Heat and Work

Ability to acquire and explain the basic

concepts in thermodynamics

The student should be able to:

� Define the concept of heat and the terminology associated with energy transfer by heat.

� Discuss the three mechanisms of heat � Discuss the three mechanisms of heat transfer: conduction, convection, and radiation.

� Define the concept of work and different forms of mechanical work.

3.1 Heat and energy transfer by heat

3.2 Work and energy transfer by work

� Heat = A form of energy that is transferred across the boundary of a system by virtue of a temperature difference.

• Heat is energy in transition. It is recognized only as it crosses the boundary of a system.

Adiabatic process A process during which there is no

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Adiabatic process A process during which there is no heat transfer

� How ? (1) System is well insulated

(2) Both system & surroundings

are at same T

�Mechanisms of heat transfer:

(a)(a)(a)(a) ConductionConductionConductionConduction

Transfer of energy from more energetic particles of a

substance to the adjacent less energetic ones as a result of interaction between particles.

Fourier's law of heat conduction is

&Q A kdT

= −

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&Q A kdx

cond t= −

&Qcond

dT

dx

= heat flow per unit time (W)

kt = thermal conductivity (W/m⋅K)A = area normal to heat flow (m2)

= temperature gradient in the direction of heat flow

(°C/m)

A flat wall is composed of 20 cm of brick having a

thermal conductivity kt = 0.72 W/m⋅K. The right

face temperature of the brick is 900°C, and the left

face temperature of the brick is 20°C. Determine

the rate of heat conduction through the wall per

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the rate of heat conduction through the wall per

unit area of wall.

Tright = 900°C

Tleft = 20°C

20 cm

T∆

9

23168

2.0

)90020(72.0

m

W

m

K

Km

W

x

Tk

A

Q

x

TAkQ

tcond

tcond

=

−

⋅=

∆

∆−=

∆

∆−=

&

&

(b)(b)(b)(b) ConvectionConvectionConvectionConvection

Transfer of energy between a solid surface & adjacent fluid that is in motion & involves the combined effects of conduction & fluid motion.

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& ( )Q h A T Tconv s f= −

&Qconv = heat transfer rate (W)

A = heat transfer area (m2)

h = convective heat transfer coefficient (W/m2⋅K)

Ts = surface temperature (K)

Tf = bulk fluid temperature away from the surface (K)

The rate of heat transfer by convection is determined from Newton's law of cooling,

expressed as

&Qconv

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The convective heat transfer coefficient is an experimentally determined parameter that

depends upon the surface geometry, the nature of the fluid motion, the properties of the

fluid, and the bulk fluid velocity. Ranges of the convective heat transfer coefficient are given

below.

h W/m2⋅⋅⋅⋅K

free convection of gases 2-25

free convection of liquids 50-100

forced convection of gases 25-250

forced convection of liquids 50-20,000

convection in boiling and condensation 2500-100,000

(c) Radiation

Transfer of energy due to the emission of electromagnetic

waves(or photons)

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( )44

surrsrad TTAQ −= εσ&

= heat transfer per unit time (W)

A = surface area for heat transfer (m2)

σ = Stefan-Boltzmann constant, 5.67x10-8 W/m2K4 and 0.1713x10-8 BTU/h ft2 R4

ε = emissivity

Ts = absolute temperature of surface (K)

Tsurr = absolute temperature of surroundings (K)

&Qrad

Consider a person standing in a

breezy room at 20oC. Determine

the rate of heat transfer

from this person by radiation if

the exposed surface area and the

average outer surface temperature

of the person are 1.6 m2 and 29oC,

respectively, and the emissivity of

human skin is 0.95

� If energy crossing the boundary of a closed system is not heat, it must be workworkworkwork.

� Work is the energy transfer associated with a force acting through a distance.e.g. rotating shaft, rising piston, electric wiree.g. rotating shaft, rising piston, electric wire

� Work done per unit mass of a system is denoted by w & expressed as

� Work done per unit time, W/t = Power ( ) Unit: kJ/s or kW

•

W

� Heat & work requires the specification of both magnitude & direction.

� Formal sign convention:

Heat transfer to a system, Qin Heat transfer from a system, Qout

Work done by a system, Wout Work done on a system, Win

(+ve) (-ve)

� Use subscripts in & out to indicate

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� Use subscripts in & out to indicate

direction

� e.g a work input = Win

a heat loss = Qout

� Can assume a direction for interaction

� +ve assumed direction is right

� -ve direction of interaction is the opposite

Heat vs. Work• 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.

• Unlike properties, heat or work has no meaning at a state.

• Both are path functions (i.e., their • Both are path functions (i.e., their magnitudes depend on the path followed during a process as well as the end states).

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Properties are point functions; but

heat and work are path functions

(their magnitudes depend on the

path followed).Properties are point functions have

exact differentials (d ).

Path functions

have inexact

differentials (δ )

Electrical Work

Electrical work

Electrical power

When potential difference and

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Electrical power in terms of resistance R,

current I, and potential difference V.

