^{1}

^{2}

^{*}

Previous analytical results on flow splitting are generalized to consider multiple boiling channels systems. The analysis is consistent with the approximations usually adopted in the use of systems co des (like RELAP5 and TRACE5, among others) commonly applied to perform safety analyses of n uclear power plants. The problem is related to multiple, identical, parallel boiling channels, connected through common plena. A theoretical model limited in scope explains this flow splitting without reversal. The unified analysis performed and the confirmatory computational results found are summarized in this paper. New maps showing the zones where this behavior is pre dicted are also shown considering again twin pipes. Multiple pipe systems have been found not easily amenable for analytical analysis when dealing with more than four parallel pipes. However, the particular splitting found (flow along N pipes dividing in one standalone pipe flow plus N -1 identical pipe flows) has been verified up to fourteen pipes, involving calculations in systems with even and odd number of pipes using the RELAP5 systems thermal-hydraulics code.

In nuclear power plants reactors, natural circulation is one of the mechanisms for decay heat removal after a transient or accident. In this type of installations, multiple parallel tubes are a common system configuration. The complexity of the flow pattern may be exemplified by simply mentioning that after a very small break loss of coolant accident (SBLOCA), the natural circulation flow map shows different regimes, from single or two- phase natural circulation to reflux condensation regime (see e.g. [

The behavior of a set of parallel tubes or inverted U-tubes under natural or forced circulation systems in single and two-phase flow with common inlet an outlet may become complex. It is well known that these configurations can develop several types of instabilities, classified as static or dynamic. In the first case, only steady- state conservation laws are required for instability prediction.

In a previous study (see [

Other authors have studied flow rate distribution (see [

In this paper, previous results from the present authors are extended with a unified analysis to consider multiple but still small (up to fourteen) number of N identical boiling pipes systems. Results have been verified through RELAP5 calculations. The interest in increasing the number of channels came from the possible existence of instabilities during transients in nuclear reactor designs under analysis with flow driven by gravity coming in (twelve) virtual fluid channels from a core exit cross section. As a consequence, results have been obtained for thermodynamic conditions corresponding to nuclear reactor designs. When considering multiple pipes, the possible flow splitting became even more complicated and above four, and the behavior was only studied using RELAP5. The peculiarity of the results is that under certain conditions, the splitting consisted in an increased fraction of mass flow in one pipe and the corresponding remaining fraction distributed in equal mass flow rates in the remaining N − 1 pipes. No reverse flow exists. The conditions for such a behavior has been established and represented in maps numerically computed with an in-house developed code and verified, as mentioned before, using the systems thermal-hydraulics code RELAP5. Not surprisingly, modification of simulation options (such as user options on models or numerical schemes) with the systems codes may lead to different predictions, including code failure.

It may be of interest to introduce here a comment on the role of the scope of the analytical model: even when the hypothesis may look restrictive, for the intended application they are appropriate. This has been verified using a code with no approximations regarding friction losses and with two-fluid models. The agreement was very satisfactory providing confirmation of the appropriateness of the hypothesis made for this analysis. However, it should be noted that depending on boundary and geometrical consideration in the analytical models as presented in this work and in other references, flow recirculation in the inlet or outlet plenums may not be captured. This is due to model simplifications but these peculiarities may be captured with system code such as RELAP5. Again, it is pointed out that the results herein shown are directly applicable to nuclear reactors because of the working conditions considered. This is coherent with the previous results of the authors as shown in what follows.

To introduce the analysis and due to its simplicity, the results of [

describes this curve and analyze the channel behavior depending on boundary conditions and may be found in [

Different approximations to analyze this problem are considered in what follows.

Let us consider a single boiling pipe and homogeneous, equilibrium fluid model (HEM) as shown in

The pressure drop along such a boiling channel may be written as:

where Z_{B} is the non-boiling length, W is the mass flow rate, A is the channel flow area, g is the gravity acceleration, L is the channel length, f is the friction factor (assumed constant in this analysis and equal in single and two phase flow and denoted as f_{TP}), ρ_{f} is the liquid density at saturation and ρ_{m} is a mean density in the two phase region. For simplicity, as the mean density, the outlet density was used. Parameters k_{i} and k_{e} are the concentrated friction values at the inlet and outlet of the individual channels, respectively.

where ρ_{e} is the mixture density at the channel outlet.

Dry steam zone in the pipe is not considered for these stability studies because, as it will be shown latter, instabilities are expected to occur when fluid becomes saturated. The acceleration term may be disregarded for this analysis because it has low relevance on the instability threshold. In passing, it facilitates finding the analytical solution.

