Theoretical and experimental analysis of bed-to-wall heat transfer in heat recovery processing

Powder Technology 249 (2013) 186–195

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Powder Technology journal homepage: www.elsevier.com/locate/powtec

Theoretical and experimental analysis of bed-to-wall heat transfer in heat recovery processing Ruiqing Zhang, Hairui Yang, junfu Lu, Yuxin Wu ⁎ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 1 April 2013 Received in revised form 25 June 2013 Accepted 10 August 2013 Available online 20 August 2013 Keywords: Heat transfer model Packed bed Heat recovery Experimental test

a b s t r a c t The accurate prediction of heat transfer behavior in packed bed is of great importance to the design and operation of this kind of facility. In this paper, heat transfer between packed bed and adjacent wall in heat recovery processing is investigated theoretically and experimentally. A bed-to-wall heat transfer model is presented with consideration of two essential components including particle-to-wall contact heat transfer and heat conduction through the thermal penetration layer. Details of model parameter determination are provided. Through comprehensive sensitivity analysis of those parameters, it is shown that particle temperature, particle residence time and particle diameter are critical parameters to heat transfer coefficient. Besides, surface roughness, solid thermal conductivity and environmental gas species are all influencing factors of the overall heat transfer coefficient. In particular, a dimensionless particle residence time is introduced to characterize the combining effect of residence time, effective thermal conductivity and wall contact heat transfer. Meanwhile, a revised simplified formula for calculating particle contact heat transfer coefficient is correlated as well, which provides better prediction than the one widely adopted in previous studies. Furthermore, experimental heat transfer test on an industrial packed bed char cooler is performed for validating the model predicting capability in heat recovery process. The systematical test on a bench scale packed bed heat exchanger has also been carried out. Parametric analysis of the experimental data is consistent with the conclusions drawn from the model analysis section. The model provides good predicting accuracy on the overall heat transfer coefficient from the bench scale facility as well as data from the field test by correlating 95% of the measured data within ± 20%. The implementation of the proposed model shows good prediction of heat transfer coefficient in packed bed for heat recovery purpose. It is recommended that the model can be applied in industrial design. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Packed bed reactor is widely used for the purposes of cooling, heating and drying of solid materials. The accurate prediction of heat transfer behaviors in a packed bed is of great importance to the design and operation of this kind of facility. Heat transfer modes in packed bed basically include: (1) the convective heat transfer between the wall and the fluid; (2) the convective heat transfer between the fluid and the particles; (3) the conductive heat transfer between the wall and the particles; (4) the conductive heat transfer between the individual particles; (5) radiant heat transfer; and (6) heat transfer by mixing of the fluid [1]. In specific applications, it is found that these modes may take place simultaneously but their contribution tends to be different, which indicates that some play leading roles in the heat transfer mechanism while the rest can be ignored in order to simplify the analysis. When the gas flow rate is relatively small, (3)–(5) should be mainly

⁎ Corresponding author. Tel./fax: +86 10 62781743. E-mail address: [email protected] (Y. Wu). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.08.017

considered. In the case of drying of the solid material, (2) and (4) should be taken into account; (1)–(4) are the dominant factors for the case of heat transfer from hot wall-to-cold bed while only (3) and (4) are active for the converse situation when heat is transported from the hot bed to the cold wall. The last case described above is broadly found in heat recovery engineering. In the coal chemical industry, a packed bed heat exchanger is applied for the sake of collecting heat from the hot semi-char. The packed bed type ash cooler and external heat exchanger is also developed for heat recovery purpose in utility boiler facilities. The heat transfer between the hot slag and the wall of a rotary ash cooler can be analyzed using a similar approach as well [2]. Generally, compared to the Newtonian fluids, flowing and heat transfer characteristics of non-continuum granular flow, such as packed bed, are substantially different, which means the continuum theory is inappropriate for predicting heat transfer rates in these systems [3]. In the literature, there has been massive research focusing on packed bed related heat transfer problems. Wakao and Kaguei [4] provide a comprehensive review of the evaluation of the particle-to-fluid heat transfer coefficient. Ranz and Marshall [5] correlated Nusselt number

