Condensation of low volatile components in lean natural gas transmission pipelines. S.T.A.M. de Wispelaere Report number WPC 2006.

Transcription

1 Condensation of low volatile components in lean natural gas transmission pipelines S.T.A.M. de Wispelaere Report number WPC Supervisors: dr. ir. C.W.M. van der Geld dr. ir. J. Schmidt Eindhoven University of Technology Mechanical Engineering Department Division Thermo Fluids Engineering Section Process Technology

2 Abstract Condensation inside natural gas transport pipelines is highly undesirable as it can lead to corrosion and extra pressure drop. Even for very lean gases accumulation of condensate over periods of years can lead to considerable amounts of liquid in the pipes. As it is difficult and expensive to mechanically clean the pipes from liquid deposit, methods to avoid or decrease condensate flow rate are sought-after. To do so, the condensation process has to be better understood. A theoretical and numerical study of condensation in long distance natural gas pipelines is done. Possible condensation phenomena are discussed: filmwise or dropwise wall condensation and fog formation. Due to uncertainties in the modeling of fog and droplet condensation, it is assumed that the dominant process is filmwise wall condensation. To model this type of condensation, the film theory can be used. The film theory models diffusive mass transfer, assuming a binary mixture in its derivation. Problems arise when applying the film theory to natural gases at high pressure, where the difference between condensable and non-condensable gases is not clear. Phase equilibria theory is useful if fixed conditions for pressure, temperature and composition can be assumed, but is not sufficient in the case of finite mass transfer between the two phases. Therefore a combinational model is proposed. The mass transfer rate is diffusion controlled and modeled by film theory, the amount and composition of condensate however is determined with flash calculations using phase equilibria theory. Pressure and pipe wall temperature set the phase behaviour of a multicomponent mixture. Inner pipe wall temperatures were calculated using measured (p,t) profiles. At the highly turbulent flow conditions, the heat resistance of the bulk gas is small, causing small difference between bulk gas and wall temperatures. Radial temperature profiles are investigated to calculate inner pipe wall temperatures based on known axial (p,t) profiles. At the high gas velocities inside the pipe, high Nusselt numbers and heat transfer coefficient from bulk to pipe wall cause the difference between bulk gas temperature and inner wall temperature to be very small, reaching maximum values of a few Kelvin. Numerical studies were done with the combination model for a Russian natural gas of 32 components. For constant conditions of temperature and pressure, the growth rate of the film thickness proves to be constant in time. A considerable growth rate is observed, ranging from maximum values of 4.5 to 6 mm/year, depending on soil temperature. Variation of the inlet temperature does not prove to influence the location nor the value of the maximum growth. 1

3 The condensation rate of individual components is examined. Indeed, the low volatile component reaches an early maximum along the length of the pipe. Other components start to condense more downstream, all at the same location. It proves that the rate of condensation of octane up to hexa-decane is the highest, showing maxima for tri- and tetra-decane. Typical condensation rates for these two components reach maxima of about 0.02 kmol /(m year). The presence of the low volatile component in the gas increases the condensation rates of the other components slightly. 2

4 Contents List of symbols 5 1 Introduction Background Objectives and assumptions Report lay-out Phenomena of condensation Wall condensation fog formation Film theory Physical model Derivations Mass transfer Selection of variables Correction factor for mass transport Correction factor for heat transport Heat and mass transfer coefficients Phase Equilibria Phase diagram of a natural gas Equations of state Phase equilibria with Equations of State Inner pipe wall temperature Introduction Governing equations Conductive heat loss Axial velocity Density and property data Results Combination model Introduction Construction of control volumes Bulk volume equations

5 6.2.2 Wall control volume equations Mass transfer term Numerical results Temperature and pressure profiles Phase diagram Flash calculation Time dependent film thickness profile Film thickness growth rate Condensation rate of individual components Conclusions and recommendations Conclusions Recommendations for further research Bibliography 45 A Derivations for mass transfer 47 A.1 Concentration Equation A.2 Alternative variables B Equations of state 49 B.1 Peng-Robinson expressions B.2 Flash algorithm C Gas composition 51 D Property data 53 E Numerical programming 54 E.1 Matlab E.2 Aspen Dynamics

6 List of symbols Variables c p specific heat capacity at constant pressure [J/(kgK)] c mass fraction [kg/kg] d p pipe diameter [m] D binary diffusion coefficient [m 2 /s] D i,m effective diffusion coefficient [m 2 /s] j q heat flux field due to conduction [W/m 2 ] k thermal conductivity [W/(mK)] L length [m] L f molar liquid fraction [kmol/kmol] LV C the low volatile component - M molar mass [kg/kmol] M mass in control volume [kmol] ṁ mass flux [kg/(m 2 s)] ṅ molar flux [kmol/(m 2 s)] p pressure [P a] q heat flux [W/m 2 ] q R heat flux field due to radiation [W/m 2 ] r radial coordinate [m] R universal molar gas constant [J/(kmolK)] R pipe inner radius [m] t time [s] T temperature [K] u, v, w bulk velocity in x, y, z-direction [m/s] ṽ molar volume [m 3 /kmol] v velocity vector field [m/s] V volume [m 3 ] V f molar vapour fraction [kmol/kmol] x molar fraction in the liquid phase [kmol/kmol] ỹ molar fraction in the gas phase [kmol/kmol] z molar fraction in total mixture [kmol/kmol] z axial coordinate in pipe [m] Z compressibility factor [ ] 5