Some useful (and commonly used) definitions are:

where N_{p}, N_{S} and N_{fr} are the phase change, subcooling and friction dimensionless numbers; v_{fg} is the difference between the gas and liquid specific volumes and h_{f} and h_{fg} are the saturation and latent heat, respectively.

In two parallel and identical channels the pressure drop through then are equal; i.e. ΔP_{1} = ΔP_{2}. It is well known that this system may have several stable solutions. This means that for equal pressure drop, different mass flow rate for each tube come up.

In previous references, like in [_{1} + W_{2} = W_{T} (constant).

Defining:

Then

The dimensionless magnitude N_{pM} is the phase change number corresponding to a channel with same imposed heat as the single channel but with the total system mass flow rate. The channel non-boiling length Z_{B}_{1} and Z_{B}_{2} are defined as the positions at which the fluid in each channel becomes saturated, that is

Each pressure drop term will be evaluated in what follows. Recall that the expression of ΔP for channel 2 will be the same as for channel 1, exchanging f by (1 − f). The friction terms in (1), using the definition of the dimensionless numbers, reads

where

The gravity contribution, results,

Once again, the value ρ_{m}_{1} will be calculated at the exit quality turning out ρ_{e}_{1}. Adding (5) and (6), the total pressure drop along channel 1 is obtained, based on the hypothesis of the model.

The pressure drop ΔP_{2} comes from to ΔP_{1} changing f by 1 − f. Then

For simplicity, k_{em} and k_{im} denote k_{e} and k_{i} divided by N_{fr}. The objective is to find the fixed points of the equation

where Fo is the Froude number defined as

The velocity v is the inlet velocity and it was expressed as function of the total mass flow rate W_{T}. The area corresponds to one channel flow area.

When gravity is added to the pressure drop term, the function

As shown in _{i} with i = 1 to 3. When two pipes are joined together, then up to nine steady states may arise combining all possible the mass flow rates.

Then, at most, 3^{N} different mass flow rates may arise, with N the number of pipes. When identical pipes are used, several solutions are the same and are disregarded. For two identical pipes, at most 6 possible solutions may be obtained. Straightforward application of the lumped parameter model considering the values shown in

1) For each inlet pressure within a range, all possible mass flow rates are determined.

2) Since both channels have the same mass flow rate, then for each inlet pressure the total mass flow rate is determined as the sum of individual channels mass flow.

3) The inlet pressure is plotted for each mass flow as it is shown in

Recalling the fraction of flow splitting f_{1} = W_{1}/W_{T}, a flow splitting map as function of the total mass flow rate is depicted in

In

Nomenclature | Value | Variable/units |
---|---|---|

D | 0.0124 | Inner channel diameter (m) |

k_{i} | 23 | Inlet concentrated pressure losses (-) |

k_{e} | 5 | Outlet concentrated pressure losses (-) |

F | 0.01 | Friction factor (-) |

L | 3.66 | Total length of the U-tube (m or ft) |

ρ_{f} | 739.86 | Liquid saturated density (kg/m^{3}) |

ρ_{g} | 36.6 | Vapor saturated density (kg/m^{3}) |

µ | 9.46 × 10^{−5} | Liquid Dynamic viscosity at saturation (Pa s) |

W_{i} | 0.1 | Mass flow rate per channel i (kg/s) |

P | 7 | Absolute pressure (MPa) |

T | 20˚C | Inlet temperature (˚C) |

Q | 120 - 200 kW | Power delivered to each pipe |

The momentum conservation equation for two channels may be written as

Being m_{k} = L_{k}/A_{k}, W_{T} and W_{k} are the total and the k (i.e. 1 or 2) channel mass flow rates, respectively. Recalling that W_{1} + W_{2} = W_{T}, and m = m_{1} = m_{2} (from hereinafter all channels are assumed identical among them). Perturbing the mass flow rate in both channels and the total as W_{k} = W_{k}_{0} + δW_{k}, where W_{k}_{0} corresponds to the steady state solution and δW_{k} are the infinitesimal perturbations.

Now, considering that the perturbation may be written as δw_{k} = β_{k}e^{λ}^{t}. Calling

Finally, the above system equations could be written in matrix form. The determinant of the system of equations is:

The characteristic equation is

The system is stable to perturbations when the real part of the characteristic equation roots are negative, i.e.

