise.Equations for blowers and compressorsBecause of the change in density during compressible flow, the integral form of the Bernoulli equation is inadequate. In blowers and compressors the
33、mechanical, kinetic, and potential energies do not change appreciably, and the velocity and static head terms can be dropped. Also, on the assumption that the compressor is frictionless, =1.0 and hf =0. With these simplifications, mechanical balance equation becomesIntegration between the suction pr
34、essure p1 and the discharge pressure p2 gives the work of compression of an ideal frictionless gas Adiabatic compression For uncooled units, the fluid follows an isentropic path. For ideal gases, the relation between p and is given as follow Isothermal compression When cooling during compression is
35、complete, the temperature is constant and the process is isothermal. The relation between p and then is simply For a given compression ratio and suction condition, the work requirement in isothermal compression is less than that for adiabatic compression. This is one reason why cooling is useful in
36、compressors Polytropic compression In large compressors the path of the fluid is neither isothermal nor adiabatic. The process may still be assumed to be frictionless, however. It is customary to assume that the relation between pressure and density is given bywhere k is a constant. Use of this equa
37、tion in place of Eq.（2.3-3）, obviously yields Eq.（2.3-4）with the replacement of by k. The value of k is found empirically by measuring the density and pressure at two points on the path of the process, for example, at suction and discharge. The value of k is calculated by the equationcentrifugal com
38、pressorThe gain in momentum is due to the transfer of momentum from the high-speed impeller blades to the gas confined between the blade passages. The increase in static pressure is due to the self-compression caused by the centrifugal action. This is analogous to the gravitational effect, which cau
39、ses the fluid at a higher level to press the fluid below it due to gravity. If we assume the impeller blades to be radial and the inlet diameter of the impeller to be small, then the static head, h developed in the impeller passage for a single stage is given by:u= peripheral velocity of the impelle
40、r wheel or tip speed, m/sHence increase in total pressure, P as the gas flows through the passage is given by:Thus it can be seen that for a given gas with a fixed density, the pressure rise depends only on the tip speed of the blade. The tip speed of the blade is proportional to the rotational spee
41、d of the impeller and the impeller diameter. The maximum permissible tip speed is limited by the strength of the structural materials of the blade and the sonic velocity of the gas. Under these limitations, the maximum achievable pressure rise (hence maximum achievable temperature lift) of single st
42、age centrifugal compressor is limited for a given gas. Hence, multistage centrifugal compressors are used for large temperature lift applications.In multistage centrifugal compressors, the impeller diameter of all stages remains same, but the width of the impeller becomes progressively narrower in t
43、he direction of flow as gas density increases progressively. Analysis of centrifugal compressorsApplying energy balance to the compressor , we obtain from steady flow energy equation:Neglecting changes in kinetic and potential energy, the above equation becomes:In a centrifugal compressor, the heat
44、transfer rate Q is normally negligible compared to the other energy terms, hence the rate of compressor work input for adiabatic compression is given by:The above equation is valid for both reversible as well as irreversible adiabatic compression, provided the actual enthalpy is used at the exit in
45、case of irreversible compression.In case of reversible, adiabatic compression, the power input to the compressor is given by:hence using the thermodynamic relation, Tds=dhvdp; the isentropic work of compression is given by:Thus the expression for reversible, isentropic work of compression is same fo
46、r both reciprocating as well as centrifugal compressors. However, the basic difference between actual reciprocating compressors and actual centrifugal compressors lies in the source of irreversibility.In case of reciprocating compressors, the irreversibility is mainly due to heat transfer and pressu
47、re drops across valves and connecting pipelines.However, in case of centrifugal compressors, since the gas has to flow at very high velocities through the impeller blade passages for a finite pressure rise, the major source of irreversibility is due to the viscous shear stresses at the interface bet
48、ween the fluid and the impeller blade surface.In reciprocating compressors, the work is required to overcome the normal forces acting against the piston, while in centrifugal compressors, work is required to overcome both normal pressure forces as well as viscous shear forces. The specific work is h
49、igher than the area of P-V diagram in case of centrifugal compressors due to irreversibilities and also due to the continuous increase of specific volume of gas due to fluid frictionpolytropic efficiencyTo account for the irreversibilities in centrifugal compressors, a polytropic efficiency pol is d
