## Abstract

In the present study, an improvised design over circular stepped cascade (CSC) and pooled circular stepped cascade (PCSC) aerator, named the perforated pooled circular stepped cascade (PPCSC) aerator, has been conceptualized and tested for its suitability as an aerator for small intensive aquaculture ponds. Based on dimensional analysis, dimensionless geometric parameters – ratio of width of consecutive steps (W_{i}/W_{i+1}) and ratio of perforation diameter to bottom-most radius (d/R_{b}) and dimensionless dynamic parameters – Froude (Fr) and Reynolds (Re) number were proposed. Initially, aeration experiments were conducted to optimize the geometric parameters, keeping the dynamic parameters constant. Keeping the optimized values of W_{i}/W_{i+1} = 1.05 and d/R_{b} = 0.0027 as constants, aeration experiments were further conducted at different discharges (Q) and different bottommost radius (R_{b}) to study the characteristics of oxygen transfer and power consumption of PPCSC aerator at different dynamic conditions. Based on the optimized results, four prototype PPCSC aerators with R_{b} = 0.75 m, 0.90 m, 1.05 m and 1.20 m were fabricated for their aeration performances. The results showed that the standard aeration efficiency (SAE) values of the prototype PPCSC aerators based on brake power ranged between 3.36 and 4.98 kg O_{2}/kWh, with the average being 4.45 ± 0.741 kg O_{2}/kWh. This shows that the SAE of the PPCSC aerator is many more folds higher than that of the other available cascade aerators, viz., PCSC (SAE: 2.873 ± 0.342 kg O_{2}/kWh) and CSC (2.470 ± 0.256 kg O_{2}/kWh) aerators. The study clearly indicates that this PPCSC aerator may very well be used as pre-aeration or post-aeration units in water or wastewater treatment plants and small-scale intensive aquacultural ponds, replacing the other existing aerators.

## HIGHLIGHTS

Design characteristics of perforated pooled circular stepped cascade (PPCSC) aerator were evaluated.

Simulation equations for oxygen transfer and power consumption for PPCSC aerators were developed.

Prototype PPCSC aerators were designed, developed and tested for aeration performance.

Aeration efficiency of the developed prototype PPCSC aerator was found to be significantly higher (about 1.5–1.8 times more) than the other existing cascade aerators.

### Graphical Abstract

## INTRODUCTION

Dissolved oxygen (DO) is a very important parameter in aquaculture water as it directly affects the growth and survival of aquatic species. It is therefore necessary to maintain the desired oxygen level in such waters through artificial aerators. Different types of aerators, viz., mechanical, diffused-air, gravity and so on have been developed over the years and are currently in use in aquacultural operations (Boyd 1998, 2009; Moulick *et al.* 2002, 2005; Moulick & Mal 2009; Jayraj *et al.* 2018; Cheng *et al.* 2019). The cascade aerator, a type of gravity aerator, is very popularly used in water treatment plants to remove iron and manganese or various dissolved gases or volatile organic compounds (Mahmad *et al.* 2015). In aquacultural ponds with water volume ≤1,000 m^{3}, cascade aeration systems (circular stepped cascade aerator (Singh 2010) and pooled circular stepped cascade aerators (Kumar *et al.* 2013a)) were found to be the most economical ones (Kumar *et al.* 2013b).

In a circular stepped cascade (CSC) aerator, the setup consists of a CSC along with a pump. The pump lifts the water to the top of the CSC and allows the water to fall over the steps of the aerator. This arrangement creates turbulence in the water and thus breaks the air-water interface, which leads to aeration. The aeration efficiency depends on the flow rate of the water and the exposure time of the water over the steps. Singh (2010) found the standard aeration efficiency (SAE) of the CSC aerator to vary between 2.16 and 2.70 kg O_{2}/kWh. In pooled circular stepped cascade (PCSC) aerators, partial enclosures are attached at the boundary of each of the circular steps of the CSC aerator in a zigzag manner in order to increase the time of exposure of the water. Kumar *et al.* (2013a) found the SAE (2.43–3.23 kg O_{2}/kWh) of the PCSC aerator to be higher than the CSC aerator.

The idea of this present study is to further increase the aeration efficiency of the PCSC aerator by adopting some changes in its design. When water is allowed to fall over the steps, initially a larger step width is required for proper contact of the water layer with the steps. In the PCSC aerator, the width of the circular steps was kept the same. It was felt that by varying the step width in decreasing order from top to bottom of the setup, better aeration may be achieved. In addition, if holes or perforations are provided in the steps, it will allow a part of the flow to fall vertically through the holes in the form of fine spray. It is strongly believed that this modified design of PCSC aerator, hereafter named the perforated pooled circular stepped cascade (PPCSC) aerator, will have a better aeration efficiency than the other existing aerators and may have potential application in small intensive aquacultural ponds and also as pre-aeration or post-aeration units in water or wastewater treatment plants.

Based on the above discussion, the following objectives are taken up in the present study:

- I.
Optimization of the geometric design parameters – ratio of width of consecutive steps (W

_{i}/W_{i+1}) and ratio of perforation diameter to bottom-most radius (d/R_{b}) of the PPCSC aerator. - II.
Formulation of predictive model equations for evaluation of the aeration performances of the PPCSC aerator at different dynamic conditions.

- III.
Development and performance assessment of prototype PPCSC aerators.

- IV.
Comparison of the performance of the PPCSC aerator with the other existing cascade aerators.

## THEORETICAL ANALYSIS

### General

The performances of aerators are mainly assessed in terms of standard oxygen transfer rate (SOTR) and standard aeration efficiency (SAE).

_{2}/h), is the overall oxygen transfer coefficient at 20 °C (h

^{−1}) = (Shelton & Boyd 1983; ASCE 2007), is the overall oxygen transfer coefficient at T °C (h

^{−1}),

*θ*is the temperature correction factor for clean water (1.024), C* is the DO saturation at experimental condition (mg/L), C

_{0}is the DO in the aeration basin and is assumed to be zero (mg/L) and V is the volume of water in the tank (m

^{3}).

The brake power (kW) can be estimated by dividing the wire power by the overall efficiency (η) of the pump-motor assembly.

_{s}is the saturation concentration of pond water (mg/L) at T (°C), C

_{P}= initial DO concentration in pond water (mg/L) at T (°C), α = pond water/ tap water and β = DO saturation concentration of pond water/DO saturation concentration of tap water. Aeration efficiency (AE) of aerators for pond conditions can be estimated by the following equation (Boyd 1998):

### Dimensional analysis

The experimental setup for testing of the PPCSC aerator is shown schematically in Figure 1. The various geometric, material and process variables affecting oxygen transfer for the PPCSC aerator are listed in Table 1.

Class of variables . | Name of parameters . | Symbol . | Dimension . |
---|---|---|---|

Geometric | Radius of bottom-most circular step | R_{b} | L |

Ratio of width of consecutive steps | W_{i}/W_{i} + 1 | M^{0}T^{0} L^{0} | |

Total number of steps in the cascade | n | M^{0}T^{0} L^{0} | |

Total height of the perforated circular stepped cascade | H | L | |

Volume of water under aeration | V | L^{3} | |

% coverage of circumference of each step by enclosure | P_{e} | M^{0}T^{0} L^{0} | |

No. of enclosure in each step | N_{e} | M^{0}T^{0} L^{0} | |

Perforated diameter of the stepped cascade | d | L | |

Material | Mass density of air | ρ_{a} | ML^{−3} |

Mass density of water | ρ_{w} | ML^{−3} | |

Kinematic viscosity of water | ν_{w} | L^{2}T^{−1} | |

Surface tension of water in contact with air | σ_{w} | MT^{−2} | |

Process | Pump discharge | Q | L^{3}T^{−1} |

Acceleration due to gravity | g | LT^{−2} |

Class of variables . | Name of parameters . | Symbol . | Dimension . |
---|---|---|---|

Geometric | Radius of bottom-most circular step | R_{b} | L |

Ratio of width of consecutive steps | W_{i}/W_{i} + 1 | M^{0}T^{0} L^{0} | |

Total number of steps in the cascade | n | M^{0}T^{0} L^{0} | |

Total height of the perforated circular stepped cascade | H | L | |

Volume of water under aeration | V | L^{3} | |

% coverage of circumference of each step by enclosure | P_{e} | M^{0}T^{0} L^{0} | |

No. of enclosure in each step | N_{e} | M^{0}T^{0} L^{0} | |

Perforated diameter of the stepped cascade | d | L | |

Material | Mass density of air | ρ_{a} | ML^{−3} |

Mass density of water | ρ_{w} | ML^{−3} | |

Kinematic viscosity of water | ν_{w} | L^{2}T^{−1} | |

Surface tension of water in contact with air | σ_{w} | MT^{−2} | |

Process | Pump discharge | Q | L^{3}T^{−1} |

Acceleration due to gravity | g | LT^{−2} |

_{i}/W

_{i+1}= x) of the circular steps were assumed to be constant. The width and radius of i

^{th}step, i.e. W

_{i}and R

_{i}respectively, can be expressed as follows:where n is the total number of steps, R

_{n}is the radius of the n

^{th}step or the bottom-most step = R

_{b}

*SOTR*as expressed in Equation (8) is replaced by

*SAE*. Thus, the functional relationship between and the key variables may be expressed as follows:

Based on dimensional analysis, the non-dimensional SAE, i.e., NDSAE was expressed as a function of geometric and dynamic parameters.

Where, non-dimensional standard aeration efficiency (NDSAE) = ;

= Weber number, W;

= Reynolds numbers, Re; and

= Froude number, Fr.

