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Define Duty, delta and Base period. Derive the relationship between them.
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Duty:

The term duty means the area of land that can be irrigated with unit volume of irrigation water. Duty represents the irrigating capacity of a unit. It is the relation the between the area of a crop irrigated and the quantity of irrigation water required during the entire period of the growth of that crop. For example, if 3 cumecs of water supply is required for a crop sown in an area of 5100 hectares, the duty of irrigation water will be 5100/3 = 1700 hectares/cumecs, and the discharge of 3 cumecs will be required throughout the base period.

Delta:

It is the total depth of the water required by a crop during the entire period the crop is in the field and is denoted by the symbol ∆. For example, if a crop requires about 12 watering at an interval of 10 days, and a water depth of 10 cm. If the area under the crop is A hectares, the total quantity will be 1.20 X A = 1.2 A hectare-metres in a period of 120 days.

Base Period:

Base Period for a crop refers to the whole period of cultivation from the time when irrigation water is first issued for preparation of the ground for planting the crop, to its last watering before harvesting.

Relation between Duty and Delta

D= duty in hectares/cumec

∆= total depth of water supplied in metres

B= base period in days

i. If we take a field of area D hectares, water supplied to the field corresponding to the water depth ∆ metres will be = ∆ x D hectares-metres = D x ∆ x10$^4$ cubic-metres. .... (1)

ii. Again for the same field of D hectares, one cumec of water is required to flow during the entire base period. Hence, water supplied to this field. = (1) x (B x 24 x 60 x 60) m$^3$ .... (2)

Equating Equations (1) and (2), we get D x ∆ x 10$^4$ = B x 24 x 60 x 60

∆ = $\frac{B \times 24 \times 60 \times 60}{D \times 10^4} = 8.64 \text{ B/D metres}.$

1 hectare = 104 sq metres

cumec-day = 8.64 hectare-metres