January 2000

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Trey Walters, P.E. Applied Flow Technology

A Publication of Chemical Week Associates CoverCover Story Story -flowGas-flow Calculations:Calculations:

Trey Walters, P.E. Don'tDon't ChokeChoke Applied Flow Technology Assuming simplifies the math, when can be ignored. : Equation (1) is but introduces error. Always know how much not strictly applicable to compress- ible flow because, as already noted, the density and velocity change along low of in pipe systems is velocities, compressible-flow methods the pipe. Sometimes, engineers apply commonplace in chemical-pro­ will clearly be required. In practice, Equation (1) to gas flow by taking the cess plants. Unfortunately, the many gas systems fall between these average density and velocity. But, be- design and analysis of gas-flow extremes, and it is difficult to assess cause the variation of each of these systems are considerably more the error that will result from using parameters along a pipe is nonlinear, complicatedF than for liquid (incom- incompressible methods. the arithmetic averages will be incor- pressible) flow, due mainly to - A major purpose of this article is to rect. The difficult question — How induced variations in the gas-stream offer guidelines for assessing the im- seriously incorrect? — is discussed in density and velocity. Here, we review portance of compressibility effects in a detail later in this article. practical principles and present some given case. First, however, we set out Individual length of pipe: More strictly key equations governing gas flow, and relevant equations, and discuss some applicable than Equation 1 to gas flow assess several assumptions and rules key aspects of gas-flow behavior.1 in a pipe are Equations (2)–(6) [1–3], of thumb that engineers sometimes developed from fundamental fluid- apply in order to simplify gas-flow The underlying equations flow principles and generalized from analysis and calculations. Incompressible flow: An apt start- perfect gas equations [4] to apply to ing point for discussing gas flow is real gases: Compressible, incompressible an equation more usually applied to Mass: In a broad sense, the appropriate term liquids, the Darcy-Weisbach equation for gas flow is compressible flow. In a (see Nomenclature box, next page): (2) stricter sense, however, such flow can be categorized as either incompress- (1) : ible or compressible, depending on the (3) amount of pressure change the gas un- where f is the Moody factor, dergoes, as well as on other conditions. generally a function of Reynolds num- Energy: Accurately calculating truly com- ber and pipe roughness. This equation pressible flow in pipe systems, espe- assumes that the density, , is (4) cially in branching networks, is a for- constant. The density of a liquid is midable task. Accordingly, engineers a very weak function of pressure : often apply rules of thumb to a given (hence the substance is virtually (5) design situation involving gas flow, to incompressible), and density changes : decide whether the use of (simpler) in- due to pressure are ignored in compressible-flow calculations can be practice. The density varies more (6) justified. Such rules of thumb are help- significantly with tem-perature. In ful, but they can lead one astray when systems involving transfer, the Several things should be noted used without a full understanding of the density can be based on the about Equations (2)–(6): underlying assumptions. arithmetic average, or, better, the log • They assume that the pipe diameter Sometimes, the case is clear-cut. For mean . When the ap- is constant instance, if the engineer is designing propriate density is used, Equation (1) • They are applicable not only to indi- a near-atmospheric-pressure venti- can be used on a large majority of liq- vidual gases but also to mixtures, so lation system, with pressure drops uid pipe-flow systems, and for gas flow long as appropriate mixture proper- measured in inches of water, incom- 1. The quantitative compressible- and incom- ties are used pressible-flow results in this article were pressible-flow methods are perfectly obtained using, respectively, AFT Arrow and AFT • Equation (1) is a special case of the suitable. Conversely, for design or Fathom. Both are commercially available momentum equation, Equation (3). If software for pipe system modeling. A simplified specification of a pressure-relief sys- but highly useful utility program, the third term on the left-hand side tem that is certain to experience high Compressible Flow Estimator (CFE), was of the latter (commonly called the ac- developed specifically for this article, and was used in several cases. Nomenclature Flow chokes at exit into atmosphere or tank A cross-sectional flow area of a pipe s D diameter of a pipe T temperature, static

e pipe wall roughness T0 temperature, stagnation f friction factor V velocity Flow chokes at expansion in pipe area Ff, g, γ, T0 parameters in Equation (14) x length – F arithmetical average of F over z elevation computing section Z compressibility factor g acceleration (usually gravita- specific heat ratio tional) γ Flow chokes at resriction in pipe h , static θ angle from horizontal ρ density h0 enthalpy, stagnation L length of a pipe Subscripts M Mach number 1 Location 1 in pipe Figure 1. Any of these piping configu- • m mass flowrate 2 Location 2 in pipe rations can result in sonic choking P pressure i junction at which solution is sought P0 pressure, stagnation j junctions with pipes connecting Sonic choking R gas constant to junction i In almost all instances of gas flow in pipes, the gas accelerates along the length of the pipe. This behavior can celeration term) is neglected, the two nection are 1the same. 1 be understood from Equations (2), (3) equations become identical If gas streams of different composi- and (5). In Equation (3), the pressure

0.8 0.8 Mach/number • Equation (4), the energy equation, tion mix at a branch connection, a bal- falls off, due to friction. As the pressure 0.6 0.6 includes the conventional thermody- ance equation will also be needed for drops, the gas density will also drop namic enthalpy plus a velocity term each individual0.4 species present. Ad-0.4 (Equation [5]). According to Equation that represents changes in kinetic ditional discussionp stag/P stag inlet 0.2 of species balance 0.2 (2), the dropping density must be bal- energy. The sum of these two terms is can be found in Reference [3]. Use of anced by an increase in velocity to 0 0 known as the stagnation enthalpy (see these network-calculation0.20 0.4 0.6 principles 0.8 1 maintain mass balance. discussion of stagnation properties, is discussed in more detailx/L later. It is not surprising, then, that gas below). The thermodynamic enthalpy Besides the use of the basic equa- flow in pipelines commonly takes place is referred to as the static enthalpy tions set out above, gas-flow designs at velocities far greater than those for (even if it pertains to a moving fluid). and calculations also frequently in- liquid flow — indeed, gases often ap- Similarly, temperature in a non-stag- volve two concepts that are usually of proach sonic velocity, the local speed of nation context is referred to as static lesser or no importance with incom- sound. A typical sonic velocity for air temperature pressible flow: stagnation conditions, is 1,000 ft/s (305 m/s). • Equation (5), as shown, includes a and sonic choking. When a flowing gas at some location compressibility factor to correct the in the pipeline experiences a local ve- equation for real-gas behavior. Stagnation conditions locity equal to the sonic velocity of the In general, however, the real-gas prop- At any point in a pipe, a flowing gas gas at that temperature, sonic choking erties can instead be obtained from a has a particular temperature, pres- occurs and a forms. Such thermophysical property database sure and enthalpy. If the velocity of choking can occur in various pipe con- Piping networks: In situations in- the gas at that point were instanta- figurations (Figure 1). volving a gas-pipe network, Equations neously brought to zero, those three The first case, which can be called (2)–(6) are applied to each individual properties would take on new values, endpoint choking, occurs at the end of pipe, and boundary conditions be- known as their stagnation values and a pipe as it exits into a large vessel or tween the pipes are matched so that indicated in the equations of this ar- the atmosphere. In this situation, the mass and energy are balanced. The fol- ticle by the subscript 0. gas pressure cannot drop to match lowing equations describe this balance Three important stagnation condi- that at the discharge without the gas at any branch connection: tions can be calculated, for real as well accelerating to sonic velocity. A shock Mass balance: as ideal gases, from the velocity and wave forms at the end of the pipe, re- the specific heat ratio (ratio of specific sulting in a pressure discontinuity. (7) heat at constant pressure to that at The second case, which might be constant ) by Equations (9 a, b called expansion choking, crops up Energy balance: and c). As is frequently the case in gas when the cross-section area of the flow, the velocity is expressed in terms pipe is increased rapidly; for example, (8) of the Mach number: if the system expands from a 2-in. pipe to one of 3-in. pipe. This can also hap- In Equation (8) (in essence, a state- (9a) pen when a pipe enters a flow splitter ment of the First Law of Thermodynam- such that the sum of the pipe areas on ics), energy is balanced by summing (for the splitting side exceeds the area of (9b) each pipe at the branch connection) the the supply pipe. A shock wave forms at mass flowrate multiplied by the stag- the end of the supply pipe, and a pres- nation enthalpy. Elevation effects drop (9c) sure discontinuity is established. out, because all elevations at the con- (Continues on next page) 1 1 1 1

