Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
CM3110 Transport/Unit Ops IPart II: Heat Transfer
1
Complex Heat Transfer โDimensional Analysis
Professor Faith Morrison
Department of Chemical EngineeringMichigan Technological University
(Forced convection heat transfer)
ยฉ Faith A. Morrison, Michigan Tech U.2
(what have we been up to?)1D Heat Transfer
Examples of (simple, 1D) Heat Conduction
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ 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?
ยฉ Faith A. Morrison, Michigan Tech U.
inQonsW ,
Process scale
1T
2T
1T 2Tcold less cold
less hot
hot
1T
2T
1T 2Tcold less cold
less hot
hot
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Complex Heat Transfer
Lecture 17 CM3110 Heat Transfer 12/17/2019
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Engineering Modeling
ยฉ Faith A. Morrison, Michigan Tech U.
โขChoose an idealized problem and solve it
โขFrom insight obtained from ideal problem, identify governing equations of real problem
โขNondimensionalize the governing equations; deduce dimensionless scale factors (e.g. Re, Fr for fluids)
โขDesign experiments to test modeling thus far
โขRevise modeling (structure of dimensional analysis, identity of scale factors, e.g. add roughness lengthscale)
โขDesign additional experiments
โขIterate 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 โ Dimensional Analysis
5
Experience with Dimensional Analysis thus far:
ยฉ Faith A. Morrison, Michigan Tech U.
โขRough pipes
โขNon-circular conduits
โขFlow around obstacles (spheres, other complex shapes
Solution: Navier-Stokes, Re, Fr, ๐ฟ/๐ท, dimensionless wall force ๐; ๐ ๐ Re, ๐ฟ/๐ท
Solution: Navier-Stokes, Re, dimensionless drag ๐ถ ; ๐ถ ๐ถ 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
โขFlow in pipes at all flow rates (laminar and turbulent)
โขBoundary layers
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
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
ยฉ Faith A. Morrison, Michigan Tech U.
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
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
Now, move to heat transfer:โขForced convection heat transfer from fluid to wall
โขNatural convection heat transfer from fluid to wall
โขRadiation heat transfer from solid to fluid
Solution: ?
Solution: ?
Solution: ?
solid wallbulk fluid solid wall
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Complex Heat Transfer โ Dimensional Analysis
solid wallbulk fluid
ยฉ Faith A. Morrison, Michigan Tech U.
Now, move to heat transfer:โขForced convection heat transfer from fluid to wall
โขNatural convection heat transfer from fluid to wall
โขRadiation heat transfer from solid to fluid
Solution: ?
Solution: ?
Solution: ?
We have already started using the results/techniques
of dimensional analysis through defining the heat
transfer coefficient, โ
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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solid wallbulk fluid
ยฉ Faith A. Morrison, Michigan Tech U.
Now, move to heat transfer:โขForced convection heat transfer from fluid to wall
โขNatural convection heat transfer from fluid to wall
โขRadiation heat transfer from solid to fluid
Solution: ?
Solution: ?
Solution: ?
We have already started using the results/techniques
of dimensional analysis through defining the heat
transfer coefficient, โ
(recall that we did this in fluids too: we used the ๐ Re correlation
(Moody chart) long before we knew where that all came from)
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Complex Heat Transfer โ Dimensional Analysis
12
x
bulkT
wallT
wallx
solid wallbulk fluid
wallbulk TT
The temperature variation near-wall region is caused by complex phenomena that are lumped together into the heat
transfer coefficient, h
ยฉ Faith A. Morrison, Michigan Tech U.
Handy tool: Heat Transfer Coefficient
๐ ๐ฅ in solid
๐ ๐ฅ in liquid
Lecture 17 CM3110 Heat Transfer 12/17/2019
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13
xbulk wall
qh T T
A
The flux at the wall is given by the empirical expression known as
Newtonโs Law of Cooling
This expression serves as the definition of the heat transfer coefficient.
ยฉ Faith A. Morrison, Michigan Tech U.
๐ depends on:
โขgeometryโขfluid velocityโขfluid propertiesโขtemperature difference
Complex Heat Transfer โ Dimensional Analysis
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๐ depends on:
โขgeometryโขfluid velocityโขfluid propertiesโขtemperature difference
ยฉ Faith A. Morrison, Michigan Tech U.
