The HRSG primarily absorbs heat from the hot flue gases by convection heat transfer. In the convection section, heat is transferred by both radiation and convection. The convection transfer coefficients for 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 the HRSG Design is divided into five main areas, which can be selected from the jump links below to go to a section.
Uo = Overall heat transfer coefficient, Btu/hr-ft2-F
Rto = Total outside thermal resistance, hr-ft2-F/Btu
And,
Rto = Ro + Rwo + Rio
Ro = Outside thermal resistance, hr-ft2-F/Btu
Rwo = Tube wall thermal resistance, hr-ft2-F/Btu
Rio = Inside thermal resistance, hr-ft2-F/Btu
And the resistances are computed as,
Ro = 1/he
Rwo = (tw/12*kw)(Ao/Aw)
Rio = ((1/hi)+Rfi)(Ao/Ai)
Where,
he = Effective outside heat transfer coefficient, Btu/hr-ft2-F
hi = Inside film heat transfer coefficient, Btu/hr-ft2-F
tw = Tubewall thickness, in
kw = Tube wall thermal conductivity, Btu/hr-ft-F
Ao = Outside tube surface area, ft2/ft
Aw = Mean area of tube wall, ft2/ft
Ai = Inside tube surface area, ft2/ft
Rfi = Inside fouling resistance, hr-ft2-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
he = 1/(1/(hc+hr)+Rfo)
Where,
hc = Outside heat transfer coefficient, Btu/hr-ft2-F
hr = Outside radiation heat transfer coefficient, Btu/hr-ft2-F
Rfo = Outside fouling resistance, hr-ft2-F/Btu
Outside film heat transfer coefficient, hc:
The bare tube heat transfer film coefficient, hc, can be described by the following equations.
For a staggered tube arrangement,
hc = 0.33*kb(12/do)((cp*mb)/kb)1/3((do/12)(Gn/mb)))0.6
And for an inline tube arrangement,
hc = 0.26*kb(12/do)((cp*mb)/kb)1/3((do/12)(Gn/mb)))0.6
Where,
hc = Convection heat transfer coefficient, Btu/hr-ft2-F
do = Tube outside diameter, in
kb = Gas thermal conductivity, Btu/hr-ft-F
cp = Gas heat capacity, Btu/lb-F
mb = Gas dynamic viscosity, lb/hr-ft
Gn = Mass velocity of gas, lb/hr-ft2
We can use this typical superheater coil as a sample bare tube bank. We will assume the first two rows are bare:
Process Conditions:
Gas flow, lb/hr = 800,000
Gas temperature in, °F = 980
Gas temperature out, °F = 968.9
Compostion, moles
N2, % = 72.55
O2, % = 12.34
CO2, % = 3.72
H2O, % = 10.52
Ar, % = 0.87 Mechanical Conditions:
Tube Diameter, in = 2.000
Tube Spacing, in = 4
Number Tubes Wide = 28
Tube Effective Length, ft = 28.000
Number Of Tubes = 56
Tube Arrangement = Inline Pitch
Gas Properties
For the gas properties, we can use the JavaScript we previously used to get the properties of the gas at the average temperature.
From this program, we get the following properties,
kb, Btu/hr-ft-F = 0.0324
cp, Btu/lb-F = 0.2805
mb, cp = 0.0359 = 0.0359*2.42 = 0.0869 lb/hr-ft
To calculate the mass velocity, Gn, 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 = Nwide*tspc/12*tlgth-Nwide*tOd/12*tlgth = 28*4/12*28-28*2/12*28 = 130.667 ft2
Therefore,
Gn = Wgas / NFA = 800000/130.667 = 6122.448
And using our formula for hc,
hc = 0.26*0.0324 (12/2)((0.2805*0.0869 )/0.0324)1/3((2/12)(6122.448/0.0869 )))0.6 = 12.7152
To get a feel for the values of the coeffcient, use the following script to run various designs.
The radiation transfer coefficient, hr is described later in this section. Fouling resistances, Rfi and Rfo 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 he 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, Uo:
Uo = 1/Rto
Where,
Uo = Overall heat transfer coefficient, Btu/hr-ft2-F
Rto = Total outside thermal resistance, hr-ft2-F/Btu
And,
Rto = Ro + Rwo + Rio
Ro = Outside thermal resistance, hr-ft2-F/Btu
Rwo = Tube wall thermal resistance, hr-ft2-F/Btu
Rio = Inside thermal resistance, hr-ft2-F/Btu
And the resistances are computed as,
Ro = 1/he
Rwo = (tw/12*kw)(Ao/Aw)
Rio = ((1/hi)+Rfi)(Ao/Ai)
Where,
he = Effective outside heat transfer coefficient, Btu/hr-ft2-F
hi = Inside film heat transfer coefficient, Btu/hr-ft2-F
tw = Tubewall thickness, in
kw = Tube wall thermal conductivity, Btu/hr-ft-F
Ao = Total outside surface area, ft2/ft
Aw = Mean area of tube wall, ft2/ft
Ai = Inside tube surface area, ft2/ft
Rfi = Inside fouling resistance, hr-ft2-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:
he = ho(E*Afo+Apo)/Ao
Where,
ho = Average outside heat transfer coefficient, Btu/hr-ft2-F
E = Fin efficiency
Ao = Total outside surface area, ft2/ft
Afo = Fin outside surface area, ft2/ft
Apo = Outside tube surface area, ft2/ft
And, Average outside heat transfer coefficient, ho:
ho = 1/(1/(hc+hr)+Rfo)
Where,
hc = Outside heat transfer coefficient, Btu/hr-ft2-F
hr = Outside radiation heat transfer coefficient, Btu/hr-ft2-F
Rfo = Outside fouling resistance, hr-ft2-F/Btu
Outside film heat transfer coefficient, hc:
hc = j*Gn*cp(kb/(cp*mb))0.67
Where,
j = Colburn heat transfer factor
Gn = Mass velocity based on net free area, lb/hr-ft2
cp = Heat capacity, Btu/lb-F
kb = Gas thermal conductivity, Btu/hr-ft-F
mb = Gas dynamic viscosity, lb/hr-ft
Colburn heat transfer factor, j:
j = C1*C3*C5(df/do)0.5((Tb+460)/(Ts+460))0.25
Where,
C1 = Reynolds number correction
C3 = Geometry correction
C5 = Non-equilateral & row correction
df = Outside diameter of fin, in
do = Outside diameter of tube, in
Tb = Average gas temperature, F
Ts = Average fin temperature, F
Reynolds number correction, C1:
C1 = 0.25*Re-0.35
Where,
Re = Reynolds number
Geometry correction, C3:
For segmented fin tubes arranged in,
a staggered pattern,
C3 = 0.55+0.45*e(-0.35*lf/Sf)
an inline pattern,
C3 = 0.35+0.50*e(-0.35*lf/Sf)
For solid fin tubes arranged in,
a staggered pattern,
C3 = 0.35+0.65*e(-0.25*lf/Sf)
an inline pattern,
C3 = 0.20+0.65*e(-0.25*lf/Sf)
Where,
lf = Fin height, in
sf = Fin spacing, in
Non-equilateral & row correction, C5:
For fin tubes arranged in,
a staggered pattern,
C5 = 0.7+(0.70-0.8*e(-0.15*Nr^2))*e(-1.0*Pl/Pt)
an inline pattern,
C5 = 1.1+(0.75-1.5*e(-0.70*Nr^2))*e(-2.0*Pl/Pt)
Where,
Nr = Number of tube rows
Pl = Longitudinal tube pitch, in
Pt = Transverse tube pitch, in
Mass Velocity, Gn:
Gn = Wg/An
Where,
Wg = Mass gas flow, lb/hr
An = Net free area, ft2
Net Free Area, An:
An = Ad - Ac * Le * Nt
Where,
Ad = Cross sectional area of box, ft2
Ac = Fin tube cross sectional area/ft, ft2/ft
Le = Effective tube length, ft
Nt = Number tubes wide
And,
Ad = Nt * Le * Pt / 12
Ac = (do + 2 * lf * tf * nf) / 12
tf = fin thickness, in
nf = number of fins, fins/in
Surface Area Calculations:
For the prime tube,
Apo = Pi * do (1- nf * tf) / 12
And for solid fins,
Ao = Pi*do(1-nf* tf)/12+Pi*nf(2*lf(do+lf)+tf(do+2*lf))/12
And for segmented fins,
Ao = Pi*do(1-nf* tf)/12+0.4*Pi*nf(do+0.2)/12+Pi*nf (do+0.2)((2*lf-0.4)(wn+tf)+ws*tf)/(12*ws)
And then,
Afo = Ao - Apo
Where,
ws = Width of fin segment, in
In the O-Frame Evaporator example we are using here, we should point out, that some of the tube rows can be used for downcomers instead of risers. Or , alternatively, the downcomers may be outside the gas pass. If they are part of the tube bank, they normally would not be finned, even if the riser tubes are finned. The sketch shown here would indicate the center two rows are downcomers since they are outside the collection baffle which directs the water/vapor mixter, coming from the riser tubes, through the primary separators. In this example, we are not going to consider any of the tubes as downcomers, but if we did, they would absorb heat, but would need to be rated separately since they would have a different inside rate and overall heat transfer coefficient. We can describe a sample fin tube bank as follows:
Process Conditions:
Gas flow, lb/hr = 800,000
Gas temperature in, °F = 898
Gas temperature out, °F = 533
Average fin temperature, °F = 701
Compostion, moles
N2, % = 72.55
O2, % = 12.34
CO2, % = 3.72
H2O, % = 10.52
Ar, % = 0.87
Mechanical Conditions:
Tube Diameter, in = 2.00
Tube Spacing, in = 4
Number Tubes Wide = 28
Tube Effective Length, ft = 28.00
Number Of Tubes = 392
Tube Arrangement = Inline Pitch
Fin Height, in = 0.75
Fin Thickness, in = 0.049
Fin Density, fins/in = 6
Fin Type = Segmented
Fin Segment Width, in = 0.3125
Gas Properties
For the gas properties, we can use the JavaScript we used previously to get the properties of the gas at the average temperature.
From this program, we get the following properties,
kb, Btu/hr-ft-F = 0.0278
cp, Btu/lb-F = 0.2719
mb, cp = 0.0314 = 0.0314*2.42 = 0.0760 lb/hr-ft
To calculate the mass velocity, Gn, 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, An:
Ad = 28*28*4/12 = 261.333
Ac = (2+2*0.75*0.049*6)/12 = 0.2034
So,
hc = 0.0044*7853.3466*0.2719(0.0278/(0.2719*0.0760))0.67 = 11.475
To get a feel for the values of the coeffcient, use the following script to run various designs.
The radiation transfer coefficient, hr is described later in this section. Fouling resistances, Rfi and Rfo are allowances that depend upon the process or service of the HRSG 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(df / do) * (y - 1) + 1)
Where,
y = x * (0.7 + 0.3 * x)
And,
x = tanh(m * B) / (m * B)
Where,
B = lf + (tf /2)
For segmented fins,
m = (ho (tf + ws) / (6 * kf * tf * ws))0.5
And for solid fins,
m = (ho / (6 * kf * tf))0.5
Fin Tip Temperature, Ts:
The average fin tip temperature is calculated as follows,
Uo = Overall heat transfer coefficient, Btu/hr-ft2-F
Rto = Total outside thermal resistance, hr-ft2-F/Btu
And,
Rto = Ro + Rwo + Rio
Ro = Outside thermal resistance, hr-ft2-F/Btu
Rwo = Tube wall thermal resistance, hr-ft2-F/Btu
Rio = Inside thermal resistance, hr-ft2-F/Btu
And the resistances are computed as,
Ro = 1/he
Rwo = (tw/(12*kw))(Ao/Aw)
Rio = ((1/hi)+Rfi)(Ao/Ai)
Where,
he = Effective outside heat transfer coefficient, Btu/hr-ft2-F
hi = Inside film heat transfer coefficient, Btu/hr-ft2-F
tw = Tubewall thickness, in
kw = Tube wall thermal conductivity, Btu/hr-ft-F
Ao = Outside surface area, ft2/ft
Aw = Mean area of tube wall, ft2/ft
Ai = Inside tube surface area, ft2/ft
Rfi = Inside fouling resistance, hr-ft2-F/Btu
Effective outside heat transfer coefficient, he:
For staggered and inline pitch,
he = (hso*E*Afo+ht*Apo)/Ao
Where,
ht = Base tube outside heat transfer coefficient, Btu/hr-ft2-F
hso = Stud outside heat transfer coefficient, Btu/hr-ft2-F
Ao = Total outside surface area, ft2/ft
Afo = Stud outside surface area, ft2/ft
Apo = Tube outside surface area, ft2/ft
Inline pitch correction,
he = he*(do/Pl)0.333
Where,
do = Outside tube diameter, in
Pl = Longitudinal pitch of tubes, in
Base tube outside heat transfer coefficient, ht:
ht = (0.717/do0.333)(Gn/1000)0.67(Tb+460)0.3
And the stud coefficient,
hs = 0.936*(Gn/1000)0.67(Tb+460)0.3
With fouling,
hso = 1/(1/hs+Rfo)
Where,
hs = Stud outside heat transfer coefficient, Btu/hr-ft2-F
Gn = Mass velocity of flue gas, lb/hr-ft2
Tb = Average gas temperature, F
Stud efficiency, E:
E = 1/((ex+e-x)/1.950)
Where,
X = Ls/12((2*hso)/(ks*Ds/12))0.5
And,
Ls = Length of stud, in
Ds = Diameter of stud, in
ks = Conductivity of stud, Btu/hr-ft-F
The following script will allow us calculate the coeffcient for stud tubes.
Short Beam, Reflective Radiation
The gas radiation factor, hr, 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,
hr = 2.2*gr*(pp*mbl)0.50
And for fin tubes,
hr = 2.2*gr*(pp*mbl)0.50(Apo/Ao)0.75
Where,
hr = Average outside radiation heat transfer coefficient, Btu/hr-ft2-F
gr = Outside radiation factor, Btu/hr-ft2-F
pp = Partial pressure of CO2 & H2O, , atm
mbl = Mean beam length, ft
Apo = Bare tube exposed surface area, ft2/ft
Ao = Total outside surface area, ft2
Outside radiation factor, g r:
The outside radiation factor can be described by the following curves:
Convection Section Design
The following calculator will allow you to calculate the overall heat transfer coefficient for fin tubes, stud tubes, or bare tubes. This calculator uses the methods described above.
