Convection Transfer, Bare Tubes  
Convection Transfer, Fin Tubes  
Convection Transfer, Stud Tubes  
Short Beam, Reflective Radiation  
Convection Section Design 
U_{o} = Overall heat transfer coefficient, Btu/hrft^{2}F 
R_{to} = Total outside thermal resistance, hrft^{2}F/Btu 
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 
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}) 
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 
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 
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} 
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 
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.
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} = Overall heat transfer coefficient, Btu/hrft^{2}F 
R_{to} = Total outside thermal resistance, hrft^{2}F/Btu 
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 
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}) 
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 
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 
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 
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 
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 
R_{e} = Reynolds number 
l_{f} = Fin height, in 
s_{f} = Fin spacing, in 
N_{r} = Number of tube rows 
P_{l} = Longitudinal tube pitch, in 
P_{t} = Transverse tube pitch, in 
W_{g} = Mass gas flow, lb/hr 
A_{n} = Net free area, ft^{2} 
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 
w_{s} = Width of fin segment, in 
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 
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:T_{sm} = Maximum Fin Tip Temperature, F 
T_{gm} = Maximum Gas Temperature, F 
T_{wm} = Maximum Tube Wall Temperature, F 
U_{o} = Overall heat transfer coefficient, Btu/hrft^{2}F 
R_{to} = Total outside thermal resistance, hrft^{2}F/Btu 
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 
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}) 
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 
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 
d_{o} = Outside tube diameter, in 
P_{l} = Longitudinal pitch of tubes, in 
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 
L_{s} = Length of stud, in 
D_{s} = Diameter of stud, in 
k_{s} = Conductivity of stud, Btu/hrftF 
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} = 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} 
The outside radiation factor can be described by the following curves: