Studies in Furnace Analysis: Prediction of the Variation of the Specific

Energy & Fuels 1997, 11, 447-456

447

Studies in Furnace Analysis: Prediction of the Variation of the Specific Exhaust Enthalpy with Output Robert H. Essenhigh* E. G. Bailey Professor of Energy Conversion, Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210

Andre L. Boehman Fuel Science Program, Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received July 17, 1996X

Prediction of the overall performance of furnaces and engines in terms of the operational and intrinsic thermal efficiencies (η and R00) has been limited in the past by need to assume empirically a linear dependence of the specific exhaust enthalpy, hg, on the useful output (heat or work), Hs. In this paper, a theoretically-based relation between hg and Hs is developed using an integral approach, following Hottel and Thring, by treatment of furnace enclosures as heat exchangers. The expression developed is shown to be supported by comparison with experiment, using measurements on a boiler obtained by Prengle at the University of Houston. The new, theoretically-based and experimentally-supported relation shows that the variation between hg and Hs is nonlinear: it also depends on a new parameter, R0, that is the ratio, at idle, of the exhaust gas temperature above ambient to the average flame temperature above ambient. The hg-Hs curve is near-linear for values of R0 near unity, and the nonlinearity increases as R0 decreases. The near-linear case is expected to be valid for hot wall furnaces which have high exhaust temperatures, with the nonlinear case valid for systems with high heat recovery and low exhaust temperatures, as in boilers. In this study, the numerical value of R0 obtained for the boiler is small (0.09) and the hg-Hs curve, as expected, is correspondingly highly nonlinear over the full potential range of output. As a general result of the analysis, the dependence of the linear or nonlinear pattern of behavior on R0 thus creates a clear target for future test by prediction from detailed mechanistic models.

Introduction In this paper we address the long-standing problem in furnace and engine analysis studies1,2 of prediction of the dependence of the specific exhaust enthalpy (hg) on output enthalpy (Hs). This extension is required for adequate closure of the furnace analysis model structure that, based on the integral energy equation,3 predicts the three furnace and engine performance equations. These performance equations describe the dependence on output, Hs, of the firing rate, Hf, in the firing equation; the operational thermal efficiency, η; and the Thring and Reber4 heat utilization factor (HUF), R, where the standard definitions for η and R4 are

η ) Hs/Hf

(1)

R ) Hs/(Hf - Hf0)

(2)

and where Hf0 is the idle heat (energy supply rate at zero output). * Corresponding author. Tel: (614) 292-0403. Fax: (614) 292-3163. E-mail: [email protected]. X Abstract published in Advance ACS Abstracts, February 15, 1997. (1) Essenhigh, R. H.; Thekdi, A. C.; Malhouitre, G.; Tsai, Y. Combustion Technology: Some Modern Developments; Academic press: New York, 1974; Chapter 13.

S0887-0624(96)00111-9 CCC: $14.00

Since the integral energy equation equates the thermal input (Hf) to the sum of the useful output (Hs), the wall loss (Hw), and the exhaust loss (Hg), this represents one equation with four unknowns. The closure problem this creates, to obtain the firing equation, showing Hf as a function of Hs, is the need for supplementary relations to provide a match between degrees of freedom and constraints, to be able to eliminate two of the defined properties. The two properties eliminated in the standard analysis1,2 are the wall loss and the exhaust loss that, independently, must be defined in terms of Hs. Of these two, the more important on account of its relative magnitude is the exhaust loss, most generally given in the specific formulation, hg (thermal units per unit of fuel fired). The variation of hg with Hs most commonly adopted in the past was the assumption of a linear dependence. This was generally introduced as a semiempirical, first-approximation assumption though partly based on limited experimental data. Using this linear assumption, the first-approximation form of the firing equation thereby obtained is1 (2) Essenhigh, R. H. Comparative Thermal Behavior of Furnaces and Engines: Prediction of Thermal Efficiency in Real Time; In Thermal Analysis and Improvement of Thermal Systems; International Academic Press Publishers (Pergamon-CNPIRC): Pergamon Press, Oxford, UK, 1990; pp 116-125. (3) Moran, M. J.; Shapiro, H. N. Fundamentals of Engineering Thermodynamics; John Wiley: New York, 1995. (4) Thring, M. W.; Reber, J. W. J. Inst. Fuel 1945, 18, 12.

© 1997 American Chemical Society

448 Energy & Fuels, Vol. 11, No. 2, 1997

Hf ) Hf0 + Hs/[R0(1 - Hs/Hsm)]

Essenhigh and Boehman

(3)

and, at a first approximation, this equation now has substantial experimental support, as reviewed elsewhere.1,2 Notably, at that first approximation, the parameter R0 (the intrinsic efficiency factor) is a constant. With improved experimental data, however, the weakness of the linear assumption for the dependence of hg on Hs, as a general result, has now become evident, and this provided the incentive and context for the studies reported in this paper. What we show from the revised development is that the general structure of eq 1 can be retained as a useful formulation, but with R0 shown to be substantively dependent on output as a secondorder perturbing factor. A limiting intrinsic efficiency value, written as R00, can also be defined and, independently, it can be shown2 that this corresponds to the theoretical thermodynamic efficiency obtained at the classic limit of infinite time. Context of the Problem The problem of the dependence of hg on Hs has a long history. Furnace analysis originated with studies on a boiler by Hudson in 18905 which, with extensions by Orrock in 1926,6 led to the classic Hudson-Orrock (HO) equation6 as the first version of the firing equation. Those early approaches did use an external energy balance akin to that represented by the integral energy equation, but the development was strictly empirical, and the nature of the closure problem was not seen as an issue at that time. It was recognized as an issue, however, in all the subsequent theoretical developments, initiated in the 1930s and 1940s, and, particularly, by Hottel7,8 and by Thring.4,9 This recognition was substantially implicit with Hottel but was stated explicitly by Thring. For the wall loss, Hottel assumed with an emphasis primarily on boilers that it was so sufficiently small that it could be approximated as zero, and this, within limits, is a reasonable first approximation for boilers. Correspondingly, the idle heat, Hf0, is then so small that, over much of the higher output range, we have, from eqs 1 and 2, that η ≈ R. Importantly, however, the two parameters deviate sharply at low outputs: as output drops to zero, η also goes to zero, but R rises to its maximum, the intrinsic efficiency value (R0). Conversely, Thring in analyzing mostly steel-plant (hotwall) furnaces where the wall loss is not small, included the wall loss, and its inclusion in the analysis was a significant development as this introduced the concept of the heat utilization factor, R, as defined by eq 2, with its limiting value at zero output of the intrinsic efficiency, R0. For the exhaust loss, hg, both investigators implicitly identified the need for a degree of internal analysis as a requirement to obtain necessary closure. Thring used a one-dimensional conduction-type of internal model and (5) Hudson, J. G. Engineer 1980, 70, 449, 483, 523. (6) Orrock, G. A. Trans. Am. Soc. Mech. Eng. 1926, 48(3), 218. (7) Hottel, H. C. Heat Transmission; John Wiley: New York, 1942. (8) Hottel, H. C.; Sarofim, A. F. Radiative Transfer, McGraw-Hill: New York, 1967. (9) Thring, M. W. The Science of Flames and Furnaces; Wiley: New York, 1962.

obtained solutions in terms of the exhaust temperature, Tg. Hottel used a radiation model, first assuming a substantially uniform flame temperature as an average throughout the furnace volume and, subsequently, incorporated the furnace load uniformly in the walls (the “speckled” furnace) to be able, in the absence at that time of adequate computing machinery, to obtain working solutions to the equations. Hottel also introduced a temperature difference between the exhaust temperature, Tg, and the average flame temperature in the furnace, Tf. Elements of all these assumptions are used in the treatment in this paper. With the increasing availability of computing machinery, in the 1960s/1970s, this predominantly phenomenological/integral approach was largely abandoned in favor of direct computation of the device performance. The initial approach to this, because of the recognized complexity of the systems being modeled, was a middle ground procedure. This was the use of partial modeling in conjunction either with experimental profiles measured internally or with (external) experimental performance data used to calibrate the model. The modelcalibration approach was used particularly successfully by Bueters et al.10 in the 1950s/1960s for purposes of both design and performance prediction of (CE) utility boilers. The experimental profile approach was used by Enomoto et al.11 to compare predicted heat transfer to a water-cooled load with experimental measurements in a (laboratory) furnace using a detailed radiation (zone-method) model, with data input from experimentally-measured (internal) temperature and gas concentration (3-D) profiles. More recently,12 a similar approach has been used by the addition of coal to a DNS flow program, again in conjunction with (3-D) profiles of temperature and oxygen concentration (from an IFRF furnace); this predicts particle combustion behavior in a furnace enclosure that is also at a greater level of detail than is obtainable from commonly-used modeling approximations such as the k- procedure. In the more recent work, however, particularly of the last two decades, the preference has been to pursue increasingly detailed computer simulations, without direct incorporation of experimental data. Smoot,13 for example, lists over a dozen different model programs for coal-fired boilers; and for other systems, such as internal combustion engines, there are specific programs such as SPEED,14 adaptable for engines using approximate submodels such as KIVA.15 There are also general fluid dynamics programs, such as FLUENT (e.g., ref 16), or statistical alternatives such as the coalescence-dispersion (CD) modeling17 that combined (10) Bueters, K. A.; Cogoli, J. G.; Habelt, W. W. Proceedings of the 15th Symposium (International) on Combustion; Combustion Institute: Pittsburgh, PA, 1975; pp 1245-1260. (11) Enomoto, H.; Tsai, Y.-W.; Essenhigh, R. H. Heat Transfer in a Continuous Model Furnace: A Comparison of Theory and Experiment; ASME paper No. 75-HT-5 (1975). (12) Bender, C.; Mittal, M.; Essenhigh, R. H. Proc. SIAM 6th Int. Conf. Numerical Combust., New Orleans 1996. (13) Smoot, L. D., Ed. Fundamentals of Coal Combustion; Elsevier: New York, 1993. (14) Adamson, B.; Gosman, A. D.; Marooney, C. J.; Nasseri, B.; Theodoroupoulos, T. A New Unstructured Mesh Method for Flow Prediction in Internal Combustion Engines; Proc. Int. Symp. Commodia 1990, 90, 431-436. (15) Amsden, A. A.; O’Rourke, P. J.; Butler, T. D. KIVA-II: A Computer Program for Chemically reacting Flows with Sprays, Los Alamos National Laboratory, LA-11560-MS (1989).