R. Zhang et al. / Powder Technology 249 (2013) 186–195

of particle-to-fluid heat transfer with other dimensionless numbers for packed bed consisting of both single and multiple particles. An attempt at investigations at relatively high temperature was made by Chen and Churchill [6] to compare the contribution of radiation and conduction to heat transfer in a packed bed. Based on the assumption that on a macroscopic scale the bed can be described by a continuum, one can define the effective thermal conductivity [7] of the packing to further describe the heat transfer characteristics. Effective thermal conductivity is a continuum property that depends on temperature, bed material and structure. Many theoretical and experimental studies have been carried out on estimation of the effective conductivity, which broadly consider two approaches: assuming unidirectional one-dimensional heat flow or two-dimensional heat flow inside the packing. Depending on the temperature difference between packed particles and flowing fluids, these models can be further classified into on-phase homogenous model, two-phase heterogeneous model [8] and pseudo-homogenous model which is recently very popular [9]. Based on experimental and modeling studies, different forms of correlations on bed-to-wall heat transfer coefficient have been proposed [10,11]. In these studies, fluid mostly serves as an important heat transfer media while it is unacceptable to ignore the contribution of the fluid, which is why these correlations cannot be extrapolated to the case when the gas contribution is relatively smaller and can be neglected compared to the solid contribution. In a typical heat recovery process where the hot solid materials are cooled by the wall, there is nearly no macroscopic gas, making the existing correlations not applicable. Therefore, specified heat transfer model for this kind of heat transfer process is needed. In this paper, the heat transfer mechanism from hot packing to cold wall is analyzed in detail. Based on the integration of related heat transfer component, a onedimensional heat transfer model for packed bed in heat recovery process is presented. Sensitivity analysis of key parameters is also carried out to further investigate the prediction capability. Moreover, a series

187

of experiment on packed bed heat transfer is performed and the result is used to validate the model. 2. Bed-to-wall heat transfer model Fig. 1 shows the heat transfer schematic between the packed bed consisting of hot particles and the covered wall. Heat is directly transferred from the bed surface to the wall through conduction, leading to temperature decrease of the particles adjacent to the wall. This layer of particles then absorbs heat from the particle layer next to it and this process goes as far as it reaches the core area. As the heat is transported from the bulk bed to the wall, a thermal penetration layer located near the wall is formed. The mechanism of heat transfer inside the thermal penetration layer is different from the boundary layer in Newtonian flow. In a packed bed, heat is conducted between adjacent particles through contact points while gas serves as gap filling, whose thermal conductive properties is considerably different from solid. So the continuum theory of thermal boundary layer formed by continuum fluid is not applicable for predicting the heat transfer characteristics inside the thermal penetration layer. In previous studies, two parameters are chosen to describe the features of the heat transfer in a packed bed. One is the wall heat transfer coefficient, which is influenced by the contact resistance between the covered wall and bed surface, the other is the effective thermal conductivity of the packing which is substantially different from that of the solid material or the gas as it is related to many factors. In some industrial packed bed facilities, the particle is not stagnated but descending along the wall surface mainly due to gravity. Then a third parameter, residence time, needs to be introduced to capture the influence of the contact time on the heat transfer coefficient. In the following sections, three aspects that control the overall bed-to-wall heat transfer coefficient will be investigated and the determination of related parameters will be provided as well. 2.1. Contact heat transfer between the wall surface and the bed Considering the contact mode of a single particle with the wall, if the particle is perfectly shaped, absolutely smooth sphere, the contact is then point-to-point. Actually, the particles in an industrial packed bed are always arbitrary-shaped and the roughness of both the particles and the wall is not neglectable. Then the heat is transferred through a gas layer existing between the particle and the wall instead of the single contact point. Taking into account the discontinuity effect at the interfaces, one may get the local wall-to-particle heat transfer coefficient due to heat conduction [12]: hwp;loc ¼

λg δloc þ σ g þ r

ð1Þ

where, λg is the gas thermal conductivity, δloc is the local gap width between the surfaces of wall and sphere, σg is the modified free path of the gas molecules, r is the sum of roughness of both surfaces. Integration of Eq. (1) gives the average particle-to-wall heat transfer coefficient [13]:

hwp


1 2 82 9 3 = 2 σg þ r 4λg < dp 4 A 4 5  −1 ln 1 þ
¼ 1þ ; dp : dp 2 σ þr

ð2Þ

g

where, dp is the average particle diameter. The modified free path of the gas molecules is expressed as [12]:  Fig. 1. Heat transfer schematic between the packed bed and the covered wall.

σ g ¼ 2Λ 0

   T 101325 2 −1 273:15 p γ

ð3Þ

188

R. Zhang et al. / Powder Technology 249 (2013) 186–195

where, Λ0 is free path of the gas molecules at T0 = 273.15 K and p = 101,325 Pa, T is temperature, p is pressure, γ is gas accommodation coefficient. Λ0 is a function of molecule diameter η: Λ0 ¼

8:4  10−27 η2

ð4Þ

the average gas-to-wall heat transfer coefficient is written as [14]: λg hwg ¼ pffiffiffi 2 d þ σg 2 p

