Complex Heat Transfer – Dimensional Analysisfmorriso/cm310/lectures/2019 heat lecture 17 old heat 04.pdfComplex Heat Transfer –Dimensional Analysis 𝜕𝑇∗ 𝜕𝑡∗ 𝑣

Lecture 17 CM3110 Heat Transfer 12/17/2019

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© Faith A. Morrison, Michigan Tech U.

CM3110 Transport/Unit Ops IPart II: Heat Transfer

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Complex Heat Transfer –Dimensional Analysis

Professor Faith Morrison

Department of Chemical EngineeringMichigan Technological University

(Forced convection heat transfer)

© Faith A. Morrison, Michigan Tech U.2

(what have we been up to?)1D Heat Transfer

Examples of (simple, 1D) Heat Conduction


2

© Faith A. Morrison, Michigan Tech U.3

But these are highly simplified

geometries

Examples of (simple, 1D) Heat Conduction

1D Heat Transfer

How do we handle complex geometries, complex flows, complex machinery?


inQonsW ,

Process scale

1T

2T

1T 2Tcold less cold

less hot

hot

1T

2T

1T 2Tcold less cold

less hot

hot

4

Complex Heat Transfer


3

Engineering Modeling


•Choose an idealized problem and solve it

•From insight obtained from ideal problem, identify governing equations of real problem

•Nondimensionalize the governing equations; deduce dimensionless scale factors (e.g. Re, Fr for fluids)

•Design experiments to test modeling thus far

•Revise modeling (structure of dimensional analysis, identity of scale factors, e.g. add roughness lengthscale)

•Design additional experiments

•Iterate until useful correlations result

inQonsW ,

Process scale

1T

2T

1T 2Tcold less cold

less hot

hot

1T

2T

1T 2Tcold less cold

less hot

hot

(Answer: Use the same techniques we have been using in fluid mechanics)

Complex Heat Transfer – Dimensional Analysis

5

Experience with Dimensional Analysis thus far:


•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

6



4


Turbulent flow (smooth pipe) Rough pipe

Around obstaclesNoncircular cross section

f

Re

Spheres, disks, cylinders

7

McC

abe

et a

l., U

nit O

ps o

f Che

mE

ng, 5

th e

diti

on, p

147


Turbulent flow (smooth pipe) Rough pipe

Around obstaclesNoncircular cross section

f

Re

Spheres, disks, cylinders

These have been exhilarating victories

for dimensional analysis

8

McC

abe

et a

l., U

nit O

ps o

f Che

mE

ng, 5

th e

diti

on, p

147


5


Now, move to heat transfer:•Forced convection heat transfer from fluid to wall

•Natural convection heat transfer from fluid to wall

•Radiation heat transfer from solid to fluid

Solution: ?

Solution: ?

Solution: ?

solid wallbulk fluid solid wall

9


solid wallbulk fluid





Solution: ?

Solution: ?

Solution: ?

We have already started using the results/techniques

of dimensional analysis through defining the heat

transfer coefficient, ℎ

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6






Solution: ?

Solution: ?

Solution: ?

We have already started using the results/techniques

of dimensional analysis through defining the heat

transfer coefficient, ℎ

(recall that we did this in fluids too: we used the 𝑓 Re correlation

(Moody chart) long before we knew where that all came from)

11


12

x

bulkT

wallT

wallx


wallbulk TT

The temperature variation near-wall region is caused by complex phenomena that are lumped together into the heat

transfer coefficient, h


Handy tool: Heat Transfer Coefficient

𝑇 𝑥 in solid

𝑇 𝑥 in liquid


7

13

xbulk wall

qh T T

A

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

Newton’s Law of Cooling

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


𝒉 depends on:

•geometry•fluid velocity•fluid properties•temperature difference


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𝒉 depends on:

•geometry•fluid velocity•fluid properties•temperature difference


To get values of ℎ for various situations, we need to measure

data and create data correlations (dimensional analysis)

xbulk wall

qh T T

A

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

Newton’s Law of Cooling

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



8


•Forced convection heat transfer from fluid to wall



Solution: ?

Solution: ?

Solution: ?

• The functional form of ℎ will be different for these three situations (different physics)

• Investigate simple problems in each category, model them, take data, correlate

Complex Heat transfer Problems to Solve:

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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

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9

General Energy Transport Equation(microscopic energy balance)

V

n̂dSS

As for the derivation of the microscopic momentum balance, the microscopic energy balance is derived on an arbitrary volume, V, enclosed by a surface, S.

STkTvtT

Cp

2ˆ

Gibbs notation:

see handout for component notation


17


General Energy Transport Equation(microscopic energy balance; in the fluid)

see handout for component notation

rate of change

convection

conduction (all directions)

source

fluid velocity must satisfy equation of motion, equation of continuity

(energy generated

per unit volume per

time)

STkTvt

TCp

2ˆ


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10

Note: this handout is also on the

web


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


R1

Example: Heat flux in a cylindrical shell

Assumptions:•long pipe•steady state•k = thermal conductivity of wall•h1, h2 = heat transfer coefficients

What is the steady state temperature profile in a cylindrical shell (pipe) if the fluid on the inside is at Tb1 and the fluid on the outside is at Tb2? (Tb1>Tb2)

Cooler fluid at Tb2

Hot fluid at Tb1

R2

r

** REVIEW ** REVIEW **

Fo

rced

-co

nve

ctio

n h

eat

tran

sfer


20


11

Consider: Heat-transfer to from flowing fluid inside of a tube – forced-convection heat transfer

T1= core bulk temperatureTo= wall temperature

T(r,,z) = temp distribution in the fluid


In principle, with the right math/computer tools, we could calculate the complete temperature and velocity profiles in the

moving fluid.

Now: How do develop correlations for h?

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How is the heat transfer coefficient related to the full

solution for T(r,,z) in the fluid?

What are governing equations?

Microscopic energy balance plus Navier-Stokes, continuity

Scale factors?

Re, Fr, L/D plus whatever comes from the rest of the analysis

Quantity of interest (like wall force, drag)?

Heat transfer coefficient

The quantity of interest in forced-convection heat

transfer is h


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12

r

R

1T

0Tpipe w

allfluid

),,( zrT

Unknown function:


𝑇 𝑟 is an unknown function

𝑇 𝑟

Assume: • 𝜃 symmetry• Long tube

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𝒬 𝑛 ⋅ 𝑞 𝑑𝑆

ℎ 𝑇 𝑇

𝒬 2𝜋𝑅𝐿 ℎ 𝑇 𝑇

𝑞𝑞𝐴

𝑘𝜕𝑇𝜕𝑟

At the boundary, (Newton’s Law of Cooling is the boundary condition)

We can calculate the total heat transferred from 𝑇 𝑟 in the fluid:

We need 𝑇 𝑟in the fluid

Total heat conducted to the

wall from the fluid

Total heat flow through (at) the wall

in terms of h

24



13


2𝜋𝑅𝐿 ℎ 𝑇 𝑇 𝒬 �̂� ⋅ 𝑞 𝑑𝑆

Equate these two: Total heat flow through the wall

2𝜋𝑅𝐿 ℎ 𝑇 𝑇 𝒬 𝑘𝜕𝑇𝜕𝑟

𝑅𝑑𝑧𝑑𝜃

Total heat flow at the wall in terms of h

Total heat conducted to the wall from the fluid

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2𝜋𝑅𝐿 ℎ 𝑇 𝑇 𝒬 �̂� ⋅ 𝑞 𝑑𝑆

Equate these two: Total heat flow through the wall

2𝜋𝑅𝐿 ℎ 𝑇 𝑇 𝒬 𝑘𝜕𝑇𝜕𝑟

𝑅𝑑𝑧𝑑𝜃

Now, non-dimensionalizethis expression

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14

non-dimensional variables:

position:

o

o

TTTT

T

1

*

temperature:

Dr

r *

Dz

z *


Non-dimensionalize

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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)

DL

TNuNu ,*

one additional dimensionless group


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15

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)

DL

TNuNu ,*

one additional dimensionless group


This is a function of Rethrough fluid 𝑣 distribution

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𝜕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

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

Quantity of interest

ddzr

T

DLNu

DL

r

2

0 0

*

21

*

*

*/2

1

ˆPr pC

k


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16

no free surfaces

D

LNuNu ,FrPr,Re,


According to our dimensional analysis calculations, the dimensionless heat transfer coefficient should be found to

be a function of four dimensionless groups:

Now, do the experiments.

Peclet number

Pe ≡

Prandtl number

Pr ≡

31

three



Forced Convection Heat Transfer

Now, do the experiments.

• Build apparatus (several actually, with different D, L)

• Run fluid through the inside (at different 𝑣; for different fluids 𝜌, 𝜇,𝐶 , 𝑘)

• Measure 𝑇 on inside; 𝑇 on inside

• Measure rate of heat transfer, 𝑄

• Calculate ℎ: 𝑄 ℎ𝐴 𝑇 𝑇

• Report ℎ values in terms of dimensionless correlation:

Nuℎ𝐷𝑘

𝑓 Re, Pr,𝐿𝐷 It should only be a function of

these dimensionless numbers (if our Dimensional Analysis is

correct…..)32



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© Faith A. Morrison, Michigan Tech U.Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)33

Complex Heat Transfer – Dimensional AnalysisRe

Re

Sieder and Tate equation

(laminar flow)

Sieder and Tate equation

(turbulent flow)

Effect of viscosity ratio

© Faith A. Morrison, Michigan Tech U.Sieder & Tate, Ind. Eng. Chem. 28 227 (1936) 34

Complex Heat Transfer – Dimensional AnalysisSieder and Tate equation(laminar flow)

Re


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© Faith A. Morrison, Michigan Tech U.Sieder & Tate, Ind. Eng. Chem. 28 227 (1936) 35


Sieder and Tate equationeffect of viscosity ratio


Correlations for Forced Convection Heat Transfer Coefficients

1

10

100

1000

10000

10 100 1000 10000 100000 1000000

Nu

Pr = 8.07 (water, 60oF)viscosity ratio = 1.00L/D = 65

104 105 106

14.03

1

PrRe86.1

w

b

L

DNu

14.0

3

18.0 PrRe027.0

w

bNu

Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)Geankoplis, 4th ed. eqn 4.5-4, page 260

36


Sieder and Tate equation(laminar flow)

Re

Sieder and Tate equation(turbulent flow)


19



1

10

100

1000

10000

10 100 1000 10000 100000 1000000

Nu


104 105 106

14.03

1

PrRe86.1

w

b

L

DNu

14.0

3

18.0 PrRe027.0

w

bNu

37Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)Geankoplis, 4th ed. eqn 4.5-4, page 260

Re



If dimensional analysis is right, we should get a

single curve, not multiple different curves

depending on: 𝐷, 𝐿, 𝜇, 𝑒𝑡𝑐.




1

10

100

1000

10000

10 100 1000 10000 100000 1000000

Nu


104 105 106

14.03

1

PrRe86.1

w

b

L

DNu

14.0

3

18.0 PrRe027.0

w

bNu

38Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)Geankoplis, 4th ed. eqn 4.5-4, page 260

Re



If dimensional analysis is right, we should get a

single curve, not multiple different curves

depending on: 𝐷, 𝐿, 𝜇, 𝑒𝑡𝑐.


Dimensional Analysis

WINS AGAIN!


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14.031

PrRe86.1

w

baa L

DkDh

Nu

Heat Transfer in Laminar flow in pipes: data correlation for forced convection heat transfer coefficients

Geankoplis, 4th ed. eqn 4.5-4, page 260

𝑅𝑒 2100, 𝑅𝑒𝑃𝑟 100, horizontal pipes; all physical properties

evaluated at the mean temperature of the bulk fluid except 𝜇𝑤 which is evaluated at the (constant) wall temperature.

the subscript “a” refers to the type of average temperature used in

calculating the heat flow, q

2

bowbiwa

aaTTTT

T

TAhq


39


Sieder-Tate equation (laminar flow)Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)

Heat Transfer in Turbulent flow in pipes: data correlation for forced convection heat transfer coefficients

Geankoplis, 4th ed. section 4.5

the subscript “lm” refers to the type of average temperature used in

calculating the heat flow, q


40


Sieder-Tate equation (turbulent flow)Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)

N𝑢ℎ 𝐷𝑘

0.027Re . Pr𝜇𝜇

.

𝑞 ℎ 𝐴Δ𝑇

Δ𝑇Δ𝑇 Δ𝑇

lnΔ𝑇Δ𝑇


21


Forced convection Heat Transfer in Turbulent flow in pipes

•all physical properties (except 𝜇 ) evaluated at the bulk mean temperature•Laminar or turbulent flow

Physical Properties 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

41

bulk mean temperatureFine print

matters!