50、efined. It is given by:The polytropic work of compression is usually obtained by the expressionwhere k is the index of compression, f is a correction factor which takes into account the variation of k during compression.Normally the value of f is close to 1 (from 1.00 to 1.02), hence it may be negle
51、cted in calculations, without significant errors.If the gas is assumed to behave as an ideal gas, then it can be shown that the polytropic efficiency is equal to:Though gas do not strictly behave as ideal gases, the above simple equation is often used to obtain the polytropic efficiency of the centr
52、ifugal compressors by replacing by isentropic index of compression, , For actual gas the polytropic efficiency is estimated from the equation:For actual centrifugal compressors, the polytropic efficiency is found to lie in the range of 0.7 to 0.85. The index of compression k is obtained from actual
53、measurements of pressures and specific volumes at the inlet and exit of the compressor and then using the equation Pv = constant. This procedure usually gives fairly accurate results for gas made of simple molecules such as water, ammonia. The deviation between actual efficiency and polytropic effic
54、iency evaluated using the above equations can be significant in case of heavier molecules such as R 22, R 134a Capacity controlThe capacity is normally controlled by adjusting inlet guide vanes (pre-rotation vanes). 1. guide vanesAdjusting the inlet guide vanes provide a swirl at the impeller inlet
55、and thereby introduces a tangential velocity at the inlet to the impeller, which gives rise to different gas flow rates.Figure shows the performance of the compressor at different settings of the inlet guide vanes. Use of inlet guide vanes for capacity control is an efficient method as long as the a
56、ngle of rotation is high, i.e., the vanes are near the fully open condition. When the angle is reduced very much, then this method becomes inefficient as the inlet guide vanes then act as throttling devices.2. diffuserIn addition to the inlet guide vanes, the capacity control is also possible by adj
57、usting the width of a vaneless diffuser or by adjusting the guide vanes of vaned diffusers.Using a combination of the inlet guide vanes and diffuser, the capacities can be varied from 10 percent to 100 percent of full load capacity.3. rotating speedCapacity can also be controlled by varying the comp
58、ressor speed using gear drives. For the same pressure rise, operating at lower speeds reduces the flow rate, thereby reducing the gas capacity.Performance aspects of centrifugal compressorFigure shows the pressure-volume characteristics of a centrifugal compressor running at certain speed.As shown i
59、n the figure, the relation between pressure and volume is a straight line in the absence of any losses. However, in actual compressors losses occur due to eddy formation in the flow passages, frictional losses and shock losses at the inlet to the impeller. The entry losses are due to change of direc
60、tion of gas at the inlet and also due to prerotation. These losses can be controlled to some extent using the inlet guide vanes.Figure also shows an optimum point. The optimum point at which the losses are minimum is selected as the design point for the compressor.SurgingSurge occurs in a turbo comp
61、ressor when discharge head cannot be sustained at the available suction flow.for example, A centrifugal compressor is designed to operate between a given evaporator and condenser pressures. Due to variations either in the heat sink or refrigerated space, the actual evaporator and condenser pressures
62、 can be different from their design values. For example, the condenser pressure may increase if the heat sink temperature increases or the cooling water flow rate reduces. If the resulting pressure difference exceeds the design pressure difference of the compressor, then refrigerant flow reduces and
63、 finally stops.Further increase in condenser pressure causes a reverse flow of refrigerant from condenser to evaporator through the compressor. As a result the evaporator pressure increases, the pressure difference reduces and the compressor once again starts pumping the refrigerant in the normal di
64、rection. Once the refrigerant starts flowing in the normal direction, the pressure difference increases and again the reversal of flow takes place, as the pressure at the exit of compressor is less than the condenser pressure.This oscillation of refrigerant flow and the resulting rapid variation in
65、pressure difference gives rise to the phenomenon called “surging”.Surging is most likely to occur Load is low (i.e. evaporator pressure is low) and/or higher external pressure(the condensing temperature is high). Surge is usually accompanied by the following:Increase in discharge temperature Reducti
66、on in discharge pressure Increase in vibration Sharp rise in inlet temperatureIt can be seen from these figures that beyond a certain condenser pressure and below a certain evaporator pressure, the refrigerant capacity of centrifugal compressor decreases rapidly unlike reciprocating compressors wher
