Particle velocity and stationary layer height analysis for modification and validation of particulate Plug-2 pressure drop model

Highlights

• Properties on which classical plug-2 pressure drop depends established independently.

• Using the properties, classical plug-2 pressure drop model is modified and validated.

• Classical plug-2 exists for Re < 2300.

• New type of plug flow namely ‘Plug-2*’ is identified.

Abstract

For the plug flow mode of dense-phase pneumatic conveying, three different types of plugs have been defined, and models for predicting the pressure drop in conveying pipeline have been established. In literature, the model for Plug-2 pressure drop is mechanistic and depends on several properties of the materials forming the plug. Therefore, modification of the model with experimentally corrected material properties is considered in this study. The parameters tested in this study are plug velocity, particle velocity, and the stationary layer height. Using the new functional relationships, the Plug-2 pressure drop model is modified and its predictions are compared with experimentally measured pressure drops obtained in conveying Plug-2. It is found that the modified Plug-2 pressure drop model matches experimental values well within the variation range of ±20%. Further, a detailed analysis of various types of related velocities is presented and a new type of plug, called ‘Plug-2*’, is developed.

Introduction

Plug conveying is a type of dense-phase pneumatic conveying, which is gaining popularity because of its advantage in overall power saving and its capability to decrease pipe wear and particle attrition. Scientific works on plug conveying, both experimental and numerical, started years ago. Literature [[1], [2], [3], [4], [5], [6], [7], [8], [9]] presents a number of basic models for predicting the pressure drop while conveying the particulate plugs. Muschelknautz and Krambrock [1] suggested that the friction force between the bulk material and pipe wall is the main cause of energy loss. Thereafter, they developed a suitable model for the associated friction force. Konrad et al. [2] used force balance on a single plug by combining the Jannsen model [3] and the hydrostatic gravity force. Similarly, Jianglin [4] developed a three-layer model for predicting transition zone boundaries for slug conveying by using both the unstable flow-forming mechanism and the mass, force, and momentum balances. Using the same equations, Mi and Wypych [5] and Pan and Wypych [6] developed another model for predicting wall pressure for slug flows.

As in all previous studies, Hong and Klinzing [7], Borzone and Klinzing [8], and Aziz and Klinzing [9] developed pressure drop models for the dense-phase slug conveying using force balance equations, considering a pressure drop across the plug/slug given by $∆P\frac{\pi D^2}{4}$. From the force balance obtained in these studies, Shaul and Kalman [10] recently based their analysis of forces acting on a differential slice (control volume) of particulate material plug by employing ideas from solid mechanics. The forces considered for the force balance are as follows: pressure stresses (ΔP), normal stresses (σ_a), and wall shear stresses (τ_w). Shaul and Kalman [10] indicated that the previously used basic differential equations for the force balance were inaccurate. They claimed that the pressure force should be divided into two forces: the pressure force over the voids, causing a drag force because of the airflow through the permeable particulate plug (ε$∆P\frac{\pi D^2}{4}$), and the pressure force acting directly on particles [(1 − ε)$∆P\frac{\pi D^2}{4}$]. The void pressure force was applied to the differential equation of the force balance, whereas the direct pressure force was considered the boundary condition of the stress within the plug. Thus, Shaul and Kalman [10] modified the force balance equation to Eq. (1):

$$\epsilon \frac{\mathit{dp}}{\mathit{dx}}+\frac{d{\sigma }_a}{\mathit{dx}}+\frac{4{\tau }_w}{D}+{\rho }_{b}g=0$$

where ε is the void fraction, p is the pressure (Pa) through the plug, σ_a is the axial stress (Pa), τ_w is the wall shear stress (Pa), D is the internal pipe diameter (m), ρ_b is the bulk density of the material (kg/m³), and g is the gravitational acceleration (m/s²). By applying boundary conditions, using the exact direction of forces, and considering the plug front angles Shaul and Kalman [11] developed the Plug-1, Plug-2 and Plug-3 pressure drop models. Where, first type (Plug-1) is the one in which the plug covers the whole pipe cross-section and does not leave any stationary particle either behind or in front. However, it may contain a slope of particles in front and rear of the horizontal plug. In the second type (Plug-2) a stationary layer of particles is left at the rear of the plug, while the next plug picks it up. Whereas, third type of plugs (Plug-3) are the small plugs that move over a stationary bed of particles.