Sieder-Tate equation (laminar flow)

Sieder-Tate equation (turbulent flow)


14.031

PrRe86.1

w

baa L

DkDh

Nu

In our dimensional analysis, we assumed constant 𝜌, k, 𝜇, etc. Therefore we did not predict a viscosity-temperature dependence. If viscosity is not assumed constant, the dimensionless group shown below is predicted to appear in correlations.

?


(reminiscent of pipe wall roughness; needed to modify dimensional analysis to correlate on roughness)

42




Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)


22

Viscous fluids with large Δ𝑇

ref: McCabe, Smith, Harriott, 5th ed, p339Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)

heating

cooling

lower viscosity fluid layer speeds flow near the wall higher h

higher viscosity fluid layer retards flow near the wall 𝑙ower h

14.0

w

b

wb

wb

empirical result:


43


Why does 𝑳

𝑫appear in laminar flow correlations and

not in the turbulent flow correlations?

h(z)

Lh

10 20 30 40 50 60 70

Less lateral mixing in laminar flow means more variation in ℎ 𝑥 .

LAMINAR


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23

h(z)

Lh

10 20 30 40 50 60 70

7.0

1

D

L

h

h

L

In turbulent flow, good lateral mixing reduces the variation in ℎ along the pipe length.

TURBULENT


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Complex Heat Transfer – Dimensional Analysis Forced convection Heat Transfer in Turbulent flow in pipes

Sieder & Tate eqn, turbulent flow

Example:Water flows at 0.0522 𝑘𝑔/𝑠 (turbulent) in the inside of a double pipe heat exchanger (inside steel pipe, inner diameter 0.545 𝑖𝑛𝑐ℎ𝑒𝑠,length unknown, physical properties given on page 1); the water enters at 30.0 𝐶 and exits at 65.6 𝐶. In the shell of the heat exchanger, steam condenses at an unknown saturation pressure. What is the heat transfer coefficient, ℎ (based on log mean temperature driving force) in the water flowing in the pipe? You may neglect the effect on heat-transfer coefficient of the temperature-dependence of viscosity. Please give your answer in 𝑊/𝑚 𝐾.


46

(Exam 4 2016)

Physical properties of steel: thermal conductivity 𝟏𝟔.𝟑 𝑾/𝒎𝑲heat capacity 𝟎.𝟒𝟗 𝒌𝑱/𝒌𝒈 𝑲density 𝟖𝟎𝟓𝟎 𝒌𝒈/𝒎𝟑


24

laminar flowin pipes

14.03

1

PrRe86.1

w

baa L

D

k

DhNu

Re<2100, (RePrD/L)>100,horizontal pipes, eqn 4.5-4,page 238; all propertiesevaluated at the temperature ofthe bulk fluid except w whichis evaluated at the walltemperature.

turbulent flowin smooth

tubes

14.0

3

18.0 PrRe027.0

w

blmlm k

DhNu

Re>6000, 0.7 <Pr <16,000,L/D>60 , eqn 4.5-8, page 239;all properties evaluated at themean temperature of the bulkfluid except w which isevaluated at the walltemperature. The mean is theaverage of the inlet and outletbulk temperatures; not validfor liquid metals.

air at 1atm inturbulent flow

in pipes2.0

8.0

2.0

8.0

)(

)/(5.0

)(

)/(52.3

ftD

sftVh

mD

smVh

lm

lm

equation 4.5-9, page 239

water inturbulent flow

in pipes

2.0

8.0

2.0

8.0

)(

/011.01150

)(

/0146.011429

ftD

sftVFTh

mD

smVCTh

olm

olm

4 < T(oC)<105, equation 4.5-10, page 239

Example of partial solution to Homework 6 (bring to final exam)


(𝑇mean)

47



Sieder-Tate equation (turbulent flow)


Complex Heat transfer Problems to Solve:

•Forced convection heat transfer from fluid to wall



Solution: ?

Solution: ?

Solution: ?

We started with a forced-convection pipe problem, did dimensional analysis, and found the dimensionless numbers.

To do a situation with different physics, we must start with a different starting problem.

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Next:

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Documents

Complex Heat Transfer – Dimensional Analysisfmorriso/cm310/lectures/2019 heat lecture 17 old heat 04.pdfComplex Heat Transfer –Dimensional Analysis 𝜕𝑇∗ 𝜕𝑡∗ 𝑣