Air Only Data 1 INTRODUCTION Although few reliable or universal models currently exist for predicting the pressure drop for gas-solid flows in pipelines, models for the single phase flow of a gas are well established. Once again, although discussion will generally be in terms of air, the models presented will work equally well with the appropriate value of the specific gas constant for the particular gas being considered. Empty conveying pipeline pressure drop values, for air only, will provide a useful datum for both the potential capability of a system for conveying material and the condition of the pipeline. Air only pressure drop values for the conveying pipeline also provide a basis for some first approximation design methods for the conveying of materials. Air supply and venting pipelines can be of a considerable length with some systems, whether for positive pressure or vacuum systems, particularly if the air mover or the filtration plant is remote from the conveying system. In these cases it is important that the air only pressure drop values in these pipeline sections are evaluated, rather than just being ignored, for they could represent a large proportion of the available pressure drop if they are not sized correctly. Air flow control is also important, particularly if plant air is used for a conveying system, or if the air supply to a system needs to be proportioned between that delivered to a blow tank and that directed to the pipeline, for example.
2 PIPELINE PRESSURE DROP The pressure drop in the empty pipeline is a major consideration in the design of a pneumatic conveying system. If a positive displacement blower is used in combination with a long distance, small bore pipeline, for the suspension flow of a material, for example, it is quite possible that the entire pressure drop would be utilized in blowing the air through the pipeline and that no material would be conveyed. The pressure drop for air only in a pipeline is significantly influenced by the air velocity that is required for the conveying of the material. Bends and other pipeline features also need to be taken into account. The value of the empty line pressure drop for any pipeline will provide a useful indicator of the condition of the pipeline. If a pressure gauge is situated in the air supply or extraction line, between the air mover and the material conveying pipeline, this will give an indication of the conveying line pressure drop. With an empty pipeline it will indicate the air only pressure drop. If this value is higher than expected it may be due to the fact that the line has not been purged clear of material. It may also be due to material build-up on the pipe walls or a partial blockage somewhere in the pipeline. 2.1 Flow Parameters and Properties In order to be able to evaluate the pressure drop for the air flow in the empty pipeline, various properties of the air and of the pipeline need to be determined. Mathematical models and empirical relationships are now well established for this single phase flow situation, and so conveying line pressure drops can be evaluated with a reasonable degree of accuracy. 2.1.1 Conveying A ir Velocity This is one of the most important parameters in pneumatic conveying, as discussed earlier, with the air velocity at the material feed point being particularly important. If the conveying air velocity is not specified, therefore, it will usually have to be evaluated from the volumetric flow rate, pipeline bore, and the conveying line pressure and temperature, as outlined in the previous chapter. 2.7.2 Air Density The density, p, of air, or any other gas, is given simply by the mass of the gas divided by the volume it occupies: m p = lb/ft 3 V where m = mass of gas - Ib and V = volume occupied - ft 3
The Ideal Gas Law, presented earlier in Equation 5.4, applies equally to a constant mass of a gas, as to a constant mass flow rate of a gas, and so: P = m V 144 p RT ib/fr (1) where R = characteristic gas constant - ft Ibf/lb R P RT kg/rri (1SI) Gas constants for a number of gases were presented earlier in Table 5.1. A particular reference value is that of the density of air at free air conditions: For air R = 53-3 ft Ibf/lb R and so at free air conditions of p,, = 14-7 lbf/in 2 and T 0 = 519 R its density p = 0-0765 lb/ft 3 It will be seen from Equation 1 that air density is a function of both pressure and temperature, with density increasing with increase in pressure and decreasing with increase in temperature. The influence of pressure and temperature on the density of air is given in Figure 6.1 by way of illustration. 0-10 -5 0 5 10 20 Air Pressure - Ibf7in 2 gauge 40 Figure 6.1 The influence of pressure and temperature on air density.
