MAE182A
Summer2011
Homework2
DueJuly8,2011
Obtainthegeneralsolutionforthefollowingdifferentialequations
a)
b)
1
c)
cos
d)
sin
cos
e)
0
Hint:Usepolarcoordinates.
f)
2
g)
4
3
2
1

SOLUTIONS
MAE 105A
Introduction to Engineering Thermodynamics
Fall 2013
Problem Set 2
1. [Problem 3.44 in MSBB] Using the tables for water, determine the specified property data
at the indicated states. In each case, locate the state by hand on sketches o

MAE 105A
Introduction to Engineering Thermodynamics
Fall 2013
Problem Set 2
1. [Problem 3.44 in MSBB] Using the tables for water, determine the specified property data
at the indicated states. In each case, locate the state by hand on sketches of the p-v

MAE 105A Fundamentals of Engineering Thermodynamics
MAE 105A Fundamentals of Engineering Thermodynamics
Prof. R. Amar
HW #2
Summer 2011
Prof. Ravnesh
HW2 2011
Spring
Due: 7/7/11
Due in discussion (4/15/11)
3.5 (Except part e)
Determine the phase or phases

sxt
cx
k
ti
vi
,
~
~
~
2
2
L[ n]
~
Reconstruct
at solution
and PDE centres
Has
converged to
within NL ?
Yes
No
n = n +1
n
t = t + t
Set n =~
Provide initial
T
- data (
n=0
)
Fig. 2. Schematic description of the nonlinear solution procedure
this, the cur

beams were fully loaded up
to 40kN and subjected to
the ISO 834 (ISO, 1975) fire
standard
curve. Eurocode 2
(Eurocode, 1992) and ACI440 (ACI, 2008) procedures
were used to
design the beams. The
beam reinforced with the
steel stirrups achieved a
128min fir

r2 + c 2
_ m2
m 2Z+ 1 (1)
The multiquadric RBF is a conditionally positive definite function of order m, which requires
the addition of a polynomial term of order m 1, together with a homogeneous constraint
condition, in order to obtain an invertible inte

12
m111113,
6XX
(9)
AnalyticalMethodsforEstimatingThermalConductivity
ofMultiComponentNaturalSystemsinPermafrostAreas
237
at 1
22
2
122
1 1 2 1 1 1 m 1 4arcsin .
6 X 16X X 2 X
(10)
The increase in the volume fraction of the solids due to the contac

as the size of the dataset grows, and also with increasingly flat interpolating functions. This
is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as
demonstrated in Driscoll & Fornberg (2002), and is described by Rob

b b( a
)
c DT c T V T k T w c T T Q
Dt t
(9)
The partial differential equations for thermal ablation or hyperthermia are discretized at the
grid point by using the finite difference approximation using Pennes equation.
2 b b( a )
cTkTwcTTQ
t
(10)

breaks down slowly so that the drug is released gradually over several days. The third
element is a coating layer of polyethylene glycol, which is used to prevent from attacking by
antibodies and macrophage cells of the human immune system. The final elem

practice on the use of FRP
materials
in building and other types
of structures. In addition,
such studies would draw a
better
understanding on the
behavior of FRP materials
under fire actions that
would enhance the
available documentation
and literature t

k is the conductivity
h is the convective heat
transfer coefficient in
(W/m2K), typical vale is
25W/m2K
T is the temperature
difference between the
solid surface and fluid in (
C or K)
is a configuration or view
factor depends on the area
(A) of the emit

carbon CFRP plates
subjected to
bottom and top fire
loading. The models
predicted with reasonable
accuracy the experimental
results of Williams et al.
(Williams et al., 2008). It
was concluded the
developed models can
serve as a valid alternative
tool to

Fig. 1. Particle shapes in the soil thermal conductivity model at different semi-axes ratios of
ellipsoids = a/R: 1 faceted (> 1); 2 spherical (= 1); 3 worn (< 1); 4 cruciate (< 1)
All calculations are made in terms of the parameter = a/R, which is a uniq

when irregular datasets are required. Traditional polynomial-based finite difference methods
are difficult to implement on irregular datasets, and RBF collocation allows a natural
generalisation of the principle to irregular data. The inclusion of arbitra

or vibrations of free
electrons
Heat Transfer Analysis of
Reinforced Concrete
Beams Reinforced with
GFRP Bars
301
and atoms. On the other
hand, convection transfers
the heat from the source
to the RC beam
via cycles of heating and
cooling of the surroundi

The local system size varies slightly, however the modal number of boundary and solution
centres present in each local system is 14. Additionally, PDE centres are added to each local
system. A tetrahedralisation is performed on each local system, using th

t
= L [u (x, t)] + S2 (x, t) in
u (x, 0) = f (x) in
B [u (x, t)] = g (x, t) on
(14)
a finite difference approximation is made to the time-derivative
un un1
t
= L [un] + (1 ) L
_
un 1
+ S2 (x, tn) (15)
From this approximation, a modified PDE operator is o

conventional reinforcing
steel bars, the FRP bars
seem to have a high
strength to weight
ratio, moderate modulus of
elasticity and resistance to
chemical and electrical
corrosion.
Although FRP materials
were shown to have a
brittle failure, due to their
n

2
x2i
+ s (xi, t)
_
(21)
The transformation of the nonlinear diffusion term into Kirchhoff space reduces the equation
from a strongly nonlinear to a weakly nonlinear form, by removing the multiplication of first
derivatives in the diffusive term:
1
cv (T)

conducted in the previous
years on
the fire performance of RC
beams reinforced with FRP
bars due to the high costs
of such
tests, tremendous amount
of preparation, and
shortage of specialized
facilities (Franssen et
al., 2009).
Sadek et al. (Sadek et al.,

were placed in two layers
at the
tension side and two
serving as compression
reinforcement. In addition,
9 mm stirrups
were used as shear
reinforcements spaced at
160mm center to center.
The concrete
compressive strength was
42MPa. The first beam
specimen

structures have been
invulnerable to
harsh environmental
exposures, with little or no
maintenance. Furthermore,
such structures
are experiencing larger
amount of loads than their
original capacities due to
the increase
number of users over the
years (Bisb