Laboratory Investigations of Heat Transfer in Packed Beds

Chapter II

Laboratory Investigations of Heat Transfer in Packed Beds THE exceptional complexity of the mechanism of convective heat exchange is well known. Only an experiment, designed and oper­ ated on similarity principles, will give the values of the heat transfer coefficients required for practical calculations, and, at the same time, reveal the mechanism of the process. Thus a full description of the experiments on heat transfer in packed beds is essential, not only for the practical applications, but also in order to gain a deeper understanding of the process.

1. Research Carried Out by Furnas Furnas carried out work at the U.S. Bureau of Mines, as part of a general study of blast furnaces, and he began to publish his results in 1930.(26) His work is especially interesting, both for its methods and for its large scale. The history of the work is in­ structive and we shall consider not only the second method of research (which alone gave satisfactory results) but also the first (the results of which Furnas rejected). The first experimental method was based on the direct measure­ ment of gas-solid temperature differences, and cast iron spheres were used in the arrangement shown in Fig. 29. Air from reser­ voir 1 is passed by the blower 2 into the gas-fired furnace 3, where it passes through a steel tube (8 m long X 254 mm diameter) and is thereby heated to 900 °C. The packed bed is contained in a shaft (steel pipe, 15 cm o.d., 105 cm height), insulated by 14 cm of Sil-o-cell. On leaving the shaft the air passes through a pipe filled with copper filings (to reduce the oxidation of the cast iron 52

Laboratory Investigations of Heat Transfer in Packed Beds spheres), through the condenser 4 and back to vessel 1. The 0 2 content of the gases was thereby maintained at 6-10 per cent. The gas flow was measured by an orifice. Gas temperatures were measured by a suction pyrometer (gas velocity 7 m/sec). The same thermocouples were used to give the solids temperature, by not inducing gases via valves 5, 6 and 7. The author assumed Copper filings

FIG. 29. Furnas' apparatus for heat transfer coefficients (first method). 1 — reservoir; 2 — blower; 3 — furnace; 4 — condenser; 5-12 — valves; 13 — orifice.

that he was thereby measuring the solids temperature, but it seems to us that, rather, he was measuring an average temperature (between the gas temperature and the surface temperature of the spheres). The experiment was conducted in the following manner. Heated gases were passed through the bed at a constant rate (and direction) and the temperatures of gases and solids were 53

Heat Exchange in Shaft Furnaces recorded alternately for approximately 5 min periods by opening and shutting the cocks 5, 6 and 7. A typical graph of observations is given in Fig. 30. Let us work out a sample calculation from this diagram. 600

i

r—r-

i

i

r

7 m/sec gas velocity over thermocouple 500

I 400 300

200

120

130

140

150

160

FIG. 30. Typical gas and solids temperatures. DATA

(a) Diameter of spheres (b) Velocity of gases (c) Area of shaft

0-0317 m; 0-1875 m/sec; 0-0167 m2.

The calculation is for a 1 m shaft height or 0-01675 m3 volume of bed—in which volume are located 15 kg of cast iron spheres. From Fig. 30, for the time interval between 65 and 75 min from the beginning of the experiment, the rate of temperature rise for the spheres was 30-25/600 = 0-0504 °C/sec or 181-1 °C/hr. For the middle of the selected interval of time the gas temperature is 400 °C and the solids temperature 358-6 °C. Thus heat transferred from the gases to the spheres in one hour is 115X0-14X181-1 = 2920kcal/hr (where specific heat of cast iron at 355 °C is 0-14). The volumetric heat transfer coefficient is defined by hv =

Q/At'Vs,

= 2920/(400-358-6)0-01675, = 4210 kcal/m 3 hr °C. 54

Laboratory Investigations of Heat Transfer in Packed Beds Furnas, in the calculation of the heat transferred to the spheres made an erroneous correction for heat loss to atmosphere. (26) In these experiments spheres of 18-5,31 -7 and 48.6 mm were used, with gas velocities (referred to shaft cross-section) of 0-1-0-53 m/sec and temperatures up to 700 °C. Logarithmic plots give straight lines and hence the relation: hv = Kw™.

(135)

Results are given in Table 4—Furnas considered that the re­ sults obtained were possibly too high. It would appear that he did not establish the dependence of the heat transfer coefficient on the diameter of the spheres, or work out volumetric coeffi­ cients. TABLE 4.

Results of Furnas* experiments Diameter of spheres (mm)

ha—surface heat transfer coefficient (kcal/m2 hr °C)

18-5 31-7 48-6

331-5 172-8 82-8

When Furnas applied the above method to pieces of lower con­ ductivity than cast iron, he experienced complete failure. The results of these experiments (carried out with great thorough­ ness) could not be called erroneous, but there are certain irregularities, and it seemed to Furnas that he had failed to over­ come the experimental difficulties. However, in 1929 a helpful paper by Schumann (23) appeared, and in it Furnas saw new ex­ perimental possibilities, and so approached the problem by an­ other method. This second method was no longer tied to the measurement of gas-solid temperature differences, but was based on the solution of the problem of heating a fixed bed with a flow of gas or liquid.(23) Furnas' second apparatus is shown in Fig. 31. The hot com­ bustion products from burner 3 and furnace 1 were passed through 55

Heat Exchange in Shaft Furnaces the bed of lump materials 2 and their temperature was con­ tinuously monitored before and after the bed. The combustion chamber 1 could be raised by a screw jack to give a good fit with the shaft. With the aid of the fan 4 the products of combustion passed through the dust collector 5, the nichrome orifice 7 and through the water cooled condenser (a pipe 3 mX 10 cm dia.) and

FIG. 31. Furnas' apparatus (second method). 1 — combustion chamber; 2 — packed bed; 3 — burner; 4 — fan; 5 — dust collector 6, 7, 9 and 10 — orifices; 8 — cooler.

finally to atmosphere. The test material was placed in a steel cylinder insulated by a 150 mm layer of Sil-o-cell. The experi­ ments were carried out alternately in two cylinders, one 500 mmX 150 mm dia. and the other 1 mX230 mm dia. For greater accuracy in gas temperature measurements the gases were passed through carbofrax orifices, dia. 5 cm in the openings of which were thermocouples (lower T1 and upper T2). Details of the ther­ mocouple assembly is given in Fig. 32. The amount of gas sucked through these thermocouples was measured by orifices 9 and 10. The gas temperature at the orifice 7 was measured by two thermo­ couples, Tz and J^. Orifices 9 and 10 were 7-93 mm dia. in a 52-4 mm pipe, and orifice 7 was 50-8 mm dia. in a 103 mm pipe. 56

Laboratory Investigations of Heat Transfer in Packed Beds In this method the gas temperatures are the fundamental data of the experiment. In order to reduce errors due to losses through the walls of the cylinder, the apparatus was heated until a constant T2 was achieved. The furnace 1 was then removed and the bed cooled by atmospheric air. The thermal insulation maintained a temperature roughly equal to the average bed temperature be­ tween the beginning and end of the test, and this helped to reduce the error from heat loss through the walls. j V s S W s S S S S ^ ^SSSSWSSSSSSSSSSysgSl From bed

-e

ssssssssss V>sssssss^s

FIG. 32. Suction pyrometer.

The heating of the bed was then carried out. Temperature Tl9 below the bed, was maintained constant. For the sake of accuracy the gases were aspirated through the suction pyrometers Tx and T2 at a minimum speed of 10 l./sec. Similar data were recorded for bed cooling, which took place immediately after heating. In the experiments with iron ore the gas temperature reached 1100 °C. With coke and coal temperatures did not exceed 300 °C in order to avoid ignition. The combustion products had approximately the composition 5% C0 2 , 10% H 2 0, 10% 0 2 and 75% N 2 and their density, viscosity, specific heat and thermal conductivity were approxi­ mately as for air. The gas velocity, referred to the shaft cross-section ranged from 0-6 to 1-8 m/sec and the lump diameters from 4 to 70 mm. The experimental data were plotted with coordinates tg/tg against log r (time) as in Fig. 33. Schumann's curves (coordinate tglt'g against log Z, for different values of F, where Y and Z are proportional to heat transfer coefficient and to time respectively, as in eqns. (Ill) and (112)) were also plotted. 57

Heat Exchange in Shaft Furnaces The two sets of curves were superimposed, and moved along the axis of the abscissa (log r and log Z) until they coincided. The curves coincided for a particular value of 7, and hence eqn. ( I l l ) could be evaluated hv = YCgw0/H. (136) Figure 33 gives a typical example of coincidence.

vo -10 -0-9 0-8 r0-8 -07 06 ,0 6 ."I*?

Los

0-4 Lo-A

Lo-3 02 Lo-2

Lo-i 1

2 KX)

3 200

A 300

5 400

I6

7 I

I

Z 1 *J X1

500 600 700 800900 X)00t

FIG. 33. Superimposition of Schumann's curve (for Y—6) on experimental points for tg (Furnas' second method).

In systems involving large lumps of low thermal conductivity the expression "solids temperature" requires special explanation, since there is here a noticeable temperature difference within the lump during the entire heating process. Furnas understands by the temperature of the solid the average (calorimetric) tempera­ ture of the lump. It also proved possible from the experimental data to deter­ mine not only the heat transfer coefficient hv, but also the specific heat of the lump material, C s . The heat transfer coefficient is determined by the value of Ffor the particular tg/t'g versus log Z curve, without reference to the 58

Laboratory Investigations of Heat Transfer in Packed Beds displacement of the experimental curve, and depending only on the form of the curve. On the other hand, the duration of heating necessary to attain a given bed temperature is determined by the specific heat of the solids; it is this latter which determines the position of the curve along the abscissa. The value of Cs is found fromeqn. (113):

c

« = z(fer-

137

<>

Let us determine hv and Cs according to the data of Fig. 33, which referred to Mesabi iron ore (13-3-18-9 mm, average 16 mm). The bed height was 0-445 m, the bed section 0-0183 m2, and the gas velocity referred to the total bed cross-sectional area was w0 = 1-11 m/sec. Products of combustion = 5% C0 2 , 10% H 2 0, 10% 0 2 and 75 % N 2 . Specific heat of gases at average temp. 550 °C is 0-324 kcal/m3 °C. According to Fig. 33 the experimental points corre­ spond to Y = 6-0. The we have w^ ,T* 6-0X0-324X1-11X3600 hv = YCg w0/H = Q ^ , = 17-440 kcal/m3 hr °C. The bed porosity is e = 0-625. According to Fig. 33, time r = 172 sec corresponds to Z = 2-0. Hence from eqns. (113, 137) we have Cs

=

17-440X0-0478 2-0(1-0-625)

111A1 17 3 0 ^ =111 kcal/m C

°

The specific heat of the shaft walls is roughly 60 kcal/m3 °C. For the ore, then, Cs = 1110 - 6 0 = 1050 kcal/m3 °C. The weight of the ore is 3990 kg/m3, and hence we have Cs = 1050/3990 = 0-263 kcal/kg °C. This value agrees well with direct determination. From these experiments the following formula was derived: K = As

W

°d0.9

59

M.

(138)

Heat Exchange in Shaft Furnaces where As = coefficient, characteristic of the material; w0 = gas velocity, referred to 0 °C and the shaft cross-sec­ tional area (m/sec). T = average gas temperature (°K), d = lump diameter (m); M = coefficient, dependent only on bed porosity. This equation may be used for all cases except one—the cooling of coke, which Furnas considered a strange anomaly. For this u>27-r0'3

(139)

Values of M are given in Fig. 34 and Furnas considers these to be independent of the shape of the lump, the condition of the surface and of other factors.

1-5 2 c

/ f

S 10

/

4

y r

«*

o

0-5

n 0

0-1

0-2

0-3

04 0-5 06 Bed porosity, c

0-7

08

0-9

10

FIG. 34. Value of M (a coefficient dependent on porosity of bed).

In order to use the formulae it is necessary to know As, and experimental values are given in Table 5. There is as yet no means other than the experimental one for finding As. It should be noted that Furnas' data, published at various times, are sometimes at variance. (28) Furnas examined beds of cast iron spheres by both first and second methods, comparing results. 60

Laboratory Investigations of Heat Transfer in Packed Beds TABLE 5.

Coefficient As (for hv in kcal/m3 hr °C) Material Iron ore Sinter Limestone Coke Bituminous coal Anthracite Chamotte brick (fragments) Dinas brick (fragments) Magnesite brick (fragments) Cast iron spheres

Heating

Cooling

114 171 65 56 48 50 87 113 137 67

137 206 74 137 64 71 137 206 70

In both methods the errors are such that they tend to cancel each other out, and the smaller values are doubtless more reli­ able (Table 6): TABLE 6.

hv, for cast iron spheres by the two methods First method

Diameter (mm)

Gas velocity (m/sec)

Second method

Heat transfer coefficient (kcal/m3 hr °C)

Gas velocity (m/se c)

At 500°C

18-5 18-5 31-7 31-7

10

46,800

10

25,200

1-30 1-83 1-35 1-90

Heat transfer coefficient (kcal/m 3 hr °C) Corrected At100°C to conditions of first method 8,640 43,200 11,500 16,200

29,180 30,220 12,250 10,800

Furnas gives his final formulae in terms of volumetric coeffi­ cients, according to the following considerations: 1. No accurate methods for surface area determination are available. 61

Heat Exchange in Shaft Furnaces 2. It is not possible to determine the active surface — i.e. which part comes into contact with the gas flow. Previous work by the author showed that only a quarter of the bed cross-section is in­ volved in the flow of gas, so that it is fully possible that only a small part of the surface takes part in heat exchange. Thus Furnas considers that the surface heat transfer coefficient hs can only be approximately determined. Checks on the laboratory data were carried out on a blast fur­ nace, by the second experimental method (experiments 121^4 and 1215, table 1(26)), and values of the coefficient ^fswere determined. The velocity and temperature of the gases were measured at different vertical positions in the shaft. The heat transfer coeffi­ cients for the top of the shaft were determined by the equations for countercurrent exchange, and a comparison of these data is given in Table 7: TABLE 7.

Comparison of hv, from laboratory and full-scale experiments

Blast furnace

Holt Chicago Provo

Gas velocity w0 (m/sec)

1-2 1-5 1-2

Average lump temperature in shaft (°C)

500 500 500

Heat transfer coefficient hv (kcal/m3 hr °C) Observed in blast furnace

By formula

4320 5760 5400

5400 6120 5400

Furnas considers that the agreement strengthens the case for the validity of the laboratory work. In estimating the error in Schu­ mann's solution with respect to real materials Furnas considered that if \p = -^- —r- = constant for the heating period, where At is the difference between the gas temperature and the mean solids temperature, then the Schumann theory would be 62

Laboratory Investigations of Heat Transfer in Packed Beds applicable. This arises from the equation dg = hvAt dr.

