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Gaspipe Model Theory

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SAFER, SMARTER, GREENER THEORY Pipebreak Model The model PIPEBREAK models the two-phase discharge from a long pipeline. This report documents the theory underlying the model and includes also validation results Reference to part of this report which may lead to misinterpretation is not permissible. No. Date Reason for Issue Prepared by Verified by 1 Aug 2001 PHAST 6.1 Webber thermodynamically they are partial derivatives along the saturation curve, and are assumed known for whatever substance is being considered. The choke condition ( 17 ) therefore relates the choke pressure implicitly to G through φφφ =     −−+− 2)( Gv dp dv dp dh dp dvv LL L ( 23 ) | THEORY | Pipebreak Model | www.dnvgl.com/software © DNV GL AS. All rights reserved Page 9 This can be solved for known thermodynamic properties and using v(p,G2) as given above to give pchoke(G2) or G2(pchoke). A special case is the initial flow rate which corresponds with a choke pressure equal to the initial pressure p0=psat(T0) and with the specific volume being that of the saturated liquid. In this case we find explicitly         +− = L L L init vdTdvTTc G φ φ 22 ( 24 ) where cL=dhL/dT is the liquid specific heat (modified as above by the heat-transfer term when appropriate). This constitutes a prediction of the initial mass ejection rate (per unit pipe area) which depends only on the initial temperature and the thermodynamic properties of the fluid in the pipe. Numerical note: At the critical point dvL/dT and cL = dhL/dT are both singular but if we define the combined quantity LL hv −≡ φψ it appears that dψ/dT is finite everywhere. For numerical purposes close to the critical point it is convenient to recast these as φψφ =     −     − 21 G dp d dp dv ( 25 ) so that     −     − = dT dT TdT dTvG L init ψφφ φ 22 ( 26 ) Allowance is made for heat transfer, as before, by making the replacement s L s c D Y dT d dT d 4 ρ ρψψ −→ Evolution of the flow Our quasi-steady model for the flow evolution in regime ii is as follows. We start with pure saturated liquid at T0, p0 and the initial flow rate defined above. The length of the two-phase zone, Zone 2 in Figure 1, is initially zero. The flow rate will drop in time. Consider the situation when it has dropped to an arbitrary value G. The upstream pressure, temperature, and specific volume are unchanged. The new choke pressure can be computed from ( 23 ) with v(p) given by ( 18 ) and with a stagnation enthalpy 2/)()( 0 22 0 TvGThE LL += ( 27 ) defined by the upstream end of Zone 2, and the exit pressure is then given by ( 15 ) and the flow will unchoke when G falls to a point where the choke pressure is atmospheric pressure. The volume profile with pressure ( 18 ) can be used in ( 20 ) to define the profiles in x. In particular ( 20 ) can be used to yield the distance to the point (p0,T0,vL(T0)) giving a prediction of the length L2 of the two-phase zone when the given value of G is achieved. | THEORY | Pipebreak Model | www.dnvgl.com/software © DNV GL AS. All rights reserved Page 10 Finally the total mass of fluid (per unit cross-sectional area) in the pipe is found from       ⌡⌠+ −−+−= ⌡⌠+ −= ⌡ ⌠= 0 0 22 00 2 0 2 0 111 2)( )( )( )( p peLl p pl L e e v dp Gvvf D Tv LL v dp dp dx Tv LL v dxM ( 28 ) and the time taken to arrive at this value of G2 can be evaluated from ( 10 ). Let us note that this quasi steady approximation is formally justified as long as the liquid erosion velocity, dL2/dt, which is of order GvL(T0), is much smaller than the fluid velocity Gve at the exit, that is to say as long as vevL. For the most part this is satisfied well enough. Flow in Regime iii - Further Evolution 2.5 The time-dependent flow problem In the case of zero inflow the flash front will encounter the upstream end of the pipe and initiate regime iii. In regime iii the upstream pressure and temperature will also drop and the flow is intrinsically time-dependent, and so we start again from equations ( 6 ), ( 7 ) and ( 8 ). Regime iii does not have the distinct liquid zone and in fact can be thought of as being in many ways very similar to the time-dependent single-phase gas pipeline problem analysed by Fanneløp and Ryhming (1982)4 but with a rather different equation of state. It is therefore useful to follow the method of Fanneløp and Ryhming as far as possible in developing our regime iii model. The method involves estimating the approximate profiles p(x,t), G(x,t) along the pipe, specified as given functions of x (but with unknown time dependence at this stage) to satisfy appropriate boundary conditions, using them to integrate the above equations over x, and obtaining ordinary differential equations in time, which can then be solved for the pressure and flow rates at the ends of the pipe. Encouragingly, Fanneløp and Ryhming showed that the results do not depend sensitively on the details of the chosen