ð5Þ

The average radiative heat transfer coefficient between the packed bed and the wall is:

hr ¼


 σ T 2s þ T 2w ðT S þ T W Þ 1 1 þ −1 εw εso

ð6Þ

through the centers of two contiguous spheres. Zehner [17] then expanded this model to include the influence of particle shape and porosity. These models are specifically developed for determining thermal conductivity of the packing without gas flow or with relatively stagnant gas flow, which makes the model valid for the prediction of the effective thermal conductivity of the thermal penetration layer. The following correlations from the above model are employed: ( ) pffiffiffiffiffiffiffiffiffiffiffi λso
ψ λr ¼ 1− 1−ψ þψ ðψ−1Þ þ λg =λD λg λg ( 0 ) pffiffiffiffiffiffiffiffiffiffiffi λ λ þ 1−ψ φK s þ ð1−φK Þ so λg λg

where, λso is the effective thermal conductivity of the packing, λD is the equivalent thermal conductivity due to molecular flow, λr is the equivalent thermal conductivity due to radiation. φK is the relative contact surface area between contiguous particles. λso′ is expressed as: 0

where, σ is Stefan–Boltzmann constant, Ts and Tw is the temperature of the particle and the wall, εso and εw are the emissivity of packed bed and the wall. Then the average bed-to-wall heat transfer coefficient is achieved:

λso 2 ¼ K λg

(

 
B λs =λg þ λr =λg −1 λg =λD λg =λs

where,

where, φA is the relative wall coverage by particles, which is correlated to bed void fraction ψ:

λg K¼ λD

2

φA ¼ ð1−ψÞ3

ð8Þ

K2

( 1þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λso ρso C p;so hsc ¼ 2 πt c

ð9Þ

where, λso is the effective thermal conductivity of the packing, ρso is the density of the packing, Cp,so is the effective thermal capacity of the packing, tc is the residence time of the particle along the wall. It should be noted that the properties employed in the above equation is not equivalent to the solid properties. The density and thermal capacity can be determined by:

 C p;so ρso ¼ ψ  C p;p ρp þ ð1−ψÞ  C p;g ρg

ð10Þ

where, ρp and Cp,p are the density and thermal capacity of the solid materials. In this section, the effective thermal conductivity refers to that of the thermal penetration layer where the gas flow is considered as stagnant media. As stated above, the effective thermal conductivity of the packing is the main concern of many previous investigations on heat transfer in packed bed. As early as 1933, Hengst [16] introduced a method for calculating the effective thermal conductivity of spherical particle packing based on the assumption that direction of the heat flow passes

λr λ −B D λg λg

!

) !   λg λg λr λg −B −1 1þ λs λD λg λs

ð12Þ

ð13Þ

where, B is the deformation factor of the particle which essentially determines the intensity with which the particles touch each other, or the amount of heat transfer due to conduction [18]:

2.2. Heat conduction inside the thermal penetration layer As shown in Fig. 1, temperature decreases gradually from Tb,∞ in bulk area to Tb,0 near the wall surface. Similar to the boundary layer of Newtonian flow, there is a thermal penetration layer near the wall surface, whose thickness is then defined as the distance from the wall at which the temperature of particle reaches 99% of the bulk temperature. The heat transfer inside the thermal penetration layer is achieved by contact conduction between adjacent particles. The overall heat transfer coefficient through the thermal penetration layer can be described as a one-dimensional transient conduction problem. The analytical result for this problem is [15]:



 λs =λg þ λr =λg λg =λD h ih i  ln
B 1 þ λg =λD −1 λs =λg þ λr =λg

" )   #! λg B þ 1 λr λg λr B−1 λg − þ −B 1 þ −1 2B K λD λg λD λD λg

ð7Þ

hws ¼ φA hwp þ ð1−φA Þhwg þ hrad

ð11Þ

B ¼ C Form

  1−ψ 10=9 ψ

ð14Þ

where, CForm is an experimental-determined particle shape factor, for spheres, CForm = 1.25, and for particles with arbitrary shapes, CForm = 1.4[14]. In contrast to heat conductivity, the amount of radiative heat transfer is a function of the packing temperature as well as the radiation characteristics and the geometry of the packing. For small temperature gradients, the radiative heat transfer inside the packing can be described by equivalent thermal conductivity λr [19]: λr 4σ 3 Rform dp T ¼ λg 2=ε so ‐1 s λg

ð15Þ

where, RForm is experimental-determined shape factor for the interstitial energy transport by radiation for spheres, RForm = 1. In a region around the particle contact points in a packing, the normal movement of the gas molecules is so inhabited that an additional wall heat transfer resistance appears. The heat flow through the packing therefore must overcome not only the resistance of the solid and gas phases but also the transfer resistances at their boundary surfaces. Considering the additional resistance is directly proportional to the gas mean free path length while the distance between the contiguous particle surfaces is always on the magnitude of the mean free path length, the statement by Chapman and Cowling [20] is applied to describe the equivalent thermal conductivity λD:      λg 2 2 T 101325 −1 Λ 0 ¼1þ DForm dp γ 273:15 p λD

ð16Þ

where, DForm is experimental-determined shape factor for the interstitial energy transport by molecular flow. For spheres, DForm = 1.

