Jet mixing is a widely used process that utilises kinetic energy from a pumped stream to blend miscible fluids in tanks or reactors [
1,
2]. It is particularly effective for large storage tanks where mechanical agitation is impossible to implement [
3,
4]. The process entails the high-speed recirculation of tank fluid in the form of a jet stream, which is directed through a nozzle and reintroduced into the bulk fluid of the tank. The jet stream is subjected to progressive characteristics modifications across the shear layer, starting from the nozzle inlet and the entire downstream sections. Laboratory observations reveal that fluid jets entering ambient fluid consistently adopt an approximately conical shape with a universal opening angle of 11.80°, regardless of the fluid type, nozzle diameter, and injection speed [
5]. The shear flow between a turbulent jet fluid and the surrounding ambient fluid can be divided into three regions [5,6]: the initial region, consisting of a potential core, a small conical-shaped core formed by mixing the jet and ambient fluid. The potential core possesses the flow characteristics of the nozzle inlet and spans a length of approximately six times the diameter of the jet nozzle,
dj; the transition region occurs where vortex cores form due to shear flow, subsequently combining to generate larger eddies that break down into smaller ones downstream. This process facilitates mixing and entrainment of the ambient fluid, commencing at approximately eight times the nozzle diameter along the jet axis. Lastly, the fully-developed region is characterised by similarity laws for mean velocity, spread angle, and entrainment. The fully-developed zone to typically commences at 30
dj, along the jet axis [
5,
27].
Equation 1 reveals that at a distance of 100
dj from the nozzle along the jet axis, a sharp decrease in the velocity of the central streamline occurs over a short span [
7]. Specifically, the centreline velocity is observed to decline from its initial value to a mere 5% of the jet inlet velocity at this location[
7]. Moreover, the turbulent jet mixing effect becomes negligible beyond a distance of roughly 400
dj [
7]. It is important to note that the jet also loses its characteristic properties upon impact with the tank wall, bottom, or liquid surface.[
5,
6].
>Mixing Time
Steady jets have been used for industrial mixing applications for several years, mainly for mixing low-viscosity, single-phase liquids. Mixing time is a fundamental parameter used to evaluate the efficiency of mixing operations, indicating the duration needed to achieve a satisfactory level of homogeneity[
4,
7,
8]. In the context of jet mixing, various empirical equations have been suggested in the literature to estimate mixing time. Wasewar[
3] provides an overview of multiple correlations linking diverse jet mixing parameters with the mixing time, as documented in the existing literature. However, the initial correlation proposed by Fosset [29] (Equation 2) is limited in its ability to interpret mixing time due to its empirical nature, with limited accuracy depending on specific tank geometries, jet configurations and operating conditions.
Fox and Gex [
9] found a strong dependence of mixing time on Reynolds number in the laminar regime but less so in the turbulent regime (Reynolds numbers ≥ 7000 (Equation 3). However, the applicability and accuracy of their correlation is limited due to its empirical nature, as it is based on experimental data and does not consider all the physical processes involved in mixing. Consequently, the correlation may not always provide accurate mixing time predictions, especially in complex flow geometries. Okita and Oyama[
10] aimed to address this issue, and they reported correlations of mixing time for axial jet and side entry nozzle configurations (Equations 4-5). They concluded that mixing time is independent of Reynolds number for values above 5000 in the turbulent jet regime.
The two primary models with high reliability commonly employed for designing jet-mixed vessels are the circulation and the turbulent jet models since they are not just empirical fit of data but are based on physical models.
Circulation flow arises due to the entrainment of bulk fluid by the jet (
) and the flow rate through the nozzle,
Q. The mean circulation time can be expressed in terms of the liquid volume (
V) contained within the tank and the total volumetric flow rate of bulk liquid (
QT =
+
Q) entrained by the jet at the point where it terminates (Equation 7).
zmax is the free jet path length from the nozzle to the point where the jet impinge on the tank wall or liquid surface. The total flow rate at the end of
zmax: is expressed [
11,
12], [
25,
27] as follows:
Ricou and Spalding [
24] reported a value of 0.32 for
k in the context of a free jet. The value of
k was found to strongly depend on the Reynolds number of the jet (
Rej) within the range of 100 to 2000. However, as
Rej exceeds 2000, the correlation between
k and
Rej weakens [
7]. Maruyama et al. [
15] demonstrated a significant relationship between
k and several factors, including nozzle clearance, liquid height, and elevation angle. The range of values for
k was observed to be between 0.48 and 1. This variation is attributed to the distortion of the conical shape of the jet, which leads to circulation of smaller variance of circulation time compared to that of a free jet. This suggests that the jet in a tank is confined to some extent, depending on the jet orientation.
Maruyama et al.[
11] presented a model that establishes a relationship between the mixing time and the mean circulation time in tanks. Equation 8 presents the dimensionless mean circulation time (
tc), which is constant for each jet nozzle configuration [
11]. Expressing
tR (mean resident time ) as the ratio of the liquid volume in the tank and the jet flow rates (
Q) at the nozzle and substituting
tc (Equation 6) Equation 7 can be written as:
By expressing
V in Equation 7, in terms of the tank geometry and the total flow entrained according to Equation 8,
tC becomes:
Plotting the mixing time,
tm as a function of
tC, and conducting a regression analysis on the resulting data, it was possible to incorporate Equation 6 into the regression equation and derive an expression for the mixing time. The resultant expression for the mixing time is presented as follows:
The previous study conducted by Grenville & Tilton[
13] had reported φ as 9.34 for jet nozzle vertical inclination, α > 15°, and jet nozzle horizontal orientation,
β = 0°, and 13.8 for α < 15°,
β = 0° for 0.4
H/
T ≤ 1. It is worth noting that irrespective of the specific value of
k selected, φ remains constant for a given jet configuration.
- ii.
Jet Turbulence Model
Corrsin’s [
26] works demonstrated that, in contrast to the circulation model, the mixing time for a passive scalar in a low-viscosity fluid is a function of the integral scale of concentration fluctuations (
L) and the turbulent kinetic energy dissipation rate (ε). The turbulence jet model was introduced by Grenville and Tilton[
14], who postulated that the effectiveness of jet mixing in storage vessels is determined by the rate of turbulent energy dissipation at the end of the free jet path length. Grenville and Tilton[
14] expressed the turbulent kinetic energy dissipation rates at the jet nozzle,
as Equation 11 and at the end of free jet pathlength,
as Equation 12, respectively.
From the conservation of momentum,
uz is related to the velocity at the jet nozzle
uj and the diameter of the nozzle
dj as:
where
and
are the jet velocity and diameter at
zmax, respectively.
Grenville and Tilton[
14] found that the mixing time is proportional to (z
2/
)
1/3. They conducted their experiments using tanks of different aspect ratios (
H/
T) between 0.2 and 3.0, for jet nozzle positions with vertical inclinations corresponding to the free jet pathlength spanning the diagonal of the fluid volume. By substituting the values of
uz and
dz from Equations 1 and 12, respectively, they were able to express
in terms of the jet properties. Using this expression, they derived a correlation for the mixing time.
Despite the abundance of literature on the subject of jet as a method for mixing Newtonian fluids, the available correlations are case specific [
3], highlighting the need for further investigation. Past studies [
7,
10,
11,
13,
14,
15,
16,
17] have produced conflicting results on the optimal jet angle for the shortest mixing time, with some suggesting an angle of 45°[
16] and others proposing a range of angles with local maximum mixing time occurring at α = 0° and the local minimum mixing time falls within the range of α = 25°–30° [
11]. Consequently, further experimental studies are necessary to clarify the effect of jet angle on mixing time. Previous research conducted by Grenville and Tilton [
13,
14,
17] has successfully established a correlation between mixing time and jet characteristics for jet nozzles with a vertical inclinations that correspond to angles where the jet intercepted the liquid surface at the opposite wall, denoted as α
UL, that span the free jet pathlength corresponding to the diagonal of the fluid volume. However, to the best of our knowledge, none of the prior experimental studies have investigated the impact of the horizontal orientation of jet nozzles on the mixing time performance, despite the fact that this installation position has shown potential for effective mixing in wastewater treatment[
18]. Therefore, it is essential to conduct further experimental studies to evaluate the effect of jet nozzle angle on the mixing time performance in such cases.
In this study, the duration necessary to achieve a state of 95% complete mixing is defined as the mixing time. The effectiveness of different nozzle positions will be evaluated by assessing the measured mixing time, and the results obtained will be discussed on the basis of the existing correlations and physical models.