The first seven dimensionless quantities in function f_{2} represent geometric variables, the eighth and ninth one represent the material variables and finally the last two represent the process variables.

In the case of the PCSC aerator, Kumar *et al.* (2013a) established that the dimensionless geometric parameters: n, H/R_{b}, P_{e} and N_{e} should be 5, 0.25, 20% and 9 respectively to achieve the maximum SAE. In the present study, these geometric parameters (n, H/R_{b}, P_{e} and N_{e}) were kept constant at the above stated values and accordingly were removed from function f_{2}. Further, it was found that V/R_{b}^{3} does not influence SAE significantly (Moulick *et al.* 2002, 2010; Kumar *et al.* 2013a). Therefore, V/R_{b}^{3} was also omitted from the function f_{2}.

The dimensionless material variables – – remained constant as all the experiments were carried out involving air and water only. Therefore, these two dimensionless material variables were also excluded from the function f_{2}.

Studies conducted by several investigators (Schmidtke & Horvath 1977; Rao 1999; Moulick *et al.* 2005, 2010; Moulick & Mal 2009; Kumar *et al.* 2013a; Roy *et al.* 2017) showed that Froude (Fr) and Reynolds (Re) number mainly affect the aeration process. Accordingly the Weber number, W was neglected.

In the above equation, the first two dimensionless quantities (W_{i}/W_{i+1} and d/R_{b}) represent the geometric similarity and the last two (Fr and Re) represent the dynamic similarity.

^{3}).

## MATERIALS AND METHODS

### Testing facility of PPCSC aerator

A testing facility for the PPCSC aerator was developed as shown in Figure 2. The testing facility includes a reinforced cement concrete tank of dimension 4 m × 4 m × 1.5 m for conducting the aeration tests, open well submersible pump set (5 HP) with a control valve for pumping and discharging water at the top of the cascade setups and different setups of PPCSC aerator with metallic stand as per the experimental design. An Electromagnetic Flow Meter (Full Bore) (Make: Manas; Series: SROAT 1000; Liner: PTFE; Meter size: DN 100; Flow rate: 0–40 L/s) was used to measure the discharge.

Different PPCSC setups were fabricated locally using 0.64 mm thick galvanized iron (G.I.) sheets consisting of perforations of different diameters (d). A funnel with top diameter equal to twice the diameter of the topmost step and bottom diameter equal to 100 mm was fabricated using the same G.I. sheet for each cascade setup. One side of the funnel was welded with the topmost step and the other end with one 100 mm pipe piece having threads at the other end to facilitate its joining with the control valve of the pump.

^{2}) of different d/R

_{b}(0.0013, 0.0020 and 0.0027) were 643, 435 and 279 respectively. The area of annular steps was calculated using Equation (14):where A

_{p}is the area of annular steps (m

^{2}), R

_{i}is the topmost radius of the cascade (m) and W

_{i}is the width of the cascade (m).

### Aeration test

_{2}) and 10 mg/l of sodium sulphite (Na

_{2}SO

_{3}) for each 1 mg/l of DO present in the water to follow the standard conditions (ASCE 2007). The dissolved oxygen (DO) concentrations, temperature and atmospheric pressure were measured using three DO meters (Two YSI Pro 20 and one YSI 55) by inserting their DO probes at three different positions in the water body at a depth of approximately 0.20 m from the surface of the water (Baylar

*et al.*2007). The average values of dissolved oxygen (DO) concentration, temperature and atmospheric pressure were considered for each experiment. The aerator was operated until the DO in the water body reached typically greater than 90% of saturation at a specific time (1 min) interval (Jiang & Stenstrom 2012). At least 45 DO readings at equal time intervals were taken. The three parameters, C

_{0}, C* and

*K*

_{L}a_{T}were determined simultaneously using Equation (15) (ASCE 2007; Jiang & Stenstrom 2012): was then converted to , and accordingly the values of

*SOTR*and

*SAE*were computed using Equations (1) and (2) respectively. The dimensionless numbers (NDSAE, Re, Fr and Ne) were then evaluated using Equations (10), (11) and (13).

### Experimental design

#### Optimization of geometric parameters (W_{i}/W_{i+1}, d/R_{b})

Aeration experiments were conducted (Table 2) by varying the ratio of width of consecutive steps (W_{i}/W_{i+1}) from 1.00 to 1.25 at an interval of 0.05 and ratio of perforation diameter to total bottom-most radius (d/R_{b}) from 0.0013 to 0.0027 at an interval of 0.0007. A full factorial design was considered with a total of 54 experiments (first input with 6 levels × second input with 3 levels × 3 repetitions = 54 experiments) to evaluate the performance of the PPCSC aerator for an optimal solution. Keeping Q and R_{b} as 0.019 m^{3}/s and 0.75 m respectively, experiments were conducted such that the dimensionless dynamic parameters (Fr = 0.0124 and Re = 30864) remained constants. The commercial statistical software package Design Expert (Stat-Ease, 2008 and version-7.1.6) was used for FFD (full factorial design), analysis of variance (ANOVA) and regression analysis. ‘Numerical Optimization’ process was adopted to optimize the geometric parameters (W_{i}/W_{i+1}, d/R_{b}), keeping Fr and Re constants.