R. Zhang et al. / Powder Technology 249 (2013) 186–195

1000

In the core area of the packed bed, the solid is mixed so well that the temperature field is nearly uniform. Then the heat resistance due to solid mixing can be neglected, 1/hsx ≈ 0. 2.4. Overall bed-to-wall heat transfer coefficient Based on the above analysis of related heat transfer component, the overall bed-to-wall heat transfer coefficient for packed bed in heat recovery process is in the following form:

¼

1 1 1 1 þ þ hws hsc hsx

1

1 1 þ φA hwp þ ð1−φA Þhwg þ hr 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πt c λso ρso C p;so

Heat Transfer Coefficient,W/m2K

2.3. Heat resistance due to solid mixing in core area

hoverall ¼

189

ð17Þ

100

10

1 400

h-overall h-ws h-sc h-wp h-wg h-rad 500

600

700

800

900

Particle Temperature,K Fig. 2. Relationship between heat transfer coefficient and temperature.

3. Sensitivity analysis

3.1. Particle temperature Fig. 2 shows that the overall heat transfer coefficient as well as each heat transfer components increase with the increasing particle temperature. As the radiation intensity is proportional to T4 according to Stefan–Boltzmann law, the radiative heat transfer coefficient is very sensitive to the increasing temperature. The thermal conductivity of both solid material and gas is increasing function of the temperature, making heat resistance in each component smaller at higher temperature. It can be seen from Fig. 1 that the restrictive factor of the overall heat transfer is the conduction resistance in the thermal penetration layer, 1/hsc, which is about three times the contact resistance between the bed surface and the wall, 1/hws. The contact heat transfer coefficient, hws, is of the magnitude of the particle-to-wall heat transfer coefficient, hwp, which indicates that the heat conduction through the gas layer between surfaces of the particle and the wall is the controlling mechanism of the heat transfer from the packed bed to the wall. 3.2. Particle residence time While the particles descend along the wall, the overall heat transfer coefficient is closely related to the particle–wall contacting time, or say particle residence time. As described above, heat is directly transferred from the bed surface to the wall through conduction, making a temperature decrease of the adjacent particles. This first layer of particles then absorbs heat from the particles next to it and this process goes as far as it reaches the core area. In this way, if the particles descend very fast and Table 1 Baseline parameters for sensitivity analysis. Parameters

Value

Parameters

Value

Tw Tso p γ λg λs r

400 K 800 K 101325 Pa 0.9 0.058 W/(m2-K) 1.0 W/(m2-K) 0.05 μm

ψ dp t ρp Cp,p εw εso

0.4 1 mm 20 s 2700 kg/m3 780 J/(kg-K) 0.4 0.6

the residence time is very short, the particles cooled by the wall will be renewed by fresh hot particles so quickly that at a specific location, a larger temperature difference between the bed and the wall could be maintained. This will limit the thickness of the thermal penetration layer, which means the conduction resistance through this layer will be diminished, making the overall heat transfer coefficient increase consequently. When the residence time becomes longer, the first particle layer will be cooled further as a result of relatively weaker replacement by fresh particles, compared to the short contact time case. Then a smaller temperature difference between the wall and adjacent particles as well as a fully developed and thicker thermal penetration layer will decrease the overall heat transfer coefficient. Figs. 3 and 4 show the influence of residence time on overall heat transfer coefficient under different temperatures and environmental pressures, respectively. The model predicts that the overall heat transfer coefficient decreases with the increasing residence time, which is consistent with the above analysis. Apparently, this relationship is only valid for a moderate range of residence time. When the contact time varies, there are actually upper and lower limits for the overall heat transfer coefficient [12], which are presented clearly in Fig. 5. When the residence time is extremely short, a maximum value is achieved based on Eq. (2) which is

1000

Overall Heat Transfer Coefficient, W/m2K

A one-dimensional heat transfer model has been presented above. In this section, this model will be employed to investigate the effect of various parameters on the heat transfer process. Table 1 shows the baseline parameters of the model.

ParticleTemperature

100

400K

500K

600K

700K

800K

900K

10

1 5

50

500

5000

Residence Time,s Fig. 3. Influence of residence time on overall heat transfer coefficient (under different temperature).

190

R. Zhang et al. / Powder Technology 249 (2013) 186–195

The environmental pressure also shows an impact on the heat transfer by influencing the free path of the gas molecule directly. It can be observed from Fig. 4 that when the residence time is constant, the overall heat transfer coefficient decreased with the decrease of the pressure. When the pressure gradually drops from atmospheric pressure to the micro-pressure, the overall heat transfer becomes less sensitive to the change of the residence time. This phenomenon is consistent with the experimental results published in the literature [21].

101325Pa 10132.5Pa 1013.25Pa 101.325Pa 10.1325Pa 1.01325Pa 0.101325Pa

100

3.3. Particle diameter 10

1 5

50

500

5000

Residence Time,s Fig. 4. Influence of residence time on overall heat transfer coefficient (under different pressure).

independent of residence time (zone A). For the moderate time range in zone B, the overall heat transfer coefficient follows Eq. (17). For a long residence time in zone C, the minimum overall heat transfer coefficient is solely determined from the geometrical parameters and thermal properties of the packed bed system. As discussed above, in previous studies, two parameters including the heat transfer coefficient at the wall hws and the effective thermal conductivity of the packing λso are used to represent the heat transfer characteristic. An additional parameter, residence time is introduced in this paper to fully describe the packed bed heat transfer. Moreover, a universal form is proposed by reforming Eq. (17): 1 hoverall

¼

pffiffiffiffiffiffi pffiffiffi 1 1 2 þ πτ π pffiffiffi 1 ¼ f ðhws ; τ Þ τ þ ¼ hws hws hws 2 2

ð18Þ

where τ is a dimensionless residence time, which is defined as: τ¼

hws 2 t c λso ρso C p;so

ð19Þ

The dimensionless residence time integrates the comprehensive impact of hws, tc and λso on the overall heat transfer. When all the properties are fixed in a specific case, the overall heat transfer is the sole function of hws and τ.

h

(a)

(b)

(c)

hmax,, equ.(2)

p=const

hmin

log t Fig. 5. Dependence of heat transfer coefficient on residence time.

There have been plenty of academic studies showing that in a packed bed, the heat transfer coefficient will decrease as the particle diameter increases [3,22]. Fig. 6 shows the predicted relationship between heat transfer coefficient and particle diameter under different particle temperatures. At a certain temperature, the overall heat transfer coefficient decreases with the increase of the particle diameter, which is the same conclusion from previous experimental results. As for the two heat transfer components, with the increasing particle diameter, hws decreases while hsc increases. Usually, the finer the particle, the greater is the proportion of particle contact area per unit wall area and the smaller is the thermal resistance of contact for each particle. Meanwhile, when the particles are packed in the same way, finer particles will bring larger voidage, which makes a smaller effective thermal conductivity of the gas–solid mixture [14]. A combined effect of both components makes the relation between the overall heat transfer coefficient and the particle diameter quite flat, as shown in Fig. 6. In some related model development work on packed bed and even rotary ash cooler, the similar heat transfer model of hws is proposed [15,23]. Although in these literatures, the heat transfer mechanism is analyzed with the similar approach presented in this paper, a simplified relation is always adopted instead for the implementation of the original model: hws ¼

1 λg β dp

ð20Þ

where, β is an experimental-determined parameter. The above relation is derived based on dimensionless analysis of experimental data in the literatures [3,14], therefore, the scope of application can be very limited. Prediction results by Eq. (21) with different values of parameter β and the model recommended in this paper are compared in Fig. 7. The relationship of hws with dp represented by the simplified expression is 1000

Heat Transfer Coefficient,W/m2K

Overall Heat Transfer Coefficient, W/m2K

1000

100

10

Particle Temperature

1 5E-05

h-400K

h-600K

h-800K

hws-400K

hws-600K

hws-800K

hsc-400K

hsc-600K

hsc-800K

0.0005

0.005

0.05

Particle Diameter,m Fig. 6. Relationship between heat transfer coefficient and particle diameter (different particle temperatures).

R. Zhang et al. / Powder Technology 249 (2013) 186–195

10000

hws-dp

equation(7) β=0.085

1 λg β dp

10000

Heat Transfer Coefficient,W/m2K

Contact Heat Transfer Coefficient,W/m2K

100000

191

β=0.096 β=0.14

1000

β=0.198

1 λg χdm p

100

equation(21)

Tp=800K

10

1000

100

10

Particle temperature

2 g=0.043W/m -K

1 5E-05

0.0005

0.005

1 5.00E-07

0.05

Particle Diameter,m

h-400K

h-500K

h-600K

h-700K

h-800K

h-900K

5.00E-06

5.00E-05

5.00E-04

Surface Roughness,m

Fig. 7. Comparison of the simplified correlation with the original model.