7 α heat transfer coefficient [W/(m 2 K)] β mass transfer coefficient [m/s] δ boundary layer thickness [m] ζ resistance factor [ ] η dynamic viscosity [P a s] µ JT Joule-Thompson coefficient [K/P a] Θ correction factor [ ] ρ mass density [kg/(m 3 )] ρ molar density [kmol/(m 3 )] τ tension tensor [N/(m 2 )] ω acentric factor [ ] Subscripts bulk gas 0 condensate film surface, pipe inlet c thermodynamically critical cv control volume on pipe wall i, j components L liquid phase m mass n non-condensable gas T temperature V gas phase W pipe wall Superscripts L V pipe cross-sectional average of variable liquid phase gas phase Dimensionless numbers Nu P r Re Sc = η ρ D Sh = α d k = η cp k = v d ρ η = β d D 6

8 Chapter 1 Introduction 1.1 Background After producing lean natural gas from a well, a number of operations are necessary to make it suitable for further transport trough pipelines to the consumers. Apart from filtering the gas, it is important to remove condensable components to prevent the condensation further in the transport pipelines, which causes corrosion and extra pressure drop. At the high pressures (typically bars) and low temperatures (typically K) at which the gas is delivered, most of the heavier components are not present anymore. Before entering the pipeline, a low volatile component (hereafter occasionally abbreviated as LVC) is added, that absorbs the remaining water content. Although the chemical deposits with the water before entering the pipeline, small quantities (in the order of 1 ppm) of the component remain in the gas and can condense further in the pipeline. This is possible because the gas drops in temperature and pressure, due to frictional pressure drop and heat loss. These can be considerable for long distance pipelines. For a multicomponent mixture like natural gas, it is then possible to enter the two-phase region, a phenomenon known as retrograde condensation. Other low volatile components in the gas influence the condensation behaviour of the total system and must be taken into account. Although the gas is lean and the rate of condensation is expected to be very small, accumulation of condensate over long periods of time (in the order of years) can result in large amounts of liquid in the pipeline. As it is difficult and expensive to pause gas transport for mechanical removal of condensate, it is useful to search for methods to avoid condensation, or to evaporate existing condensate back into the gas. A number of difficulties are present. The concentration of the low-volatile components is extremely small and so is the expected rate of condensation. Furthermore, the conditions over a year change significantly. Ambient temperature changes, causing different temperature profiles. Locally, the ambient temperatures can differ strongly. Finally, the composition of the bulk gas changes, as the water absorbing chemical is only added in winter, when the larger temperature drop makes water condensation more critical. The challenge is to predict the condensation process accurately for these conditions. 1.2 Objectives and assumptions Experimental data from existing pipelines on the subject of condensation are sparse. Some important variables, like axial temperature and pressure profiles are well known. Measuring 7

9 the amount and composition of condensate is possible but time consuming. The amount of condensate can be extremely small and difficult to detect. Accurate methods to predict condensation rates, and to understand what influences it are therefore desirable. The approach of this study is purely theoretical. The objective is to develop a model to predict the condensate mass flow rate in time and place in the pipeline. The following assumptions are made: The object under study is a pipe of 338 km length and 1.2 m diameter. Inclination is not considered. The inlet pressure is 90 bars. Axial pressure profiles are known from measurements or are calculated using appropriate software (Aspen Plus). The inlet bulk gas temperature can vary between 20 C and 50 C. The pipe is buried in a sand bed with a temperature of 2 C in winter and 10 C in summer. The volume flow through the pipe lies between 1 and 3 million m 3 /hr at standard conditions (p=1 atm, T=0 C). The gas is Russian natural gas with 32 components. The condensate layer does not move in axial direction. Mass transport inside the condensate layer is not considered as the film thicknesses are thin and the condensate is assumed perfectly mixed The assumptions are set up to reflect a realistic long distance natural gas pipeline. Some simplifications are taken here and more will be taken in the rest of this study. The main goal is to set up a functioning model with these assumptions. Later on, other effects should be added to the model to produce more accurate simulations. 1.3 Report lay-out In chapter 2 it is assessed which phenomena of condensation are relevant to a natural gas pipeline. The assumption is made that condensation occurs only on the pipe wall in a purely filmwise manner. To model film condensation, the film theory can be used. This is a diffusion based mass transfer model that regards condensation as the diffusion of a vapour through a non-condensable gas. Although its derivation is based on a binary mixture of a condensable vapour and a arbitrary number of non-condensable gases, it can be applied to a multicomponent mixture with multiple condensable gasses as well. The derivation is given in chapter 3, leading to the main equation with which mass transfer can be modelled. 8

10 Two important variables remain unknown to determine the condensation mass transfer rate, namely the gas-side phase-boundary concentrations, and the temperature at the phase boundary. Therefore, in chapter 4 an introduction to phase equilibrium thermodynamics is given, using cubic equations of state. It is shown how flash models can be constructed, which are useful in situations where thermodynamic equilibrium can be assumed between a liquid and gas phase. Radial and axial temperature profiles are discussed in chapter 5. It will be shown that at the given conditions, the temperature difference between bulk gas and wall is small, given that the condensate film is very thin. Where the film model provides an equation with which mass transfer can be modelled, its assumption of a binary mixture of condensable and noncondensable gases can not be applied as such to a natural gas at high pressure. In these conditions, even low volatile components like methane can solve to a certain degree in the condensate and can not be considered completely noncondensable. Phase equilibria are useful in static situations, but are not sufficient to describe situations where there is no thermodynamic equilibrium, as is the case with a bulk gas flowing over a condensate film, with a finite mass transfer between the two phases. Therefore, in chapter 6 a combination model is proposed. Here the mass transfer between condensate and bulk gas is modelled using film theory, but the condensation itself is calculated in a small control volume on the pipe wall, where flash calculations using phase equilibria theory are applied to calculate the amount and composition of condensate. It is time-consuming to program flash calculations for a large number components. Therefore, a multiphase computer program, Aspen Dynamics, was used to implement the model. Chapter 7 gives the results of the numerical simulations. First, results are given for the typical use of a pipeline with a 30 component natural gas at conditions of inlet temperature, pressure, etc that are constant in time. This is compared to results that would be obtained if the standard two-phase calculation method of Aspen Dynamics is used, which uses only flash calculations. A few parameter studies are done, to show the influence of inlet pressure and of seasonal ambient temperature. The condensation rate of individual components is discussed. Finally, chapter 8 summarizes the results. Uncertainties due to restrictions of the model are discussed. Recommendations for further development of the model is given, together with possible parameter studies. 9

