CHEMICAL ENGINEERING TRANSACTIONS

VOL. 81, 2020

A publication of

The Italian Association
of Chemical Engineering
Online at www.cetjournal.it

Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš

ISBN 978-88-95608-79-2; ISSN 2283-9216

Experimental Analysis of Stagnant Effective Thermal

Conductivity in Packed Beds: Bulk Region and Near Wall

Region

Shanshan Bua,*,Jie Wangb, Bo Chena, Zhengzhong Lia

aKey Laboratory of Low-Grade Energy Utilization Technologies & System, Chongqing University, Chongqing 400044, P.R.

China

bReactor Engineering Research Sub-institute，Nuclear Power Insititute of China, Chengdu 610041, China

shanshanbu@cqu.edu.cn

Effective thermal conductivities in four different packed beds are experimentally studied in this paper, namely

two random graphite packed beds with porosity ɛ=0.40 and ɛ=0.35, a simple cubic (SC) graphite packed bed

and a random stainless steel packed bed. It is found that the effective thermal conductivity of all packed beds

has a downward trend as the wall is approached. Based on the effective thermal conductivity distributions, the

near wall region can be defined within four pebble diameters from walls for random packed beds and one pebble

diameter for SC packed beds. As a result of wall effect, the effective thermal conductivities in the near wall

regions are generally smaller than those in the bulk regions at the same temperature. It is found that the ZBS

correlation gives best predictions in bulk regions and in near wall regions with the proper empirical parameter

selected.

1. Introduction

Based on the large surface area to volume ratio, packed beds have excellent performance in heat transfer and

mass transfer and are widely used in nuclear reactors (Bu et al., 2017), adsorption process (Haddadi et al.,

2016), magnetic refrigeration (Shi et al., 2018) and chemical combustion process (Diglio et al., 2017).

Understanding heat transfer in packed beds is essential to the design and operation of industrial packed bed

applications. The heat transfer in a packed bed is complicated and then the heat transfer mechanisms are

usually lumped into an important parameter called the effective thermal conductivity.

The effective thermal conductivity is dominated by the fluid and solid phase thermal conductivities as well as

the microstructure of packed bed. The microstructure of a packed bed is heterogeneous and the local porosity

varies sharply, especially near a wall since there the geometry of the packing is altered (Van Antwerpen et al.,

2010). This is called wall effect, mainly caused by the relatively high void fraction near the wall. So the wall effect

on hydrodynamic, heat and mass transfer performances for a packed bed is significant. The effective thermal

conductivity is also affected by the wall effect (Dixon, 2012). Therefore, the effective thermal conductivity in the

near wall region and bulk region of a packed bed would be different. The effective thermal conductivity in the

bulk region has been widely studied, and some correlations were proposed, such as correlations by Zehner and

Schlünder (ZS) (1970), Bauer and Schlünder (ZBS) (1978), and Hsu et al. (1995). Some of these correlations

were found to be limited for the prediction of the effective thermal conductivity in the near wall region. For

example, Van Antwerpen et al. (2012) found the IAEA ZS correlation and the ZBS correlation to overpredict the

heat transfer in the near wall region and then presented a new model to better predict the effective thermal

conductivity in the near wall region. measured and studied The local effective thermal conductivity distributions

in a packed bed were studied experimentally (De Beer et al., 2018) and numerically (De Beer et al., 2017) and

they found an obvious decrease in the magnitude of the effective thermal conductivity in the near wall region. In

the previous work, the bulk effective thermal conductivities were consistently greater than the near-wall effective

DOI: 10.3303/CET2081078

Paper Received: 23/04/2020; Revised: 05/06/2020; Accepted: 10/06/2020
Please cite this article as: Bu S., Wang J., Chen B., Li Z., 2020, Experimental Analysis of Stagnant Effective Thermal Conductivity in Packed
Beds: Bulk Region and Near Wall Region, Chemical Engineering Transactions, 81, 463-468 DOI:10.3303/CET2081078

463

thermal conductivities at the same temperature (Bu et al., 2020a). The natural convection was also found to

playe a contrary part in the near wall region and the bulk region (Bu et al., 2020b).

Based on the above analysis, the effective thermal conductivity in packed beds, especially in the near wall region

should be discussed more to gain a better prediction of thermal behavior of a packed bed. Therefore, the

effective thermal conductivities in four different packed beds, namely two random graphite packed beds with

porosity ɛ=0.40 and ɛ=0.35, a simple cubic (SC) graphite packed bed and a random stainless steel packed bed,

are experimentally studied in the present paper. The effective thermal conductivity distributions are analyzed in

detail. Meanwhile, the experimental results in bulk regions and near wall regions are separately compared with

three semi-empirical correlations.

2. Experimental system

As shown in Figure 1, the experimental system consists of six parts: a test section, a nitrogen supply, an

evacuation pump, an electricity heater, a cooling loop, and a temperature data acquisition system. In the test

section, pebbles are packed in a graphite box. The heater is located on the left side of the test section to heat

the packed bed. A DC power supply (precision within ±0.01 V) is used to regulate power supply. A water jacket

is placed on the right side of the test section to cool the packed bed. An electromagnetic flow meter (precision

within ±0.5 %) is used to measure the flow rate of cooling water in water jacket. The inlet and outlet temperatures

of cooling water in water jacket are monitored by two thermocouples (precision within ±0.5 K). The fluid filling in

the pores of packed beds is nitrogen. The evacuation pump is used to extract the air out of the bed before

charging nitrogen. The measured data including temperatures and the input power are collected by the NI data

acquisition card.

Figure 1: Schematic representation of the experimental system

Pebbles fit into a rectangular box made of graphite in the test section. The length of the graphite box is 210 mm

and the width and height are both 90 mm. Four different packed beds are built, namely a random graphite

packed bed with porosity ɛ=0.40, a random graphite packed bed with porosity ɛ=0.35, a SC graphite packed

bed and a random 304 stainless steel packed bed (SS packed bed) with porosity ɛ=0.39.

As shown in Figure 2, the hot wall of the test section heated by the heater is on one side and the cold wall cooled

by the flowing water is on the other side. The other sides of the test section are wrapped with the insulated

materials. Accordingly, the heat flow passes through the heater, the hot wall, the pebble bed, the cold wall, and

is finally taken out by the cooling water in the water jacket. Some type K thermocouples (1.5 mm diameter,

precision within ±0.5 K) are inserted into the selected pebbles to measure the temperature distribution along the

heat transfer direction (z-coordinate direction). There are 12 and 7 internal pebbles equipped with

thermocouples for the random packed beds and the SC packed bed (see Figure 2). Thermocouples are fixed

by passing through a series of holes of pebbles to minimize the disruption to the packing structure of the bed.

Another two thermocouples (precision within ±0.5 K) are installed to measure the hot wall and cold wall

temperature. The insulated walls and several pebbles adjacent to insulated pebbles are also equipped with

thermocouples to estimate the heat loss through the insulated walls.

As shown in Table 1, five experimental tests are carried out for each packed bed. Thot is the hot wall temperature

and Tcold is the cold wall temperature. For each test, the input power and temperature are monitored until it

reaches the steady state. The Fourier’s law of one-dimensional heat conduction is used to calculate the effective

thermal conductivity using the following equation:

464

e
( ) ( )

T
Q z Ak z


= −


(1)

So the calculation of the effective thermal conductivity needs the value of the heat transfer area A, the

temperature gradient, ΔT/Δz and the heat transfer rate Q(z). The temperature gradient is obtained by the

derivation of fitting curves of the measured temperatures.The detailed data reduction and the uncertainty

analysis are according to the previous work (Bu et al., 2020a).

Figure 2: Schematic representation of a vertical section of test sections: (a) random graphite packed bed and

(b) SC graphite packed bed

Table 1: Tests for four different packed beds

Number Thot, [K] Tcold, [K] Packed bed Number Thot, [K] Tcold, [K] Packed bed

Case 1 378.0 307.6 random

graphite
dp=15mm

=0.40

Case 1 378.0 301.8 random

graphite
dp=15mm

=0.35

Case 2 483.0 336.7 Case 2 483.0 327.3

Case 3 583.0 370.8 Case 3 583.0 354.1

Case 4 708.0 410.6 Case 4 708.0 412.5

Case 5 803.0 440.2 Case 5 803.0 442.0

Case 1 378.0 291.0 SC

graphite
dp=30 mm

=0.48

Case 1 378.0 307.6 random

stainless

steel
dp=15mm

=0.40

Case 2 484.8 316.7 Case 2 483.0 336.7

Case 3 591.0 338.0 Case 3 583.0 370.8

Case 4 705.0 355.0 Case 4 708.0 410.6

Case 5 809.0 413.6 Case 5 803.0 440.2

3. Experimental results and discussion

3.1 Definitions of the near wall region and bulk region

The local effective thermal conductivity distributions of the two random graphite packed beds with different