To get values of โ 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
Newtonโs Law of Cooling
This expression serves as the definition of the heat transfer coefficient.
Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
โขForced convection heat transfer from fluid to wall
โขNatural convection heat transfer from fluid to wall
โขRadiation heat transfer from solid to fluid
Solution: ?
Solution: ?
Solution: ?
โข The functional form of โ will be different for these three situations (different physics)
โข Investigate simple problems in each category, model them, take data, correlate
Complex Heat transfer Problems to Solve:
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Complex Heat Transfer โ Dimensional Analysis
ยฉ Faith A. Morrison, Michigan Tech U.
Following procedure familiar from pipe flow,
โข What are governing equations?
โข Scale factors (dimensionless numbers)?
โข Quantity of interest?
Answer: Heat flux at the wall
Chosen problem: Forced Convection Heat TransferSolution: Dimensional Analysis
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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General Energy Transport Equation(microscopic energy balance)
V
nฬdSS
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
2ห
Gibbs notation:
see handout for component notation
ยฉ Faith A. Morrison, Michigan Tech U.
17
Complex Heat Transfer โ Dimensional Analysis
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
2ห
ยฉ Faith A. Morrison, Michigan Tech U.
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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Note: this handout is also on the
web
ยฉ Faith A. Morrison, Michigan Tech U.
https://pages.mtu.edu/~fmorriso/cm310/energy.pdf
Complex Heat Transfer โ Dimensional Analysis
R1
Example: Heat flux in a cylindrical shell
Assumptions:โขlong pipeโขsteady stateโขk = thermal conductivity of wallโขh1, 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
ยฉ Faith A. Morrison, Michigan Tech U.
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Lecture 17 CM3110 Heat Transfer 12/17/2019
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Consider: Heat-transfer to from flowing fluid inside of a tube โ forced-convection heat transfer
T1= core bulk temperatureTo= wall temperature
T(r,,z) = temp distribution in the fluid
ยฉ Faith A. Morrison, Michigan Tech U.
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
ยฉ Faith A. Morrison, Michigan Tech U.
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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r
R
1T
0Tpipe w
allfluid
),,( zrT
Unknown function:
ยฉ Faith A. Morrison, Michigan Tech U.
๐ ๐ is an unknown function
๐ ๐
Assume: โข ๐ symmetryโข Long tube
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Complex Heat Transfer โ Dimensional Analysis
ยฉ Faith A. Morrison, Michigan Tech U.
๐ฌ ๐ โ ๐ ๐๐
โ ๐ ๐
๐ฌ 2๐๐ ๐ฟ โ ๐ ๐
๐๐๐ด
๐๐๐๐๐
At the boundary, (Newtonโs Law of Cooling is the boundary condition)
We can calculate the total heat transferred from ๐ ๐ in the fluid:
We need ๐ ๐in the fluid
Total heat conducted to the
wall from the fluid
Total heat flow through (at) the wall
in terms of h
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
2๐๐ ๐ฟ โ ๐ ๐ ๐ฌ ๏ฟฝฬ๏ฟฝ โ ๐ ๐๐
Equate these two: Total heat flow through the wall
2๐๐ ๐ฟ โ ๐ ๐ ๐ฌ ๐๐๐๐๐
๐ ๐๐ง๐๐
Total heat flow at the wall in terms of h
Total heat conducted to the wall from the fluid
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Complex Heat Transfer โ Dimensional Analysis
ยฉ Faith A. Morrison, Michigan Tech U.
2๐๐ ๐ฟ โ ๐ ๐ ๐ฌ ๏ฟฝฬ๏ฟฝ โ ๐ ๐๐
Equate these two: Total heat flow through the wall
2๐๐ ๐ฟ โ ๐ ๐ ๐ฌ ๐๐๐๐๐
๐ ๐๐ง๐๐
Now, non-dimensionalizethis expression
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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non-dimensional variables:
position:
o
o
TTTT
T
1
*
temperature:
Dr
r *
Dz
z *
ยฉ Faith A. Morrison, Michigan Tech U.
Non-dimensionalize
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Complex Heat Transfer โ Dimensional Analysis
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
ยฉ Faith A. Morrison, Michigan Tech U.
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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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
ยฉ Faith A. Morrison, Michigan Tech U.
This is a function of Rethrough fluid ๐ฃ distribution
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Complex Heat Transfer โ Dimensional Analysis
๐๐โ
๐๐กโ๐ฃโ๐๐โ
๐๐โ๐ฃโ
๐โ๐๐โ
๐๐๐ฃโ๐๐โ
๐๐งโ1
Pe1๐โ
๐๐๐โ
๐๐๐โ
๐๐โ1๐โ
๐ ๐โ
๐๐๐ ๐โ
๐๐ง๐โ
**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
หPe PrRe pC VD
k
Quantity of interest
ddzr
T
DLNu
DL
r
2
0 0
*
21
*
*
*/2
1
หPr pC
k
ยฉ Faith A. Morrison, Michigan Tech U.
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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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no free surfaces
D
LNuNu ,FrPr,Re,
ยฉ Faith A. Morrison, Michigan Tech U.
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 โก
Prandtl number
Pr โก
31
three
Complex Heat Transfer โ Dimensional Analysis
ยฉ Faith A. Morrison, Michigan Tech U.
Forced Convection Heat Transfer
Now, do the experiments.
โข Build apparatus (several actually, with different D, L)
โข Run fluid through the inside (at different ๐ฃ; for different fluids ๐, ๐,๐ถ , ๐)
โข Measure ๐ on inside; ๐ on inside
โข Measure rate of heat transfer, ๐
โข Calculate โ: ๐ โ๐ด ๐ ๐
โข Report โ values in terms of dimensionless correlation:
Nuโ๐ท๐
๐ Re, Pr,๐ฟ๐ท It should only be a function of
these dimensionless numbers (if our Dimensional Analysis is
correctโฆ..)32
Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)33
Complex Heat Transfer โ Dimensional AnalysisRe
Re
Sieder and Tate equation
(laminar flow)
Sieder and Tate equation
(turbulent flow)
Effect of viscosity ratio
ยฉ Faith A. Morrison, Michigan Tech U.Sieder & Tate, Ind. Eng. Chem. 28 227 (1936) 34
Complex Heat Transfer โ Dimensional AnalysisSieder and Tate equation(laminar flow)
Re
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.Sieder & Tate, Ind. Eng. Chem. 28 227 (1936) 35
Complex Heat Transfer โ Dimensional Analysis
Sieder and Tate equationeffect of viscosity ratio
ยฉ Faith A. Morrison, Michigan Tech U.
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
Complex Heat Transfer โ Dimensional Analysis
Sieder and Tate equation(laminar flow)
Re
Sieder and Tate equation(turbulent flow)
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
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
37Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)Geankoplis, 4th ed. eqn 4.5-4, page 260
Re
Sieder and Tate equation(turbulent flow)
Sieder and Tate equation(laminar flow)
If dimensional analysis is right, we should get a
single curve, not multiple different curves
depending on: ๐ท, ๐ฟ, ๐, ๐๐ก๐.
Complex Heat Transfer โ Dimensional Analysis
ยฉ Faith A. Morrison, Michigan Tech U.
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
38Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)Geankoplis, 4th ed. eqn 4.5-4, page 260
Re
Sieder and Tate equation(turbulent flow)
Sieder and Tate equation(laminar flow)
If dimensional analysis is right, we should get a
single curve, not multiple different curves
depending on: ๐ท, ๐ฟ, ๐, ๐๐ก๐.
Complex Heat Transfer โ Dimensional Analysis
Dimensional Analysis
WINS AGAIN!
Lecture 17 CM3110 Heat Transfer 12/17/2019
20
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
๐ ๐ 2100, ๐ ๐๐๐ 100, horizontal pipes; all physical properties
evaluated at the mean temperature of the bulk fluid except ๐๐ค which is evaluated at the (constant) wall temperature.
the subscript โaโ refers to the type of average temperature used in
calculating the heat flow, q
2
bowbiwa
aaTTTT
T
TAhq
ยฉ Faith A. Morrison, Michigan Tech U.
39
Complex Heat Transfer โ Dimensional Analysis
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 โlmโ refers to the type of average temperature used in
calculating the heat flow, q
ยฉ Faith A. Morrison, Michigan Tech U.
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Complex Heat Transfer โ Dimensional Analysis
Sieder-Tate equation (turbulent flow)Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)
N๐ขโ ๐ท๐
0.027Re . Pr๐๐
.
๐ โ ๐ดฮ๐
ฮ๐ฮ๐ ฮ๐
lnฮ๐ฮ๐
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
Forced convection Heat Transfer in Turbulent flow in pipes
โขall physical properties (except ๐ ) evaluated at the bulk mean temperatureโขLaminar or turbulent flow
Physical Properties evaluated at:
N๐ขโ ๐ท๐
0.027Re . Pr๐๐
.๐ , ๐ ,
2
May have to be estimated
Forced convection Heat Transfer in Laminar flow in pipes
N๐ขโ ๐ท๐
1.86 RePr๐ท๐ฟ
๐๐
. ๐ , ๐ ,
2
41
bulk mean temperatureFine print
matters!
Sieder-Tate equation (laminar flow)
Sieder-Tate equation (turbulent flow)
Complex Heat Transfer โ Dimensional Analysis
14.031
PrRe86.1
w
baa L
DkDh
Nu
In our dimensional analysis, we assumed constant ๐, k, ๐, 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.
?
ยฉ Faith A. Morrison, Michigan Tech U.
(reminiscent of pipe wall roughness; needed to modify dimensional analysis to correlate on roughness)
42
Forced convection Heat Transfer in Laminar flow in pipes
Complex Heat Transfer โ Dimensional Analysis
Sieder-Tate equation (laminar flow)
Sieder & Tate, Ind. Eng. Chem. 28 227 (1936)
Lecture 17 CM3110 Heat Transfer 12/17/2019
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Viscous fluids with large ฮ๐
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 ๐ower h
14.0
w
b
wb
wb
empirical result:
ยฉ Faith A. Morrison, Michigan Tech U.
43
Complex Heat Transfer โ Dimensional Analysis
Why does ๐ณ
๐ซappear 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 โ ๐ฅ .
LAMINAR
ยฉ Faith A. Morrison, Michigan Tech U.
44
Forced convection Heat Transfer in Laminar flow in pipes
Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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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 โ along the pipe length.
TURBULENT
ยฉ Faith A. Morrison, Michigan Tech U.
45
Complex Heat Transfer โ Dimensional Analysis Forced convection Heat Transfer in Turbulent flow in pipes
Sieder & Tate eqn, turbulent flow
Example:Water flows at 0.0522 ๐๐/๐ (turbulent) in the inside of a double pipe heat exchanger (inside steel pipe, inner diameter 0.545 ๐๐๐โ๐๐ ,length unknown, physical properties given on page 1); the water enters at 30.0 ๐ถ and exits at 65.6 ๐ถ. In the shell of the heat exchanger, steam condenses at an unknown saturation pressure. What is the heat transfer coefficient, โ (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 ๐/๐ ๐พ.
ยฉ Faith A. Morrison, Michigan Tech U.
46
(Exam 4 2016)
Physical properties of steel: thermal conductivity ๐๐.๐ ๐พ/๐๐ฒheat capacity ๐.๐๐ ๐๐ฑ/๐๐ ๐ฒdensity ๐๐๐๐ ๐๐/๐๐
Lecture 17 CM3110 Heat Transfer 12/17/2019
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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)
ยฉ Faith A. Morrison, Michigan Tech U.
(๐mean)
47
Complex Heat Transfer โ Dimensional Analysis
Sieder-Tate equation (laminar flow)
Sieder-Tate equation (turbulent flow)
ยฉ Faith A. Morrison, Michigan Tech U.
Complex Heat transfer Problems to Solve:
โขForced convection heat transfer from fluid to wall
โขNatural convection heat transfer from fluid to wall
โขRadiation heat transfer from solid to fluid
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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Complex Heat Transfer โ Dimensional Analysis
Lecture 17 CM3110 Heat Transfer 12/17/2019
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ยฉ Faith A. Morrison, Michigan Tech U.
Next:
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