0-044 0-042 > Ui < 0-040.- 475 500 525 Air Temperature - R 550 Figure 6.2 The influence of temperature on the viscosity of air. 2.7.3 Air Viscosity The viscosity, ft, of gases can usually be obtained from standard thermodynamic and transport properties tables. In general the influence of pressure on viscosity can be neglected. The influence of temperature on the viscosity of air is given in Figure 6.2 [1]. 2.1.4 Friction Factor The friction factor,/ for a pipeline is a function of the Reynolds number, Re, for the flow and the pipe wall roughness, e. Note: Reynolds number Re = 5 p Cd where p C d and fj = density of air - lb/ft 3 = velocity of air - ft/min = pipeline bore - in = viscosity of air - lb/ft h Alternatively, by substituting p from Equation 1 and C from a combination of Equation 5.4 into Equation 5.3 gives an alternative form:
Re = 2880 m.. n d JLI where rh a = air mass flow rate - Ib/min Note: The substitution of the volumetric flow rate, V, from Equation 5.4 into Equation 5.3 gives: 4 m a R T C = ft/min (2) n d p This provides a useful alternative expression for the evaluation of conveying air velocity. Typical values of wall roughness, 8, are given in Table 6.1 [2] and values of friction coefficient can be obtained from a Moody chart, a copy of which is given in Figure 6.3. It should be noted that Figure 6.3 is a UK version of the Moody diagram and gives values of friction coefficient that are one quarter the value of those on an equivalent US version of the diagram. This difference is explained, and compensated for, in the next equation to be presented. It will be seen from Figure 6.3 that an accurate value of a surface roughness is clearly not critical, for a 100% error in relative roughness will only result in a 10% error in friction coefficient. Table 6.1 Typical Values of Pipe Wall Roughness Pipe Material (new) Surface Roughness - S in Glass 'smooth' Drawn tubing 0-000,05 Commercial steel and wrought iron pipes 0-002 Asphalted cast iron 0-005 Galvanized iron 0-006 Cast iron 0-0! Concrete 0-01 - 0-1 Riveted steel 0-05 - 0-5
0-02 Relative Roughness s I co 0-01 0-006 0-003 0-002 0-001 Smooth Pipes 10 J 10" 10" Reynolds Number - Re 000,01 Figure 6.3 Friction coefficients for flow in circular pipes. 2.2 Pressure Drop Relationships The pressure drop for straight pipeline, regardless of orientation, is derived in terms of the pipeline friction coefficient. The pressure drop for bends and other pipeline fittings and features is obtained in terms of a loss coefficient. For the total pipeline system the two are added together. 2.2.7 Straight Pipeline The pressure drop, Ap, for a fluid flowing in a straight pipeline can be determined from Darcy's Equation: Ap = 4/1 pc 2 x d (3a) This is the UK version of Darcy's Equation, which is in terms of d/4. This is derived in terms of a hydraulic mean diameter, to allow application to non circular pipes and open channels. Hydraulic diameter is the ratio of the flow section area to the wetted perimeter, which for a circular pipeline running full is equal to d/4. This is the reason for the value of the pipeline friction coefficients on Figure 6.3 being four times lower than those on similar US charts.
If the pressure drop, Ap, is to have units of lbf/in 2, and the other parameters in the equation are as follows: / = friction coefficient - - L = pipeline length - ft d = pipeline bore - in p = density - lb/ft 3 C = velocity - ft/min and g c = gravitational constant - ft Ib/lbf s 2 = 32-2 ftlb/lbfs 2 then Darcy's Equation will appear as follows: fl p C 2 lbf/in 01 21,600 Ann d A g c... (3b) For a compressible fluid such as air, the equation in this form is rather inconvenient, particularly if there is a large pressure drop, for average values of both density and velocity need to be specified, as they are both very pressure dependent. Both density and velocity, however, can be expressed in terms of constants and air pressure, which means that the expression can be easily integrated. From Equation 1 : 144 p P ^ - and from Equation 2: 4 m a R T C = - j - ft/min n d p Substituting these into Equation 3 and expressing in differential form gives: frhl RT p dp = - r -- dl -... (4) / i * r\ T-J r 2 /5 ^ 9-375 n d g c. Integrating gives:
Pi - P2 0-213 f Lm] R T (5) where subscripts, and 2 refer to pipeline inlet and exit conditions This can be used to obtain the air only pressure drop for any straight pipeline since: Ap = p } - p 2 and noting that: if T = p] - p\ then Ap a = p l - ^f - F 0-5 / \fl.s and Ap a = (p 2 + F For a positive pressure system p 2 will be specified (usually atmospheric pressure) and so a more useful form of Equation 5, which eliminates the unknown Pi\s: 0-213 f Lm] R T xo-5 -p 2 lbf/in 2 (6) Similarly for a negative pressure system p t will be specified (usually atmospheric) and so an alternative form of Equation 5, which eliminates the unknown p 2 is: 0-213 R T P^ lbf/in' (7) 2.2.1.1 The Influence of Air Flow Rate The velocity of the conveying air will be approximately proportional to the air flow rate, whether on a mass or volumetric flow rate basis. From Equation 3 it will be seen that pressure drop is proportional to the square of the velocity, and so air flow rate will have a very significant effect on conveying line pressure drop. The influence of velocity is considered in conjunction with pipeline length and bore below.
