Chemical Engineering 140

It is useful to utilize an exponential dilution flask upstream of the photometer to examine the dynamic response of an inline or flow-through spectrophotometer. The photometer receives the waste from a tiny continually stirred tank, or flask. The tank is initially filled with a weak dye at concentration C. Then, while keeping the same inlet and output flow rates, clear solvent—i.e., one devoid of dye—is added to the flask. If the transmitted absorbance measured by the photometer (i.e. minus the logarithm of the intensity) is linear with dye concentration, find an expression for the spectrophotometer absorbance, A, in terms of a steady flow rate Q, and the steady volume of liquid in the flask, V. Explain why the device is called exponential dilution flask and why it is useful in testing the dynamic behavior of the spectrophotometer

Solution

Given that this is a decaying potential concept, we have Absorbance as follows:

















The expression:





The device is known as exponential dilution because when the dye’s concentration is plotted against time, an exponential graph is found. In which case, let us say that the starting concentration associated with the dye is C molecules/litre; With the tank drained off to half its level and then a clear solution is used as a replacement in the process of stirring, the concentration will reduce by a half, that is 0.5C molecules/litre. The halving process of the concentration continues to occur with a constant time range and at the same time, the rate of flow is also fixed. This is what results to an aspect of decaying exponential hence the name, exponential dilution flask.

The flask is significant in testing of dynamic behavior of spectrophotometer because of its novel structure which allows for the instantaneous blending for the incoming clear solvent and the content of the flask. This follows the concepts that exponential dilution takes the nature of a process that is continuous and dynamic that is accompanied by a steady influx of the solvent and a corresponding efflux of the resulting mixture.



*15. (17.18) A large tank is connected to a smaller tank by means of a valve. The large tank

contains N2 at 700 kPa while the small tank is evacuated. If the valve leaks between the two tanks and the rate of leakage of gas is proportional to the pressure difference between the two tanks (P1 – P2), how long does it take for the pressure in the small tank to be one-half its final value? The instantaneous initial flow rate with the small tank evacuated is 0.091 kgmol/hr.



Tank 1 Tank 2 Initial pressure (kPa) 700 0

Volume (m3) 30 15



Assume that the temperature in both tanks is held constant and is 20 °C.

Solution

For any ideal gas we have:



Further, we also know that 1 kg mole for an ideal gas at standard condition is 22.4 m3

In this case, the temperatures are held constant at 20 °C = 293 K

Based on the question, the rate of gas leakage is proportional to the pressure difference existing between the tanks. The following is a good illustration giving us equation 1:



Whereby K is considered to be the constant of proportionality and is the instantaneous initial flow rate associated with the tank and is given to be 0.91 kgmol/hr



Initially we are given that p1 = 700kPa and p2 = 0

Substituting this values in the equation above gives us:









In which case, we can use the ideal gas equation to calculate the initial moles



We then substitute the known values





With the ideal gas laws, we can get the first and second tanks:



We can substitute the known values as shown







Also:













But we know that

This indicates that equation 1 can be written as follows:









Therefore:



In which case, the final number associated with the number of moles in second tank is as shon below:









The times is computed on the basis of half of the ending number of moles found in the second tank









Integration of equation 2









Substitution

Hence:







Hence, 21.8 is the number of hours that the pressure in the second tank will take to be half of its value