Lake inflow-outflow problem

Question

The evaporation from a lake is to be calculated by the water balance method. Inflow to the lake occurs through three small rivers, A, B and C. The outflow occurs through the river D. Calculate the evaporation from the lake surface during summer (May – August) if the water level was +571.04 m on May 1 and +571.10 m on August 31. The lake surface area is 100 km$^2$. The precipitation P during the period was 100 mm. Average inflows and outflow are given below:

              River      Catchment(km2) Average Q(m3/s)
                A             150        15
                B             120        20
                C             130        17
                D                        45

This is the way I have attempted it:

• Total Inflow ( by A, B, C) $= 52\ \mathrm{m^3/s}$.

• Outflow (by D, assuming that D is fed by the lake) $= 45\ \mathrm{m^3/s}$.

• Inflow - Outflow = Change in storage (which is in turn change in elevation × area)

• Net Inflow into the lake = Inflow by 3 rivers + Precipitation $$ = (52\ \mathrm{m^{3}/s} \times 122 \times (30\ \mathrm{days}+31+30+31)\times24\times60\times60 + 0.1\ \mathrm{m}\times100\times10^3 \mathrm{m}^2) $$

• Total Outflow = Outflow + Evaporation $$ = (45\times122\times24\times60\times60 + E), $$ where E is the component to be found.

• Change in storage = Change in elevation × Area $$ = (571.10 - 571.04)\times100 \times 10^6 E = 0.68\ \mathrm{m} $$

Answer 1

Everything you did looks reasonable. There is one error in the 'net inflow into lake' line where you take the area of the lake to be $100\cdot10^3\ \text{m}^2$ instead of $100\cdot10^6\ \text{m}^2$, but that might just be a typo.

In general, this type of problem is much easier to work (especially when you think you've made an error) by keeping everything symbolic. For your problem I would define the variables

• $i$: inflow into the river in $\frac{\text{m}^3}{\text{s}}$
• $o$: outflow from the river in $\frac{\text{m}^3}{\text{s}}$
• $r$: inflow from rain in $\text{m}$
• $e$: evaporation in $\text{m}^3$
• $\Delta h$: change in height of the lake in $\text{m}$
• $A$: area of the lake in $\text{m}^2$
• $\Delta t$: time period in $\text{s}$

The equilibrium equation is then

$$\text{sum of inflows}-\text{sum of outflows}=\text{change in storage}$$

which is written in terms of the defined variables as

$$i\ \Delta t+r\ A-o\ \Delta t-e=\Delta h\ A.$$

You can check that this equation at least makes sense in terms of units by observing that all of the terms have units of volume, i.e. $\text{m}^3$. Finally, you can rearrange the equation to solve for the variable of interest;

$$e=(i-o)\Delta t+(r-\Delta h)A.$$

At this point you can plug in the numbers.