-
Sketch the geometric
interpretation of the mean value theorem of integral calculus.
-
Sketch the geometric
interpretation of the mean value theorem of differential calculus.
-
What is an Eigen value?
-
What is an Eigen vector?
-
What is the
characteristic equation of a matrix and how is it used?
-
What are the definitions
of rank/transpose/adjoint/inverse/determinant of a matrix?
-
What is the Exponential
Matrix?
-
What is the Matrizant?
-
How does one split a
second order DE into two first order DE’s and proceed to solve the
two first order DE’s by the matrix exponential method?
-
How is the Laplace
transformation used in solving PDE’s?
-
What is the Gauss Newton
Method for Non-Linear parameter estimation?
-
What are sensitivity
equations and sensitivity coefficients?
-
Give the definitions of
erf(x) and erfc(x).
-
What is a Bessel
function?
-
What is the finite
difference formula used for? Explain with a figure the finite
differences formula for:
-
Explain in words the
first law of thermodynamics.
-
Explain in words the
second law of thermodynamics for:
-
A reversible process.
-
An irreversible
process.
-
Draw the diagram for a
heat engine and for a refrigerator and explain how it relates to the
second law.
-
Draw a P-T diagram for a
pure component and identify the one, two, and three phase regions.
-
Draw a P-V diagram for a
pure component and identify the one, two, and three phase regions.
-
Draw a P-x-y diagram at
constant T for a binary system. Identify the one and two-phase
regions. From this diagram obtain the corresponding x-y diagram.
Explain.
-
Same as (6) starting
from a T-x-y diagram at constant P.
-
Define an azeotrope.
Draw P-x-y, T-x-y and x-y diagrams for a binary system that forms
azeotropes.
-
Write the equilibrium
equations for a VLE binary system. Explain how you would model the
vapor and liquid phases for a (an):
-
System at low pressure.
-
System at high
pressure.
-
Liquid containing a
polymer.
-
Aqueous salt solution.
-
Write the equilibrium
equations for a LLE binary system. Draw the T-x diagram for a:
-
Partial miscibility with
an upper critical solution point.
-
Total miscibility at low
temperatures, partial miscibility at intermediate temperatures, and
total miscibility at high temperatures.
-
In a P-T diagram for a
pure component, identify the curves of saturation pressure for a
liquid and for a solid.
-
Explain the physical
meaning of fugacity and fugacity coefficient. What are the units
for these two quantities? What is the value of the fugacity and
fugacity coefficient for a single component at low pressures?
-
Write the Gibbs phase
rule and calculate the number of degrees of freedom for a single
component:
-
Along the vaporization
curve.
-
At the triple point.
-
At the critical point.
Does the Gibbs phase rule apply at the critical point? Explain your
answer.
-
Chemical
potential:
-
Define with an equation
and give units.
-
Explain its physical
meaning.
-
Is there a unique
definition? Explain.
-
Explain the physical
meaning of an activity coefficient and describe a situation where
you would use one. Provide the simplest model for a liquid phase
activity coefficient.
-
Define the Gibbs free
energy and the Helmholtz free energy. Why are they called 'free’
energies?
-
What is osmotic
pressure? Give an example of an application that takes advantage of
the osmotic pressure.
-
Define and give an
example of a partial molar property.
-
In electrolyte
solutions, why is it necessary to define a “mean” activity
coefficient?
-
Draw
a P-V phase diagram for a pure component and identify regions of
supersaturated vapor and compressed liquid.
-
Explain in
words the meaning of the continuity equation. Describe a physical
situation and state in words how you would apply the continuity
equation.
-
Explain in
words the meaning of the Navier-Stokes equation. Describe a
physical situation and state in words how you would apply the
continuity equation.
-
In simple
terms, what is a Newtonian fluid and a non-Newtonian fluid? Give
examples of both.
-
Write the
stress tensor for a Newtonian fluid.
-
Explain in
words Newton’s second law of motion. Does this apply to all fluids,
whether Newtonian or not? Explain.
-
What is the
difference between a partial derivative and a material derivative?
-
For the
following cases, simplify the continuity equation and the equation
of motion (in the appropriate coordinate system), write the boundary
conditions, and sketch the velocity profiles and the shear stress
profiles. Clearly state what you are assuming and justify all
assumptions. For each case, consider water and air.
-
Unidirectional flow between infinite parallel plates due only to the
movement of one of the plates.
-
Unidirectional flow between infinite parallel plates due only to a
pressure difference.
-
Unidirectional flow inside a circular pipe due to a pressure
difference.
-
Unidirectional flow outside a cylinder rotating in a large pool of
fluid.
-
Unidirectional flow outside a sphere rotating in a large pool of
fluid.
-
Flow near
an infinite rotating cylinder.
-
Falling
film down an incline plane.
-
What is
vorticity? Give examples of flows with and without vorticity.
-
What is a
stream function? Define a stream function for the case when there
is flow only in the x and y directions (Cartesian coordinates).
-
What is a
boundary layer? Simplify the continuity equation and the equation
of motion for the laminar boundary layer of a flat plate placed
parallel to a fluid moving at a uniform velocity.
-
How does
method of separation of variables work? Give an example of a flow
where this method is used to solve the equations of fluid
mechanics.
-
How does
method of similarity solutions work? Give an example of a flow
where this method is used to solve the equations of fluid
mechanics.
-
Describe
how the flow develops when a pressure gradient is applied to a fluid
that was previously stationary inside a pipe.
-
For a
steady state situation (no change in time), describe how the
velocity profile changes along a pipe starting from its entrance,
which is connected to a large reservoir from which the fluid enters
due a pressure difference.
-
Why is the
stress tensor symmetric? Does it have to be symmetric? Explain.
-
Describe
the assumptions behind the “lubrication approximation”. Describe a
flow situation in which the analysis is simplified using this
approximation.
-
How would
you solve the problem of fluid flow in a tapered tube under a
pressure gradient?
-
How
would you solve the problem of the development of the thickness of a
fluid film falling down an inclined surface?
