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Heat Transfer to Solids in a Flowing Fluid
The heat transfer is dependent on:
1. Geometry of the body.
2. The position or orientation of the body (parallel, perpendicular to flow).
3. Proximity of other bodies.
The heat transfer coefficient varies across the surface of the object. But the average heat transfer coefficient can be determined from an equation of the form:
1/ 3Re Pr
Re
Pr
where: is the Nusselt number,
, are constants
is the Reynolds number
is the Prandtl number
mNu
Nu
N CN N
hDN
kC m
N
N
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Flow Parallel to a Flat Plate
L
0.5 1/ 3 5Re, Pr Re, Pr
Re
For heat transfer along the entire length of the surface, ,
for laminar flow:
0.664 3 10 0.7
where the Reynolds number is computed from
For turbulent
Nu L L
L
N N N N N
LvN
5Re, Pr
0.8 1/ 3Re, Pr
flow, 3 10 0.7
0.0366
L
Nu L
N N
N N N
Use fluid properties at average film temperature = (Ave. temp. of Wall + Ave. temp. of fluid)/2
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Flow Perpendicular to a Single Cylinder
D
Re
1/ 3Re Prm
Nu
DvN
N CN N
Use properties at the film temperature. Velocity is free field velocity of fluid.
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Flow Past a Single Sphere
Re
0.5 1/ 3Re Pr
Re Pr
2.0 0.60
1 70,000 0.6 400Nu
DvN
N N N
N N
Use properties at film temperature.
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Flow Thru Tube BanksVery important for heat exchanger design!
Flow around the first bank is essentially the same as for a single tube. For subsequent rows, flow depends on the tube bank arrangement. The convection coefficient of a row increases with increasing row number until about the 5th row, after which there is little change. For aligned tubes, the front row shields the back rows, particularly for short distances between tubes. In general, heat transfer is encouraged by the staggered arrangement.
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Flow Thru Tube Banks
In-Line Tubes Staggered Tubes
'
Tube spacing parallel to flow.
Tube spacing normal to flow.
Diagonal tube spacing for staggered rows.
P
n
P
S
S
S
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Flow Thru Tube Banks
max
maxRe
1/ 3Re Pr
ratio of minimum flow area to total front area, between tubes
maximum velocity in tube banks.
n
n
n
n
mNu
S
S D
vSv
S D
DvN
N CN N
Only for more than ten rows. Tables are available for non-equal ratios.
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Flow Thru Tube Banks
Correction factors for banks of less than ten tubes.
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Flow Thru Tube Banks
Procedure for solving tube bank problems:
Given: tube geometry, inlet temperature, tube surface temp., fluid velocity.
1. Assume an outlet temperature.
2. Determine properties of the fluid at the average temperature.
3. Calculate max. velocity based on geometry.
4. Calculate Reynolds number based on max. velocity.
5. Determine average heat transfer coefficent.
6. Determine overall q from total area of all tubes using temperature difference between tube wall and average fluid temperature.
7. Determine mass flow rate from:
8. Use to determine temperature drop.
9. Continue until guessed = calculated.
open areat tm vA A
Pq mC T
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Heat Transfer for Flow in Packed Beds
2 / 3
' 0.35Re Re
'
2.876 0.3023
where: is the void fraction
is the superficial velocity based on the cross section
of the empty container in m/s
PH
fP
ChJ
k NC v N
v
'
Re
' '
is called the Colburn J factor
where the subscript means the property is evaluated at the
film temperature. All other properties are at the bulk flow temp.
H
P
f
J
D GN
G v
f
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Convective Heat Transfer
Natural convection occurs when a quiescent fluid is exposed to a hot or cold surface.
If the surface is hot, the fluid next to the surface will be heated, its temperature will increase and its density will decrease. Due to the decreased density of the fluid next to the surface, it will rise due to buoyancy.
If the surface is cold, then the temperature of the fluid will be colder than the bulk fluid, its density will decrease and will fall due to buoyancy.
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Convective Heat Transfer
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Convective Heat Transfer
Typical chemical engineering problems involving convective heat transfer:
1. If a hot fluid is transported thru a pipe from process A to process B, how much will its temperature drop?
2. If a hot fluid is stored in a storage vessel, how much will the temperature drop each day?
3. What are the convective heat losses from my process unit, i.e., distillation column?
4. If a hot solid is cooled in the open, how long will it take to cool the solid to room temperature?
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Convective Heat Transfer
Natural convection heat transfer involves an additional dimensionless parameter called the Grashof number. The Grashof number represents the buoyancy force.
3 2
r 2
where: is the characteristic length, i.e. length of a heated plate
is the density of the fluid, at the film temperature.
is the coefficient of volumetri
fG
f
L g TN
L
c expansion.
is the temperature difference between surface and bulk fluid
is the fluid viscosity, evaluated at the film temperature
Remember:
2wall bulk
film
T
T TT
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Convective Heat Transfer
The volumetric expansion coefficient is defined as:
-1
1
where: is the volume
is the temperature
is the volumetric expansion coefficient, (deg)
dV
V dT
V
T
Ethyl alcohol: 112 x 10-5 /deg. C
Methyl alcohol: 120 “
Benzene: 124 “
Glycerin: 51 “
Air: 3 “
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Convective Heat Transfer
2
For ideal gases
1
1 1But and
and it follows that
1
For an ideal gas, and it follows that
1
g
dV
V dT
dV d
d
dT
P
R T
T
Ideal gas only
True for any material
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Convective Heat TransferMost natural convection geometries are represented by the equation:
Pr
m
Nu GrN a N N
The physical properties are evaluated at the film temperature.
For vertical and horizontal plates and cylinders use Table 4.7.1 (handout).
For horizontal plates the length, L, is used.
For cylinders L is replaced by D.
For horizontal rectangles the average of the two dimensions is used.
For a horizontal circular disk, the diameter is multiplied by 0.9.
Simplified equations for various types of surfaces are provided in Table 4.7-2 (handout).
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Convective Heat Transfer
For natural convection at pressures other than 1 atm, the heat transfer coefficients are multiplied by a correction factor:
4 9 1/ 2Pr
9 2 / 3Pr
For from 10 to 10 multiply by
For > 10 multiply by
where is in atmospheres.
Gr
Gr
N N P
N N P
P
4 9Pr
3 3 3 o
The correlations in the tables apply mostly for
10 10
This generally holds for
4.7 m or 300 ft F
GrN N
L T K
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Example - Convective Heat Transfer
4.7-2. A vertical cylinder 76.2 mm in diameter and 121.9 mm high is maintained at 397.1 K at its surface. It loses heat by natural convection to air at 294.2 K. Heat is lost from the sides and top – the bottom is insulated. Calculate the total heat loses neglecting radiation.
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Example - Convective Heat Transfer
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Example - Convective Heat Transfer
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Example - Convective Heat Transfer
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Example - Convective Heat Transfer
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Example - Convective Heat Transfer