packing porosity (0.40 and 0.35) are shown in Figure 3a and Figure 3b. The local effective thermal conductivity

at the same position decreases as the temperatures rise (from Case 1 to Case 5) for both the random packed

beds. The reason is that the heat mainly transfers through the internal pebble conduction and contact

conduction. The graphite thermal conductivity decreases with the increasing temperature. The near wall region

could be recognized on the basis of the effective thermal conductivity distributions. The local effective thermal

conductivity begins to drop at a distance about four pebble diameters (4 dp) from the cold wall for both random

graphite packed beds. Symmetrically, the local effective thermal conductivity also has a downward trend within

a distance about 4 dp from hot wall. Overall, the near wall region can be defined within 4 dp from walls.

The local effective thermal conductivity distributions in the SC packed bed of graphite pebbles are shown in

Figure 3c. Similarly to the random packed beds, it has a downward trend near the cold wall and the hot wall. At

a distance about one pebble diameter (1 dp) from cold wall, the local effective thermal conductivity begins to

drop. The local effective thermal conductivity increases more rapidly at a distance about 1 dp from hot wall when

compared to that at the middle positions of the bed. Compared to the random packed bed, wall effect in SC

packed bed seems to be weaker, because the near wall region in the SC packed bed is within 1 dp distance

while for the random packed bed within 4 dp distances.

The effective thermal conductivity distributions in the random packed bed of stainless steel pebbles are shown

in Figure 3d. Contrary to the graphite packed beds, the effective thermal conductivity at the same position

increases as temperatures rise (from Case 1 to Case 5) because the stainless steel and nitrogen thermal

conductivities both increase with increasing temperatures. Similarly to the graphite packed beds, it has a

465

downward trend near the hot wall and cold wall. The local effective thermal conductivity begins to drop faster at

a distance of about four pebble diameters (4 dp) from the cold wall. Local effective thermal conductivity increases

within a distance of about 3 dp from the hot wall when compared to that at the middle positions of the bed. The

near hot-wall region and the near cold-wall region are not symmetrically distributed, which may be caused by

the complicated packing structure. However, the near wall region is still defined at a distance less than 4 dp from

walls in a conservative perspective. This is almost consistent to the random graphite packed beds.

-2 0 2 4 6 8 10 12 14
3

random graphite packed bed

 =0.40

Case1

Case2

Case3

Case4

Case5

near

cold-wall

region

E
ff

e
c
ti

v
e
t

h
e
rm

a
l

c
o

n
d

u
c
ti

v
it

y
,

W
/(

m
K

)

near

hot-wall

region
cold

wallhot

wall

Distance from hot wall, d
p

bulk region

(a)

-2 0 2 4 6 8 10 12 14

random graphite packed bed

 =0.35

near

hot-wall

region
hot

wall

cold

wall

Distance from hot wall, d
p

E
ff

e
c
ti

v
e
t

h
e
rm

a
l

c
o
n
d
u

c
ti

v
it

y
,
W

/(
m

K
) Case1

Case2

Case3

Case4

Case5

bulk region
near

cold-wall

region
(b)

-1 0 1 2 3 4 5 6 7 8

15 SC graphite packed bed

near hot-wall

region

near cold-wall

region

bulk region

Distance from hot wall, d
p

E
ff

e
c
ti

v
e
t

h
e
rm

a
l

c
o
n
d
u
c
ti

v
it

y
,
W

/(
m

K
)

cold

wall

hot

wall

Case1

Case2

Case3

Case4

Case5

(c)

-2 0 2 4 6 8 10 12 14
0

7
random stainless steel packed bed

bulk region
near

cold-wall

region

hot

wall

cold

wall

near

hot-wall

region

Case1

Case2

Case3

Case4

Case5

Distance from hot wall, d
p

E
ff

e
c
ti

v
e
t

h
e
rm

a
l

c
o
n
d
u
c
ti

v
it

y
,
W

/(
m

K
)

(d)

Figure 3: Effective thermal conductivity distributions in four different packed beds

3.2 Separate analysis of effective thermal conductivity in bulk regions and near wall regions

The effective thermal conductivity in the near wall regions varying with temperature is shown in Figure 4.The

effective thermal conductivity in the near wall regions is scattered widely at the same temperature, especially

for the random packed beds. This is because the porous structure varies sharply near walls. This indicates that

the effective thermal conductivity is related to not only the temperature but the local pore characteristics (local

positions). Nearly all measured values of the effective thermal conductivity in the near wall regions of all packed

beds are generally smaller than those in the bulk regions at the same temperature, which is also a result of the

wall effect.

200 300 400 500 600 700 800 900
4

random graphite packed bed

 =0.40

Temperature, K

E
ff

e
c
ti

v
e
t

h
e
rm

a
l

c
o

n
d

u
c
ti

v
it

y
,

W
/(

m
K

)

bulk region

near wall region

(a)

200 300 400 500 600 700 800 900
4

E
ff

e
c
ti

v
e
t

h
e
rm

a
l

c
o
n
d
u
c
ti

v
it

y
,
W

/(
m

K
)

SC packed bed

Temperature, K

bulk region

near wall region

(b)

200 300 400 500 600 700 800 900
0

random SS packed bed

Temperature, K

E
ff

e
c
ti

v
e
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h
e
rm

a
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c
o

n
d

u
c
ti

v
it

y
,

W
/(

m
K

)

bulk region

near wall region

(c)

Figure 4: Effective thermal conductivity in near wall regions varying with temperature: (a) graphite packed beds

(ɛ=0.40); (b) sc packed beds;(c) stainless steel packed beds

466

Total 65 measured values of the effective thermal conductivity in the bulk regions are compared with the three

semi-empirical correlations considering contact conduction. Here the ZS+K correlation combines the ZS (Zehner

and Schlünder,1970) correlation (neglecting contact conduction) and the Kaviany model (considering the

contact conduction) (2012). Experimental values of the effective thermal conductivity in the bulk regions versus

the predicted values by the ZS+K correlation are illustrated in Figure 5a. As a whole, the ZS+K correlation could

describe 89.2 % of experimental values in the bulk regions within error of ±20 % and shows a 9.2 % average

deviation rate and a 36.2 % maximum deviation rate. Figure 5b shows the experimental values of the effective

thermal conductivity in the bulk regions versus predicted values by the ZBS correlation (Bauer and

Schlünder,1978). The predicted values by the ZBS correlation agree rather well with the experimental values.

About 92.3 % of the experimental values in the bulk regions are within an error of ±20 % when compared with

the predicted values. The average deviation rate is 8.2 % and the maximum deviation rate is 20.7 %. All

experimental values are within an error of ±30 %. As shown in Figure 5c, the Hsu correlation could describe

90.8 % of the total experimental values in the bulk regions within an error of ±20 % and shows an 11.4 %

average deviation rate and a 38.9 % maximum deviation rate.

0 4 8 12 16 20
0

-20%

+20%

-30%

E
x
p
e
ri

m
e
n
ta

l
v
a
lu

e
s,

W
/(

m
K

) graphite bed,  =0.40

graphite bed,  =0.35

SC packed bed

SS packed bed

Predicted values by ZS+K correlations, W/(mK)

+30%

(a)

0 4 8 12 16 20

-30%

+30%

-20%

+20%

Predicted values by ZBS correlations, W/(mK)

graphite bed,  =0.40

graphite bed,  =0.35

SC packed bed

SS packed bed

E
x

p
e
ri

m
e
n

ta
l

v
a
lu

e
s,

W
/(

m
K

)

(b)

0 4 8 12 16 20

Predicted values by Hsu correlations, W/(mK)

E
x
p
e
ri

m
e
n
ta

l
v
a
lu

e
s,

W
/(

m
K

)

graphite bed,  =0.40

graphite bed,  =0.35

SC packed bed

SS packed bed

+30%

+20%

-20%

-30%

(c)

Figure 6: Experimental values in the bulk regions versus predicted values by different correlations: (a) ZS+K

correlation; (b) ZBS correlation; (c) Hsu correlation

Total 165 measured values of the effective thermal conductivity in the near wall regions are compared with the

three semi-empirical correlations. When compared with the predicted values by the ZS+K correlation, most

experimental points (about 98.2 %) are within an error of ±40 %, as shown in Figure 6a. As a whole, the ZS+K

correlations shows a 14.6 % average deviation rate and a 48.5 % maximum deviation rate. Measured values of

effective thermal conductivity in the near wall regions of packed beds are compared with the ZBS correlation in

Figure 6b. About 98.8 % of measured values are located within error of ±40 %. As a whole, the ZBS correlation

shows a 17.4 % average deviation rate and a 41.9 % maximum deviation rate. Figure 6c shows comparisons

between experimental values in near wall regions and predicted values by the Hsu correlation. The Hsu

correlation could describe 93.3 % of total experimental values in the near wall regions within an error of ±40 %

and shows a 20.8 % average deviation rate and a 50.3 % maximum deviation rate. Generally, the effctive

thermal conductivities in the near wall regions are scattered, which poses a real challenge for the capability of

the correlations.

0 4 8 12 16 20
0

20
graphite bed,  =0.40

graphite bed,  =0.35

SC packed bed

SS packed bed

E
x
p
e
ri

m
e
n
ta

l
v
a
lu

e
s,

W
/(

m
K

)

Predicted values by ZS+K correlations, W/(mK)

+40%

-40%

-30%

+30%

(a)

0 4 8 12 16 20
0

+40 %

-40 %

-30 %

+30 %

graphite bed,  =0.40

graphite bed,  =0.35

SC packed bed

SS packed bed

Predicted values by ZBS correlations, W/(mK)

E
x

p
e
ri

m
e
n

ta
l

v
a
lu

e
s,

W
/(

m
K

)

(b)

0 4 8 12 16 20
0

20
graphite bed,  =0.40

graphite bed,  =0.35

SC packed bed

SS packed bed

E
x
p
e
ri

m
e
n
ta

l
v
a
lu

e
s,

W
/(

m
K

)

Predicted values by Hsu correlations, W/(mK)

+40%

-40%

-30%

+30%

(c)

Figure 6: Experimental values in the near wall regions versus predicted values by different correlations: (a)

ZS+K correlation; (b) ZBS correlation; (c) Hsu correlation

4. Conclusions

In this paper, the effective thermal conductivities in four different packed beds are experimentally studied to

discuss the different behaviours in the near wall region and bulk region. It is found that the effective thermal

467

conductivity has a downward trend as the wall is approached, which is caused by wall effect. On the basis of

effective thermal conductivity distributions, the near wall region and bulk region could be determined. The near

wall region can be defined within four pebble diameters from walls for random packed beds and one pebble

diameter for SC packed beds, indicating wall effect in SC packed bed is weaker.

Nearly all measured values of the effective thermal conductivity in the near wall regions are generally smaller

than that in the bulk regions at the same temperature. The ZBS correlation is found to give better predictions of

effective thermal conductivity in the bulk region for all packed beds. In the near wall regions, the experimental

results are scattered widely at the same temperature. Thus, the ZS+K correlation, the ZBS correlation and the

Hsu correlation can not predict the effective thermal conductivity in the near wall regions as well as that in the

bulk regions. Even the ZBS correlation describes 98.8 % of measured values in the near wall regions within

error ±40 % while it describes all experimental values in the bulk regions within error of ±30 %.Generally, the

ZBS correlation gives best predictions in the bulk regions and in the near wall regions with the proper empirical

parameter selected. It should be noted that empirical parameters related to contact area used in the correlations

significantly influence the predicted values. Further efforts should be paid to improve the prediction capability of

the effective thermal conductivity correlation in the near wall regions and give a quanatitive method to determine

the empirical parameters.

Acknowledgements

We would like to acknowledge fifinancial supports for this work provided by the National Natural Science

Foundation of China (No.51606016, No.51806023), China Postdoctoral Science Foundation

(No.2016M602650) and Chongqing Postdoctoral Science Special Foundation (No. Xm2016024).

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