DEMOCRITUS UNIVERSITY OF THRACE SCHOOL OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING

DEMOCRITUS UNIVERSITY OF THRACESCHOOL OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING XANTHI - GREECE Submerged Radial Intrusion of Wastewater Field from a Diffuser in a Motionless Stratified Sea Lecturer P. B. Angelidis Professor N. E. Kotsovinos

The flow from a sewage outfall in a stratified sea rises as high as its momentum and buoyancy will carry it, then it spreads horizontally at its neutral level, forming an intruding axisymmetric gravity current (or patch or slug). The hydraulic engineer possesses various models to calculate the dynamics of the effluent plume from its exit to its neutral level (dilution, mass flow rate, height of rise), e.g. CORMIX, VPLUMES, VISJET or semi empirical non dimensional equations.

However, the engineer can not use CORMIX, or other models to predict quantitatively the increase of the radius of the axisymmetric gravity current as a function of time. The basic objective of this paper is to present dimensionless relationships for the growth of the radius of a patch of the diluted effluent resulting from a submerged intrusion.

Configurationof the problem • The ambient fluid is stagnant and is stratified. • The buoyant plume rises as high as its momentum and buoyancy will carry it, and then it spreads horizontally at its neutral level. • We assume that the spreading gravity current is axisymmetric. This is true assuming motionless environment and outflow of sewage from only one circular pipe. However, considering the large horizontal scales of spreading, we may use the results of this analysis for intrusions originated from outflows from diffusers. • The buoyant plume volume flux at its exit is Qo, which increases to Q at the level of its neutral stability.

It is reasonable to distinguish two regions: • i) the impingement region within the control volume ABCD where the flow is in general turbulent and is characterized by significant entrainment. • ii) the main spreading region which is outside the control volume ABCD.

On the average the entrainment in the region ABCD is proportional to the input flow rate Q so that in general the radial (horizontal) volume flux out of the box ABCD is cQ where c is a constant larger than one which depends on the flow and stratification parameters. • The constant c clearly tends to one, when the buoyant plume impinges with very small vertical momentum on its neutral density level and increases with increasing the vertical impinging momentum.

Due to interfacial mixing it is expected that the volume of the slug increases, and it may be argued that approximately the volume of the slug at time t is given by the following equation: Volume of slug outside the control volume ABCDcQt1+α cQ(1+α)t where α a global entrainment coefficient for the submerged spreading, much smaller than one i.e. α<<1. If it is assumed that the typical vertical and horizontal extent of the intruding fluid are H and R respectively, then the equation gives: HR2 Qt1+α (1+ α)Qt

By integrating the vertical component of the momentum equation over the spreading patch, neglecting small terms, and assuming that the density of the fluid in the spreading slug is constant and equal to ρs, we obtain (Kotsovinos, 2000) the following relationship between the ambient and slug densities:

Since the intrusion layer is neutrally buoyant, at the height where it spreads horizontally, it is, strictly speaking, the "squeezing” pressure pu , pl exerted on the upper and lower surfaces of the slug which oblige the slug to spread. For linear ambient stratification and constant density ρs within the slug, we find that the pressure inside the slug is greater than the pressure outside.

A balance of the forces that drive and retard the entrapped wastewater flow intrusion indicates the existence of two basic regimes: • a regime at “small times” where the pressure (buoyancy) force is balanced by the inertia force • a regime for “large times” where the pressure force is balanced by the interfacial drag (“viscous force”). • For linear ambient stratification and constant density ρs within the slug, we find that the pressure inside the slug is greater than the pressure outside, and therefore the excess horizontal pressure (usually called "buoyancy “) force Fp, which drives the spreading (intrusion) is given by • with • Using HR2 Qt1+α (1+ α)Qt, we have

The force which drive the flow is the pressure (or buoyancy) force Fp. • The forces, which retard (or resist) the flow are two: • the inertia of the intruding gravity current Fi • the drag Fsh which is exerted by the ambient fluid on the intruding fluid. • The scaling of these two forces is: • Fi = rate of change of the inertia of the fluid within the slug • = O(ρs R3 H t-2)=O(ρs R Q t-1 ) • Fsh = interfacial shear force = O(μ U H-1 R2) = O(μR3 H-1t-1) • = O(μR5Q-1 t-2 ) • For most of the practical problems of the dispersion of outfall effluents, the initial radial momentum can be neglected and therefore the growth rate follows two regimes: the inertia - buoyancy and viscous - buoyancy regime, (Kotsovinos, 2000).