As far as plug-1 model was concerned this model was related to many independent properties from which Plug-1 density, Plug-1 void fraction, and plug front angles were defined in terms of the operating parameters (pressure drop) by Rawat and Kalman [12]. Using those functions, the Plug-1 pressure drop model was further successfully modified and validated by Rawat and Kalman [12].

Shaul and Kalman [11] considered a mechanistic approach also for Plug-2, which is the most common type of plug. Plug-2 has a moving core (the same as plug I) and a stationary layer of particles in-between consequent plugs. To keep the plug length constant, each plug picks up the stationary layer at the front and releases particles onto the stationary layer at the rear. Therefore, in addition to the pressure drop required to move the plug core (in the same way as Plug-1), three extra sources of pressure drops are considered. These are the axial boundary stresses from the plug front slope, the momentum exchange from the particles accelerating from zero (at the stationary layer) to those travelling within the plug, and the stationary layer resistance. These result in the following expression for the pressure drop over a single Plug-2 (Eq. (2)).

$${\mathit{∆P}}_{\mathbf{Type}\mathbf{-}\mathbf{2}}\mathbf{=}\left(\frac{\mathbf{8}{\mathbf{\mu }}_{\mathbf{e}}\mathbf{g}{\mathbf{\rho }}_{\mathbf{bst}}\mathbf{h}}{\left.\mathbf{\pi }{\mathbf{D}}^{\mathbf{2}}(\mathbf{D}\mathbf{-}\mathbf{h}\right)}\right)\underset{{\mathbf{y}}_{\mathbf{cm}}}{\overset{\mathbf{D}\mathbf{-}{\mathbf{y}}_{\mathbf{cm}}}{\int }}\sqrt{\mathbf{y}\left(\mathbf{D}\mathbf{-}\mathbf{y}\right)}\left(\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{cm}}\right)\mathbf{dy}\mathbf{+}\frac{\left(\frac{\mathbf{4}{\mathbf{\mu }}_{\mathbf{wK}}}{\mathbf{D}}\left(\frac{{\mathbf{\mu }}_{\mathbf{w}}{\mathbf{\rho }}_{\mathbf{b}}\mathbf{g}}{\mathbf{2}\mathbf{tan\theta }}\mathbf{+}\mathbf{\alpha }{\mathbf{\rho }}_{\mathbf{bst}}\left(\mathbf{1}\mathbf{-}\mathbf{\alpha }\frac{{\mathbf{\rho }}_{\mathbf{bst}}}{{\mathbf{\rho }}_{\mathbf{b}}}\right){\mathbf{U}}_{\mathbf{plu}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{\rho }}_{\mathbf{bst}}\mathbf{g}\left[\frac{\mathbf{D}}{\mathbf{2}}\left(\frac{\mathbf{\varnothing }}{\mathbf{2}}\mathbf{+}\frac{\mathbf{Sin}\mathbf{2}\mathbf{\varnothing }}{\mathbf{4}}\mathbf{-}\mathbf{Sin}\mathbf{\varnothing }\right)\mathbf{+}\mathbf{hSin}\mathbf{\varnothing }\right]\right)\mathbf{+}{\mathbf{\mu }}_{\mathbf{w}}{\mathbf{\rho }}_{\mathbf{b}}\mathbf{g}\mathbf{+}\left(\frac{\mathbf{4}{\mathbf{C}}_{\mathbf{w}}}{\mathbf{D}}\right){\mathbf{e}}^{\frac{\mathbf{4}{\mathbf{\mu }}_{\mathbf{w}}\mathbf{KL}}{\mathbf{D}}\mathbf{-}\frac{\mathbf{4}{\mathbf{C}}_{\mathbf{w}}}{\mathbf{D}}\mathbf{-}{\mathbf{\mu }}_{\mathbf{w}}{\mathbf{\rho }}_{\mathbf{b}}\mathbf{g}}\right.}{\frac{\left(\mathbf{1}\mathbf{-}\mathbf{\epsilon }\right)\mathbf{4}{\mathbf{\mu }}_{\mathbf{w}}\mathbf{K}}{\mathbf{D}}\mathbf{+}\frac{\mathbf{\epsilon }}{\mathbf{L}}\left({\mathbf{e}}^{\frac{\mathbf{4}{\mathbf{\mu }}_{\mathbf{wKL}}}{\mathbf{D}}}\mathbf{-}\mathbf{1}\right)}$$

