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1) A fluid in a pipe or channel has an axial velocity profile described by a function of a radial position or the cross-flow direction

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1) A fluid in a pipe or channel has an axial velocity profile described by a function of a radial position or the cross-flow direction. Let vz = vz(r), where vz is the local axial velocity and r is the cross-flow direction. Derive an expression for the average velocity at that plane.

2) A fluid enters a vessel through an inlet pipe of cross-sectional area A1 and leaves via a pipe of area A2. Find a relation between the average velocities in the inlet and exit pipes. Assume steady state.A tank has a volume V of liquid at time zero and is draining from the bottom. An expression for the height change with time is needed.

3) The simplified case of Eq. (2.8) with no reaction (Da = 0) is important and provides a very useful experimental method to determine whether and to what extent a reactor is backmixed. The method is known as tracer analysis, also known as stimulus–response experiments. The stimulus represents a change in the inlet conditions, and the change in concentration in the outlet as a function of time is measured and represents the response curve of the system. The exit response can be predicted from the above analysis and matched with the experimental data to test some of the model assumptions, for example, whether the assumption that the system is well mixed holds or not. Two types of tracer injection are common: (i) step injection and (ii) pulse or bolus injection. The outlet responses for these two cases are presented below. This section is also useful for reaction engineering studies, where stimulus–response experiments are common.

4) Calculate the number of gas molecules in a cube of size 1 μm in each dimension at 1 atm pressure and temperature 298 K, and examine the validity of the continuum assumption. Also calculate the density of the gas, if the gas is air, based on an equation of state for the gas.

5) Other electrophoretic methods: a case-study problem. In the text we considered only the zone electrophoresis where a zone of a solute-rich layer is created by the action of an electric field. Other techniques are isotochorphoresis and iso-electric focusing. Review what these techniques are. In particular, emphasize the transport modeling issues involved in each of these techniques. The book by Westermeier (2005) would be a good starting point.

6) Separation distance for proteins. It is required to separate two proteins with the mobilities of μ1 = 8×10−5 m · C/(N ·s) and μ2 = 6×10−5 m · C/(N ·s), The diffusion coefficient is 6 × 10−11 m2/s. The flow velocity is 0.2 mm/s. The applied field is 2000 V/m. Find the trajectory of the “plumes” of the two proteins and the distance at which the two proteins can be separated.

7) Consider the settling of charged spherical particles with a zeta potential of 50 mV in a solution of 0.0 5 M NaCl. The particle density is 2060 kg/m3 and the particle holdup is 0.1. Calculate and plot the sedimentation electric field as a function of the particle diameter. Also calculate and plot the particle settling velocity and show that the correction to Stokes’ law is negligible for this case.