67、e the capacity drop under these conditions is more gradual. This is unlike reciprocating compressors, which continue to pump gas, albeit at lower flow rates (when the condenser temperature increases and/or the evaporator pressure falls).A centrifugal compressor cannot pump the refrigerant when the c
68、ondensing pressure exceeds a certain value and/or when the evaporator pressure falls below a certain point. Effect of condensing temperature on power inputIt can be seen that while the power input increases with condensing temperature for a reciprocating compressor, it decreases with condensing temp
69、erature for a centrifugal compressor. This is due to the rapid drop in refrigerant mass flow rate of centrifugal compressor with condensing temperature. This characteristic implies that the problem of compressor overloading at high condensing temperatures does not exist in case of centrifugal compre
70、ssors.damage of surgingSurging produces noise and imposes severe stresses on the bearings of the compressor and motor, ultimately leading to their damage. Hence, continuous surging is highly undesirable, even though it may be tolerated if it occurs occasionally.One or more of the following can resul
71、t from surge:Unstable operation Partial or total flow reversal through the compressorDisrupted process Mechanical damage to the compressorsurging controlFrom a surge control standpoint, the challenge is to keep the compressor out of surge without wasting energy on excessive recycling. This requires
72、that the surge point be precisely computed from measurable, compressor-operating conditions. This goal will be addressed in the construction of the Surge Line. In the gas compressor section, surge can be avoided by(1) recycling a controlled portion of the discharge flow back to the suction through a
73、 recycle valve. Gas Compressor Anti-Surge ValveRecycling raises the suction pressure and lowers the discharge pressure, which artifitically increases flow and moves the operation away from surge.(2) Raising speed also moves the compressor away from surge. This is a temporary solution because it also
74、 raises Pd and lowers Ps, which tends to drive the machine back towards surge.(3) A blow off valve is used to vent the compressor discharge to atmosphere. Air Compressor Anti-Surge Valve This does not affect the suction conditions, but it reduces discharge pressure and increases flow, which moves th
75、e operating point away from surge.The problem with both and performance curves is that they both show the compressor surge characteristics for specific conditions only.Ideally, a curve and a method which accurately defines surge for all gas and suction conditions, is needed.surge marginTo prevent su
76、rge, the system must accurately predict surge and begin to open the recycle valve before surge can occur. The safety margin between the predicted surge point,and the point that recycle flow is initiated, is called the surge margin.The surge margin is implemented by modifying the surge line to produc
77、e a Control Line. Referring to FigureThe Control Line lies to the right of the surge line by an amount equal to the safety margin.Depending on the application, either a constant (parallel), or progressive(meeting at the origin), characteristic can be configured.The control line provides these point
78、to the surge controller,which opens the recycle valve to prevent flow from falling below the Control LineSurge Control Block DiagramThe TRISENTM surge control system can be best described in the form of individual modules. Each module has a readily defined functionality, and interacts with other mod
79、ules through input and output signals, which are given tag names.problemIf a centrifugal pump operates on air, then, liquid from an initially empty suction line。Positive-displacement pumps air binding.Before a centrifugal pump starts up, close the valve in discharge in other to ( ) the start-up powe
80、r requiredproblemThe difference between the theoretical and actual curves results from: ; ; .When higher developed head is required, the best selection is .A) operation in series B) multistage centrifugal pump C) throttled by valve problem ( )One pump can develop more H than that two same pumps whic
81、h work in series. The methods of adjusting volume flow rate of centrifugal pump are (1) ; (2) ; (3) ; The commonest way in practice is ( ). Questions & answers:1.Which of the following statements concerning centrifugal compressors are true?a) Centrifugal compressors are subjected to less vibration a
82、nd noise as they rotate at very high speedsb) Pressure rise in centrifugal compressor is due to the continuous conversion of angular momentum into static pressurec) The stagnation enthalpy of gas remains constant everywhere, except across the impeller bladesd) Conversion of dynamic pressure into sta
83、tic pressure takes place in the volute casing due to its convergent shape Ans.: b) and c)2. Which of the following statements concerning centrifugal compressors are true?a) Centrifugal compressors with vaneless diffusers are compact compared to vaned diffusers b) In multistage centrifugal compressor
84、s, the width of the blades reduces progressively in the direction of flow c) In multistage centrifugal compressors, the width of the blades increases progressively in the direction of flow d) Multistaging in centrifugal compressors is commonly used for high refrigerant capacity applicationsAns.: b)3
