^{1}

^{*}

^{2}

^{3}

The carrier fluid (air or water) is used to transport solid material from the source place to its destination point through pipeline. Using air as carrier fluid to transport solid material through pipeline is called Pneumo transport, whereas transporting material with water or any other liquid through pipeline is called as hydraulic transport. A large number of installations are now available globally to transport solid materials to short, medium, and long distances using water/air as carrier fluid. However , the design of such system of pipeline is still an empirical art. In the present investigation, one generalized mathematical model developed by Shrivastava and Kar (SK Model) and CFD models were used and compared with experimental results for pneumatic and hydraulic transport of granular solids. The motivation of present work is to find the accuracy of SK model based on analytical, empirical and semi- empirical for the prediction of pressure drop and comparing the result with CFD based on mathematical equation for the mixture flow in the horizontal and vertical pipe lines. The comparison of pressure drop results obtained by using SK model and CFD model were validated with the experimental results for pneumatic and hydraulic transport of solids through. From the comparison results, it was observed that the results of pressure drop predicted by SK model are more accurate than the CFD models for all the cases considered.

Fluid-solid mixture flow is form of multiphase flow which is simultaneous flow of two or several phases through a system such as pipe in the present investigation. From the literature survey [

Many investigators have carried out experimental on prediction of pressure drop and minimum conveying velocity on fluid-solid mixture flow through pipes. Titus et al. [

Kelessidis et al. [

An experimental investigation related to frictional pressure loss in a horizontal pipeline on the effect of various particle size was examined by Nabil et al. [

Investigation is done on hydraulic transport of coarse solid particles to demonstrate as to how a generalized mathematical model (SK model), developed earlier by Shrivastava and Kar [

For any fluid-solid transport system the most important parameters required to design the system are the pressure drop and minimum transport velocity. Estimation of pressure drop is very complicated phenomena in multiphase flow. The techniques for analyzing two phase flow include correlations, phenomenological models, simple analytical model, integral analysis, differential analysis, computational fluid dynamics (CFD) and artificial neural network (ANN) [

The physics of two-phase flow is more complex than for single flow due to the presence of dispersed (solid) phase so that this work deals with investigation on fluid (gas/liquid)-solid mixture flow through pipes. Such two-phase flow is widely encountered in pipe flows. One example of the gas-solid mixture through pipe in textile factory (air-cotton ball) that found nearby Mikelle city, Ethiopia in eastern Africa is presented in the application of this investigation. The data used for the study is based on both primary and secondary sources. The primary data are collected through MAA Garment and Textile Factory at Mekelle city, Tigray region, Ethiopia. The secondary sources of data that the researcher used are from different relevant books and Journals. In this study an attempt has been made to apply and compare the two methods of prediction pressure drop i.e. Shrivastava and Kar (SK model) [

The total head loss ( h t ) per unit pipe length, for conveying heterogeneous mixture of fluid and solid through an horizontal, vertical, inclined and bending pipe under fully-accelerated and suspended conditions of flow, is the summation of head loss ( h f ) due to fluid-pipe friction and additional head loss h s due to the presence of solid particles in the flow, Shrivastava and Kar [

h t = f V f 2 2 g D + { 0.5 C d ρ f A p V T 2 + m P g ( 1 − ρ f ρ p ) sin θ } M s M f 1 m P g (1)

The Darcy-Weisbach friction factor (f) of the above Equation (1) may be determined by using the Blasius equations as given below:

f = 0.3164 R e 0.25 (2)

where, Re-Reynolds numbers defined as R e = V f D v f . The values of terminal

velocity ( V T ) and drag coefficient ( C d ) of the solid particles to be transported with a carrier fluid through a pipe may be determined either experimentally or analytically by using the properties of the carrier fluid and the solid particles and taking the mean equivalent spherical diameter (d) of the particles in the following Equations (3) to (6):

V T = 4 g d ( ρ p − ρ f ) 3 C d ρ f (3)

C d = 18.5 R e p − 0.6 , 0.1 < R e p < 500 (4)

C d = 0.44 , 500 < R e p < 2 × 10 5 (5)

where, R e p ―Reynolds number of particles is defined as

R e p = V p d v f (6)

M s = C v A ( V g − V T ) ρ p (7)

And Mass flow rate of the fluid is defined as: M f = ρ f × A × V f

The mathematical expression developed for critical velocity ( V C ) in the SK model by applying the well-known techniques of optimization to Equation (1) is as given below:

V c = [ { 0.5 × C d × p f × A p × V t 2 + m p × g ( 1 − ρ f p p ) sin θ } × { 2 × M s × D C + 1 m p × K × V f C ( 2 − C ) A × ρ f } ] 1 3 − C (8)

where values of constants K and C may be determined experimentally from the plots of friction factor (f) against Reynolds number (Re) for the flow of fluid alone in pipe, which results into a relation of the type:

f = K R e C (9)

The value of density of particle ( ρ p ) can be determined by using the relationship between mass of particle ( m p ) which is directly measured and volume of the particle as shown below equation:

ρ p = m p V (10)

The value of density of fluid is expressed by the equation below:

ρ f = P R s p e c f i c T (11)

Advances in computational fluid mechanics have provided the basis for further insight into the dynamics of multiphase flows. Currently there are two approaches for the numerical calculation of multiphase flows: the Euler-Lagrange approach and the Euler-Euler approach.

The fluid phase is treated as a continuum by solving the Navier-Stokes equations, while the dispersed phase is solved by tracking a large number of particles, bubbles, or droplets through the calculated flow field. The dispersed phase can exchange momentum, mass, and energy with the fluid phase. This approach is made considerably simpler when particle-particle interactions can be neglected, and this requires that the dispersed second phase occupies a low volume fraction, even though high mass loading ( m p ≥ m f ) is acceptable as reported by Bartosik and Shook [

Manoj Kumar and Kaushal [

The input data used for prediction pressure drop using the SK model are collected

S.No | Parameters | Items |
---|---|---|

1 | Type of pipe | Steel based |

2 | Material of the pipe | Galvanized steel |

3 | Total length of pipe | 16.6 m |

4 | Geometry of pipe―Vertical, Horizontal & Bending | Circular pipe |

5 | Inner Diameter of the pipe 1 | 250 mm |

6 | inner diameter of the pipe 2 | 650 mm |

7 | Solid phase material | Cotton seed(waste) |

8 | Mass flow rate pipe 1 | 0.167 kg/s |

9 | Capacity of Fan | 5.5 kW (4Nos) |

from the Manual of MAA garment and Textile factory [

The equations given in the SK model predicts pressure drops more accurately than any other correlations available. However, many work in this area has been done utilizing the numerical methods and CFD techniques during last five decades to predict the values of pressure drop for conveying solids through pipes using water or air as carrier fluids. Hence it was decided to compare the performance of SK model with that of the CFD models by comparing the calculated results of pressure drop with the experimental values of the investigators [

In this section the objective is to examine the accuracy of SK model in predicting the values of pressure drops by comparing the results with the experimental values and those obtained utilizing the CFD model for hydraulic and pneumatic conveying of granular solids.

Experimental investigations have been conducted by Tamer Nabil et al. [_{50} = 0.2 mm, 0.7 mm and 1.4 mm) at different volumetric concentrations with water through a horizontal pipes of diameter 26.8 mm. These results of pressure drop have been compared with the predicted values using CFD model and also the experimental results. All these comparisons of pressure drops obtained SK model, CFD model and experimental results have been plotted in Figures 1-3 for sands of three different sizes d_{50} = 0.2 mm, 0.7 mm and 1.4 mm respectively. It may be observed from Figures 1-3, the SK model results are more closely and it was matching with experimental results than the predictions made by using CFD model for all the sizes of sand and all volumetric concentrations. The properties

of solid material and water along with calculated values of C_{d} and V_{t} given in the Tables 2-4.

It may be concluded from the curves obtained using SK and CFD model that the pressure drop is inversely proportional to particle size, i.e pressure drop is minimum for highest size of particle and as the size reduces pressure drop also increases throughout the velocity of water spectrum. However it is evident from these comparisons that, the results of pressure drop obtained by SK model match very closely with experimental values than the predictions made by CFD model.

The deviation from the experimental in the case of SK model was found to be ±7.54% whereas for CFD model it was observed to be ±27% as clearly reflected in the graph. The Properties of PVC granules and water along with calculated values of C_{d} and V_{t} given in

In another work made with by Bartosik and Shook et al. [

Conveying of Sand | ||
---|---|---|

D = 26.8 mm | d 50 = 2 × 10 − 4 m | ρ p = 2650 kg / m 3 |

ν f = 1.004 × 10 − 6 m 2 / s | ρ f = [ 2 3 4 5 ] | ρ f = 1000 kg / m 3 |

C v = [ 0.05 0.1 0.25 0.3 ] | e = 1.5 × 10 − 6 | |

V t = 0.0246 m / s | C d = 7.1131 |

Hydro-transport | ||
---|---|---|

D = 26.8 mm | d 50 = 7 × 10 − 4 m | ρ p = 2650 kg / m 3 |

ν f = 1.004 × 10 − 6 m 2 / s | ρ f = [ 2 3 4 5 ] | ρ f = 1000 kg / m 3 |

C v = [ 0.05 0.1 0.25 0.3 ] | e = 1.5 × 10 − 6 | |

V t = 0.1030 m / s | C d = 1.4234 |

Hydro-transport (Sand-water mixture flow through Horizontal pipe) | ||
---|---|---|

