Lecture 1

ARO-401-2

Heat, mass and Momentum Transfer

April 1, 2014

California State Polytechnic University, Pomona

Aerospace Engineering Department

ARO 401 Heat, Mass & Momentum Transfer Spring 2013

Instructor: Dr. Tony C. Lin

Office: Room 13-1229

Telephone: (909) 528-0493

(909) 869-2470 (Dept. office)

E-mail: tclin@csupomona.edu; lin.tony98@yahoo.com

Office hours: Tuesday/Thursday 11:30AM to 1:00Pm

Prerequisite: C or better in ARO 301.

Textbook: Theodore L. Bergman, A. S. Lavine, F. P. Incropera, and D. P. DeWitt (2011),

Fundamentals of Heat and Mass Transfer, 7th ed., John Wiley & Sons, New York, NY.

Course Outline:

1. Introduction

a. Why study heat transfer

b. Modes of heat transfer

c. Heat transfer rate equations

d. Conservation of energy

e. Surface energy balance

2. Introduction to conduction

a. Fourier's law of conduction

b. Thermal properties of matter

c. Derivation of the heat equation

d. Boundary and initial conditions

3. One-dimensional steady conduction

a. Planar wall

b. Thermal resistance

c. Contact resistance

d. Alternative conduction analysis

e. Cylindrical and spherical walls

f. Composite walls

g. Walls with internal heat generation

h. Heat transfer from fins

i. Pin, straight and annular fins

j. Fin effectiveness and efficiency

4. Two-dimensional steady conduction

a. Analytical approach - method of separation of variables

b. Graphical approach

c. Computational approach

d. Derivation of finite-difference equations

e. Matrix solution methods

f. Gauss-Seidel method

5. Transient conduction

a. Lumped capacitance method

b. Biot and Fourier numbers

c. Exact solutions for planar, cylindrical and spherical problems

d. The one-term approximation

e. Graphical solution - Heisler and Grober charts

f. Exact solutions for semi-infinite solids

g. Multi-dimensional problems

h. Discretization of the unsteady heat equation

i. Explicit formulation and stability

j. Implicit formulation and solution method

6. Introduction to convection

a. Heat, mass and momentum transfer

b. Velocity, thermal and concentration boundary layers

c. Boundary layer equations

d. Similarity parameters

e. Functional form of solutions

f. Heat and mass transfer analogy

g. Reynolds analogy

h. Chilton-Colburn analogies

7. Forced convection, external flows

a. Flat plate in parallel flow

b. Skin-friction coefficient, Nusselt no. and Sherwood no.

c. Average values of the boundary-layer parameters

d. Laminar, turbulent and mixed-flow cases

e. Cylinder in cross flow

f. Sphere in a flow

8. Forced convection, internal flows

a. Entrance and fully-developed regions

b. Critical Reynolds number

c. Darcy's friction factor and the Moody diagram

d. Constant surface heat flux case

e. Constant surface temperature case

f. Convection coefficient for laminar and turbulent flows

9. Introduction to radiation heat transfer

a. Fundamental concepts

b. Radiation intensity

c. Blackbody radiation

d. Stefan-Boltzmann law

e. Emissivity

f. Absorptivity

g. Reflectivity

h. Transmissivity

i. Kirchhoff's law

References:

1. Holman, J.P. (1990), Heat Transfer, 7th ed., McGraw-Hill, New York, NY.

1. White, F.M. (1984), Heat Transfer, 3rd ed., Addison Wesley, Reading, MA.

1. Rohsenow, W.M., Choi, H. (1961), Heat, Mass, and Momentum Transfer, Prentice-Hall,

Englewood Cliffs, NJ.

Grading: Based on a weighted average of the following items.

Homework 10%

Course notebook 5%

Course conduct 5%

Computer assignment 15%

Quizzes 25%

Midterm Exams (2) 20%

Final Exam 20%

Quizzes will be closed-book, closed-notes, emphasizing basic concepts. Exams will be

open-book, closed-notes, emphasizing quantitative- and derivation-type problems.

What is “Heat” ?

