Gas Pipeline Hydraulics: Pressure Drop

Transcription

1 COURSE NUMBER: ME 423 Fluids Engineering Gas Pipeline Hydraulics: Pressure Drop Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1

2 FLOW EQUATIONS Several equations are available that relate the gas flow rate with gas properties, pipe diameter and length, and upstream and downstream pressures. These equations are listed as follows: 1. General Flow equation 2. Colebrook-White equation 3. Modified Colebrook-White equation 4. AGA equation 5. Weymouth equation 6. Panhandle A equation 7. Panhandle B equation 8. IGT equation 9. Spitzglass equation 10. Mueller equation 11. Fritzsche equation Acomparison of these equations will also be discussed using an example pipeline. 2

3 The General Flow equation, also called the Fundamental Flow equation, for the steady-state isothermal flow in a gas pipeline is the basic equation for relating the pressure drop with flow rate. The most common form of this equation in the U.S. Customary System (USCS) of units is given in terms of the pipe diameter, gas properties, pressures, temperatures, and flow rate as fll follows. It must be noted tdthatt for the pipe segment from section 1 to section 2, the gas temperature T f is assumed to be constant (isothermal flow). In SI units, the General Flow equation is stated as follows: 3

4 Sometimes the General Flow equation is represented in terms of the transmission factor F instead of the friction factor f. This form of the equation is as follows. where the transmission factor F and friction factor f are related by and in SI units 4

5 When there is elevation difference between the ends of a pipe segment, the General Flow equation needs further modification. AVERAGE PIPE SEGMENT PRESSURE In the General Flow equation, the compressibility factor Z is used. This must be calculated ltdat the gas flowing temperaturet and average pressure in the pipe segment. Therefore, it is important to first calculate the average pressure in a pipe segment. Consider a pipe segment with upstream pressure P 1 and downstream pressure P 2. An average pressure for this segment must be used to calculate the compressibility factor of gas at the average gas temperature T f. As a first approximation, we may use an arithmetic average of (P 1 +P 2 )/2. However, it has been found that a more accurate value of the average gas pressure in a pipesegment pp is Another form of the average pressure in a pipe segment is 5

6 VELOCITY OF GAS IN A PIPELINE Unlike a liquid pipeline, due to compressibility, the gas velocity depends upon the pressure and, hence, will vary along the pipeline even if the pipe diameter is constant. The highest velocity will be at the downstream end, where the pressure is the least. Correspondingly, the least velocity will be at the upstream end, where the pressure is higher. Consider a pipe transporting gas from point A to point B. Under steady state flow, at A, themassflowrateofgas is designated as M and will bethesame as the mass flow rate at point B, if between A and B there is no injection or delivery of gas. We can write the following relationship for point A: The volume rate Q can be expressed in terms of the flow velocity u and pipe cross sectional area A as follows: Therefore, combining i Equation 2.16 and Equation 2.17 and applying li the conservation of mass to points A and B, we get If the pipe is of uniform cross section between A and B, the velocities at A and B are related by the following equation: 6

7 Since the flow of gas in a pipe can result in variation of temperature from point A to point B, the gas density will also vary with temperature and pressure. If the density and velocity at one point are known, the corresponding velocity at the other point can be calculated using Equation If inlet conditions are represented by point A and the volume flow rate Q at standard conditions (60 F and 14.7 psia) are known, we can calculate the velocity at any point along the pipeline at which the pressure and temperature of the gas are P and T, respectively. The mass flow rate M at section 1 and 2 is the same for steady-state state flow. Therefore, where Q b is the gas flow rate at standard conditions and ρ b is the corresponding gas density. Therefore, simplifying Equation 2.20, Applying the gas law, we get or 7

8 where P 1 and T 1 are the pressure and temperature at pipe section 1. Similarly, at standard conditions, 8

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10 EROSIONAL VELOCITY As flow rate increases, so does the gas velocity. How high can the gas velocity be in a pipeline? As the velocity increases, vibration and noise are evident. In addition, higher velocities will cause erosion of the pipe interior over a long period of time. The upper limit of the gas velocity is usually calculated approximately from the following equation: 10

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13 In the preceding Examples 1 and 2, we have assumed the value of compressibility factor Z to the constant. t A more accurate solution will be to calculate the value of Z using CNGA or Standing-Katz method. The inlet and outlet gas velocities then will be modified. 13

14 REYNOLDS NUMBER OF FLOW The Reynolds number is a function of the gas flow rate, pipe inside diameter, and the gas density and viscosity it and is calculated ltdfrom the following equation: In gas pipeline hydraulics, using customary units, a more suitable equation for the Reynolds number is as follows: In SI units, the Reynolds number is 14

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16 COMPARISON OF FLOW EQUATIONS In Figure 2.5, we consider a pipeline 100 mile long, NPS 16 with in. wall thickness, operating at a flow rate of 100 MMSCFD. The gas flowing temperature is 80 F. With the upstream pressure fixed at 1400 psig, the downstream pressure was calculated using the different flow equations. By examining Figure 2.5, it is clear that the highest pressure drop is predicted by the Weymouth equation and the lowest pressure drop is predicted by the Panhandle B equation. It must be noted that we used a pipe roughness of 700 µin. for both the AGA and Colebrook equations, whereas a pipeline efficiency of 0.95 was used in the Panhandle and Weymouth equations. 16

17 Figure 2.6 shows a comparison of the flow equations from a different perspective. In this case, we calculated the upstream pressure required for an NPS 30 pipeline, 100 miles long, hldi holding the dli delivery pressure constant tat 800 psig. The upstream pressure required for various flow rates, ranging from 200 to 600 MMSCFD, was calculated using the five flow equations. Again it can be seen that the Weymouth equation predicts the highest upstream pressure at any flow rate, whereas the Panhandle A equation calculates the least pressure. We therefore conclude that the most conservative flow equation that predicts the highest pressure drop is the Weymouth equation and the least conservative flow equation is Panhandle A. 17

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25 The wall thickness required for this pipe diameter and pressure will be dictated by the pipe material 25

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30 EQUIVALENT LENGTH METHOD: Pipes in Series The equivalent length method can be applied when the sameuniform flow exists throughout the pipeline consisting of pipe segments of different diameter, with no intermediate deliveries or injections. Consider the same flow rate Q through all pipe segments. The first pipe segment has an inside diameter D 1 and length L 1,followedbythesecond segment of inside diameter D 2 and length L 2 and so on. We calculate the equivalent length of the second pipe segment based on the diameter D 1 such that the pressure drop in the equivalent length matches that in the original pipe pp segment of diameter D 2. The pressure drop in diameter D 2 and length L 2 equals the pressure drop in diameter D 1 and equivalent length Le 2. Thus, the second segment can be replaced with a piece of pipe of length Le 2 and diameter D 1. Similarly, the third pipe segment with diameter D 3 and length L 3 will be replaced with a piece of pipe of Le 3 and diameter D 1. Thus, we have converted the three segments of pipe in terms of diameter D 1 as follows: 30

31 We now have the series piping system reduced to one constant-diameter (D 1 ) pipe of total equivalent length given by For the same flow rate and gas properties, neglecting elevation effects, the pressure difference (P 12 P 22 ) is inversely proportional to the fifth power of the pipe diameter and directly proportional to the pipe length. Therefore, we can state that, approximately, From Equation 3.2 we conclude that the equivalent length for the same pressure drop is proportional to the fifth power of the diameter. Therefore, in the series piping discussed in the foregoing, the equivalent length of the second pipe segment of diameter D 2 and length L 2 is or 31

32 Similarly, for the third pipe segment of diameter D 3 equivalent length is and length L 3,the Therefore, the total equivalent length Le for all three pipe segments in terms of diameter D 1 is It can be seen from Equation 3.6 that if D 1 =D 2 =D 3, the total equivalent length reduces to (L 1 +L 2 +L 3 ), as expected. 32