profiles. Approximating the energy equation In attempting this here, we rapidly discovered that there is one feature of single phase gas flow which gave Fanneløp and Ryhming an advantage that we do not share in the two-phase analysis. It turns out that the gas flow case can be considered isothermal - the temperature is approximately constant along the pipe. Fanneløp and Ryhming knew this from experience with actual gas pipelines, and in the course of this work we have now also found a formal mathematical derivation of this. Essentially heat transfer into the fluid ensures that any temperature change (from a compressor) near the end is compensated by a return to ambient temperature within a finite distance. For a sufficiently long pipeline, then, the temperature variations occur over a very small fraction of the total length and contribute minimally to the profile integrals along the whole length of the pipe. The specific volume is simply proportional then to 1/p from the isothermal gas equation of state, and the energy equation reduces to T=T0. In the two phase case, not only is the form of v(p) rather more complicated, but the energy equation cannot reduce to anything like T=T0. This last is very clear if homogeneous equilibrium holds (as indicated by Figure 3) as a pressure drop along the pipe must be accompanied by a temperature drop as p=psat(T) holds at every point along the pipe. If we are to make further progress with an analytic model of time-dependent two-phase flow, we first need an analogously simple approximation of the energy equation, but one which can support homogeneous equilibrium flow. The key to generalising the model of Fanneløp and Ryhming to two- phase flow is to observe that for gas flow, one might just has well have specified a constant specific | THEORY | Pipebreak Model | www.dnvgl.com/software © DNV GL AS. All rights reserved Page 11 enthalpy h. Because, for simple gases, h~T it makes no difference whether one chooses constant h or constant T. It makes a big difference for two phase flow however. And because the reason for the constant T is largely to do with heat transfer, then constant enthalpy is in fact a more physical way of thinking of it. We can adopt this uniform enthalpy hypothesis as an estimate of the behaviour of two- phase flow down long pipelines where any heat transfer may keep the overall specific enthalpy relatively constant along the pipe. In fact we'll go one step further and write EvGh =+ 2/22 ( 29 ) for some uniform E. This makes little difference compared with assuming uniform h as the velocity (Gv) term is generally small compared with h (except perhaps very close to a choke) especially by the time we're in régime iii, as by then the outlet velocity will have decreased very much from its initial value. Adopting this equation also conveniently gives us (with the homogeneous equilibrium thermodynamics) exactly the same v(p) profile ( 18 ) as we had before, except that we are now considering that G may vary along the flow. However, consistently with the above remarks on the velocity terms in the simplified energy equation, we expect v(p) to be very close to its limiting (G→0) form throughout régime iii, and so any variation of G along the flow will make little difference to v(p). For the same reason, in practice the stagnation enthalpy E varies only very slightly in time throughout régime ii, and so, consistently both with régime ii and with the gas flow analysis we shall take it to be constant in time through régime iii of the two-phase flow, and give it whatever value it adopts at the end of régime ii. It is also important to note that h really is just the fluid enthalpy here. The arguments used in régime ii to include heat transfer by means of a pipe term added to the enthalpy only apply to quasi-steady flow and break down for fully time dependent flow. On the other hand we are allowing for heat transfer here as it is only the existence of heat transfer which allows us to argue for constant specific enthalpy in the gas flow case, and by extension for constant E here. Approximating the momentum equation Fannelop and Ryhming (1982)4 neglect the ∂G/∂t in the momentum equation, arguing for a balance of pressure gradient and friction. With this approximation they integrate the momentum equation ( 7 ) along the length of the pipe. This gives ∫=⌡⌠  + Lp p dxGDfvdpdpvGd e 0 2 2 2)( 1 0 ( 30 ) The profile method can be illustrated by approximating the mass flux profile as [ ] )/(2 0 22 0 2 LxKGGGG e −+= ( 31 ) for some function K with K(0)=0, K(1)=1. And so the integral so the right hand side of ( 30 ) can be simply estimated. In fact we shall consider the case G0=0 as being the only one of interest for régime iii and define k as the integral of K(z)dz from 0 to 1. The constant k will be of order 1. It is worth noting in fact that Fannelop and Ryhming also neglected the d(G2v)/dp term on the left of the momentum equation. It tends to be small except near the choke and in the gas