D = 26.8 mm | d 50 = 1.4 × 10 − 3 m | ρ p = 2650 kg / m 3 |

ν f = 1.004 × 10 − 6 m 2 / s | ρ f = [ 2 2.5 3 3.5 4 ] | ρ f = 1000 kg / m 3 |

C v = [ 0.05 0.1 0.25 0.3 ] | e = 1.5 × 10 − 6 | |

V t = 0.2275 m / s | C d = 0.5836 |

Hydro-transport (liquid-solid mixture flow through vertical pipes) | ||
---|---|---|

D = 40 mm | d 50 = 3.4 × 10 − 3 m | ρ p = 1000 kg / m 3 |

ν f = 1.004 × 10 − 6 m 2 / s | material = PVC granules | ρ f = 1000 kg / m 3 |

V m i x = [ 3.06 3.55 4.09 4.55 4.85 5.67 6.36 ] | For C v = 10 % | |

V m i x = [ 2.97 3.55 4.03 4.58 5.13 5.52 5.81 ] | For C v = 20 % | |

V t = 0.2011 m / s | C d = 0.44 |

pipe of 40 mm inner diameter at two volume concentration of 10% and 20%. In this work the SK model was also applied to calculate pressure drops and the results were compared with experimental values and the prediction by CFD model in

An exhaustive numerical simulation using the CFD methodology and taking Euler-Euler approaching was performed for developing CFD model by Pandaba Patro et al. [

In the present investigation the SK model was applied to calculate pressure drops for conveying solids of different mass loading ratio pneumatically at different volumetric concentrations and results of pressure drop have been compared with the predicted values using CFD model and also with the experimental

results of Tsuji et al. [_{d} and V_{t} for plastic solid material given in

Besides the above investigations numerical simulation was also conducted by Brundaban Patro [

Pneumatic transport (Gas-solid) | ||
---|---|---|

D = 30 mm | d = 200 μ m | ρ p = 1000 kg / m 3 |

ν f = 1.46122 × 10 − 5 m 2 / s | β = 1, 2 & 3 and | ρ f = 1.225 kg / m 3 |

R e = [ 12540 20794 31270 41905 ] | SC = 0, 0.04 & 0.08 | |

V t = 0.6559 m / s | C d = 4.9574 |

Pneumatic transport (Gas-solid) | |||
---|---|---|---|

D = 30 mm | d = 200 μ m | ρ p = 1000 kg / m 3 | |

ν f = 1.46122 × 10 − 5 m 2 / s | β = 1, 2 & 3 | ρ f = 1.225 kg / m 3 | |

V f = [ 10 15 20 ] m / s | SC = 0.005 | ||

V t = 0.6559 m / s | C d = 4.9574 | ||

results than the predicted values using CFD model. The properties of air and calculated value of C_{d} and V_{t} for 200 µm given in

・ From the above experimental literatures for prediction of pressure drop, the deviation between the experimental result and calculated value is found to be good agreement as similar with Shrivastava Kaushal. The validation of experimental results of pressure drop by using of empirical and semi-empirical correlation of SK model was concluded as more accurate.

・ The SK model is good agreement for prediction of pressure drop as the volume concentration is less than 25%. Because this model is not considered the friction caused due to collision of the particle.

・ The pressure drop does not depend on particle size with constant internal diameter of pipe and mass flow rate of mixture rather it depends on the critical velocity.

We acknowledge our thanks to MAA garment for providing the necessary data and extraordinary support in this research work.

The authors declare no conflicts of interest regarding the publication of this paper.

Jothi, M., Haimanot, R. and Kumar, U. (2019) Investigation on Pressure Drop of Fluid-Solid Mixture Flow through Pipes Using CFD and SK Model. Journal of Applied Mathematics and Physics, 7, 218-232. https://doi.org/10.4236/jamp.2019.71018

Symbol Description [Unit]

A: Area, [m^{2}]

A p : Projected area [m^{2}]

C d : Drag coefficient [−]

C v : Volume fraction [%]

D: Diameter of the pipe [m]

d p : Diameter of the particle [m]

e: Roughness of pip [m]

f: Darcy friction factor [−]

R e p : Particles Reynolds number [−]

m f : Mass flow rate of fluid [kg/s]

M s : Mass flow rate of solid particles [kg/s]

h s : Head loss due to the solid particles [−]

h f : Head loss of fluid [−]

h t : Total head loss [−]

ν f : Kinematic viscosity of fluid [m^{2}/s]

ρ f : Density of fluid [kg/m^{3}]

ρ p : Density of particles [kg/m^{3}]

β: Mass loading ratio [−]

SC: Sepecularity coefficient [−]