January 2000 www.che.com Trey Walters, P.E. Applied Flow Technology A Publication of Chemical Week Associates CoverCover Story Story Gas-flowGas-flow Calculations:Calculations: Trey Walters, P.E. Don'tDon't ChokeChoke Applied Flow Technology Assuming incompressible flow simplifies the math, when compressibility can be ignored. Compressible flow: 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 gases 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, Fcomplicated than for liquid (incom- incompressible methods. the arithmetic averages will be incor- pressible) flow, due mainly to pressure- 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) Momentum: ible or compressible, depending on the (3) amount of pressure change the gas un- where f is the Moody friction 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 Equation of State: often apply rules of thumb to a given (hence the substance is virtually (5) design situation involving gas flow, to incompressible), and density changes Mach number: 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 heat 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 temperature. 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 entropy 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 enthalpy, 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 0.6principles 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- ideal gas 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 shock wave 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 volume) 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.