ESE 501 Mathematics of Modern Engineering I
Lecture 2
Vector spaces and their subspaces. Dimension and a basis of a vector
space.
Here we are going to discuss some elements from the general theory of
vector spaces that are of practical importance, also fo
ESE 319 Engineering Mathematics B
Lecture 3 2?
Laplace equation. The potential
Let P0 = (mg,y0,zg) and P = (1t,y,z) be two arbitrary points from IRB'
where P0 is considered to be xed and P is variable. The distance between
P0 and P is dened as
'r 2 Mac, y
ESE 501 Mathematics of Modern Engineering I
Lecture 14
Surfaces and their representation. Surface integrals and their evaluation.
Previously, we discussed the curves in IR3 and defined the notion of a line
integral which had a very practical meaning (work
ESE 501 Mathematics of Modern Engineering I
Lecture 11
Further properties of the gradient: direction of maximal increase of a
function and normal vector to a tangent plane. Divergence and curl.
Now, we look at some other important properties of the gradie
ESE 501 Mathematics of Modern Engineering I
Lecture 1
Vectors and matrices.
The course will consists of three parts:
- Part I: Some elements of linear algebra: Vectors and vector spaces;
- Part II - Calculus of vector-valued functions;
- Part III: Solving
ESE 501 Mathematics of Modern Engineering I
Lecture 6
Orthogonality in a general vector space. Fourier approximation.
One more application: Finding the best fitting line (statistics)
The problem: We do have two variables of interest, Y and X, that
descri
ESE 501 Mathematics of Modern Engineering I
Lecture 15
Surface integrals and their connection to triple and line integrals: theorems of Gauss and Stokes
We start first with
Theorem of Gauss (Divergence theorem). In short, it states that
Z Z
Z Z Z
F ndS =
ESE 501 Mathematics of Modern Engineering I
Lecture 9
Vector-valued functions. Continuity and differentiability of vector functions. Gradient and directional derivative of a scalar function
Now we turn our attention to the calculus whose primary goal - a
ESE 501 Mathematics of Modern Engineering I
Lecture 13
Connection between line and double integrals: the Greens theorem.
Line integrals taken over a closed curve called a contour are, in particular,
very important in many applications. Here we will discus
superheated region. Thus ultimately there may not be any improvement in system
COP due to this arrangement. It is easy to see that this modification does not result in
significant improvement in performance due to the fact that the refrigerant vapour at
t
replacement of isothermal heat rejection process of Carnot cycle by isobaric heat
rejection in case of VCRS.
Since the heat rejection increases and refrigeration effect reduces when the Carnot cycle
is modified to standard VCRS cycle, the net work input t
vapour, since reciprocating compressors are most widely is refrigeration, traditionally dry
compression (compression of vapour only) is preferred to wet compression.
ii. The second practical difficulty with Carnot cycle is that using a turbine and extract
Q. What is meant by IAQ and what does it involve?
Ans.: IAQ stands for Indoor Air Quality and it refers to the ways and means of reducing
and maintaining the pollutants inside the occupied space within tolerable levels. IAQ
involves specifying suitable le
8. An ammonia based refrigeration system with a refrigeration capacity of 100TR
(1TR=3.5167 kW) operates at an evaporating temperature of 36oC (saturation
pressure = 0.8845 bar) and a condensing temperature of 30oC (saturation
pressure = 11.67 bar). Assum
.
.
Q c = m r (h 2 h 3 )
(10.25)
where h3 and h2 are the specific enthalpies (kJ/kg) at the exit and inlet to the condenser,
respectively.
The condenser pressure Pc is the saturation pressure corresponding to evaporator
temperature Tc, i.e.,
Pc = Psat (Tc
change of angular momentum of the system. This equation is applied for hydraulic
machines such as pumps, turbines, compressors etc.
6.1.3. Bernoullis equation:
The Bernoullis equation is one of the most useful equations that is applied in a wide
variety o
a) For fixed heat rejection temperature (T3) and fixed refrigeration temperature
(T1), the COP of reverse Brayton cycle is always lower than the COP of
reverse
Carnot
cycle
(Fig.
9.3),
that
is
T4
T1
< COPCarnot =
COPBrayton =
T3 T4
T3 T1
Fig. 9
12
1Q2
= ( U 2 U1 )+ 1W2
1W2
= P.dV
2
(5.21)
1
If the working fluid behaves as an ideal gas and there are no phase changes, then, the work done,
heat transferred and entropy change during the isothermal process are given by:
1 Q 2 = 1W2
2
( U = f (T)
v
P
diagram. As shown in the figure, the ideal cycle consists of the following four
processes:
Process 1-2: Reversible, adiabatic compression in a compressor
Process 2-3: Reversible, isobaric heat rejection in a heat exchanger
Process 3-4: Reversible, adiabat
14
6. 2 kg of ice at -10 oC and 3 kg of water at 70 oC are mixed in an insulated container. Find a)
Equilibrium
temperature
of
the
system
b)
Entropy
produced.
( Cice = 2.0934 kJ / kg K , L fusion = 334.944 kJ / kg , C water = 4.1868 kJ / kg K ) (Solution)
8
From 1st T ds equation , Eq. (5.1):
T ds = du + P dv
(5.1a)
If the liquid is assumed to be incompressible then dv = 0 and
T ds = du
(5.8)
For liquids, the internal energy may be assumed to be function of temperature alone, that is,
u a = u a' , because
Tg
Wp
Pump
Qg
Cycle
System
Qa + Qc
Te
Qe
TT
o
Fig.14.3: Various energy transfers in a vapour absorption refrigeration system
From first law of thermodynamics,
Q e + Q g Q c + a + Wp = 0
(14.4)
where Qe is the heat transferred to the absorption system at e
Ans.:
a) Single compressor: The P-h diagram for the above system is shown below:
P
3
2
4
-6.7oC
5
o
-34.4 C
6
7 1 8
h
The required mass flow rate through the low temperature evaporator (mr,l) is given by:
mr,l = Qe,l/(h7 h5) = (10 X 3.517)/(1417 330.4) =
The vapour pressure data of water-lithium bromide solutions can be very
conveniently represented in a Dhring plot. In a Dhring plot, the temperature of the
solution is plotted as abscissa on a linear scale, the saturation temperature of pure
water is plot
10.4. Standard Vapour Compression Refrigeration System (VCRS)
Figure 10.5 shows the schematic of a standard, saturated, single stage (SSS) vapour
compression refrigeration system and the operating cycle on a T s diagram. As shown in
the figure the standar
compressor is taken from the cold space. In such a case, the low side pressure will be
atmospheric. In closed systems, the same gas (air) flows through the cycle in a closed
manner. In such cases it is possible to have low side pressures greater than
atmo
10
Thermal efficiency for a heat engine, HE is defined as:
HE =
Wcycle
QH
=1
QC
QH
(4.12)
where Wcycle is the net work output, QC and QH and are the heat rejected to the low temperature
reservoir and heat added (heat input) from the high temperature rese
10
Therefore if ln (P) is plotted against 1/T, the saturated vapour line will be a straight line.
Also, it has been observed that for a set of similar substances the product of Mhfg/Tnb called
Trouton number is constant. Here M is the molecular weight of
6
some property must be same for the three systems. This property is temperature. Thus this law is
the basis for temperature measurement. Equality of temperature is a necessary and sufficient
condition for thermal equilibrium, i.e. no transfer of heat.
Th