^{1}and Julia M. Rees

^{2}

### Abstract

Double diffusion of a viscous fluid is simulated for heat leakage driven by buoyant convection under cryogenic storage conditions in a cylindrical tank with laminar flow. If the tank is stably stratified, there is a potential instability due to the inability of the fluid in the lower layer to release heat to the top vapor space, whereas the upper liquid layer can exchange heat and mass through sensible heat transfer and evaporation with the vapor space. Eventually, the lower layer becomes less dense due to thermal expansion and is no longer constrained in the stratification. The rapid rise and overturning of the fluid is termed rollover, and can be accompanied by a potentially explosive release of vapor. In this paper, hydrodynamics and heat and mass transport are used to study the stability characteristics of rollover. The transient state is used as a base state for a linear stability analysis which shows the transition from a “corner eddy” mode spinning down to spinning up is the driver for the rollover instability. Four different vapor-liquid interfacial boundary conditions are tested, with similar results for the time to rollover. Surprisingly, the long time prerollover state is dominated in the laminar flow regime by heat conduction and diffusion, as the expected double roll structure is suppressed and advection plays a small roll in the majority of the prerollover period. Scalings are suggested for controlling dimensionless groups on this prerollover basis that can be used as a guideline to determine the regime of double diffusion—a single roll or a double roll stratification, as well as the severity of the eventual rollover event. An energy analysis demonstrates the switch from practically advection free to free convection regimes.

W.B.Z. would like to thank the longstanding support from Geoff Clegg, a LNG safety expert, of UMIST in encouraging this work. Frank Ramsay provided helpful discussions and understanding. Kiran Deshpande also supported the understanding of modelling in this field. Financial support from the UK Health and Safety Executive (CASE Award with Robin Hankin), Department of Trade and Industry (Contract No. KTP0000181), and MHT Technology Ltd. is acknowledged. Bob Whiting and Richard Parker Smith of the KTP programme and Malcolm Tennant and Marcus Webster of MHT Technology are thanked for practical insight into the LNG processing sector.

I. INTRODUCTION

II. METHODS

A. Modelling double diffusion in a stable stratification driven by heat leakage

B. Domain and boundary conditions

C. Vapor-liquid interfacial conditions

D. Initial conditions

E. Numerical methods

III. RESULTS

A. Case study I: Constant flux (Neumann) BC model with , , ,

B. Case study II: Fixed temperature (Dirichlet) BC model with the same parameters as case study I

C. Case study III: Dirichlet and Biot models—, , ,

D. Summary of results of parametric variation of the thermal Rayleigh number

E. Linear stability analysis

F. Energy stability analysis

G. Shannon entropy analysis

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Boundary value problems
- 37.0
- Diffusion
- 31.0
- Convection
- 29.0
- Thermal convection
- 21.0
- Eddies
- 16.0

## Figures

Typical configuration of a cylindrical storage vessel for liquid natural gas. There are four practically constant heat fluxes at the bottom , sidewalls (, ), and top of the container . Evaporation across the vapor-liquid interface is described by an evaporation rate and an associated heat flux which includes the sensible heat flux across the interface and the latent heat of vaporization lost from the liquid. “HT” is shorthand for sensible heat transfer and represents enthalpy change by evaporation. Treating is the key challenge for a hydrodynamics-only model.

Typical configuration of a cylindrical storage vessel for liquid natural gas. There are four practically constant heat fluxes at the bottom , sidewalls (, ), and top of the container . Evaporation across the vapor-liquid interface is described by an evaporation rate and an associated heat flux which includes the sensible heat flux across the interface and the latent heat of vaporization lost from the liquid. “HT” is shorthand for sensible heat transfer and represents enthalpy change by evaporation. Treating is the key challenge for a hydrodynamics-only model.

Axisymmetric coordinate system with a standard mesh used for many of the simulations; 12 352 triangular elements with 106 077 degrees of freedom in the Galerkin finite element expansions for radial and axial velocities, pressure, temperature, and concentration fields. The initial concentration profile is a steep change from unit concentration in the lower layer to zero concentration in the upper layer. Kinematic arguments suggests two pseudosteady buoyant convention cells shaped like tori should evolve (see Ramsay, Ref. 7).

Axisymmetric coordinate system with a standard mesh used for many of the simulations; 12 352 triangular elements with 106 077 degrees of freedom in the Galerkin finite element expansions for radial and axial velocities, pressure, temperature, and concentration fields. The initial concentration profile is a steep change from unit concentration in the lower layer to zero concentration in the upper layer. Kinematic arguments suggests two pseudosteady buoyant convention cells shaped like tori should evolve (see Ramsay, Ref. 7).

Constant flux BC model with , , , and , showing 20 contours of temperature, gray scale concentration, and arrows for velocity vectors for times .

Constant flux BC model with , , , and , showing 20 contours of temperature, gray scale concentration, and arrows for velocity vectors for times .

Fixed temperature BC model with , , , and , showing 20 contours of temperature, gray scale concentration, and arrows for velocity vectors for times .

Fixed temperature BC model with , , , and , showing 20 contours of temperature, gray scale concentration, and arrows for velocity vectors for times .

Biot boundary condition simulation with , , , showing the phase averaged temperature difference, , for times .

Biot boundary condition simulation with , , , showing the phase averaged temperature difference, , for times .

Same simulations as Fig. 5 showing the time history of the maximum point temperature difference, .

Same simulations as Fig. 5 showing the time history of the maximum point temperature difference, .

Fixed temperature BC simulation showing the induced average Reynolds number , which is the domain averaged velocity magnitude.

Fixed temperature BC simulation showing the induced average Reynolds number , which is the domain averaged velocity magnitude.

Same simulation as Figs. 5 and 6 demonstrating the Nusselt number Nu calculated as the dimensionless average heat flux, averaging Eq. (13).

Same simulation as Figs. 5 and 6 demonstrating the Nusselt number Nu calculated as the dimensionless average heat flux, averaging Eq. (13).

Time to rollover for which the volume averaged velocity magnitude is maximum: for the constant flux BC simulation with varying and fixed , , .

Time to rollover for which the volume averaged velocity magnitude is maximum: for the constant flux BC simulation with varying and fixed , , .

Time to rollover for which the volume averaged velocity magnitude is maximum: for the fixed temperature BC simulation with varying and fixed , , .

Time to rollover for which the volume averaged velocity magnitude is maximum: for the fixed temperature BC simulation with varying and fixed , , .

Constant flux BC simulation for the same conditions as Fig. 9 with three measures of velocity.

Constant flux BC simulation for the same conditions as Fig. 9 with three measures of velocity.

Fixed temperature BC simulation for the same conditions as Fig. 10 with three measures of velocity.

Fixed temperature BC simulation for the same conditions as Fig. 10 with three measures of velocity.

The decay rate vs time for the most dangerous mode for a range of . The crossing of the -axis corresponds to the most dangerous mode becoming unstable. , , for all simulations for the Dirichlet boundary condition.

The decay rate vs time for the most dangerous mode for a range of . The crossing of the -axis corresponds to the most dangerous mode becoming unstable. , , for all simulations for the Dirichlet boundary condition.

The most dangerous mode computed for Dirichlet boundary condition with , , , just before the neutral stability time . Apparently, the velocity field switches sense with the lower right-hand corner eddy switching from counterclockwise to clockwise rotation. However, care should be taken in recognizing that the eigenvector of the unknown is determined only up to a multiplicative constant in the linear theory. Thus, the most dangerous eigenmodes, before and after the critical time, are indistinguishable. Both temperature and concentration profiles consistently flip signs as well.

The most dangerous mode computed for Dirichlet boundary condition with , , , just before the neutral stability time . Apparently, the velocity field switches sense with the lower right-hand corner eddy switching from counterclockwise to clockwise rotation. However, care should be taken in recognizing that the eigenvector of the unknown is determined only up to a multiplicative constant in the linear theory. Thus, the most dangerous eigenmodes, before and after the critical time, are indistinguishable. Both temperature and concentration profiles consistently flip signs as well.

Constant flux BC simulation reporting the energy stability functional for , , , .

Constant flux BC simulation reporting the energy stability functional for , , , .

Shannon entropy evolution for , , , and for , , , with the constant flux BC model.

Shannon entropy evolution for , , , and for , , , with the constant flux BC model.

## Tables

Time averaged temperature differences with each boundary condition: phase averaged and fixed geometry temperature difference . The averaging is taken over for and 15 000 and for and 5000.

Time averaged temperature differences with each boundary condition: phase averaged and fixed geometry temperature difference . The averaging is taken over for and 15 000 and for and 5000.

Summary statistics for velocity (induced Reynolds number) for the constant flux boundary condition with , , and . is the time at which the simulation was ended. is the average velocity in the tank at . is the time at which the average tank velocity was achieved, which is taken as the quantity distinguishing rollover. is the maximum velocity instantaneously achieved in the tank during the entire simulation at any point.

Summary statistics for velocity (induced Reynolds number) for the constant flux boundary condition with , , and . is the time at which the simulation was ended. is the average velocity in the tank at . is the time at which the average tank velocity was achieved, which is taken as the quantity distinguishing rollover. is the maximum velocity instantaneously achieved in the tank during the entire simulation at any point.

Summary statistics for velocity (induced Reynolds number) for the fixed temperature boundary condition with , , and . is the time at which the simulation was ended. is the average velocity in the tank at . is the time at which the average tank velocity was achieved, which is taken as the quantity distinguishing rollover. is the maximum velocity instantaneously achieved in the tank during the entire simulation at any point.

Summary statistics for velocity (induced Reynolds number) for the fixed temperature boundary condition with , , and . is the time at which the simulation was ended. is the average velocity in the tank at . is the time at which the average tank velocity was achieved, which is taken as the quantity distinguishing rollover. is the maximum velocity instantaneously achieved in the tank during the entire simulation at any point.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content