
In the convection
section, heat is transferred by both radiation and convection. The
convection transfer coefficients for fin and stud tubes are explored
here as well as bare tube transfer. The short beam radiation is
treated separately from the convection transfer below.
This section of Fired Heater Design is divided into five main areas,
which can be selected from the subject drop down box above or you
may use the jump links below to go to a section. You may also use
your browser "back button" to return to where you were.
Overall Heat Transfer Coefficient,
U_{o}:
U_{o} =
1/R_{to}
Where,
U_{o} = Overall heat transfer coefficient,
Btu/hrft^{2}F 
R_{to} = Total outside thermal resistance,
hrft^{2}F/Btu 
And,
R_{to} = R_{o} + R_{wo} + R_{io} 
R_{o} = Outside thermal resistance,
hrft^{2}F/Btu 
R_{wo} = Tube wall thermal resistance,
hrft^{2}F/Btu 
R_{io} = Inside thermal resistance,
hrft^{2}F/Btu 
And the resistances are
computed as,
R_{o} = 1/h_{e} 
R_{wo} =
(t_{w}/12*k_{w})(A_{o}/A_{w}) 
R_{io} =
((1/h_{i})+R_{fi})(A_{o}/A_{i}) 
Where,
h_{e} = Effective outside heat transfer coefficient,
Btu/hrft^{2}F 
h_{i} = Inside film heat transfer coefficient,
Btu/hrft^{2}F 
t_{w} = Tubewall thickness, in 
k_{w} = Tube wall thermal conductivity, Btu/hrftF 
A_{o} = Outside tube surface area, ft^{2}/ft 
A_{w} = Mean area of tube wall, ft^{2}/ft 
A_{i} = Inside tube surface area, ft^{2}/ft 
R_{fi} = Inside fouling resistance,
hrft^{2}F/Btu 
Inside film heat
transfer coefficient, hi: The inside heat transfer coefficient
calculation procedure is covered in detail, elsewhere in this
course.
Effective outside heat transfer coefficient, he
h_{e} =
1/(1/(h_{c}+h_{r})+R_{fo})
Where,
h_{c} = Outside heat transfer coefficient,
Btu/hrft^{2}F 
h_{r} = Outside radiation heat transfer coefficient,
Btu/hrft^{2}F 
R_{fo} = Outside fouling resistance,
hrft^{2}F/Btu 
Outside film heat
transfer coefficient, hc: The bare tube heat transfer film
coefficient, h_{c}, can be described by the following equations. For
a staggered tube arrangement,
h_{c} =
0.33*k_{b}(12/d_{o})((c_{p}*m_{b})/k_{b})^{1/3}((d_{o}/12)(G_{n}/m_{b})))^{0.6}
And for an
inline tube arrangement,
h_{c} =
0.26*k_{b}(12/d_{o})((c_{p}*m_{b})/k_{b})^{1/3}((d_{o}/12)(G_{n}/m_{b})))^{0.6}
Where,
h_{c} = Convection heat transfer coefficient,
Btu/hrft^{2}F 
d_{o} = Tube outside diameter, in 
k_{b} = Gas thermal conductivity, Btu/hrftF 
c_{p} = Gas heat capacity, Btu/lbF 
m_{b} = Gas dynamic viscosity,
lb/hrft 
G_{n} = Mass velocity of gas,
lb/hrft^{2} 
We can describe a sample bare
tube bank as follows:
Process Conditions: Gas flow, lb/hr = 100,000 Gas
temperature in, °F = 1000 Gas temperature out, °F = 868 Compostion,
moles N_{2}, % = 71.5779 O_{2}, % =
2.8800 CO_{2}, % = 8.6404 H_{2}O, % = 16.4044 Ar,
% = 0.8609 Mechanical Conditions: Tube Diameter, in =
4.500 Tube Spacing, in = 8 Number Tubes Wide = 8 Tube Effective
Length, ft = 13.000 Number Of Tubes = 48 Tube Arrangement =
Staggered Pitch


Gas Properties For the gas
properties, we can use the script we used in the radiant section design to get
the properties of the gas at the average temperature.
From
this program, we get the following properties, k_{b}, Btu/hrftF =
0.0315 c_{p}, Btu/lbF = 0.2909 m_{b}, cp = 0.0340 = 0.0340*2.42 = 0.0823
lb/hrft To calculate the mass velocity, G_{n}, we need to first
calculate the net free area of the tube bank. For these calculations, we are
going to assume the tube rows are corbelled, so the net free area, NFA: NFA =
N_{wide}*t_{spc}/12*t_{lgth}N_{wide}*t_{Od}/12*t_{lgth}
= 8*8/12*138*4.5/12*13 = 30.333 ft^{2} Therefore, G_{n}
= W_{gas} / NFA = 100000/30.333 = 3296.739 And using our formula for
h_{c},
h_{c} = 0.33*0.0315 (12/4.5)((0.2909
*0.0823 )/0.0315 )^{1/3}((4.5/12)(3296.739/0.0823 )))^{0.6} =
8.1115
To get a feel for the values of the coeffcient, use
the following script to run various designs.
The radiation transfer coefficient, h_{r} is described later in this
section. Fouling resistances, R_{fi} and R_{fo} are allowances
that depend upon the process or service of the heater and the fuels that are
being burned.
Convection Transfer, Fin Tubes
You will notice that the heat transfer equations for the fin tubes are
basically the same as for the bare tubes untill you reach the h_{e}
factor, where a new concept is introduced to account for the fin or extended
surface. The procedure presented herein are taken from the Escoa manual which
can be downloaded in full from the internet.
Overall Heat Transfer
Coefficient, U_{o}:
U_{o} =
1/R_{to}
Where,
U_{o} = Overall heat transfer coefficient,
Btu/hrft^{2}F 
R_{to} = Total outside thermal resistance,
hrft^{2}F/Btu 
And,
R_{to} = R_{o} + R_{wo} + R_{io} 
R_{o} = Outside thermal resistance,
hrft^{2}F/Btu 
R_{wo} = Tube wall thermal resistance,
hrft^{2}F/Btu 
R_{io} = Inside thermal resistance,
hrft^{2}F/Btu 
And the resistances are
computed as,
R_{o} = 1/h_{e} 
R_{wo} =
(t_{w}/12*k_{w})(A_{o}/A_{w}) 
R_{io} =
((1/h_{i})+R_{fi})(A_{o}/A_{i}) 
Where,
h_{e} = Effective outside heat transfer coefficient,
Btu/hrft^{2}F 
h_{i} = Inside film heat transfer coefficient,
Btu/hrft^{2}F 
t_{w} = Tubewall thickness, in 
k_{w} = Tube wall thermal conductivity, Btu/hrftF 
A_{o} = Total outside surface area, ft^{2}/ft 
A_{w} = Mean area of tube wall, ft^{2}/ft 
A_{i} = Inside tube surface area, ft^{2}/ft 
R_{fi} = Inside fouling resistance,
hrft^{2}F/Btu 
Inside film heat
transfer coefficient, hi: The inside heat transfer coefficient
calculation procedure is covered in detail, elsewhere in this
course. Effective outside heat transfer coefficient,
h_{e}:
h_{e} =
h_{o}(E*A_{fo}+A_{po})/A_{o}
Where,
h_{o} = Average outside heat transfer coefficient,
Btu/hrft^{2}F 
E = Fin efficiency 
A_{o} = Total outside surface area, ft^{2}/ft 
A_{fo} = Fin outside surface area, ft^{2}/ft 
A_{po} = Outside tube surface area,
ft^{2}/ft 
And, Average outside heat
transfer coefficient, h_{o}:
h_{o} =
1/(1/(h_{c}+h_{r})+R_{fo})
Where,
h_{c} = Outside heat transfer coefficient,
Btu/hrft^{2}F 
h_{r} = Outside radiation heat transfer coefficient,
Btu/hrft^{2}F 
R_{fo} = Outside fouling resistance,
hrft^{2}F/Btu 
Outside film heat
transfer coefficient, hc:
h_{c} =
j*G_{n}*c_{p}(k_{b}/(c_{p}*m_{b}))^{0.67}
Where,
j = Colburn heat transfer factor 
G_{n} = Mass velocity based on net free area,
lb/hrft^{2} 
c_{p} = Heat capacity, Btu/lbF 
k_{b} = Gas thermal conductivity, Btu/hrftF 
m_{b} = Gas dynamic viscosity,
lb/hrft 
Colburn heat transfer factor,
j:
j =
C_{1}*C_{3}*C_{5}(d_{f}/d_{o})^{0.5}((T_{b}+460)/(T_{s}+460))^{0.25}
Where,
C_{1} = Reynolds number correction 
C_{3} = Geometry correction 
C_{5} = Nonequilateral & row correction 
d_{f} = Outside diameter of fin, in 
d_{o} = Outside diameter of tube, in 
T_{b} = Average gas temperature, F 
T_{s} = Average fin temperature,
F 
Reynolds number correction,
C_{1}:
C_{1} =
0.25*R_{e}^{0.35}
Where,
Geometry
correction, C_{3}: For segmented fin tubes arranged in, a
staggered pattern
C_{3} =
0.55+0.45*e^{(0.35*lf/Sf)}
an inline pattern,
C_{3} =
0.35+0.50*e^{(0.35*lf/Sf)}
For solid fin tubes
arranged in, a staggered pattern
C_{3} =
0.35+0.65*e^{(0.25*lf/Sf)}
an inline pattern
C_{3} =
0.20+0.65*e^{(0.25*lf/Sf)}
Where,
l_{f} = Fin height, in 
s_{f} = Fin spacing,
in 
Nonequilateral & row correction,
C_{5}: For fin tubes arranged in, a staggered pattern
C_{5} =
0.7+(0.700.8*e^{(0.15*Nr^2)})*e^{(1.0*Pl/Pt)}
an inline pattern
C_{5} =
1.1+(0.751.5*e^{(0.70*Nr^2)})*e^{(2.0*Pl/Pt)}
Where,
N_{r} = Number of tube rows 
P_{l} = Longitudinal tube pitch, in 
P_{t} = Transverse tube pitch,
in 
Mass Velocity, G_{n}:
G_{n} =
W_{g}/A_{n}
Where,
W_{g} = Mass gas flow, lb/hr 
A_{n} = Net free area,
ft^{2} 
Net Free Area,
A_{n}:
A_{n} = A_{d}  A_{c} *
L_{e} * N_{t}
Where,
A_{d} = Cross sectional area of box, ft^{2} 
A_{c} = Fin tube cross sectional area/ft,
ft^{2}/ft 
L_{e} = Effective tube length, ft 
N_{t} = Number tubes wide 
And, 
A_{d} = N_{t} * L_{e} * P_{t} /
12 
A_{c} = (d_{o} + 2 * l_{f} * t_{f} *
n_{f}) / 12 
t_{f} = fin thickness, in 
n_{f} = number of fins,
fins/in 
Surface Area Calculations: For
the prime tube,
A_{po} = Pi * d_{o} (1 n_{f} *
t_{f}) / 12
And for solid fins,
A_{o} = Pi*d_{o}(1n_{f}*
t_{f})/12+Pi*n_{f}(2*l_{f}(d_{o}+l_{f})+t_{f}(d_{o}+2*l_{f}))/12
And
for segmented fins,
A_{o} = Pi*d_{o}(1n_{f}*
t_{f})/12+0.4*Pi*n_{f}(d_{o}+0.2)/12+Pi*n_{f}
(d_{o}+0.2)((2*l_{f}0.4)(w_{n}+t_{f})+w_{s}*t_{f})/(12*w_{s})
And
then,
A_{fo} = A_{o} 
A_{po}
Where,
w_{s} = Width of fin segment, in 
We can
describe a sample fin tube bank as follows:
Process Conditions: Gas flow, lb/hr = 100,000 Gas
temperature in, °F = 1000 Gas temperature out, °F = 591 Average fin
temperature, °F = 755 Compostion, moles N_{2}, % =
71.5779 O_{2}, % = 2.8800 CO_{2}, % =
8.6404 H_{2}O, % = 16.4044 Ar, % = 0.8609


Mechanical Conditions: Tube Diameter, in = 4.500 Tube
Spacing, in = 8 Number Tubes Wide = 8 Tube Effective Length, ft =
13.000 Number Of Tubes = 48 
Tube Arrangement = Staggered Pitch Fin Height, in = 0.75 Fin
Thickness, in = 0.05 Fin Density, fins/in = 6 Fin Type =
Segmented Fin Segment Width, in = 0.3125

Gas
Properties For the gas properties, we can use the script we used in the
radiant section design to get the properties of the gas at the average
temperature. From
this program, we get the following properties, k_{b}, Btu/hrftF =
0.0290 c_{p}, Btu/lbF = 0.2858 m_{b}, cp = 0.0317 = 0.0317*2.42 = 0.0767
lb/hrft To calculate the mass velocity, G_{n}, we need to first
calculate the net free area of the tube bank. For these calculations, we are
going to assume the tube rows are corbelled, so the net free area,
A_{n}: A_{d} = 8*13*8/12 = 69.333 A_{c} =
(4.5+2*0.75*0.05*6)/12 = 0.4125 So,
A_{n} = 69.333  0.4125 * 13 * 8 = 26.4333
And,
G_{n} = 100000 / 26.4330 = 3783.1069
Now we can
calculate the reynolds number, R_{e},
R_{e} = 3783.1069*4.5/(12*0.0767) = 18496.2854
And,
C_{1} = 0.25*18496.2854^{0.35} =
0.0080
For,
s_{f} = 1/6.05=0.1167
C_{3} = 0.55+0.45*e^{(0.35*0.75/0.1167)} =
0.5975
And, P_{l} =
(8^{2}8/2^{2})^{0.5}=6.9282
C_{5} =
0.7+(0.700.8*e^{(0.15*6^2)})*e^{(1.0*6.9282/8)} =
0.9929
Now we can calculate the Colburn factor,
j =
0.0080*0.5975*0.9929(6/4.5)^{0.5}((795.5+460)/(755+460))^{0.25}
= 0.0055
And finally,
h_{c} = 0.0055*3783.1069*0.2858 (0.0290
/(0.2858*0.0767))^{0.67} = 7.1732
To get a feel
for the values of the coeffcient, use the following script to run various
designs.
The radiation transfer coefficient, h_{r} is described later in this
section. Fouling resistances, R_{fi} and R_{fo} are allowances
that depend upon the process or service of the heater and the fuels that are
being burned.
Fin Efficiency, E: For segmented fins,
E = x * (0.9 + 0.1 * x)
And for solid fins,
E = y * (0.45 * ln(d_{f} / d_{o}) * (y  1) +
1)
Where,
y = x * (0.7 + 0.3 * x)
And,
x = tanh(m * B) / (m * B)
Where,
B = l_{f} + (t_{f} /2)
For segmented fins,
m = (h_{o} (t_{f} + w_{s}) / (6 * k_{f}
* t_{f} * w_{s}))^{0.5}
And for solid fins,
m = (h_{o} / (6 * k_{f} *
t_{f}))^{0.5}
Fin Tip Temperature,
T_{s}: The average fin tip temperature is calculated as
follows,
T_{s} = T_{g} + (T_{w}  T_{g}) *
1/((e^{1.4142mB}+e^{1.4142mB})/2)
Maximum Fin
Tip Temperature, T_{fm}: The maximum fin tip temperature is
calculated as follows,
T_{sm} = T_{wm} + q(T_{gm}  T_{wm})
Where,
T_{sm} = Maximum Fin Tip Temperature, F 
T_{gm} = Maximum Gas Temperature, F 
T_{wm} = Maximum Tube Wall Temperature,
F 
And, The value for theta, q, can be described by the following curve.
Convection Transfer, Stud Tubes
For
studded tubes, the correlations used are as provided by Birwelco,
Ltd.