6 5 4 JD 3 1 2 CO CO 1 ol- 0 Conveying Line Exit Air Velocity - ft/min 1000 2000 3000 4000 5000 o 20 40 60 Air Mass Flow Rate - Ib/min,-1 80 6000. ' i ' '. ' i t S *- r- 100 Figure 6.4 pressure drop. The influence of pipeline length and air flow rate on the empty pipeline 2.2.1.2 The Influence of Pipeline Length From Equation 3 it will be seen that pressure drop is directly proportional to pipeline length. Typical values of pressure drop for a 6 in bore pipeline are given in Figure 6.4. This is a plot of conveying line pressure drop for the air against the air mass flow rate. Pipeline lengths of 500, 1000 and 1500 ft have been considered. Conveying line exit air velocity values are also given on the air flow rate axis of Figure 6.4. This clearly shows the adverse effect of velocity, and hence air flow rate, on pressure drop. It also shows that if a material has to be conveyed over a long distance, the proportion of the total system pressure drop due to the air only in the pipeline could be very significant. 2.2.1.3 The Influence of Pipeline Bore From Equation 3 it will also be seen that pressure drop is inversely proportional to pipeline bore. Typical values of conveying line pressure drop for a 500 ft long pipeline are given in Figure 6.5. This is a similar plot to that of Figure 6.4. The air mass flow rate axis is proportional to pipe section area, hence the (d/3) 2 term, and so conveying line exit air velocities are constant in each case. It can be clearly seen from this plot that the air only pressure drop reduces with increase in pipeline bore. If an air mover with a pressure limitation, such as a positive displacement blower, has to be used to convey a material over a long distance, therefore, it should be possible to achieve reasonably high flow rates with a large bore pipeline.
188 5 i_ I 3 22 Q Conveying Line Exit Air Velocity - ft/min 20 25 1000 2000 3000 4000 5000 6000 7000-4- 5 10 15 Air Mass Flow Rate - Ib/min x (d/3) 2 Figure 6.5 The influence of pipeline bore on the empty pipeline pressure drop. 2.2.2 Bends The pressure drop for bends in a pipeline can be expressed in terms of the 'velocity head': Ap = k x where k pc -.2 2 g c (8a) = the number of velocity heads lost for the particular bend geometry and configuration If the pressure drop, Ap, is to have units of lbf/in 2, the other parameters will be as follows: k = constant - dimensionless p = air density - lb/ft 3 C = air velocity -ft/min g c = gravitational constant - ft Ib/lbf s 2 and the bend loss equation will appear as follows: 4? = k 1,036,800 g c lbf/in" (8b)
20 Ratio of D/d Figure 6.6 Head loss for 90 degree radiused bends. The pressure loss in such a bend will depend upon the ratio of the bend diameter, D, to the pipe bore, d, and the surface roughness. Typical values are given in Figure 6.6 [3]. Data for radiused bends, showing the influence of bend angle, is presented in Figure 6.7. 0-3 0-2 T- 0-1 - 0 0 30 Angle 90 Figure 6.7 Head loss for radiused bends.
