heat_work - lecture notes

8/9/2019 heat_work - lecture notes

1/20

Energy AnalysisHeat, Work & Mass

Ng Tin Yau (PhD)

Department of Mechanical Engineering

The Hong Kong Polytechnic University

Jan 2013

By Ng Tin Yau (PhD) 1/20


2/20

Heat and Work Transfer

Energy can cross the boundary of a system by means of heatandwork. Itis important to distinguish between these two forms of energy.Heat is defined as the form of energy that is transfered between twosystems by virtue of a temperature difference. That is, an energyinteraction is heat only if it takes places because of a temperaturedifference. A process during which there is no heat transfer is called an

adiabatic process.Work, like heat, is an energy interaction between a system and itssurroundings. In thermodynamics analysis, we normally decompose thetotal work into reversible work and irreversible work (or dissipative work).That is, for a given process the total amount of work is given by

Wtot=Wrev+Wirrev (1)

The amount of heat and work done transferred during the process betweentwo states (states 1 & 2) is denoted by Q12 and W12, respectively.



3/20

Heat and Work Sign Convention

Heat and work are directional quantities, and thus the completedescription of a heat or work interaction requires the specification of both

the magnitude and direction. One way of doing that is to adopt a signconvention.

Formal Sign Convention

Heat transfer to a system and work done by a system are positive; heat

transfer from a system and work done on a system are negative.

Heat and work are energy transfer mechanisms between a system and itssurroundings, and there are many similarities between them:

Both are recognized at the boundaries of a system as they cross the

boundaries. That is, both heat and work are boundary phenomena.Systems possess energy, but not heat or work.

Both are associated with a process, not a state.

Both are path functions (i.e. their magnitudes depend on the path

followed during a process as well as the end states.). By Ng Tin Yau (PhD) 3/20


4/20

Path and Point Functions

Path functions have inexact differentials designated by the symbol .Therefore, a differential amount of heat or work is represented by Q orW, respectively. For instance, the total work done during process 1 2is

W12=

21

W (2)

Properties, however, arepoint functions, and they have exact differentialsdesignated by the symbol d. For example, a small change in volume isrepresented by dVand the total volume change during a process betweenstates 1 and 2 is

V =V2

V1

= 21

dV (3)

Clearly, we have dV = 0 (4)



5/20

The First Law & Internal Energy

The First Law (Joule)

For all adiabatic processes between two specified states of a closed system,the net work done is the same regardless of the nature of the closedsystem and the details of the process.

In the case where only microscropic energy is involved, then we have

Wad12 =U1 U2=U (5)

Since only heat and work can cross the boundary of a closed system,therefore, one can define the heat transfer into the system Q12 as

Q12=W12Wad12 (6)

which yieldsQ12W12= U (7)

This is the common mathematical form of the first law of thermodynamics. By Ng Tin Yau (PhD) 5/20


6/20

A Direct Consequence of the First Law

Consider the differential form of the first law:Q12 W12=dU (8)

Since internal energy is a property, therefore,

dU= 0 and hence, for a

cycle, we have Q12=

W12 (9)

Recall from our sign convention, if both the quantitiesQ12 and

W12

are positive, the result represents the net heat transfer to the system and

net work done by the system, respectively. In other words, for a closedsystem the net amount of heat transfer into the system is equal to the netamount of work done by the system.



7/20

Free Expansion of Gases

An insulated rigid tank is divided into two compartments with different

equal volumes. Initially, side 1 is filled up with gas, then the wallseparating the two compartments is removed and the gas is free to expanduntil the gas is equally distributed on both sides.

During this process, Q12=W12= 0, hence U= 0. Note that in thiscase V = 0 and pressure is not necessary zero but the total work done ofthe process is zero. This situation is called gas free expansion.



8/20

Moving Boundary Work

Consider a closed system subjected to a quasi-static volume change fromV1 and V2, then the work done due to this volume is given by

Wb=

2

1

PdV (10)

This work done is called moving boundary work. An example of movingboundary work is the piston-cyclinder device.



9/20

Boundary Work for a Polytropic Process

For a polytropic process PV = C where ,C R. For = 1, theboundary work is given by

21

CVdV =CV1

1

2

1

= C

1

V2

V

2

V1

V

1

=P2V2 P1V1

1

For = 1 we have 21

CV1dV =P1V1ln

V2

V1

In summary

Wb=

P2V2P1V1

1= 1

PiViln

V2V1

= 1 and i= 1 or 2

(11)


S ifi H C i


10/20

Specific Heat Capacity

The specific heat capacity or just specific heat is defined as the heatenergy required to raise the temperature of a unit mass of a substance by

1 degree and is denoted as c.

c q

T(12)

Consider a fixed mass in a stationary closed system undergoing a constant

volume process, then dv= 0 and q=du, hence

cv =

u

T

v

(13)

which is known as constant volume specific heat. Since the differential

of enthalpy dh is du+Pdv+vdPand for constant presure, dP= 0 whichimplies q=dh. Hence, the constant pressure specific heat is

cp=

h

T

p

(14)


C fi i (R ibl ) W k


11/20

Configuration (Reversible) Work

Up to this point we have encountered two types of reversible work, namely,the adiabatic work and moving boundary work. In general, one may dividethe work into two types: reversible and irreversible work. In this case, wehave

Wrev=yidZi (15)

where yi and Ziare intensive and extensive properties, respectively. Anexample ofy and Z are pressure Pand volume V, respectively. Thenaccording to the first law, we have

Q12=dU+yidZi (16)

which further implies that Q12 must be reversible. The extensiveproperties Ziare said to determine the configuration of the system andhence, the reversible work

yidZi is also called configuration work.

Finally, irreversible work is somtimes called dissipative work.


A E l f Di i ti W k El t i l W k


12/20

An Example of Dissipative Work Electrical Work

The electrical work is the work needed to maintain an electric current I ina resistor of resistance R. According to Ohms law V=IRwe have thework done on the resistor

We=

21

Wedt=

21

VIdt (17)

where V and Iare the potential difference and current, respectively.



13/20

A B k D f T s i E B l E ti


14/20

A Break Down of Terms in Energy Balance Equation

Energy transfer involves heat, work and mass, that is

Ein Eout= (Qin+Win+Emass in) (Qout+Wout+Emass out) (19)

Rewrite it as

Ein Eout=Qnet inWnet out+ (Emass in Emass out) (20)

where Qin Qout Qnet in and WoutWin Wnet out.In the case of simple compressible system, we assume

Esys= U+ KE + PE (21)

The kinetic energy is given by KE = 12m(V V) where V is the velocity of

the system. The potential energy is written as PE = mgz where z is theelevation of the center of gravity of a system relative to some arbitrarilyselected reference level.



15/20

Mass Transfer


16/20

Mass Transfer

Generally speaking, for open system, one must include the flow energy intothe energy term. As a result, the total energy of a flowing fluid on a

unit-mass basis, denoted by is given by

Pv+u+ ke + pe =h+ ke + pe (24)

In this case, the energy transfer due to mass transport is given byEmass=m. Hence, the total energy transported by mass through the

boundary is obtained by integration:

Emass=

m (25)

In the case of uniform properties1

, that is, is constant throughout theboundary surface, and hence,

Emass=

m=m (26)

1

Uniform means no change with location over a specificed region. By Ng Tin Yau (PhD) 16/20

Unsteady Flow Analysis


17/20

Unsteady-Flow Analysis

Many processes of interest, however, involve changes within the controlvolume with time. Such processes are called unsteady-flow processes.Unlike steady-flow processes, unsteady-flow processes start and end oversome finite time period instead of continuing indefinitely.Most unsteady-flow processes, however, can be represented reasonably well

by the uniform-flow process, which involves the following idealization:

Uniform-Flow Approximation

The fluid flow at any inlet or exit is uniform and steady, and thus the fluidproperties do not change with time or position over the cross section of an

inlet or exit. If they do, they are averaged and treated as constants for theentire process.


General Analysis for Multi Inlet Outlet Systems


18/20

General Analysis for Multi-Inlet-Outlet Systems

In general unsteady-flow analysis, conservation of mass principle gives

in

mout

m= msys (27)

and the energy balance is given by

Qnet,inWnet,out+

inm

outm= Esys (28)

In the case of a closed system,

in

m= out

m= 0msys

= 0 (29)

Similarly, we haveQnet,inWnet,out= Esys (30)


An Example


19/20

An Example

Consider a uniform-flow system which involveselectrical, shaft and boundary work with one

inlet. Suppose that the kinetic and potentialchanges associated with the control volumeand fluid stream are negligible, then the massbalance equation becomes

mi =m2m1 (31)

where the subscript i represents the inlet ofthe system.

On the other hand, the energy balance equation is given by

Qnet,in+

21

VIdt+Wsh

21

PdV+mii = (m2u2m1u1)sys (32)


Steady Flow Analysis


20/20

Steady Flow Analysis

In the case of steady flow, we have msys= 0 and Esys= 0.

Equations for Steady Flow Analysis

in

m=out

m (33)

andQnet,inWnet,out+

in

m out

m= 0 (34)


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heat_work - lecture notes