^{1}

^{1}

Here we suggest an algorithm for calculation of the process parameters and design of a vertical cooler with inclined, gas-permeable blades and with a vibrating, suspended layer of granules on them (Vibrating Fluidized Bed—VFB). The algorithm is based on the use of the equations of heat and material balance, taking into account the influx of moisture into the layer with cold air and dust—as a carryover. Mode entrainment of dust particles and moisture from the VFB is described by using empirical formulas and
*Π*-theorem. To calculate the cooling time of granules a model of the dynamics of a variable mass VFB was built, which linked the geometrical and physical process parameters to a single dependency. An example showed that mass flow of granules of 248 kg/h and a volume flow of air of 646 m
^{3}/h with temperature of 30
℃ to cool the zeolite granules from 110
℃ to 42
℃ for 49 s required a vertical apparatus of rectangular shape with four chambers and with volume of 0.2 m
^{3}. A comparative analysis of technological parameters of the projected cooler with the parameters of typical industrial apparatuses showed that for all indicators: the cooling time of granules, the flow rate of gas (air) and the heat flow, a 4-chambered, vertical apparatus of rectangular shape with VFB was the most effective.

Currently, a widely used method of materials production in several industries is in the form of granules. The granules come out of the press or granulator at a relatively high temperature of 90˚C - 120˚C and must be cooled to 30˚C - 50˚C―depending on the season and the geographic position of the enterprise. The temperature and humidity of the granules at the output must be reduced to the limits for the conservation of their properties and persistent storage. For cooling the granules two basic types of coolers are used: refrigerating vertical columns and horizontal conveyor coolers. The former have a preferential distribution [

towards the falling granules, forming at the same time the VFB. The layer flows from one chamber of the cooler to another, thereby cooling the granules proceeds in continuous mode. In the apparatus, due to vibrations, there is an intensive heat exchange of granules with a stream of cold air and short-term contact with each other. Dust particles, together with droplets of moisture that are formed, are removed from the apparatus through openings in the side walls of the cooler (

・ Problem 1: An algorithm for calculation and design of the cooler with the vibrating fluidized bed of granules.

・ Problem 2: Determining the number of cooling chambers n and technological parameters which ensure cooling hot granules with a predetermined flow rate

・ Problem 3: Comparatively analyzing the efficacy of the investigational cooler with the typical industrial apparatuses which is used to cool the granules in the fluidized bed.

By using the scheme of convective heat transfer [

where

there

where

To create an algorithm for calculation of the final values

Mode of fluidization (by the criteria

For the pressure drop

Speed entrainment of dust particles and moisture from the layer of granules

We imagine the mass flow of dust and moisture carryover from the layer of granules

Then, using p-theorem [

The factor in (9), comprising a heat transfer coefficient

Here

For the geometric and technological parameters of the cooling process of the granules we will use a formula similar to that of [

The height of the humidified layer of granules on the blade

The height and volume of chambers of cooler are

where

The average mass flow rate of cooling granules

The average heat flow

The total mass of moisture, introduced into the layer

For the relationship of geometrical and physical parameters of the cooling process of granules below we propose a model of a dynamics of the vibrating, suspended bed of variable mass. The model is based on the representation VFB of cooled granules, as a system of material points of variable composition, that move with its center of mass under the influence of external and internal forces (

Consider the layer of granules, located above the perforated blade, and write the differential equation of motion of the mass center VFB in vector form:

In equation (16) the forces

is the power of the impact of the vibrator on the gas-permeable blade with the layer of granules on it.

is the resistance force layer granules, which takes into account the interaction of discrete granules with the flow of cold air [

moisture, carried away from the bed of granules

For the axis x-along the lines of movement of the layer (

For the axis z:

where

From Equation (24) under

Equation (23) after the substitution of (25) can be represented in the form of a linear differential equation:

Solution of differential Equation (26) for the velocity of the center of mass

where

From (28) we obtain the equation to determine the residence time of the granules in the layer on the blade in a chamber of the cooler

Below is a procedure-algorithm for calculating a vertical rectangular cooler with VFB of granules (

The algorithm is based on the above formulas and is reduced to determining the geometrical and technological parameters of the cooler, which provide the specified mode of cooling the granules and dust-moisture carryover from the VFB.

Tables 1-3, referenced in the algorithm, contain the initial data: geometry of permeable to gas, vibrated blade and layer of granules on it; kinematic, heat and mass transfer parameters, involved in the calculations, and the critical parameters-limitations, which are dependent on the requirements by the consumers. One example is cooling of granules of zeolite introduced specific numerical values of design parameters.

The geometrical parameters of a gas-permeable blade and layer of granules on it | ||
---|---|---|

Symbols | Values | Units |

0.45 | m | |

0.15 | m | |

0.01 | m | |

78 | degree | |

0.2 | dimensionless | |

5 × 10^{−10} | m^{2} | |

3.75 × 10^{−5} | m | |

0.09 | m | |

208 | s^{−1} | |

1.2 × 10^{−3} | m | |

2.675 × 10^{−3} | m | |

0.6 | dimensionless | |

0.0242 | m^{2} | |

9 × 10^{−4} | m^{2} | |

0.027 | m^{2} | |

6 × 10^{−4} | m^{2} | |

1.68 × 10^{−}^{4} | m |

Initial physical parameters | ||
---|---|---|

Symbols | Values | Units |

75.2 | dimensionless | |

2100 | kg/m^{3} | |

0.2 | dimensionless | |

9.81 | m/s^{2} | |

0.7275 | N/m | |

0 | Kg/s | |

0 | Pa | |

0 | m/s | |

16,948 | Pa | |

1.15 | Kg/m^{3} | |

1.7 × 10^{−}^{5} | Kg/m・s | |

1.48 × 10^{−}^{5} | m^{2}/s | |

0.962 | kJ/kg・˚C | |

2.45 × 10^{−}^{5} | kW/m・˚C | |

1 | kJ/kg・˚C | |

4.2 | kJ/kg・˚C | |

125.61 | kJ/kg | |

1,568,827 | dimensionless | |

300.6 | dimensionless | |

1.66 | m/s | |

0.0516 | kg/s | |

388.6 | dimensionless | |

1.14 | m/s |

Critical parameters―limitations | ||
---|---|---|

Symbols | Values | Units |

250 | kg/h | |

110 | ˚C | |

45 | ˚C | |

0.01 | kg/kg | |

30 | ˚C | |

60 | % | |

2.5 | kg/h |

Chamber 1 | Chamber 2 | ||||||
---|---|---|---|---|---|---|---|

Symbols | Values | Units | Symbols | Values | Units | ||

60 | % | 60 | % | ||||

0 | % | 5 | % | ||||

110 | ˚C | 87.345 | ˚C | ||||

0.0154 | kg/kg | 0.0154 | kg/kg | ||||

0.0326 | kg/kg | 0.02017 | kg/kg | ||||

0.89 × 10^{−}^{3} | kg/s | 0.25 × 10^{−}^{3} | kg/s | ||||

0.0367 | m/s | 0.0369 | m/s | ||||

12.255 | s | 12.204 | s | ||||

0.60 × 10^{−}^{3} | kg/s | 0.76 × 10^{−}^{3} | kg/s | ||||

0.0697 | kg/s | 0.0689 | kg/s | ||||

0.03592 | kJ/s | −0.0646 | kJ/s | ||||

2707.7 | kJ/kg | 2663 | kJ/kg | ||||

0.0225 | kg/kg | 0.026 | kg/kg | ||||

87.345 | ˚C | 56.54 | ˚C | ||||

Chamber 3 | Chamber 4 | ||||||

Symbols | Values | Units | Symbols | Values | Units | ||

60 | % | 60 | % | ||||

20 | % | 30 | % | ||||

56.54 | ˚C | 49.79 | ˚C | ||||

0.0154 | kg/kg | 0.0154 | kg/kg | ||||

0.02129 | kg/kg | 0.02198 | kg/kg | ||||

0.30 × 10^{−}^{3} | kg/s | 0.34 × 10^{−}^{3} | kg/s | ||||

0.0364 | m/s | 0.0362 | m/s | ||||

12.347 | s | 12.416 | s | ||||

1.16 × 10^{−}^{3} | kg/s | 1.31 × 10^{−}^{3} | kg/s | ||||

0.0685 | kg/s | 0.0684 | kg/s | ||||

−0.10775 | kJ/s | −0.12144 | kJ/s | ||||

2602.4 | kJ/kg | 2589.1 | kJ/kg | ||||

0.03024 | kg/kg | 0.035 | kg/kg | ||||

49.79 | ˚C | 42.1 | ˚C | ||||

Technological parameters | ||
---|---|---|

Symbols | Values | Units |

0.44 × 10^{−}^{3} | kg/s | |

0.96 × 10^{−}^{3} | kg/s | |

0.0366 | m/s | |

0.022 | kg | |

0.047 | kg | |

46.4 | % | |

12.3 | s | |

49.2 | s | |

646 | m^{3}/h | |

8662 | kJ/h | |

248 | kg/h | |

118.4 | kJ | |

0.252 | kg/m^{3} |

Geometrical parameters | ||
---|---|---|

Symbols | Values | Units |

0.01 | m | |

0.608 | m | |

0.708 | m | |

2.832 | m | |

0.0408 | m^{3} | |

0.187 | m^{3} |

Coefficients of Equation (29) | ||
---|---|---|

Symbols | Values | Units |

0.1532 | s^{−}^{1} | |

0.0217 | dimensionless | |

1860.6 | m/s | |

16.224 | m/s^{2} | |

−3.574 | m/s^{2} | |

0.0188 | m/s | |

2.76 × 10^{−}^{7} | m | |

0.239 | m | |

12 | s |

Type of devices, the starting materials | ||||
---|---|---|---|---|

Type 1―Tower rectangular shape. Material-Ammonium nitrate. | Type 2―Vertical rectangular unit. Material-Urea granules. | Type 3―The test apparatus ( | ||

Symbols | Values | Values | Values | Units |

11 × 8 × 15 | 4.65 × 4.65 × 1.5 | 0.4 × 0.15 × 2.83 | m3 | |

(1/5) × 10^{−}^{3} | (1/4) × 10^{−}^{3} | 2.675 × 10^{−}^{3} | m | |

≥60 | 60 | 60 | % | |

25/30 | 28/30 | 30 | ˚C | |

0.1/0.15 | 0.15 | 0.01 | m | |

80/110 | 70/80 | 110 | ˚C | |

40/50 | 40/50 | 42 | ˚C | |

13.7 | 8.35 | 4 | - | |

2.5 | kg/h |

In order to implement the given mode of the cooling granules that takes into account dust-moisture carryover, the most effective shape is the vertical device, with a rectangular shape with sloping, gas-permeable blades and with the vibrating fluidized bed of granules on them (

The proposed mathematical model of the dynamics of a vibrating, suspended layer, allows us to estimate the effect of geometrical and physical parameters on the time of cooling the granules in the cooling chamber

A comparative analysis of the technological parameters of the study device (

The proposed algorithm for calculation of coolers with the VFB is based on the joint use of the equations of heat and mass balances (1) and the mathematical model of dynamics of a vibrating fluidized bed (Equation (16) and Equation (29)).

The algorithm takes into account the massive influx of moisture into the layer of granules

The authors would like to thank the head of the laboratory of special fertilizers (Haifa Chemicals Ltd.), Dr. B. Gordonov, for the invitation to the solution of problems for cooling the granules in the Chemical Industry of Israel.

Type of devices | ||||
---|---|---|---|---|

Type 1―Tower rectangular shape. | Type 2―Vertical rectangular unit. | Type 3―The test apparatus | ||

Symbol | Values | Values | Values | Units |

2/3 | 4/5 | 0.82 | Min | |

(75/80) × 10^{3} | (22/84) × 10^{3} | 646 | m^{3}/h | |

2 × 10^{6} | 0.6 × 10^{6} | 8662 | kJ/h | |

6864 | 4176 | 248 | kg/h | |

11.3 | 5.6 | 2.6 | m^{3}/kg | |

291 | 144 | 35 | kJ/kg |

Valery Katz,Slava Katz, (2016) Cooling of Granules in Vibrating, Suspended Bed: Engineering Simulation. Modern Mechanical Engineering,06,76-90. doi: 10.4236/mme.2016.62009

^{2};^{2};

pressure drop of the heat carrier (air), Pa;^{3}/h;

heat flow, kJ/h;^{2};^{2};

Greek symbols:^{2}・C;^{2}/s;^{3};^{2};

Superscript: *: Critical values-limitations.

Subscripts:

Dimensionless groups-Criteria: