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# Energy dissipation rate

Turbulent flow consist of eddies of various size range, and the size range increases with increasing Reynolds number. The kinetic energy cascades down from large to small eddies by interactional forces between the eddies. At very small scale, the energy of the eddies dissipates into heat due to viscous forces. Energy dissipation rate is the parameter to determine the amount of energy lost by the viscous forces in the turbulent flow. Different approaches are used to calculate the energy dissipation rate, depending on the type of restrictions the fluid passes through.

## Turbulent flow

Turbulent flow is a complex phenomenon, which may seem highly unpredictable. Turbulence, though, has several common characteristics such as nonlinearity, vorticity, diffusivity and energy dissipation. Based on these features of the flow, turbulence can be defined as a dissipative flow state characterized by nonlinear fluctuating three-dimensional vorticity[1].

The dissipation property of the turbulent flow describes conversion of kinetic energy into the heat due to large velocity gradients created by eddies of different scales. Kinetic energy experience cascading effect, where it transfers from large scale eddies to smaller scale eddies, until it convertes into heat by viscous dissipation[2]. Therefore, if no external energy is added to turbulent flow, with time the intesity of the flow will diminish and lose its turbulent characteristics.

The Bernoulli equation can be used to do derive practical equation for calculating the turbulent energy dissipation rate. The Bernoulli equation models a fluid moving from location (1) to location (2). For given flow, it can be written as[3]

,..............(1)

where

ρ = density of the fluid (kg/m3)

P1 = pressur at upstream location (N/m2)

P2 = pressur at downstream location (N/m2

u1 = superficial velocity of the fluid at the upstream location (m/s)

u2​ = superficial velocity of the fluid at the downstream location (m/s)

z1,z2 = elevation of fluid relative to reference points upstream and downstream locations, (m)

h = energy dissipation from point (1) to point (2) (J/kg)

g = gravity acceleration constant (m/s2

The equation shows that energy is conserved as the fluid flows from location (1) to location (2). Any conversion of mechanical energy into thermal energy is accounted for the energy dissipation term h.

Energy dissipation rate is the rate of energy loss due to fluid flow from location (1) to location (2). The rate of energy loss is given by

,..............(2)

where ε = energy dissipation rate per unit mass (m2/s3 or W/kg)

= time required for fluid to travel from (1) to (2) (seconds)

Vorticity property describes turbulence as a numerious set of structures appearing in the flow in shape of streaks, strain regions and swirls of various size. Most characteristics structures in a turbulent flow are called eddies[1]. The large scale eddies create anisotropic behavior of the turbulent flow. Due to the cascade effect, decreasing eddy sizes will become less dependent on the mean flow. At very small scale, the turbulence can be considered isotropic. Kolmogorov[4] suggested that the size of the small scale eddies, which contribute to the viscous dissipation, is only dependent on those parameters that are relevant for the smallest eddies. These parameters are the energy dissipation rate and the kinematic viscosity. Through the dimensional analysis and Reynolds Number, Kolmogorov showed that energy is dissipated by eddies of microscale at which inertial and viscous effects are balancing each other.

## Energy dissipation rate in the duct flow

For estimation of energy dissipation rate in the turbulent pipe flow, the well-known empirical relationship can be used[3]

,................(3)

where

D = pipe diameter (m)

f = Fanning friction factor

The turbulent energy dissipation will occur whether the fluid is single phase, a dispersion of oil droplets in water, or multiphase flow. In the case of oil droplets dispersed in water, not all of the turbulent energy dissipates into heat. The fluid friction will be experienced by eddies, which occurs over all sizes of eddies, but the greatest dissipation occurs at the small-scale eddies. These eddies break the droplets of the dispersed phase, which would commonly be described as shearing. At the same time, the process of coalescence is also influenced by energy in the turbulent flow. Droplets are transported by eddies equal to or larger than their size. The energy of these eddies contributes to the process of droplet collision and coalescence.

Coalescence and break-up of droplets determine the droplet size distribution in an oil-water mixture. Van der Zande[5] although points out that under certain conditions, e.g. at low oil concentration and high energy dissipation rate, coalescence can be neglected.

## Energy dissipation rate in the flow passing through a restriction

When fluid flows through a restriction, it experiences pressure drop. It is due to the energy dissipation that take place when large velocity gradients are present in the flow[5]

By applying the conservation laws in the integral format to a suitable control volume, Kundu[6] derived that in a duct flow the energy dissipation rate is

,.............(4)

where

E = energy dissipation rate (W)

ΔPperm = permanent pressure drop (N/m2)

Q = volumetric flow rate (m3/s)

Since most of the energy dissipation takes place where large velocity gradients are present, the description of turbulent flow is often simplified by using the mean energy dissipation rate per unit mass. Most of the dissipation occurs in the region immediately downstream the restriction that produces pressure drop. This region is often referred to as the dissipation zone[5]. The mass of the fluid in the dissipation zone is given by Consequently the mean energy dissipation rate per unit mass is equal to

,..................(5)

where

ρc = density of continuous phase (kg/m3)

Vdis = volume used for energy dissipation (m3

In cases where the flow rate is a given parameter, the energy dissipation rate per unit mass can be defined by the time period which most of the dissipation takes place

,...............(6)

where

tres  = mean residence time of the fluid in the dissipation zone (seconds)

## Nomenclature

 D = pipe diameter, E = energy dissipation rate f = Fanning friction factor g = gravity acceleration constant h = energy dissipation rate Q = volumetric flow rate t = travel time tres = residence time u = flow velocity Vdis = volume used for energy dissipation ΔPperm = permanent pressure drop z = elevation point above a reference plane ε = energy dissipation rate per unit mass ρ = fluid density ρc = density of continuous phase

## References

1. Kundu, P.K., Cohen, I.M., Dowling, D.R. 2012. Fluid Mechanics, fifth edition. Academic Press. Cite error: Invalid `<ref>` tag; name "r1.0" defined multiple times with different content
2. Richardson, L.F. 1922. Weather Prediction by Numerical Process. Cambridge: Cambridge University Press.
3. Walsh. J. 2016. The Effect of Shear on Produced Water Treatment. The Savvy Separator Series: Part 5. Oil and Gas Facilities.[1] Cite error: Invalid `<ref>` tag; name "r3.0" defined multiple times with different content
4. Kolmogorov, A.N. 1941. Dissipation of energy in locally isotropic turbulence. Compt. Rend. Acad. Sci. USSR 32 (1).
5. M. van der Zande. 2000. Droplet Break-p in Turbulent Oil-in-Water Flow Through a Restriction. PhD thesis, Delft University of Technology, Delft, the Netherlands (June 2000) Cite error: Invalid `<ref>` tag; name "r5.0" defined multiple times with different content Cite error: Invalid `<ref>` tag; name "r5.0" defined multiple times with different content
6. Kundu, P.K. 1990. Fluid Mechanics. Academic press.