(140)

In order to find the errors involved in applying Schumann's solution to real bodies (A ^ 0) Furnas calculated ip values for spheres of various diameters (10-100 mm), with A = 1-44 kcal/m hr °C, with a constant central temperature and with a constant hv. The results are given in Fig. 35. For the second half of the heat­ ing period ip may be, for example, 20 per cent less than for the first half. According to Furnas this 20 per cent is the possible ex­ perimental error.

O «, nn

i*

R3T1

'o to

se

80

Nk^*L

U-SNM o\F1

"5 v -m\

*

fin

-d=i 1 °1 /t '

1

1 1 10 20

d=50

f~1M

■

I

Mil

UJ^^J^^ =^= I I I

I I I I I J 1 1 1 701 1 1 1 1 1 1 30 40 50 60 Total heat transferred, 7o

80

90

FIG. 35. Significance of total coefficient of heat transfer (as a percen­ tage) in terms of total heat absorbed by a sphere.

Judging the results and anomalies of his own experiments, Furnas started from the fact that if the gas side resistance is con­ trolling, then one may expect a strong velocity dependence. If, on the other hand, the internal resistance controls, then the effect of gas velocity will be weak. In other work on heat transfer the coefficient hv is usually proportional to velocity raised to the 0-8 power. In Furnas' work a slightly lower power (0-7) was obtained and Furnas thought that this proved that the internal resistance did not have a great influence on heat transfer. A decrease in bed porosity increases the real speed of the gas and makes its path more tortuous. It is, therefore, completely natural that a decrease in porosity should increase hv. Furnas assumed that hs would not change with the size of the lump. Thus the surface per unit volume of spheres increases inversely with 63

Heat Exchange in Shaft Furnaces their diameter, and so hv should vary inversely with respect to d10. According to the experimental data the power is a little smaller, d0'9. It seems amazing that the lump size does not have more influence, since the internal heat resistance increases with size. According to Furnas the paradox may be explained in that, as the diameter of the pieces increases, so the active surface decreases. The value of As changes broadly from 48 for bituminous coal to 171 for sinter, and there must be some reason for this fourfold discrepancy. If the character of the surface were of moment then As for coke would, at the least, not be less than for chamotte and other materials, since the surface of coke is extremely rough. The thermal conductivity could also, as Furnas thought, not be the controlling factor. Most probably the shape of the lumps was the principal influence on As, and on this also depends the actual surface area of the bed. Up to this moment no means for finding As by calculation existed and Furnas had to experiment with each material. He gave no explanation for the coke cooling anomaly. 2. Research Carried Out by Saunders and Ford The work of Saunders and Ford is of great interest in that in setting up their experiments and in working out the results the authors (29) tried to observe the principles of similarity. Experiments were carried out with spheres of various diameters, placed in three geometrically similar cylinders of diameter 203*2, 101-6 and 50-8 mm. The corresponding sphere sizes were 6-35, 3-17 and 1-6 mm. The spheres were of steel (ball bearings), lead and glass; the last-named were tested only in the 203-2 mm vessel. Details of the experiment are given in Table 8. Heating was by hot air, the temperature of which was not high— 84 °C. The velocity of the air referred to the total shaft crosssection was, in most experiments, 0-61 m/sec. After heating came bed cooling. The temperatures of air entering and leaving the bed, tfg and tg\ were measured by the usual thermocouples. The results of the experiments have been plotted with the ratio as (^-'M)/^-'^) ordinates. 64

Laboratory Investigations of Heat Transfer in Packed Beds TABLE 8.

Experiments of Saunders and Ford Diameter of cylinder (mm) Item

Diameter of steel balls (mm) Diameter of lead balls (mm) Diameter of glass balls (mm) Thermal capacity of steel balls at Hid = 41 (kcal/kg °C) Thermal capacity of supporting grid Thermal capacity of cylinder Thickness of Al foil insulation (mm)

203-2

101-6

50-8

6-35 5-9 50

3-17 -

1-6 -

41,050

5113

660

84 39

12 13

1-9 1

50-8

25-4

25-4

In Fig. 36 are shown the results of heating two geometrically similar charges, whose similarity constant was 2, i.e. D'/D" = d'/d" = H'/H" = 2 Both experiments were carried out at the same velocity, w0 = = 1-22 m/sec. On graphs (a) and (b) the results are plotted with abscissa r, and two different graphs are obtained. On graph (c) the abscissa is xjd and the curves now coincide, indicating the relationship tg-*M

\dj

If two more experiments are performed, with sphere diameter and bed height unchanged, but with another gas velocity, two new graphs are obtained:

If we plot the dimensionless function w0r/d on the abscissa all four experimental curves coincide, as in Fig. 37:

V-'iM

'(=?)•

t

g~tM

65

Heat Exchange in Shaft Furnaces

10

0-8

(a)

10

— — * - j —

(b)

°

0-8

o o 0-6

0-6

•

I

o 0

04

0-2

c) o o

02

200 r, sec

400

200 T,

(c)

1-0

J?

0-8

•

400 sec

o#o c)

o

600

!

•P

o

8

0-6

-3 ~ at • oi

'

0-4

% o

0-2

o*

20

30

.1.10-3 a

40

50

60

FIG. 36. Comparison of heating in geometrically similar apparatus (dia. 50-8 and 101-6 mm), with air velocity 1-22 m/sec; from Saunders and Ford. a — H~65 mm, £> = 50-8 mm, d—1'6 mm; b - H~ 130 mm, b = 101-6 mm, d—3-2 mm: c - H = 65 and 130 mm, 2) = 50-8 and 101-6 mm, d=V6

65

and 3 2 mm.

Laboratory Investigations of Heat Transfer in Packed Beds

t ^ r ^ H c + ef»o+—c?-»—9

10

Jcft &

0-8 A

*K

'I

*p

r

0-6

4£ <3

0,1.0)

** '*" 0-4

0-2

ff

' „Xk1

0

.4*ft *&

2

8

A

10

12 14 ^x10"4

20

16

22

FIG. 37. Saunders' and Ford's data for different gas velocities.

In order to be able to include various materials the authors extend­ ed the criterion wQr/d to (w0t/d)(Cg/Cs). They were then able to describe the behaviour of steel, lead and glass balls in one formula: l

M M

tg—tn

\ d

(141)

Cs)

and finally, to include the effect of bed height (measured in ball diameters H/d): ta-t

M

tg—tM

(WatCg\

= 0 \dCs)9(d)

(H\

(142)

as in Fig. 38. The authors maintain that they did not find any influence of t -t' thermal conductivity of the materials on the ratio -JL—~, alt

g'~tM

though the curves for the glass balls do show some divergences. In order to derive a heat transfer coefficient from their data (in which, as in Furnas' later work, the surface temperatures were not defined) Saunders and Ford had to turn, once again, to Schumann's curves. Instead of using the function Z on the abscissa (as did Schu­ mann) they used the function Z^

Y

WptCg

HCs(l-e)* 67

Heat Exchange in Shaft Furnaces which is very convenient for calculation and does not include the heat transfer coefficient hs. The graphs of Fig. 38 were recalculated to this new criterion, as in Fig. 39, by the following method:

FIG. 38. Saunders' and Ford's data for small glass, lead and steel balls. (The points represent average data observed, and several are corrected by the authors according to considerations set forth in the text.)