11 Chapter 2 Phenomena of condensation In this chapter phenomena of condensation are discussed that are relevant to pipelines. Condensation of gases in a pipe can principally occur in two ways: condensation on the wall and fog formation. 2.1 Wall condensation If a gas flows over a cold wall, condensation occurs if locally the dew point of the gas, depending on both temperature and pressure, is reached. If the condensate forms a continuous film it is called film condensation. The condensate film can be quiescent, or be in laminar or turbulent flow. Instead of a film the condensate can also exist in the form of droplets, as shown in fig 4.2. This type of condensation is called drop condensation. Whether film or drop condensation is present depends on whether the wall is completely or incompletely wetted. The decisive factor for this are the forces acting on a liquid droplet, which are illustrated in Figure 2.1. The interfacial tensions at the edge of a droplet are shown, σ SW is the tension of the liquid against its own vapour, σ SW is the tension of the wall against the liquid and σ SW the tension of the wall of the vapour. The contact angle is thus found by [1] σ W G σ W L = σ LG cos β 0 (2.1) Finite values of this contact angle imply incomplete wetting and droplet formation. If on the other hand β = 0 the droplet will spread out over the entire surface and form a continuous film. To determine the contact angle for, at least the the composition of the liquid and the wall have to be known. If the gas has components that are immiscible in the liquid phase, a combination of both types can be observed. In the following, it will be assumed that wall condensation is filmwise. 2.2 fog formation When condensation starts in the gas itself, fog is formed. The main criterion for the occurrence of fog formation is the rate of supersaturation in the gas. By definition it is the ratio of partial to saturation pressure S = p i p sat,i (2.2) 10

12 σ LG β 0 σ SG σ WL Figure 2.1: Interfacial tension at a condensate droplet s edge. The indices W, L, G denote the wall, liquid and gas, the contact angle is β 0 If there are sufficient alien particles in the gas, low degrees of supersaturation suffice for condensation to occur on those particles, leading to heterogenic nucleation, minimum needed values of 1.02 are often given. Much higher values of the supersaturation are needed for homogeneous nucleation, when no, or few particles are present, usually a range of about 5 to 10 is given [9]. The rate of supersaturation is dependent on the speed at which temperature and pressure drops. As we will see later on, axial pipeline profiles show very slow temperature and pressure drops. Among others, Brouwers [2] and Schaber [17] investigated the possibility of fog formation in the film flowing over a condensate layer. During condensation towards a liquid film, a velocity is induced. Gas flowing towards the film can be cooled quickly, leading to fog formation. A criterion is that temperature difference between wall and gas is relatively high. As will be shown later, this is not the case for the pipe under study. Since the composition of the condensate is not known beforehand, an evaluation on the occurrence of either type of wall condensation can not be made yet. Due to too many uncertainties, in the modelling of the fog formation (e.g. number of particles), the assumption is made that condensation occurs in a purely filmwise manner. 11

13 Chapter 3 Film theory When filmwise condensation occurs in a pipe, molecules of the condensing component are transported from the bulk gas to the wall. The rate of condensation is limited by the diffusive mass transfer through the bulk gas. Multiple approaches are possible to model this mass transfer. The general approach is solving the coupled concentration, momentum and energy equations for multiple components. For simple geometries like pipes, theories that are based on a dimensional analysis, using experimentally validated characteristic numbers, provide an accurate alternative. Examples of these are are film theory, boundary layer theory and penetration and surface renewal theory [1]. All of these assume the mass transfer to take place in a small film lying on the surface to which is diffused to. In the film theory the concentration of the diffusing component and the velocity only vary in the direction, normal to the surface. The boundary layer theory additionally considers the direction in which the bulk gas flows. Penetration and surface renewal theories take non-steady flow into account as well. The axial variation in the pipe of temperature and pressure and consequently density and velocity are very small. This is shown in chapter 5 where typical measured axial pressure and temperature profiles of the pipe are given, together with calculated axial velocity profiles. Variations in time can be considered small as well, changes occur in the time span of minimally a few hours. It is therefore assumed that of the simplified theories, the film theory suffices to describe the mass transfer. In this chapter, the assumptions and derivations of the film theory will be given. 3.1 Physical model In the following derivations, a local cartesian coordinate system is used, where x denotes axial coordinate in the pipe, and y is the coordinate perpendicular to the wall. The coordinate system is located on the phase boundary, i.e. y = 0 corresponds with the condensate film surface. y x The derivations are made for a steady state, one-dimensional process. The assumptions for the film theory are [6]: The profiles for pressure, velocity and temperature are only dependent on the y-coordinate and become constant at their respective boundary layer thicknesses. 12

