ACTS- P00510

NUMERICAL STUDY ON HEAT TRANSFER CHARACTERISTICS OF

SLUSH NITROGEN IN A HORIZONTAL PIPE

Y. J. Li, S. Q. Wu, T. Jin*

Institute of Refrigeration and Cryogenics / Key Laboratory of Refrigeration and Cryogenic

Technology of Zhejiang Province, Zhejiang University, Hangzhou 310027, China

Presenting Author: zjulyj@zju.edu.cn

*Corresponding author: jintao@zju.edu.cn

ABSTRACT

Slush nitrogen is considered as a potential coolant for high-temperature superconductors with decreased

nitrogen consumption and storage cost, thanks to its lower temperature, higher density and higher heat

capacity than those of liquid nitrogen. The heat transfer characteristics are key factors for application of

slush nitrogen as cooling fluid. In the present study, a three-dimensional Euler-Euler two-fluid numerical

model based on the PBM (population balance model) has been built to predict the heat transfer

characteristics of slush nitrogen in a horizontal pipe. The PBM was ever used to predict the evolution of

the particle size distribution for breakage due to the particle-particle interactions and effects between

particles and wall in the pipe. Meanwhile, an effective viscosity of mixture which takes the particle shape

and size into consideration is introduced into the model and adopted for the drag law and the mass and

heat transfer, since the slush nitrogen usually has non-spherical and coarse particles with an average size

of 0.5 mm - 2 mm. The improved model is then used to predict the particle size distribution and to

analyze the effects of inlet velocity, solid concentration and heat flux on the heat transfer characteristics

to discuss the heat transfer deterioration.

KEYWORDS: slush nitrogen, multiphase flow, heat transfer, population balance model

1. INTRODUCTION

Slush nitrogen is the cryogenic suspension of solid nitrogen particles and subcooled liquid nitrogen and

has been considered as a potential coolant for high-temperature superconductors with decreased nitrogen

consumption and storage cost, thanks to its lower temperature, higher density and higher heat capacity

than those of liquid nitrogen[1]. The flow and heat transfer characteristics are key factors for application

of slush nitrogen as cooling fluid. Earlier studies on slush nitrogen focus on the production, measurement

and transfer characteristics[2-4]. Pressure drop reduction and heat transfer deterioration phenomena (i.e.

pressure drop for subcooled liquid nitrogen is greater than that of slush nitrogen and heat transfer

coefficient for slush nitrogen is smaller than that of liquid nitrogen) has been found in the experimental

study on slush nitrogen flow in a horizontal pipe[3]. Previous CFD (Computational Fluid Dynamics)

models are mainly applicable to the ambient slurry flow like sand-water suspension and oil slurry and the

solid phase is usually treated as even particles with constant size[5-10]. However, the model for

cryogenic slurries should also consider the interphase mass and heat transfer. Also, the effects of particle

size distribution should be concerned. QMOM (the quadrature method of moments), one of the solution

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Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS March 26-30, 2017, Jeju Island, Korea

method for PBM, is considered to be a computationally affordable and also accurate method for tracking

the moments of the particle size distribution (PSD) in CFD simulation of multiphase flow[11-13].

A three-dimensional Euler-Euler two-fluid CFD model based on the PBM (population balance model)

has been built to predict the heat transfer characteristics of slush nitrogen in a horizontal pipe in the

present study. The model is adopted to calculate and analyze the effects of inlet velocity, solid

concentration and heat flux on heat transfer characteristics of slush nitrogen.

2. MATHEMATICAL MODEL

A 3-D (three-dimensional) Euler-Euler two-fluid CFD model that takes into account the changes in mass,

momentum and energy exchange between the solid and liquid phases has been built to predict the flow

and heat transfer characteristics of slush nitrogen in a horizontal pipe. The turbulent quantities are

predicted by the standard k-ε model, the per-phase type. The flow pipe modeled in this study is a

horizontal circular pipe with the diameter D of 15 mm and the length L of 1 m which should meet the

demand of for full flow development.

The method for the effective viscosity of mixture used in this study is an exponential formula

developed by Cheng and Law[14], as given by

2.5 1exp 1

1m l

s

(1)

where is the unique parameter to account for the influence of particle condition, and =1.5 in

this study. The effective viscosity of mixture takes the particle shape and size into consideration and is

adopted for the drag law and the mass and heat transfer.

The PBM is used to predict the evolution of the particle size distribution for breakage due to the

particle-particle interactions and effects between particles and wall in the pipe. The QMOM approach,

which is based on the internal coordinate of particle length, is implemented to solve the population

balance equation for tracking the moments of the particle size distribution[15],

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n( )dN

k k

k i

i

m L L L L w

(2)

where n( )L is the particle size distribution, iw are the weights, and iL are the abscissas. The

quadrature approximation can be calculated by its first 2N moments of the particle size distribution. In

the present study, N=3 and six moments are tracked, and the quadrature approximation is given by =3

1 2 3

1 2 3

1

=N

k

k i

i

m L w L w L w L w

(3)

The values of iw and kL are introduced to close the transport equations for the moments, and to

include the influence of particle breakage on the particle size distribution.

For the wall boundary conditions, no slip is assumed for the liquid phase. The transport equation derived

for the solids phase is based on the granular temperature and the boundary conditions for the solid phase

are based on the Johnson-Jackson model which can involve the friction and collision between the particle

and the wall[16]. The specularity coefficient, particle-particle and particle-wall restitution coefficients

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based on the wall boundary conditions adopted for the granular phase of slush nitrogen are 0.01 ,

0.9sse and 0.7we , respectively[10].

3. RESULTS AND DISCUSSION

The CFD results of Nusselt number for subcooled liquid nitrogen, together with comparative results

for the Sieder-Tate equation, are shown in Fig. 1. The Sieder-Tate equation is given by

0.14

0.8 1/30.027 Re Pr l

lw

Nu

(4)

where l is the viscosity of bulk fluid, while lw is the viscosity of near-wall fluid. The numerical

results agree well with those of Sieder-Tate equation with a maximum deviation of 10%, which is

acceptable if considering that the heat flux in the CFD calculations is 30 kW/m2 and the Sieder-Tate

equation is an empirical equation fitted with the results for different working conditions. The CFD

model is considered to be effective for predicting the heat transfer characteristics of cryogenic flow in

a horizontal circular pipe.

Fig. 1. Comparison of Nusselt number for CFD data and Sieder-Tate equation

Fig. 2 and Fig. 3 present the numerical results of the heat transfer coefficients of slush nitrogen and

subcooled liquid nitrogen at the triple point for heat flux of 10 kW/m2 and 30 kW/m2, respectively,

comparing to the experimental data from the literature [4]. The calculation ranges are 1-5 m/s for inlet

flow velocity, and 5-35% for solid fraction. The local heat transfer coefficients are obtained by h=q /

(Twall - Tbulk), where wallT is the temperature of near-wall fluid, bulkT is the temperature of bulk fluid,

and q is the wall heat flux. The heat transfer coefficients for slush nitrogen rise with the increased

flow velocity and solid fraction on the whole and the CFD results agree well with the reference

experimental data. However, the heat transfer coefficients obtained by CFD analysis are lower than

that from the experimental results with the data misfit within 12% at high velocity. Further attention

should be paid to the effects of turbulence on the heat transfer characteristics. The results for the ratios

between the heat transfer coefficients of slush nitrogen and that of subcooled liquid nitrogen /sl subh h at

the heat flux of 10 kW/m2 are presented in Fig. 4. As indicated in the early researches, the heat transfer

deterioration phenomena appear at some cases of high velocity, which has been confirmed in the

present numerical results. Moreover, the numerical and reference results for bulk temperature increase

in the flow pipe at the heat flux of 10 kW/m2 are given in Fig. 5, which also shows good agreement.

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Fig. 2. Variation of heat transfer coefficient of

slush nitrogen with flow velocity for heat flux of

10 kW/m2

Fig. 3. Variation of heat transfer coefficient of

slush nitrogen with flow velocity for heat flux of

30 kW/m2

Fig. 4. Heat transfer coefficient ratio of slush

nitrogen for different flow velocity at 10 kW/m2.

Fig. 5. Bulk temperature increase of slush

nitrogen for different flow velocity at 10 kW/m2.

4. CONCLUSIONS

A 3-D Euler-Euler two-fluid model based on the population balance model has been developed to

study the flow and heat transfer characteristics of slush nitrogen in a horizontal circular pipe. The

improvements of the model lie in: the effective viscosity of slush mixture, which takes the particle

shape and size into consideration, is introduced into the model to modify the correlations for

interphase momentum exchange and heat transfer. Also, the particle size distribution is introduced

based on the PBM to include the breakage due to particle-particle interactions and effects between

particles and wall in the pipe. The model is proved to be effective for predicting the heat transfer

characteristics of slush nitrogen flow in the horizontal pipe. Extensive cases at various working

conditions (inlet solid fraction of 5-35%, inlet velocity of 1-5 m/s, heat flux of 10 kW/m2 and 30

kW/m2) are calculated to analyze the heat transfer coefficients and the CFD results agree well with the

experimental data from the literatures with a deviation of 12%. In addition, the heat transfer

deterioration phenomena are obtained in the numerical analysis.

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ACKNOWLEDGMENT

This work is financially supported by the Zhejiang Provincial Natural Sciences Foundation (LZ14E060001).

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