Studies in Furnace Analysis

with a Monte-Carlo treatment18 can be applied to a variety of combustion systems (e.g., ref 19). A more recent approach for computing fluids behavior is the lattice-gas or lattice-Boltzman procedure20 that shows particular promise, although use in combustion calculations is evidently minimal at this time. The value in those more detailed approaches is the increased gain in specificity of prediction, but with the consequence that such analyses are increasingly restricted to limited groups or subsets of furnaces or engines and it is necessary to develop individual computer or predictive models for each subset. Additionally, an operational problem with all such detailed programs is the question of the accuracy and/or validity of the subroutines contained in the full programs which may be untested in the parameter range used in particular model applications. Consequently, the increased gain in specificity of prediction can be at the expense, as is increasingly being recognized, of loss of generality; likewise, the accuracy of prediction may be arguable if subroutines are not fully tested. In short, this identifies a dichotomy, as the boundaries of options in treating the overall problem of describing furnace and engine behavior, between detailed but application-limited analyses set against less detailed but more generally applicable models. Although the trend in the last two decades has been, as noted, in favor of the detailed, mechanistic-type or differential models, in preference to the more general integral model, the need for a degree of generality is not thereby voided, and part of the incentive to revisit the integral analytical approach is the value of such generality. This is illustrated, for example, in a recent treatment to determine the performance characteristics of a furnace fired with coalwater fuel (CWF).21 Accordingly, in parallel with the more detailed partialmechanistic approaches as in refs 11 and 12, the option being pursued here is to investigate the properties of a very general internal analysis treatment of the thermaldevice behavior with the objective of preserving generality, even at the expense of some potential loss of predictive accuracy. This point is revisited in the Discussion. The approach, following both Thring and Hottel, is to regard the furnace or engine as a heat exchanger. The extent to which this is an acceptable generality is to be determined by comparison with experiment, and an example selected at random is included in this paper. Theory Model and Primary Assumptions. The theoretical analysis, as outlined in the Introduction, is based on an energy balance, using the integral energy equation3 on a control volume enclosing a thermal device (furnace or engine), as detailed in relevant reviews.1,2 In sum(16) Sharifi, R.; Scaroni, A. W.: The Combustion Aerodynamics of a Pulverized Coal Low NOX Swirl Burner in an Industrial Boiler. Presented at the International Joint Power Generation Conference, Houston, Oct 1996. (17) Curl, R. C. I. Chem. E. J. 1963, 9, 175-181. (18) Speilman, L. A.; Levenspiel, O. Chem. Eng. Sci. 1965, 20, 247254. (19) Pratt, D. T. J. Energy 1979, 3, 177, 178. (20) Doolan, G. D., Ed. Lattice Gas Methods: Theory, Applications, and Hardware; Bradford Books/MIT Press: Cambridge, MA, 1996. (21) Essenhigh, R. H.; Obloza, J. J.; Hammond, T. K. Energy Fuels 1994, 8, 179-187.

Energy & Fuels, Vol. 11, No. 2, 1997 449

mary, for an entering thermal (fuel) stream supplying energy at a rate Hf (thermal units/h), a balance on the sole resultant leaving streams (useful output, Hs; exhaust loss, Hg; and wall loss, Hw) gives

Hf ) Hw + Hg + Hs

(4a)

with a similar expression for conditions at idle (when Hs ) 0):

Hf0 ) Hw0 + Hg0

(4b)

Using the standard conversion1 of Hg into exhaust specific enthalpy, hg (per unit of fuel fired; see Nomenclature), written as Hg ) Hf(hg/hf); we have after substitution and subtraction of eq 4b from eq 4a and rearrangement

Hf(1 - hg/hf) - Hf0(1 - hg0/hf) ) Hs + (Hw - Hw0) (5a) ) (1 + Kw)Hs

(5b)

where Hw is assumed in this presentation, on the basis of substantial experimental evidence,1,2,22 to increase linearly with output, thus

Hw ) Hw0 + KwHs

(5c)

Using the same transformation for Hg0 as for Hg, we also get from eq 4b the important condition at idle that

Hw0 ) Hf0(1 - Hg0/Hf0) ) Hf0(1 - hg0/hf) ) R00Hf0 (5d) where R00 ) (1 - hg0/hf), and is the limiting intrinsic efficiency.1 This is used later to get a value of Hf0 from Hw0. To reduce eq 5b to a function of the general form required for the firing equation, Hf ) f(Hs), requires elimination of the (normalized) specific enthalpy (hg/hf) by substitution of hg as a function of output, Hs; this procedure also identifies the significance of this hg-Hs relation in the general development. Using the semiempirical but common first approximation for (hg/hf) that it varies linearly with output, we get after appropriate algebraic rearrangement the first approximation form of the firing equation, eq 3, as set out elsewhere.1,2 The two other performance equations are then obtained using the standard definitions, eqs 1 and 2, for the operational efficiency and the HUF (R). The linearization of rising (hg/hf) with output also results in a linearly-declining variation of R with output, thus:

R ) Hs/(Hf - Hf0) ) R0(1 - Hs/Hsm)

(6)

where the parameter R0, as noted earlier, is a constant in this first approximation solution and is the first approximation to the limiting intrinsic efficiency, R00. As noted in the Introduction, however, the linearity of hg with Hs under most conditions may only be valid for many devices for a limited range of values of Hs; and over the full range, 0 < Hs < Hsm, the variation of hg with Hs may have significant curvature. Adequate (22) Prengle, H. W. University of Houston, 1975, private communication.



prediction of this result, however, has been lacking in the past and it is now developed here (below) by treating the thermal device, following Hottel and Thring, as a heat exchanger. Our starting point is, substantially, the first approximation analysis used by Thring as this is valuable in setting up the approach; this also recovers the initial empirical linear result, thus providing a theoretical basis for that result, and this also provides important comparisons with the more general extension. The Furnace as Heat Exchanger: First Approximation. In the Thring and Reber approach,4 the furnace is regarded as a heat exchanger between hot gases at a temperature represented by an average temperature, Tg, the exhaust gas temperature, and a load at an average temperature, Ts. The heat exchanged is between the flame and the load and walls. Introducing a general heat exchange factor, β, for this process, we write for the general condition and for the Idle condition:

Hs + Hw ) β(Tg - Ts)

(7a)

Hw0 ) β(Tg0 - Ts)

(7b)

Subtracting eq 7b from eq 7a eliminates Ts; and determining the conditions at which Hs ) Hsm, and Tg ) Tad, this procedure leads to two equations that when ratioed eliminates β with the temperature differences in the form Tg - Tg0. As shown in Appendix A, the equations then transform into the linear relation of eq 8a (below) by reforming the temperature differences as [(Tg - T0) - (Tg0 - T0)], and dividing by Tad - T0. The final equation, using the definitions in the Nomenclature is

hg/hf ) hg0/hf + R00(Hs/Hsm)

(8a)

R00 ) 1 - hg0/hf

(8b)

into their analysis to take this into account. We now use this extension in a more general form in this treatment by rewriting eq 7, in the more general form:

Hs + Hw ) β(Tf - Ts)

(9a)

Hw0 ) β(Tf0 - Ts)

(9b)

where Tf is an average flame temperature throughout the furnace enclosure. The procedure for equation transformation is then substantially as before, summarized in Appendix B, first subtracting eq 9b from eq 9a to eliminate Ts, but now introducing the temperature differences between flame and exit using, first, [(Tf T0) - (Tf0 - T0)], and then multiplying the difference relations by the unit ratio, [(Tg - T0)/(Tg - T0)], so that, in the next step of division by the term Tad - T0, the term groups that are obtained then have the form {[(Tf - T0)/(Tg - T0)](hg/hf)} as obtained in the eq B4 (Appendix B). This treatment shows that the specific exhaust enthalpy ratio, hg/hf, is now multiplied by the temperature ratio term involving the ratio of the average flame temperature (difference from ambient) and the exhaust temperature (difference from ambient). Hottel and Sarofim set this at a nominally fixed value (since its variability was not an essential concern in their treatment); but the ratio does, in fact, converge to unity with increasing output since, from eq 3, the firing rate at maximum output is infinite, and the system is then at the adiabatic flame temperature: i.e., Tf ) Tg ) Tad. The difference, Tf - Tg, declines from a maximum at idle (Hs ) 0), to zero at maximum output. The relation can most conveniently be written:

(Tf - T0)/(Tg - T0) ) (Tf0 - T0)/(Tg0 - T0)[1 - (1 - R0)(Hs/Hsm)] (10a)

where

where

Equation 8a is the standard, linear (first approximation) form of the required variation of specific exhaust enthalpy with output that has been in use for over two decades, and this treatment provides a theoretical basis in place of the empirical/experimental basis of the past. There are numerous approximations implicitly invoked in this treatment, however. First, the load and walls are at approximately the same temperature, and second, the heat exchange factor, β, is substantially independent of temperature. These assumptions are reviewed in the Discussion. The most important assumption, however, is that the temperature of the exhaust gases is an adequate stand-in for the average temperature of the flame gases; this is considered next. The Furnace as Heat Exchanger: General Formulation. The obvious objection to the first approximation treatment given above was, as pointed out by Hottel and Sarofim,8 that it assumed that the flame temperature, Tf, was so sufficiently well approximated by the exhaust gas temperature, Tg, that the heat transfer as given by eq 5a could be expressed in terms of Tg. In careful studies of experiment and theoretical analysis, Hottel and Sarofim concluded that the excess of Tf over the exhaust, Tg, could be as much as 400 °C. Accordingly, a difference term (Tf - Tg) was introduced

R0 ) (Tg0 - T0)/(Tf0 - T0)

(10b)

With these conversions, completing the algebraic procedure paralleling that used in the first approximation treatment (Appendix A) yields the extended expression (below), as shown in Appendix B, and written for convenience of interpretation in two different forms:

hg/hf ) (hg0/hf) + R00R0(Hs/Hsm)/ [1 - (1 - R0)(Hs/Hsm)] (11a) ) [(1 - R00) + (R0 - (hg0/hf))(Hs/Hsm)]/ [1 - (1 - R0)(Hs/Hsm)] (11b) In the interpretations, eq 11a is more conveniently used in examining the conditions when R0 f 1, and eq 11b for R0 f 0, as discussed next. Comparisons and Reductions. Comparison of eqs 11 with eq 8a now broadly identifies the conditions under which the original linear approximation given by eq 8a is approximately valid, and also it identifies the conditions under which the approximation can fail. (1) Clearly, if the flame temperature and exhaust temperature at Idle are high enough that Tg0 ≈ Tf0, then from eq 10, R0 is approximately 1, and eq 11 reduces


immediately to eq 8a. At maximum output, hg f hf, and the linearity of hg with output from idle to maximum output is expected and predicted. (2) At low output, the denominator in eq 11 can be expanded, and taking the first two terms of the expansion, eq 11 approximates to

hg/hf ) (hg0/hf) + R00R0(Hs/Hsm) + R00R0(1 - R0)(Hs/Hsm)2 (12) If R0 is close to but less than unity, as it often is in hot wall furnaces, the third rhs term has little influence until (Hs/Hsm) is quite large. Consequently, over much of the experimental range of measurements on actual furnaces and engines, a reduced eq 12 apparently has the same linear form as eq 8a. The difference, however, is in the presence of the additional factor, R0, in the second rhs term of eq 12 that reduces the slope of the data, even for values of R0 approaching (but not equal to) 1, so that a linear fit will be a good approximation to experimental data, but the fitting line does not go through the point hg/hf ) 1 at Hs/Hsm ) 1. This identifies what has been a continuing source of confusion in data evaluation for a long time because data appeared to support eq 8a when, in fact, in the experimental range, at low values of output, they were following the approximate linearity of eq 12 which is also, in fact, an approximation to eq 11a, but which failed to extrapolate to the appropriate value of the maximum output. (3) If R0 is small, which is evidently the case for cold wall devices (including the example following), eq 11 can simplify, with the second term in the numerator becoming very small, tending to zero. At low values of Hs, hg/hf is then almost constant but, as Hs approaches Hsm, it increases very rapidly and nonlinearly. This is more fully evident in the experimental evaluation following. Experimental Comparisons Data Source. The data used to test the equations are boiler data provided by Dr. William Prengle,22 measured at the University of Houston (UH) heating plant. This is a particularly valuable set of data since all four terms in eq 4 (Hf, Hw, Hg, and Hs) were determined independently, and the closure on the four measurements was to better than 99.5%. For comparison, the majority of published data are limited to input and output (Hf and Hs), and/or the wall loss is determined by difference and contains all the errors of measurement, and closure to 95% using averaged wall loss values is often considered good. Numerically, the UH parameter values are typical for boilers, with the exhaust loss at about 10% of input, and the wall loss in the range 1-2%. Such small wall losses are generally within the overall measurement of error if evaluated by difference and are then unreliable. In this case, the wall losses were determined directly; nevertheless, in spite of the careful measurements and the close closure, it will be seen in the data presentations following that there is still a degree of variation in the data, particularly in the values of hg. This illustrates the difficulty of obtaining accurate values of operational data; this has been one of the past sources of difficulty in validating any of the performance equations.


