Week 1 Practical: Understanding Residence Time using Salinity

Introduction
Residence time, denoted by the Greek letter τ (tau), is an extremely useful concept for understanding
and calculating how systems and their components respond to disturbances away from steady-state
conditions.

Residence time is the average time that a substance (or energy) spends within a reservoir that is at
steady state; that is the amount of substance or energy in the reservoir is not changing, a condition that
occurs when the input flux of substance (or energy) into the reservoir equals the output flux. Residence
time is calculated using the formula

𝜏 = 𝐴/𝐹

where A is the amount of substance in a reservoir and F is the input or output flux of the substance
(energy) into and out of the reservoir.

Consider a simple water tank that has an input flow of water and an outlet flow. If the water level in
the tank is constant (unchanging) then the input and outlet flows of water must be equal, and the tank
is at steady state.

If a tank contains a constant volume of 1000 litres and the input flow (Fin) is 10 litres/min we can
calculate the residence time to be 100 minutes as follows:

𝐴 1000 𝑙𝑖𝑡𝑟𝑒𝑠
𝑇 = 𝐹𝑖𝑛 = 10 𝑙𝑖𝑡𝑟𝑒𝑠/𝑚𝑖𝑛 = 100 𝑚𝑖𝑛

Note how the units of amount, in this case, litres of water, are cancelled out by the units of flux litres
per min to give units of time (minutes).

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Knowledge of residence time allows very useful predictions to be made, for example:

• How long carbon is likely to be stored in soils and plants

• How long carbon is likely to be stored in the deep ocean before returning to the surface ocean
where it can exchange with the atmosphere.
• How long different negative emission technology options would take to draw down CO2 from
the atmosphere.

Conductivity
Electrical conductivity (EC) is commonly used to measure the amount of salt (salinity) that is dissolved
in water. The saltier the water the greater its electrical conductivity. In this practical you will be
measuring electrical conductivity in standard units of micro Siemens per centimeter (µS/cm). The digital
meters can nominally measure EC up to 9999 µS/cm but are more stable at values <6000 µS/cm.

Experiment

Aims

• To measure residence time, τ, through a series of experiments that introduce a flux of fresh
water (inflow) into a reservoir (tank) that holds a constant volume of salty, dyed water.
• To set up a spreadsheet of measured data, to graph the data in a variety of ways, and then to
fit equations to the measured data to estimate residence time (τ).
• To compare the measured τ with the theoretical τ for the experiment.
• To set up and operate experimental apparatus, including pumps.

Equipment
Each group will have:

• a Plexiglas tank with an over-flow outlet and hose

• a small low voltage aquarium pump with an output hose
• a digital conductivity (salinity) meter
• food dye
• a spoon/plastic ruler (for stirring)
• a measuring cylinder
• four plastic buckets
• a balance
• a jug

In an emergency (e.g. tank overflows) or in the event of equipment not working (e.g. salinity meter),
turn off the pump and pause the timer, If you are unable to rectify the problem please seek help from
a demonstrator or lecturer to solve the problem.

Experimental Procedure
1. Position the tank with overflow hose draining into a bucket on the floor.
2. Fill the tank with water (until it just overflows). Determine and record the volume of water in
the tank.

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 3. Add 20 g of salt to the tank and dissolve it by stirring well. Measure the conductivity and record
the value. If you get an error message (Err) on the salinity meter, you will need to pump fresh
water in to reduce the salinity to a value that can be measured.
4. Add 10 drops of food dye to the tank and stir.
5. Before adding fresh water into the tank at a constant flow rate (by pumping from the bucket),
measure and calculate the pumps water flow rate (in mL/s) using a measuring cylinder and a
timer. Depending on the pump, you will be able to start and stop the pump by inverting the
float switch or by using the power switch. Adjust the flow rate to around 50 mL/s (or 5×10-5
m3/s) or as high as possible. Note: Litre (symbol, L) is a non-SI metric system unit of volume but
is commonly used: 1 L = 10-3 m3 and 1 mL = 1 cm3 = 10-6 m3.
6. Prepare a timer. You will need to record the EC of the water in the tank at regular time intervals
(at least once per minute) after you start adding the fresh water. Note that the EC readings may
fluctuate at high values; you may only be able to obtain values to a precision of ±10 µS/cm).
Record the beginning of fresh water inflow as time (t) = 0, and the initial conductivity of the
salty water, C0, as the conductivity reading for time = 0.
7. Start pumping fresh water into the tank at a constant flow rate. Stir the box gently throughout
the experiment. Assign a team member to keep the bucket topped up with fresh water, and
ensure the overflow bucket does not spill.
8. Note what happens to the dye.
9. After the conductivity of the water in the tank has reached an almost constant value (or the
conductivity has reduced by a factor of 10 to the initial conductivity) stop the flow, and record
this constant value as final conductivity C1 (this should be similar to the conductivity of the input
fresh water). Enter the data into a spreadsheet and analyse the results as outlined below.

