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Conservation Equations

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.

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Conservation Equations

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  1. 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.

  2. 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)

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

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

  5. 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.

  6. 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.

  7. Q in Volume • Area (ha) x10000(meter2/ha) x depth ( meter)= value in cubic meter • Eg 2ha x10000m2/ha x 200mmx 1m/1000mm) 4000 cubic meter • 4000 000 liter • 4 megaliter • (1 cubic meter = 1000 liter)

  8. 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.

  9. 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.

  10. 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.

  11. 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

  12. 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

  13. 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

  14. Concept of Generation/Consumption of extensive property of System. 6 CO2 + 12 H20 + LIGHT  C6H12O6 + 6 O2 + 6 H20 Consumption ( REACTANTS) Generation ( PRODUCTS) TOTAL MASS REMAIN THE SAME

  15. Accounting and Conservation Equations • ALGEBRAIC • DIFFERENTIAL • INTEGRAL

  16. ALGEBRAIC ACCOUNTION STATEMENTS • Algebraic equations can be applied when discreate quantities or “chunks” of extensive property are involved. • Ψin – Ψout + Ψgen – Ψcons= Ψacc • Ψf – Ψ0 = Ψacc Ψ

  17. 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. Ŷin- Ŷout +Ŷgen -Ŷcons = Ŷacc = dŶ/dt

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

  19. Accumulation between to and t1

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