W_{i}/W_{i+1}
. | d/R_{b}
. | Constants . |
---|---|---|

1.00 | 0.0013–0.0027 at an interval 0.0007 | Q (0.019 m^{3}/s) and R_{b} (0.75 m) |

1.05 | ||

1.10 | ||

1.15 | ||

1.20 | ||

1.25 |

W_{i}/W_{i+1}
. | d/R_{b}
. | Constants . |
---|---|---|

1.00 | 0.0013–0.0027 at an interval 0.0007 | Q (0.019 m^{3}/s) and R_{b} (0.75 m) |

1.05 | ||

1.10 | ||

1.15 | ||

1.20 | ||

1.25 |

#### Prediction of aeration characteristics of PPCSC aerator at different dynamic conditions

Experiments were conducted on different-sized PPCSCs (R_{b}: 0.75 to 1.20) and at different discharges (Q: 0.007 to 0.022 m^{3}/s) to assess the effect of dynamic conditions on aeration characteristics (Table 3). In this set of experiments, optimized values of geometric parameters: W_{i}/W_{i+1} and d/R_{b} were kept as constants.

R_{b} (m)
. | Q (m^{3}/s)
. | Similarity conditions . |
---|---|---|

0.75 | 0.007–0.022 at an interval of 0.003 m^{3}/s | (W_{i}/W_{i+1}) = (W_{i−1}/W_{i})^{a} and (d/R_{b}) = (d/R_{b})^{a} |

0.90 | ||

1.05 | ||

1.20 |

R_{b} (m)
. | Q (m^{3}/s)
. | Similarity conditions . |
---|---|---|

0.75 | 0.007–0.022 at an interval of 0.003 m^{3}/s | (W_{i}/W_{i+1}) = (W_{i−1}/W_{i})^{a} and (d/R_{b}) = (d/R_{b})^{a} |

0.90 | ||

1.05 | ||

1.20 |

^{a}denotes optimum values.

#### Performance assessment of prototype PPCSC aerators

Prototype PPCSC setup was fabricated by attaching the cascade setup to a propeller pump (0.18 kW; Make – Kirloskar Electric Co. Ltd, India) along with the motor assembly (Figure 3). The outlet of the propeller pump was kept above the top surface of the PPCSC setup to allow the discharge of water to fall all over the cascade.

Standard aeration experiments were conducted with four different sized (R_{b}: 0.75, 0.90, 1.05 and 1.20 m) prototype PPCSC setups, keeping the water flow rate (Q) at 0.007 m^{3}/s (Table 4).

R_{b} (m)
. | Q (m^{3}/s)
. | Constants . |
---|---|---|

0.75 | 0.007 | Optimize values of the geometric parameters:(W_{i}/W_{i+1}) and (d/R_{b}) |

0.90 | ||

1.05 | ||

1.20 |

R_{b} (m)
. | Q (m^{3}/s)
. | Constants . |
---|---|---|

0.75 | 0.007 | Optimize values of the geometric parameters:(W_{i}/W_{i+1}) and (d/R_{b}) |

0.90 | ||

1.05 | ||

1.20 |

The wire power (0.14 kW) was recorded using MECO 3 phase 4 wire power analyser. The power consumption without any load was determined as 0.09 kW. Therefore, the brake power was calculated as (0.14–0.09) kW = 0.05 kW. The SAE in each case was reported on the basis of wire power as well as brake power.

#### Comparison of aeration performance of prototype PPCSC aerator with PCSC and CSC aerators

The aeration performance, in terms of brake power SAE, of the prototype PPCSC aerator was compared with PCSC and CSC aerators (Singh 2010; Kumar *et al.* 2013a).

## RESULTS AND DISCUSSION

### Optimization of geometric parameters of PPCSC aerator

*SOTR*,

*SAE*and NDSAE at different geometric parameters (W

_{i}/W

_{i+1}and d/R

_{b}) of the PPCSC aerator are presented in Table 5. As Q and R

_{b}were kept constants at 0.019 m

^{3}/s and 0.75 m respectively,

*SAE*becomes directly proportional to NDSAE. The NDSAE ranged from 0.858 × 10

^{−5}to 2.21 × 10

^{−5}for different combinations of W

_{i}/W

_{i+1}and d/R

_{b}. The ANOVA of the experimental results is presented in Table 6. It can be noted from the table that the model is highly significant with a high R

^{2}value of 0.908. The adjusted R

^{2}value of 0.863 also implies that the model is highly significant. A low value of coefficient of the variation (C.V. = 10.17%) shows a very high degree of precision for the model. A third order regression equation with NDSAE as the response and W