Fig. 8. Relationship between heat transfer coefficient and surface roughness.

deviated from the one that is calculated by Eq. (7). By simply adjusting the value of β, the deviation cannot be corrected, which indicates that the Eq. (21) is not the fittest correlation of Eq. (7), or say, hws is not proportional to 1/dp. Considering the results are plotted in logarithmic coordinate system in Fig. 7, the alternative simplified formula should be in the form of:

Solid thermal conductivity is also a major parameter determining the heat transfer coefficient. It is mainly the function of temperature for a specific material. Fig. 9 presents the relation between the overall heat transfer coefficient and the solid thermal conductivity at different temperatures. The conclusion has been drawn above that he contact heat transfer is affected by the gas layer properties while the conduction through the penetration layer is influenced by the comprehensive effect of many factors including gas and solid characteristics. So the overall heat transfer coefficient increases relatively slightly when the solid thermal conductivity increases exponentially. For instance, when the particle is 800 K, 100 times increase on solid thermal conductivity only brings a raise of 30% on the overall heat transfer coefficient. When the gas species changes, the gas properties that have impact on heat transfer, including thermal conductivity λg, gas molecule diameter η, accommodation coefficient γ and free path of gas molecule σg, will change consequently. The curves of the overall heat transfer coefficient changing with particle temperature under different atmospheres are shown in Fig. 10. The accommodation coefficient γ and molecule diameter η of different gas species are also provided. In the packed bed heat transfer, the gas thermal conductivity is so much lower than that of the solid that it restricts the heat transfer. If a kind of gas with higher thermal conductivity serves as the gaseous medium, the heat transfer

1 λg χ dp m

ð21Þ

By correlating the curve by Eq. (7) in Fig. 7, the parameters in Eq. (21) are obtained: −4

m ¼ −1:2  10 −5

χ ¼ −10

T þ 0:6362

T þ 0:0168

ð22Þ ð23Þ

The prediction by the new proposed correlation is also presented in Fig. 7. Obviously, the new formula fits the original model better and can be used as a substitute if necessary, but direct applying the simplified Eq. (20) may bring potential error. Since the detail of determination of model parameter has been provided in this paper, for the sake of better prediction, it is recommended that one should considering adopting the original model first. 3.4. Other parameters In addition to the parameters investigated above, there are some other parameters whose influences on the heat transfer need to be further examined, such as the surface roughness, solid thermal conductivity and gas species. When the particle contacts with the wall surface, as a result of the geometrical irregularity and the presence of surface roughness, a gas layer exists between the particle and the wall, from which the contact resistance primarily comes. So the surface roughness plays an important role in the contact heat transfer coefficient. The dependence of the overall and wall contact heat transfer coefficient on the surface roughness is shown in Fig. 8. Particularly, the surface roughness in this model integrates the surface roughness of both the particle and the wall. The contact heat transfer coefficient decreases rapidly with the increase of the roughness while the overall heat transfer experiences moderate decrease, which can be attributed to the fact that the restrictive factor of the total heat transfer is mainly the thermal conduction through the penetration layer.

120

Overall Heat Transfer Coefficient,W/m2K

hws ¼

100

80

60

40

Particle Temperature 20

400K

500K

600K

700K

800K

900K

0 0.2

2

20

200

Solid Thermal Conductivity,W/m-K Fig. 9. Relationship between heat transfer coefficient and solid thermal conductivity (different temperature).

192

R. Zhang et al. / Powder Technology 249 (2013) 186–195

Overall Heat Transfer Coefficient,W/m2K

250 M/g/mol

γ

η /nm

Air

29

0.9

0.35

O2

32

0.93

0.346

N2

28

0.88

0.364

CO2

44

0.98

0.33

H2

2

0.15

0.289

He

4

0.4

0.26

1 H2

200

150

He

3

2 100

O2

Air

N2

CO2

50

p=101325Pa 0 400

500

600

Air

O2

N2

CO2

H2

He

700

800

4 900

Particle Temperature,K Fig. 10. Overall heat transfer coefficient under different atmospheres (different particle temperature).

5

5

will be strengthened. The heat transfer coefficient appears significantly larger in the atmosphere of hydrogen and then helium as the thermal conductivity of hydrogen and helium is 10 times that of the rest species. The heat transfer coefficient is similar under oxygen, nitrogen and air atmosphere as the properties of these gasses are very similar. In addition, the carbon dioxide atmosphere brings slight larger heat transfer coefficient than the three gas species above.

This model is presented for the purpose of predicting heat transfer of packed bed in heat recovery process. This kind of heat transfer problem always involves hot particles and cold wall. In previous studies, most of the experimental data on bed-to-wall heat transfer was obtained under different heat transfer conditions, usually the case of hot surface heating cold particles [3,22]. This is the actually the reversed heat transfer problem to what is concerned in this paper. As there is nearly no suitable experimental data available for validating this model in published literatures, verification is achieved by experimental measurement on both an industrial packed bed type heat recovery facility and a laboratory test rig. The schematic of the industrial facility is shown in Fig. 11. The test section has a square cross section with diameter of 1.5 m and height of 3.2 m. The particle is hot char from coal pyrolysis with diameter range of 0.5–5.4 mm. The residence time is determined by the particle descending speed, which is maintained about 0.03 m/s. This speed is controlled by adjusting the discharge appliance at the bottom of the facility. The particle temperature varies from 300–800 K. The tube is cooled by the forced flowing water. The flow rate of water is measured by a rotameter. Thermocouples are mounted into the packed bed to measure the in/out temperature of the particles, Tp,in/Tp,out, while the in/out temperature of the fluid, Tf,in/Tf,out, is measured by thermal resistance installed in the tube. Then the heat transfer coefficient is calculated by the following equation:


h¼

7 Fig. 11. Schematic of the packed bed heat transfer test rig (1. hot particles, 2. insulation, 3. thermocouples, 4. cooling water, 5. thermal resistance, 6. discharge valve, and 7. cooled particles).

Fig. 12, the model provides good predicting accuracy on the overall heat transfer coefficient by correlating the measured data within ± 20%. Specifically, the predicted data is calculated based on the experimental conditions and operating parameters. Due to the difficulties in performing systematical field test, detailed information is always incomplete while fixing one specific parameter is always infeasible making experimental parametric analysis impossible.

80

60

+20% 40

-20%

20



Q m C p; f T f;out −T f;in
 Atot  T p;ave −T f;ave

6

Prediction,W/m2K

4. Experimental verification

3

ð24Þ 0 0

where, Qm is the mass flow rate of the fluid, Cp,f is the heat capacity of the fluid, A is the total heat transfer area, Tp,ave and Tf,ave is the average temperature of the particle and the fluid, respectively. As presented in

20

40

60

Measurement,W/m2K Fig. 12. Model prediction accuracy (predicted vs. measured).

80

R. Zhang et al. / Powder Technology 249 (2013) 186–195

Up ¼

Qm Ac ψρp

(a) dp=200μm 300

Heat Transfer Coefficient,W/m2K

Then a series of experimental test has been performed on a bench scale packed bed heat exchanger for validating the model and studying experimental parametric response as supplement to the former field test. The test rig has a similar configuration to the one shown in Fig. 11. The cross section is 0.1 m ∗ 0.04 m, the water-cooled tube has an inner diameter of 0.01 m. Quartz sand with different sizes is preheated to a certain range and then poured into the test tube continuously. There are three K-type thermocouples mounted 0.01 m above and below the central plane of the tube cross section, respectively. Pt100 RTDs are mounted upstream and downstream the water-cooled tube. All the temperature signals are monitored and recorded by an online data acquisition system (YOKOGAWA -MV2020). The solid mass flow rate is controlled by a rotating valve while an electronic scale is arranged below to capture the mass flow rate. Then the solid descending velocity in the test tube is obtained from:

193

ð24Þ

Particle Temperature(C) 250

226 380

200

461 553

150

100

50

dp=200µm

0 0

5

Then the particle residence time is determined: ð25Þ

where Lp is the characteristic length of the descending particles. Considering the effect of the surface curvature, this characteristic length is set to be the tube outer diameter in this paper. As the thermal resistance through the thin tube wall is neglectable and the fluid-side heat transfer coefficient is very high, a proper residence time is determined based on test rig configuration (Eq. (25)). The measured heat transfer coefficient from Eq. (24) is considered to be the bed-wall heat transfer coefficient, which is comparable with the prediction from Eq. (17). Fig. 13 shows the effect of particle temperature on the relationship between heat transfer coefficient and residence time. It is clear that the overall heat transfer coefficient decreases with the increasing residence time while elevated temperature increases the overall heat transfer coefficient when the residence time is constant. Fig. 14 shows the effect of particle diameter on the relationship between heat transfer coefficient and residence time. Under the condition of different temperatures, it is shown that the increasing particle diameter always decreases the overall heat transfer coefficient when the residence time is constant. All these conclusions are consistent with the remarks in Section 3 where model sensitivity analysis is implemented. In other words, the influence of the relevant parameters on the overall heat transfer coefficient has been thoughtfully analyzed before. Fig. 15 reveals that the model provides good predicting accuracy on the overall heat transfer coefficient from the bench scale facility by correlating 95% of the measured data within ±20%. In conclusion, for a heat recovery process, the model presented in this paper is reliable and can be used on industrial design. 5. Conclusions A bed-to-wall heat transfer model for packed bed in heat recovery process is proposed based on theoretical analysis and integration of two essential heat transfer components including particle-to-wall contact heat transfer and heat conduction through the thermal penetration layer: hoverall

15

20

25

1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ 1 1 1 1 1 πt c þ þ þ hws hsc hsx φ h þ ð1−φ Þh þ h 2 λso ρso C p;so r A wp A wg

