Heat transfer measurements and simulation of KATAPAK-M® catalyst supports

Chemical Engineering Science PERGAMON

Chemical Engineering Science 54 (1999) 1375-1381

Heat transfer measurements and simulation ofKATAPAK-M® catalyst supports C. von Scala Sulzer Chemtech Ltd., Catalyst Technology, CH-8404 Winterthur, Switzerland

M.Wehrli Sulzer Innotec Ltd., Fluid Dynamics Laboratory, CH-8401 Winterthur, Switzerland

G. Gaiser Institut fUr Chemische Verfahrenstechnik, Stuttgart Univer!ljty, D-70 199 Stuttgart, Germany

Abstract - The heat transfer from a heated wall to a cold fluid in a tube filled with KATAPAK-M elements is investigated. The overall heat transfer coefficient, the local wall heat transfer coefficient and the effective radial conductivity are determined from the experimental results using the pseudohomogeneous plug flow model. The same heat transfer is also simulated with three-dimensional numerical flow simulations (CFD). The simulation results agree well with the experimental fmdings and reproduce the characteristic flat radial temperature profiles. The overall heat transfer coefficients obtained by numerical simulations support the experimental results. It is shown that the local wall heat transfer coefficient and the effective radial conductivity depend strongly on the experimental resolution of the temperature profiles close to the tube wall since the pseudohomogeneous model does not adequately represent the relatively flat temperature profiles typical for KATAPAK-M. With the numerical results presented in this paper, a first reliable basis for a better understanding of catalytic reactors using KATAPAK-M is created. © 1999 Elsevier Science Ltd. All rights reserved.

Keywords: catalyst support, CFD, flow simulations, heat transfer coefficients, pseudohomogeneous plug flow model, random packing, structured packing INTRODUCTION For reactions which are strongly exothermic, such as the partial oxidation of n-butane to maleic anhydride, or strongly endothermic, such as the dehydrogenation of ethylbenzene to styrene, a good heat transfer is important to guarantee a homogeneous radial temperature profile throughout the reactor. In exothermic reactions, a poor heat transfer can induce hot spots, which favor side reactions, diminishing the selectivity.of the desired product. Hot spots can also lead to fast deactivation or sintering of the active catalytic components and even to temperature runaway. In endothermic reactions, a poor heat distribution leads to a decrease in the conversion. Catalyst supports are frequently used in two basic forms, as random packings or as monoliths structures with parallel channels. Both types have their limitations. Using random packings, transverse mixing is restricted and .the fluid motion through them is usually characterized by channeling and stagnant zones. Another disadvantage is the high pressure drop. The honeycomb structure offers a much lower pressure drop; the absence of radial mixing, however, results in very poor mass and heat transfer. This type of structure is therefore practically not used in reactions where the heat transfer is a main issue. The KATAPAK-M washcoated structure is a promising alternative to the commonly used random packings, offering a homogeneous radial temperature profile combined with a low pressure drop.

FUNDAMENTALS The KATAPAK-M catalyst support structure from Sulzer Chemtech consists of superimposed individual corrugated metal sheets, with the corrugations in opposed orientation such that the resulting unit is characterized by an open cross-flow structure pattern. Such patterns are well proven as distillation packings, permitting optimal gas-liquid contact and high separation performance at low pressure drop; and as static mixers, improving the radial mixing, the heat and the mass transfer together with the advantage of a narrow residence time distribution (Stringaro et al., 1998). A picture of KATAPAK-M catalyst support structure is shown in Figure 1.

Figure 1: KATAPAK-M catalyst support structure.

0009-2509/99/$-see front matter © 1999 Elsevier Science Ltd. All rights reserved. PII: S0009-2509(99)00077-9

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C. von Scala et al.lChemical Engineering Science 54 (1999) 1375-1381

Earlier flow visualization experiments (Gaiser, 1990) with cross-flow structures similar to KATAPAK-M revealed two flow types within such structures, see Figure 2. Part of the fluid follows the valleys, is reflected at the side wall and returns through a valley of the opposite plate, see regime (a). Some fluid follows the main flow direction and crosses the valleys, regime (c). The relative amount of flow in each regime will depend on the channel height, channel angle and channel width. The ideal combination of both flow types, regime (b), allows optimal heat and mass transfer. The first flow type (a) is of high importance for the heat transfer to and from the tube wall.

The radial heat flux q, can be characterized by the following equation:

aT

qr = -A r &;

The local heat transfer from the wall to the fluid qw is described by: qw = a w(T,.=R - Tw) (2) T""R is the temperature of the fluid near the tube wall and Tw the tube wall temperature. The overall heat transfer coefficient atol is defmed from the total heat flux q by following equation:

q=atot(T-Tw ) (3) where 't is the mean temperature over the cross sectional area. Due to continuity, the relationship q = q, = qw holds next to the wall. Provided that the tempera~e profIle is parabolic, the overall heat transfer coeffiCient a tot is given by: R 1 1 -=-+(4)

a tot

(a) (b) (c) Figure 2: Flow types within cross-flow structures: (a) flow following the valleys, (c) flow crossing the valleys, (b) ideal mixture. For construction reasons there is a cleavage between the structures and the tube wall, thus creating a third flow type, a bypass. The flow coming through the valleys is not directly reflected to the opposite plate, but tends to stay for a certain time in this bypass before leaving it again through another valley further downstream. Hence, to ensure a good radial mixing, this cleavage should be kept as small as possible. Definition of the heat transfer parameters Strongly exo- or endothermic processes are generally carried out in multitube reactors to permit heat removal, respectively heat supply, through a fluid medium that surrounds the tubes. The heat transfer in tubes filled with catalytic random packings is usually calculated using the two-dimensional pseudohomogeneous plug flow model (Dixon, 1988, Logtenberg and Dixon, 1998) with two parameters describing the overall heat transfer resistance: a) an effective radial thermal conductivity A, lumping together all heat transfer mechanisms within the fluid (conduction and convection) while neglecting temperature differences between the liquid and the solid phase, and b) a wall heat transfer coefficient a w first introduced by Coberly and Marshall (1951). The latter is used in conjunction with the near wall temperature jump, which may be observed experimentally. It describes the heat transport at the boundary between the fixed bed and the tube wall and stands for the complex interplay between fluid convection and conduction close to the heat exchange surface and the conductive transport by the solids of the fixed bed.

(1)

a

w

4A r

The total heat flux is therefore dependent on both contributions; the same amount of heat can be transferred if a w is small and Ar is large or if a w is large and A, is small. Figure 3 illustrates the effect of varying A, and a w on radial temperature profIles, see also Eigenberger et al. (1991).

Tw

"... ...'"

T.-R

E

T.-R

'(;j

"0.

.!l

case 2: small r= 0

t.., and large

(l w

r= R

Figure 3: Qualitative representation of the effect of varying heat transfer coefficients on radial temperature profIles. Large values for A, are given if the heat transport within the packing is good, resulting in an almost uniform radial temperature profIle. Small values of A, lead to significant radial temperature variations. Large values for a w correspond to a good heat transfer between the wall and the packing resulting in a small temperature jump near the wall. Small a w values allow only a poor heat transfer from the wall and lead to a large temperature difference between the wall and the fluid. With increasing radius the influence of the heat transfer coefficient a w decreases and the radial heat transfer coefficient A, gains more importance. HEAT TRANSFER EXPERIMENTS Experiments were carried out to determine the heat transfer efficiency of the KA TAPAK-M structures.

C. von Scala et al.lChemical Engineering Science 54 (1999) 1375-138/

Experimental set-up Four KATAPAK-M elements were filled in a tube of D = 0.05 m diameter, the length of each element being 0.1 m, giving a total length of L = 0.4 m. The elements were rotated by 90° along the axis relative to the previous one. Cool air entered the test section at To = 20°C and was heated through the tube wall which had a constant wall temperature of Tw = 280°C. At the exit the radial temperature profile of the flow leaving the structures was measured using a thermocouple which could be moved freely over the cross sectional area with the aid of microscrews. Determination of the heat transfer parameters The heat transfer parameters a w and Ar were determined by a least square fit of the average radial temperature profile based on the two-dimensional pseudohomogeneous steady plug flow model (Dixon, 1988):

with the following boundary conditions: Inlet (z = 0): T = To Outlet (z = L): aTI& = 0 Center (r = 0): aTlar = 0 Wall (r=R):

ar

Ar

the bypass flow becomes more important and mixing is less efficient, decreasing the overall heat transfer coefficient. The high effective radial conductivity is a typical feature of the KATAPAK-M structure. It is at least twice as big as corresponding data of random packings with a similar specific surface area.