In the inertia – buoyancy regime the driving pressure force is much larger than the radial momentum flux, and the resisting inertia force is much larger than the drag force. We have therefore a balance of the driving pressure (buoyancy) force Fp and the resisting inertial force Fi: Fp = Fi==> R3 = C3 (ρ'gQ/ρ2 ) 1/4 t3/4 where C3 is an experimental constant.. The above equation is valid for times t<Ttr, where Ttr is the time at which the spreading changes from the inertia - buoyancy regime to viscous – buoyancy regime. The experimental constant C3 according Kotsovinos (2000) is equal to 0.9.

The dimensionless radius R(t) of spreading in the inertia – buoyancy regime as a function of the dimensionless time t/Tc. For comparison the results from two field experiments of the Lemckert and Imberger (1993) are plotted (continuous line). From Kotsovinos (2000).

At large times, i.e. at times t>Ttr, the spreading is governed by a balance of the buoyancy driving force Fp and the resisting interfacial (viscous or eddy viscous) shear force Fsh, i.e. it is obtained the buoyancy – viscous regime. Fp= Fsh ==> for t>Ttr: where C4 is an experimental constant, which strictly speaking is in general a function of the gross Richardson number and of the gross Reynolds number Res of the intrusion. From the next figure, we observe that for Res >400 (which is true for most practical problems), we may adopt for the experimental constant C4 = 0.5.

The dimensionless radius R(t) of spreading in the viscous – buoyancy regime as a function of the time in seconds. In the same Figure we plot the experimental results of Zatsepin and Shapiro (1982). From Kotsovinos (2000).

We define the Brunt - Vaissala frequency N by the relation The mean height H of the slug is given by the relation (Kotsovinos, 2000):

Using the Brunt - Vaissala frequency N we find that the density ρ΄ that is needed to calculate the spreading radius R(t), can be estimated from the following relation: Combining the above equation we can derive the following explicite equations for the radius of spreading: for t<Ttr (inertia - buoyancy regime) for t>Ttr (viscous - buoyancy regime)

A basic question is at which radius and at which time the submerged spreading changes from the inertia - buoyancy regime to viscous - buoyancy regime. On the basis of dimensional arguments and the experimental results of Kotsovinos (2000) we deduce that for most of the practical cases the transition radius Rtr from the inertia buoyancy regime to viscous – buoyancy regime is given by the following relation: The transition time Ttr from the inertia - buoyancy regime to viscous – buoyancy regime is given by the relation:

In a practical application we know the effluent discharge from the diffuser Q0, the vertical distance h of the diffuser from the pycnocline, and the density profile. The above mentioned equations for the spreading radious R(t),the transition radius Rtr,and transition time Ttr need the spreading discharge Q and not the initial effluent discharge from the diffuser Q0. We may calculate the discharge Q at the basis of pycnocline using the CORMIX expert system, or an other model (VPLUMES, VISJET).

For simplicity, for the case of sewage effluent out of diffuser in the sea, we may assume that the flow is a two dimensional buoyant jet, so that we may use the explicit equation proposed by Kotsovinos (1978): where

The typical equivalent width of many diffusers varies between w=0.0015 to 0.003 m (for example, w=0.002m for the Hyperion Outfall - Los Angeles, w=0.0015m for the San Diego – California, w=0.0014m for Honolulu). Therefore, for a typical value of h=30m, the dimensional distance h/w varies between 10000 and 20000 It is observed that the spreading discharge is 100 to 800 times larger than the diffuser discharge.

We plot the transition radius Rtr as a function of the discharge Q, and various gradients of density profile (or various values of the Brunt - Vaissala frequency).

We plot transition time Ttr as a function of the discharge Q, and various gradients of density profile (or various values of the Brunt - Vaissala frequency).

We plot here the radius R of the intruding effluent slug for both regimes as a function of the time in logarithmic scale, for Q=100 m3/s at the neutral density level time. The Brunt-Vaissala frequency is in this diagram N=0.04.It is interesting to observe the transition between the two regimes take place at Ttr=10 hours and which time the radius is about 1000m.

We plot the radius of the intruding effluent slug for both regimes as a function of the time and for various values of the discharge at the neutral density level time, and for a certain Brunt-Vaissala frequency N.

Conclusions The scaling analysis indicates that the intrusive gravity currents are characterized in general by the appearance of two regimes, depending on the relative magnitude of the forces that drive or retard the intrusion. The driving pressure (buoyancy) force increases with time, and therefore the predominant driving force is the pressure force. The predominant retarding force is the drag force. The theoretical analysis and the experimental results of this paper indicate that, for the inertia – buoyancy regime R  t3/4. The characteristic length and time scale, which determine the transition from the inertia – buoyancy regime to viscous – buoyancy regime, were correlated with experimental results and the appropriate scaling coefficients were determined. Analytical equations and diagrams were presented, which predict with reasonable accuracy the spreading rates and facilitate the quantitative predictions of the spreading patch.

DEMOCRITUS UNIVERSITY OF THRACE SCHOOL OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING