7. semester chemical engineering civil engineeringhomes.et.aau.dk/mma/transport/lek8 transport...
TRANSCRIPT
![Page 1: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/1.jpg)
Transport processes
7. Semester
Chemical Engineering
Civil Engineering
![Page 2: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/2.jpg)
Course plan
1. Elementary Fluid Dynamics2. Fluid Kinematics3. Finite Control Volume Analysis4. Differential Analysis of Fluid Flow5. Viscous Flow and Turbulence6. Turbulent Boundary Layer Flow7. Principles of Heat Transfer8. Internal Forces Convection9. Unsteady Heat Transfer10. Boiling and Condensation11. Mass Transfer12. Porous Media Flow13. Non-Newtonian Flow
![Page 3: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/3.jpg)
Today's lecture
• Internal Forced Convection– The thermal boundary layer
– Thermal entrance length
– Forced convection in pipes
![Page 4: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/4.jpg)
Newtons law of cooling
• Calculation of convective heat transfer
– h depends on:• Geometry
• Fluid properties
• Flow properties
( ) [ ]conv s sQ hA T T W∞= −
![Page 5: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/5.jpg)
Thermal boundary layer
• Thermal boundary thickness– 99% of the free stream temperature, that is when:
( )0.99s sT T T T∞− = −
![Page 6: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/6.jpg)
Thermal boundary layer
• Turbulence enhances momentum and heat transfer
• Similar relations apply
![Page 7: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/7.jpg)
Thermal boundary layer
• The thermal boundary layer does not necessarily develop at the same rate as the momentum b. layer
Molecular diffusivity of momentumPrMolecular diffusivity of heat
pckµν
α= = =
![Page 8: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/8.jpg)
Solution to the energy equation for boundary layer flow
• We can get the similar solution as Blasius for the thermal b. layer
• For Pr=1, exactly the same
3 20.332w Uxµρτ = 1 30.332Prx
uhx
ρµ
∞=
Momentum transfer Heat transfer
![Page 9: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/9.jpg)
Thermal boundary layer
• The heat transfer rate is proportional to the temperature gradient
0conv cond
y
Tq q ky =
∂= = −
∂
![Page 10: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/10.jpg)
Nusselt number
• Heat transfer is by conduction when the fluid is motionless and by convection when the fluid moves
• Taking there ratios:
• For Nu=1 we have pure conduction
convq h T= ∆ condTq kL∆
=
Nu/
conv
cond
q h T hLq k T L k
∆= = =
∆
![Page 11: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/11.jpg)
Entrance length
![Page 12: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/12.jpg)
Entrance region
• Larger pressure drop
• Larger heat flux
• Entrance lengths– Laminar flow
– Turbulent flow
0.05RevelocityL D=
0.05Re Pr Prthermal velocityL D L= =
10velocity thermalL L D≈ ≈
![Page 13: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/13.jpg)
Intermezzo
• Use the next 2 minutes to discus with the person next to you what kind of heat transfer enhancement mechanisms are employed in this plate heat exchanger
![Page 14: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/14.jpg)
Boundary conditions
• For empirical relations we have two options:– Constant heat flux
– Constant surface temperature
![Page 15: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/15.jpg)
General thermal analysis
• From energy-balance:
• From surface heat flux
– If ts is constant, qs must change
– if qs is constant, ts must change
( ) (W)p e iQ mC T T= −
( ) 2(W/m )x s mq h T T= −
![Page 16: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/16.jpg)
Bulk mean temperature, Tm
• Note: the mean temperature of the fluid must change during heating or cooling
• The energy transported by the fluid through a cross section in actual flow must be equal to the energy that would be transported through the same cross section if the fluid were at a constant temperature Tm
– Use either inlet temperature or arithmetic average, (Te+Ti)/2, to determine fluid properties
– Use logarithmic mean temperature difference if constant surface temperature (will be show next)
![Page 17: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/17.jpg)
Constant surface flux
• If qs= constant then:
thus
( ) (W)s s p e iQ q A mC T T= = −
s se i
p
q AT TmC
= +
Constant heat flux = Constant temperature gradient
perimeter constants
p
qdTdx mC
×= =
![Page 18: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/18.jpg)
Constant temperature
• If Ts=constant then:
• The logarithmic mean temperature:
• The mean fluid temperature at exit:
( ) ln (W)s s m saveQ hA T T hA T= − = ∆
( ) ( )ln lni e
s e s i
T TTT T T T
−∆ =
− −
( ) ( )expe s s i s pT T T T hA mC= − − −
![Page 19: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/19.jpg)
Laminar flow in pipes
• Fully developed flow, circular pipe, constant surface flux:
• Fully developed flow, circular pipe, constant surface flux:
4.36hDNuk
= =
3.66hDNuk
= =
![Page 20: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/20.jpg)
Laminar flow in pipes – empirical correlations
• Its basically a matter of finding an appropriate formulae in a book– Constant heat flux / constant surface temperature
– Non-circular ducts/ circular pipes / flat plate
– Entrance region / fully developed
• It could look something like: (Sieder and Tate)
( )0.141
3av b
Nu re Pr Reavw
= =1.86 <2100h D DN N N Nk L
µµ
![Page 21: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/21.jpg)
Turbulent flow in pipes – empirical correlations
• Again, mostly it is a matter of finding a usable expression..
