Resistances

We can use concepts of mass transfer resistance to make the analysis of mass transfer with combined mechanisms much easier.

Definition of resistance

A resistance is the driving potential (or driving force) divided by the transfer rate:

resistance=driving potentialtransfer rate

We are probably familiar with Ohm’s law, in which the driving potential is the voltage and the transfer rate is the current, leading to

�elec=voltagecurrent

For mass transfer, our driving potential is an appropriate concentration difference, Δ�, and the transfer rate is the molar transfer rate, �˙�.

Exercise:

For both diffusion and mass convection,

• what is the driving potential (concentration difference)?
• what is the molar transfer rate equation?
• what is the resulting expression for the mass transfer resistance?

Solution

Like in electrical circuits, resistances can be added directly if they are in series:

�total=∑���

For the above mass transfer device, there are five resistances between our concentration measurements (��,1, ��,2)

�total=�flowing fluid 1+�membrane 1+�stagnant fluid+�membrane 2+�flowing fluid 2

The resistances �flowing fluid 1 and �flowing fluid 2 are for convective processes, while those for �membrane 1, �stagnant fluid, and �membrane 2 are for diffusive processes.

So, to compute the rate of mass (molar) transfer, we can use the following:

Resistance equations

�˙�=��,2−��,1�total

where, for resistances in series, we have

�total=∑���=∑��diffusion,�+∑��convection,�

and

�diffusion,�=Δ�����,����convection,�=1ℎ�,���

Note

The equations we have developed are applicable only to systems being analyzed in rectangular Cartesian coordinate systems. When we analyze heat transfer resistances, we will examine changes that are needed for other coordinate systems (e.g., cylindrical coordinates)

Exercise: Mass transfer resistances

You have been assigned to improve the efficiency of a chemical separations device. The team that improves the efficiency most will win an all-expense-paid trip to the destination of their choice. A schematic of the device is as follows:

Baseline characteristics

�A/hydrogel=5×10−4cm2/s	�A/membrane=7×10−5cm2/s
�1=5mm	�2=1mm
ℎ�=0.002cm/s	�=1m2

Your team has developed technologies leading to three design options:

1. improved hydrogel diffusivity: 4 time the baseline value
2. improved membrane diffusivity: 3 times the baseline value
3. increased mass transfer coefficient: 6 times the baseline value

Assuming that ��,1>��,2, make a sketch of the expected concentration of � in the device.

Which single option will you choose to achieve the best improvement of device performance?

One useful measure of performance is the mass transfer rate (moles/min).