A Form of Energy

• Heat or thermal energy flows from a region of higher temperature to a

Region of lower temperature

Heat

•Heat is defined as the transfer of energy across the boundary of a

system due to a temperature difference between the system and its

surroundings.

•Heat, internal energy, and temperature are all different quantities.

– Be sure to use the correct definition of heat.

– You cannot talk about the “heat of a system,” you can refer to heat only when energy has been transferred as a result of a temperature difference.

9

• Energy can exist in numerous forms such as:

– thermal,

– mechanical,

– kinetic,

– potential,

– electrical,

– magnetic,

– chemical,

– nuclear.

• Their sum constitutes the total energy E (or e on a unit mass basis) of a system.

• The sum of all microscopic forms of energy is called the internal energy of a system.

HEAT AND OTHER FORMS OF ENERGY

Thermodynamics M. D. Eastin

Forms of Energy

Energy comes in a variety of forms…

Potential

Mechanical Chemical Electrical

Internal Kinetic

Heat

Herein, we will study internal, mechanical, kinetic, and heat energy

What is “Work” ?

A Form of Energy

• Mechanical work = Force * distance

sFW

Internal Energy

•Internal energy includes matter molecular motions

– Random translational motion

– Rotational motion

– Vibrational motion

•Internal energy also includes potential energy between molecules.

13

THERMODYNAMICS AND HEAT TRANSFER

• Heat: The form of energy that can be transferred from one system

to another as a result of temperature difference.

• Thermodynamics is concerned with the amount of heat transfer as

a system undergoes a process from one equilibrium state to

another.

• Heat Transfer deals with the determination of the rates of such

energy transfers as well as variation of temperature.

• The transfer of energy as heat is always from the higher-

temperature medium to the lower-temperature one.

• Heat transfer stops when the two mediums reach the same

temperature.

• Heat can be transferred in three different modes:

conduction, convection, radiation

Internal Energy and Enthalpy

• In the analysis of systems that involve fluid flow, we frequently encounter the combination of properties u and Pv.

• The combination is defined as enthalpy (h = u + Pv).

• The term Pv represents the flow energy of the fluid (also called the flow work).

1

Historical Background

•Thermodynamics and mechanics (mass/momentum transfer) were considered to be distinct branches of physics.

– Until about 1850

– Experiments by James Joule and others showed a connection between them.

•A connection was found between the transfer of energy by heat in thermal processes and the transfer of energy by work in mechanical processes.

•The concept of energy was generalized to include internal energy.

•The principle of conservation of energy emerged as a universal law of nature.

Thermodynamics – Historical Background

•Thermodynamics and mechanics were considered to be distinct branches of physics.

– Until about 1850 – Experiments by James Joule and others showed a

connection between them.

•A connection was found between the transfer of energy by heat in thermal processes and the transfer of energy by work in mechanical processes. •The concept of energy was generalized to include internal energy and the conversion of work to heat. •The principle of conservation of energy emerged as a universal law of nature.

wqe

James Prescott Joule

•1818 – 1889

•British physicist

•Largely self-educated

– Some formal education

from John Dalton

•Research led to establishment

of the principle of conservation

of energy

•Determined the amount of

work needed to produce one

unit of energy

Unification of dynamics (F =ma) and heat transfer/thermal science

Mechanical Equivalent of Heat

•Joule established the equivalence between mechanical energy and internal energy.

•His experimental setup is shown at right.

•The decrease in potential energy associated of the system as the blocks fall equals the work done by the paddle wheel on the water.

Mechanical Equivalent of Heat

•Joule found that it took approximately 4.18 J of mechanical energy to raise the water 1oC.

•Later, more precise measurements determined the amount of mechanical energy needed to raise the temperature of water from 14.5oC to 15.5oC is 1 cal

•1 cal = 4.186 J

– This is known as the mechanical equivalent of heat.

• A more proper name would be the equivalence between mechanical energy and internal energy, but the historical name is well entrenched.

– 1 Joule = 1(kg*m/s2)*m=1 N* m= 1Watt*s

• Joule developed the absolute scale of temperature, made observation on magnetrostriction.