14 The boundary layer thicknesses are constant and independent of the mass transfer rate The flow is laminar The property data viscosity, diffusion coefficient and thermal conductivity are constant There are no chemical reactions in the gas. Viscid dissipation and radiation energy are neglected 3.2 Derivations Mass transfer In the film model, boundary layers near the surface for velocity temperature and mass transfer are defined. The resistance against heat and mass transfer for the condensation process is dominant in the gas phase, and the resistance in the liquid phase can often be neglected. Brouwers [2] made his derivation for a mixture of inert gases (index n) and condensable vapours (index v). The derivation starts with the three-dimensional full Fickian diffusion equation, as is derived in appendix A as (A.9), with no chemical reactions, assuming a constant diffusion coefficient, written in mass fractions c i : ( ci ρ t + u c i x + v c i y + w c ) ( i 2 ) c i = ρd z x c i y c i 2 (3.1) z Neglecting the time dependent term and assuming the variations to be small in the axial and tangential directions gives: ρv c i y = ρd 2 c i y 2 (3.2) The bulk velocity v is a convective flow, induced by the diffusion of vapour through the total mixture. An expression for this velocity is now derived. For every component, a steady state mass balance can be written in the y direction, giving for vapour and inert components respectively: (ρ v v v ) y = 0, (ρ n v n ) y = 0 (3.3) Since no inert gases disappear at the phase boundary due to condensation, it holds that v n (y = 0) = 0. Applying to (3.3) and integrating gives v n (y) = 0, meaning that the inert gases do not move in y direction. Inserting this in the general expression for the bulk velocity gives: v = 1 ρ n ρ i v i = 1 ρ (ρ vv v + ρ n v n ) = 1 ρ (ρ vv v ) = c v v v (3.4) i=1 Fick s first law states that diffusive molar flux is equal to its mass density gradient multiplied with the diffusion constant. In appendix A, it is shown that for a constant ρ mixture, the equation can be written in mass fraction c too. Another definition of diffusive mass flux is the velocity of a component, relative to its bulk velocity, see (A.6). Combining for the vapour gives: 13

15 ρ v (v v v) = ρd c v y (3.5) Combining (3.5), (3.4) and (3.3) gives the expression for the bulk flow, induced by diffusion of vapour: v = 1 D c v 1 c v y (3.6) This is often called Stefan flow. Substitution of (3.6) in (3.2) gives the diffusion equations from which the radial profile of c v can be found: With the boundary conditions ρd 2 ln(1 c v ) y 2 = 0 (3.7) this gives as a concentration profile: c v (y = 0) = c v,0 and c v (y = δ) = c v, (3.8) ( ) y δ c v (y) = 1 (1 c v,0 )e ln 1 cv, 1 c v,0 (3.9) The mass transfer in the film is the bulk flow at the phase boundary multiplied with the total mass density: ṁ = ρv(y = 0) = ρd 1 c v 1 c v y (3.10) y=0 Taking the first spatial derivative of (3.9) with respect to y, setting y = 0 and substituting in (3.10) gives: ṁ = ρ D ( ) 1 δ ln cv, (3.11) 1 c v,0 This expression for the mass transfer in a film is the original result of Stefan (1873). The fraction D/δ is commonly replaced with β and is called the mass transfer coefficient, with unit [m/s] Selection of variables The above derived equation for the mass transfer in the film model can be made for different variables: mass density ρ [kg/m 3 ], molar density ρ [kmol/m 3 ], mass fractions c [kg/kg] and molar fractions ỹ [kmol/kmol]. For the last two, the implicit assumption is made that the ratio of mass density to molecular weight is constant in the flow. The validity of this assumption will be derived here. Consider the Navier-Stokes equation for an incompressible inviscid flow: v t + ( v ) v = p + g (3.12) Neglecting gravity and the derivative with respect to time and selecting the y-direction gives: ρu v v + ρv x y = p y 14 (3.13)

16 In the film model the so-called boundary layer approximation is taken, velocity variations u v in axial x-direction are assumed negligible: x = 0 and x = 0. If the flow is sufficiently subsonic, the gas flow can be considered incompressible, giving for the continuity equation v = 0. This gives v = 0. Then it follows from (3.13) that the pressure is constant in the y-direction: p y y = 0. The general form of the equation of state for gases gives: ρ M = p (3.14) ZRT The compressibility factor Z is only dependent on temperature and pressure. If the temperature variations in y-direction are small and Z T is small1, Z does not vary in y-direction either. It follows that ρ/m is constant. As molar fraction and partial density are related with: ( ) M ỹ i = ρ i (3.15) ρm i it follows that the mass transfer equations in terms of mass density ρ i can be rewritten in terms of molar fraction ỹ i without the production of additional terms. The mass transfer equation (3.11) in terms of molar fractions thus becomes: ( ) 1 ỹv, ṅ = ρβ ln 1 ỹ v,0 (3.16) Correction factor for mass transport The general film model mass transfer equation (3.11) can be simplified by linearization 2 : ṁ = ρ D δ c v, c v,0 1 c v,0 (3.17) This means that the induced flow at the phase boundary is neglected. This can be seen if v = 0 is substituted in (3.2) and applying boundary conditions (3.8), which produces the same result. Comparing (3.11) with (3.17) and taking the ratio between the mass flow with and without induced flow gives the Stefan correction factor: Θ m = φ m e φm 1 with φ m = ṁδ ρd (3.18) It is important to note that ṁ is calculated with the uncorrected mass transfer expression (3.11). Baehr [1] gives for the correction factor: Θ m = φ m e φ with φ m m = ṅ 1 cβ (3.19) It easy to see that it is equal to (3.18) because Baehr defines the mass transfer as negative for condensation ṅ = ṁ/ρ and the definition of the mass transfer coefficient is: 1 For a temperature and pressure range of bars and K, Z has typical values of T for pure methane, as shown in figure 4.2 ( ) 2 cv,0 c in (3.17) the first order Taylor approximation of (3.11) is used: ln v, 1 c v,0 + 1 c v,0 c v, 1 c v,0, valid if c v,0 c v, 1 c v,0 0 15