For a constant mass flow rate or a vertical external characteristic curve,

It could easily verified that the system is stable if

or

Since ΔP_{1} y ΔP_{2} are functions of f (the mass flow rate fraction) then

For twin parallel boiling channels, using the lumped parameter model disregarding the gravity contribution the stability region could be easily obtained, given by

with the constrain that N_{pm} ≥ N_{s}/f. For_{pM} outside the shaded area in

that shows the stability map for a two twin boiling parallel channel with vanishing gravity force contribution; the inner area corresponds to the instability area. It has to be remarked that the horizontal axis is N_{pM} variable defined as the phase change number of channel with total system flow rate and the power of individual channel, i.e. N_{p}_{1} = 2N_{pM} since f = 0.5. The red lines indicate the thermodynamic quality of one in a single channel.

As stated above, a stability map may be built using the analytical expression (22) for twin parallel channels. However, in what follows a larger number of channels will be considered, so an analytical expression may be complex to obtain. Another, but equivalent way for plotting the stability map for twin channels, is evaluating the pressure drop over the N_{s} − N_{pM} plane for a constant total mass flow rate f = 0.5 ± Δ, for each channel. This corresponds to use (21). Here D is a small and arbitrary parameter used to calculate a finite difference around 0.5. The difference of pressure drop in each channel divided by 2D gives the plotted discrete approximation of the derivative.

It is usual to perform parametric studies to check the effects of different system parameters on its overall behavior. For instance, in _{s} values. The outlet concentrated pressure losses tend to destabilize the system, so increasing the outlet concentrated pressure coefficient k_{e} to 10 or 50 as is shown respectively in

In

This oscillation shows a sign change indicating intermittent flow reversal. Static instabilities and dynamic instabilities in boiling channel may come up together, and both could be distinguished due to the oscillation characteristic time. For instance, density wave instabilities have larger oscillatory frequency in relation to pressure drop oscillation. Another example of this behavior will be shown later.

The procedure performed in [

In this section a general procedure to get the stability regions for multiple identical boiling pipes is outlined. A stability map is built, as above, evaluating the stability condition related with pressure drop over the N_{s} − N_{pM} plane for a constant total mass flow rate.

For an arbitrary number of parallel boiling channels, the procedure becomes somewhat cumbersome. Writing the momentum conservation equation for each channel, perturbing their flow rates with respect to the steady state values and rearranging, N differential equations are obtained. Writing the set of equation in matrix form:

where m_{k} is quotient between length and channel area,

mass flow rate. Stability regions are determined by finding the conditions where the real part of the characteristic equation roots is negative. Analytical conditions could be written based on the coefficient of the characteristics equations but are difficult to apply. From Hurwitz theorem, it could be derived that a necessary condition for having negative real roots is that all characteristics coefficients should be negative. However, the reverse is not true.

The conditions for flow stability for N parallel identical channels then become

Recalling that the fraction of mass flow though a channel is:

when bifurcation or the conditions for several multiple steady states are met in N parallel channels, the splitting will develop in pairs. The latter means that N − 1 channel will have the same flow rate and only one different. The flow ratio for N − 1 is equal, and hence

and the fraction of the individual channel as function of the other N − 1 channel results:

Since

Considering that

The latter expression is the necessary condition for N parallel tubes with the condition that splitting is channel to channel. The quantity _{1} is evaluated at expression between brackets. In addition the stability area is also delimited by

For three parallel channels, we follow the same procedure as in Section 2.3, namely: writing the momentum conservation equation for each channel, perturbing their flow rates with respect to the steady state values and rearranging, the system characteristic equation is determined from the vanishing determinant of the following matrix.

It was assumed, as above, that the three channels have the same length and flow area and its ratio is denoted by m. The characteristic equation has the form:

To determine the stability regions, the conditions to get a negative real part of the characteristic equation roots must be found. Applying the Hurwitz theorem said conditions are:

Recalling that

The third condition could be analyzed in the following way. Replacing

Condition (35), neglecting the first term in brackets, may be rewritten

or

The above expression is a necessary condition to get the stability region for static instabilities for three boiling parallel channels. It must be noted that stability depends on a single channel stability condition since if just one of the

The stability conditions for four channels are:

It may be shown using a similar procedure performed previously for three channels that the necessary conditions are:

and

Both Equations (31) and (32) are necessary conditions for stability.

Stability maps for any number of channels may be constructed by using expression (23) and (28). In the latter equations no assumptions should be made, however; more computational time is required if the same number of points have to be evaluated. For the sake of simplicity, only cases with HEM model and gravity are shown in this section and the same conclusions could be applied for HEM neglecting gravity or considering the two fluid model. _{pM} to consider any number of channels in the same graph. When the number of channels is increasing, the left boundary of the instability region increases its slope and the area becomes smaller while keeping constant the concentrated pressure drop coefficients at the inlet and outlet. It may be observed that unstable regions overlap for all the configurations shown. Further refinement of the sample grid to eliminate the ripple was not attempted.

