steady heat transfer review

CM3120 Module 1 Lecture 2 1/20/2021

1

Module 1: Intro and Prerequisite Material

© Faith A. Morrison, Michigan Tech U.1

Professor Faith A. Morrison

Department of Chemical EngineeringMichigan Technological University

www.chem.mtu.edu/~fmorriso/cm3120/cm3120.html

Steady Heat Transfer Review

Module 1: Intro and Prerequisite Material

© Faith A. Morrison, Michigan Tech U.www.chem.mtu.edu/~fmorriso/cm3120/cm3120.html

Steady Heat Transfer Review

CM3120: Module 1


Introduction and Prereq Material

I. IntroductionII. Review of Prerequisite Material

a. Microscopic energy balancesb. Fourier’s law of heat conduction (𝑘, homogeneous)c. Newton’s law of cooling (ℎ, at a boundary)d. Resistances due to 𝑘 and ℎe. Solving for the steady temperature field 𝑇(𝑥,𝑦,𝑧)f. Dimensional analysis in heat transfer for ℎg. ℎ Data correlations for forced and free convectionh. ℎ For radiation heat transfer


2


Why study transport/unit ops?

Where are we in our study of

transport/O.U.?How far along did we get in CM3110 and other prerequisite courses?


We begin with a review:

• Microscopic energy balance• Fourier’s law of heat conduction (𝑘, homogeneous)• Newton’s law of cooling (ℎ, at a boundary)• Resistances due to 𝑘 and ℎ• Solving for the steady temperature field 𝑇 𝑥,𝑦, 𝑧• Dimensional analysis in heat transfer for ℎ• ℎ Data correlations for forced and free convection• ℎ For radiation heat transfer

CM3120: Unsteady State Heat Transfer/Mass Transfer/Unit Operations

CM3120 builds on these topics from the

prerequisites.CM3110 REVIEW


3

© Faith A. Morrison, Michigan Tech U.

Microscopic Energy Balance Review CM3110 REVIEW

We begin our prereq review

here


6

Equation of Thermal Energy

V

n̂dSS

Microscopic energy balance written on an arbitrarily shaped volume, V, enclosed by a surface, S

Gibbs notation:general conduction

Gibbs notation: Only Fourier conduction

Microscopic Energy Balance:

𝜌𝜕𝐸𝜕𝑡

𝑣 ⋅ 𝛻𝐸 𝛻 ⋅ 𝑞 𝑆

𝜌𝐶𝜕𝑇𝜕𝑡

𝑣 ⋅ 𝛻𝑇 𝑘𝛻 𝑇 𝑆

(incompressible fluid, constant pressure, neglect 𝐸 ,𝐸 , viscous dissipation )

The microscopicenergy balance is an expression of the law of conservation of energy.

http://pages.mtu.edu/~fmorriso/cm310/energy_equation.html

CM3110 REVIEW

Microscopic Energy Balance Review

It includes consideration of unsteady energy flows.


4


What physicsdetermines how rapidly (the rate) the heat transfers from one location to another?

𝑞𝐴

𝑘𝑑𝑇𝑑𝑥

Fourier’s Law of Heat Conduction

(the driving physics of Fourier’s law is Brownian motion: energy transports down 𝛻𝑇 due to Brownian motion)

heat flux=energy/area time)

𝒌 – thermal conductivity

temperature gradient

(for a homogeneous phase)

Energy Transport law

CM3110 REVIEW


𝑞𝐴

𝑘𝑑𝑇𝑑𝑥

8


Heat Transfer Rate law:

•Heat flows down a temperature gradient

•Flux is proportional to the magnitude of temperature gradient

Makes reference to a coordinate system

Allows you to solve for temperature profiles (also known as temperature distributions or fields)