In time dr the gases give out heat nd2 dQ = WCg~^-tg)dt

(143) (144)

68

Laboratory Investigations of Heat Transfer in Packed Beds

HC s (1-e)

FIG. 39. The total heat transferred in a period of time from 0 to T

HCs(1-e)

FIG. 40. Method of comparing Saunders' and Ford's data with the Schumann curves.

69

Heat Exchange in Shaft Furnaces The total heat which can be transferred from the gases to the bed in infinite time will be:

Q = w C , ^ ' - ^ ) j"~ (l-^E^f-) dT- (145> This may also be defined in terms of bed parameters: jid2

Q = H—(l-e)Cs^-t'M).

(146)

Hence, from (145) and (146) (and dividing by d):

^rm)*-^-

<4i7)

f(x) dx will represent

o

the area between the curve and the x-axis, and

Px

( 1 - / ( * ) ) dx

the area between the curve and the j-axis. On this basis the left-hand side of eqn. (147) is the area S be­ tween the curve corresponding to a given H/d (Fig. 38) and the j>-axis, since d

(H!C£L)

\dCsJ

=

jgg

dt.

(148)

Hence the coordinate on the abscissa of Figs. 391" and 40 is ex­ pressed thus: wC X

° dCs

^ — H(\-e) 9

(149)

that is, as the product of w Cgr/dCs and l/S. The authors, like Furnas, superimposed Schumann's curves on to their own data, and noticed how the curves coincided for a ^Editor's note: Kitaev makes no reference to the ordinates of Fig. 39, and we have changed the notation slightly to clarify matters. The ordinate is QxlQoo—the proportion of the total heat which can be transferred (in time oo) which is actually transferred in time T. Theoretically, the curves should be asymptotic to QrlQ^ = 1*0, and a small experimental error in fixing the moment at which the experiment commences would account for the observed discrepancy.

70

Laboratory Investigations of Heat Transfer in Packed Beds given H/d. Thanks to the substitution Z/Y for Z it was not necessary to displace the curves along the abscissa in order to achieve a fit. The results are given in Table 9. TABLE 9.

The results of Saunders and Ford H d

Cgw0

6-2 13-7 20-5 27-3 410

4 9 12 18 25

M Wo

1-88X10- 4 1-91 1-75 1-82 1-78 1-85X10- 4

Mean

Thus, for spheres, the authors obtained hv= 185 wQjd kcal/m3 hr °C.

(150)

We may recall that Furnas obtained a somewhat different value for spheres: hv = 17-2 w0/dh*5 kcal/m3 hr °C. (151) Having worked out Furnas' data by their own methods the au­ thors remain silent on the differences in the equations, and merely note that in two cases Furnas' results are less than their own. In regard to this it may be noted that the divergencies between the work of Saunders and Ford and of Furnas for spheres are greater than they indicated, as may be seen in Fig. 41. In judging this work one may be permitted the following com­ ments : 1. The influence of thermal conductivity should have mani­ fested itself, but would probably not have been noted by the authors because of the very small glass balls used (6*35 mm). The diameter d should be raised to a power above unity. 2. The low temperature range (up to 84 °C) makes it difficult to extrapolate the authors' data to high temperatures. 71

Heat Exchange in Shaft Furnaces 3. The index of wQ (unity) is doubtful, since it disagrees with other experimental transfer data. This doubt is confirmed in the authors' own approach from dimensional analysis. 20000 15000 10000

8000 7000 6000 5000 4000 ■*: 3000

1 £

2000

S

1500

L hv=Wb w/d TSaunders and Ford

"Si o 1000

Z 800 g

x

700 600 500 400 300

hv=17-2/d RJ rnas

135

/

/ /

200 150 100

001 0-015 002 0-03004005006008 010 0-15 0-20 0-30 0-400-50060 Diameter of small balls, m

FIG. 41. Comparison of Saunders' and Ford's with Furnas' data.

The authors chose the following independent variables; their dimensions are given in the order — length, time, temperature and quantity of heat: d GgCg

Cs A hs t

— characteristic linear dimension of burden (1,0,0,0); thermal capacity of gas flow ( 0 , - 1 , - 1 , 1 ) where Gg is the weight of gas per unit of time and Cg is the specific heat; — specific heat of burden material (—3,0,-1,1); — thermal conductivity of burden material ( - 1 , - 1 , - 1 , 1 ) ; — surface heat transfer coefficient ( — 2 , - 1 , - 1 , 1 ) ; — time (0,1,0,0). 72

Laboratory Investigations of Heat Transfer in Packed Beds The authors did not include the gas velocity, supposing it to influence the dependent variable (tg) indirectly via Gg and Cg. Equally they ignored gas viscosity, which influences tg via hs, The required function is assumed to have the form: dXl(GgCgfzCxs3hXiXX5tx\

(152)

The indices xl9 x29 etc., must be found such that the function (152) is dimensionless, and hence we have four equations, correspond­ ing to the four units: for d: x±- -3x$- -2x 4 - * 5 = 0; for r: x2""■ * 4 ~- *5 + * 6 - 0 ; for t: -x2- ■ x$-" * 4 ~ * 5 = 0; forg: x2+ x 3 + x4 + x5 = 0. In fact there are only three equations, since the last two are identical. There are six unknowns, and hence three dimensionless groups determine the function (152). Solving the equations for xl9 x2 and x2 = — x4 —x5-f-x6; x 3 = x6, and hence we have the function

Supposing that hs is independent of temperature, and that1* GgCg = wd2Cg9 Saunders and Ford obtained: <

S

JWCg - < M =

t

g~tM

/

/ WdCg ^ ^ J ^WtCg\ \

\K~9~lT'~dc^)

(154)

Now, for convective heat exchange in turbulent flow, the NusseltReynolds relation holds: *£=Cffi

or

Nu=CRe»,

(155)

t Editor's note: Saunders and Ford said, rather, that for geometrically similar beds GgCg is proportional to wdzCg. Also, in Kitaev's ensuing treat­ ment there is a danger of confusing gas and solids conductivities, and this section follows Saunders' and Ford's original more closely than Kitaev's account.

73

Heat Exchange in Shaft Furnaces where Xg and vg are the thermal conductivity and kinematic viscosity of the gas, and n is a constant usually between 0-8 and 1*0. Supposing this to hold for gas flow in a packed bed we have wCg ^ k

S

'

/wdy \Vg)

\

*g

)

Since for all gases Cgvg/X is practically constant, then the group wCg/hs is equivalent to the group wd/vg. In particular, if n = 1, then wCg/hs is a constant, and eqn. (154) may be written

As we have said, Saunders and Ford did not detect the influence of the term wdCJl in their experiments.1" If the experimental data is to be extended to other geometrically similar systems, then two more dimensionless groups must be included. For spheres of diameter d, in a cylinder of height H and diameter Z>, then, for similarity, H/d and D/d must be constant, and one finally obtains: t'o-t'M

J

[dCs

'

X

> d'

d)-

^ °

For all their experiments D/d remained constant ( = 22). Finally, we give a numerical example, which is interesting in that it shows how Saunders and Ford solve problems of heat transfer without knowing the heat transfer coefficient explicitly: Cast iron balls, 63 mm dia. are heated in a bed of height 1-27 m, by air at t'8 = 205 °C. The balls enter at tM = 38 °C and move at a speed of 0-72 m/sec. The specific heat of the balls in Cs = 880 kcal/m 3 °C. Porosity of bed e = 0-38. Determine the exit temperature of the gases, one hour after the commencement of heating. Solution: find the group w0tCg HCs(l-e)

0-72X3600X0-32 1-27X880X0-62

1-28

t Editor's note: Saunders and Ford point out that if d is small or A large, then this term becomes small.