• Joule’s law: Heat generated on a electric circuit: Q = (I2R)t

( where I =current, R = circuit resistance, t =time)

• Joule led to the theory of conservation of energy, which led to the development of the first law of thermodynamics

Units of Heat

•Historically, the calorie was the unit used for heat.

– One calorie is the amount of energy transfer necessary to

raise the temperature of 1 g of water from 14.5oC to

15.5oC.

• The “Calorie” used for food is actually 1 kilocalorie.

•In the US Customary system, the unit is a BTU (British Thermal

Unit).

– One BTU is the amount of energy transfer necessary to

raise the temperature of 1 lb of water from 63oF to 64oF.

•The standard in the textbook is to use Joules.

• 1 cal = 4.186 J.

• 1 Btu (British thermal unit) = 1055.06 J = 2.931 10-4 kWh = 0.252 kcal

• 1 BTU (British Thermal unit) = 778.16 ft.lbf = 1.0551010 ergs = 0.293 watt-hours

• The most common units for heat are

-------- BTU (Btu) - British Thermal Unit

-------- Calorie

--------- Joule

• The calorie was first defined by Nicolas Clément in 1824 as a unit of heat,

The word comes from Latin calor meaning "heat".

• A calorie (cal) is the amount of heat required to raise the temperature of 1 gram

of water by 1° C (at sea level).

• Joule is a unit of energy equal to the work done when a force of one newton acts

through a distance of one meter

• the amount of heat required to raise the temperature of one pound of water

through 1oF (58.5oF - 59.5oF) at sea level (30 inches of mercury).

Units of Heat

Temperature and Heat

• Kinetic energy of the molecules is the

“energy” of motion

• “Temperature” is the measure of the average

kinetic energy of molecules’ motions

Kelvin scale

• The Kelvin scale is a metric temperature scale

measured in Kelvin units (K)

• Formula (273+ºC)= Kelvin

The important parameters

• All of thermodynamics can be expressed in

terms of four quantities

– Temperature (T)

– Internal Energy (U)

– Entropy (S)

– Heat (Q)

• These quantities will be defined as we progress

through the lesson

The basic Laws in Thermodynamics

(3 laws+ a - zeroth law)

According to British scientist C. P. Snow, the

three laws of thermodynamics can be

(humorously) summarized as

1. You can’t win (1st law)

2. You can’t even break even (2nd law)

3. You can’t get out of the game (3rd law)

FIRST LAW OF THERMODYNAMICS

∆E = q + w

heat energy transferred

energy change

work done by the system

Energy is conserved!

You can’t win (1st law)

• The first law of thermodynamics is an

extension of the law of conservation of energy

• The change in internal energy of a system is

equal to the heat added to the system minus the

work done by the system

ΔU = Q - W

Slide courtesy of NASA

First Law of Thermodynamics)

Process Terminology

• Adiabatic – no heat transferred

• Isothermal – constant temperature

• Isobaric – constant pressure

• Isochoric – constant volume

Heat Capacity

• The amount of heat required to raise a certain

mass of a material by a certain temperature is

called heat capacity

Q = mcxΔT

• The constant cx is called the specific heat of

substance x, (SI units of J/kg·K)

You can’t break even

(2nd Law of Thermodynamics)

• Think about what it means to not “break

even”. Every effort you put forth, no matter

how efficient you are, will have a tiny bit of

waste.

• The 2nd Law can also be stated that heat flows

spontaneously from a hot object to a cold

object (spontaneously means without the assistance of external work)

– This implies there is a direction of heart flows (one-way street)

Second Law of Thermodynamics

S2 = S1 S2 > S1

Concerning the 2nd Law

• The second law of thermodynamics introduces the notion of entropy (S), a measure of system disorder (messiness)

Direction of a Process

• The 2nd Law helps determine the preferred

direction of a process

• A reversible process is one which can change

state and then return to the original state

• This is an idealized condition – all real

processes are irreversible

A restatement of the Second Law which turns out to be

equivalent:

Heat will not flow from a colder body to a hotter body unless

some other process (which does work) is also involved.

Another restatement to be discussed next time:

The entropy of an isolated system can only increase or

remain constant. Its entropy cannot decrease.