The detail of determination of model parameters has been provided. A comprehensive sensitivity analysis is performed to investigate the effect of various parameters on the heat transfer process. Increase of

(b) dp=330μm 300

Heat Transfer Coefficient,W/m2K

Lp d ¼ t Up Up

Particle Temperature(C) 226

250

380

200

461 150

553 100

50

dp=330µm

0 0

5

10

15

20

25

Residence Time, s

(c) dp=500μm 250

Heat Transfer Coefficient,W/m2K

t¼

10

Residence Time, s

Particle Temperature(C) 200

226 380

150

461 553 100

50

dp=500µm 0 0

5

10

15

20

25

Residence Time, s Fig. 13. Effect of particle temperature on the relationship between heat transfer coefficient and residence time.

194

R. Zhang et al. / Powder Technology 249 (2013) 186–195

(a) Tp=220oC

(b) Tp=380oC 300

Particle Diameter(µm)

Heat Transfer Coefficient,W/m2K

Heat Transfer Coefficient,W/m2K

300

250

200 330

200

500 150

100

50

Tp=220 oC

0

Particle Diameter(µm) 250

200 330

200

500 150

100

50

Tp=380 oC

0 0

5

10

15

20

25

0

5

Residence Time, s

10

(c) Tp=460oC

20

(d) Tp=550oC

300

300

Particle Diameter(µm)

250

Heat Transfer Coefficient,W/m2K

Particle Diameter(µm)

Heat Transfer Coefficient,W/m2K

15

Residence Time, s

200 330

200

500

150

100

50

Tp=460 oC

0

250

200 330

200

500

150

100

50

Tp=550 oC

0 0

5

10

15

20

Residence Time, s

0

5

10

15

20

Residence Time, s

Fig. 14. Effect of particle diameter on the relationship between heat transfer coefficient and residence time.

300

Particle Diameter Model Prediction,W/m2K

250

200µm

+20%

330µm 200

550µm

150

-20%

100

50

0 0

50

100

150

200

250

300

Measurement,W/m2K Fig. 15. Model prediction accuracy of different particle diameter (Predicted vs. Measured).

particle temperature will raise heat transfer coefficient. The overall heat transfer coefficient decreases with the increase of particle residence time and this is only valid for moderate time range. When the residence time is very short or long, there is actually an upper or lower limit respectively. The concept of a dimensionless residence time is put forward to characterize the combining effect of hws, tc and λso. Particle diameter is also an influencing factor to heat transfer, whose increase will bring a decrease of the overall heat transfer coefficient. A new simplified correlation of contact heat transfer coefficient with particle diameter is suggested, which predicts better than the one that is widely adopted in literature. In addition, surface roughness, solid thermal conductivity and gas species also affect heat transfer. Furthermore, an experimental heat transfer test on an industrial packed bed char cooler is performed for validating the model predicting capability in heat recovery process. The systematical test on a bench scale packed bed heat exchanger has also been carried out. Parametric analysis of the experimental data is consistent with the conclusions drawn from the model analysis section. The model provides good predicting accuracy on the overall heat transfer coefficient from the bench scale facility as well as data from the field test by correlating 95% of the measured data within ±20%. It is recommended that the model can be applied in industrial design.

R. Zhang et al. / Powder Technology 249 (2013) 186–195

List of symbols

A B CForm, Cp DForm d h L m p Qm RForm r T t

Area (m2) deformation factor of the particle particle shape factor heat capacity (J/kg-K) particle shape factor for the interstitial energy transport by molecular flow diameter (m) heat transfer coefficient (W/m2-K) characteristic length (m) parameter in Eq. (21) pressure (Pa) mass flow rate of the fluid (kg/s) particle shape factor for the interstitial energy transport by radiation roughness (m) temperature (K) particle residence time

Greek letters β parameter in Eq. (20) γ gas accommodation coefficient δ gap width between the surfaces (m) ε emissivity η gas molecule diameter (m) Λ0 free path of the gas molecules at 273.15 K, 101325 Pa (m) λ thermal conductivity (W/m-K) ρ density (kg/m3) σ Stefan–Boltzmann constant, (5.67 × 10−8 Wm−2 K−4) σg the modified free path of the gas molecules (m) τ dimensionless residence time φA the relative wall coverage by particles φK the relative contact surface area between contiguous particles χ parameter in Eq. (21) ψ bed void fraction

Subscripts ave averaged b bed c cross-sectional D molecular flow f fluid g gas loc local overall overall heat transfer coefficient p particle r radiation s solid sc conduction through thermal penetration layer so packed bed

sx t tot w wg wp ws

195

solid mixing water-cooled tube total heat transfer surface wall gas to wall particle to wall bed to wall

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