1-2 1-0 0-8 o u 0-6 0-4 0-2 0 Figure 6.8 30 60 Angle - 0 - degrees Head loss for mitered or sharp angle bends. From Figure 6.6 it can be seen that very short radius bends will add significantly to the empty line pressure drop. Minimum pressure drop occurs with bends having a D/d ratio of about 12. This is not a critical value, however, for a reasonably low value of head loss will be obtained with a D/d range from about 5 to 40. A similar plot for sharp angled or mitered bends is given in Figure 6.8 [3]. This shows that the mitered bend will result in the highest value of air only pressure drop for a ninety degree bend, particularly for smooth pipes. In terms of pressure drop, therefore, such bends should be avoided. 2.2.2.1 Equivalent Length The head loss for straight pipeline, as will be seen from Equation 3a, is given by 4 fl d The equivalent length of straight pipeline, L c, of a bend, with a head loss of k, will therefore be: kd (9)
Taking a typical pipeline friction coefficient, / of 0-005, the equivalent length of a 6 in bore 90 mitered bend of smooth pipe, for which k= 1-1, will be about 27-5 ft. If there are a number of such bends in a short pipeline, the bends will add significantly to the total air only pressure drop value. 2.2.3 Other Pipeline Features Other pipeline features, such as branches and section changes, are treated in exactly the same way as any of the above pipeline bends, with the use of Equation 8b. In Figures 6.9 to 6.11 similar head loss values are given for various pipeline fittings. 2.2.3.1 Expansion Fittings Expansion fittings are required in stepped pipelines, where the diameter of a line is increased part way along its length in order to reduce the conveying air velocity. Figure 6.9 shows that the air only pressure drop will be a minimum if a tapered section is used having an included angle of about six degrees. Expansion and contraction sections often occur in association with pipeline feeding systems such as rotary valves and screws. In venturi feeders the expansion section is an integral part of the design. Where expanded bends are fitted into a pipeline both expansion and contraction section are required. At the discharge from a pipeline into a reception vessel the expansion is effectively infinite. Figures 6.9 and 6.10 show the importance of careful design in such devices. 30 60 90 180 Total Angle - 6 - degrees Figure 6.9 Head loss for enlarging pipeline sections.
Abrupt Entrance k = 0-5 Abrupt exit k = 1-0 l/y i /l i 111 I 11 I I / Gradual Entrance k = 0-05 Gradual Exit k K = 0-2 u-z y II III 11II I 111 11 \\ \\\ \\\\\ V\AA\ \J^ Figure 6.10 Entrance and exit head losses. The head loss for various diverter sections, fabricated bends and 'dog-leg' sections, that are often used in air supply and exhaust pipelines, are given in Figure 6.11. A comparison of the two 'dog-leg' sections shows just how important careful pipeline design and layout are in minimizing pressure drop. k = 3-0 k = 040 smooth 0-53 rough k = 1-0 k = 0-40 smooth = 0-60 rough = 0-16 smooth = 0-30 rough 30" Figure 6.11 Head loss for various pipe fittings.