Gibbs notation:𝑞

𝐴𝑘𝛻𝑇

𝑞𝑞

𝐴

𝑘𝜕𝑇𝜕𝑥

𝑘𝜕𝑇𝜕𝑦

𝑘𝜕𝑇𝜕𝑧

Fourier’s law in three

dimensions

CM3110 REVIEW


Fourier’s law of Heat Conduction


5


𝑣 ⋅ 𝛻𝑇 𝑘𝛻 𝑇 𝑆

Equation of Energy(microscopic energy balance)

see handout for component notation

rate of change

convection

conduction (all directions)

source

velocity must satisfy equation of motion, equation of continuity

(energy generated per unit volume per time)

© Faith A. Morrison, Michigan Tech U.http://pages.mtu.edu/~fmorriso/cm310/energy_equation.html

CM3110 REVIEW


Due to: electrical current; chemical reaction


10http://pages.mtu.edu/~fmorriso/cm310/energy_equation.html


6



Front side: • Micro E-balance in

terms of flux 𝑞 ≡• Fourier’s law, 𝑞 𝑘𝛻𝑇



Front side: • Micro E-balance in

terms of flux 𝑞 ≡• Fourier’s law, 𝑞 𝑘𝛻𝑇

Back side: • Micro E-balance in terms

of temperature (Fourier’s law incorporated)


7


Microscopic Energy Balance

http://pages.mtu.edu/~fmorriso/cm310/energy_equation.html

CM3110 REVIEW


Fourier’s Law of Heat Conduction

https://pages.mtu.edu/~fmorriso/cm310/energy.pdf

𝑞𝑞𝐴

CM3110 REVIEW


8



Now, BoundaryConditions and Resistances

16

We will needboundary conditionson temperature to solve the microscopic balances for the temperature distribution.

What is the steady state temperature profile in a rectangular slab if one side is held at T1 and the other side is held at T2?

Assumptions:•wide, tall slab•steady state

Example 1: Heat flux in a rectangular solid – Temperature BC

CM3110 REVIEW

Microscopic Energy Balance Review—Boundary Conditions

We may know the temperature at the boundary.

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A. M

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U.


9

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What is the steady state temperature profile in a rectangular slab if one side is held at T1 and the other side is held at T2?

Example 1: Heat flux in a rectangular solid – Temperature BC

CM3110 REVIEW


What if we don’t know the wall temperature?

We will needboundary conditionson temperature to solve the microscopic balances for the temperature distribution.

Assumptions:•wide, tall slab•steady state

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A. M

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CM3110 REVIEW


There may exist a resistance to heat transfer at the boundary, due to fluid characteristics

The interface between the solid and the fluid calls for a new type of boundary condition,Newton’s Law of Cooling.

Temperature offset is evidence of heat-transfer

resistance at the wall

solid wallbulk fluid

𝑇 𝑥𝑇

𝑇

𝑇 𝑥 in solid

𝑇 𝑇

𝑥𝑥

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Tech

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10

The temperature difference at the fluid-wall interface is caused by complex fluid phenomena that are

lumped together into the heat transfer coefficient, ℎ

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A. M

orris

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gan

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19

CM3110 REVIEW


Temperature offset is evidence of heat-transfer

resistance at the wall


𝑇 𝑥𝑇

𝑇

𝑇 𝑥 in solid

𝑇 𝑇

𝑥𝑥

𝒉

The interface between the solid and the fluid calls for a new type of boundary condition,Newton’s Law of Cooling.

This expression serves as the definition of the heat transfer coefficient.

𝑞𝐴

ℎ 𝑇 𝑇

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The heat flux at the wall is given by the empirical expression known as

Newton’s Law of Cooling

(a linear driving-force model for interphase heat transport)

𝒉 depends on:

•geometry•fluid velocity field•fluid properties•temperature

homogeneous solid

bulk fluid

bT

wallT

wallb TT What is the flux at the wall?

𝑣 𝑥,𝑦, 𝑧 0


© F

aith

A. M

orris

on, M

ichi

gan

Tech

U.


11

© F

aith

A. M

orris

on, M

ichi

gan

Tech

U.

21


rate of change

convection

conduction (all directions)

source• Microscopic

energy balance

• Fourier’s law of heat conduction

• Newton’s law of cooling (ℎ, at the phase boundary)

• Resistances due to 𝑘 and ℎ

This expression serves as the definition of the heat transfer coefficient.

Fourier’s Law of Heat Conduction(for a homogeneous phase)

Next

Review so far…

© F

aith

A. M

orris

on, M

ichi

gan

Tech

U.

22

Microscopic Energy Balance Review—Resistance to Heat Transfer

The language of resistance to describe the physics of heat transfer will be handy in our study of unsteady state temperature profiles. We encountered this language in CM3110, and we review and summarize now.

𝓡 Resistance to Heat Transfer

1. Limited conductivity within the homogeneous phase 𝑘

2. Limited heat transfer between phases at a boundary ℎ

Two limitations create resistance:

1. Are affected by geometry (rectangular versus radial)

2. Can be stacked (that is, added together like electrical resistances)

Also, resistances:

𝑞𝐴

driving force resistances∑

Note: Geankoplisuses a slightly different definition of resistance; we follow Bird et al. 2002.


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23

Thermal conductivity 𝑘 and heat transfer coefficient ℎ may be thought of as sources of resistance ℛ to heat transfer.

These resistances ℛ stack up in a logical way, allowing us to quickly and accurately determine the effect of adding insulating layers, encountering pipe fouling, and other applications.

𝑇

𝑘

r

𝑘

𝑇2𝑅

2𝑅

2𝑅

𝐵

x

0

𝑇

𝑇

𝐵/2 𝐵

𝑘

𝑘

𝑇

Using the microscopic energy balance on a test problem, we can solve for the temperature profile and then the heat flux, which is the driving force/resistance.

𝑞𝐴

driving force resistances∑

We can then identify the resistances for each test case considered.

1D Heat Transfer – Resistance

1D Rectangular: Door 𝒌𝟏 𝒌𝟐 , and Composite Door

𝐵

x

0

𝑇

𝑇

𝐵/2 𝐵𝑘 𝑘

1D Rectangular: Slab with Newton’s law BC

𝑞𝐴

𝑇 𝑇𝐵/2𝑘

𝐵/2𝑘

𝑞𝐴

𝑇 𝑇1ℎ

𝐵𝑘

1ℎ

𝑇

𝑘

r

𝑘

𝑇2𝑅

2𝑅

2𝑅

1D Radial: Pipe 𝒌𝟏 𝒌𝟐 and

Composite Pipe

𝑞𝐴

𝑇 𝑇1𝑘 ln

𝑅𝑅

1𝑘 ln

𝑅𝑅

1𝑟

1D Radial: Pipe with Newton’s law BC

𝑞𝐴

𝑇 𝑇1

ℎ 𝑅1𝑘 ln

𝑅𝑅

1ℎ 𝑅

1𝑟

RESISTANCE SUMMARY:



13

25© Faith A. Morrison, Michigan Tech U.

𝑞𝐴

𝑇 𝑇ℛ

1𝑟

driving forceresistance

Let: ℛ ≡ ln

𝑞𝐴

𝑇 𝑇ℛ


Let: ℛ ≡

Note: Geankoplis uses a different resistance. For rectangular heat flux:

𝑅 ℛ/𝐿𝑊

Note: Geankoplis uses a different resistance. For radial heat flux:

𝑅 ℛ/2𝜋𝐿

𝐵

x

𝑇

𝐵

𝑘

0

𝑇

𝑇 𝑥

R1

R2

r𝑘

Hot wall at 𝑇

Cooler wall at 𝑇

1D Rectangular

1D Radial

Temperatu

re Boundary C

onditio

ns


26© Faith A. Morrison, Michigan Tech U.

