A23604 Advanced Transport Processes 2019
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School of Chemical Engineering
04 20545
Advanced Transport Processes
Summer Examinations 2019
1. Figure 1 illustrates an apparatus consisting of a horizontal duct with a triangular cross section. Figure 2 shows the transversal cross section of the duct and the corresponding coordinate system.
In order to understand the performance of the apparatus it is necessary to analyse the steady state isothermal flow of water through the duct. The general mass and momentum balance equation (z-component) are given below (Equation 1 and Equation 2).
Figure 1 (left); Figure 2 (right)
+ + = 0
( + + + ) = − + ( + +
(1)
) +
(2)
(a) Assume that the flow is laminar, fully developed and stationary. Simplify
the general continuity equation and momentum equation for the problem considered, justifying any further assumption made.
[20%]
(b) Give the physical interpretation (including units) of every term of the
simplified continuity and momentum equation obtained in (a) above
[20%]
(c) A colleague has told you that the velocity distribution and pressure gradient are given by Equation 3 and Equation 4 respectively:
= ( − )(32 − 2) (3)
= −1 (4)
Does the velocity distribution satisfy the no-slip boundary conditions and the momentum equation? Justify your answer.
[30%]
(d) Starting from the velocity distribution provided by Equation 3, express the viscous stress acting on the top wall of the duct. Assuming that the length of the duct is 1 m, calculate the viscous force acting on the top wall of the duct
[30%]
DATA for Question 1:
= 1000 kg m−3, = 0.001 Pa s , = 0.03 m, 1 = 5 Pa m−1
2. A long solid cylinder is immersed in a fluid bath. The fluid is a diathermic oil.
The cylinder has a constant surface temperature 0 (see Figure 3) and radius R0 . The cylindrical tank containing the diathermic oil has a constant wall temperature ∞ and radius ∞ .
It is of interest to study the heat transfer phenomenon in the diathermic oil surrounding the cylinder in the absence of fluid motion.
The general form of energy balance equation is given by Equation 5 .
Figure 3
( + + + ) = ( ( ) + + ) + (5)
(a) Give the physical interpretation (including units) of every term in the
general form of the energy equation (Equation 5).
[20%]
(b) Assuming steady state conditions, simplify Equation 5, justifying all the
steps, and develop the expression of temperature distribution for the diathermic oil surrounding the cylinder in absence of fluid motion.
[30%]
(c) Starting from temperature distribution, express and calculate the total energy flux at the following positions:
r1 = 0.3 m; r2 = 0.9 m
[30%]
(d) Calculate the energy transfer rate across the outer surface of the cylinder and across the wall of the tank. Assume both the cylinder and
tank are 2.5 m long.
DATA for Question 2
Properties of diathermic oil:
= 1000 kg m−3, = 0. 12 W m−1 K −1 ,
[20%]
= 1700 J kg −1K −1
Dimensions:
0 = 0.02 m;
0 = 100°C;
∞ = 1 m
∞ = 20°C
3
(a) Briefly describe the three-layer model for a turbulent boundary layer
above a horizontal surface using a well labelled diagram, and sketch the anticipated velocity and shear stress distributions within such a boundary layer.
[20%]
(b) Consider the incompressible flow of a fluid above a smooth flat plate.
The local velocity u at a vertical distance y in the turbulent boundary layer above the plate can be described by:
u y
u
where ′ is the velocity at = ′ and ′ is the thickness of the laminar sub-layer. In addition, where required, the wall shear stress is denoted by o , the fluid density is denoted by , and the fluid kinematic viscosity is denoted by .
Show that the velocity profile within the boundary layer can be written in the following dimensionless form:
u* 5.34 2.55 ln y*
where
∗ = and ∗ = .
[50%]
(c) A smooth rectangular plate of negligible thickness having dimensions
1 m · 12 m is submerged in a fluid stream with its longer edges parallel to the flow. If the free stream velocity, U, outside the boundary layer is
25 m s-1:
i. Calculate the wall shear stress at the trailing edge;
[15%]
ii. Locate the point above the plate at which the fluid velocity reaches
12 m s-1 , explaining clearly the reasoning behind your calculation
[15%]
The data for question 3 is overleaf
DATA for Question 3:
The local skin-friction coefficient, cf , may be estimated from the Prandtl- Schlichting correlation: cf 0.65 0.87ln Rx 2.3 , in terms of the Reynolds number Rx .
The thickness of the laminar sub-layer can be estimated from the relationship: 11.6 .
The physical properties of the fluid are
= 15 x 10-6 m2 s-1 , and = 1.3 kg m-3 .
4
(a) Explain what is meant by a Maxwell liquid and show that the rheological
behaviour of such a liquid is described by the Maxwell constitutive equation:
where is the shear stress and the shear rate at time t; and ,
where is the Newtonian viscosity and G is the modulus of rigidity.
[20%]
(b) Consider an experiment where a Maxwell liquid is subjected to the
sudden application of a shear strain o , which is then held constant at that value. Show that the stress will then relax according to the following exponential decay function:
t
oe
where is the shear stress in the liquid at time t = 0.
[15%]
(c) In a test, starting from a state of zero strain and zero stress, a Maxwell liquid is strained at a steady rate C . Show that the shear stress
response is described by the following function:
t
C 1 e
[20%]
(d) In 4(c) above, calculate in terms of the time it takes the stress to rise to 90% of its final value.
[15%]
(e) The Maxwell model often fails to describe the relaxation behaviour of
most real viscoelastic materials because it does not incorporate an equilibrium stress. This difficulty can be circumvented by coupling a Maxwell model with a spring in parallel which accounts for an equilibrium stress e . On this basis, establish the general stress relaxation equation for this model explaining clearly your reasoning.
[30%]
DATA for Question 4:
The general solution of the Maxwell constitutive equation can be
t t t
2022-01-13