2.2.4 Total Pipeline The pressure drop for the total pipeline system is simply given by a summation of all the component pressure drop values, so that: ^ ^21,600 d l,036,800j g c where 2k = the sum of the head losses for all the bends and fittings in the pipeline Substituting p from Equation 1 and C from Equation 2 gives: ( f L 2 k\ ml R T {9-375 d tjuy n a' p g c For convenience the head loss for the pipeline, bends and fittings can be grouped together using the term y/, such that: fl 9-375 d S k (dimensionless) - - - (11) Substituting and integrating, as with Equation 4, gives: ' ' pl '- 2i// m 2 R T This can be used to obtain the air only pressure drop in any pipeline situation. 2.2.4.1 Positive Pressure Systems For a positive pressure system p 2 will be specified, as mentioned earlier in connection with Equation 6, and so a more useful form of Equation 12 is: I m 2 D r\ ' 5 a R T] P2 + - r~t4- - P2 lbf/i " 2 - - (13) " Sc ) For air R = 53-3 ft Ibf/lb R and if T = 519 R
this gives: and taking p 2 substituting g, and for jf 14-7 Ibf/in" (atmospheric pressure) 32-2 ft Ib/lbf s 2 216-1 + 174^ ml 14-7 M/in 2 - - (14) 1-0 + xo-5 x 10 5-1-0 bar (14SI) In many cases a value of the conveying line exit air velocity, C 2, can be determined, by using Equation 11, for example. A substitution of C? for m a can be made from Equation 2: m,. n d 2 C 2 p 2 4 R T Ib/min (15) Substituting this into Equation 12 gives: Pi ~ Pi from which: t~i~> ~> Q p; 8 g c (16), 0-5 Ap 8 R T 2 g c Ibf/in 2 (17) Thus in a situation where the downstream pressure, p 2, is known (commonly this would be atmospheric pressure in a positive pressure system) and the conveying line exit air velocity can be determined, this expression allows the pressure drop for the air alone to be estimated quite easily. Alternatively, if the conveying line inlet air velocity, C/, is known, this can be used instead. A substitution of C, for m a, from Equation 2, in Equation 12 gives:
P\ ~ P\ R T t g c (18) from which: Ap a = 8 R T, g c.0-5 g c - -1 lbf/in 2 - (19) Note: The velocity, C t, in Equations. 18 and 19, is not the conveying line inlet air velocity that is specified for gas-solid flows in pneumatic conveying. It is the conveying line inlet air velocity that will result when no material is conveyed. C 2 in Equations 16 and 17, of course, is the same whether material is conveyed or not, since the pressure will always be the same at the end of the pipeline. 2.2.4.2 Negative Pressure Systems For a negative pressure system, p/, will be specified (usually atmospheric). A rearrangement of Equation 18 gives: 0-5 1-1 - Ibf/in 2 (20),0-5 = Pi 1 - R r, gc N/m 2 - (20si) Note: In this case the conveying line inlet air velocity, C/, will be the same whether the material is conveyed or not, since the pressure, p t, will be atmospheric in both cases. This is similar to Equations 16 and 17 for positive pressure systems. 2.3 Air Only Pressure Drop Datum The empty pipeline pressure drop relationships for a pipeline, such as those shown in Figures 6.4 and 6.5, provide a datum for material conveying characteristics and capability. At a given value of air flow rate the pressure drop available must be greater than the air only pressure drop value, otherwise it will not be possible to convey material.
At any value of conveying line pressure drop there will be a corresponding value of air flow rate at which the air only pressure drop will equal the conveying line pressure drop. This value can be determined from Equation 13 by making m a the subject of the equation. Such a re-arrangement gives: m. 2 R T Ib/min (21) (PI 16 i/ RT 0-5 kg/s - - (21 si) This is quite a useful relationship, for it allows an estimate to be made of where the various lines of constant conveying line pressure drop on material conveying characteristics will reach the horizontal axis. For air R = 53-3 ft Ibf/lb R and if T = 519 R and with g, = 32-2 ftlb/lbfs 2 0-5 m,. = 0-0758 P\ Pi Ib/min (22) 2.4 Venturi Analysis Particular advantages of using venturi feeders for positive pressure conveying lines are that minimum headroom is required, there are no moving parts and, if the device is correctly designed, there need be no air leakage from the feeder, as there is with nearly all other types of feeder. A venturi basically consists of a controlled reduction in pipeline cross-section in the region where the material is fed from the supply hopper, as shown in Figure 6.12 and first presented in Figure 2.12. A consequence of this reduction in flow area is an increase in the entraining air velocity, and a corresponding decrease in pressure, in this region. With a correctly designed venturi the pressure at the throat should be just a little lower, or about the same, as that in the supply hopper which, for the majority of applications, is atmospheric pressure. This then encourages the material to flow readily under gravity into the pipeline, and under these conditions there will be no leakage of air from the feeder in opposition to the material feed.