74

Laboratory Investigations of Heat Transfer in Packed Beds Find also H/d = 1-27/0-063 = 20. From Fig. 40 t

g~tM

t

g~tM

' ■ ^ ? _ = 0.73. 205-38

Hence tg = 160 °C. 3. Research Carried Out by Tsukhanova and Shapatina Tsukhanova and Shapatina (30) performed experiments with small steel balls, chamotte fragments and bronze particles, under the conditions set out in Table 10. TABLE 10.

Experiments of Tsukhanova and Shapatina

Material

Small steel balls Chamotte fragments Bronze particles

Particle size (mm)

Linear gas velocity (m/sec)

Bed height (mm)

Gas entry tem­ perature (°Q

Density of bed (kg/m3)

3-15 f 2-3 2-3 4-5 6-7

0-64 1 1 1 1

10-250 100 50-150 50-150 50-150

200 250, 600 240 240 240

4705 922 922 922 922

0-64 1-38

70 70

240 300

4680 4680

2 2

The method used was analagous to that of Saunders and Ford. The authors formulate their experimental findings for the situation where wCg/h < 4 as follows: 1. Heat exchange in a stationary packed bed is determined by WQXCJHCQ which characterizes the ratio of the heat carried by the gas flow to the heat accumulated in the solids. 2. The value tgjt'g in the bed depends only on the thermal capacity per unit volume of burden, and not on the size, shape factor or roughness of the particle. 75

Heat Exchange in Shaft Furnaces 3. The temperature profile of the gas and the heat transferred to the bed can be determined by the generalized curve of Fig. 42. 4. In the unsteady-state regime the influence of radiation and contact heat transfer was not detected. 10 0-9 0-8 07 06

0A 0-3 02 0-1 0

1

3x10"3

2 WTC9

HCS

FIG. 42. Generalized curve of t,/t^ as a function of WTCJHC8 according to the experiments of Tsukhanova and Shapatina. 1 - # / < / = 4 1 ; 2 - Hld=2Q5;

3 - Wd = &2; 4 - generalized curve.

In Fig. 42 the generalized curve is given for comparison with Saunders' and Ford's data. The conclusion that heat transfer is independent of particle size contradicts all previous work, and results only because the authors used pieces of approximately the same diameter (2-3 mm). For practical calculations the authors propose the method of Saunders and Ford. Perhaps because it proved possible to solve problems without using the heat exchange coefficient the authors did not attempt its definition. This lack narrows the field of practical application, for without the coefficient it is not possible to solve the problem of heat exchange in other configurations, e.g. parallel and counterflow.

76

Laboratory Investigations of Heat Transfer in Packed Beds 4. Research Carried Out by Shapovalov The earliest attempt to prove Furnas' data was undertaken by M. A. Shapovalov(31) using iron ore in three size fractions, 12-30 mm, 11-19 mm and 2-5-4 mm. Air was the heating medium, passed through a stationary bed at various velocities (related to the speed of movement of gases in a blast furnace). It is assumed that the coefficient is defined by:

••-lag-

(158

>

In order to find the heat loss, q0, the experiment was run with the containing cylinder at different surface temperatures. The heat Q transferred to the bed, was calculated in terms of the amount of air, its specific heat, and the temperature difference between entry and exit. The average temperature difference Afgs was determined by the difference between the arithmetic mean gas and burden temperatures. The coefficients calculated in this way are two to three times smaller than those given by Furnas' formula for a blast furnace burden. Apart from this, Shapalov gives hv in terms of gas velocity: hv = aw (159) but did not take the matter further—evidently for lack of observa­ tions. Unfortunately, the description of the experimental method, and of the means of measuring solids temperatures are incomplete and do not permit an evaluation of the results. The absence of experimental data makes it impossible to repeat the calculations. 5. Research Carried Out by Chukhanov and His Fellow Workers To reduce the heat loss error Chukhanov kept the temperature of the containing cylinder constant (at the entry temperature of the gas). This method, as was shown by the experimental control achieved, meant that the "danger of heating or cooling the bed and the gas from the walls of the container" could be "practically 77

Heat Exchange in Shaft Furnaces excluded". (5) In practice these conditions were achieved as follows. A bed of the experimental material of given height was heated by hot air and electrical windings, at a rate appropriate to the experiment, until the moment when the temperatures of the gas at entry and exit of the bed seemed to be equal both to each other and to the value selected for the experiment. With the help of the ammeter and the thermocouple the heating system of the apparatus was fixed, the blast switched off, and the hot burden removed and rapidly replaced by a fresh charge of the same material. The blast was then turned on and the thermocouple readings taken. For the most part the particle size of the burden was one-fifteenth that of the cylinder diameter. Chukhanov carried out experiments with beds of small steel balls, of particles of damp coal from the Moscow region, peat, bronze cylinders and chamotte fragments. In the published article(5) only the steel ball results are quoted (for Bi = 0-010-0034). This condition enabled the use of Schumann curves in working out the results. Bed height varied widely — from 10 to 125 mm. Air temperature was about 230 °C, and the air velocity (on the total cross section of the container) varied from 0-69 to 2-0 m/sec. The balls were all of 3-15 mm diameter. In working out the experimental results the curve t'g' = / ( r ) was compared with theoretical curves computed for different hs values by Schumann's methods. The coincidence of the experi­ mental and calculated curves at once gave the heat transfer coefficient. In order to obtain an average value for hs, close to the true value, experiments were made using a correspondingly nar­ row range of temperatures t"/t'g = 0-4-0-6. The results of these experiments of Z. F. Chukhanov serve as an example of the fortunate coincidence of experimental and calculated data. (5) Figure 43 refers to the heating of 3-15 mm steel balls, 10 cm bed height, average gas velocity 1-965 mm/sec, gas inlet tempera­ ture 212 °C. All the data were worked out in the form Nu = / ( R e ) . For Re = 100-140 the following equation results: Nu = 0-24 Re 0 8 3 . (160) 78

Laboratory Investigations of Heat Transfer in Packed Beds The coefficients obtained by Chukhanov were less than those determined by Furnas using his first method. These results led Chukhanov to two basic conclusions: 1. Heat exchange in packed beds is distinguished by its high intensity. 2. Heat exchange is significantly influenced by gas velocity.

0

25

50 75 100 125 150 Time from beginning of test % sec

175

FIG. 43. Comparison of experimental data (Z. F. Chukhanov) with theoretical curves. 1 - hs = 620 kcal/m2 hr °C; 2 - hs = 310 kcal/m* hr °C; 3 - hs = 186kcal/m2hr°C.

6. Research Carried Out by Bernstein, Paleev and Fedorov This work is interesting in that it allows us to determine the purely external heat transfer coefficient. The internal resistance is fully excluded, in contrast to all other researches - which were forced to choose experimental conditions (small d9 high conduc79

Heat Exchange in Shaft Furnaces tivity, etc.) such that the internal resistance could be neglected, often introducing significant errors of up to 17 per cent of the actual value of h. It is well known that, in drying, there is an initial constant rate period. Heat transfer and mass (moisture) transfer will proceed simultaneously at the surface, and for the constant rate period the temperature of the bed material coincides with the wet bulb temperature. The temperature of the gas (effecting drying) was found to be constant at the dry bulb value. To calculate the heat transferred it is also necessary to know the amount of evaporated moisture, determined simply by weighing the bed material before and after the experiment. The amount of heat transferred from the gas is determined by the following equation : (2) Q = C?w(595 + 0 - 4 6 ^ w b - ^ ) + ( g l C s + g 2 ) ( r w b - ^ ) ,

(161)

where Gw

= amount of evaporated moisture over the duration of the experiment (kg); Cs = specific heat of the material (kcal/kg °C); 'wb = w e t bulb temperature (°C); t'M = initial temperature of solids (°C); gx = weight of dry material in bed (kg); g2 = weight of unevaporated moisture (kg).

Under these conditions the temperature of the solids can be considered constant. The average difference in temperature between the particle surface and the gas flow is therefore given by At

**-

In Ah/At2

'

°62)

where Afx and At2 refer to temperature differences at entry and exit. If we calculate the surface of the bed S, then, knowing the duration of the experiment (Ar) we can determine the heat transfer coefficient (hs): Q_ >h =SAxAt o Ae • 063) *gs The above method was the basis of Bernstein's work. (2) 80

Laboratory Investigations of Heat Transfer in Packed Beds For the experiments gypsum, with a 10 per cent (by weight) addition of cement, was used, in the form of 16, 21 and 31 mm balls, and also lumps of arbitrary form measuring from 12 to 18 mm. Bernstein conducted his experiments over a wide range of Reynolds numbers (20-1850) and worked out his results in the form: Nu = A Ren, (164) where A is a coefficient, depending on bed porosity, and the exponent n depends on the form of the particles. Thus for balls n = 0-6 and for arbitrary forms n = 1-0. The dependence of the coefficient on the porosity is complex, and is described by a curve with a maximum at 45-55 per cent porosity. As V. N. Timofeev showed,(27) this maximum was determined on erroneous data by Furnas (thus the point plotted for M = 0-295 is inaccurate). Bernstein himself explains the improvement of heat exchange with porosity in that the influence of lower gas velocities is to a significant degree countered by the increase in that part of the surface of the particles involved actively in heat exchange.(2) This effect should be most marked with a voidage of 100 per cent (a single ball), but the data of D. N. Vyrubov(32) cited by Bernstein, says the opposite—in this case heat transfer deterio­ rates, which follows naturally from the decrease in gas velocity. Bernstein also discusses the dependence of the coefficient on the porosity of beds of particles of arbitrary form. For a com­ parison with his own data Bernstein uses Furnas' experimental material for coke and iron ore (70 and 40 mm diameter respec­ tively). Under these conditions the internal resistance begins to play an important part. Finally, if we use Furnas' data corrected by V. N. Timofeev and plot them on Bernstein's graph, Nu =/(Re, M), for particles of irregular shape, they fall on the same straight line obtained for gypsum particles with another value of M, and hence no clear dependence of hv on M is to be found. The analogy between heat and moisture transfer is also used in the work of Fedorov/33, 34) Like Bernstein, he used moistened 81

Heat Exchange in Shaft Furnaces materials, and related all his observations to the constant rate period. During the experiment the rate of loss of weight of the material during the drying process was recorded by weighing. This was related to 1 m2 of the evaporative surface, defining this (for the irregular particles with which Fedorov was concerned) as the surface of balls with volume equal to the average volume of the particles. The heat transfer coefficients from material to air was deter­ mined by: hs =

WeVe

kcal/m3 hr °C,

(165)

'db~~ 'wb

where we = rate of evaporation (kg/m2 hr); re = heat of evaporation (kcal/kg); / db and fwb = respectively, dry and wet bulb temperatures (°C). Fedorov used this method for coal from the Moscow area, size 3-4-4, 6*6-8 and 8-12 mm, and also for mixtures of these size fractions. He represented his results in the form: Nu = 0-23 Re0'863 .

(166)

This equation should give accurate values of h for Re between 15 and 160; for larger Re the dependence is broken. It is also necessary to note that the experiments were carried out at temperatures of only 20-40 °C, and in a few cases at a tempera­ ture of 80 °C. In a comparison with the work of Saunders and Ford, Furnas and others, Fedorov(34) notes a satisfactory agreement (within 15-20 per cent) between his data and those for small balls; he therefore considers that the form of the particles, and the resulting porosity, have little influence on the heat transfer coefficient. Equations (163) and (165) make it possible to determine the "pure" heat transfer coefficient as a function of the temperature difference between gas and solids surface, since, in the first period of drying, the inside temperature of the lump does not change. This method, based on the analogy between heat and moisture transfer, differs from the direct method, and allows the carrying 82

Laboratory Investigations of Heat Transfer in Packed Beds out of experiments in great number, so that the required results can be obtained despite the low accuracy (±20 per cent). One must point out a defect of the method—the low temperatures of the bed material (below 100 °C), at which the influence of radi­ ation is negligible. At the temperatures normally found in in­ dustrial practice it is impossible to ignore radiation. 7. Research Carried Out by Timofeev In Timofeev's paper<27) most of the published experimental data for heat exchange in packed beds is reviewed. Furnas car­ ried out experiments in which Bi was equal to 1-0 or less, so that the internal resistance consisted of about 17 per cent of the total. In order to use Furnas' results with metal balls Timofeev cal­ culated anew the heat transfer coefficients, since Furnas' method of calculatio (for his first experimental method) was inaccurate. The point is that Furnas used the formula dt h

*=

At(S-S>)

'

(167)

where q0 is the heat loss to the surrounding atmosphere. As Timofeev points out, for an instantaneous measurement of tM and tg> q0 should not be taken into account in the calculations. After making this correction, Furnas' data coincides with that of other workers. The results of his experiments, and those of I. M. Fedorov, L. Z. Chukhanov, R. S. Bernstein, and others, are united by V. N. Timofeev in the equation Nu = 0-61 Re 067 , (168) from which it follows that h

* = 0-61

ffff,3

kcal

M 2 hr °C,

(169)

where Xg = thermal conductivity of the gas (kcal/m hr °C); r\ = viscosity of gas (kg/m2 sec); 2 W0Q8 = mass flow of solid per unit area of bed (kg/m sec); d = diameter of lump (m). 83

Heat Exchange in Shaft Furnaces This reduced formula is correct for Re between 200 and 2000. Timofeev analysed the influence of porosity on the surface heat transfer coefficient. He noticed that in Furnas' work changes in porosity took place together with changes in ball diameter, so that the influence of each factor could not be shown separately. What is more, Furnas' derivation of the function hv did not satisfy the theory of similarity. Having reworked Furnas' data adhering to the principle of similarity Timofeev showed that, with experiments accurate to within 5 per cent, the porosity has no effect upon the surface coefficient, but it does influence the volume coefficient. Unfortunately, Timofeev derived his formulae for the case of small internal resistance (1/Bi > 1-0-1-5). Experiments with large lumps of ore, limestone and coke were not taken into account. Thus the conditions of heat exchange in beds of large lumps for the larger Bi values are not reflected in the formulae of V. N. Timofeev. More recent work of V. N. Timofeev and I. V. Dubrovin(25) was devoted to a study of the cooling of coke with inert gases, and not only was Schumann's well-known problem solved more simply, but a series of experiments was carried out for the deter­ mination of heat transfer coefficients. The experimental method consisted of cooling a coke bed, pre­ heated electrically. The gas temperature and the coke temper­ ature were recorded, to give curves, tg = /i(r) and tM =/ 2 (t), which could be superimposed on the corresponding theoretical curves (as with Furnas and Saunders). The authors showed that the coke temperature measurements were more reliable, since the thermal inertia of the thermocouple was practically zero, whereas the gas temperature measurements at the beginning and end of the experiment, due to thermocouple inertia, did not coincide with the theoretical curves. This discrep­ ancy between the methods of working out the data shows itself in the accuracy of the experiment (between 11-0 and 13-5 per cent). Research was carried out on coke beds of 10-15, 15-25 and 25-40 mm fractions. 84

Laboratory Investigations of Heat Transfer in Packed Beds The gas velocity (on the shaft area) lay between 0-298 and 0-566 m/sec, and Re between 325 and 915. Figure 44 shows the experimental results, which agree with eqn. (168) (plotted as a straight line on the figure). logNu

1-8

^f

T

r

M

10

24

2-6

2-5

2-7 log Re

2-8

2-9

30

FIG. 44. Experiments of Timofeev and Dubrovin for cooling coke.