_{i}/W

_{i+1}and d/R

_{b}as independent parameters is presented as follows:

W_{i}/W_{i+1}
. | d/R_{b}. | ||||||||
---|---|---|---|---|---|---|---|---|---|

0.0013 . | 0.0020 . | 0.0027 . | |||||||

SOTR
. | SAE
. | NDSAE × 10^{5}
. | SOTR
. | SAE
. | NDSAE × 10^{5}
. | SOTR
. | SAE
. | NDSAE × 10^{5}
. | |

1.00 | 0.256 ± 0.030 | 3.68 ± 0.176 | 1.289 ± 0.061 | 0.367 ± 0.042 | 5.286 ± 0.416 | 1.847 ± 0.145 | 0.413 ± 0.148 | 5.782 ± 1.218 | 1.915 ± 0.293 |

1.05 | 0.232 ± 0.058 | 3.292 ± 0.338 | 1.150 ± 0.118 | 0.267 ± 0.068 | 3.787 ± 0.395 | 1.323 ± 0.138 | 0.410 ± 0.100 | 5.819 ± 0.526 | 2.033 ± 0.184 |

1.10 | 0.233 ± 0.037 | 3.330 ± 0.128 | 1.163 ± 0.044 | 0.233 ± 0.067 | 3.308 ± 0.478 | 1.156 ± 0.167 | 0.192 ± 0.040 | 2.738 ± 0.213 | 0.956 ± 0.074 |

1.15 | 0.189 ± 0.017 | 2.726 ± 0.237 | 0.952 ± 0.082 | 0.229 ± 0.067 | 3.231 ± 0.455 | 1.128 ± 0.159 | 0.205 ± 0.036 | 2.937 ± 0.072 | 1.026 ± 0.025 |

1.20 | 0.183 ± 0.035 | 2.615 ± 0.143 | 0.914 ± 0.050 | 0.251 ± 0.091 | 3.526 ± 0.736 | 1.233 ± 0.257 | 0.283 ± 0.124 | 3.963 ± 1.144 | 1.385 ± 0.399 |

1.25 | 0.214 ± 0.031 | 3.069 ± 0.210 | 1.072 ± 0.073 | 0.251 ± 0.050 | 3.580 ± 0.218 | 1.251 ± 0.076 | 0.376 ± 0.070 | 5.372 ± 0.172 | 1.877 ± 0.060 |

W_{i}/W_{i+1}
. | d/R_{b}. | ||||||||
---|---|---|---|---|---|---|---|---|---|

0.0013 . | 0.0020 . | 0.0027 . | |||||||

SOTR
. | SAE
. | NDSAE × 10^{5}
. | SOTR
. | SAE
. | NDSAE × 10^{5}
. | SOTR
. | SAE
. | NDSAE × 10^{5}
. | |

1.00 | 0.256 ± 0.030 | 3.68 ± 0.176 | 1.289 ± 0.061 | 0.367 ± 0.042 | 5.286 ± 0.416 | 1.847 ± 0.145 | 0.413 ± 0.148 | 5.782 ± 1.218 | 1.915 ± 0.293 |

1.05 | 0.232 ± 0.058 | 3.292 ± 0.338 | 1.150 ± 0.118 | 0.267 ± 0.068 | 3.787 ± 0.395 | 1.323 ± 0.138 | 0.410 ± 0.100 | 5.819 ± 0.526 | 2.033 ± 0.184 |

1.10 | 0.233 ± 0.037 | 3.330 ± 0.128 | 1.163 ± 0.044 | 0.233 ± 0.067 | 3.308 ± 0.478 | 1.156 ± 0.167 | 0.192 ± 0.040 | 2.738 ± 0.213 | 0.956 ± 0.074 |

1.15 | 0.189 ± 0.017 | 2.726 ± 0.237 | 0.952 ± 0.082 | 0.229 ± 0.067 | 3.231 ± 0.455 | 1.128 ± 0.159 | 0.205 ± 0.036 | 2.937 ± 0.072 | 1.026 ± 0.025 |

1.20 | 0.183 ± 0.035 | 2.615 ± 0.143 | 0.914 ± 0.050 | 0.251 ± 0.091 | 3.526 ± 0.736 | 1.233 ± 0.257 | 0.283 ± 0.124 | 3.963 ± 1.144 | 1.385 ± 0.399 |

1.25 | 0.214 ± 0.031 | 3.069 ± 0.210 | 1.072 ± 0.073 | 0.251 ± 0.050 | 3.580 ± 0.218 | 1.251 ± 0.076 | 0.376 ± 0.070 | 5.372 ± 0.172 | 1.877 ± 0.060 |

Units: SOTR (kgO_{2}/h), SAE(kgO_{2}/kWh).