Validation Procedure. Test for validation of the structure of an equation classically requires reformulation of the equation in a linear form without, if possible, the a priori need to adopt values of relevant constants. The suite of equations involved in this development, however, does not permit such direct test, except as an approximation. In each case, at least one of the necessary parameters must be included and determined by optimization fitting. The starting point in evaluating the boiler data was eq 11. This can be reformed in the alternative forms:

1/[(hg/hf) - (hg0/hf)] ) -(1 - R0)(R00R0) + (Hsm/R00R0)(1/Hs) (13a) 1/[1 - (hg/hf)] ) (1 - R0)/R00 + (R0/R00)/[1 - (Hs/Hsm)] (13b) Expansion of the term [1 - (Hs/Hsm)] in eq 13b yields

1/[1 - (hg/hf)] = (1/R00) + (R0/R00)(Hs/Hsm)

(13c)

Test of eq 13a requires adoption of a value for hg0/hf, the idle load intercept on the specific enthalpy axis; using eq 8b, this also gives the value of R00. In this form of eq 13, if valid, it shows that the lhs is then linear with 1/Hs. The intercept gives the value of Ro, and the slope gives the value of the maximum output, Hsm. Alternatively, test of eq 13b requires adoption of a value for Hsm, the maximum output. Again, the slopes and intercepts yield values of the additional constants. Equation 13c requires no a priori constant parameter values, but it is an approximation, and the approximation will be sufficiently accurate only if Hs/Hsm is sufficiently small. For the wall loss, there is no phenomenological prediction of trend with output at this time and, accordingly, it is only possible to use empirical trends. For simplicity a linear variation is assumed, as given by eq 5c. On this assumption, when supported by the data, the slope gives the value of Kw, and the intercept gives the idle wall loss, Hw0, which, in turn, is used to get the idle heat, Hf0, using eq 5d. The empiricism required at this point is a concern, but it is essentially minor since the wall losses are only about 2% of the total input so that errors in the trends will have little or no significant influence on the dominant performance characteristics. The final equation validated was the firing equation. This is developed below (see also Appendix C). This has the form of eq 1, but with the intrinsic efficiency factor, R0, given by the output-dependent expression

R0 ) R00/[(1 + Kw + R0Kr) (1 + Kw)(1 - R0)(Hs/Hsm)] (14) where Kr ) (R00(Hf0/Hsm)). This expression shows that, as R0 f 1, R0 converges to a constant, the original assumption. The firing equation can then only be linearized by using the heat utilization factor form (eq 6), which requires adoption either of the idle heat (Hf0) or the maximum output (Hsm), or requires adoption of one, with sensitivity iteration to determine the other. Specific Exhaust Enthalpy: Eq 11. Figure 1 illustrates the test of eq 11, using the Houston boiler


Figure 1. Test of eq 11 in eq 13 reformulation: plot of [(hg/ hf) - (hg0/hf)] against output, Hs, to test linearity of reformulated equation. Value of (hg0/hf) [)intrinsic efficiency, R00] is obtained by optimization. Slope and intercept give values of other firing parameters: maximum output, Hsm, the temperature ratio, R0. [R00 ) 0.9125; Hsm ) 241 MMBtu/h; R0 ) 0.093].

data, in the reformulation as eq 13a, showing support of the linearity of the eq 13a form, within the limits of statistical error. Test of the equation in this form required adoption of a value for hg0/hf, the Idle load intercept on the specific enthalpy axis, and the fit was obtained by adjustment of the intercept value (hg0/hf) to get a maximum R2 (variance) in the linear regression. The variance was found, in fact, to be relatively insensitive to the intercept value adopted, and likewise also the parameter coefficients calculated from the slopes and intercepts of Figure 1. The optimum value obtained for the variance (R2) was 0.956. In the parameters in the slope and intercept groups, there are three unknown coefficients and, in the coefficients reduction, these were evaluated using the three bounding conditions of the explicit slope and intercept values; and the implicit value of the lhs term at Hs ) Hsm. The parameter values thus obtained were hg0/hf ) 0.085; R00 ) 0.9125; Hsm ) 241 MMBtu/h; and R0 ) 0.093. This last parameter value is of particular interest since the difference R0 - hg0/hf is then small, resulting in substantial reduction of the influence of the second term in the numerator of eq 11b. In this data reduction the results were found to be sensitive to the values of R00 at four decimal places. This was on account of the difference term, 1 - R00, appearing in an intercept form in eq 11b. Rounding out R00 to 0.91 or 0.92 then introduced nearly a 3% error in 1 - R00. Using these values, Figure 2 is a backplot of the variation of specific exhaust enthalpy (hg/hf) with output. This figure shows two lines. The (solid) curved line is calculated according to eq 11. The linear fit (dashed line) is a least-squares fit to the data in accordance with the first-approximation eq 8a. The linear-fit parameters were hg0/hf ) 0.081; R00 ) 0.919; and Hgm ) 1681 MMBtu/h. In visual inspection of the data plots and fitted graphs, taken at face value and lacking any other information, there is apparently little to choose between the two lines within experimental error and over the range of the experimental data; and the values of hg0/hf and R00 are almost identical to the nonlinear fit values. The extensions of the two lines to high output, however, are very different. This is shown


Figure 2. Variation of (normalized) specific exhaust enthalpy (hg/hf) with output (Hs); showing also backplots of nonlinear (solid) and linear (dashed) curve-fitted lines: eqs 11 and 8a.