0.8 0.8 0.8 Mach/number CoverCover Story Story 0.8 Mach/number 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 p stag/P stag inlet The third case, which may be called p stag/P stag inlet 0.2 0.2 0.2 0.2 restriction choking, occurs when the 0 0 0 0 gas flows through a restriction in the 0.20 0.4 0.6 0.8 1 0.20 0.4 0.6 0.8 1 1 pipe, such as an orifice or valve. In x/L x/L such a case, the flow area of the gas Figure 2. These stagnation-pressure and Mach-number profiles are for (left) ex- is reduced, causing a local increase in pansion choking, involving a 2-in. pipe expanding to 3 in., and (right) restriction chok- velocity, which may reach the sonic ing at a 0.6-area-ratio orifice in a 2-in. pipe velocity. A shock wave forms at the re- 5 4 /s /s m striction, with a pressure discontinu- m 4 ity similar to the first two cases. 3 Figure 2 shows stagnation-pressure 3 2 and Mach-number profiles for expan- 2 sion choking and restriction choking; 1 1 both involve supply air at 100 psia Choked-flow rate, lb Choked-flow rate, Choked-flow rate, lb Choked-flow rate, and 1,000°R discharging to 30 psia. 0 0 0 0.2 0.4 0.6 0.8 1 0 100 200 300 400 500 600 Endpoint-choking behavior appears in dP stag/P stag inlet Supply pressure, psia Figure 7, discussed later. For a given process situation, the Figure 3. In this adiabatic flow of 100- Figure 4. Increasing the supply pres- choked flowrate can be determined psia, 70°F air, sonic choking occurs at sure raises the choked flowrate (shown 63.6-psia or lower discharge pressure here for an adiabatic flow of steam) from Equation (10a), by inserting a Mach number of 1 into Equation (10b): a 2-in. pipe carrying air that is sup- another shock wave. In fact, there is plied at 100 psia. Despite containing no limit to the number of choke points (10a) no physical restrictions, this system in a pipe, other than the number of experiences endpoint choking at any possible geometric configurations that where: discharge pressure below 63.6 psia. permit shock waves. The three mecha- Some engineers misapply the con- nisms that cause choking can all occur cept of sonic choking and conclude in the same pipeline, in any combina- that the sonic flowrate is the maxi- tion. References [2] and [3] discuss (10b) mum possible through a given system calculation procedures for multiple- These equations can be derived from for all conditions. In fact, however, the choking systems. the continuity equation [4, p.97]. flowrate can be increased by raising In practice, it is difficult to apply the supply pressure. Indeed, the in- Single-pipe adiabatic flow these equations to choked conditions, creased choked-flowrate presumably Before presenting compressible-flow because the local conditions, P0 and increases linearly with increased sup- equations that are generally applica- T0, are not known at the point of chok- ply pressure (Figure 4). ble (Equations [13] and [14]), we con- ing. For instance, to apply the equa- The pressure drop across the shock sider two special cases: adiabatic and tions to endpoint choking, one must wave in choked flow cannot be calcu- . Both are important calculate the and lated directly.2 The only recourse is in their own right. What’s more, analy- temperature at the end of the pipe, up- to use the choked flowrate as a new sis of the two (see below) leads to the stream of the shock wave — but these boundary condition on the pipe down- guidelines that can help the engineer two variables depend on the flowrate, stream of the shock wave (assuming decide whether compressibility (with which is not yet known. that one is not dealing with endpoint its far more-complex calculations) The only way to solve such a prob- choking) and to apply Equations (2) – must be taken into account in a given lem accurately is by trial and error: (6) in the remaining pipes. The shock- process situation. first, assume a flowrate and march wave process is not truly isenthalpic, The a gas down the pipe; if M reaches 1 before but (in accordance with Equation [4]) undergoes in constant-diameter adia- the end of the pipe, repeat the proce- instead entails constant stagnation batic flow can be viewed in terms of dure with a lower assumed flowrate; enthalpy. entropy and static enthalpy. This repeat until M reaches 1 right at the Be aware that a given pipe can choke process traces out a curve called the pipe endpoint. Obviously, this calcula- at more than one location along its Fanno line3 (Figure 5). The Fanno line tion sequence is not practical without length. This occurs when the choked neglects elevation changes, a safe as- a computer. flowrate set by the upstream choke sumption in most gas systems. From the standpoint of pipe design point is applied to the pipes beyond According to the Second Law of Ther- or system operation, sonic choking the upstream shock wave, and the modynamics, the entropy increases as sets a limit on the maximum possible gas at this flowrate cannot reach the the gas flows through the pipe. Thus, flowrate for a given set of supply con- end of the pipe without experiencing depending on the initial state of the ditions. In particular, lowering the 2.“Normal shock tables” (perhaps more familiar gas (either subsonic or supersonic), the to aeronautical engineers than to chemical engi- discharge pressure does not raise the neers) apply only to supersonic flows, and are of 3. Some authors show the Fanno line as a plot of flowrate. Figure 3 illustrates this for no use for sonic or subsonic pipe flow. temperature (rather than enthalpy) vs. entropy. h ho = constant V2 M < 1 (subsonic) 2 Figure 6. Adiabatic Heated Isothermal 130 and isothermal flow do Adiabatic Cooled not bracket gas flowrates. Sonic point In four situations shown 110 Case Mass flow (lb/s) here, 100-psia, 111°F air is fed into a 1-in. pipe 20 ft 90 Cooled 0.9197 long. Outlet pressure is 60 Adiabatic 0.8994 psia. Cooled flow has 30°F M > 1 (supersonic) ˚F Temperature, 70 Isothermal 0.8903 ambient temperature; Heated 0.8776 heated flow, 220°F. The 50 heat-transfer coefficients 0 5 10 15 20 are 100 Btu/(h)(ft2)(°F) smax s Distance, ft.

Figure 5. Fanno lines, such as the one presented here, show enthalpy vs. en- consider the adiabatic case, where no tropy for adiabatic flow in a pipe heat is added but the gas cools. If heat is removed, the cooling will exceed process will follow either the upper or (11b) that in adiabatic flow. Next consider lower portion of the curve. Very few isothermal flow, where the addition of process situations entail supersonic heat keeps the gas static temperature flow in pipes, so we will focus on the constant. If more heat is added than subsonic (i.e., upper) portion. Single-pipe isothermal flow required to maintain isothermal flow, The stagnation enthalpy, h0, is con- In the second special case, isothermal the static temperature will increase. stant because the system is adiabatic. flow, the static temperature of the gas In summary, the heat-transfer envi- However, the gas is accelerating, which remains constant. As already noted, ronment plays a critical role in deter- causes the static enthalpy to decrease, the tendency is for gas to cool as it mining whether the gas flow is closer in accordance with Equation (4). If the flows along a pipe. For the tempera- to adiabatic or isothermal. It is also proper conditions exist, the gas will ture to remain constant, an inflow of the mechanism that can cause the gas continue to accelerate up to the point heat is required. flow to exceed the limits of the two at which its velocity equals the sonic When temperature is constant, special cases. Figure 6 demonstrates velocity, where sonic choking begins. Equations (2)–(6) become somewhat the different situations. As Figure 5 shows, the enthalpy simpler. In Equation (5), for instance, approaches the sonic point asymp- density becomes directly proportional General single-pipe equations totically. Accordingly, the thermody- to pressure, and a perfect-gas analyti- For the general (neither adiabatic nor namic properties experience intensely cal solution can be obtained: isothermal) case, in situations when rapid change at the end of a sonically the compressibility of the gas can- choked pipe. Examples of such change (12a) not be ignored, Equations (2)–(6) can arise later in this article. be combined and, through calculus The gas static temperature usu- where the T subscript on L empha- and algebra [3, 4], represented in dif- ally decreases as it travels along the sizes that the system is isothermal. ferential form by Equations (13) and pipe, due to the decreasing pressure. Integrating from 0 to L gives: (14). Equation (13a) [1-3] is based on Under certain conditions, however, the a fixed-length step between Locations reverse is true. The governing param- 1 and 2 along the pipe. The terms in- eter in this regard is the Joule-Thomp- (12b) volving  and Z account for the real- son coefficient [5, 8]. The points made gas effects: in this article are (unless otherwise noted) applicable for either the cool- To truly maintain isothermal flow ing or heating case if the appropriate up to the sonic point would require an words are substituted, but we assume infinite