Source . | Sum of Squares . | df . | Mean square . | F value . | Probability > F . | . |
---|---|---|---|---|---|---|

Model | 4.401 × 10^{−10} | 6 | 7.336 × 10^{−11} | 15.52 | <0.0001 | significant |

W_{i}/W_{i+1} | 3.890 × 10^{−11} | 1 | 3.890 × 10^{−11} | 8.23 | <0.0001 | significant |

d/R_{b} | 2.683 × 10^{−12} | 1 | 2.683 × 10^{−12} | 0.57 | <0.0001 | significant |

(W_{i}/W_{i+1}) × (d/R_{b}) | 1.047 × 10^{−13} | 1 | 1.047 × 10^{−13} | 0.022 | 0.8824 | |

(W_{i}/W_{i+1})^{2} | 1.734 × 10^{−10} | 1 | 1.734 × 10^{−10} | 36.68 | <0.0001 | |

(W_{i}/W_{i+1})^{2} × (d/R_{b}) | 5.679 × 10^{−11} | 1 | 5.679 × 10^{−11} | 12.01 | 0.0012 | |

(W_{i}/W_{i+1})^{3} | 1.340 × 10^{−11} | 1 | 1.340 × 10^{−11} | 2.83 | 0.0991 | |

Residual | 2.175 × 10^{−11} | 46 | 4.728 × 10^{−12} | |||

Lack of fit | 1.572 × 10^{−10} | 11 | 1.429 × 10^{−11} | 8.29 | <0.0001 | significant |

Pure error | 6.032 × 10^{−11} | 35 | 1.724 × 10^{−12} | |||

Corrected total | 6.576 × 10^{−10} | 52 |

Source . | Sum of Squares . | df . | Mean square . | F value . | Probability > F . | . |
---|---|---|---|---|---|---|

Model | 4.401 × 10^{−10} | 6 | 7.336 × 10^{−11} | 15.52 | <0.0001 | significant |

W_{i}/W_{i+1} | 3.890 × 10^{−11} | 1 | 3.890 × 10^{−11} | 8.23 | <0.0001 | significant |

d/R_{b} | 2.683 × 10^{−12} | 1 | 2.683 × 10^{−12} | 0.57 | <0.0001 | significant |

(W_{i}/W_{i+1}) × (d/R_{b}) | 1.047 × 10^{−13} | 1 | 1.047 × 10^{−13} | 0.022 | 0.8824 | |

(W_{i}/W_{i+1})^{2} | 1.734 × 10^{−10} | 1 | 1.734 × 10^{−10} | 36.68 | <0.0001 | |

(W_{i}/W_{i+1})^{2} × (d/R_{b}) | 5.679 × 10^{−11} | 1 | 5.679 × 10^{−11} | 12.01 | 0.0012 | |

(W_{i}/W_{i+1})^{3} | 1.340 × 10^{−11} | 1 | 1.340 × 10^{−11} | 2.83 | 0.0991 | |

Residual | 2.175 × 10^{−11} | 46 | 4.728 × 10^{−12} | |||

Lack of fit | 1.572 × 10^{−10} | 11 | 1.429 × 10^{−11} | 8.29 | <0.0001 | significant |

Pure error | 6.032 × 10^{−11} | 35 | 1.724 × 10^{−12} | |||

Corrected total | 6.576 × 10^{−10} | 52 |

Coefficient of determination (R^{2}) = 0.908; Adjusted (R^{2}) = 0.863; S.D. = 0.131 × 10^{−5}; Mean = 1.291 × 10^{−5}; Coefficient of variation (C.V.) = 10.17%; Adequate precision = 7.724; Press = 1.364 × 10^{−10}.

Optimization of the dimensionless geometric parameters (W_{i}/W_{i+1} and d/R_{b}) was carried out using ‘Numerical Optimization’ technique with the Design Expert software. The optimum values of the dimensionless geometric parameters at which the maximum SAE can be achieved for the PPCSC aerator are presented in Table 7.

Sl. no. . | Geometric parameters . | Optimum values . | Maximum predicted response NDSAE . |
---|---|---|---|

1 | W_{i}/W_{i+1} | 1.0500 | 1.625 × 10^{−5} |

2 | d/R_{b} | 0.0027 |

Sl. no. . | Geometric parameters . | Optimum values . | Maximum predicted response NDSAE . |
---|---|---|---|

1 | W_{i}/W_{i+1} | 1.0500 | 1.625 × 10^{−5} |

2 | d/R_{b} | 0.0027 |

### Effect of dynamic conditions on PPCSC aerator

Aeration experiments were conducted at different discharges (Q = 0.007 to 0.022 m^{3}/s) and different sizes of PPCSC (R_{b} = 0.75 to 1.20 m). Based on the experimental results, simulation equations for prediction of oxygen transfer and power consumption were formulated.