Model Limitations The model expressed by equation (5) is a well established, relatively uncomplicated tool for the calculation of heat transfer in random packings, provided the heat transfer coefficients are known. It is therefore reasonable to apply the same model for the KATAPAK-M structure. The heat transfer coefficients a w and Ar are, however, intimately linked with this model. Their numerical values are very much dependent on how well the model profiles fit the experimental temperature profiles. It can be shown that a parabolic radial temperature profile, expressed by equation (6), approximates well the solution for the equation (5), applying the boundary conditions at a given longitudinal coordinate z with a local average temperature T and a total heat transfer coefficient a to':

~-

w

The heat capacity cp and the density p were considered as constants at T = 20°C. The effective radial conductivity Ar as well as the porosity E were considered independent of the radius r. The last boundary condition accounts for the temperature jump at the wall, according to equation (2). For the purpose of flexibility the equations were solved numerically with a fmite difference method although an analytical solution exists for the most general case even with axial dispersion but without chemical reaction (Dixon, 1988). The resulting heat transfer coefficients are shown in Table 1. With increasing flow rate the heat transfer parameters increase due to a) improved mixing inside the structure and b) reduced thickness of the boundary layers.

=

[mls]

[W/mK]

[W/m2K]

[W/m2K]

0.5 1.0 1.5 0.5 1.0 1.5

1.02 2.22 2.78 0.85 1.32 1.94

63.3 69.9 85.0 57.4 67.7 77.6

45.6 58.4 71.4 40.4 51.3 62.1

The influence of the cleavage Ie between structure and wall is clearly seen in Table 1. With increasing cleavage

a tot aw

_

a totD Ar

[(!:)D _.!.]4

(6)

2

While such a function is very close to the temperature profile in random packings, this does not apply for the relatively flat KATAPAK-M temperature profile. In fact, the pseudohomogeneous model may not be adequate for the cross-flow structures. In the context of random packings the question of adequacy has already been risen by Gunn et a/. (1987) or more explicitly by Tsotsas and Schliinder (1990). The main reason for the problem is the local wall heat transfer coefficient which, dependent on the real thickness of the thermal sublayer, is either a value with physical meaning or merely a mathematically useful lumping parameter. As an example consider Figure 4 with a normalized experimental temperature profile (left-hand side of equation 6). 1.2

1.1

Table 1: Heat transfer coefficients experimentally determined for different cleavages Ie between elements and tube wall. aw a,ot Vo Ar

2

Tw

T - Tw

aT=~(T._T)

1377

...• ~

····i·

I

....

.

.

experimental speculative -curve (a) -_. curve (b)

1t,0.9

~0.8 t,0.7 0.6

....,

curve c

0.5 0

0.2

0.4

0.6

0.8

r/R

Figure 4: Least square fits of the radial temperature profile in KATAPAK-M (vo = 1.0 mis, Ie = 1 mm). Twelve points have been measured experimentally, a speculative thirteenth point close to the wall has been

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C. von Scala et al.lChemical Engineering Science 54 (1999) 1375-1381

added at rlR = 0.968 (according to results from CFD flow simulations). The lines represent least square fits of equation (6) using different numbers of points near the wall. Curve (a) is based on all 12 experimental points, curve (b) would be obtained if a thirteenth point could have been measured experimentally, curve (c) is obtained if the last three near wall points are boldly omitted. In the flat profile region inside the structure the last curve undoubtedly represents the best fit, it is however unsatisfactory close to the wall. Curves (a) and (b) are bad everywhere, while (a) is perhaps the best compromise. Since the curves end at different near wall temperatures Tr : R• the conclusion is straight forward that the values of the corresponding wall heat transfer coefficients a w depend on the position of the last data points closest to the wall. Indeed, the coefficients for the three curves differ very strongly, see Table 2, although they belong to the same a tat • Table 2: Heat transfer coefficients for the three different curves in Figure 4. Computed using the solution of equation (5) expressed by equation (6) and experimental overall heat transfer coefficients. Curve Number of A.r points [W/rnK] 71 2.5 (a) 12 77 1.9 (b) 13 66 3.5 (c) 9 Due to this sensitivity it is advisable not to use the effective heat transfer coefficients to qualify the heat transfer efficiency ofKATAPAK-M. The same applies when comparing different types of bed configurations, experimental data from different authors or experimental and computational data. Instead, heat balances or temperature profiles should be used. Despite these shortcomings it is relatively save to use these coefficients within the pseudohomogeneous model, since they satisfy the overall heat balance. Whether values from curves (a), (b) or (c) should be inserted is up to the user. Curve (c) with a large effective radial conductivity emphasizes the flat form of the temperature profiles in KATAPAK-M structures. COMPUTATIONAL FLUID DYNAMICS Computational Fluid Dynamics (CFD) provides data like temperature and velocity profiles all over the computational domain. Hence, heat transfer parameters may be calculated more accurately than with the above experiment relying only on the temperature profile at the exit of the test section.