Figure 3. Extrapolation to maximum output of (normalized) specific exhaust enthalpy (hg/hf) with output (Hs), showing also nonlinear (solid) and linear (dashed) curve-fitted lines eqs 11 and 8a.

in Figure 3. The maximum output (Hsm) predicted by the linear fit, at 1680 MMBtu/h, is over 6 times greater than the eq 11 fit. This identifies the major difference in outcome between the two equations and the two theoretical approaches. Use of eq 13b provided substantially similar results, including the values of the component parameter constants. Use of eq 13c also provided good support for linearity and provided an additional estimate of the idle exhaust enthalpy value (hg0/hf) from the equation intercept, and thus the value of R00, by use of eq 8b. Evaluation of the other two parameters, R0 and Hsm, only from the slope, was not possible, however, and this represents a limitation in the use of this form of the equation. Wall Loss Data. This is illustrated in Figure 4. In the absence, as noted above, of any (phenomenological) theoretical model to predict the trend of Hw with output (Hs), a linear variation is assumed at this time. This is also in accordance with data available from past work.1,2,11,21 If the trend was nonlinear, it is clear from the data that the graph would still be quite flat. The values are also in the range, as noted, of about 2% of input so that they have little influence overall on the total energy balance and, therefore, on the efficiency curves. The exact trend can be significant, however,



Figure 4. Wall loss. Variation of wall loss with output; linear variation is adopted empirically; see text.

with respect to the extrapolation to the intercept on the y-axis (at Hs ) 0) since this determines the value of the idle wall loss (Hw0) and hence, using eq 5d, a determination of the idle firing rate, Hf0. This last is needed for test of the firing curve equation unless it is obtained by parameter adjustment. It is clear, however, that the intercept value must be quite limited with a range that can be set at 0.75 ( 0.05 MMBtu/h. Using the linear extrapolation value from Figure 4 of Hw0 ) 0.752 MMBtu/h, and the value of 0.92 determined for R00, the value obtained for Hf0 was 0.815 MMBtu/h. This was the value used in the firing curve evaluations. Firing Curve. Derivation of the firing curve equation using the nonlinear forms of the exhaust enthalpy relation is given in Appendix C. The firing equation can be independently validated since the firing data were obtained independently of the exhaust loss data (Figures 1, 2, and 3). This validation was carried out in two ways: first, as a test of the first approximation form of the equation, eq 3, with R0 ) constant; and second, as a test of the extended form of the equation with R0 given by eq 14. Both methods required formation of the heat utilization factor (HUF), R [)Hs/(Hf Hf0)], so that both required adoption of a value for the idle heat, Hf0 (which is standard procedure; see refs 1, 2, and 19). The starting value adopted was that determined from the wall loss data (Figure 4) and then, as before, this was adjusted in both cases for maximum R2 in the operational plot against the relevant function of Hs [f(Hs)]. Again, the relevant linear regressions were found to be substantially insensitive to the value adopted. The function of Hs [f(Hs)] used in the first case was the first-order dependence of R on Hs given above as eq 6. The function in the second case, invoking eq 14, had the form

1/R ) (Hf - Hf0)/Hs ) (1 + Kw)(1 - R0) + R0(1 + Kr + Kw)/ (1 - Hs/Hsm) (15a) This shows linearization in terms of the ordinate calculated as the reciprocal of the heat utilization factor (1/R) and an abscissa calculated as [1/(1 - Hs/Hsm)], which required initial use of the value of Hsm previously

Figure 5. Firing curve. Variation of thermal input from fuel, Hf, with output, Hs, backplotted using eq C3, illustrating the almost linearity of the curve and very small idle heat, typical of boilers.

Figure 6. Operational thermal efficiency and heat utilization factor curves. Variation of η and R with output, Hs; also showing curve fits using the linear-based (dotted) and nonlinearbased (solid) backplots.

obtained (Figure 1), followed by sensitivity analyses for R2 optimization, as before. As in the case of eq 13, expansion of the term 1/(1 Hs/Hsm) yields a first approximation alternative:

1/R ) (1 + Kw + R0Kr) + R0(1 + Kr + Kw)(Hs/Hsm) (15b) To a first approximation, this expression is also consistent with or supported by the experimental data, but as in the case of eq 13c there are more parameters than constants, and a full determination of all parameter values is not possible. The plot of eq 15a is not shown. The firing curve is shown in Figure 5, based on eqs 3 and 14, and back plotted using the determined values. The linear-fit result (eq 3 with R0 constant) is not included on this figure as the differences between predictions from the two procedures were negligible in this instance. Figure 5 reemphasizes the very small idle heat which at a value (adopted) of 0.815 MMBtu/h is ∼1% of the values in the experimental range, which is standard for boilers. Operational Thermal Efficiency and Heat Utilization Factor. These plots are shown in Figure 6 calculated from eqs 1 and 2. The full lines are calculated from eq 5b using the nonlinear eq 11 for the


exhaust specific enthalpy. The dotted lines are calculated from the eq 8a linear assumption. Again, over the experimental range, and within the limits of the experimental variations (which are small for measurements of this type) there is no evident difference to the eye between the two pairs of curves. The differences are evident, however, in the intercept values of the curves, for R at idle (Hs ) 0), and for both R and η at maximum output. This is elaborated in the section following. The scientific significance of the difference lies in precision of knowledge. The engineering significance lies in the potential for comparisons using independent prediction of the maximum output, which is determined by the adiabatic flame temperature, as a target for a detailed mechanistic modeling analysis. Linear and Nonlinear Approximation Comparisons In the figures shown above, where the curve fits from the two different equations for hg/hf are used (eqs 8a and 11), it may appear that there is no evident superiority of one method of equation fit or calculation over the other. This basis for judgment is misleading, however, since the lines shown in the three figures (Figures 1-3) are those optimized individually for each particular graph, and this optimization required adjustment of certain parameter values from one graph to another. This refers to the values used in each graph for the three firing coefficients: R00 (or hg0/hf), Hf0, and Hsm, for the linear fit case, with the additional coefficient, R0, for the nonlinear fit case. In the case of the nonlinear fit, based on equations using the nonlinear eq 11 for (hg/hf), the coefficients used for the curves in the graphs all had the same value. In the case of the linear fits (using eq 6), however, the curves shown were separately optimized in selection of the coefficients so that, although the visual agreement clearly is good (and the variance in the curve fit was high in each case, generally greater than 0.95), at least one of the three coefficients was inconsistent with the values used in the other graphs. The present conclusion is that, using the linear fit of eq 8a, it is impossible to get a common set of parameter values that will also give good graphical or correlative agreement with the experimental data sets. From this it was concluded that, using eq 8a, there were too many constraints for the degrees of freedom. Consequently, this provides support for the validity of eq 11 by demonstration of the inadequacy of eq 8a. Discussion and Conclusions What we have provided in this paper is a theoretical basis for a new (nonlinear) relation (eq 11) between the specific exhaust enthalpy (hg) and output (Hs) that is supported by experimental data. The support by a single data set represents more general application than a single test might indicate since the data set used for the validation testing was chosen at random and the measurements were made independently. By the same argument, the failure of the linear eq 8a by test against the same data set can also be seen to be potentially general. To the extent that these results have general applicability, it would seem to provide adequate preliminary validation of the equation developed, and also


to represent, in furnace and engine analysis developments, a useful advance on the previously accepted basis of the linear relation for the dependence of the specific exhaust enthalpy with output. This clarifies the position outlined in the Introduction regarding the previous uncertainty of the structure of the exhaust-enthalpy relation. To the extent that the result is valid and general, it closes a major gap in phenomenological analysis of thermal system performance. This also emphasizes the weaknesses and strengths of such an approach in that the method of analysis is able to predict the qualitative existence of the various constants involved, notably the idle heat, the maximum output, and the intrinsic efficiency; but it is not able to predict (quantitatively) their numerical values. Conversely, however, the method of analysis is able to predict the broad structure of the performance characteristics identified as the firing curve, the heat utilization factor, and the operational efficiency, where this structure is independent of the actual system whether it is a first law or a second law device. In this latter regard, a study on similar lines of an internal combustion engine (second law device) has been initiated. Preliminary results have been reported elsewhere;23 extension to complete the evaluation of the exhaust properties on similar lines as presented here is in progress. We now consider briefly some of the assumptions required for the theoretical development given. The most significant of these is the assumption that the heat exchange between the flame (at Tf) and the load (at Ts) is linear with temperature (i.e., the heat exchange coefficient, β, is independent of or is insensitive to temperature). This was the assumption used by Thring,9 although a radiation-based 4th power dependence has been the more common assumption in integral formulations for obvious reasons (e.g., ref 8). In support of the linear result, however, there are two aspects to be considered. The first is that the heat exchange parameter, β, is eliminated by division so that the ratio β(Tf - Ts)/β(Tf0 - Ts), as indicated by eq 9, will be intrinsically less sensitive to temperature than the heat exchange parameter itself. This will be particularly the case if the differences Tf - Ts and Tf0 - Ts are both relatively close values (as in furnaces as compared with boilers). More direct support is provided by a comparison of experiment with prediction21 on the effect of stoichiometry on the maximum output (Hsm), which has been shown in those studies21 to be linear; and since the maximum output is essentially determined by the heat transfer from the flame at the adiabatic flame temperature for the given stoichiometry, this is contrary to a simple analysis using a fourth power assumption unless corrections are included for radiative interactions. This linearity with stoichiometry was also obtained in extensive calculations reported elsewhere,24 using STANJAN,25 showing almost linear decline of the adiabatic flame temperature with stoichiometry. (23) Boehman, A. L.; Essenhigh, R. H. Proc. 1995 Fall Int. Symp. Combust. Res. Ind. Practice, Am. Flame Res. Comm., Monterey, CA, Oct. 1995. (24) Boehman, A. L.; Simmons, J. W.; McCarty, J. G. Topical Report No. 4 from SRI International on “Analysis of Heterogeneous Reactions, Heat Transfer, and Thermal Stresses in Combustion Catalyst Monoliths, Gas Research Institute, Feb 1994. (25) Reynolds, W. C. STANJAN (v.3), Department of Mechanical Engineering, Stanford University, Jan 1986.


This linear result is further supported by the detailed breakdown of the energy flows in a small test furnace described by Enomoto et al.11 which shows that, in a hot wall furnace, there can be a major trade-off between increased flame emissivity and reduced energy supply to the load from the walls with the result that the normal radiative 4th power influence can be somewhat or substantially modified or suppressed. The theoretical basis for the trade-off is that radiation from one segment of the flame and/or the (radiating) walls is attenuated in transfer through the (absorbing/emitting) flame so that, if flame emissivity is increased, the direct contribution from the near flame is also increased, but this is offset by reduction from far flame (or wall) segments due to increased (near-flame) absorption. Consequently, as also supported by other independent studies, as noted elsewhere,1 the influence of flame emissivity can be second order or less. In a large boiler with cold walls it would appear that the same effect is provided by the radiation of one segment of the flame through the other segments. This evident modification of the normal radiative T4 relation is not unexpected, arguing from the prior work, and notably from that of the effect of flame emissivity.1 It is also a substantial simplification for analysis of furnace performance data. Further, it identifies a major target for test by solution of the detailed mechanistic models listed earlier. At a more general level, the analytical development presented here, based on the integral energy equation, also identified several characteristics of performance behavior that provide precise targets for detailed mechanistic analysis. These, first, include independent prediction of the values of the three performance equation constants: the idle heat (Hf0), the idle exhaust enthalpy (hg0), and the maximum output (Hsm). Second, it includes prediction of the structures of the hg-Hs curve and the temperature ratio parameter, R0. Third, it includes prediction of the linearity with temperature of the heat exchange between the flame and load. To the extent that these parameters, parameters values, and related equations can be supported or validated, within reasonable accuracy, by such independent studies and analyses, that validation, likewise, provides support both for the integral analysis presented in this paper and for the relevant detailed mechanistic analyses. In summary, these results now define future targets for both experimentalists and modellers to determine how far either experiments or the detailed models, available or under development, support or deny the general pattern presented here. Acknowledgment. We have particular pleasure in acknowledging provision by Dr. William Prengle of the boiler data measured on the University of Houston boiler, and permission to use those data in this paper. Glossary cp F Gs hf hg hg/hf

