Steady-state Flow Characteristics

Chapter 2: Review of Literature

2.2. Steady-state Flow Characteristics

Steady-state implies a condition where the system parameters are invariant with time. Assessment of steady-state characteristics is important both for system design and performance comparison point of view and hence various theoretical models have been proposed Chatoorgoon, 2001 was probably the first one to develop an analytical model of single-channel SCNCL, as he presented numerical results using SPORTS (Special Predictions Of Reactor Transients and Stability) code (Chatoorgoon, 1986). He studied both distributed and point heat source and sink, with the point source being a simplified representation to eliminate nonlinear effects. Solving conservation equations for mass, momentum and energy, steady-state flow rate was found to increase with power supply till a maxima and decrease afterwards (Figure 2-2). He postulated that the stability threshold for such system can be determined through steady-state analysis using the following criterion.

𝜕𝐺

𝜕𝑄= 0 (2-1)

Therefore the location of maximum flow rate was identified as the threshold. The value of power corresponding to that maxima was termed as the bounding power. Integration of steady-state momentum equation around the loop also provided a closed-form expression of mass flux having the following form,

𝐺²= 2𝐷𝑔𝐻(𝜌₁− 𝜌₁) 𝑓₁𝐿₁+ 𝑓₃𝐿₃

𝜌₁ +𝑓₂𝐿₂ 𝜌₂

(2-2) Here 1 refers to the section between entrance and heat source, 2 is the hot leg between heat source and sink and 3 is the cold leg of the loop. 𝑓_𝑖 and 𝜌_𝑖 indicates the friction factor (inclusive of obstruction coefficient) and fluid density in the 𝑖^th section. A simple equation of state was considered with density being a sole function of enthalpy and no pressure dependence. Couple of non-dimensional parameters were introduced as,

𝑄_𝑏^∗ = 𝑄𝑏

𝐴𝐺_𝑚𝑎𝑥ℎ₁ 𝐺_𝑚𝑎𝑥^∗ =𝐺𝑚𝑎𝑥

𝜉𝜌₁ (2-3)

Here 𝐺_𝑚𝑎𝑥 is the maximum steady-state mass flow rate, 𝑄_𝑏 is the bounding power corresponding to maxima of mass flow rate and 𝜉 =

√2𝐷𝑔𝐻 𝑓⁄ 2𝐿₂. 𝐿_𝑡 refers to the total length of the loop, whereas ℎ₁ is the fluid enthalpy in the cold leg. These parameters were found useful for scaling analysis and experimental design, and were also used for stability analysis.

Figure 2-2. Effect of power on steady-state flow rate with distributed heat source and sink Chatoorgoon, 2001.

The same methodology was extended in the subsequent numerical studies of Chatoorgoon et al., 2005a, 2005b, as they analyzed the relative influence of various geometrical parameters (heater and cooler lengths, loop height, inlet and outlet restriction coefficients) on the stability behavior with different fluids like water, carbon dioxide and hydrogen. Steady-state profiles of flow rate for different fluids were found to be quite similar in nature, but with considerable magnitude difference. It was also observed that the flow rate corresponding to the stability threshold can be predicted using the peak

steady-state flow rate with about 95% accuracy. Another computational model named FIASCO based on implicit finite difference scheme was proposed by Jain and Rizwan-uddin, 2006 using constant pressure drop boundary condition, along with constant inlet conditions. Predicted steady-state profile of mass flow rate with heater power was found to be quite similar to the observations of Chatoorgoon et al., 2005b.

Vijayan et al., 2010, proposed a general 1-D theoretical model of a closed rectangular NCL of uniform diameter with horizontal-heater- horizontal-cooler (HHHC) configuration along with adiabatic connecting pipes to compare the steady-state and stability performances of geometrically- identical single-phase, two-phase and supercritical loops, all using water as the working fluid. Boussinesq approximation was used for single-phase loops, whereas equivalent expansion coefficients were defined for two-phase and supercritical loops. Finite difference discretization was followed for supercritical systems, with an equation of state defining density as a function of enthalpy and pressure. Single-phase flow rate monotonically increased with power, whereas two-phase loop exhibited a maxima after initial increase.

Steady-state flow behavior of SCNCL was found to be similar to single-phase loops at lower power. But, with increase in power supply, profile behaved akin to the two-phase system. The reduction in flow rate corresponded to the pseudo-critical zone, as thermal expansion coefficient showed a rapid reduction. That approach was extended by Sharma et al., 2010c, 2010b on a similar model, but having different dimensions, to develop two contrasting computer codes, namely, SUCLIN and NOLSTA. The first one employed frequency-domain approach to perform linear stability analysis, whereas the second one followed time-domain non-linear. Both predicted nearly identical steady-state performances. Effects of various parameters like pipe diameter, system pressure, heater inlet temperature and loop height were investigated.

Loop flow rate was found to increase rapidly with the increase in diameter and height (Figure 2-3) and moderately with system pressure, whereas it decreased significantly with higher heater inlet temperature. In subsequent studies (Sharma et al., 2013, 2010a), NOLSTA was applied to an open loop in HHHC orientation with 13.88 mm inner diameter and 2 m height. Steady- state results exhibited decent comparison with the experimental study of Lomperski et al., 2004 performed with CO2 at 8 MPa and 297 K. Effect of loop orientation on mass flow rate was numerically evaluated. HHHC configuration

was found to have the maximum flow rate for any given power input, whereas the vertical-heater-vertical-cooler (VHVC) configuration experienced the least.

(a) Effect of loop diameter (b) Effect of loop height

Figure 2-3. Influence of (a) loop diameter and (b) height on steady-state mass flow rate (Sharma et al., 2010a).

A 3D computational model was developed by Yadav et al., 2012a using Fluent for steady-state simulation of a rectangular NCL with specified source and sink temperatures. Implicit finite volume algorithm was followed with second-order upwind discretization for convective terms and PRESTO for the pressure term. PISO scheme was used to solve the coupling model between velocity and pressure and RNG 𝜅 − 𝜀 turbulence model was employed. Both water and CO2 were considered as the working fluids at three different pressures and temperatures to compare their heat transfer and fluid flow aspects. Comparison of velocity contours with water at 1.013 bar and CO2 at 100 bar at some selected locations of the loop are presented in Figure 2-4. The asymmetry in velocity profile at any cross-section is clearly evident, which originates due to local buoyancy effects, more on which is discussed later on.

CO2 continually exhibited greater velocity magnitude than water at any given power condition. Analyzing the simulated results, they also proposed correlations for estimating friction factor (𝑓) and Reynolds number (Re) in terms of Grashof number (Gr) in the following form.

𝑓 = (0.7907 𝑙𝑛(𝑅𝑒) − 1.868)⁻²

(2-4) 𝑅𝑒 = 2.066 (𝐺𝑟_𝑚 𝐷

𝐿_𝑡)

1⁄2.77

The proposed relations are applicable in the range of 27,000 ≤ 𝑅𝑒 ≤ 180,000, with an R²-value of 0.9387, thereby showing a good regressed fit. A maximum deviation of 6.5% was observed with all the simulated results.

Figure 2-4. Velocity contour plot for water and CO2 respectively at the centre of (a) source, (b) riser, (c) sink and (d) downcomer (Yadav et al., 2012b).

Swapnalee et al., 2012 used their experimental data to propose a general correlation to estimate the steady-state flow in SCNCL in terms of a relationship between dimensionless density and dimensionless enthalpy. The definitions proposed by Ambrosini and Sharabi, 2008 were used for the dimensionless quantities, as those can provide a unique profile of dimensionless density against dimensionless enthalpy for CO2. The correlation was found to be applicable for CO2, H2O and a few other fluids.

The relationship was proposed by a sigmoidal curve relationship in the following form.

𝜌^∗= 𝐴 + 𝐵

1 + 𝑒𝑥𝑝(1.5181 ℎ^∗+ 0.5689) (2-5) Here 𝐴 = 0.15704 and 𝐵 = 2.4785. The dimensionless quantities are defined as,

𝜌^∗= 𝜌

𝜌_𝑝𝑐 ℎ^∗= 𝛽_𝑝𝑐

𝐶_{𝑝,𝑝𝑐} (ℎ − ℎ_𝑝𝑐) (2-6) The proposed relationship is independent of system pressure and applicable for different supercritical fluids.