Heat Engine

• A device which transforms heat into work is

called a heat engine

• This happens in a cyclic process

• Heat engines require a hot reservoir to supply

energy (QH) and a cold reservoir to take in the

excess energy (QC)

– QH is defined as positive, QC is less than QH

A schematic representation of a

heat engine. Heat is taken in at

high temperatures, TH. Some

heat is converted to work, and

the remainder is released at a

lower temperature, TC.

0CQ

WQQ CH CH TT

CH TT

You can’t get out

(3rd Law of Thermodynamics ) • No system can reach absolute zero temperature

• This is one reason we use the Kelvin temperature scale. Not only is the internal energy proportional to temperature, but you never have to worry about dividing by zero in an equation!

• There is no formula associated with the 3rd Law of Thermodynamics

– The 3rd law defines the “zero” temperature (0K)

Third Law of Thermodynamics

• The third law of thermodynamics is sometimes stated

as follows:

The entropy of a perfect crystal at absolute zero

is exactly equal to zero.

• In 1912 Nernst stated the law as : "It is impossible for

any procedure to lead to the isotherm T = 0 in a finite

number of steps."

Implications of 3rd Law

• MIT researchers achieved 450 picokelvin in 2003

(less than ½ of one billionth!)

• Molecules near these temperatures have been called

the fifth state of matter: Bose-Einstein Condensates

– Awesome things like super-fluidity and super-

conductivity happen at these temperatures

• K 0 (thermal conductivity)

• 0 (gas viscosity)

– Exciting frontier of research

Helium II will "creep" along surfaces (anti-gravity and anti-surface tension)

Superfluidity of liquid helium

The Third Law of Thermodynamics

• The third law: The entropy of a perfect crystal at

0 degree Kelvin is zero.

• The third law provides the reference state for use

in calculating absolute entropies.

What is a Perfect Crystal?

Perfect crystal at 0 K Crystal deforms at T > 0 K

Zeroth Law of Thermodynamics (thermodynamic equilibrium)

Thermodynamics M. D. Eastin

Energy Conservation

The First Law of Thermodynamics states that total energy is conserved for any

thermodynamic system → energy can not be created nor destroyed

→ energy can only change from one form to another

constant)( EEnergy

constantelectricalchemicalheat

mechanicalpotentialkineticinternal

EEE

EEEE

Our main concern in this course are : internal, mechanical, and heat

Thermodynamics M. D. Eastin

Internal Energy = Kinetic Energy + Potential Energy

(of the molecules in the system)

• Depends only on the current system state (p,V,T)

• Does not depend on past states

• Does not depend on how state changes occur

• Changes are the result of external forcing

on the system (in the form of work or heat)

First Law of Thermodynamics

tenvironmentenvironmeninternal Heat WorkE

dQ dW dU

dQ pdVdU

Thermodynamics M. D. Eastin

Joule’s experiments

Valve

Closed

Air Vacuum

Thermally Insulated System

Thermodynamics M. D. Eastin

Joule’s experiments

Thermally Insulated System

Valve

Open

Air Air

Joule’s experiments

dQ pdVdU

Valve

Open

Air Air

• Air expanded to fill the container

• Change in volume

• Change in pressure

• No external work was done

• Air expanded into a vacuum

within the system

• No heat was added or subtract

• Thermally insulated system

• No change in internal energy

• No change in temperature

What does this mean?

This is essence of 1st law of

thermodynamics 0dU

Thermodynamics M. D. Eastin

First Law of Thermodynamics

Valve

Open

Air Air

What energy

transformations occur as

air parcels move around

within thunderstorms?

Heat Transfer

• The movement of heat is from a warmer object to a

colder one

• Heat transfer process can be quantified by appropriate

“rate equation”

----- These equations are used to compute the

amount of energy being transferred per unit time

Forms of heat transfer

• Three forms (Modes) of heat transfer:

– Conduction

– Convection

– Radiation

53

Modes of heat transfer

• Conduction: diffusion of heat due to temperature gradients. A

measure of the amount of conduction for a given gradient is the

heat conductivity.

• Convection: when heat is carried away by moving fluid. The

flow can either be caused by external influences, forced

convection; or by buoyancy forces, natural convection.

