Boiler systems contain many heat exchangers. In these devices, the fluid temperature changes as the fluids pass through the equipment. With an energy balance specified between two locations, 1 and 2:
q = mc T p
(
2 −T1)
(63)the change in fluid temperature can be calculated:
T2 = T1 +
(
q mc/ p)
(64)It is therefore appropriate to define a mean effective temperature difference governing the heat flow. This difference is determined by performing an energy bal- ance on the energy lost by the hot fluid and that en- ergy gained by the cold fluid. An equation of the form:
q =UA F T∆ LMTD (65)
is obtained where the parameters U, A and F define the overall heat transfer coefficient, surface area, and ar-
Fig. 23 Heat transfer depth factor for number of tube rows crossed
in convection banks. (Fd = 1.0 if tube bank is immediately preceded by a bend, screen or damper.)
rangement correction factor, respectively. The term
∆TLMTD, known as the log mean temperature difference,
is defined as: ∆ ∆ ∆ ∆ ∆ T T T n T T LMTD =
(
1 − 2)
1 2 / (66)∆T1 is the initial temperature difference between the hot
and cold fluids (or gases), while ∆T2 defines the final
temperature difference between these media. The pa- Fig. 22 Arrangement factor, Fa, as affected by Reynolds number for various in-line tube patterns, commercially clean tube conditions for crossflow of ash-laden gases.
2
F = Arrangement Factor for In-Line Tube Banks
0.9 0.8 0.7 0.6 0.5 0.9 0.8 0.7 0.6 0.5 0.4 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 2 - 3 1.5 1.5 1.25 1 3 1.25 1.8 1.25 - 1.5 Reynolds No. = 40,000 Reynolds No. = 20,000 1 2 3 1 1 Reynolds No. = 8,000 1.8 2 3 Reynolds No. = 2,000 2 1.25 - 1.5 3
Tube Spacing Transverse to Gas Flow Outside Tube Diameter
Tube Spacing in Direction of Gas Flow Outside Tube Diameter Curves Denoted By:
Fig. 21 Arrangement factor, Fa, as affected by Reynolds number for various in-line tube patterns, clean tube conditions for crossflow of air or natural gas combustion products.
3
F = Arrangement Factor for In-Line Tube Banks
1.1 1.0 0.9 0.8 0.7 0.6 0.5 1.1 1.0 0.9 0.8 0.7 0.6 0.5 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 2 - 3 1 1.5 1.8 3 1.25 Reynolds No. = 2,000 Reynolds No. = 8,000 Reynolds No. = 20,000 Reynolds No. = 40,000 2 1.8 1.5 1.25 1 1 3 1.25 - 1.5 3 1 2 2 1.25 - 1.5 Tube Spacing in Direction of Gas Flow
Outside Tube Diameter Curves Denoted By:
Tube Spacing Transverse to Gas Flow Outside Tube Diameter
rameter U in Equation 65 defines the overall heat trans- fer coefficient for clean surfaces and represents the unit thermal resistance between the hot and cold fluids:
1 1 1
UA h Ai i Rw h A
o o
clean
= + + (67)
For surfaces that are fouled, the equation is written:
1 1 UA R A UA R A f i i f o o = , + + , clean (68)
where Rf,i is the reciprocal effective heat transfer co- efficient of the fouling on the inside surface, ( l / UA) is the thermal resistance and Rf,o is the reciprocal heat transfer coefficient of the fouling on the outside sur- face. Estimates of overall heat transfer coefficients and fouling factors are listed in Tables 10 and 11. Actual fouling factors are site specific and depend on water chemistry and other deposition rate factors. Overall heat transfer coefficients can be predicted using: 1) the fluid conditions on each side of the heat transfer surface with either Equation 56 or 59, 2) the known materials of the heat transfer surface, and 3) the foul- ing factors listed in Table 11. Often the heat exchanger tube wall resistance (Rw) is small compared to the sur- face resistances and can be neglected, leading to the following equation for a clean surface:
U h h h h D D i o i o o i = +
(
/)
(69)This equation assumes that area, A in product UA, is based on the outside diameter of the tube, Do.
The difficulty in quantifying fouling factors for gas-, oil- and coal-fired units has led to use of a cleanliness
factor. This factor provides a practical way to provide
extra surface to account for the reduction in heat trans- fer due to fouling. In gas-fired units, experience indi- cates that gas-side heat transfer coefficients are higher as a result of the cleanliness of the surface. In oil- and coal-fired units that are kept free of slag and deposits, a
Table 11
Selected Fouling Factors
Type of Fluid h ft2 F/Btu m2 K/W
Sea water above 125F (50C) 0.001 0.0002 Treated boiler feedwater
above 125F (50C) 0.001 0.0002
Fuel oil 0.005 0.0010
Alcohol vapors 0.0005 0.0001
Steam, non-oil bearing 0.0005 0.0001
Industrial air 0.002 0.0004
lower value is used. For units with difficult to remove deposits, values are reduced further.
There are three general heat transfer arrange- ments: parallel flow, counterflow and crossflow, as shown in Fig. 24. In parallel flow, both fluids enter at the same relative location with respect to the heat transfer surface and flow in parallel paths over the heating surface. In counterflow, the two fluids enter at opposite ends of the heat transfer surface and flow in opposite directions over the surface. This is the most efficient heat exchanger although it can also lead to the highest tube wall metal temperatures. In crossflow, the paths of the two fluids are, in general, perpendicu- lar to one another.
Fig. 24 shows the flow arrangements and presents Equation 66 written specifically for each case. The ar- rangement correction factor, F, is 1.0 for parallel and counterflow cases. For crossflow and multi-pass ar- rangements, the correction factors are shown in Figs. 25 and 26.
Extended surface heat transfer
The heat absorption area in boilers can be increased using longitudinally and circumferentially finned tubes. Finned, or extended, tube surfaces are used on the flue gas side. In regions prone to fouling, the fins must be spaced to permit cleaning. Experimental data on actual finned or extended surfaces are preferred for design purposes; the data should be collected at conditions similar to those expected to be encountered. However, in place of these data, the method by Schmidt15 generally describes the heat transfer across
finned tubes. It is based on heat transfer to the un- derlying bare tube configuration, and it treats the tube as if it has zero fin height. Schmidt’s correlation for the gas-side conductance to tubes with helical, rect- angular, circular, or square fins is as follows:
h h Z S S f c f f = − −
(
)
1 1 η (70)where hc is the heat transfer coefficient of the bare tubes in crossflow defined by Equations 59 and 60, and Z is the geometry factor defined as:
Z L L h t = − 1 0 18 0 63 . . (71) Table 10
Approximate Values of Overall Heat Transfer Coefficients
Physical Situation Btu/h ft2
F W/m2
K
Plate glass window 1.10 6.20
Double plate glass window 0.40 2.30 Steam condenser 200 to 1000 1100 to 5700 Feedwater heater 200 to 1500 1100 to 8500 Water-to-water heat
exchanger 150 to 300 850 to 1700 Finned tube heat exchanger,
water in tubes, air
across tubes 5 to 10 30 to 55 Water-to-oil heat exchanger 20 to 60 110 to 340 Steam-to-gas 5 to 50 30 to 300 Water-to-gas 10 to 20 55 to 110
Sf represents the fin surface area including both sides and the peripheral area, while S represents the ex- posed bare tube surface between the fins plus the fin surface, Sf. The ratio Lh/Lt is the fin height divided by the clear spacing between fins. Fin efficiency, ηf, is shown in Fig. 27 as a function of the parameter X, defined as:
X = Lh 2Zhc/
( )
k Lf t (72)for helical fins, and
X = r Y 2Zhc/
( )
k Lf t (73)for rectangular, square or circular fins. The param- eter Y is defined in Fig. 28.
The overall conductance can be written:
1 1 1
UA C A hf o f o Rw A h
i c i
= + +
, , (74)
The parameter Cf is the surface cleanliness factor. NTU method
There are design situations for which the perfor- mance of the heat exchanger is known, but the fluid temperatures are not. This occurs when selecting a unit for which operating flow rates are different than those Fig. 25 Arrangement correction factors for a single-pass,
crossflow heat exchanger with both fluids unmixed.
Fig. 26 Arrangement correction factors for a single-pass,
crossflow heat exchanger with one fluid mixed and the other unmixed (typical tubular air heater application).
previously tested. The outlet temperatures can only be found by trial and error using the methods previously presented. These applications are best handled by the net transfer unit (NTU) method that uses the heat exchanger effectiveness (see Reference 16).
Heat transfer in porous materials
Porosity is an important factor in evaluating the ef- fectiveness of insulation materials. In boiler applica- tions, porous materials are backed up by solid walls or casings, so that there is minimal flow through the pores. Heat flow in porous insulating materials occurs by conduction through the material and by a combina- tion of conduction and radiation through the gas-filled voids. In most refractory materials, the Grashof- Prandtl (Raleigh) number is small enough that neg- ligible convection exists although this is not the case in low density insulations [< 2 lb/ft3 (32 kg/m3)]. The
relative magnitudes of the heat transfer mechanisms depend, however, on various factors including poros- ity of the material, gas density and composition fill- ing the voids, temperature gradient across the mate- rial, and absolute temperature of the material.
Analytical evaluation of the separate mechanisms is complex, but recent experimental studies at B&W have shown that the effective conductivity can be approximated by:
keff = a bT+ +cT3 (75)
Experimental data can be correlated through this form, where a, b and c are correlation coefficients. The heat flow is calculated using Equation 1; k is replaced by keff and T is the local temperature in the insulation.
Fig. 28 Coefficient Y as a function of ratio R/r for fin efficiency.
In high temperature applications, heat transfer across the voids occurs mainly by radiation and the third term of Equation 75 dominates. In low tempera- ture applications, heat flow by conduction dominates and the first two terms of Equation 75 are controlling. Film condensation
When a pure saturated vapor strikes a surface of lower temperature, the vapor condenses and a liquid film is formed on the surface. If the film flows along the surface because of gravity alone and a condition of laminar flow exists throughout the film thickness, then heat transfer through this film is by conduction only. As a result, the thickness of the condensate film has a direct effect on the quantity of heat transferred. The film thickness, in turn, depends on the flow rate of the condensate. On a vertical surface, because of drainage, the thickness of the film at the bottom will be greater than at the top. Film thickness increases as a plate surface is inclined from the vertical position. As the film temperature increases, its thickness decreases primarily due to increased drainage veloc- ity. In addition, the film thickness decreases with in- creasing vapor velocity in the direction of drainage. Mass diffusion and transfer
Heat transfer can also occur by diffusion and mass transfer. When a mixture of a condensable vapor and a noncondensable gas is in contact with a surface that is below the dew point of the mixture, some conden- sation occurs and a film of liquid is formed on the sur- face. An example of this phenomenon is the conden- sation of water vapor on the outside of a metal con- tainer. As vapor from the main body of the mixture diffuses through the vapor-lean layer, it is condensed on the cold surface as shown in Fig. 29. The rate of condensation is therefore governed by the laws of gas diffusion. The heat transfer is controlled by the laws of conduction and convection.
The heat transferred across the liquid layer must equal the heat transferred across the gas film plus the latent heat given up at the gas-liquid interface due to condensation of the mass transferred across the gas film. An equation relating the mass transfer is:
h T Tδ
(
i− δ)
= h Tg(
g−Ti)
+ K H Yy fg(
g−Yi)
(76)where T and Y define the temperatures and concen- trations respectively identified in Fig. 29, hδ is the heat transfer coefficient across the liquid film, hg is the heat transfer coefficient across the gas film, and
Ky is the mass transfer coefficient. Hfg is the latent heat of vaporization.
Heat transfer due to mass transfer is important in de- signing cooling towers and humidifiers, where mixtures of vapors and noncondensable gases are encountered. Evaporation or boiling
The phenomenon of boiling is discussed in Chap- ters 1 and 5, where the heat transfer advantages of nucleate boiling are noted. Natural-circulation fossil fuel boilers are designed to operate in the boiling range. In this range, the heat transfer coefficient var- Fig. 27 Fin efficiency as a function of parameter X.
ies from 5000 to 20,000 Btu/h ft2 F (28,392 to 113,568
W/m2 K). This is not a limiting factor in the design of
fossil fuel boilers provided scale and other deposits are prevented by proper water treatment, and provided the design avoids critical heat flux (CHF) phenomena. (See Chapter 5.)
In subcritical pressure once-through boilers, water is completely evaporated (dried out) in the furnace wall tubes which are continuous with the superheater tubes. These units must be designed for subcooled nucleate boiling, nucleate boiling, and film boiling, depending on fluid conditions and expected maximum heat absorption rates.
Fluidized-bed heat transfer
The heat transfer in gas-fluidized particle beds used in some combustion systems is complex, involving particle-to- surface contact, general convection and particle-to-surface thermal radiation. Correlations for heat transfer to tube bundles immersed in fluidized beds are summarized in Chapter 17.