Ch5 Joint probability distribution
TWO DISCRETE RANDOM VARIABLES
The joint probability mass function of the discrete random
variables X and Y, denoted as fXY (x, y), satises
Marginal Probability Distributions
If X and Y are discrete random variables with
Chapter 6 Random Sampling and Data Description
6-1 DATA SUMMARY AND DISPLAY
The standard deviation is the positive square root of the variance
We can characterize the location or central tendency in the data by the ordinary
arithmetic average or mean.
Sam
CHAPTER 2 PROBABILITY
Events
SAMPLE SPACES AND EVENTS
An event is a subset of the sample space of a random
experiment.
Random Experiment
An experiment that can result in different outcomes, even
though it is repeated in the same manner every time, is
call
Ch4 CONTINUOUS RANDOM VARIABLES
Denition
For a continuous random variable X, a probability density
function is a function such that
(1) f(x) 0
(2)
f ( x) dx 1
b
(3) P(a X b) =
f ( x)dx
a
For a continuous random variable X and any value x.
If X is a conti
Ch3 DISCRETE RANDOM VARIABLES
PROBABILITY DISTRIBUTIONS AND PROBABILITY MASS
FUNCTIONS
Definition For a discrete random variable X with
possible values, a probability mass function is a function
such that
(1) f ( xi ) 0
n
(2) f ( xi ) 1
i 1
(3) f ( xi ) P
7-31 The radiator of a steam heating system is initially filled with superheated steam. The valves are
closed, and steam is allowed to cool until the temperature drops to a speciﬁed value by transferring heat to
the room. The entropy change of the steam d
Refrigeration Cycles
Refrigerator and Heat Pump
1. An ideal refrigeration cycle uses ammonia as the working fluid between
saturation temperature of -40C and 10C. If the refrigerant mass flux is 1
kg/s, determine the rate of refrigeration and the coefficie
Power Gas Cycle
1. The Air-Standard Cycle
In this section we introduce engines that utilize a gas as the working fluid.
Spark-ignition engines that burn gasoline and compression-ignition (diesel)
engines that burn fuel oil are the two most common engines
The Second Law of Thermodynamics
Heat engine and thermal efficiency.
1. A steam power plant receives heat from a furnace at a rate of 280 GJ/h. Heat
losses to the surrounding air from the steam as it passes through the pipes and
other components are estim
Power Gas Cycles
Otto Cycle
1. A six-cylinder engine with a compression ratio of 8 and a total volume at TDC
of 600 mL intakes atmospheric air at 20C. The maximum temperature during
a cycle is 1500C. Assuming an Otto Cycle, calculate
a) The heat supplied
Refrigeration Cycles
1. Refrigerator and Heat Pump
q out
T
2
condenser
2
3
compressor
w
com
3
1
4
4
1
evaperator
q in
s
2. Gas Refrigeration Cycle
Q
4
heat
exchanger
out
3
T
3
turbine
compressor
W
net
4
2
1
heat
exchanger
Q
in
1
2
s
q out
6
5
1
internal h
Power Cycle
1. Consider a 300 MW steam power plant that operates on a Rankine cycle.
Steam enters the turbine at 10 MPa and 500C and is cooled in the condenser
at a pressure of 10 kPa. Determine
a) The quality of steam at the turbine exit. (79.3%)
b) The
Entrnpy Changes Bf Pure Substances T—34 A well-insulated rigid tank contains 2 kg of a satu-
lated liquid—vapor mixture of water at ltlﬂ kPa. Initially,
lhree-quarters of the mass is in the liquid phase. An electric
lesistance heater placed in the tank is
Power Cycle
1. Rankine Cycle
Boiler
Turbine
pump
Condenser
w
tur
2. Reheat Rankine Cycle
Boiler
Turbine
pump
Condenser
w
tur
3. Regenerative Rankine Cycle
Boiler
Turbine
w
pump
open
FWH
pump
Condenser
tur
Boiler
Turbine
w
pump
pump
Condenser
tur
1. Two Carnot engines operate in series between two reservoirs maintained at
527C and 40C, respectively. The first engine rejects 90 MJ of energy every
hour which is utilized as energy input to the second engine. Assuming the
thermal efficiency of both en
Ex. Steam at 6 MPa and 500C enters the high-pressure turbine and is
expanded to state of saturated dry steam. The steam is then reheated to
500C and expanded in the second stage (low pressure turbine) to 10 kPa.
Assuming the cycle to be ideal; determine
a
Power Cycle
Rankine Cycle
1. Consider a 300 MW steam power plant that operates on a Rankine cycle. Steam enters the turbine at 10
MPa and 500C and is cooled in the condenser at a pressure of 10 kPa. Determine
a) The quality of steam at the turbine exit. (
1. Steam 6 MPa, 500C enters in a high-pressure turbine (in a reheat cycle) and
expand to state of saturated dry steam. The steam is reheated to 500C and
expanded in the second stage (low pressure) at a turbine to 10 kPa. Assume
the cycle to be ideal, dete
1. Steam at 10 MPa and 550C enters the first-stage turbine of an ideal Rankine
cycle with reheat. Steam enters the second-stage turbine after being reheated to the
same temperature as of the superheated steam. Saturated vapor exits the secondstage turbine
Discharging Process
1. A 0.1 m3 rigid tank contains saturated refrigerant 134a at 800 kPa. Initially, 40
percent of the volume is occupied by liquid and the rest by vapor. A valve at the
bottom of the tank is now opened, and liquid is withdrawn from the t
Adiabatic efficiency of turbine and compressor
1. Steam enter turbine at 2 MPa and 400C and leave at 10 kPa determine
temperature at exit when turbine efficiency is 85%.
2. Steam enters turbine at 400C and 2 MPa and leaves at 200 kPa. If the mass flow rat
Work
1. A mass of 5 kg of saturated water vapor at 200 kPa is heated at constant pressure
until the temperature reaches 300C. Calculate the work done by the steam
during this process. (430.5 kJ)
2. A frictionless piston-cylinder device initially contains
Work
1. A mass of 5 kg of saturated water vapor at 200 kPa is heated at constant pressure
until the temperature reaches 300C. Calculate the work done by the steam
during this process.
2. A frictionless piston-cylinder device initially contains 200 L of sa
Properties of Pure Substance
Thermodynamics: the science that includes the study of energy transformation and the relationship
among the physical properties of substance when are affected by these
transformations.
System: is defined as a portion of the ma
3—54 _ﬁ~.(J.5-m3 vessel contains Ii] kg of refrigerant-l34a at 3—59 A piston—cylinder device initiaIJy contains SCI L of
-HJ‘C. Determine (a) the pressure, (in) the tota] interna] liquid water at 43": and 233 kPa. Heat is transferred to the
energy, and (c
3—54 A {LS-n1.J vessel contains 1U kg of refrigerant—l34a at 3—59 A piston—cyiinder device initiaily contains 51'} L of
-3]”C. Detennine (a) die pressure, U3} die total internal 1iquid water at 4D=C and 20f] kPa. Heat is transferred to the
energy. and [c]
Phase
Compressed
liquid
Sat. liquid
Sat. mixture
Sat. vapor
Superheated
vapor
For sat mixture
v = vf + xvfg
u = uf + xufg
h = hf + xhfg
s = sf + xsfg
Know P&T
P>Psat for given T
T<Tsat for given P
P=Psat for given T
T=Tsat for given P
P<Psat for given T
T
Properties of Pure Substance
Thermodynamics: the science that includes the study of energy transformation and the relationship
among the physical properties of substance when are affected by these
transformations.
System: is defined as a portion of the ma