SVHIM

Consider the iBow of air from a large reservoir vessel at 1.101 MPa and 20°C through 10 m of horizontal Schedule 40 steel pipe (inside diameter = 52.5 mm), three standard elbows and a control valve to the atmosphere, assumed at 0.101 MPa. The length of pipe upstream of the valve is 4 m and contains one standard elbow. The pipe inlet is abrupt. 

The control valve is a 2-inch globe valve, with a linear characteristic, a full-travel liquid sizing coefficient Cy = 65.3 US gall/min/psi and a full-travel gas sizing coefficient Cq = 2280 scf/h/psia. The control valve travel is initially 100%, but this is decreased by a 5% per second ramp starting at time = 5 seconds to 5% open at time = 24 seconds. The control valve is then maintained in this position to the end of the transient at time = 30 seconds. 

Calculate the mass flow and the thermodynamic conditions at various points in the pipe as functions of time. 

SVHIM is considered to be the most accurate of the methods presented, and has been selected on this basis for the first analysis of the problem so as to bring out the most important features. 

Assume that the Reynolds number is greater than 100000 at all times, and so the Fanning friction factor may be taken as / = 0.0045 from equations (4.28) and 

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The remarks made in Section 8.1 translate across to the gas flow cases more or less word for word, except that the methods of Chapter 9 must now be allied to those of Chapter 6 in order to calculate gas flow through line and valve. But the more complicated equations for both line flow and valve flow render explicit solutions to the full set of equations impossible. Two implicit methods, the Velocity-Head Implicit Method (VHIM) and the Smoothed Velocity-Head Implicit Method (SVHIM), will be presented, where the solution process has been reduced to iteration on a single variable. The SVHIM is judged to be more accurate because it deals with the compressible-flow valve equations at all times. 

An approximate meth, the Average Specific Volume Approximation Method (ASVAM), is also presented, based on the Long-Pipe Approximation described in Chapter 6. This approximate method retains much of the accuracy of the SVHIM, but has the advantage of yielding a direct estimate of flow. 

Gas flow through an installed valve - Smoothed Velocity-Head Implicit Method (SVHIM)... 

SVHIM seeks to circumvent these problems by attacking the full set of line and valve equations at the outset. It is again necessary to solve the equations for subsonic and sonic flow in the valve separately, but the transition should now be smooth. The necessary equations for the subsonic case will now be listed in a logically consistent solution sequence. The basis of the method is first to guess and then to iterate on the value of the ratio of specific volumes Vi, /V2. 

Allowing for sonic flow in the valve using SVHIM... 

Figure 10.4 Pressures at various locations in the pipe (SVHIM).

The pressure transient calculated by VHIM is shown in Figure 10.8, which matches closely that plotted in Figure 10.4 for SVHIM. In particular, it will be seen that the pipe outlet pressure, p3, has reduced to atmospheric, p4, by 20 seconds, in agreement with SVHIM, indicating that the outlet flow is subsonic at this time. 

Figure 10.11 compares the flow transient calculated by ASVAM with the standard transient calculated by SVHIM. ASVAM, like VHIM, relies implicitly on the C value to characterize valve flow in the subsonic region via the calculation of AT, and then Kp. As a result, ASVAM underestimates the flow by about 3% at the beginning of the transient in the same way as VHIM. But ASVAM produces essentially the same value as SVHIM for flow by time = 15 seconds. In fact, ASVAM predicts sonic flow in the valve at time = 16 seconds, a second in advance of SVHIM, but the difference in flow is very small. 

We shall assume that the values calculated by SVHIM for a 70% valve travel are available, namely ... 

The agreement with the standard case of SVHIM is generally very good. The flow when the valve is fully open is 2.26 kg/s, within 1% of the SVHIM value. SASVAM predicts the onset of sonic flow in the valve at 16 seconds, which is close to the SVHIM value of just after 17 seconds. 

SVHIM provides the most complete treatment of both valve and pipe, and is considered to have given the most accurate calculation of flow and thermodynamic conditions. However, this method requires an iterative solution, which is a disadvantage in a dynamic simulation, where the flow is likely to need calculating many hundreds or thousands of times, possibly over a wide range of conditions. Non-conveigence is an everpresent danger for which the modeller must always be on his guard when the solution is iterative. 

VHIM is based on a simple transfer to the pipe-plus-valve case of the method outlined in Section 6.4 for calculating flow in a pipe. It will be less accurate than SVHIM in the subsonic-valve region because only the liquid valve coefficient, C , is used in this flow regime, rather than the more representative gas coefficient, Cg. This causes a small discontinuity to occur when sonic flow conditions are met in the valve, and the C characterization is superseded by a characterization based on Cg. The loss in accuracy compared with SVHIM is 3% or less, but VHIM retains the disadvantage that it requires an iterative solution. 

All the accuracy figures above have been quoted against the standard of SVHIM, and these may be put in context by the consideration that no method can... 

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