𝑞𝐴

𝑇 𝑇ℛ ℛ ℛ

1𝑟


𝑞𝐴

𝑇 𝑇ℛ ℛ ℛ


Let:

ℛ ≡ for 𝑖 1,2

ℛ ≡

1D Rectangular

1D Radial

New

ton’s Law

of C

oolin

g Boundary C

onditio

ns

𝐵

x

𝑇

𝐵

𝑘

0

𝑇𝑇 𝑥

ℎ

ℎ

Let:

ℛ ≡ for 𝑖 1,2

ℛ ≡ ln

𝑇

𝑇ℎ inside

ℎ outsideR1

R2

r𝑘

𝜃



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Now, “Slash and Burn”


28

For review, let’s carry out an example of 1D, steady heat transfer

CM3110 REVIEW

Microscopic Energy Balance—Solve for Temperature Field


15

Example 3: Heat Conduction with GenerationWhat is the steady state temperature profile in a wire if heat is generated uniformly throughout the wire at a rate of Se W/m3 and the bulk fluid surrounding the wire is at 𝑇 ? What is the heat flux?

R

r 𝑇

long wire

Se = energy production per unit volume




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Example: Heat conduction with generation

Let’s try.


ℎ heat transfer coefficient


16


31

In class solution will be posted with the other “hand notes.”

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

r/R or r/R1

Te

mp

era

ture

ra

tio

Conduction in pipe

Wire with generation

1

2

121

1 lnln

1

R

r

RRTT

TT

2

21

4/

R

r

kRS

TT

e

w

Compare radial conduction solutions


Radial conduction through pipe wallRadial conduction in wire with generation



17


Steady Heat‐Transfer Review Summary (thus far):

• Microscopic energy balance• Fourier’s law of heat conduction (𝑘, homogeneous)• Newton’s law of cooling (ℎ, at the boundary between two phases)• Resistances due to 𝑘 and ℎ; vary with boundary conditions (BC)

and geometry

• Solving for the steady temperature field 𝑇 𝑥,𝑦, 𝑧 , a.k.a. “Slash and Burn”

Unsteady State Heat Transfer

1𝑘

ln𝑅𝑅

𝐵𝑘

1𝑅ℎ

1ℎ

1D radial

1D rectangular

𝑻 BC 𝒉 BC

CM3110 REVIEW


Steady Heat‐Transfer Review Summary (thus far):

• Microscopic energy balance• Fourier’s law of heat conduction (𝑘, homogeneous)• Newton’s law of cooling (ℎ, at the boundary between two phases)• Resistances due to 𝑘 and ℎ; vary with boundary conditions (BC)

and geometry

• Solving for the steady temperature field 𝑇 𝑥,𝑦, 𝑧 , a.k.a. “Slash and Burn”

Unsteady State Heat Transfer

1𝑘

ln𝑅𝑅

𝐵𝑘

1𝑅ℎ

1ℎ

1D radial

1D rectangular

𝑻 BC 𝒉 BCSneak peak: The ratio of 𝑇 (internal) and ℎ(external) resistances is the Biot number:

𝑩𝒊𝐵/𝑘1/ℎ

𝒉𝑩𝒌

This is important in unsteady heat transfer.

CM3110 REVIEW


18



Finally, Dimensional analysis and data correlations for heat transfer coefficient 𝒉



For complex systems, we turn to data correlations based on dimensional analysis

The engineering quantity of interest in heat transfer is the amount of heat 𝒬 transferred

Nusselt number, a dimensionless heat transfer coefficient, is a dimensionless amount of heat 𝒬transferred

Dimensional Analysis?🤔


19

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37

Microscopic Energy Balance Review—Dimensional Analysis


𝑇 𝑥𝑇

𝑇

𝑇 𝑥 in solid

𝑇 𝑇

𝑥𝑥

𝒉

Heat Transfer Coefficient:• Linear driving force model

• Heat transfer between phases

CM3110 REVIEW

𝑞𝐴

ℎ 𝑇 𝑇

Review: What is Dimensional Analysis?


•Rough pipes

•Non-circular conduits

•Flow around obstacles (spheres, other complex shapes

Solution: Navier-Stokes, Re, Fr, 𝐿/𝐷, dimensionless wall force 𝑓; 𝑓 𝑓 Re, 𝐿/𝐷

Solution: Navier-Stokes, Re, dimensionless drag 𝐶 ; 𝐶 𝐶 Re

Solution: add additional length scale; then nondimensionalize

Solution: Use hydraulic diameter as the length scale of the flow to nondimensionalize

Solution: Two components of velocity need independent lengthscales

•Flow in pipes at all flow rates (laminar and turbulent)

•Boundary layers

38

Complex Heat Transfer – Dimensional Analysis CM3110 REVIEW


20


39

Complex Heat Transfer – Dimensional Analysis CM3110 REVIEWData Correlations:


40

Complex Heat Transfer – Dimensional Analysis CM3110 REVIEWData Correlations:

Dimensional analysis allows us to capture and

engineer around or with complex behavior


21

Following procedure familiar from pipe flow,

• What are governing equations?

• Scale factors (dimensionless numbers)?

• Quantity of interest?

Answer: Heat flux at the wall

Chosen problem: Forced Convection Heat TransferSolution: Dimensional Analysis

Complex Heat Transfer – Dimensional Analysis


41

First try: Forced Convection

How does Dimensional Analysis work in Heat Transfer?

CM3110 REVIEW

z-component of the Navier-Stokes Equation:

z

vv

v

r

v

r

vv

t

v zz

zzr

z

zzzz gz

vv

rr

vr

rrz

P

2

2

2

2

2

11

Choose:

𝑫 = characteristic length𝑽 = characteristic velocity𝑫/𝑽 = characteristic time𝝆𝑽𝟐 = characteristic pressure


42

• Choose “typical” values (scale factors)• Use them to scale the equations• Deduce which terms dominate

Dimensional Analysis in Forced Convection Heat Transfer

Pipe flow CM3110 REVIEW

FORCED CONVECTION


22

V

vv zz *

non-dimensional variables:

D

tVt *

D

zz *

D

rr *

2*

V

PP

g

gg zz *

time: position: velocity:driving force:

V

vv rr *

V

vv *


43


Pipe flow


CM3110 REVIEW

FORCED CONVECTION

Microscopic energy balance:


44


position:

oo

TTTT

T

1

*

temperature:

Choose:𝑇 use a

characteristic interval (since distance from absolute zero is not part of this physics)

𝑆 use a reference source, 𝑆

𝑟∗ ≡𝑟𝐷

𝑧∗ ≡𝑧𝐷

source:

𝑆∗ ≡𝑆𝑆

Energy


𝑣𝜕𝑇𝜕𝑟

𝑣𝑟𝜕𝑇𝜕𝜃

𝑣𝜕𝑇𝜕𝑧

𝑘1𝑟𝜕𝜕𝑟

𝑟𝜕𝑇𝜕𝑟

1𝑟𝜕 𝑇𝜕𝜃

𝜕 𝑇𝜕𝑧

𝑆

𝑆 ≡


FORCED CONVECTION

𝑇 surface𝑇 bulk


23

Complex Heat Transfer – Dimensional Analysis Forced Convection


45

Substitute all these definitions,

(𝑡∗, 𝑟∗, 𝑧∗, 𝑝∗,𝑔∗, 𝑣∗, 𝑣∗ , 𝑣∗,𝑇∗, 𝑆∗)

into the governing equations and simplify…

V

vv zz *


D

tVt *

D

zz *

D

rr *

2*

V

PP

g

gg zz *

time: position: velocity:driving force:

V

vv rr *

V

vv *

Forced Convection Heat Transfer

CM3110 REVIEW


Pipe flow

Microscopic energy balance:


CM3110 REVIEW


position:

oo

TTTT

T

1

*

temperature:

Choose:𝑇 use a

characteristic interval (since distance from absolute zero is not part of this physics)

𝑆 use a reference source, 𝑆

source:

Energy


𝑣𝜕𝑇𝜕𝑟

𝑣𝑟𝜕𝑇𝜕𝜃

𝑣𝜕𝑇𝜕𝑧

𝑘1𝑟𝜕𝜕𝑟

𝑟𝜕𝑇𝜕𝑟

1𝑟𝜕 𝑇𝜕𝜃

𝜕 𝑇𝜕𝑧

𝑆

𝑆 ≡

CM3110 REVIEW

FORCED CONVECTION

FORCED CONVECTION

𝜕𝑇∗

𝜕𝑡∗𝑣∗𝜕𝑇∗

𝜕𝑟∗𝑣∗

𝑟∗𝜕𝑇∗

𝜕𝜃𝑣∗𝜕𝑇∗

𝜕𝑧∗1

Pe1𝑟∗

𝜕𝜕𝑟∗

𝑟∗𝜕𝑇∗

𝜕𝑟∗1𝑟∗

𝜕 𝑇∗

𝜕𝜃𝜕 𝑇∗

𝜕𝑧𝑆∗

**2*

** 1Re1

gFr

vz

PDtDv

zz

Non-dimensional Energy Equation

Non-dimensional Navier-Stokes Equation

0*

*

*

*

*

*

z

v

y

v

x

v zyx

Non-dimensional Continuity Equation

ˆPe PrRe pC VD

k

ˆPr pC

k


Complex Heat Transfer – Dimensional Analysis Forced Convection

46𝐷𝑣𝐷𝑡

≡𝜕𝑣𝜕𝑡

𝑣𝜕𝑣𝜕𝑟

𝑣𝑟𝜕𝑣𝜕𝜃

𝑣𝜕𝑣𝜕𝑧

**2*

** 1Re1

gFr

vz

PDtDv

zz

*

𝑇∗ 𝑇∗ Re, Pr𝑣∗ 𝑣∗ Re, Fr

CM3110 REVIEW

FORCED CONVECTION

FORCED CONVECTION


24


47


For complex systems, we turn to data correlations for heat transfer coefficients based on dimensional analysis

The engineering quantity of interest is the amount of heat transferred

Nusselt number, a dimensionless heat transfercoefficient, is also a dimensionless amount of heat transferred

Dimensional Analysis?🤔

Next?


2𝜋𝑅𝐿 ℎ 𝑇 𝑇 𝒬 �̂� ⋅ 𝑞 𝑑𝑆


2𝜋𝑅𝐿 ℎ 𝑇 𝑇 𝒬 𝑘𝜕𝑇𝜕𝑟

𝑅𝑑𝑧𝑑𝜃

Now, non-dimensionalizethis expression as well.

48

𝑞𝐴

ℎ 𝑇 𝑇Linear driving force model

Apply in the fluid, at the surface:

CM3110 REVIEW

FORCED CONVECTION


25


position:

oo

TTTT

T

1

*

temperature:

Dr

r *

Dz

z *


Non-dimensionalize

49

Complex Heat Transfer – Dimensional AnalysisCM3110 REVIEW

𝑇 surface𝑇 bulk

ddzr

T

D

L

k

hDDL

r

2

0 0

*

21

*

*

*

2

ddzD

DTT

r

TkTTDLh

DLo

r

o

2

0 0

*2

1

21*

*

1 2*

Nusselt number, Nu(dimensionless heat-

transfer coefficient; dimensionless amount of heat transferred)

DL

TNuNu ,*



50

CM3110 REVIEW


26

𝜕𝑇∗

𝜕𝑡∗𝑣∗𝜕𝑇∗

𝜕𝑟∗𝑣∗

𝑟∗𝜕𝑇∗

𝜕𝜃𝑣∗𝜕𝑇∗

𝜕𝑧∗1

Pe1𝑟∗

𝜕𝜕𝑟∗

𝑟𝜕𝑇∗

𝜕𝑟∗1𝑟∗

𝜕 𝑇∗

𝜕𝜃𝜕 𝑇∗

𝜕𝑧𝑆∗

**2*

** 1Re1

gFr

vz

PDtDv

zz

**2*

** 1Re1

gFr

vz

PDtDv

zz



0*

*

*

*

*

*

z

v

y

v

x

v zyx

Non-dimensional Continuity Equation

ˆPe PrRe pC VD

k

Quantity of interest

ddzr

T

DLNu

DL

r

2

0 0

*

21

*

*

*/2

1

ˆPr pC

k


51


FORCED CONVECTION

FORCED CONVECTIONFORCED CONVECTION


According to our forced convection dimensional analysis calculations, the dimensionless heat transfer

coefficient should be found to be a function of fourdimensionless groups:

Now, do the experiments.

Peclet number

Pe ≡

Prandtl number

Pr ≡


52

(fluid properties)

CM3110 REVIEW

FORCED CONVECTION

(tentative)

Nu Nu Re, Pr, Fr,𝐿𝐷


27

no free surfaces


According to our forced convection dimensional analysis calculations and follow-up experiments, the

dimensionless heat transfer coefficient should be found to be a function of these four dimensionless groups:

Peclet number

Pe ≡

Prandtl number

Pr ≡


53

(fluid properties)

CM3110 REVIEW

FORCED CONVECTION

FORCED CONVECTION

Nu Nu Re, Pr, Fr,𝐿𝐷

,𝜇𝜇

Sometimes𝜇 𝑇 seen to be important

“it turns out…”


Forced convection Heat Transfer in Turbulent flow in pipes

Physical Properties (except 𝜇 ) evaluated at:

N𝑢ℎ 𝐷𝑘

0.027Re . Pr𝜇𝜇

.𝑇 , 𝑇 ,

2

*May have to be estimated

Forced convection Heat Transfer in Laminar flow in pipes

N𝑢ℎ 𝐷𝑘

1.86 RePr𝐷𝐿

𝜇𝜇

.

𝑇 , 𝑇 ,

2

54

Bulk mean temperature

Sieder-Tate equation (laminar flow)

Sieder-Tate equation (turbulent flow)

Data Correlations for Forced Convection Heat Transfer

𝑞 ℎ 𝐴Δ𝑇

Δ𝑇𝑇 𝑇 𝑇 𝑇

2

𝑞 ℎ 𝐴Δ𝑇

Δ𝑇Δ𝑇 Δ𝑇

lnΔ𝑇Δ𝑇

CM3110 REVIEW


28


55

Complex Heat Transfer – Dimensional Analysis—Forced Convection

Exam 1 Handout, Forced Convection Data Correlations


56

When the physics of the heat transfer changes, the correlations and dimensionless numbers change.

Free Convection i.e. hot air rises

•heat moves from hot surface to cold air (fluid) by radiation and conduction•increase in fluid temperature decreases fluid density•recirculation flow begins•recirculation adds to the heat-transfer from conduction and radiation

coupled heat and momentum transport

Complex Heat Transfer – Dimensional Analysis—Free Convection

new physics: Natural Convection

The methods don’t change,

however

CM3110 REVIEW


29

𝜕𝑇∗

𝜕𝑡∗𝑣∗𝜕𝑇∗

𝜕𝑥∗𝑣∗𝜕𝑇∗

𝜕𝑦∗𝑣∗𝜕𝑇∗

𝜕𝑧∗1Pr

𝜕 𝑇∗

𝜕𝑥∗

𝜕 𝑇∗

𝜕𝑦∗𝜕 𝑇∗

𝜕𝑧∗



0*

*

*

*

*

*

z

v

y

v

x

v zyx

Non-dimensional Continuity Equation Quantity of interest


57


NATURAL(FREE) CONVECTION

NATURAL CONVECTION

≡Grashof number

Gr ≡𝑔𝐿 �̅� �̅� 𝑇 𝑇

𝜇


𝐷𝑣∗

𝐷𝑡∇ 𝑣 ∗ 𝑔𝐿 �̅� �̅� 𝑇 𝑇

𝜇𝑇∗

Nu𝑑𝑇∗

𝑑𝑦∗ ∗𝑑𝑥∗𝑑𝑧∗

CM3110 REVIEW


According to our natural convection dimensional analysis calculations, the dimensionless heat transfer

coefficient should be found to be a function of twodimensionless groups:

Now, do the experiments.

Grashof number

Prandtl number

Pr ≡


58

(fluid properties)

CM3110 REVIEW

NATURAL CONVECTION

Nu Nu Gr, Pr Gr ≡

𝑔𝐿 �̅� �̅� 𝑇 𝑇𝜇



30

mmak

hLPrGrNu

Example: Natural convection from vertical planes and cylinders

•a,m are given in Table 4.7-1, page 278 Geankoplis for several cases•L is the height of the plate•all physical properties evaluated at the film temperature, Tf

2bw

f

TTT

FREE CONVECTION


Gr ≡𝑔𝐷 �̅� �̅�Δ𝑇

𝜇

Free convection correlations use the

film temperature for calculating the physical properties

Experimental Results:

59



60


Exam 1 Handout, Natural Convection Data Correlations


31

© F

aith

A.

Mor

rison

, M

ichi

gan

Tech

U.

61ref: BSL1, p581, 644

Dimensional analysis will be a key tool in the third transport field, diffusion/mass transfer (Module 3)

Dimensional Analysis

Dimensionless numbers from the Equations of Change

mom

entu

men

erg

yThese numbers tell us about the relative importance of the

terms they precede.

(microscopic balances)


ma

ss

Non-dimensional Continuity Equation (species A)




Radiation


32


Summary:Radiation

• Absorptivity, αgray body: 𝛼 constantblack body: 𝛼 1

• Emissivity, 𝜀𝑞 𝜀𝑞 ,

• Kirchoff’s law: 𝛼 ε

• Stefan-Boltzman law𝑞 ,

𝐴𝜎𝑇

𝜎 5.676 10𝑊

𝑚 𝐾

Geankoplis 4th ed., eqn 4.10-10 p304

ℎ𝜀| 𝜎 𝑇 𝑇

𝑇 𝑇

General properties:

Heat transfer coefficient:

Heat shields:

𝑞𝐴

1𝑁 1

𝜎 𝑇 𝑇2𝜀 1

Always use absolute temperature (Kelvin)

in radiation calculations.

NET Radiation energy going from surface 1 to surface 2:

𝑞 𝑞𝐴

𝜎 𝑇 𝑇1𝜀

1𝜀 1

𝑞𝐴

𝜀 𝜎 𝑇 𝑇

Net heat transfer to a body:

Stefan-Boltzman constant:

CM3110 REVIEW


Summary:

• Absorptivity, αgray body: 𝛼 constantblack body: 𝛼 1

• Emissivity, 𝜀𝑞 𝜀𝑞 ,

• Kirchoff’s law: 𝛼 ε

• Stefan-Boltzman law𝑞 ,

𝐴𝜎𝑇

𝜎 5.676 10𝑊

𝑚 𝐾

Geankoplis 4th ed., eqn 4.10-10 p304

ℎ𝜀| 𝜎 𝑇 𝑇

𝑇 𝑇

General properties:

Heat transfer coefficient:

Heat shields:

𝑞𝐴

1𝑁 1

𝜎 𝑇 𝑇2𝜀 1

Always use absolute temperature (Kelvin)

in radiation calculations.

NET Radiation energy going from surface 1 to surface 2:

𝑞 𝑞𝐴

𝜎 𝑇 𝑇1𝜀

1𝜀 1

𝑞𝐴

𝜀 𝜎 𝑇 𝑇

Net heat transfer to a body:

Stefan-Boltzman constant:

CM3110 REVIEW

Radiation


33


Microscopic Energy Balance Review CM3110 REVIEWModule 1: Intro and Prerequisite material

DONE!

CM3120: Module 1


Introduction and Prereq Material

I. IntroductionII. Review of Prerequisite Material

a. Microscopic energy balancesb. Fourier’s law of heat conduction (𝑘, homogeneous)c. Newton’s law of cooling (ℎ, at a boundary)d. Resistances due to 𝑘 and ℎe. Solving for the steady temperature field 𝑇(𝑥,𝑦,𝑧)f. Dimensional analysis in heat transfer for ℎg. ℎ Data correlations for forced and free convectionh. ℎ For radiation heat transfer

DONE!


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67

Module 2:

steady heat transfer review

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