Data Analysis
1. Record your data for each experimental run. A suitable spreadsheet template can be found on
Wattle) which is setup to record:
• Experiment number #
• The calculated volume V of water held by the tank
• The fresh water flow rate, measured before starting the inflow to the tank (F).
• Initial conductivity of the water in the tank, C0
• A column of measurement times t
• A column of corresponding conductivity readings C(t) (i.e., C as a function of time)
• Final conductivity of the water in the tank, C1
2. Calculate the theoretical residence time τ ≈ V/F (where V is your measured volume of water in
the box and F is your measured input flow rate of the pump). 10497.6/33.3=315.343
3. Construct a graph of conductivity C (µS/cm) versus elapsed time t (in seconds) using linear axis
scales. You should observe that conductivity decays exponentially with time after the fresh
water inflow begins.
4. In a separate column, calculate values for the conductivity difference C(t)-C1 by subtracting the
final conductivity C1 from your conductivity readings at each time, C(t).

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 5. Re-plot the conductivity difference C(t)-C1 vs. elapsed time t again on a separate graph, this
time using a logarithmic scale on the vertical (conductivity) axis. This will allow ready
comparison with the theoretically predicted exponential change:

C(t ) − C1 = (C0 − C1 )e(−t /  )

6. Fit an exponential curve to the data (this will be a straight line on the log-linear plot) to estimate
a more accurate value for the residence time, τ. In the spreadsheet, right-click on your data
points, select ‘Add Trendline’, then select ‘Exponential’ from the Trendline options and ‘Display
equation on chart’ (and Display R2 value if you like) from the checkboxes at the bottom of the
dialogue box.
7. Extract the value of the residence time, τ, from the equation fit you obtain in (6) above.

An example of how you might set up your spreadsheet for this practical is available on Wattle and at
the end of the practical.

Lab book discussion question 1

Does your exponential curve fit to the data (6 above) provide a more reliable result than the theoretical
value for τ (Step 2 in Data Analysis)? Explain your answer?

i would say that the data that was taken is less reliable from the theoretical mesument as there is
many outlires off the data causes inacuracy and a unsteady decline in the data. this was due to
the conductivity device showing inacurate mesure where it held on 2105 for 3 mintes before
changing. it then increased and decrease many times causing a fether inaccurate decline.

Online questions – to be answered in wattle

Question 2

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The first experiment you performed is an analogue for Earth’s ocean: the Plexiglass tank represents a
seawater-filled ocean into which rivers provide the main input (fluxes, F) of water from land and
dissolved salts. The residence time of water in the ocean (in years) with respect to river inputs can be
calculated from the volume of water in the ocean (Vocean = 1.29 x 1018 m3) divided by the the annual
volume flux of rivers to the ocean (Friver = 3.7 x 1013 m3/yr):
Vocean
 (years) =
Friver
Similarly, the residence times of different salt components in the ocean can be calculated from the total
amount of a particular element in the ocean divided by the flux of that element carried to the ocean
via rivers each year.

The amount of an element in the ocean (moles) = Vocean  Cocean

where Vocean is the volume of the ocean (m3) and Cocean is the concentration of the element in the ocean
(moles/m3).

The annual flux of an element via rivers to the ocean (moles/yr) = Friver  Criver
where Friver is the river volume flux (m3/yr) and Criver is the element concentration in river water
(moles/m3).

The residence time (τ in years) for any particular element is:

amount in ocean (moles)

 (years) =
flux to ocean (moles/yr)
(V  C )
 (years) = ocean ocean
(Friver  C river )

Using this equation, calculate the residence times for sodium (Na), chlorine (Cl), calcium (Ca), silicon
(Si) and aluminium (Al) in the ocean given the values in Table 1 below.

Table 1: Concentration of elements in river and ocean.

Element Criver Cocean Residence time, τ
(moles/m3) (moles/m3) (years)
Na 0.3 480
Cl 0.22 560
Ca 0.35 10.5
Si 0.193 0.084
Al 0.0019 0.00001

Question 3

The residence time of water in the deep ocean is about 1500 years (that is, the volume amount of water
in the deep ocean divided by the volumetric flow rate of the ocean’s thermohaline circulation. If you

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think of this time scale as being characteristic of the time needed to stir and mix the ocean, which of
the elements of those in the above table would you expect to be:

a) well mixed in the ocean (stirred around many times) and therefore have the most uniform
composition)

b) least well mixed in the ocean (most variable composition)

Question 4
In this experiment, our ‘ocean’ (the tank) ended up with a similar salinity to the ‘rivers’ (inflow). What
must be occurring to balance the inflows from the Earths’ rivers from causing the ocean to become
significantly less saline (fresher)

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