Computational mesh While computational models of random pac kings with realistic amounts of particles are still difficult to generate, cross-flow structures have now come into reach of today's CFD techniques. Based on a unit cell the interior 3D grid was built semi-automatically. The cleavage Ie between the irregular boundary of the structure and the tube wall was then added manually. The resulting structural mesh for one KATAPAK-M element with a

length of approximately twice the diameter consists of 108'000 hexahedral cells. It is shown in Figure 5.

Figure S: Computational grid for one KATAPAK-M element. The final geometry consists of 4 such elements and empty inlet and exit sections of a few millimeters on both sides. Between the elements a short gap is. consisting of two cell layers, was required in order to take advantage of the unmatched mesh capability of the CFD software. The dimensions of the computational domain are given in Table 3. Table 3: Geometric dimensions of the KATAPAK-M test section used in the CFD simulations. total length L 433mm 6mm inlet section length Linlet 12mm exit section length Lexit element length LK 103mm gap between elements i, lmm tube diameter D 50mm element diameter 48mm and46mm lmm cleavage between structure and2mm and tube wall

Physicai model The momentum equation for the three velocity components, the mass continuity equation and the heat transport equation were solved numerically for laminar flow. The following boundary conditions were applied: Inlet (z = 0): Outlet (z = L): Wall (r=R):

T=To,v=v o

(7)

BTIBz = 0, BvIBz = 0 T=Tw>v=O

The inlet temperature and the wall temperature were set to To = 20°C and Tw = 280°C, in analogy to the experiments. Four different inlet velocities with uniform velocity profiles were considered, namely Vo = 0.5, 1, 1.5 and 5 mls. Temperature dependence of air density, viscosity, conductivity and the heat capacity were calculated from the ideal gas law, second order polynomials and a fifth order polynomial respectively fitted to the data of the VDI Warmeatlas (1991). The heat conduction of the solid structure was neglected, only heat transfer due to fluid flow and fluid heat conduction were taken into account.

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C. von Scala et al.lChemical Engineering Science 54 (1999) 1375-1381

Temperature [0C] 260.0 230.0 200.0 170.0 140.0 110.0 80.00 50.00 20.00

Figure 6: Complete set-up of the four element test section (above) and temperature profile on the longitudinal cut for Va = 1.0 mls (below) computed with CFD. The flow is from the left to the right.

Computational method The fmite volume method as provided in the commercial CFD program STAR-CD was applied. For pressurevelocity coupling the SIMPLE algorithm and the formally third order spatial discretization scheme QUICK were used. Despite experimental evidence for a transition from the laminar to the turbulent regime in the Reynolds number range considered here, no transient effects could be captured during the simulation. They may have been detected if a fmer computational mesh had been used. We assume, however, that the heat transfer coefficients are not much affected by turbulence at the given Reynolds numbers. In the present set-up it is more important that the large scale radial mixing within the structure is correctly simulated.

280 Vo

I

l ~ eXJlenmentl ~ simulation

= 1.0 m/s

260

G 240 e.... ... 220 200 180 0.005

0

0.01

0.015

0.Q2

0.Q25

rIm)

Figure 7: Comparison of simulated and measured average radial temperature profiles at z = 0.42 m for Va = 1.0 mls. 280

CFDRESULTS In Figure 6 the temperature profile on a longitudinal section through the axis is given. It can be seen how the fluid temperature smoothly increases from the entry to the exit on the right-hand side. The contour plot suggests that the radial temperature profiles are flat inside the structure and exhibit a strong gradient close to the wall. Due to the rotation of 90° between two elements the figure shows two kinds of temperature pattern, namely on cuts parallel to the KATAPAK-M sheets (first and third element) and on perpendicular cuts (second and fourth element). Obviously, the radial temperature profiles are a function of the azimuthal coordinate. However, the following discussion is restricted to the averaged temperature profiles. This simplification is justified, since most applied engineering simulations rely on the two-dimensional pseudohomogeneous model, which does not take into account azimuthal variations.

Temperature profiles Figures 7 and 8 show the average radial temperature profiles at the end of the four element test section for two different flow rates. Experimental and numerical results agree very well, and consequently the same applies for the energy balance.