17 β = D δ (3.20) The form of (3.17) can be further simplified if it is assumed that c v,0 is very small. Numrich [13] does this in a form for molar fractions, that rewritten for mass fractions is: The resulting Stefan correction factor now becomes: Θ m = ṁ = ρβ(c v, c v,0 ) (3.21) 1 1 c v,0 φ m e φm 1 with φ m = ṁδ ρd (3.22) The question arises why the the correction factor for mass transfer is introduced in the first place. The first argument is historical, as the logarithmic expression of (3.11) was considered time-consuming to calculate. The second argument is qualitative as the mass transfer equation, especially in the form of (3.21) has similarity with the one for heat transfer. The mass transfer is made proportional to a concentration gradient times a mass transfer coefficient. Compare this to the well known equation for convective heat flux: q = α(t hot T cold ) (3.23) which is known as Newton s law of cooling, with α the overal heat transfer coefficient [m 2 /s]. Forcing the mass transfer problem into a form that is similar to the heat transfer problem is attractive, because it provides an analogy between heat and mass transfer. Nevertheless, if the induced flow at the phase boundary is significant, it is easier to simply use the form of (3.11) instead of applying corrections Correction factor for heat transport Analogously to the way the correction factor for induced flow on the mass transport, a correction of this mass flow on the heat transfer can be derived. This will not be shown here, but the result for the correction factor is [2]: Θ T = φ T e φ with φ T = ṁc p,v T 1 α (3.24) This is called the Ackermann correction factor. By definition it is the ratio of the heat flux with induced velocity and the heat flux without. Note that, again the mass transfer ṁ is calculated with (3.11) Heat and mass transfer coefficients It is not possible to use (3.11) directly to calculate mass transfer rates, as the boundary layer thickness δ m is unknown. A mass transfer coefficient has to be calculated, for which the Sherwood number has to be known: β = ShD d The following equation for Nusselt and Sherwood numbers for turbulent flow in circular pipes is used, Gnielinski [8], recommended by [1]: 16

18 With resistance factor: [ (ζ/8)(re 1000)P r Nu = ζ/8(p r 2/3 1) [ (ζ/8)(re 1000)Sc Sh = ζ/8(sc 2/3 1) ( ) ] d 2/3 L ( ) ] d 2/3 L (3.25) (3.26) Valid in the region: ζ = 1 (0.79 ln Re 1.64) 2 (3.27) 2300 Re , 0.5 P r 2000 and L/d > 1 This equation is chosen because it is valid over a wide range of Reynolds numbers. The term (d/l) 2/3 in (3.25) and (3.26) can be neglected if d L, as is the case for very long pipes. A perfectly smooth pipe is assumed, as the resistance factor ζ does not depend on the friction factor of the pipe wall. 17

19 Chapter 4 Phase Equilibria In the following chapter, phase equilibria for natural gases are discussed. 4.1 Phase diagram of a natural gas The phase diagram of a multicomponent mixture like natural gas differs from that of a pure component. Figure 4.1 shows a typical (p,t) phase diagram for a natural gas at fixed composition. Figure 4.1: phase diagram for a natural gas mixture, from Voulgaris [18] At point E, the mixture is in the gas phase. When temperature is decreased at constant pressure from point E to D, the dew line is reached at point X and the two phase region is 18

20 entered. Lowering the temperature further from the dew point will not cause the mixture to condense completely, but in increasing amounts of liquid up to 100% when the bubble point is reached. Where a pure component can only have two phases in equilibrium on the dew line, a multicomponent mixture has a region of possible (p,t) combinations where two phases are present. Now consider a pressure change at a constant temperature above the critical point (denoted by C). If pressure is increased from A, the two-phase region is entered on the dew line on point Z. Interestingly, a maximum of liquid fraction is reached at point R. Increasing pressure further will cause the liquid to vaporize again until at the dew line at point W, only pure gas is present. This phenomenon is known as retrograde condensation. It is typical for natural gases at high pressures. Summarizing, two main differences appear when comparing the phase diagram of a multicomponent mixture, with that of a pure component: A multicomponent mixture can be in a two phase region in the (p,t) diagram. For a pure component, this is only possible on the saturation line for a fixed set of pressuretemperature combinations. For a multicomponent mixture at high pressure it is possible to enter and leave the two phase region at a constant temperature by changing the pressure only. Whether condensation will occur in a natural gas is therefore dependent on whether the dew line is crossed, the amount of possible condensation is dependent on how much it is crossed. This emphasizes the importance of both temperature and pressure in the modelling of the condensation process. In chapter 7, dew lines are shown in a phase diagram that is constructed for the natural gas under study. 4.2 Equations of state The relation of pressure, temperature and volume of a gas can generally be written as pṽ = ZRT (4.1) where ṽ is the molar volume, Z is the compressibility factor and R is the universal gas constant. By definition, if the limit for infinite volume, or equivalently zero pressure is taken, the compressibility factor Z = 1 and the ideal gas law is found. At high pressures and low temperatures real gas PVT behaviour deviates strongly from that of an ideal gas. Therefore, more accurate equations are needed to describe natural gases. Equations of state are a science in its own right, as they can be used to qualitatively an quantitatively describe many thermodynamical and chemical variables. Thermodynamic quantities as compressibility factor, Joule-Thompson coefficient as well as phase equilibria can be determined with equations of state. A good summary of the possible equations is given by Wei and Sadus [19]. In the following, the assumption is made that the equation of Peng-Robinson is accurate at describing natural gases. In a pressure formulation the equation is written as where the parameters a and b are given by P = RT ṽ b aα ṽ(ṽ + b) + b(ṽ b) (4.2) 19

21 a = R2 Tc 2 [1 + S(1 Tr 0.5 )] 2, b = RT c, S = ω ω 2, p c p c (4.3) For a pure component, the equation is completely set if only three parameters are known: reduced pressure p r = p/p c, reduced temperature T r = T/T c and the acentric factor ω. The pressure form of (4.2) can be rewritten to obtain an expression for the compressibility factor Z 3 + f 1 (a, b)z 2 + f 2 (a, b)z + f 3 (a, b) = 0 (4.4) The complete form of (4.4) is given in appendix B. This form of Peng-Robinson shows why this is called a cubic equation of state. Other cubic equations of state are the van der Waals and Soave-Redlich-Kwong equation. Interestingly, even if for a pure component, these equations can demonstrate two phase behaviour. If (4.4) is solved for Z at fixed pressure and temperature three solutions are possible. For a thermodynamically supercritical temperature, the one non-complex solution gives the real gas factor for the sole possible phase. For subcritical temperatures, two real solutions are possible. For coexisting vapour and liquid phases in thermodynamic equilibrium, the maximum value of Z will correspond to the vapour phase and the minimum to the liquid phase. Cubic equations of state are therefore capable of predicting the density of both liquid and gas phases. In Figure 4.2, values for the compressibility factor as calculated with Peng-Robinson for pure methane are given in a temperature and pressure range, typical for natural gas pipelines. The derivative of the compressibility factor with respect to temperature is also given. 4.3 Phase equilibria with Equations of State A phase equilibrium of coexisting phases is defined by the equality of pressure, temperature and chemical potential in the phases: p = p L = p V (4.5) T = T L = T V (4.6) µ i = µ L i = µ V i (4.7) An equivalent of the equality of chemical potential is the equality of fugacity in both phases f L i = f V i (4.8) Equations of state can be used to calculate phase equilibria. To understand how many independent properties are needed to do so, consider the following. Gibbs phase rule predicts the degree of freedom F, in a system consisting of π phases and N components: F = 2 + N π (4.9) If only liquid and gas phases are considered, the degree of freedom is equal to the number of components. In the case of a two-phase, pure component system, there is one degree of freedom and setting one intensive property (for example temperature or pressure) fixes all other 20

22 Z [ ] p=30 bar p=60 bar p=90 bar p=120 bar T [K] 5 x dz/dt [K 1 ] T [K] Figure 4.2: Temperature dependent Z and Z/ T, with the Peng-Robinson equation of state for pure methane properties for the liquid-vapour saturation state. For a two-phase multicomponent system, there are a number of possible combinations of properties. For example, setting one intensive property like pressure or temperature leaves N 1 properties to be chosen. These can be all the molar fractions in a single phase 1. Fixing the composition of one phase, and fixing one intensive property of the system thus completely fixes the composition of the other phase, as well as all the other intensive properties. Possible combinations of unknown variables and variables to be found are summarized in the table below, together with the name the problem is commonly given. Known variables Variables to be found Name p, x T, ỹ Bubble point T p, ỹ T, x Dew-Point T T, x p,ỹ Bubble-Point p T, ỹ p, x Dew-Point p p, T, z x,ỹ, L f flash As will be discussed, the combination of interest is the flash problem 2. Here the temperature, 1 This requires choosing only N 1 molar fractions, since ỹ i = 1, and x i = 1 2 The validity of the phase rule for the flash problem is explained in reversed order. If N 1 z i s are fixed in the N equations of (4.10),the number of unknowns is N 1 for ỹ i, N 1 for x i, and 1 for V f, giving 2N 1 unknowns. This makes the system under specified with degree N 1. Since only N 1 z i s have to be fixed 21

23 pressure and overal molar fraction z are known and the molar fractions of liquid and vapour phase are to be found as well as the molar liquid fraction L f (unit [kmol/kmol]). The relation between x, ỹ, z and L f is per component: z i = L f x i + (1 L f )ỹ i (4.10) The first 4 methods, given in the table are mainly useful in constructing phase diagrams for multicomponent mixtures. One of the variables to be found is always pressure or temperature. This only tells at which point the saturation state is reached, but not how much condensation will occur. Solving the flash problem has a more practical purpose. For an input feed of total composition z i, not only the composition z i of both liquid and gas phases can be predicted, but also the amount of condensate with the liquid fraction. To solve any of the problems given in the table, a relation between molar liquid and gas molar fraction has to be known. The fugacity factor Φ i relates molar fraction of liquid and molar phases to the fugacities in both phases, Φ V i = f V i ỹ i p Φ L i = f L i x i p (4.11) so that the iso-fugacity condition of (4.8) gives for the ratio of molar liquid to gas fraction y i x i = ΦL i Φ V i (4.12) Equations of state provide expressions for the fugacity factors, see appendix B for the full expression for Peng-Robinson. With the combination of (4.10)with (4.12), flash algorithms can be written. An example of such an algorithm is given in appendix B.2. and the total amount of degrees of freedom is N, this leaves 2 properties to be chosen freely, temperature and pressure 22