amount of heat addition. This (13a) the cooling case for the sake of discus- leads to the strange but mathemati- sion. For more on see Ref- cally correct conclusion that for iso- Integration yields: erences [4, 6, 7]. thermal flow, sonic choking occurs at a From Equations (2)– (6), the follow- Mach number less than 1. Practically (13b) ing equation can be derived for adia- speaking, it is not feasible to keep a where: batic flow of a perfect gas [4, p. 209]: gas flow fully isothermal at high veloc- ities. For a more-complete discussion of isothermal flow in pipes, see Refer- ence [4], pp. 265–269. (11a) One occasionally finds a misconcep- (13c) tion among engineers designing gas systems: that adiabatic and isother- Integrating from 0 to L along the mal flow bracket all possible flow- length of the pipe gives: rates. However, this is not true. First, Conditions at Location 1 are known; 1 1 0.9 1.0 0.9 0.8 CoverCover Story Story 0.8 0.8 0.7 0.7 0.6 0.6

T/T inlet 0.5 Tambient/Tinlet = 0.4 0.6 T /T = 0.4 0.6 ambient inlet Mach number 0.4 0.5 0.8 0.3 1.0 the goal is to find those at Location 2 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 that satisfy the equations. There are x/L x/L multiple unknowns at Location 2, and much iteration is required. 1.1 1 1 0.9 In addition, some expression for the 1.0 0.8 1.0 0.9 0.8 heat-transfer process is required in 0.7 0.8 0.8 0.6 0.6 order to apply the energy equation, T /T = 0.4 0.7 0.5 ambient inlet Equation (4). In a convective applica- 0.6 0.6 0.4 T /T = 0.4 0.3 tion, this will usually require a con- T stag/T stag inlet ambient inlet 0.5 P stag/P stag inlet vection coefficient. For more details, 0.2 0.4 0.1 see Reference [3]. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Another formulation of these equa- x/L x/L tions is better suited for systems that Figure7. These typical, dimensionless property variations were taken with respect incur either endpoint or expansion to air supplied at 100 psia and 1,000R into a 2-in. pipe 100 ft long, under conditions sonic choking. This method takes so- providing a sonically choked discharge. The curves relate to four ambient-temperature conditions. Similar curves could be drawn showing other parameters, such as density, lution steps over equal Mach-number enthalpy and static pressure, as functions of the distance along the pipe increments rather than length incre- 1.1 1 ments [1-3]: Simplification error : How big? focusing on a specific pipe diameter. 1.0 As already noted, a key question arises: For the conditions 0.9 modeled, air fol- 0.8 How much error is introduced if the en- lowed the ideal gas law0.8 closely. How- gineer sidesteps the calculational com- ever, the steam and methane0.7 conditions 0.6 0.6 (14a) plications of equations such as Equa- did not follow the , withT /T = 0.4 T stag/T stag inlet 0.5 ambient inlet tions (13) and (14) by instead making compressibility factors0.4 (corrections for where: the incompressible-flow assumption? non-ideality) ranging from0 0.920.2 to 0.97. 0.4 0.6 0.8 1 Adiabatic flow: In the fully adia- From these data, it appears that thex/L batic-flow case (that is, assuming a generalizations implied by Figure 8 can perfectly insulated pipe), Figure 8 pro- be applied to non-ideal gases. vides typical answers to that question, To extend the generalization, the with respect to three specific cases. preceding calculations were repeated They involve, respectively, the flow for air flowing in pipes with diameters of three widely used fluids: air, steam of 3, 6, 12 and 24 in., increasing the and methane (the last-having proper- pipe length each time to maintain the ties similar to those of natural gas). L/D ratios of 50, 200 and 1,000. Results The results in Figure 8 were devel- (not shown) indicate that the error is oped by building models for both com- always larger than for the 1-in.dia pipe pressible and incompressible flow. The with the same L/D. For 24-in. pipe, the latter models used the arithmetic av- error is larger by over a factor of two. erage fluid density, and assumed that Why does the incompressible-flow- the was constant. The inlet assumption error increase as the pipe Integration yields: stagnation conditions for the three diameter increases? The reason re- streams were as follows: lates to the pipe-roughness data. As Air: 100 psia, 70° F the pipe diameter increases, the abso- Steam: 500 psia, 600° F lute roughness remains constant, re- Methane: 500 psia, 100° F sulting in a decreasing relative rough- All pipes were standard steel, with ness (e/D). This leads to lower friction a roughness of 0.00015 ft. factors, which leads to larger velocities (14f) With respect to each of the three for a given pressure drop, and, thus, An increase in Mach number from gases, we compared the calculated greater