– The most simple correlation is the Colburn equation:
– This can be improved by the Dittus-Boelter equation:
– This is only around ±25% accurate and more precise but increasingly more complex equations exists.
0.8 1 30.023Re PrNu =
0.80.023Re PrnNu = n = 0.3 for coolingn = 0.4 for heating
![Page 22: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/22.jpg)
Example
• Calculate the heat loss from a oil pipe running through a icy lake.
1. Fluid properties evaluated at 20 °C initially2. Calculate the Reynolds number:
3. Calculate the entrance length:
4. Find suitable Nu correlation, calculate the Nusselt number:laminar flow, thermally developing flow, small temperature difference - use correlation by Edwards et al 1979:
Note, this is the average Nu for the pipe flow
3
Oil @ 20 C: 888 /0.145 /0.8 /1880 /
Pr 10400p
kg mk W m K
kg m sC J kg K
ρ
µ
== ⋅= ⋅= ⋅
=
888 2 0.3Re 666 Laminar0.8
UDρµ
⋅ ⋅= = = →
0.05Re Pr 0.05 666 10400 0.3 104thermalL D km= = ⋅ ⋅ ⋅ =
( )( ) 2 3
0.065 Re Pr3.66 37.3
1 0.04 Re Pr
D LhDNuk D L
= = + =+
![Page 23: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/23.jpg)
Example cont.
5. Calculate the convective heat transfer coefficient:
6. Calculate the exit temperature
Note, this makes the bulk mean temperature Tm=(20+19.71)/2=19.85°C. This low temperature difference makes it acceptable to evaluate fluid properties at 20°C
7. Calculate the logarithmic temperature difference:
3
Oil @ 20 C: 888 /0.145 /0.8 /1880 /
Pr 10400p
kg mk W m K
kg m sC J kg K
ρ
µ
== ⋅= ⋅= ⋅
=
20.145Nu 37.3 18.0 /0.3
kh W m KD
= = = ⋅
( ) ( )exp 19.71e s s i s pT T T T hA mC C= − − − =
2188.5sA DL mπ= =2
4888 0.3 2 125.5 /c meanm A U kg sπρ= = ⋅ ⋅ =
( ) ( ) ( ) ( )ln20 19.71 19.86
ln ln 0 19.71 0 20i e
s e s i
T TT CT T T T
− −∆ = = = −
− − − −
![Page 24: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/24.jpg)
Example cont.
8. Calculate the heat loss:
a) Calculate the pressure loss:
b) Calculate the required pump work:3
Oil @ 20 C: 888 /0.145 /0.8 /1880 /
Pr 10400p
kg mk W m K
kg m sC J kg K
ρ
µ
== ⋅= ⋅= ⋅
=
( )ln 18.0 188.5 19.85 67.4 WsQ hA T k= ∆ = ⋅ ⋅ − = −
2 2564 200 888 2 1.14 10 1.14
2 Re 0.3 2L UP f Pa barDρ ⋅
∆ = = = ⋅ =
16.1pumpm PW kWρ∆
= =
![Page 25: 7. Semester Chemical Engineering Civil Engineeringhomes.et.aau.dk/mma/transport/lek8 Transport processer... · 2010-09-15 · Internal Forces Convection 9. Unsteady Heat Transfer](https://reader033.vdocuments.us/reader033/viewer/2022041816/5e5af4e7ac456259e059aef3/html5/thumbnails/25.jpg)
Excercises
• Time to wake up!