# Lake inflow-outflow problem

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 $$= (571.10 - 571.04)\times100 \times 10^6 E = 0.68\ \mathrm{m}$$

• Hi Kavya. It is not clear what you are asking. Can you please expand your question with more context and details so it is clear what you are asking? – milancurcic Sep 8 '14 at 15:35
• It is a problem. Can I post it? – Kavya Sep 8 '14 at 15:36
• I have edited my question. – Kavya Sep 8 '14 at 15:42
• OK, I made a few formatting edits. Since this is a homework problem, you should further edit your question to show what you have tried already and where you got stuck, and somebody can then give you hints on how to proceed. – milancurcic Sep 8 '14 at 15:53
• OK. Even if it was not a problem assigned by an instructor, it is good to keep the homework tag to let the users know they should not post full solutions to the problem, but rather the suggestions on how to proceed. – milancurcic Sep 8 '14 at 17:30

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.