Air and Material Inlet (i) Throat (t) Figure 6.12 Basic type of venturi feeder. In order to keep the throat at atmospheric pressure, and also of a practical size that will allow the passage of material to be conveyed, a relatively low limit has to be imposed on the air supply pressure. These feeders, therefore, are usually incorporated into systems that are required to convey free-flowing materials at low flow rates over relatively short distances. Since only low pressures can be used with the basic type of venturi, a positive displacement blower or a standard industrial fan is all that is needed to provide the air. To fully understand the limitations of this type of feeder, the thermodynamic relationships are presented below. The two parameters of interest in venturi feeders are the velocity at the throat and the area, or diameter, of the throat. From the steady flow energy equation, equating between the inlet (i) and the throat (t) gives: Cp T, + C 2 CpT,+ 2 gc (23) from which: c, = 2 g e c p (T, - T,) + c? 0-5 (24a) If the velocities, C, and C,, are to have units of ft/min, and the other parameters in the equation are as follows: Cp = specific heat - Btu/lb R T = absolute temperature - R and g c = gravitational constant - ft Ib/lbf s 2
= 32-2 ftlb/lbfs 2 and noting that 1 Btu = 778 ft Ibf the equation will appear as follows: C t = [l80 Cp(T,-T,)xl0 6 +CfJ 5 ft/mm - - (24b) If an isentropic model of expansion is assumed for the venturi then: Pt - = I T, \ P, Note that this appeared earlier in Equation 3.3 (25) Substituting Equation 25 into Equation 24b gives: C t = \\WCpT, 0-5 6 2 x!0 6 +C ft/min - (26) From the continuity equation: m a = Pi AjCj = p,a,c, Ib/min where A = section area n d 2 = 4 - in d = diameter - in and p = density of gas - lb/ft 3 P RT (27) (1) Substituting Equation 1 into Equation 27 gives: x - x - x d, P, in (28) Substituting Equation 25 into Equation 28 gives:
d t = C ' Xx 1 c, 1 ; \ EL P, } 1 7 -i 0-5 d, in (29) If, for example, C, = 4000 ft/min dj = 4 in Ti = 528 R = 68 F p, = 14-7 Ibf/in 2 abs p, = 3 lbf/in 2 gauge = 17-7 lbf/in 2 abs note that for air Cp = 0-24 Btu/lb and y = 1-4 substituting into Equation 48 gives: C, = j 180x0-24x528 14-7 17-7 0-286 xlo 6 +4000 2,0-5 = 34,582 ft/min and substituting into Equation 29 gives: d, = 4000 34,582 14-7 17-7 -0-714 0-5 x 4 = 1-45 in Although Venturis capable of feeding materials into conveying pipelines with operating pressure drops of 6 lbf/in 2 are commercially available, the additional pressure drop across the venturi can be of the same order. This means that the air supply pressure will have to be at about 12 lbf/in 2 gauge and consequently, for this type of duty, it would be recommended that the air should be supplied by a positive displacement blower. 3 AIR FLOW RATE CONTROL If the air to be used for conveying is taken from a plant air supply, or some central source, it will probably be necessary to put a flow restriction into the pipeline. This will be needed in order to limit the quantity of air drawn to that of the volumetric
flow rate actually required. If this is not done an uncontrolled expansion will occur and very much more air than necessary will be used. It will only be limited by the volumetric capability of the supply, or by the increased factional resistance of the flow in the pipeline. The increased air flow rate will almost certainly result in a decrease in the material flow rate through the pipeline. It will also add significantly to problems of erosive wear and particle degradation. Flow restrictors may also be required in situations where the air supply needs to be divided, as in blow tank systems. For the control of many types of blow tank it is necessary to proportion the air supply between the blow tank and the conveying line. If the total air supply is set, a flow restrictor can be placed in one or both of the divided lines. This, however, can only be done if the blow tank is dedicated to a single material conveyed over a fixed distance. For systems handling more than one material, or conveying to a number of