Flow Instability in Two-Phase Circuits
Flow Instability in Two-Phase Circuits
In once-through boilers or evaporators generating steam at high quality, the problem of flow instability is often a concern. This is due to the nature of the two-phase pressure drop characteristics inside tubes, which can have a negative slope with respect to flow under certain conditions. The problem is felt when multiple streams are connected to a
344 Steam Generators and Waste Heat Boilers: For Process and Plant Engineers
common header in once-through boilers. Small perturbations can cause large changes in flow through a few tubes resulting in flow depletion, dry-out, or overheating of tubes. Vibration can also occur. The problem has been observed in a few low-pressure systems generating steam at high quality.
To illustrate the problem, let us take up the example of steam generation inside a tube.
A few assumptions will be made:
1. Heat flux is uniform along the tube length.
2. Steam at the exit of tube has a quality x.
3. Some subcooling of feed water is present. (The feed water enters the tube at less
than saturation temperature.) If a tube is supplied with subcooled water, the boiling process starts after the enthalpy of
the water has reached the saturated liquid state. Thus, the length of the boiler tube can be divided into two sections, the economizer and evaporator, and their lengths will be deter- mined by heat input to their respective sections.
Let W be the flow of water entering in lb/h. Let Q = total heat input to the evaporator and Q l the heat input per unit length, Btu/ft h. Let the length of economizer portion be
L 1 ft. The pressure drop ΔP 1 in the economizer section is . 3 36 10 − × 6 fLWv 2 1 f
where Δ h is the enthalpy absorbed, Btu/lb v f is the average specific volume of water in the economizer section, ft 3 /lb
d i is the tube inner diameter, in. The pressure drop in the evaporator section of length (L − L 1 ) is given by
∆P =
× − 6 f L − L W . 2 3 36 10 ( 1 ) v f +× . 5 ( v g − v f )
(6.32) Also,
xh fg ∆= ( L − L 1 ) /L 1 (6.33)
As the energy applied is uniform along the evaporator length, we are simply taking the ratio of energy absorbed in the evaporator and economizer, as being proportional to their lengths.
h fg is the latent heat in Btu/lb
v g ,v f are specific volumes of saturated liquid and vapor, ft 3 /lb
Miscellaneous Boiler Calculations 345
Substituting for x from (6.33) into Equation 6.32 and for L 1 from (6.31) and simplifying, we have
∆ P = ∆ P 1 + ∆ P 2 = kW h v 3 ∆ 2 ( g − v/Qh f ) 2 2 l fg – kW ∆ hv ( g − v /h f ) fg − v f + k W WL Q v 2 1 ( g − v/h f ) 2 fg (6.34)
or, ∆P AW 3 = – BW 2 + CW (6.35)
Though this is a simplistic analysis, it may be used to show the effect of variables involved. Equation 6.35 is depicted in Figure 6.15. It is seen that the curve of pressure drop versus flow is not monotonic but has a negative slope. This is more so if the steam pressure is low. Hence, it may lead to unstable conditions. For example, at some pressure drops, there could be three possible operating points, which may cause oscillations and large flow variations between circuits if there are multiple streams. (For the meaning of streams, refer to Appendix B.) Some tubes with low flows can reach departure from nucleate boiling conditions.
To improve the situation, one may place a restriction such as a control valve or orifice at the economizer inlet. The orifice increases the pressure drop in proportion to the square of the flow as shown by the term R in Equation 6.36. Figure 6.15a shows the effect of orifice, which makes the pressure drop monotonic:
∆P AW 3 = + ( R − BW 2 ) + CW (6.36)
Due to the large specific volume and latent heat of steam at low steam pressures, the problem is more likely at low pressures as shown in Figure 6.15b. Decreasing the inlet subcooling by
Stable Orifice P 1
P 2 Unstable P 3 Pressure drop
Pressure drop P 3 >P 2 >P 1
t=t stat t<t stat
t << t stat
Pressure drop
Effect of (a) orifice size, (b) pressure, and (c) inlet subcooling on stability of two-phase boiling circuits.
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using a higher feed water temperature also helps as shown in Figure 6.15c. If inlet subcooling is eliminated, Δh = 0, and then Equation 6.35 becomes more stable as shown here:
2 ∆P BW = + CW (6.37)