## 1 Introduction  
Increasing attention has been paid to the high temperature helium-cooled pebble bed reactor (HTR) due to its high efficiency, high levels of passive safety, and potential usage for hydrogen production [1; 2]. The Chinese high temperature gas-cooled reactor pebble bed module (HTR-PM) demonstration project, oriented by the Institute of Nuclear and New Energy Technology of Tsinghua University (INET), has been installed at the Shidaowan plant in Shandong Province, China and is scheduled to go online in 2017 [3; 4].  
HTR-PM has a cylindrical pebble bed core with a diameter of 3 m and a height of 11 m, and thousands of spherical fuel elements are randomly packed inside. In engineering practice, values of the effective thermal conductivity of the pebble bed core of HTR-PM at different temperatures are essential parameters required in safety analysis and thermal calculation of the reactor [5]. The effective thermal conductivity has been closely related to the HTR safe characteristic. When a loss-of-coolant accident happens, the residual heat must be removed from the core in time, which directly depends on the effective thermal conductivity of the pebble bed [6; 7; 8; 9]. A relevant experiment named SANA-1 [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12] designed by the Research Center Juelich has historically been carried out for validation of the afterheat removal capacity of HTR. The size of the pebble bed and the highest test temperature (below 1000 \({}^{\circ}\)C) in SANA-1 were limited. The temperature range of SANA-1 was unable to cover the whole temperature range of the safety analysis of HTR-PM (0-1600 \({}^{\circ}\)C).  
In order to support the HTR-PM project, Tsinghua University has designed and constructed a full-scale heat transfer test facility for pebble bed equivalent conductivity measurement (TF-PBEC) over the whole temperature range (0-1600 \({}^{\circ}\)C) [13; 14], as shown in Figure 1. The function of the TF-PBEC is to create a full-size pebble bed to simulate the complicated thermal transfer condition in the real reactor core of HTR-PM. The TF-PBEC is an integrated experimental system and has the structure of the internal heating type resistance furnace. About 70,000 machined graphite spheres with a diameter of 60 mm were randomly packed in an annular test zone bounded by inner and outer walls inside the TF-PBEC to simulate the fuel packing structure, as shown in Figure 1[5]. The radius of the inner wall of the pebble bed is 500 mm and that of the outer wall is set to be 2000 mm. The height of the pebble bed is set to be 1000 mm. The temperature distribution in the pebble bed under helium atmosphere conditions will be measured to determine the effective thermal conductivity values of the pebble bed with temperature up to 1600 \({}^{\circ}\)C. In order to optimize the design of the facility, a theoretical prediction of the effective thermal conductivity of the pebble bed over the whole temperature range is needed. This paper carries out detailed theoretical analysis on the effective thermal conductivity of the pebble bed in the HTR-PM heat transfer test facility, which is necessary for the design and optimization of the test facility and further experiments. In the SANA experiment, the effective thermal conductivity of a graphite pebble bed at the temperature range of 0-1000 \({}^{\circ}\)C was measured. Due to the similarity of the two reactors (Chinese HTR-PM and Germany HTR-Module), the graphite pebbles used in the TF-PBEC are similar to those in the SANA experiment. Hence, the experimental data used for analysis in this paper is from SANA's data set.