Corresponding author: Muthanna H. AlDahhan ( aldahhhanm@mst.edu ) Academic editor: Anton Smirnov
© 2018 Ibrahim A. Said, Mahmoud M. Taha, Vineet Alexander, Shaoib Usman, Muthanna H. AlDahhan.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Said IA, Taha MM, Alexander V, Usman S, AlDahhan MH (2018) Axial dispersion and mixing of coolant gas within a separateeffect prismatic modular reactor. Nuclear Energy and Technology 4(3): 167178. https://doi.org/10.3897/nucet.4.27346

Multiphase Reactors Engineering and Applications Laboratory performed gas phase dispersion experiments in a separateeffect coldflow experimental setup for coolant flow within heated channels of the prismatic modular reactor under accident scenario using gaseous tracer technique. The separateeffect experimental setup was designed on light of local velocity measurements obtained by using hot wire anemometry. The measurements consist of pulseresponse of gas tracer that is flowing through the mimicked riser channel using air as a carrier. The dispersion of the gas phase within the separateeffect riser channel was described using onedimensional axial dispersion model. The axial dispersion coefficient and Peclet number of the coolant gas phase and their residence time distribution within were measured. Effect of heating intensities in terms of heat fluxes on the coolant gas dispersion along riser channels were mimicked in the current study by a certain range of volumetric air flow rate ranging from 0.0015 to 0.0034 m^{3}/s which corresponding to heating intensity range from 200 to 1400 W/m^{2}. Results confirm a reduction in the response curve spreads is achieved by increasing the volumetric air velocity (representing heating intensity). Also, the results reveal a reduction in values of axial dispersion coefficient with increasing the air volumetric flow rate.
Prismatic modular reactor, gaseous tracer technique, axial dispersion coefficient, Peclet number
The prismatic modular reactor (PMR) is one of the next generation nuclear plants (NGNPs). One of the prismatic modular reactor (PMR) advantages is its capability to passively remove the decay heat from the reactor core through natural circulation under the loss of flow accidents (LOFA) scenario (
Multiphase Reactors Engineering and Applications Laboratory (mReal) research team at Missouri S&T designed and constructed a high pressure and temperature dualchannel facility, mimicking prismatic modular reactor (PMR), for natural circulation passive safety system heat transfer and gas dynamics investigations (
Values of the air velocity within the riser channel of the dualchannel facility and the convergent channel (separateeffect test section)
Heating intensity (W/m^{2})  Nondimensional length (Z/L)  Velocity (m/s) in the dualchannel facility (hot flow system)  Velocity (m/s) in the convergent channel – separate effect test section (cold flow system) 
200  0.178  0.72  0.75 
0.436  1.03  1.04  
0.546  1.09  1.10  
0.729  1.16  1.17  
0.867  1.31  1.33  
600  0.178  1.37  1.25 
0.436  1.66  1.73  
0.546  1.77  1.84  
0.729  1.79  1.95  
0.867  1.86  2.22  
1000  0.17 8  1.65  1.50 
0.436  2.26  2.08  
0.546  2.37  2.21  
0.729  2.42  2.35  
0.867  2.17  2.67  
1400  0.178  1.82  1.75 
0.436  2.34  2.42  
0.546  2.57  2.57  
0.729  2.70  2.74  
0.867  2.29  3.11 
The gaseous tracer (GT) is an advanced technique used to accurately measure the residence time distribution (RTD) in a complex flow structure of single and multiphase flow systems by injecting pulse or step change gas tracer and then monitoring its concentration at the exit (
The designed three measurements for the gaseous tracer technique
Measurement  Tracer injection  Tracer detection  Tracer signal  Dispersion zones 
Measurement–1 (I1S)  I1  S  C (1)  Sampling lines+ measurement system + lower plenum + test section + upper plenum 
Measurement–2 (I2S)  I2  S  C (2)  Sampling lines + measurement system + test section + upper plenum 
Measurement–3 (I3S)  I3  S  C (3)  Sampling lines + measurement system + upper plenum 
In this study, averaged resident time results were obtained from three replications of each experimental. Figure
C_{normal} = (C_{i} – C_{min}) / (C_{max} – C_{min}) (1)
For all measurements, minimum values for output tracer signals are zero as shown in Figure
C_{normal} = C_{i} / C_{max} (2)
Figure