Convective heat transfer is tightly coupled to the fluid flow

solution.

• Radiation: transfer of energy by electromagnetic waves

between surfaces with different temperatures, separated by a

medium that is at least partially transparent to the (infrared)

radiation. Radiation is especially important at high

temperatures, e.g. during combustion processes, but can also

have a measurable effect at room temperatures.

Heat Transfer

• The science of how heat flows is called heat transfer.

• There are three ways heat transfer works: conduction, convection, and radiation.

• Heat flow depends on the temperature difference.

Thermal Equilibrium

• Two bodies are in thermal equilibrium with each other when they have the same temperature.

• In nature, heat always flows from hot to cold until thermal equilibrium is reached.

Heat Conduction

Key Question:

How does heat pass through different materials?

Conduction

When you heat a metal strip at one end, the heat

travels to the other end.

As you heat the metal, the particles vibrate, these

vibrations make the adjacent particles vibrate, and so on

and so on, the vibrations are passed along the metal and

so is the heat. We call this? Conduction

Metals are different

The outer e______ of metal atoms drift, and are free to move.

When the metal is heated, this ‘sea of electrons’ gain kinetic energy and transfer it throughout the metal.

Insulators, such as wood does not have these sea of electrons’ which is why they do not conduct heat as well as metals.

lectrons

Heat Conduction

• Conduction is the transfer of heat through materials by the direct contact of matter.

• Dense metals like copper and aluminum are very good thermal conductors.

Heat Conduction

• A thermal insulator is a material that conducts heat poorly.

• Heat flows very slowly through the plastic so that the temperature of your hand does not rise very much.

Heat Conduction

• Styrofoam gets its insulating ability by trapping spaces of air in bubbles.

• Solids usually are better heat conductors than liquids, and liquids are better conductors than gases.

Heat Conduction Equation (Fourier’s Law)

q = -k A (T2 -T1)

L

Area of cross section (m2)

Length (m)

Thermal conductivity

(watts/moC)

Heat flow

(watts)

Temperature

difference (oC) Tkq In vector form:

Convection

Can moving matter carry

thermal energy?

4 processes of convection:

(a) forced convection;

(b) Free or natural convection;

(c) Boiling

(d) Condensation

Convection

• Convection is the transfer of heat by the motion of liquids and gases.

– Convection in a gas occurs because gas expands when heated.

– Convection occurs because currents flow when hot gas rises and cool gas sink.

– Convection in liquids also occurs because of differences in density.

Convection

• When the flow of gas or

liquid comes from

differences in density and

temperature, it is called

free convection.

• When the flow of gas or

liquid is circulated by

pumps or fans it is called

forced convection.

Convection

• Convection depends on

surface area.

• If the surface contacting

the fluid is increased, the

rate of heat transfer also

increases.

• Almost all devices made

for convection have fins

for this purpose.

Free Convection and Sea Breezes

• On a smaller scale near coastlines, convection is responsible for sea breezes.

• During the daytime, land is much hotter than the ocean.

• A sea breeze is created when hot air over the land rises due to convection and is replaced by cooler air from the ocean.

• At night the temperature reverses so a land breeze occurs.

Convection Currents

• Much of the Earth’s climate is regulated by giant convection currents in the ocean.

Heat Convection Equation

(Newton’s Law of cooling)

q = h A (T2 -T1)

Area contacting fluids (m2) Heat transfer coefficient

(watts/m2oC)

Heat flow

(watts)

Temperature

difference (oC)

Radiation

Key Question:

How does heat from

the sun get to Earth?

Radiation Heat Transfer (Black Body Radiation)

• Blackbody – a perfect emitter & absorber of radiation

• Emits radiation uniformly in all directions – no directional

distribution – it’s diffuse

• Joseph Stefan (1879)– total radiation emission per unit

time & area over all wavelengths and in all directions:

• =Stefan-Boltzmann constant =5.67 x10-8 W/m2K4

24 mW TEb

73

Radiation with surrounding

Stefan-Boltzman law [ W / m2 ]

qrad = hr A (Ts – Tsur) [ W ]

radiation heat transfer coefficient,

Stefan Boltzman const. , σ = 5.67 x 10-8 [ W/m2.K4 ]

ε = emissivity (grey body), =1 for black body

44

surs TTq

22

surssursr TTTTh

Conduction

• Conduction involves the transfer of

heat through direct contact

• Heat conductors conduct heat well,

insulators do not

76

Heat conduction - Fourier’s law

• The heat flux is proportional to the temperature gradient:

where k(x,y,z,T) is the thermal conductivity.