RELAP5 is a best estimate systems thermal-hydraulic code extensively used for nuclear safety evaluations related to nuclear power plants. It provides numerical results using closure correlations and steam tables. Results heavily depend on user choices. The present authors discussed in detail the effects of different code options and fluid models (see [

It is interesting to verify firstly the behavior of a system composed of three or four parallel channels. For simulating this case, additional pipes have been added in parallel to the twin pipes case and identical to the others. The inlet mass flow rate and power delivered to the channel were changed accordingly. It must be noted that the same rate of power delivered to the fluid was not used in all cases. The system behavior for three parallel channels using RELAP5 with and without gravity and the HEM or two-phase flow model is shown in

When gravity is taken into account using HEM, no flow splitting was observed at least at the expected power. Moreover, the RELAP5 code fails near to the 2000 s, corresponding to 200 kW. On the contrary, with two-fluid model and gravity, flow splitting occurs at 1300 s, corresponding to 167 kW.

RELAP5 calculations for four identical channels considering different modeling options are shown in _{e} equal to 25 comes at 119 kW, whereas for two fluid model near to 170 kW. This is consistent with the three channel models showing that two fluid model requires more power to become unstable. This difference might be due to the nucleate subcooled boiling region before fluid saturation. In all cases k_{i} was set to 23.

In addition, as in the three channels case, flow splitting appears in (1 and N − 1) pairs. It should be noted that power required for flow splitting is almost independent of the number of channels; this means that for two, three or four parallel channels the needed power is ranged between 120 to 130 kW. This supports the idea that flow splitting or flow mal-distribution in the system comes up when a single tube becomes unstable.

It is interesting to consider a 3D representation of the flow splitting for the four tubes case plotting the time variation of N_{pM} for a given value of N_{s} when power is increased (then changing the value of flow rate). This plot is shown in _{s} vs N_{pM} shows the instability zone as predicted by the theoretical analysis. The case shown corresponds with _{pM} for N_{s} =15 shows that splitting occurs when N_{pM} reaches the left limit of the instability zone as predicted by the theoretical analysis.

Some results for nine and twelve parallel identical tubes using a two fluid model are shown in

For nine parallel identical tubes, flow splitting comes again in pairs supporting the previous results for two, three and four identical channels. As above, eight channels increases their mass flow rates while the one decreases it. It is interesting to note that the channel that increases the mass flow rate after flow splitting becomes subcooled. When running the above cases with the HEM model, the fluid in the individual channel is in single phase while in the two fluid model remains in subcooled boiling with a fluid temperature closed and below to saturation. This implies that the other eight channels work in nucleate boiling with a very high flow quality.

Flow oscillations may be observed in

Despite of the behavior showed in _{e}, set to 10. In this fourteen channels configuration, two channels increase the mass flow rate while the rest decrease it. This unexpected (2 & N − 2) result may imply that analytical stability prediction is just a necessary but not sufficient condition for the 1 & (N − 1) splitting or that other effects may come up with increasing number of channels.

The flow splitting should be related to static instabilities since one channel increases mass flow rate with the other channels decrease.

Static instabilities in boiling channels with common plena, fixed inlet mass flow rate and outlet pressure have been studied. An analytical simplified model using HEM was first developed to study static instabilities in twin parallel channels. The latter analysis was extended to three and four parallel channels and, finally, for N parallel channels. Using a perturbation analysis to the momentum equation for HEM model, a stability analysis was performed for two, three and four parallel channels and a necessary condition and the stability maps based on dimensionless numbers were obtained. It was shown that when the instability boundary was crossed, the mass flow rate in the channels split in non-symmetric ways. When a larger number of channels are considered, it is found (using RELAP5 systems thermal-hydraulics code) that splitting also comes up in this particular way: one channel increases (decreases) mass flow whereas the other N − 1 channels decrease (increase) mass flow rates. From necessary stability conditions, a system with several parallel channels may be stable while some of the channels fall individually in the unstable condition.

Concluding, stability maps for flow splitting in a system of up to fourteen identical parallel pipes have been considered, showing similar behavior although the splitting occurs first in the expected way followed by another one. This stands for the latter case in the explored range of parameters. It is shown that gravity has a stabilization effect in the sense that the unstable region becomes smaller. In addition, when the outlet concentrated pressure drop coefficient increases, the system becomes more unstable, as is well known from density wave oscillations analysis. Both instabilities may appear in sequence.

Alejandro I. Lazarte,J. C. Ferreri, (2016) Analytical and Computational Analysis of Flow Splitting in Multiple, Parallel Channels Systems. World Journal of Nuclear Science and Technology,06,170-190. doi: 10.4236/wjnst.2016.63019