85、. The polytropic efficiency of a centrifugal compressor is found to be 0.85. The isentropic index of compression of the refrigerant, which behaves as an ideal gas, is 1.17. The polytropic index of compression, k is then equal to:a)1.206b) 0.829c) 0.854d) 1.141Ans.: a)4. Which of the following statem
86、ents are true:a) In reciprocating compressors, the irreversibility is mainly due to heat transfer and viscous shear stressesb) In the reciprocating compressors, the the irreversibility is mainly due to heat transfer and pressure drops across valves and connecting pipelines c) In centrifugal compress
87、ors, the irreversibility is mainly due to heat transfer and viscous shear stresses d) In centrifugal compressors, the irreversibility is mainly due to viscous shear stressesAns.: b) and d)5.Which of the following statements are true:a) Due to slip, the actual pressure rise and volumetric flow rate o
88、f a centrifugal compressor is less than that of an ideal compressorb) for a given impeller diameter, the slip factor decreases as the number of blades increasesc) For a given impeller diameter, the slip factor decreases as the number of blades decreasesd) For a given flow rate, the frictional losses
89、 decrease as the number of blades increaseAns.: a) and c)6.Which of the following statements are true:a) the capacity of a centrifugal compressor can be controlled by using inlet guide anes and by changing the width of the diffuserb) Surging in centrifugal compressors takes place as evaporator and c
90、ondenser pressures increasec) Surging in centrifugal compressors takes place as evaporator pressure increases and condenser pressure decreasesd) Surging in centrifugal compressors takes place as evaporator pressure decreases and condenser pressure increasesAns.: a) and d)7. Which of the following st
91、atements are true:a) When operated away from the surge point, the reduction in evaporator temperature with refrigeration load is smaller for centrifugal compressors compared to the reciprocating compressorsb) When operated away from the surge point, the reduction in temperature with refrigeration lo
92、ad is much larger compared to the reciprocating compressorc) The problem of compressor motor overloading due to high condenser temperature does not take place in a centrifugal compressord) Compared to reciprocating compressor, the performance of centrifugal compressor is less sensitive to speedAns.:
93、 a) and c) A centrifugal pump is to be used to extract water from a condenser in which the vacuum is 640 mm of mercury. At the rated discharge, the net positive suction head must be at least 3m above the cavitation vapor pressure of 710mm mercury vacuum. If losses in the suction pipe accounted for a
94、 head of 1.5m. What must be the least height of the liquid level in the condenser above the pump inlet?Hg problem A petroleum fraction is pumped 2 km from a distillation plant to storage tanks through a mild steel pipeline, 150 mm in diameter, at the rate of 0.04m3/s. what is the pressure drop along
95、 the pipe and the power supplied to the pumping unit if it has an efficiency of 50%? The pump impeller is eroded and the pressure at its delivery falls to one half. By how much is the flow rate reduced? ( is not changed) specific gravity of the liquid =0.705 viscosity of the liquid =0.5mPa.s roughne
96、ss of pipe surface =0.004mmA process requires 56m3/h at a total operating head of 46 m. Assume the centrifugal pump will be powered by a 500kW motor, operate for 8,000 hours annually, and transport fluid with a specific gravity of 1.0. One candidate pump has an efficiency (1) of 81% at the operating
97、 point; a second is expected to operate at 78% efficiency (2). What are the energy savings given selection of the first pump? problem EXAMPLE 2.5. A three-stage reciprocating compressor is to compress 306std m3/h of methane from 0.95 to 61.3 atm abs. The inlet temperature is 26.7C. For the expected
98、temperature range the average properties of methane areCp = 38.9 J/g mol C = 1.31(a) What is the brake horsepower if the mechanical efficiency is 80 percent? (b) What is the discharge temperature from the first stage? (c) If the temperature of the cooling water is to rise 11.1C, how much water is ne
99、eded in the intercoolers and after cooler for the compressed gas to leave each cooler at 26.7C? Assume that jacket cooling is sufficient to absorb frictional heat.Solution(a) For a multistage compressor it can be shown that the total power is a minimum if each stage does the same amount of work. By
100、Eq.（2.3-4） this is equivalent to the use of the same compression ratio in each stage. For a three-stage machine, therefore, the compression ratio of one stage should be the cube root of the overall For a three-stage machine, therefore, the compression ratio of one stage should be the cube root of th
101、e overall The power required for each stage is, by Eq. The total power for all stages is 3 19387.5= 58162.5J/s=58.2 kW (b) From Eq.（2.3-1）, the temperature at the exit of each stage is T2=1.388T1=1.388(273.2+26.7)=416.3K (C) The heat load in each cooler ismcp(T2-T1)=0.060738.9(416.3 299.9)/16 =17.18kJ/sThe total heat load is 3 17.18 = 51.53kJ/s. The cooling water requirement is