#### Variation of NDSAE with Re and Fr

It can be observed from Figure 4 that NDSAE increases with increase in Re as a power function and NDSAE values for all cascade setups can be fitted by the above single function. As the Reynolds number is the ratio of inertial and viscous forces and transfer of oxygen takes place under a turbulent condition due to comparatively high inertial force, the Reynolds number significantly influences the phenomenon. A reasonably good coefficient of determination (R^{2}) of 0.816 is obtained for the developed relationship. The uncertainty in the experimental results is represented in Figure 4 by the error bars corresponding to plus or minus one standard deviation (STD) with a 95% confidence limit.

From Figure 5, it is clear that the NDSAE increases with the increase in the values of Fr. A very high value of coefficient of determination (R^{2}) of 0.989 shows that the relationship is highly significant. In Figure 5, the uncertainty of the analysis is shown by plotting the data points using error bars with one plus or minus STD at 95% confidence limit. It may be noted that the overall fitting for the data is good and most of the experimental data lie within the domain of the model.

It may be observed that Equation (18) provides better correlation than Equation (17). This is due to the fact that the flow, in the case of the cascade aerator, occurs primarily due to gravity forces. Therefore Fr plays a dominant role under such conditions. Many researchers also found the Froude criterion to be the most influencing factor in the analysis of stepped cascade systems (Toombes & Chanson 2005; Kumar *et al.* 2013a; Bayon-Barrachina & Lopez-Jimenez 2015; Rathinakumar *et al.* 2017). Hence in this study, Equation (18) (Fr criterion) was regarded as the simulation equation depicting oxygen transfer.

#### Variation of Ne with Re and Fr

It can be noted from Figure 6 that, particularly at low values of Re, the data points representing Ne values are much away from the plotted curve. This clearly explains the fact that as the flow pattern is dependent mostly on gravity force, the effect of viscous force may not be significant. The experimental data are represented by error bars corresponding to plus or minus one standard deviation (STD) with the 95% confidence limit.

It can be seen from Figure 7 that all the points are located either on the predicted curve (Equation (20)) or are very close to it. Therefore, Equation (20) may be regarded as the simulation equation for power consumption of the PPCSC aerator. The experimental data represented in Figure 7 is plotted using error bars corresponding to plus or minus one standard deviation (STD) with the 95% confidence limit.

### Performance assessment of prototype PPCSC aerators

Four prototype PPCSC aerators of different sizes (R_{b}: 0.75, 0.90, 1.05 and 1.20 m) were fabricated and evaluated for their aeration performance at a particular flow rate of the propeller pump (*Q*: 0.007 m^{3}/s). The aeration performances of the prototype PPCSC aerators are shown in Table 8. It can be seen from the table that the *SOTR* as well as the *SAE* increases with the increase in size of the aerator as the power consumption in all the cases was nearly equal because of having the same pump discharge. It is clear from the result that, with the increase in the size of the aerator, more water will be exposed to atmosphere over the surface of the aerator, leading to higher oxygen transfer. However, a large size of aerator is not recommended as it may lead to maintenance and transport problems.

Exp. no. . | R_{b} (m)
. | Q (m^{3}/s)
. | SOTR (kg O_{2}/h)
. | Wire power SAE (kg O_{2}/kWh)
. | Brake power SAE (kg O_{2}/kWh)
. |
---|---|---|---|---|---|

1 | 0.75 | 0.007 | 0.168 ± 0.041 | 1.142 ± 0.110 | 3.360 ± 0.570 |

2 | 0.90 | 0.007 | 0.231 ± 0.038 | 1.571 ± 0.350 | 4.620 ± 0.655 |

3 | 1.05 | 0.007 | 0.242 ± 0.017 | 1.646 ± 0.192 | 4.840 ± 0.194 |

4 | 1.20 | 0.007 | 0.249 ± 0.025 | 1.693 ± 0.267 | 4.980 ± 0.328 |

Exp. no. . | R_{b} (m)
. | Q (m^{3}/s)
. | SOTR (kg O_{2}/h)
. | Wire power SAE (kg O_{2}/kWh)
. | Brake power SAE (kg O_{2}/kWh)
. |
---|---|---|---|---|---|

1 | 0.75 | 0.007 | 0.168 ± 0.041 | 1.142 ± 0.110 | 3.360 ± 0.570 |

2 | 0.90 | 0.007 | 0.231 ± 0.038 | 1.571 ± 0.350 | 4.620 ± 0.655 |

3 | 1.05 | 0.007 | 0.242 ± 0.017 | 1.646 ± 0.192 | 4.840 ± 0.194 |

4 | 1.20 | 0.007 | 0.249 ± 0.025 | 1.693 ± 0.267 | 4.980 ± 0.328 |

### Comparison of standard aeration efficiencies of PPCSC aerator with PCSC and CSC aerators

Standard aeration efficiencies based on brake power for the CSC, PCSC and PPCSC aerators are presented in Table 9. The results clearly indicate that the SAE of the PPCSC aerator is approximately 1.55 times and 1.80 times higher than the PCSC aerator and CSC aerator respectively.