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36 PARALLEL PIPING Sometimes two or more pipes are connected such that the gas flow splits among the branch pipes and eventually combines downstream into a single pipe, as illustrated in Figure 3.7. The reason for installing parallel pipes or loops is toreduce pressure drop inacertain section of the pipeline due to pipe pressure limitation or for increasing the flow rate in a bottleneck section. By installing a pipe loop from B to E, in Figure 3.7 we are effectively reducing the overall pressure drop in the pipeline from A to F, since between B and E the flow is split through twopipes. pp Applying the principle of flow conservation, at junction B, the incoming flow into B must exactly equal the total outflow at B through the parallel pipes. Therefore, at junction B, 36

37 Where, Q = inlet flow at A, Q 1 = flow through pipe branch BCE and Q 2 = flow through pipe branch BDE. Both pipe branches have a common starting point (B) and common ending point (E). Therefore, the pressure drop in the branch pipe BCE and branch pipe BDE are each equal to (P B P E ), where P B and P E are the pressuresat junctions B and E, respectively. Therefore, we can write The pressure drop due to friction in branch BCE can be calculated from Where, K 1 =aparameter that depends on gas properties, gas temperature, etc., L 1 = length of pipe branch BCE, D 1 = inside diameter of pipe branch BCE and Q 1 = flow rate through pipe branch BCE. Similarly, the pressure drop due to friction in branch BDE is calculated from In Eq and Eq. 3.11, the constants K 1 and K 2 are equal, since they do not depend on the diameter or length of the branch pipes BCE and BDE. 37

38 Combining both equations, we can state the following for common pressure drop through each branch: Simplifying further, we get the following relationship between the two flow rates Q 1 and Q 2 : In equivalent diameter method, we replace the pipe loops BCE and BDE with a certain length of an equivalent diameter pipe that has the same pressure drop as one of the branch pipes. Since the pressure drop in the equivalent diameter pipe, which flows the full volume Q, is the same as that in any of the branch pipes, from Eq. 3.10, we can state the following: where Q = Q 1 +Q 2 from Equation 3.7 and K e represents the constant for the equivalent diameter pipe of length L e flowing the full volume Q. We get, using Eq. 3.10, Eq. 3.11, and Eq. 3.14: 38

39 Setting K 1 =K 2 =K e and L e =L 1, we simplify Equation 3.15 as follows: Using Equation 3.16 in conjunction with Equation 3.7, we solve for the equivalent diameter D e as Where and the individual flow rates Q 1 and Q 2 are calculated from and 39

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43 Solving for the outlet pressure at E, we get P 2 = psia, which is almost thesameaswhatwecalculatedbefore. Therefore, using the equivalent diameter method, the parallel pipes BCE and BDE can be replaced with a single pipe 24 mi long, having an inside diameter of in. 43

44 LOCATING PIPE LOOP How do we determine where a loop should be placed for optimum results? Should it be located upstream, downstream, or in a midsection of the pipe? Three looping scenarios are presented in Figure 3.8. In case (a), a pipeline of length L is shown looped with X miles of pipe, beginning at the upstream end A. In case (b), the same length X of pipe is looped, but it is located on the downstream end B. Case (c) shows the midsection of the pipeline being looped. 44

45 For most practical purposes, we can say that the cost of all three loops will be the same as long as the loop length is the same. Which of these cases is optimum,? It is found that if the gas temperature is constant throughout, at locations near the upstream end, the pressure drops at a slower rate than at the downstream end. Therefore, there is more pressure drop in the downstream section compared to that in the upstream section. Hence, to reduce the overall pressure drop, the loop must be installed toward the downstream end of the pipe. This argument is valid only if the gas temperature is constant throughout the pipeline. In reality, due to heat transfer between the flowing gas and the surrounding soil (buried pipe) or the outside air (above-ground pipe), the gas temperature will change along the length of the pipeline. If the gas temperature at the pipe inlet is higher than that of the surrounding soil (buried pipe), the gas will lose heat to the soil and the temperature will drop from the pipe inlet to the pipe outlet. If the gas is compressed at the inlet using a compressor, r then the gas temperature will be much higher than that of the soil immediately downstream of the compressor.the hotter gas will cause higher pressure 45

46 drops. Hence, in this case the upstream segment will have a larger pressure drop compared to the downstream segment. Therefore, considering heat transfer effects, the pipe loop should be installed in the upstream portion for maximum benefit. The installation of the pipe loop in the midsection of the pipeline, as in case (c) in Figure 3.8, will not be the optimum location, based on the preceding discussion. It can therefore be concluded that if the gas temperature is fairly constant along the pipeline, the loop should be installed toward the downstream end, as in case (b). If heat transfer is taken into account and the gas temperature varies along the pipeline, with the hotter gas being upstream, the better location for the pipe loop will be on the upstream end, as in case (a). HYDRAULIC PRESSURE GRADIENT Since pressure in a gas pipeline is nonlinear compared to liquid pipelines, the hydraulic gradient for a gas pipeline appears to be a slightly curved line instead of a straight line. The slope of the hydraulic gradient at any point represents the pressure loss due to friction per unit length of pipe. This slope is more pronounced as we move toward the downstream end of the 46

47 pipeline, since the pressure drop is larger toward the end of the pipeline. If there are intermediate deliveries or injections along the pipeline, the hydraulic gradient will be a series of broken lines, as indicated in Figure A similar broken hydraulic gradient can also be seen in the case of a pipeline with variable pipe diameters and wall thicknesses, even if the flow rate is constant. Unlike liquid pipelines, the breaks in hydraulic pressure gradient are not as conspicuous in gas pipelines. In a long-distance gas pipeline, due to limitations of pipe pressure, intermediate compressor stations will be installed to boost the gas pressure to the required value so the gas can be delivered at the contract delivery pressure at the end of the pipeline. 47

48 PRESSURE REGULATORS AND RELIEF VALVES In a long-distance gas pipeline with intermediate delivery points, there may be a need to regulate the gas pressure at certain delivery points in order to satisfy the customer requirements. Suppose the pressure at a delivery point is 800 psig, whereas the customer requirement is only 500 psig. Obviously, some means of reducing the gas pressure must be provided so that the customer can utilize the gas for his or her requirements at the correct pressure. This is achieved by means of a pressure regulator that will ensure a constant pressure downstream of the delivery point, regardless of the pressure on the upstream side of the pressure regulator. This concept is further illustrated using the above example. The main pipeline from A to C has a branch pipe BE. The flowrate from A to B is 100 MMSCFD, with an inlet pressure of 1200 psig at A. At B, gas is delivered into a branch line BE at the rate of 30 MMSCFD. 48

49 The remaining volume of 70 MMSCFD is delivered to the pipeline terminus C at a delivery pressure of 600 psig. Based on the delivery pressure requirement of 600 psig at C and a takeoff of 30 MMSCFD at point B, the calculated pressure at B is 900 psig. Starting with 900 psig on the branch line at B, at 30 MMSCFD, gas is delivered to point E at 600 psig. If the actual requirement at E is only 400 psig, a pressure regulator will be installed at E to reduce the delivery pressure by 200 psig. 49

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54 LINE PACK The quantity of gas contained within the pipeline pp under pressure, measured at standard conditions (generally 14.7 psia and 60 F), is termed the line pack volume. Consider a segment of pipe, of length L, with upstream pressure and temperature of P1 and T1 and downstream values of P2 and T2, respectively. Suppose the inside diameter of the pipe is D; then the physical volume of the pipe section is This volume is the gas volume at pressures and temperatures ranging from P1, T1 at the upstream end to P2, T2 at the downstream end of the pipe length L. In order to convert this volume to standard conditions of pressure, Pb, and temperature,tb, we apply the gas law Equation as follows: 54

55 From Equation 3.31, solving for line pack Vb at standard conditions, we get Substituting the value ofvp from Equation 3.30 and simplifying, we get Equation 3.33 is modified in terms of commonly used units as follows: Where, equation in SI units is Vb is in standard ft 3, D in inch and L in mi. The corresponding Where, Vb is in standard m 3, D in mm and L in km. 55

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