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Minimum Energy Dissipation Rate Theory and Its Applications for Water Resources Engineering

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Advances in Water Resources Engineering

Part of the book series: Handbook of Environmental Engineering ((HEE,volume 14))

Abstract

Minimum energy dissipation rate principle can be derived from minimum entropy production principle. Minimum entropy production principle is equivalent to the minimum energy dissipation rate principle. The concept of minimum energy dissipation rate principle is that, when an open system is at a steady nonequilibrium state, the energy dissipation rate is at its minimum value. The minimum value depends on the constraints applied to the system. If the system deviates from the steady nonequilibrium state, it will adjust itself to reach a steady nonequilibrium state. The energy dissipation rate will reach a minimum value again. In order to verify the fluid motion following minimum energy dissipation rate principle, re-normalisation group (RNG) k -ε turbulence model and general moving object (GMO) model of Flow-3D were applied to simulate fluid motion in a straight rectangular flume. The results show that fluid motion satisfies the minimum energy dissipation rate principle. Variations of energy dissipation rate of alluvial rivers have been verified with field data. When a river system is at a relative equilibrium state, the value of its energy dissipation rate is at minimum. The minimum value depends on the constraints applied to the river system. However, due to the dynamic nature of a river, the minimum value may vary around its average value. When a river system evolves from a relative state of equilibrium to another state, the process is very complicated. The energy dissipation rate does not necessarily decrease monotonically with respect to time. When a system is at a new relative state of equilibrium, the energy dissipation rate must be at a minimum value compatible with the constraints applied to the system. Hydraulic geometry relationships can be derived from the minimum energy dissipation rate principle. Combining the minimum energy dissipation rate principle with optimization technology as the objective function under the given constraints, the optimum design mathematical models can be developed for a diversion headwork bend structure and stable channel design.

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Abbreviations

A :

Cross-sectional area of flow (m2)

\({{A}_{k}}\) :

Chemical affinity (J/mol)

B :

River width (m)

b :

Channel bottom width (m)

C :

Chezy coefficient (m1/2/s)

\({{C}_{\text{H}}}\) :

Nonsilting sediment concentration (kg/m3)

\({{C}_{\max }}\) :

Maximum permissible sediment concentration (kg/m3)

\({{C}_{\text{r}}}\) :

Criterion of circulation intensity (dimensionless)

\({{C}_{\text{s}}}\) :

Sediment concentration (kg/m3)

\({{C}_{\text{V}}}\) :

Sediment concentration by volume (dimensionless)

\({{C}_{*}}\) :

Sediment transport capacity (kg/m3)

\({{d}_{50}}\) :

Sediment median diameter (m or mm)

\({{\text{d}}_{\text{e}}}E/\text{d}t\) :

The entropy flux (W/K)

\({{\text{d}}_{\text{i}}}E/\text{d}t\) :

The entropy production (W/K)

\(E\) :

Entropy (J/K)

\({{E}_{\text{V}}}\) :

Local entropy, also known as unit volume entropy or entropy density (J/(K·m3))

e :

Internal energy (J/kg)

\(F\) :

Mass force acting on a unit of fluid mass (N/kg)

\({{G}_{\text{b}}}\) :

Cross-sectional rate of bed-load transport (kg/s)

g :

Acceleration of gravity (m/s2)

h :

Average water depth (m)

\({{J}_{i}}\) :

Generalized flows (no unique units)

\({{L}_{kl}}\) :

Phenomenological coefficients (dimensionless)

m :

Mass or bankside slope (kg or dimensionless)

n :

Roughness (s/m1/3)

\(n\) :

The outward unit vector (dimensionless)

\(P\) :

Second-order stress tensor (Pa)

P :

Entropy production (W/K)

p :

Pressure (Pa)

Q :

Water discharge (kg/m3)

\({{q}_{\lambda }}\) :

Thermal transport vector (W/m2)

\({{q}_{\text{R}}}\) :

Thermal radiation per unit mass (W/kg)

R :

Hydraulic radius (m)

S :

Slope (dimensionless)

T :

Absolute temperature (K)

t :

Time (s)

U :

Velocity (m/s)

\({{U}_{\text{c}}}\) :

Incipient velocity (m/s)

\(u\) :

Velocity vector (m/s)

\({{u}_{i}}\) :

Component of velocity (m/s)

\(V\) :

Volume (m3)

\({{X}_{i}}\) :

Generalized forces (no unique units)

\({{z}_{\text{b}}}\) :

Elevation at the bottom of cross section (m)

\(\Gamma \) :

Permissible ratio (dimensionless)

\(\gamma \) :

Specific weight of water (N/m3)

\(\delta \) :

Second-order unit tensor (dimensionless)

\(\nu \) :

Molecular viscosity (m2/s)

\({{\nu }_{t}}\) :

Turbulent viscosity (m2/s)

\(\Pi \) :

Tangential stress tensor (Pa)

\(\rho \) :

Concentration or density (kg/m3)

\({{{\rho }'}_{\text{s}}}\) :

Dry density of sediment (kg/m3)

\(\sigma \) :

Local entropy production (W/(K·m3))

\(\Phi \) :

Energy dissipation rate per unit length (W/m)

\({{\Phi }_{V}}\) :

Energy dissipation rate per unit fluid volume (W/m3)

\(\varphi \) :

Energy dissipation function per unit volume of energy in unit time (W/m3)

\(\chi \) :

Wetted perimeter (m)

\(\omega \) :

Sediment particle fall velocity (m/s or mm/s)

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Author information

Authors and Affiliations

  1. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, 92 Weijin Road, 300072, Tianjin, China

    Guobin Xu PhD & Lina Zhao

  2. Department of Civil and Environmental Engineering, Colorado State University, 80523, Fort Collins, CO, USA

    Chih Ted Yang PhD, PE, D.WRE

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  1. Guobin Xu PhD
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  2. Chih Ted Yang PhD, PE, D.WRE
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  3. Lina Zhao
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Correspondence to Guobin Xu PhD .

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Editors and Affiliations

  1. Colorado State University, Fort Collins, Colorado, USA

    Chih Ted Yang

  2. Rutgers University, New Brunswick, New Jersey, USA

    Lawrence K. Wang

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Xu, G., Yang, C., Zhao, L. (2015). Minimum Energy Dissipation Rate Theory and Its Applications for Water Resources Engineering. In: Yang, C., Wang, L. (eds) Advances in Water Resources Engineering. Handbook of Environmental Engineering, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-11023-3_5

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  • DOI: https://doi.org/10.1007/978-3-319-11023-3_5

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