It is a measure of the gas tracer response about the mean (t_{m}) and has units of (s^{2}). The magnitude of the variance depends on the degree of the dispersion within the system, the greater value of variance, the higher degree of dispersion in the system and hence, the more response curve spreads and vice versa. The variance is given by the following equation (
or we can rewrite it in a discrete form
Variance and mean residence time results confirm that as the volumetric air flow rate (heating intensity) increases, the mean residence time and variance decrease. In other words, there is a reduction in the response curve spreads which is achieved by increasing the volumetric air velocity and consequently the heating intensity. Also, is clear from Table
Mean residence time (t_{m}) and variance (σ^{2}) for the three measurements (I1S, I2S, and I3S) for selected two volumetric air flowrates
Volumetric air flow rate: 0.0032 m^{3}/s (corresponding to heat flux of 1000 W/m^{2})  
Measurement  Mean Residence time (t_{m}), s  Variance (σ^{2}), s^{2} 
Measurement–1 (I1S)  13.06  9.95 
Measurement–2 (I2S)  12.81  8.81 
Measurement–3 (I3S)  12.505  8.02 
Volumetric air flow rate: 0.0034 m^{3}/s (corresponding to heat flux of 1400 W/m^{2})  
Measurement  Mean Residence time (t_{m}), s  Variance (σ^{2}), s^{2} 
Measurement–1 (I1S)  12.87  8.08 
Measurement–2 (I2S)  12.77  8.42 
Measurement–3 (I3S)  12.43  7.478 
The main objective of integral convolution and regression methods is to properly extract the response or residence time distribution (RTD) of the test section separately from the whole system, which includes sampling lines from the system to the thermal conductivity detector (TCD) as well as external volumes of upper and lower plena. The upper and lower plena were assumed to follow an ideal continuous stirred tank reactor (CSTR) model, while the deviation of the flow of the gas phase (coolant) from plug flow characteristics in the flow channels of prismatic block reactor is described using onedimensional axial dispersion model (ADM) where such representation is valid if there is not much deviation from ideal plugflow reactor (PFR) model. It is worth mentioning that these assumptions were validated by the experimentally measured signals of the gas tracer as shown in the next sections.
An ideal CSTR model was used to assess the gas tracer mixing within the lower plenum to get the input gas tracer concentration (c_{in}) to the onedimensional axial dispersion model (IDADM). Figure
Gas tracer in – Gas tracer out = Accumulation (7)
0.0 – C_{i} V° = V (dC_{i}/dt) (8)
By separating the variables and integration with C_{i} = C_{max} at t = 0 and normalizing the concentration yields
where C_{in} is a dimensionless form for the theoretical lower plenum tracer output signal, which is the input to the IDADM for the test section, t is the instant time, and τ_{s} is the residence time within the lower plenum for the CSTR model. It is worth mentioning that the τ_{s} of the CSTR model was estimated by the regression analysis using the measured gas tracer signal in terms of residence time distribution at the lower plenum outlet. Measuremnt2 (I2S) “C(2)” was used as the same input to the lower plenum to convolute the lower plenum as an ideal CSTR to predict the input tracer concentration (C^{*}_{in} (t)) to the 1DADM (
More details regarding the convolution method can be found elsewhere (
Figures
The onedimensional axial dispersion model (ID–ADM) is used to describe the gas tracer dispersion within the test section. In this model, there is an axial dispersion of the gas tracer, which is governed by analogy to Fick’s law of diffusion. Every element of the system is transported by molecular and convective diffusions at a rate equal to “D ∙ A(dC/dZ)” in conjunction with bulk flow “UAC”. Where D is the effective dispersion coefficient within the system (m^{2}/s), A is the cross sectional area (m^{2}), U is the superficial velocity (m/s), and dC/dZ is the concentration gradient of the tracer (mole/m^{4}). A nonreactive mole balance on the gas tracer over a short length ∆Z of the test section in absence of radial variations yields (
The gas tracer concentration in the mole balance equation (Equation 15) can be rewritten in a dimensionless form as follows:
where,
Once initial and boundary conditions defined, the solution of Equation 14 will yield the effluent tracer concentration (C_{out}).
Closedclosed Danckwerts boundary conditions (
At Z = 0.0 (downstream)
The tracer concentration within Equation 16 can be rewritten in a dimensionless form as follow:
where,
At Z = L (upstream)
where C_{in} is estimated by Equation 10 with the fitted parameter of τ_{s} as shown in Figure
t = 0.0, C = 0.0 (21)
Figure
Then the convoluted 1DADM prediction, C^{*}_{out}, was compared against the measured response of the measurement–1 “C (1)” of the whole system. Then, the dispersion coefficient parameter, D, was estimated by minimizing the averaged squared error between the convoluted prediction from 1DADM (C^{*}_{out} (t)) and the experimental measured value C (1) from measurement1 as follows (
Figures
The axial dispersion coefficient (D) is used to determine the dispersive Peclet number (N_{Pe}) which represents the ratio of the rate of transport by convection to the rate of transport by dispersion as follows:
The value of N_{Pe} is used to quantify the degree to which the axial dispersion affects the performance of the separateeffect test section. A high value of the N_{Pe} corresponds to a slightly dispersed system.
The qualitative effect of the volumetric air flow rate on the axial dispersion coefficient (D) can be illustrated by noting that, as the dispersion coefficient (D) varies from 0 to ∞, the system behavior changes from ideal plug flow to perfect mixing. As shown in Figure
Gaseous dispersion experiments were conducted, for the first time, in a mimicked coldflow separateeffect riser channel of the prismatic modular reactor using advanced gaseous tracer technique. The current separateeffect facility was designed and developed considering measured local velocities in a previous study under same operating conditions along the riser channel of the dualchannel circulation loop at mReal (
The dispersion and mixing in the sampling lines and analytical system are a significant source of errors in the measured residence time distribution (RTD) and consequently values of D and N_{Pe}. Hence, the integral convolution method is implemented to account the extra dispersion and provide accurate gaseous dispersion measurements.
The results confirmed that a reduction in the residence time distribution (RTD) is achieved by increasing the volumetric air velocity (increasing heating intensity).
the RTD experiments of the current study show that Peclet number increases with the volumetric air flowrate (and consequently Reynolds number). Therefore, the high volumetric flow rate mainly has the effect of decreasing the gas axial dispersion coefficient through lowering the turbulence and eddy diffusivities
The values of the Peclet number (N_{Pe}) are found to be increased with increasing the heating intensity (as represented by volumetric air flow rate).
The obtained axial dispersion coefficient values can be used as needed model inputs in the mass transfer measurements of the flow coolant in the prismatic block reactor as well as pressure and heat correlations.
The measured gas phase axial dispersion coefficients (D) and Peclet number (N_{Pe}) of the coolant gas flow in the riser channel are useful for efficient operation and safe design of the prismatic modular reactor (PMR).
The authors acknowledge the financial support provided by the U.S. Department of EnergyNuclear Energy Research Initiative (DOENERI) Project (NEUP 134953 (DENE0000744)) for the 4th generation nuclear energy, which made this work possible.
c concentration of the tracer in the gas phase, mol/m^{3}
c_{min} minimum concentration of the tracer in the gas phase, mol/m^{3}
c_{max} maximum concentration of the tracer in the gas phase, mol/m^{3}
c_{inj} concentration of the injection tracer, mol/m^{3}
C_{in} dimensionless tracer concentration in the gas phase at the lower plenum outlet
C_{in}* dimensionless convoluted tracer concentration in the gas at the lower plenum outlet
C_{out} dimensionless tracer concentration in the gas phase at the test section outlet
C_{out*} dimensionless convoluted tracer concentration in the gas at the test section outlet
C_{normal} normalized value of the output tracer signal (dimensionless)
D effective axial dispersion coefficient of the gas phase, m^{2}/s
L length of the separateeffect test section, m
d inlet inside diameter of the separate effect test section, m
n total number of experimental data points
N_{Pe} dispersive Peclet number (V d/D), dimensionless
U superficial gas velocity, (m/s)
t time, s
t_{m} mean residence time of the bed, s
V volume of the air, m^{3}
V^{o} volumetric air flow rate, m^{3}/s
Z axial distance along the test section, m
Z/L nondimensional length, dimensionless
σ^{2} variance, s^{2}
τ_{s} CSTR parameter, s