24 Chapter 5 Inner pipe wall temperature 5.1 Introduction As explained in chapter 3, the driving force for condensation or evaporation in the film model is the concentration gradient between bulk gas and the gas at the interface. For this, the gas-side molar fractions of the condensate layer is needed. In chapter 6, a model will be presented that calculates this gradient using phase equilibrium theory. The temperature at the phase boundary is an important input variable. In this chapter it will be discussed how this temperature can be calculated. In a first approach, the interface temperature can be set equal to the inner pipe wall temperature, as condensate film thicknesses are generally very small, and its heat resistance can often be neglected. Measurements of an axial temperature and pressure profile in the pipe is available, see Figure 5.1. In this chapter, local inner pipe wall temperature will be calculated using these profiles. The goal is to evaluate if the wall temperature differs much from the known bulk gas temperature, so that it should be considered in the modelling of the condensation. 5.2 Governing equations The following conservation laws will be used in the following. Continuity equation: Energy equation in enthalpy [5]: ρ t + (ρ v) = 0 (5.1) ρ h t + ρ v h = p t + v p + τ : ( v) }{{} viscid dissipation + q }{{ R } radiation j q }{{} conduction + ρ n g }{{} gravity (5.2) Conductive heat loss An expression is sought for the heat loss due to conduction through the pipe wall over every point along the pipe length, using the p, T profiles of Figure 5.1. First consider the energy equation (5.2). When written in a cylindrical coordinate system, the radial velocity v r is zero, which is correct for very small condensation rates. Due to axial symmetry, the angular 23

25 p [bar] pipe length [km] T [K] pipe length [km] Figure 5.1: measured axial pressure and temperature profile in the bulk gas over the pipeline length velocity v θ is set zero as well. The situation is considered in steady-state, which eliminates the time term on the left hand side. On the right hand side, there are 4 possible heat loss effects for the general case. At the highly turbulent conditions, viscid dissipation can be significant, but it acts only inside the pipe and cannot cause heat loss through the pipe wall. It is therefore discounted for in the measured pressure and temperature profiles. Radiation can be fully neglected at the low temperatures of the pipe and its surroundings. The gravity term is not considered as the measurements are for a purely horizontal pipe. The only term that is considered is therefore heat loss trough conduction. Axial conduction can be neglected compared to axial convection. This leads to the following energy balance: h ρv z z = v p z z 1 (rq r ) r r First, the following cross-sectional averages are introduced: (5.3) u = 4 πd 2 p 2π d/2 0 h = 4 πd 2 p 0 2π d/2 0 v z (r) r dφ dr, ρ = 4 0 πd 2 p h(r) r dφ dr, T = 4 πd 2 p 2π d/ π d/2 0 ρ(r) r dφ dr, 0 T (r) r dφ dr (5.4) Here, u is the averaged axial velocity, ρ is the averaged mass density, h the averaged enthalpy and T the averaged temperature. Pressure needs not to be averaged, as it can be shown with 24

26 the Navier-Stokes equation that it is constant in radial direction, see section Now both sides of (5.3) can be easily integrated over the cross-sectional area to obtain: πd 2 4 ρ u h z = πd2 4 u p z π d q r r=d/2 (5.5) Gas enthalpy is defined as h = c p T, the zero value is taken at h(t = 0K) = 0. Now the energy equation can be rewritten in terms of temperature, density and pressure only. Rearranging (5.5) gives the following expression for the heat loss trough the pipe wall: q r (z) = d ( ) p 4 u(z) z ρ c T p z (5.6) With this expression the inner pipe wall temperature can be calculated in the following way. With Newton s law of cooling, q r = α(t G T W ), (5.7) an effective heat transfer coefficient α is defined that is found from an empirical Nusselt equation. The Gnielinski equation (3.25) is used. The inner pipe wall temperature is thus found with T W (z) = T G (z) q r(z) α, (5.8) where q r (z) is calculated with (5.6) Axial velocity The Reynolds number is used in the calculation of the heat transfer coefficient with the Nusselt number. Therefore, an expression for the axial velocity is needed. This is obtained with the continuity equation (5.1). Neglecting the time term and radial and tangential velocities v r and v θ, it reads: ρ v z z + v ρ z z = 0 (5.9) Again, this is integrated over the cross-sectional pipe area and written with the definitions of (5.4) to obtain: ρ u z + u ρ z = 0 (5.10) With the boundary condition u(z = 0) = u 0 this gives for the axial velocity profile: u(z) = u 0 e 1 ρ ρ z z (5.11) The last unknown is appropriate inlet velocities u 0. Typical volumetric flow rates of the pipe are known and vary between m 3 /h at standard conditions (T std = K, p std = 1 atm). It is assumed that at other conditions differing from standard, equal mass flow rates ṁ = V ρ are present. This implies, ( V ρ )p,t = ( V ρ ) p std,t std (5.12) 25

27 u 0 πd 2 4 ρ 0 = V std ρ std (5.13) The inlet velocity can thus be calculated with: Density and property data u 0 = V std 4 πd 2 ρ std ρ 0 (5.14) Mass density and its axial gradient are needed in (5.5) and (5.11). The density is calculated with the Peng-Robinson equation of state. As pure methane is considered, the temperature is always well above its critical value of K and finding the roots of the compressibility factor is straightforward. The real gas factor is discussed in chapter 4. As the natural gas is extremely high in methane content, all the physical data as heat capacity, thermal conductivity, dynamic and static viscosity were taken for pure methane. The used correlations for the heat capacity and the viscosity are given in appendix D. 5.3 Results For volume flows at standard conditions ranging from m 3 /hr to m 3 /hr the axial velocity profile is given in Figure 5.2. The (p, T ) profiles of Figure 5.1 were used V st =1.0e6 m 3 /hr V st =1.5e6 m 3 /hr V st =2.0e6 m 3 /hr V st =2.5e6 m 3 /hr V st =3.0e6 m 3 /hr 10 u [m/s] z [km] Figure 5.2: Axial gas velocity for a set of volume flows at standard condition 26