Although the surface heat transfer coefficient is a complex function of many parameters: hs

=

fi^g, d, Re, tg, tju, AM),

which can be determined only by experiment, the authors did not detect the most interesting dependence —that of hs upon As, pre­ sumably because the internal resistance was not great (Bi^l*0, and only in one experiment was Bi % 2-0). 8. Conclusions to Experimental Work Much ingenuity has been shown by research workers in the design of experiments for solving complex problems of heat ex­ change in packed beds. The work of I. I. Paleev and R. S. Bern­ stein is remarkable in that it enables purely external exchange to be observed; other researchers made interesting generalizations, and so on. However, Furnas carried out work over the widest range of the different parameters: temperatures up to about 1100 °C, veloc­ ities from 0-6 to 1-8 m/sec, and lumps of the most diverse mate­ rials, from 4 to 70 mm in diameter. The greatest deficiency in his experiments was his failure to separate external from internal transfer, and to find the influence *J

Heat Exchange in Shaft Furnaces of screen analysis on heat transfer, although, in fairness, it must be said that the last problem has, as yet, not been solved by any­ one. One would wish, above all, to notice Furnas' objectivity, in that he himself recorded almost all the deficiencies of his work, and thus lightened the work of those who followed him. Let us note, for example, the following. For otherwise constant condi­ tions (w0, t, d and M) the coefficient of heat transfer differs for different materials (see Table 5). For anthracite, for example, the coefficient is three times that for sinter. Actually, if w0, f, d and M are the same, then what individual properties of the material can so influence the value of the coeffi­ cient? According to Furnas, the coefficient involves the heat transferred by 1°C temperature difference between the gas and the average solids temperature. In this case the question posed above can be answered thus: the character of the surface (rough­ ness, degree of blackness), and the thermal conductivity of the material. A new question then arose. (8) Would it not have been more con­ venient and useful to determine the "pure" coefficient of external heat transfer ? In this case the answer to the question posed above would involve fewer variables (mainly the character of the sur­ face). Also, the tempting prospect of calculating the system would open. Pieces of material, with similar surface characteristics, would have the same hv or more accurately As. There would be no need to define hv experimentally for each new material. It would be necessary merely to find, from the voluminous experimental data available, a material of similar surface characteristics, and to take the value As for this material. Obviously it is incomparably more convenient to arrange the data according to the "pure" coefficients of heat transfer—but how are these to be obtained ? Two means are possible: 1. To set up experiments by new methods making it possible to determine the "pure" hv. As we know, Paleev and Bernstein are working in this direction, in their experiments with moist gypsum balls. 86

Laboratory Investigations of Heat Transfer in Packed Beds 2. To dissect Furnas' heat transfer coefficient (related to the average solids temperature) into two parts—internal and external. The latter is the "pure" coefficient, related to the temperature difference between the gas flow and the solids surface. A study of Furnas' experimental techniques led us to conclude that this dissection is impossible until Schumann's calculation is extended to real solids. Later we shall carry out this calculation. New Interpretation of Furnas9 Experimental Data The solution of the problem of real solids in a fixed bed, which we succeeded in finding as a result of the hydraulic analogy, allows us to determine the true heat transfer coefficients (related to surface temperature) from Furnas' original papers. Furnas' method of comparing the experimental data with Schumann's curves (plotted on a semi-logarithmic basis) is, with­ out doubt, irreproachable. Hence Furnas was able to find the total heat transfer coefficient hv9 embracing the outer and inner resistances. We have been able to rework Furnas' data, since the Biot number is known for each experiment. In Furnas' work (26) data is given on the size of the lumps, but the author unfortunately did not measure the thermal conductivities, with the exception of one value for an ore (A = 1-44 kcal/m hr °C). However, an attempt to shed light on the more puzzling as­ pects of the problem (which were formulated by Furnas himself, although he was not in a position to explain them) is not impos­ sible. Kel'berg (35) showed that the specific heat of coke, regardless of the original coal, is almost always constant and equal to 0-21. The position is less certain for other materials. What is more, thermal conductivity data for lumps are rare, and such data as we have managed to assemble are given in Table 23. The core of our recalculation of Furnas' data is contained in the three terms of the following equation:

TV = FV+-9X' 87

(169a

>

Heat Exchange in Shaft Furnaces Furnas determined the first term experimentally, and we cal­ culated the third term from the dimensions of the lumps and their conductivities. Hence unequivocal values of the external coefficient may be calculated. The external coefficients hv obey somewhat different laws, which is only natural. Calculated in this way the hv values in­ crease with increasing lump size d and on a logarithmic plot the 0015 1

V

\£.

0010 9 8 7 6 5

ik

Pi

A

^

] \2 \^

**S

-1

Nt A 00015 00010 9 8 7 6 00005

Nf ▲

Si*. r\J 3;

0-5

1

H> r11 m

2 3 A 5 6 7 8910 Lump diameter d, cm

FIG. 45. Influence of lump diameter on heat transfer coefficient. 1 - slope 0-75 (Kitaev);

2 - slope 0-90 (Furnas);

3 - slope 1-12 (Kitaev).

hv-d points lie on a more gently sloping line than that of Furnas (whose slope of 0-9 compares with our 0-75, Fig. 45). The equa­ tion for hv versus d is hv = E/d°-™ kcal/m3 hr °C

(170)

(where hv is already related to surface temperature). This equation confirms Furnas' view that the finer the particle size, the larger the proportion of surface excluded from heat exchange owing to "deterioration". We can therefore propose for the heating sur­ face per cubic metre: S = &5(l-e)/d°^. (171) 88

Laboratory Investigations of Heat Transfer in Packed Beds In connection with this it becomes clear that for spheres all authors, apart from Saunders and Ford, obtained the equation hv = E/d1'*5 kcal/m 3 hr °C,

(172)

since metal balls cannot deteriorate, and for them S = 6(1-e)/d.

(173)

Nonetheless, the exponent of d for the cooling of coke (1-2) is an anomaly, since coke can deteriorate. If it is necessary to determine the heat transfer coefficient (with reference to the average temperature of the lump as a whole) it is best to use the formula

h0 = hvl(l + ^-\.

(174)

For small ranges of hv this function can also be represented in the form K = E/dn. (175) For example, for velocities between 1-15 and 1-30 m/sec and d from 5 to 100 mm (Fig. 44): hv = Eld1'12.