Vo

= 1.5 m/s

260

G 240 e.... ... 220 200 180 0

0.005

0.01

0.015

0.02

0.Q25

rIm)

Figure 8: Comparison of simulated and measured average radial temperature profiles at z = 0.42 m for Va = 1.5 mls. At a given longitudinal coordinate z the average temperature over a cross section is computed as follows:

= f TVzdxdy

(8) fVzdxdy In Figure 9 axial profiles are plotted against the residence time for a range of flow rates. Their end points represent the temperature at the same axial coordinate (z = 0.42 m), which corresponds with the end of the four KATAPAK-M elements test section. Due to the shorter residence time the end temperature is lower at high flow rates, but the temperature increases more rapidly as high velocities enhance heat transfer. T(z)

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C. von Scala et al.lChemical Engineering Science 54 (1999) 1375-1381 temperature. As already observed in the experimental results, the heat transfer decreases with increasing cleavage. On the other hand, an additional simulation using a smaller cleavage (Ie = 0.5 mm) leads only to a slightly increased heat transfer.

250 200

V 150 2!- 100

-vO=0.5 -vO=I.O -vO=1.5 -vO=5.0

50

m/s m/s m/s m/s

Table 4: Mean values of heat transfer parameters computed with CFD.

Vo [mlS] 1.0 0.5 1.0 1.5 5.0 1.0

0

0

0.1

0.2

OJ lsI

0.4

0.6

0.5

t

0.5

Figure 9: Profile of the average axial temperature as a function of the residence time.

Heat transfer parameters Different to the experimental practice, the CFD simulations do not use a temperature jump near the wall, as described by equation (2). Instead, the radial profile ends exactly at the wall temperature Tw' If the computational mesh is fme enough the wall gradient is represented realistically. Therefore, the overall heat transfer coefficient Cl'OI may be derived directly from local heat fluxes and the temperature difference according to equation (3). Figure 10 shows how the coefficient (averaged over the periphery) evolves along the tube. There is a sudden decay at the beginning of each element, followed by a strong increase in the flrst quarter. This is caused by the 90° rotation of the consecutive KATAPAK-M elements. In its function as a static mixer the structure flrst eliminates high fluid temperatures close to the wall, while a more uniform temperature profile is achieved and new wall gradients are built up. This effect is certainly due to the azimuthal dependence of the radial profiles and is out of the reach of the pseudohomogeneous model for catalyst beds. Apart from the effect discussed before the profile in Figure 10 steadily increases in the average and eventually stops at a constant value after a longer distance from the entry. The fact that Cl'OI is not constant in short tubes is consistent with the observations made by Bird et al. (1960). Only in a sufficiently long experimental set-up Cl/OI becomes independent of the tube length. 80~------------------------~

70

Ji 60 ~

Tend

[0C]

58.9 39.3 57.3 67.2 90.5 50.5

245.3 221.0 199.6 122.4 209.0

Comparison with experimental values The simulated overall heat transfer coefficients with gas velocities between 0.5 and 1.5 mls correspond well with the experimental results (see Figure II). Differences up to 15% are due to both numerical and experimental errors. At Vo = 0.5 mls the numerical simulation probably underestimates the heat transfer because conductivity through the solid structure is neglected. There is no experimental value for the highest flow rate (vo = 5 mls), but it is very likely that the overall heat transfer coefficient obtained numerically, see Table 4, is too small. This is probably due to the insufficient mesh resolution, making it difficult to capture very large wall gradients. 80,---------------------------70 ~ 60

Ne

~ 50

J 40 30+----r--~----+_--_+----~~

1.5

2

2.5 lit [s")

3.5

4

--'-experimental, Ie - I mm _experimental, Ie = 2 mm -ir-simulation, Ie = I mm C simulation, Ie = 2 mm

Figure 11: Overall heat transfer coefficient as a function of the space velocity.