When potential difference and

current change with time

When potential difference and

current remain constant

MECHANICAL FORMS OF WORK

• There are two requirements for a work interaction between a system and its surroundings to exist:

– there must be a force acting on the boundary.

– the boundary must move.

Work = Force × Distance

When force is not constant

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The work done is proportional to the force

applied (F) and the distance traveled (s).

If there is no movement,

no work is done.

Shaft

Work

A force F acting through a

moment arm r generates a

torque T

This force acts through a distance s

The power transmitted through the shaft is

the shaft work done per unit time

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Energy transmission through rotating shafts is

commonly encountered in practice.

Shaft work is proportional to the

torque applied and the number of

revolutions of the shaft.

Determine the power transmitted through

the shaft of a car when the torque applied is

200 N m and the shaft rotates at a rate of

4000 revolutions per minute (rpm)

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Spring Work

Elongation of

a spring

When the length of the spring changes by

a differential amount dx under the influence of

a force F, the work done is

For linear elastic springs, the displacement x is

proportional to the force applied

k: spring constant (kN/m)

Substituting and integrating yield

x1 and x2: the initial and the final

displacements

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a spring

under the

influence of

a force.

The

displacement of

a linear spring

doubles when

the force is

doubled.

Moving boundary work (boundary work/ P dV work)

� Associated with the expansion & compression of a gas in a piston-cylinder

devise.

Piston is allowed to move a distance

ds in a quasi-equilib. manner, the

differential work done during process,

dVPPAdsdsFWb ===δ(valid for quasi-equilibrium process)

P = absolute pressure

dV = volume change (+ve expansion)

(-ve compression)

� To calculate the boundary work, the process by which the system changed

states must be known, i.e. functional relationship between P & V during

process.

� P = f(V) should be available: equation of the process path on a P-V diagram.

dVPPAdsdsFWb ===δ

The boundary work = Area under the process curve plotted on a PV

diagram

Total boundary work from initial

state to final state:

(kJ)∫=

2

1

PdVWb

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Note:P is the absolute pressure and is always positive.When dV is positive, Wb is positive.When dV is negative, Wb is negative

Each process gives a

different value for the

boundary work

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The boundary work done during a process depends on the path followed as

well as the end states.

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The net work done during a cycle is the

difference between the work done by the

system and the work done on the system

Some typical processes:

(a) Boundary work during a constant-volume process

If the volume is held constant, dV = 0, boundary work equation becomes

P 1

1

26

V

2

P-V diagram for V = Constant

01

2== ∫ PdVWb

(b) Boundary work for a constant-pressure process

P

V

2 1

( )12

2

1

2

1VVPdVPPdVW ob −=== ∫ ∫

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V

P-V diagram for P = constant

For the constant pressure process shown above, is the boundary work positive or negative and why?

(c) Isothermal compression of an ideal gas

If the temperature of an ideal gas system is held constant, then the equation of state provides the temperature-volume relation:

Then, the boundary work is

V

mRTP =

dVV

mRTPdVWb ∫∫ ==

2

1

2

1

28

Note: The above equation is the result of applying the ideal gas assumption for the equation of state. For real gas undergoing an isothermal process, the integral in the boundary work equation would be done numerically.

=

1

2lnV

VmRT

(d) Polytropic process

� During expansion & compression processes of gases, P & V are often related by: PVn = C where C are constants

� The value of n -∞ to + ∞ depending on process

� Some of the common values are given below:

Process Exponent nConstant pressure 0Constant volume ∞

Isothermal & ideal gas 1

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Isothermal & ideal gas 1 Adiabatic & ideal gas k = CP/CV

Cp = specific heat at constant pressure

Cv = specific heat at constant volume

Schematic & P-V diagram for a polytropic process:

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How to determine the boundary work for polytropic process?

∫=2

1PdVWb

nCVP

−=

dVV

Cn∫=

2

1

Example 4.1

A rigid tank contains air at 500 kPa and 150oC. As

a result of heat transfer to the surroundings, the

temperature and pressure inside the tank drop to

65oC and 400 kPa, respectively. Determine the

boundary work done during this process.boundary work done during this process.

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Example 4.2 A frictionless piston-cylinder device contains 5 kg of

steam at 400 kPa and 200oC. Heat is now transferred

to the steam until the temperature reaches 250oC. If

the piston is not attached to a shaft and its mass is

constant, determine the work done by the steam

during this process.during this process.

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Example 4.3

A piston-cylinder device initially contains 0.4 m3 of

air at 100 kPa and 80oC. The air is now compressed

to 0.1 m3 in such a way that the temperature inside

the cylinder remains constant. Determine the work

done during this process.done during this process.

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Example 4.4

A piston-cylinder device contains 0.05 m3 of a gas initially

at 200 kPa. At this state, a linear spring that has a spring

constant of 150 kN/m is touching the piston but exerting no

force on it. Now heat is transferred to the gas, causing the

piston to rise and to compress the spring until the volume piston to rise and to compress the spring until the volume

inside the cylinder doubles. If the cross-sectional area of

the piston is 0.25 m2, determine:

(a) The final pressure inside the cylinder

(b) The total work done by the gas

(c) The fraction of this work done against the spring to compress

it

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