error. M1 to M2 can be arbitrarily specified flowrates for the two cases. The differ- We have also developed a more (say, by increments of 0.01); then, one ence between the two is the error that widely applicable tool than Figure 8 computes the distance from x1 to x2 results from using the incompressible for assessing the error introduced by that is required to obtain this change assumption. The error is plotted in assuming incompressible flow. The in Mach number. Again, extensive Figure 8 for 1-in. pipe of three differ- more-appropriate parameter to re- iteration is required because there ent lengths. late gas-flow supply and discharge are multiple unknowns at Location The clustering of the air, steam and conditions is not the L/D ratio, but 2. This method lets the engineer fol- methane results confirms that the pipe the ratio of fL/D (a choice commonly low the rapidly changing conditions pressure-drop ratio and the ratio of employed in gas-flow tabulations, and at the end of the pipe during choking length to diameter are appropriate pa- consistent with the arrangement of (see Figure 7). rameters to use for generalization when Equations [11] and [12]). Plotting the 0.1 0.5 30 40 30 Cooling L/D = 50 Air 1.0 20 25 Sonic choking above this line Tambient/Tinlet = 0.6 L/D = 200 Steam 30 20 10 0.8 Methane 1.5 15 20 0 1.2

fL/D=3 % Error, 10 5 -10 L/D = 1000 10 Flow rate error, % Flow rate error, 10 15 1.5 5 30 -20 2.0 Heating 50 0 0 -30 0.20 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 0 0.20.1 0.3 0.4 0.5 0.6 Incompressible-flow rate error, % Incompressible-flow rate error, dPstag/Pstag, inlet dPstag/Pstag, inlet dPstag/Pstag, inlet

FIGURE 8. The pipe pressure-drop ratio FIGURE 9. This map shows the error Figure 10. When a pipe is treated as adia- and the ratio of pipe length to diameter (overprediction) in flowrate prediction for batic but actually has , the flow- are appropriate parameters for general- a single pipe due to using incompress- rate prediction error can be sizable, even with- izing about the error that is introduced ible-flow assumptions rather than an out an incompressible-flow assumption. The when assuming incompressible flow adiabatic compressible-flow calculation case here is for 100-psia, 70°F air entering an uninsulated steel pipe with L/D ratio of 200 incompressible-flow-assumption error the size of the incompressible-flow-as- (4), or using more-convenient forms against this parameter makes it possi- sumption error. However, some insight of these equations, such as Equations ble to summarize the information on a can be gained from comparing rel- (13) or (14). Realistically, this requires single curve for each fL/D value, which evant compressible-flow calculations appropriate software. applies for all pipe diameters. (setting aside for a moment our pre- Such an error map appears in Fig- occupation with the incompressible- Network complications ure 9. It is based on an iterative pro- flow-assumption error). Computer When applying the concepts in this ar- gram, Compressible Flow Estimator models were constructed to determine ticle, and in particular the use of the (CFE), developed by the author and the difference in flowrate for air at dif- CFE program that underlies Figure being made available as a free down- ferent ambient . 9, to a pipe network, the number of load at http://www.aft.com/cfe.htm. The difference in flowrate for air variables increases and the difficulty The results shown in Figure 9 are with different ambient temperatures in assessing the potential error like- of general applicability. Various spe- as compared to the compressible adia- wise increases. To investigate possible cific heat ratios, , and compressibility batic case appears in Figure 10. It can error-estimating methods, we have factors, Z, have been entered into the be seen that cooling a gas may result constructed simple flow models, one CFE, and the results always fall along in a greatly increased flowrate. In for incompressible flow and the other the lines shown in Figure 9. This error contrast, heating a gas can cause the for compressible flow, of a manifolding map is also consistent with real-sys- flowrate to decrease significantly. pipe system. For simplicity, the com- tem predictions based on more-sophis- Accordingly, if an engineer is trying pressible-flow model assumed that all ticated calculation methods. Accord- to design for a minimum flowrate, a gas flows are adiabatic. The basis is a 110- ingly, Figure 9 is recommended to the stream that is cooling works in his or psia air system that enters a header engineer for general use as a guide in her favor by causing an underpredic- and flows to three pipes at successive assessing compressibility in pipes. tion of the flowrate when using adia- points along the