(average POC) specific heat fuel supply (firing) rate air/fuel (mass) ratio specific enthalpy of fuel (heat of combustion) ) (1 + Gs/φ)cp(Tad - T0) specific exhaust gas enthalpy (per unit of fuel) ) (1 + Gs/φ)cp(Tg - T0) ) (Tg - T0)/(Tad - T0)

Energy & Fuels, Vol. 11, No. 2, 1997 455 hg0/hf Hi Hi0 Him Hg Hw Kr Kw R R0 Ti Ti0 Tf R R0 R00 β η φ

) (Tg0 - T0)/(Tad - T0) enthalpy flow rate: subscripts: f, fuel; w, wall loss; g, exhaust loss; s, output superscript 0: parameter value at idle (no load) superscript m: parameter value at maximum output ) Fhg ) Hf(hg/hf) ) Hw0(1 + const‚Hs/Hsm) (linear assumption) ) R00Hi0/Hsm ) const‚(Hw0/Hsm) ) (Tg - T0)/(Tf - T0) ) (Tg0 - T0)/(Tf0 - T0) temperature: subscripts: g, exhaust gas; s, useful load; ad, adiabatic; o, ambient temperature: superscript 0: parameter value at idle (no load) average flame temperature Thring and Reber heat utilization factor ) Hs/(Hf - Hf0) intrinsic efficiency maximum intrinsic efficiency ) (1 - hg0/hf) overall heat transfer coefficient between flame and (load + walls) operational thermal efficiency ) Hs/Hf equivalence ratio

Appendix A: Derivation of Linear Exhaust Enthalpy Relation Equations 7a and 7b are

Hs + Hw ) β(Tg - Ts)

(7a, A1a)

Hw0 ) β(Tg0 - Ts)

(7b, A1b)

Subtracting and substituting for Hw - Hw0 using eq 5c gives

Hs(1 + Kw) ) β(Tg - Tg0)

(A2a)

At maximum output (Hs ) Hsm), this becomes

Hsm(1 + Kw) ) β(Tad - Tg0)

(A2b)

Ratioing eqs A2, and rearranging

Hs/Hsm ) (Tg - Tg0)/(Tad - Tg0) or

[(Tg - T0) - (Tg0 - T0)] ) [(Tad - T0) - (Tg0 - T0)](Hs/Hsm) (A3) Dividing by Tad - T0 and using hg/hf ) (Tg - T0)/(Tad T0)], eq A3 becomes

hg/hf ) hg0/hf + (1 - hg0/hf)(Hs/Hsm) (A4, 6) which is eq 8a. Appendix B: Derivation of Nonlinear Exhaust Enthalpy Relation Paralleling eqs 7, eqs 9a and 9b transform into

Hs(1 + Kw) ) β(Tf - Tf0)

(B1a)

Hsm(1 + Kw) ) β(Tad - Tf0)

(B2b)

Ratioing and rearranging, including the expansion of



the temperature difference terms, the corresponding equation to eq A3 is

(1 - hg/hf) ) R00(1 - Hs/Hsm)/[1 - (1 - R0)(Hs/Hsm)] (C2a)

[(Tf - T0) - (Tf0 - T0)] )

(hg/hf - hg0/hf) ) R00R0(Hs/Hsm)/

[(Tad - T0) - (Tf0 - T0)](Hs/Hsm) (B3) Introducing now the defining ratios

[1 - (1 - R0)(Hs/Hsm)] (C2b) Substituting eq C2b in the rhs of eq C1, this becomes, after rearrangement

R ) (Tg - T0)/(Tf - T0) R0 ) (Tg0 - T0)/(Tf0 - T0)

rhs ) Hs[(1 + Kw + KrR0) -

Equation B3 transforms by substitution for Tg - T0 and Tg0 - T0, followed by division by Tad - T0 and rearrangement into

(hg/hf)/R ) (hg0/hf)/R0 + [1 - (hg0/hf)/R0](Hs/Hsm) (B4) Adopting the assumption stated as eq 9, and rewritten in terms of R, this becomes

1/R ) (1/R0))[1 - (1 - R0)(Hs/Hsm)]

(B5)

Eliminating R between eqs B4 and B5 then yields

hg/hf ) (hg0/hf) + R00R0(Hs/Hsm)/

((1 + Kw)(1 - R0)(Hs/Hsm)]/[1 - (1 - R0)(Hs/Hsm)] Likewise, using eq C2a, the lhs becomes

lhs ) (Hf - Hf0)R00(1 - Hs/Hsm)/ [1 - (1 - R0)(Hs/Hsm)] Canceling the common denominator term, [1 - (1 Ro)(Hs/Hsm)], on both sides, and rearranging, eq C1 becomes the firing equation:

Hf - Hf0 ) Hs[(1 + Kw + KrR0) ((1 + Kw)(1 - R0)(Hs/Hsm)]/[R00(1 - Hs/Hsm)] (C3)

[1 - (1 - R0)(Hs/Hsm)] which can be written the form of eq 1

which is eq 11.

Hf ) Hf0 + Hs/[R0(1 - Hs/Hsm)]

Appendix C: Derivation of the Firing Equation The firing equation is obtained by extensions of eq 5 by adding and subtracting the term Hf0(1 - hg/hf); eq 5 becomes 0

(Hf - Hf )(1 - hg/hf) ) 0

with

R0 ) R00/[(1 + Kw + R0Kr) (1 + Kw)(1 - R0)(Hs/Hsm)] (14, C5)

0

(1 + Kw)Hs + Hf (hg/hf - hg /hf) (C1) To eliminate the terms in hg/hf, eq 11a is rewritten in the two forms

(1, C4)

which is eq 14. Notably, if R0 ) 1, R0 is a constant. EF960111C

Studies in Furnace Analysis: Prediction of the Variation of the Specific

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