28 The resulting axial temperature profile for the inner pipe wall is given in Figure 5.3, together with the value in the bulk gas. The volume flow is m 3 /hr 305 bulk gas wall T [K] z [km] Figure 5.3: Axial temperature profile for bulk gas and inner pipe wall It can be seen that the difference between bulk gas and wall temperature is extremely small, lying in the order of a few Kelvin. It is interesting to see that at a certain point, the bulk temperature goes under the wall temperature. This can only be explained with the Joule- Thompson effect. This effect causes real gases to cool down or warm up during an expansion process at constant enthalpy. The definition of the Joule-Thompson coefficient is ( ) T µ JT = (5.15) p The value of µ JT is strongly dependent on temperature and pressure. For pure methane in a temperature range of 200 to 700 K and a pressure range of 0 to 600 bar this coefficient is always positive [3]. Natural gas with a high methane fraction is thus cooled during adiabatic expansion in this p, T range. This is clarified in Figure 5.4, where the radial heat loss flux is given separately for the pressure and the conduction term in (5.6). The conduction term decreases over the pipe length but it is always positive. The pressure term is negative over the whole range due to Joule Thompson expansion. At a certain point the sum of the two terms becomes zero, causing the temperature difference between bulk and wall to change in sign. H 27

29 80 pressure term conduction term total q r [W/m 2 ] z [km] Figure 5.4: Radial heat loss flux over pipe length 28

30 Chapter 6 Combination model In this chapter, a model will be proposed that combines the film theory with phase equilibria theory to obtain a method that is able to calculate condensation rates for a multicomponent mixture at high pressure. 6.1 Introduction As shown in chapter 3, to predict mass flow with the film model, concentrations (or equivalently molar fractions) of each component have to be known in both the bulk gas as on the gas-side of the phase boundary between gas and condensate. The axial concentration in the bulk gas is relatively easy to calculate if locally the mass transfer flow is known. The challenge is to predict the concentrations on the phase boundary. If there would be a theory that would be able to predict these concentrations perfectly, the condensation rate that the film model predicts are accurate. For simple mixtures at standard pressure and temperature conditions, this is trivial. Take for example the case of an air-water vapour mixture flowing trough a cold pipe. Air can be considered incondensable at room pressure and temperature. This leaves water as the only condensable component. At the gas-side of the phase boundary, water vapour is in thermodynamic equilibrium with its liquid and is saturated in the gas phase. Since for ideal gases the molar fraction is equal to the ratio of partial to total pressure, the molar fraction of water vapour at the phase boundary is given directly by ỹ v = p sat(t ) p (6.1) As indicated, the saturation pressure is only a function of temperature and an empirical equation like Antoine can be used. For multicomponent mixtures with multiple condensable components, the molar fractions in a saturated gas mixture do not only depend on temperature and pressure, but also on the composition of the liquid phase. At low pressures, Raoult s law gives a relation between molar fraction in gas and in liquid per component, ỹ i = γ i x p sat(t ) p (6.2) where γ i is the activity coefficient. At high pressures, the real gas behaviour must be accounted for in the equilibrium of the vapour-liquid mixture. Not only does the gas molar fraction 29

31 depend on the liquid molar fractions, it is also strongly dependent on pressure. As shown in chapter 4, the condition of iso-fugacity in the phases leads to an expression in fugacity factors: Φ L i ỹ i = x i Φ V i Equations of state provide expressions for the fugacity factors. (6.3) Problems arise when it is tried to apply the expressions (6.1) and (6.2) to the situation under study. To predict the concentrations ỹ on the gas-side of the phase boundary with these expressions, the composition of the condensate x has to be known. If there are no measurement data available, it is not possible to predict the composition based on temperature only. Estimation methods, like a flash calculation can be used. For the low pressure case where phase equilibria are not strongly influenced by pressure, this is a possibility. If the pressure is high, calculations are more difficult. It is not possible to choose temperature and pressure, and additionally choose the composition of one phase. If temperature and pressure are known, only a set of combinations of both phases will be possible. The simple relation given in (6.2) is therefore not combinable with mass transfer. 6.2 Construction of control volumes The basis of the combined model is the construction of two separate control volumes. The pipe s cross section is divided in a large control volume lying in the center of the pipe representing the bulk gas, and a small control volume on the pipe wall wherein the condensate in the form of a film is located, see Figure 6.1. To model the diffusive mass transfer between the volumes the film theory is applied. The amount and composition however is calculated with flash calculations. As the model will be implemented in Aspen Dynamics later on, some balancing equations are not necessary. Axial heat and impuls transport are not modelled, giving known pressure and temperature profiles. The model takes the following assumptions: At t = 0, only gas is present in the control volume on the wall, with the same composition as in the bulk Condensation occurs purely filmwise. Film thickness is thin and covers the wall uniformly in a pipe increment In the following, equations that relate to the control volume on the pipe wall will have index cv, the control volume in the bulk will have no index Bulk volume equations The mass conservation in the bulk gas will be derived here. For a pipe element of length dx, diameter d p and volume dv, the total inside the volume (unit [kmol]) is given by M = ρ dv With the bulk molar fraction in the volume given by ỹ i,, the mass per component is 30