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# Energy balance

Energy balance has the following meanings in several fields:

• In physics, energy balance is a systematic presentation of energy flows and transformations in a system. Theoretical basis for an energy balance is the first law of thermodynamics according to which energy cannot be created or destroyed, only modified in form. Energy sources or wave of energy are therefore inputs and outputs of the system under observation.
• In biology, total body energy balance is measured with the following equation: Energy intake = internal heat produced + external work + energy storage. The Dynamic Energy Budget theory makes explicit use of energy, mass and time balances. One Calorie (or kilogram calorie) equals the energy needed to increase the temperature of 1 kg of water by 1 °C. This is about 4.184 kJ.
• In energy economics, the energy balance of a country is an aggregate presentation of all human activities related to energy, except for natural and biological processes. National energy balances are compiled on at least an annual basis. Common methodology for compilation and presentation of energy balances allows simple addition of national energy balances to form supranational ones, such as is compiled for the European Union. United Nations compile energy balances for all member countries. International Energy Agency, a specialised agency of OECD is regularly preparing world energy balances.
• In engineering, energy balances are used to quantify the energy used or produced by a system. This can be used to build complex differential equation models to design and analyze real systems. To make an energy balance for a system is very similar to making a Mass balance but there are a few differences to remember, e.g. 1) that a specific system might be closed in a mass balance sense, but open as far as the energy balance is concerned and 2) that while it is possible to have more than one mass balance for a system there can be only one energy balance. If a balance is made for total energy, the energy balance becomes IN=OUT+ACC (where ACC stands for accumulation). Notice that there is no production (PROD) term since energy can not be produced, only converted. If instead some kinds of energy are ignored, e.g. if a heat balance is made the energy balance becomes IN+PROD=OUT+ACC (if heat is consumed the PROD term is negative, compare Mass balance.
• When comparing fuel production, energy balance is the difference between the energy produced by 1 kg of the fuel (i.e. biodiesel, petroleum, uranium ) and the energy necessary to produce it (extraction (e.g. drilling or cultivation of energetic plants), transportation, refining etc). Other factors affect fuel selection, such as portability. See also net energy gain and EROEI.
• In geography, specifically climatology and hydrology, the "'energy balance'" refers to the total of all energy inputs and outputs at any location; these include solar, atmospheric transfer, and ground conducted energy.

## Energy balance of groundwater flow

### Theory

When multiplying the horizontal velocity of groundwater (dimension, for example, m3/day per m2 cross-sectional area) with the groundwater potential (dimension energy per m3 water, or E/m3) one obtains an energy flow (flux) in E/day per m2 cross-sectional area.

Summation or integration of the energy flux in a vertical cross-section of unit width (say 1 m) from the lower flow boundary (the impermeable layer or base) up to the water table in an unconfined aquifer gives us the energy flow through the cross-section gives the energy flow through the cross-section in E/day per m width of the aquifer.

While flowing, the groundwater loses energy due to friction of flow. At the same time, energy may be added with the recharge of water coming into the aquifer through the water table. Thus one can make an hydraulic energy balance of a block of soil between two nearby cross-sections. The energy flow in the first section plus the energy added by recharge minus the energy flow in the second section must equal the energy loss due to friction of flow.

In mathematical terms this balance can be obtained by differentiating the cross-sectional integral of E in the direction of flow, taking into account that the level of the water table may change using the Leibnitz rule . The mathematics is simplified by assuming that the horizontal of the flow is constant within the section. This assumption is similar to the Dupuit assumption (the flow is horizontal), but it is more realistic because it only refers to the horizontal component of the flow and it does acknowledge the presence of non-horizontal flow.

The hydraulic friction losses can be described in analogy to the law of Joule in electricity, where the friction losses are proportional to the square value of the current (flow) and the electrical resistance of the material through which the current occurs. In groundwater hydraulics one often works with the hydraulic conductivity (permeability of the soil for water), which is inversely proportional to the resistance.

The resulting equation of the energy balance of groundwater flow can be used, for example, to calculate the shape of the water table under specific aquifer conditions. For this we can use a numerical solution, taking small steps along the impermeable base. The equation is to be solved by trial and error (iterations), because the hydraulic potential is taken with respect to a reference level for which we use the level of the water table at the water divide midway between the drains. As we are calculating the shape of the water table, its level at the water divide is initially not known. Therefore we have to assume this level beforehand, start the calculations, adjust the initial assumption according to the findings of the calculation procedure, and restart the calculations from the beginning until the level of the water table at the divide does not differ significantly from the assumed level.

To trial and error procedure is not very inviting to do the calculations by hand. Therefore, a computer program was developed to remove the burden.

A detailed article on "The energy balance of groundwater flow" can be viewed and downloaded freely from www.waterlog.info.

The same holds for an article on "The energy balance of groundwater flow applied to subsurface drainage in anisotropic soils by pipes or ditches with entrance resistance."

## Computer program, drain spacing equation

The computer program (EnDrain) can be freely downloaded from the same website.

The computer program compares the outcome of the traditional drain spacing equation based on the Darcy's law with the solution obtained by the energy balance and it can be seen that drain spacings are wider in the latter case than in the first. This is owing to the introduction of the energy supplied by the incoming recharge.

The owner of the referenced website invites the reader to apply the energy balance to the seepage surface along a sloping outlet boundary of the aquifer, or at the foot of a river embankment, or at the end of a water conduit from which the water falls down.

## References

1. ^ R.J.Oosterbaan, J.Boonstra and K.V.G.K.Rao, 1996, The energy balance of groundwater flow. In: V.P.Singh and B.Kumar (eds.), Subsurface-Water Hydrology, p. 153-160. Kluwer academic publishers, The Netherlands.