For understanding the above equation better, the equation has been divided into three parts as below:

$${∆\mathit{P}}_{\mathbf{Type}-\mathbf{2}}=\left[\frac{\mathbf{8}{\mathbf{\mu }}_{\mathbf{e}}\mathbf{g}{\mathbf{\rho }}_{\mathbf{bst}}\mathbf{h}}{\left.\mathbf{\pi }{\mathbf{D}}^{\mathbf{2}}(\mathbf{D}-\mathbf{h}\right)}\right]\left(\mathbf{A}\right)+\frac{\left[\frac{\mathbf{4}{\mathbf{\mu }}_{\mathbf{wK}}}{\mathbf{D}}\left(\mathit{B}\right)+{\mathbf{\mu }}_{\mathbf{w}}{\mathbf{\rho }}_{\mathbf{b}}\mathbf{g}+\frac{\mathbf{4}{\mathbf{C}}_{\mathbf{w}}}{\mathbf{D}}\right]{\mathbf{e}}^{\frac{\mathbf{4}{\mathbf{\mu }}_{\mathbf{w}}\mathbf{KL}}{\mathbf{D}}-\frac{\mathbf{4}{\mathbf{C}}_{\mathbf{w}}}{\mathbf{D}}-{\mathbf{\mu }}_{\mathbf{w}}{\mathbf{\rho }}_{\mathbf{b}}\mathbf{g}}}{\frac{\left(\mathbf{1}-\mathbf{\epsilon }\right)\mathbf{4}{\mathbf{\mu }}_{\mathbf{w}}\mathbf{K}}{\mathbf{D}}+\frac{\mathbf{\epsilon }}{\mathbf{L}}\left({\mathbf{e}}^{\frac{\mathbf{4}{\mathbf{\mu }}_{\mathbf{wKL}}}{\mathbf{D}}}-\mathbf{1}\right)}$$

$$\mathit{A}\mathit{=}\underset{{\mathbf{y}}_{\mathbf{cm}}}{\overset{\mathbf{D}\mathbf{-}{\mathbf{y}}_{\mathbf{cm}}}{\int }}\sqrt{\mathbf{y}\left(\mathbf{D}\mathbf{-}\mathbf{y}\right)}\left(\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{cm}}\right)\mathbf{dy}$$

$$\mathit{B}\mathit{=}\frac{{\mathbf{\mu }}_{\mathbf{w}}{\mathbf{\rho }}_{\mathbf{b}}\mathbf{g}}{\mathbf{2}\mathbf{tan\theta }}\mathbf{+}\mathbf{\alpha }{\mathbf{\rho }}_{\mathbf{bst}}\left(\mathbf{1}\mathbf{-}\mathbf{\alpha }\frac{{\mathbf{\rho }}_{\mathbf{bst}}}{{\mathbf{\rho }}_{\mathbf{b}}}\right){\mathbf{U}}_{\mathbf{plu}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{\rho }}_{\mathbf{bst}}\mathbf{g}\left[\frac{\mathbf{D}}{\mathbf{2}}\left(\frac{\mathbf{\varnothing }}{\mathbf{2}}\mathbf{+}\frac{\mathbf{Sin}\mathbf{2}\mathbf{\varnothing }}{\mathbf{4}}\mathbf{-}\mathbf{Sin}\mathbf{\varnothing }\right)\mathbf{+}\mathbf{hSin}\mathbf{\varnothing }\right]$$

The equation indicates that the pressure drop depends on a number of physical and geometrical properties of the plug's material (all of which can be characterized independently), namely the wall friction coefficient (μ_w), coefficient of internal friction (φ_e), coefficient of cohesion (C_w), plug angle (θ), and stress transmission coefficient (k). The values of μ_w, φ_e, and C_w can be measured using a Jenike shear cell apparatus following the standard operating procedure described in its operation manual. As reported by Shaul and Kalman [11] in different terms, the stress transmission coefficient (k) can be calculated for each plug length. The plug front angle (θ), plug density (ρ_b), and plug void fraction (ε) were modified by Rawat and Kalman [12] for Plug-1. Although the plug front angle of Plug-1 might also fit to Plug-2, the density and void fractions do not. Because Plug-1 is constructed with very fine powders with negligible permeability, the pressure drop compresses the plug, while for Plug-2 the air flow through the plug is expected to fluidize the particles and the density may decrease. In addition, some new properties are introduced for Plug-2, namely the stationary layer height, the fraction of the pipe area covered by the stationary layer, and the plug velocity. The first two are related to each other irrespective of the pipe diameter. However, the use of the plug velocity seems to be incorrect and should be replaced by the particle velocity in Eq. (2). It should also be noted that the model defines two bulk densities, one for the plug and the other for the stationary layer. All of the above parameters will be discussed and defined later in this paper in order to modify the model.