Overall Heat Transfer Coefficient, U_{o}:
U_{o} =
1/R_{to}
Where,
U_{o} = Overall heat transfer coefficient,
Btu/hrft^{2}F 
R_{to} = Total outside thermal resistance,
hrft^{2}F/Btu 
And,
R_{to} = R_{o} + R_{wo} + R_{io} 
R_{o} = Outside thermal resistance,
hrft^{2}F/Btu 
R_{wo} = Tube wall thermal resistance,
hrft^{2}F/Btu 
R_{io} = Inside thermal resistance,
hrft^{2}F/Btu 
And the resistances are
computed as,
R_{o} = 1/h_{e} 
R_{wo} =
(t_{w}/(12*k_{w}))(A_{o}/A_{w}) 
R_{io} =
((1/h_{i})+R_{fi})(A_{o}/A_{i}) 
Where,
h_{e} = Effective outside heat transfer coefficient,
Btu/hrft^{2}F 
h_{i} = Inside film heat transfer coefficient,
Btu/hrft^{2}F 
t_{w} = Tubewall thickness, in 
k_{w} = Tube wall thermal conductivity, Btu/hrftF 
A_{o} = Outside surface area, ft^{2}/ft 
A_{w} = Mean area of tube wall, ft^{2}/ft 
A_{i} = Inside tube surface area, ft^{2}/ft 
R_{fi} = Inside fouling resistance,
hrft^{2}F/Btu 
Effective outside heat
transfer coefficient, h_{e}: For staggered and inline pitch,
h_{e} =
(h_{so}*E*A_{fo}+h_{t}*A_{po})/A_{o}
Where,
h_{t} = Base tube outside heat transfer coefficient,
Btu/hrft^{2}F 
h_{so} = Stud outside heat transfer coefficient,
Btu/hrft^{2}F 
A_{o} = Total outside surface area, ft^{2}/ft 
A_{fo} = Stud outside surface area, ft^{2}/ft 
A_{po} = Tube outside surface area,
ft^{2}/ft 
Inline pitch correction,
h_{e} =
h_{e}*(d_{o}/P_{l})^{0.333}
Where,
d_{o} = Outside tube diameter, in 
P_{l} = Longitudinal pitch of tubes,
in 
Base tube outside heat transfer coefficient,
h_{t}:
h_{t} =
(0.717/d_{o}^{0.333})(G_{n}/1000)^{0.67}(T_{b}+460)^{0.3}
And
the stud coefficient,
h_{s} =
0.936*(G_{n}/1000)^{0.67}(T_{b}+460)^{0.3}
With
fouling,
h_{so} =
1/(1/h_{s}+R_{fo})
Where,
h_{s} = Stud outside heat transfer coefficient,
Btu/hrft^{2}F 
G_{n} = Mass velocity of flue gas, lb/hrft^{2} 
T_{b} = Average gas temperature, F 
Stud
efficiency, E:
E =
1/((e^{x}+e^{x})/1.950)
Where,
X =
L_{s}/12((2*h_{so})/(k_{s}*D_{s}/12))^{0.5}
And,
L_{s} = Length of stud, in 
D_{s} = Diameter of stud, in 
k_{s} = Conductivity of stud,
Btu/hrftF 
The following script will allow us
calculate the coeffcient for stud tubes.
Short Beam, Reflective Radiation
The gas radiation factor, h_{r}, can be calculated from the following
correlations. This factor is used in calculating the overall heat transfer
coefficient for bare tubes and fin tubes. The formulas for the stud tubes has
this factor built into the equations. For bare tubes,
h_{r} = 2.2*g_{r}*(pp*mbl)^{0.50}
And
for fin tubes,
h_{r} = 2.2*g_{r}*(pp*mbl)^{0.50}(A_{po}/A_{o})^{0.75}
Where,
h_{r} = Average outside radiation heat transfer coefficient,
Btu/hrft^{2}F 
g_{r} = Outside radiation factor,
Btu/hrft^{2}F 
pp = Partial pressure of CO_{2} & H_{2}O, ,
atm 
mbl = Mean beam length, ft 
A_{po} = Bare tube exposed surface area,
ft^{2}/ft 
A_{o} = Total outside surface area,
ft^{2} 
Outside radiation
factor, g _{r}:
The outside radiation factor can be described by the following
curves:
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