Conservation Equations Is a mathematical description of the movement and accumulation of an extensive property in a system. Conserved Property is one that is neither created nor destroyed. The conservation law: Property is neither created nor destroyed despite changes in the system or the surroundings.

Accounting and Conservation Equations An accounting Equation is a mathematical description of the movement, generation, consumption, and accumulation of an extensive property in a system. Mass, moles, Energy, charges, Momentum Value express in specific unit of measurement or expressed as proxy to real value (depth, ppm)

Mathematically Accounting Equation Input – Output + Generation – Consumption  Accumulation. (Irrigation –Drainage –Evapotranspiration  Accumulated water in field)

Mathematically Conservation Law Equation Input – Output  Accumulation…..a Input – Output  Depletion…..….b a: Input>Output; b: Input<Output

Conservation of mass Amount of water in lake/soil. Rainfall –Evaporation = Quantity (R-E=Q) Unit expression: mm/unit time/unit area. mm/hr; mm/day Depth =mm as proxy to Volume/Area.

Conservation of mass in hydrosphere (lake) R-E=Q ( mass balance equation) Calculate the amount of water in the lake after one month ( during the wet season) ; monthly rainfall is 350mm and monthly evaporation is 150 mm. Assume no runoff water flow into the lake and drainage gate is close, surface area of lake about 2ha. From the equation : 350 mm-150 mm= 200mm Answer: there is an increase of 20 cm of water level in lake. Absolute value of Q,= Depth x Area.

Q in Volume Area (ha) x10000(meter 2 /ha) x depth ( meter)= value in cubic meter Eg 2ha x10000m 2 /ha x 200mmx 1m/1000mm)  4000 cubic meter  liter  4 megaliter (1 cubic meter = 1000 liter)

System Accumulation The final and initial amounts in the system mathematically describe the accumulation term in both the accounting and conservation equation Final Condition – Initial Condition= Accumulation.

Simulation of accumulation. Bathroom accumulate 30 liter of water after you take shower. It accumulate at the rate 1.5lit/min. Your shower head spray at 5 liter/min. What is the rate of water flowing out of drainage outlet.? How long do you take your shower.

Simulation of accumulation Base on mass balance equation (LCM) Inflow-outflow = Acc 5lit/min-outflow= 1.5lit/min Outflow= 3.5 lit/min Drainage Rate = 3.5 lit/min.

Simulation of accumulation Diff between begin and final water= 30 liter To calculate time of shower. Inflow-Outflow=Acc (MBE, integral acc) 5 lit/min -3.5 lit/min = 30 lit Integrate at t=0, MBE  1.5lit/min =30lit, ; t final = 30lit/1.5lit/min  20 min

Simulation of accumulation Calculate time to drain the water. After the shower turn off. In flow=0 MBE  - Outflow rate= final volume At t=0, integrate with respect to time -3.5 lit/min at Final time= 30 lit Final time= 30lit/3.5lit/min=8.6 min

Accumulation Generation term describes the quantity of an extensive property that created by the system. Consumption term describes the quantity that is used or destroyed by the system Net Production = Generation + Consumption

Concept of Generation/Consumption of extensive property of System. 6 CO H20 + LIGHT  C6H12O6 + 6 O2 + 6 H20 Generation ( PRODUCTS) Consumption ( REACTANTS) TOTAL MASS REMAIN THE SAME

Accounting and Conservation Equations ALGEBRAIC DIFFERENTIAL INTEGRAL

ALGEBRAIC ACCOUNTION STATEMENTS Algebraic equations can be applied when discreate quantities or “chunks” of extensive property are involved. Ψ in – Ψ out + Ψ gen – Ψ cons = Ψ acc Ψf – Ψ 0 = Ψ acc Ψ

DIFFERENTIAL ACCOUNTION STATEMENTS The differential form of accounting statement is most appropriate when the extensive properties are specified as RATES eg gm/min, liter/sec. Ŷ in - Ŷ out +Ŷ gen -Ŷ cons = Ŷ acc = dŶ/dt

INTEGRAL ACCOUNTION STATEMENTS Integral balances are most useful when trying to evaluate conditions between two discrete time points

Accumulation between t o and t 1