flow problem they were considering unchoked flow for much of the time. We can do slightly better in the case of choked flow (and a relatively large range of specific volume) if we approximate dp dvG dp vGd e 2 2 )( λ≈ ( 32 ) with λ a constant of order 1. (Fannelop and Ryhming's approximation is recovered by setting λ=0.) With these approximations equation ( 30 ) reduces to | THEORY | Pipebreak Model | www.dnvgl.com/software © DNV GL AS. All rights reserved Page 12 D Lkf v v v dp G e p pe e 2ln1 0 2 0 =    − ⌡ ⌠ λ ( 33 ) In principle the free constants k, λ reflect our lack of knowledge of the profiles. However, in practice we can demand that they be such as to make the results behave as continuously as possible through the transition from régime ii to régime iii. This effectively fixes k=1 and λ=1 and is consistent with assuming that changes in v along the pipe have much more important effect on the momentum balance than changes in G2. It also means that the choke pressure may be estimated exactly as in régime ii. The momentum equation now relates the two end pressures pe and p0 to the outflow Ge. We already have v(p) and so we can make use of it without any further profile assumptions. However, as before, we need to know about v(p(x)) to evaluate the mass release rate and here an assumption about the pressure profile in x is required. Our first approximation followed the gas flow analysis exactly: we set p(x) to be quadratic in x with the appropriate values at the ends and dp/dx=0 at the closed upstream end. We discovered, however, that the isothermal gas equation of state v~1/p makes the mass integral uniquely tractable, and failed to make analytic progress to the same extent with our more complicated two-phase v(p). The maths gets complicated very rapidly here, which rather defeats the object of the approximation: choosing a “simple“ profile to get results simply. Therefore, noting Fannelop and Ryhming's observation that the results are not strongly sensitive to the precise profiles chosen, it is worth asking if there is an appropriate choice which simplifies the calculation in this case. In fact there is. Let us approximate the pressure profile p(x) by one which satisfies D vGGkf dx pvGd eee ||2)( 2 −=+λ ( 34 ) We don't have an explicit form for it and it only approximately satisfies dp/dx=0 at the closed end, but it does have the right values at each end as well as the correct analytic behaviour near the choke, and again it is very convenient in that it gives complete continuity of the mass integral (with the above values of k and λ) through the transition from regime ii to regime iii. With this profile, the mass integral is in fact       ⌡⌠+ −−=⌡⌠=⌡⌠= 00 22 00 111 2 p pee p p L ee v dp Gvvkf D v dp dp dx v dxM λ ( 35 ) and can be evaluated as before from our knowledge of the specific volume profile v(p). Evolution of the flow in régime iii As earlier we can evolve the flow by decreasing the mass flux density G to a new value. The stagnation enthalpy E is assumed not to change. The exit pressure is computed exactly as in ( 15 ). Then ( 33 ) is solved for the new upstream pressure p0, using the same volume profile v(p,Ge) as before. The mass of fluid in the pipe is given by the integral ( 35 ) and finally the time at which this new value of G is achieved is computed from ( 10 ) [with G0=0]. This completes the main description of the model. It remains only to compare it with experimental data. Thermodynamic and fluid dynamic properties 2.6 In order to evaluate solutions of the equations we need a number of thermodynamic properties of the fluid in the pipe. | THEORY | Pipebreak Model | www.dnvgl.com/software © DNV GL AS. All rights reserved Page 13 Fluid dynamic properties 2.6.1 The required fluid-dynamic (as opposed to thermodynamic) parameter is the Fanning friction coefficient. Fannelop (1994)7 p115 quotes a formula used by the American Gas Association:    = 0 10 7.3log41 z D f ( 36 ) and recommends taking a roughness length for the inner pipe wall z0=1.3 10-5m (0.0005 inches) in the absence of any better information. Following this we estimate f=0.0029. (It turns out that the results - especially the mass release rate - are not enormously sensitive to this value and so we shall not, at this stage, explore further refinements.) Thermodynamic properties 2.6.2 There is a choice of three different methods for determining thermodynamic properties which go under the names “Simple HEM“, “DNV HEM“, and “Hybrid HEM“. They were developed in this order and will be described in this order below. The last is the recommended one, and has currently been made available only. The Simple HEM The Simple HEM is as follows. We approximate the liquid specific volume vL as constant so that dvL/dT=0. The liquid specific enthalpy is modelled as hL=cLT with constant specific heat cL. The vapour pressure curve is fitted in the appropriate range by the Clausius-Clapeyron form p=Aexp(
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