• In most practical situations conduction, convection, and radiation appear in combination. Also for convection, the heat transfer coefficient is important, because a flow can only carry heat away from a wall when that wall is conducting.

x

hot wall cold wall dx

dT

temperature

profile

Qq k T

A

Tkq

vectoraisqqNote

:

77

Tbody

T

TAhTTAhAqQbody

)(

average heat transfer coefficient (W/m2-K) h

q

Newton’s law of cooling – Convection

• Newton described the cooling of objects with an arbitrary

shape in a pragmatic way. He postulated that the heat transfer

Q is proportional to the surface area A of the object and a

temperature difference T.

• The proportionality constant is the heat transfer coefficient

h(W/m2-K). This empirical constant lumps together all the

information about the heat transfer process that we don’t know

or don’t understand.

78

Convection heat transfer

• Convection is movement of heat with a fluid.

• e.g., when cold air sweeps past a warm body, it draws

away warm air near the body and replaces it with cold

air.

flow over a

heated block

tcoefficientransferheath

ThTThq

lawcoolingsNewtonTransferheatConvection

ambientwall

)()(

':

79

Forced convection example

• Developing flow in a pipe (constant wall

temperature).

T wT T wT T wT

T

wT

x

bulk fluid temperature

heat flux from wall

T

wT

80

Natural convection around a person

• Light weight warm air tends to

move upward when surrounded by

cooler air.

• Thus, warm-blooded animals are

surrounded by thermal plumes of

rising warm air.

• This plume is made visible by

means of a Schlieren optical system

that is based on the fact that the

refraction of light through a gas is

dependent on the density of the gas.

• Although the velocity of the rising

air is relatively small, the Reynolds

number for this flow is on the order

of 3000.

Radiation Heat Transfer (Black Body Radiation)

• Blackbody – a perfect emitter & absorber of radiation

• Emits radiation uniformly in all directions – no directional

distribution – it’s diffuse

• Joseph Stefan (1879)– total radiation emission per unit

time & area over all wavelengths and in all directions:

• =Stefan-Boltzmann constant =5.67 x10-8 W/m2K4

24 mW TEb

82

Radiation with surrounding

Stefan-Boltzman law [ W / m2 ]

qrad = hr A (Ts – Tsur) [ W ]

radiation heat transfer coefficient,

Stefan Boltzman const. , σ = 5.67 x 10-8 [ W/m2.K4 ]

ε = emissivity (grey body), =1 for black body

44

surs TTq

22

surssursr TTTTh

• Energy balance at the surface:

– since a control surface is a special control volume that

contains “zero” volume, energy generation and storage terms

are zero; this leaves:

0 outin EE

Surface Energy Balance

Ein

Eout

84

The Surface Energy Balance

Ėin - Ėout = 0

q″cond - q″conv - q″rad = 0

Surface Energy Balance

Since no volume or mass is encompassed by the control surface.

Conservation Energy (Instant in Time):

(1.12)

• Applies for steady-state and transient conditions

Consider surface of wall with heat transfer by conduction, convection and radiation.

0cond conv radq q q

4 41 22 2 2 0sur

T Tk T T T T

L

h

• With no mass and volume, energy storage and generation are not pertinent to the energy balance, even if they occur in the medium bounded by the surface.

THE SURFACE ENERGY BALANCE

0 outin EE

• Read Chapter 1 Introduction

• Homework Problem PS1

• Chapter 1

• Problem 19

• Problem 30

• Problem 44

• Problem 65

• Due on April 10, 2013

1.30 A spherical interplanetary probe of 0.5-m diameter contains electronics that dissipate

150 W. If the probe surface has an emissivity of 0.8 and the probe does not receive

radiation from other surface, as, for example, from the sun, what is its surface

temperature ?