(176)

In evaluating the influence of gas velocity on heat transfer coeffi­ cient Furnas' formula is significantly in error. The fact is that with greater velocities the external resistance falls, but the internal resistance is unchanged. According to eqn. (174) the dependence on velocity cannot be expressed as a power function; if this is attempted the exponent of velocity is extremely variable, and at high velocities it can fall to zero. This occurs when h -+ «>, when hv takes the value (e.g. for spheres) of hv = 5X/R (177) and does not tend to infinity. Hence the function put forward by Furnas hv = £ < 7 (178) probably holds for only a limited range of Biot numbers and ve­ locities. 89

Heat Exchange in Shaft Furnaces On correcting Furnas' data the correction grew with increasing hv and/or velocity, and the curves grew steeper but for all Bi the same function approximately holds: h0 = £ < 9 2 . (179) It is now understood that this function is not fully in agreement with other work on convective heat exchange,(20) e.g. bundles of tubes, but it does agree well with the work of Saunders and Ford on packed beds. By the way, let us note that Furnas himself obtained for spheres (small Bi) the function hv = EwQ.m) We leave the influence of temperature as before, in the absence of data on the temperature dependence of specific heat. Resulting Formulae In this way, for beds of natural materials, we obtained the following formula for the heat transfer coefficient (related to surface temperature): hv = As ^pn-

M' kcal/m3 hr °C,

(180)

where w0 = velocity relative to shaft cross-sectional area at 0 °C (m/sec). M' = coefficient depending on the voidage of the burden. For 20 per cent voidage M' — 0-5. A more accurate relation between M' and voidage has yet to be established experimentally. Below we give As values for various material, based on the revision of Furnas' data: Iron ore Sinter Limestone Coke Bituminous coal

160 150 166 170 170

Anthracite 140 Chamotte brick (fragments) 135 Dinas brick (fragments) 150 Magnesite brick (fragments) 180

There is some indication of the individuality of the material in the new data, which is natural enough. Thus coke is rougher than 90

Laboratory Investigations of Heat Transfer in Packed Beds anthracite and might be expected to have a greater hv. However, the differences are so small that one can for all practical purposes take the single value, As = 160, for all lump materials. The weak influence of the material of the lump on heat exchange was also noted in the work of Tsukhanova and Shapatina.(30) For spheres the formula hv = 12 ^° 1 3 5 kcal/m3 hr °C

(181)

holds, which brings Furnas' data into line with that of Saunders and Ford. Our results afford wide possibilities for practical calculations. It is possible to calculate hv for any material, without a special experiment. By finding hv (related to surface temperature) it is possible to calculate well-known processes of heat absorption (e.g. drying, calcination, fusion). Basing ourselves on Timofeev's considerations, we have not used Furnas' coefficient M (as a function of voidage) and in practice there are usually no data for the voidage of a furnace charge. However, experiment shows that voidage has a consider­ able influence on heat transfer, and for 20 per cent voidage the coefficient is halved. This important matter has not been studied in detail and so lowers the value of B. I. Kitaev's formula, cor­ rect for lumps of approximately equal diameter, whilst justifying the rejection of Furnas' particular M =
Heat Exchange in Shaft Furnaces ing was begun. The surface (with underheating) at first cooled more quickly than when the slab was completely heated, and at the end of cooling, cooled more slowly. When ts/t'g was plotted on a t-Z diagram quite different curves were obtained (Fig. 46). Trac­ ing cloth with the Schumann curve for Bi = 4 (and not Bi = 2, for which the experiment was carried out) was placed on the graph obtained. The lines 7 = 0 coincided. A false impression was created that the cooling had taken place with a large coeffit/tg 1-0 .

.

,

r

,

FIG. 46. Cooling of an incompletely heated lump of coke—model experiment with Bi=2.

cient of heat transfer. This impression was the greater as the degree of underheating increased (i.e. the greater the residual temperature difference t8 — tc)9 and was the less pronounced for smaller lumps and greater conductivities. The following anomalies, noted by Furnas in working out his data, are explained by these experiments and considerations: 1. The hv relationship for coke varies from heating (hv = B/d0'9) to cooling, when the exponent is raised to 1-3. 2. For cooling the coefficient As is for all solids greater than for heating; this is especially true for coke, where the difference is 100 per cent. In this connection Furnas himself showed that he had not suc­ ceeded in explaining the anomalous behaviour of coke, or in 92

Laboratory Investigations of Heat Transfer in Packed Beds understanding why, in all systems studied, As for cooling is great­ er than for heating. We cannot justifiably state that the considerations expressed (on the cooling of underheated lumps) are proof of the mecha­ nism of the coke anomaly, if we do not know, for certain, that underheating took place in Furnas' experiments. These consider­ ations are more properly to be regarded simply as a guess, which, however, is immediately converted into evidence if we can es­ tablish the fact of underheating. There is certainty that underheating did occur, for the follow­ ing reasons: 1. In heating and cooling the coke the experimenter was dealing with large pieces, 7 cm in diameter. For large values of Bi (in the coke experiments up to 12), many hours will be needed for full heating—several times longer than for the heating of lumps of infinite thermal conductivity. 2. At large values of Bi and small values of Y, the tg/t'g curves can easily mislead the experimenter, since they cross very quickly into an almost horizontal position near the value tg/tg = 1. This horizontal section can easily be taken, in error, for the end of heating of the bed, whereas the lump centres may still be far from heated. For example, this situation can be observed, with Bi = 10 and Y = 1, with tc/tg = 0-31. In describing his experiments Fur­ nas states explicitly that, when the temperature at the bed top ceased to rise, the furnace was removed and cooling of the bed commenced.(26) On the basis of the above, we reach the following conclusions: (a) Furnas' experimental data for the cooling of coke and other materials are in error, and it is recommended that they should not be used. (b) Furnas' other data is probably in error as defined by him­ self—of the order of 20 per cent. (c) Furnas' formula, eqn. (138), cannot accurately reflect a combination of the two processes (external and internal heat ex­ change) and can therefore be regarded only as an approximation. (d) The use of eqn. (180) for the external coefficient is to be re­ commended. 93

Heat Exchange in Shaft Furnaces

15000

10000 5000 Volumetric heat transfer coefficient, 3 hy, kcal/m hr °C

0

K) 20 Gas velocity (0°C) in free cross-section of bed, m/sec

FIG. 47. Nomogram to find volumetric heat transfer coefficient in relation to gas velocity w0, gas temperature and lump size d.

10000

I ^

<

Hr

1 . 1 «. R 2 h v ~ h v SX

>^ *J0000s

£ 5000

7000 — • 5000 — 2500

5000

10000 h v , kcal/m 3 hr °C

15000

FIG. 48. Nomogram to find the overall heat transfer coefficient hv as a function of the coefficient hv, the lump size R and the thermal conductivity A.

94

Laboratory Investigations of Heat Transfer in Packed Beds (e) The overall (external and internal) coefficient can be deter­ mined from the equation:1"

+

k'k ^-

(182)

where R = the radius of the lump (m); A = the thermal conductivity (kcal/m hr °C); hv = the external heat transfer coefficient, to be found graphically in Fig. 47 (kcal/m3 hr °C). The graph, Fig. 47, is a plot of eqn. (180), with the coefficient M' taken as 0»5. hv may be found from Fig. 48, given hv, R and A. This is a plot of eqn. (182). With the aid of these two graphs finding the coeffi­ cient of heat transfer is considerably simplified.

t Editor's note: Equation (182) is a useful approximation, derived by substituting eqn. (43) (with e = 0-5) into eqn. (35).

95