~ 50

J

2

Cl lot

[W/m2KJ

40

30 20+------+-----4------+-----~

o

0.1

0.2

0.3

0.4

z[m)

Figure 10: Axial profile of the overall heat transfer coefficient for Vo = I mls and Ie = I mm. Some average values of the overall heat transfer coefficients are presented in Table 4, together with the end

CFD does not provide values for the reference temperature T,_R' Instead, the radial temperature profiles are continuous and end exactly at the wall temperature Tw' By applying equation (6), effective heat transfer parameters can nevertheless be calculated based on a set of radial profiles at various axial coordinates z. There is no meaning, however, in repeating this procedure due to the shortcomings of the model discussed before. Depending on the choice of the last data point closest to

C. von Scala et al.lChemical Engineering Science 54 (1999) 1375-1381 the wall, almost any combination of effective heat transfer parameters could be obtained. A method to overcome this problem could be the tworegion model proposed by Gunn et al. (1987) and successfully applied for fixed beds with large particles. The use of such a method for KATAPAK-M will be the subject of future work. CONCLUSION The heat transfer from a heated wall to a cold fluid in a tube filled with KATAPAK-M elements was studied. Experimentally determined overall heat transfer coefficients and numerical results from CFD simulations agreed well. The typical flat radial temperature profiles already known from measurements could also be supported by CFD. It has been shown that CFD is now able to analyze complicated structures like KATAPAK-M. Given the large amount of data it provides also a sound basis for a deeper insight into mechanisms governing heat transfer in such structures. The results presented are a reliable starting point for future research including search for an adequate defmition of the local wall heat transfer coefficient, which can be used in a modified pseudohomogeneous plug flow model. In the next step CFD simulations combined with wall heat transfer will be coupled with exo- or endothermic model reactions. Based on these results, the applicability of the pseudohomogeneous model for KATAPAKM and chemical reactions can be verified. A major point of interest will certainly be to quantify the positive effect of the flat radial temperature profile on the selectivity of exothermic chemical reactions.

Tw: tube wall temperature [0C] v: Vo: z:

fluid velocity [mls] superficial velocity [mls] axial coordinate [m]

Greek letters

atot : overall heat transfer coefficient [W/m2K] aw: local wall heat transfer coefficient [W/m2K] A,: effective radial thermal conductivity [W/mK] p: 't:

fluid density [kg/m3] residence time [s]

REFERENCES Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960, Transport Phenomena, John Wiley and Sons, Inc. Coberly, C.A., Marshall, W.R. Jr, 1951, Temperature Gradients in Gas Streams Flowing through Fixed Granular Beds. Chem. Eng. Prog. 47, 141. Dixon, A.G, 1988, Wall and Particle-Shape Effects on Heat Transfer in Packed Beds. Chem. Eng. Commun. 71,217. Eigenberger, G., Kottke, V., Daszkowski, T., Gaiser, G. and Kern, H.-J., 1991, Regelmiissige Katalysatorformkorper fUr technische Synthesen. Fortschrittberichte VDI, Reihe IS: Umwelttechnik, Nr. 112, VDI Verlag, Dusseldorf. Gaiser, G., 1990, Stromungs- und Transportvorgiinge in gewellten Strukturen. Dissertation, Stuttgart University. Gunn, DJ., Ahmad, M.M., and Sabri, M.N., 1987, Radial Heat Transfer to Fixed Beds of Particles. Chemical Engineering Science 42,2163-2171.

NOTATION heat capacity [J/kgK] D: tube diameter [m] dK : KATAPAK-M element diameter [m] L: tube length [m] Luit : exit section length [m] L inl.,: inlet section length [m] LK : KATAPAK-M element length [m] Ie: cleavage between elements and tube wall [m] I,: distance between elements [m] q: total radial heat flux [W/m2] q,: radial heat flux [W/m2] qw: local wall heat transfer rate [W/m2] R: tube radius [m] r: radial coordinate [m] T: fluid temperature [0C] f: mean temperature over the cross sectional area

Tsotsas, E., Schliinder, E.-U., 1990, Heat Transfer in Packed Beds with Fluid Flow: Remarks on the Meaning and the Calculation of a Heat Transfer Coefficient at the Wall. Chemical Engineering Science 45,819-837.

[0C] To: initial temperature [0C] T,_R: temperature of the fluid near the tube wall [0C]

VDI Wiirmeatias, 1991, Chapter Da, 6. edition, VDI Verlag, Dusseldorf.

cp :

Logtenberg, S.A., Dixon, A.G., 1998, Computational Fluid Dynamics Studies of the Effects of TemperatureDependent Physical Properties on Fixed-Bed Heat Transfer. Ind. Eng. Chem. Res. 37,739-747. Stringaro, I.-P., Collins, P., Bailer, 0., 1998, Open Cross-Flow-Channnel Catalysts and Catalysts Supports. In Structured Catalysts and Reactors, Cybulski, A., Moulijn, J.A. (eds.), Marcel Dekker, Inc., New York, 393-416.

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