header, terminating Keep in mind, though, that Figure batic flow methods. When this error is in a known pressure of 90 psia. 9 assumes adiabatic flow. Additional combined with that of an incompress- For each pipe in the system, the error can result from flows involving ible-flow assumption, which overpre- predicted fL/D and pressure-drop heat transfer. The relative importance dicts the flow, these two errors in ratio have been determined from the of heat transfer is addressed in the opposite directions, in part cancelling incompressible-flow model. The re- next section. each other out. Conversely, a gas being sulting data have been entered into Finally, note that the direction of heated adds further error on top of the the CFE program for each pipe, and the incompressible-flow-assumption incompressible-flow-assumption error, an approximate error generated for error is to overpredict the flowrate. causing even more overprediction of each. Then, starting from the supply, Or, stated differently, for a given the flowrate. a path has been traced to each termi- flowrate, it will underpredict the In many gas-pipe-system designs, nating boundary (of which there are pressure drop. Unfortunately for the delivery temperature is as impor- three). The error for each pipe in the typical pipe-system applications, tant as the delivery flowrate and pres- path has been summed, and then di- neither of these conclu-sions is sure. In those cases, the heat-transfer vided by the number of pipes in the consistent with conservative design. characteristics of the pipe system take path, giving an average error. This The sequence of steps that underlie on the highest importance, and nei- average has been compared to the the CFE program are available from ther adiabatic nor isothermal methods actual difference between the results the author. Also available from him are —let alone incompressible-flow as- of the incompressible- and compress- modified sequences, for handling sumptions — can give accurate predic- ible-flow models. situ-ations in which (1) the endpoint tions. Unless the gas flow is very low Overall the comparison has proved static pressure rather than the and can be adequately calculated with favorable. However, applying CFE to stagnation pressure are known, or (2) incompressible methods, the designer this networked system underpredicts the temper-ature and flowrate are is left with no choice but to perform the actual error from the detailed mod- known but the endpoint stagnation a full compressible flow calculation. els by up to 20%. The first pipe in the pressure is not. Effect of heat This means solving Equations (2)–(6) header shows the largest error, and the transfer: The author knows of no with a suitable relationship for the last pipe the smallest. As in the single- general relationship showing the heat transfer to be used in Equation pipe calculations, the incompressible effect of heat transfer on Cover Story

method overpredicts the flowrate. However, Equation (15) breaks ible-flow methods and estimation equa- In short, extra care should be taken down for pipe-system analysis when tions in this article to such systems. when interpreting the meaning of in- pipe friction becomes a factor. The The methods discussed in this article compressible-flow methods applied to reason is that the stagnation pressure can help the engineer assess endpoint gas pipe networks. in the equation is the pressure at the sonic choking, but restriction and ex- upstream side of the shock wave. If pansion choking are somewhat more Rethinking the rules of thumb there is any pressure drop in the pipe complicated. Accordingly, the estima- The information presented up to now from the supply pressure to the shock tion methods in this article may not be provides a basis for critiquing a num- wave, then the supply pressure cannot applied to all choking situations. ber of rules of thumb upon which en- be used in Equation (15). Instead, the For new designs that require a lot of gineers often depend when dealing local stagnation pressure at the shock pipe, the engineer should consider the with gas flow. wave must be used — but this is not potential costs savings if smaller pipe Adiabatic and isothermal flow: known, unless the pressure drop is sizes can be used. If significant cost One rule of thumb is the myth that calculated using other means. savings prove to be possible, it may be adiabatic and isothermal flow bracket In short, Equation (15) cannot be prudent to invest in developing a de- all flowrates. They do not, as has al- used to predict the supply and dis- tailed model that can more accurately ready been noted. charge necessary for sonic determine the system capability over 40%-pressure-drop rule: A common choking unless the piping has negli- a range of pipe sizes. A detailed