Aerators . | SOTR (kg O_{2}/h)
. | Brake Power SAE (kg O_{2}/kWh)
. |
---|---|---|

PPCSCA | 0.222 ± 0.037 | 4.450 ± 0.741 |

PCSCA | 0.143 ± 0.017 | 2.873 ± 0.342 |

CSCA | 0.123 ± 0.012 | 2.470 ± 0.256 |

Aerators . | SOTR (kg O_{2}/h)
. | Brake Power SAE (kg O_{2}/kWh)
. |
---|---|---|

PPCSCA | 0.222 ± 0.037 | 4.450 ± 0.741 |

PCSCA | 0.143 ± 0.017 | 2.873 ± 0.342 |

CSCA | 0.123 ± 0.012 | 2.470 ± 0.256 |

In the existing cascade aerators (CSC and PCSC), the oxygen transfer occurs because of increase in interfacial area of air-water due to breaking of the water surface while falling over the circular steps. However, in the case of the PPCSC aerator, the oxygen transfer takes place not only due to breaking of the water surface over the steps, but also due to free fall of water droplets through the perorated holes in the PPCSC setup. This free fall of water droplets allows the atmospheric oxygen to enter into the water at a very fast rate as the air-water interfacial area becomes very high (El-Zahaby & El-Gendy 2016).

In the cascade aerator, the maximum turbulence is created in the top-most step and thereafter it decreases as it falls through the remaining steps. Due to this high turbulence, the air-water interfacial area increases and aeration takes place. At high flow rate, it may happen that the water flow may bypass the top-most steps without breaking the water droplets. In that case, effective aeration may not take place even through the water flows at high velocity. In PPCSC, such a phenomenon may not occur as the width of the top-most step is increased. As a result, at higher flow velocity also, breaking of water drops and resultant increase in air-water interfacial area takes place in the top-most step. The widths of the remaining steps are decreased following a constant ratio (W_{i}/W_{i+1} = 1.05) as unnecessary higher widths of lower steps increase the input cost of the cascade setup.

## SUMMARY AND CONCLUSIONS

In the present study, a new type of cascade aerator, namely the perforated pooled circular stepped cascade (PPCSC) aerator, has been developed by modifying some design features of the existing cascade aerators; that is, the circular stepped cascade aerator and pooled circular stepped cascade aerator. The modification was made in two aspects – (1) varying the step width with a decreasing rate starting from the topmost step and (2) adding perforations on the horizontal portion of the steps. The parameters affecting the aeration efficiency were non-dimensionalised and categorized as geometric [ratio of width of consecutive steps (W_{i}/W_{i+1}) and ratio of perforation diameter to bottommost radius (d/R_{b})] and dynamic [Froude (Fr) and Reynolds (Re) number] based on Buckingham *π* theorem. Keeping the dynamic parameters constant, experiments were conducted to optimize the geometric parameters based on full factorial experimental design. Further, maintaining the optimized geometric parameters (W_{i}/W_{i+1} = 1.05 and d/R_{b} = 0.0027), aeration experiments were further conducted at different dynamic conditions (varying discharges (Q) and bottom-most radius (R_{b})) to study the aeration characteristics of the PPCSC aerator. Based on the experimental results, simulation equations could be formulated to predict *SOTR*, *SAE* and P for PPCSC aerators subject to 0.00141≤ Fr ≤0.01441. Four prototype PPCSC aerators with different sizes (R_{b} = 0.75 m, 0.90 m, 1.05 m and 1.20 m) were fabricated and evaluated for their aeration performances. It was found that the standard aeration efficiency (*SAE*) values of the prototype PPCSC aerators based on brake power ranged between 3.36 and 4.98 kg O_{2}/kWh, with an average of 4.45 ± 0.741 kg O_{2}/kWh. This clearly shows that the *SAE* of the PPCSC aerator is much higher than that of the other available cascade aerators; that is, PCSC (SAE: 2.873 ± 0.342 kg O_{2}/kWh) and CSC (2.470 ± 0.256 kg O_{2}/kWh) aerators.

The specific conclusions of the study are presented as follows:

- 1.
The ratio of consecutive widths of steps (W

_{i}/W_{i+1}) and the ratio of perforation diameter (d) to bottom-most radius (R_{b}) of the PPCSC should be kept at 1.05 and 0.0027 respectively for achieving the maximum aeration efficiency. - 2.
Simulation equations for prediction of aeration efficiency and power characteristics of the PPCSC aerator were formulated based on Froude (Fr) criterion, subject to 0.00141≤ Fr ≤0.01441.

- 3.
The comparative aeration performance of cascade aerators reveals that the

*SAE*of the PPCSC aerator is at least 150% more than that of the other available cascade aerators.

Techniques for Modern Aquaculture, Proceedings of Aqua. Eng. Conf., ASAE, Michigan, USA, 511, pp. 21–23