Furthermore, it can be observed that the plug bulk density and void fraction found in Eq. (2) are related; one can be found from the other when the particle density [13] is known:

$$\epsilon =1-\frac{{\rho }_b}{{\rho }_p}$$

where ε is the average void fraction, ρ_b is the bulk density, and ρ_p is the particle density. During the conveyance of plugs, both the plug bulk density and the void fraction are expected to depend on several operating parameters. Among them, the pressure drop is the most important. Increasing the pressure drop from the initial values may change the plug density and, in turn, the void fraction of the plugs may also change. Accordingly, the void fraction and bulk density of particulate materials have been an active area of research.

Initially, for describing the flow through porous systems, three approaches have been mentioned in literature to define a relationship between pressure drop and void fraction: voids modelling [[14], [15], [16], [17], [18], [19]], imensional analysis [20], and particle force balance [21]. Furthermore, the three models of Carman–Kozeny [15,16], Ergun [17], and Molerus [21] are of great theoretical interest as far as pneumatic conveying is concerned as they describe the relationship between the pressure drop and the void fraction, as shown in Eqs. (4), (5), (6), respectively:

$$\frac{∆P}{L_{\mathit{bed}}}=180\frac{{\left(1-\epsilon \right)}^2}{{\epsilon }^3}\frac{{\eta }_f}{d_p^2{\phi }^2}{v}_f$$

$$\frac{∆P}{L_{\mathit{bed}}}=150\frac{{\eta }_f{\left(1-\epsilon \right)}^2}{d_p^2{\epsilon }^3}{v}_f+1.75\frac{{\rho }_b\left(1-\epsilon \right)}{d_p{\epsilon }^3}{v}_f^2$$

$$\frac{∆P}{L_{\mathit{bed}}}=\frac{{\rho }_f{v}_f^2}{d_p}\frac{18}{\mathit{Re}}\left(1-\epsilon \right){\epsilon }^{-4.65}$$

where, ∆P is the pressure drop, L_bed is the length of the particulate bed, ε is the void fraction, η_f is the viscosity of the fluid, d_p is the particle size of the materials, φ is the sphericity, v_f is the superficial velocity of air, and ρ_b is the bulk density of the material.

Many experimental studies were based in these models. Rizk [22] seems to be the first to investigate experimentally the void fraction using the double solenoid method for the dilute-phase pneumatic conveying. Subsequently, several researchers studied the void fraction experimentally [[23], [24], [25], [26], [27]]. Among these, the most important and most recent is the study conducted by Pahk and Klinzing [28]. They postulated two methods for determining the void fraction. The first method is for bulk solids, whereby they define the average void fraction as $\epsilon =1-\frac{{\rho }_{\mathit{plug}}}{{\rho }_p},$where ρ_plug is the bulk density of the particulate plugs, and ρ_p is the particle density of solids in plugs. In the second method, they employed the empirical Ergun equation modified by Lecreps and Sommer [29] to calculate the void fraction of the plug material. They found that the void fraction predicted by both techniques are comparable. Nevertheless, the void fraction obtained by the method for bulk solids is more accurate because it involved experiments. Similarly, a few more important experimental studies [[30], [31], [32], [33]] have been conducted for void fraction determination of horizontal slugs, which will be critically evaluated along with the appropriateness of Eqs. (4), (5), (6) in the dedicated section of plug density and void fraction in this paper.

A technique involving feeding and controlling a single plug to experimentally study its behavior was established in recent years. It was initiated by Vasquez et al. [34] to measure the friction over a single plug. It was then followed by Lecreps et al. [30,32, and 35] to investigate various aspects related to plug flow, such as porosity, different velocities, and the pressure drop. While all the above studies involved Plug-2 conveying, Rawat and Kalman [12 and 36] studied the behavior of Plug-1. However, it should be emphasized that, although single plugs are studied, the plug themselves are not the targets. The target is to understand more about real multi-plug conveying, and single plug studies elucidate some phenomena.

Furthermore, the model given in Eq. (2) for Plug-2 is a theoretical equation based on the mechanics involved. However, to date, there is no experimental work that has investigated the similarity of the model to the actual pressure drop. Accordingly, an experimental validation of the model is a further objective of the current study. The validation of the model will not only indicate the modifications necessary for the model of Shaul and Kalman for Plug-2 [11,37], but will also establish the pressure drop equation of Plug-2 for future designers of such type of conveying.