hoppers over varying distances, a variable flow control might be needed. In these cases special control valves would be required rather than fixed restrictors. Nozzles and orifice plates are most commonly used for restricting the air flow in a pipeline. Under certain flow conditions they can also be used to meter and control the air flow. 3.1 Nozzles For the single phase flow of fluids through nozzles the theory is well established, and for a gas such as air it is based on the use of many of the equations already presented. Nozzles are either of the convergent-divergent type, as shown in Figure 6.13a, or are convergent only, as shown in Figure 6.13b. Both types restrict the flow by means of a short throat section at a reduced diameter. 2 ) Direction (b) Figure 6.13 in pipeline. Nozzle types, (a) Convergent - divergent nozzle and (b) convergent nozzle
3.7.7 Flow Analysis Assuming a steady one-dimensional flow, and equating the steady flow energy equation between inlet (1) and throat (t) gives: CpT, C2 = Cp T t + 2 g c 2 g, Neglecting the inlet velocity, Ci, and re-arranging gives: (23) c, = 0-5 T, g c C P TA\ - -± T,. (30a) For consistency in units, constants of 778 and 3600 have to be applied, as with Equation 24, and this yields: C t = 13,430 T. CpTAl - ^ 0-5 ft/min (30b) Assuming isentropic flow, for which Equation 25 applies, the unknown temperature at the throat, T,, can be expressed in terms of the pressure at the throat, p,. Such a substitution gives: C, = 13,430 CpT, ft/min (31) Also for isentropic flow: v t = v, x fr/lb \P t ) - - (32) where v = specific volume - ft 3 /lb Now, from the Ideal Gas Law (Equation 5.4): 144 Pl v, = R T, and substituting this into Equation 32 gives:
R T } 144 Pl (33) From the continuity equation (Equation 27): A, C, A, C, lb/min Substituting C, from Equation 31 and v ( from Equation 33 into this gives: 4 x!3,430x- CpT, -te) T _ y-\ m,. RT 144 p, p, (pvr lb/min - (34) Re-arranging this gives: 0-5 T; lb/min - (35) where d, = nozzle throat diameter - in 3.1.2 Critical Pressure A peculiarity of the expansion of the flow of a fluid through a nozzle is that as the downstream pressure, p 2, reduces, for a given upstream pressure, p h the pressure at the throat,/),, will not reduce constantly with downstream pressure. The pressure at the throat will reduce to a fixed proportion of the inlet pressure, and any further reduction of the downstream pressure will not result in a lowering of the pressure at the throat. Under these conditions the nozzle is said to be 'choked'. When critical flow conditions exist, the velocity at the throat will be equal to the local sonic velocity. The air mass flow rate through a nozzle is a maximum under choked flow conditions and no reduction of the downstream pressure, below the critical throat pres-
sure, will result in any change of the air mass flow rate. It can be shown [eg 4] that the ratio between the throat pressure and the supply or inlet pressure is given by: (36) and so For air y = 1-4 R = 53-3 ftlbf/lbr and Cp = 0-24 Btu/lb R = 0-528 P\ 3.1.3 Nozzle Size and Capability Substituting the above data for air into Equation 35 gives: m a = 25-1 jj- Ib/min (37) -'i where p/ = inlet or supply pressure - Ibf/in 2 abs For the air flow rate in volumetric terms, Equation 5.4 gives: m a RT V = 144 p flrvmin For the volumetric flow rate at free air conditions: 0 V 0 = 0-1743 x x - ftvmin ---- (38) and substituting for R and free air conditions gives: ^1 Pa P^ d? V 0 = 328 ^ ft-vmin -... (39) Alternatively, for a given air flow rate:
m,. -05 0-5 d t = 0-2 in (40) The relationship between d,, pi and both m a and V 0, for air at a temperature, t,, of 68 F (T, = 528) is given in Figure 6.14. 3.1.4 Nozzle Types The above analysis applies to either convergent-divergent or to convergent nozzles. For convergent nozzles, however, the range of operation is limited to downstream pressures that are less than 52-8% of the upstream pressure, that is, below the critical pressure ratio. With convergent-divergent nozzles this range can be extended significantly, and for a well made nozzle the downstream pressure can be as high as 90% of the upstream pressure, with little deviation from the predicted flow rate. 3.1.4.1 Orifice Plates These are frequently used for measuring the flow rate of gases through pipelines but can also be used to choke the flow and so apply a limit to the throughput. The orifice is generally made from thin plate that is usually fitted into a flanged joint in the pipeline. It has a sharp edged opening which is concentric with the pipe. PlOO I 90 80 Nozzle Thrc(at Diameter -inch 70 3 /4 60 i fr/ntjn of Free Air 200 300 400 20 30 Air Flow Rate - Ib/min 500 600 40 50 Figure 6.14 rate for nozzles. Influence of throat diameter and air supply pressure on choked air flow
The above analysis also applies to orifice plates. There is, however, a coefficient of discharge associated with orifice plates and this has the effect of reducing the flow rate to about 61 % of the theoretical value. This means that the constants in Equations 34 to 39 would have to be multiplied by a factor of 0-61 and the constant in Equation 40 would have to be divided by VO- 61 to take account of this coefficient of discharge. As with the convergent nozzle, the range of operation is limited to downstream pressures below the critical pressure ratio. 3.1.5 Flow Rale Control It will be seen from Figure 6.14 that, for a given nozzle, the air flow rate can be varied over a wide range simply by varying the air supply pressure. In a pipeline from a service supply, a diaphragm valve could be positioned upstream of the flow restrictor, and this could be used to vary the inlet pressure and hence the air flow rate. Provided that critical flow conditions exist, only the inlet air pressure and temperature, and the throat diameter, are needed to evaluate the air flow rate, as will be seen from Equation 37. It will be noticed that, apart from including a representative coefficient of contraction for orifices, no other coefficients have been included in the analysis to allow for friction and other irreversibilities in the flow. For most pneumatic conveying applications it will not be necessary, as these losses are generally quite small. If these devices are to be used for flow measurement purposes, however, with a need for a high degree of accuracy, either the loss factors will have to be taken into account or the device will have to be calibrated. NOMENCLATURE SI A Pipe section area in 2 m 2 C Velocity ft/min m/s Cp Specific heat at constant pressure Btu/lb R kj/kg K Cv Specific heat at constant volume Btu/lb R kj/kg K d Pipe bore in m / Friction coefficient g Gravitational acceleration ft/s 2 m/s 2 = 32-2 ft/s 2 = 9-81 m/s 2 g c Gravitational constant ftlb/lbfs 2 kgm/ns 2 = 32-2 ftlb/lbfs 2 = 1-0 kgm/ns 2 k Bend loss coefficient L Pipeline length ft m m Mass Ib kg m Mass flow rate Ib/min kg/s
p Pressure lbf/in 2 R Characteristic gas constant Btu/lb R / Actual temperature F T Absolute temperature R = t + 460 V Volume ft 3 V Volumetric flow rate ftymin Greek y Ratio of specific heats = Cp/Cv (adiabatic index) 8 Pipe wall roughness in // Viscosity Ib/ft h v Specific volume ftvlb = \lp p Density Ib/ft 3 if/ Total pipeline head loss coefficient N/m 2, kn/m 2, bar (1 bar= 100 kn/m 2 ) kj/kg K C K = t H m 3 m 3 /s m kg/m s nv/kg kg/m j 273 Subscripts a c e i t o Air Constant Equivalent value - usually length Inlet conditions Throat conditions Reference conditions (free air) Po = 14-7 lbf/in 2 abs = 101-3 kn/m 2 abs T n = 519 R = 288 K 1,2 Actual conditions - usually inlet and outlet Superscripts Per unit of time ie /min Repeating value, eg 1/3 = 0-3 Prefixes A E Difference in value Sum total Non-Dimensional Groups Re Reynolds Number 5 p C d p C d
REFERENCES 1. Y.R. Mayhew and G.F.C. Rogers. Thermodynamic and Transport Properties of Fluids. Basil Blackwell. 1968. 2. J.M. Gasiorek and W.G. Carter. Mechanics of Fluids for Mechanical Engineers. Blackie and Son. 1967. 3. J.R.D. Francis. Fluid Mechanics for F^ngineering Students - 4 th Ed. Edward Arnold. 1975. 4. V.M. Faires. Applied Thermodynamics - 3 rd Ed. MacMillan. 1957.