model belief is what can be called the 40%- gible friction loss. may also help assess the wisdom of pressure-drop rule. Presented in a va- Other simplified compressible-flow making modifications proposed for an riety of handbooks, it states that if the methods: A variety of simplified gas- existing system. ■ pipe pressure drop in a compressible- flow equations, often based on assuming Edited by Nicholas P. Chopey flow system is less than 40% of the isothermal flow, crop up in the practical References: inlet pressure, then incompressible- engineering literature. These typically 1. Winters, B.A., and Walters, T.W., X-34 High flow calculation methods can be safely have several drawbacks that are not al- Pressure Nitrogen Reaction Control System Design and Analysis, NASA Thermal Fluid employed, with the average density ways acknowledged or recognized: Analysis Workshop, Houston, Tex., 1997. along the pipe used in the equations. • Most gas flows are not isothermal. 2. Walters, T.W., and Olsen, J.A., Modeling In the handbooks, it is not made In such cases, one cannot know how Highly Compressible Flows in Pipe Net- works Using a Graphical User Interface, clear whether the pressure drop ratio much error is introduced by the as- ASME International Joint Power Generation is to be based on the stagnation or the sumption of constant temperature. Conference, Denver, Colo., 1997. 3. “AFT Arrow 2.0 User’s Guide,” Applied Flow static pressures. (In the author’s expe- Related to this is the general issue of Technology, Woodland Park, Colo., 1999. rience, engineers apply the rule more the importance of heat transfer on the 4. Saad, M.A., “Compressible Fluid Flow,” 2nd frequently using stagnation-pressure gas flow, already mentioned Ed., Prentice-Hall, Englewood Cliffs, NJ, 1993. 5. Carroll, J.J., Working with Fluids that Warm ratios.) In any case, Figures 8 and 9 • Simplified equations typically do Upon Expansion, Chem. Eng., pp. 108–114, make it clear that the 40%-pressure- not address sonic-choking issues September 1999. drop rule has no validity unless as- • These equations are of no help when 6. Anderson, J.D., Jr., “Modern Compressible Flow: With Historical Perspective,” McGraw- sociated with a specific L/D ratio. Ac- the delivery temperature is important Hill, New York, N.Y., 1982. cordingly, this rule of thumb is highly • The simplified equations break 7. Shapiro, A.H., “The Dynamics and Thermo- dynamics of Compressible Fluid Flow,” 2 misleading, and should be discarded down at high Mach numbers vols., Ronald, New York, N.Y., 1953. by the engineering community. • Unrealistically, the entire pipe is 8. Barry, John, Calculate Physical Properties for Choked air flow at 50% pressure solved in one lumped calculation, Real Gases, Chem. Eng., pp. 110–114, April 1999. drop: An equation sometimes used rather than using a marching solu- 9. Flow of Fluids Through Valves, Fittings, and as a rule of thumb to assess the likeli- tion Pipe, Technical Paper No. 410, Crane Co., Jo- hood of sonic choking is as follows (see, • It is difficult to extend the equations liet, Ill., 1988. for instance, Reference [4], p 94): to pipe networks Author In summary, simplified compressible- Trey Walters, P.E., is Presi-dent and Director of (15) flow equations can be an improvement Software Development for over assuming incompressible flow, Applied Flow Technology (AFT, 2955 Professional where p* is the critical static pressure but numerous drawbacks limit their Place, Suite 301, Colorado at sonic velocity and p0 the local stag- usefulness. Springs, CO, 80904 USA; Phone: 719-686-1000; Fax: nation pressure. For air, the specific 719-686-1001; E-mail: heat ratio is 1.4, so the pressure ratio in Final thoughts [email protected]). He founded the company, a the equation works out to 0.5283. This Compressors, blowers and fans raise developer of Microsoft-Win- results in a pressure drop ratio of near the system pressure and density. These dows-based pipe-flow-simu- lation software, in 1993. Previously, he was a 47% (in other words, about 50%) to bring changes in properties inside the gas- senior engineer in cryogenic rocket design for about sonic choking. For gases with dif- flow system further limit the applicabil- General Dynamics, and a research engineer in steam-equipment design for Babcock & Wilcox. ferent specific heat ratios, the pressure ity of incompressible methods, beyond Walters holds B.S. and M.S. degrees in mechani- drop ratio will differ somewhat, in ac- the cautions already discussed. Take cal engineering from the University of California at Santa Barbara, and is a registered engineer cordance with Equation (15). special care in applying the incompress- in California.

Reprinted from the January 2000 issue of Chemical Engineering. © 2006 Access Intelligence, LLC.