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Transient heat conduction through a substrate of brine-spongy ice

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Abstract

An analytical model for heat conduction through brine-spongy ice is developed. This model fills a gap in knowledge related to transient heat conduction to a two-phase substrate which is crucial for modeling transient icing and deicing of cold surfaces in contact with salt water. The core of the model is based on the phase change of pure ice and brine pockets trapped in the structure of spongy ice. Freezing of brine pockets causes the release of the latent heat of fusion that is considered as the source of heat generation distributed throughout the brine-spongy ice. A nonlinear partial differential equation and a number of equations of state for ice, brine, and brine-spongy ice create governing equations of heat transfer through brine-spongy ice. A standard numerical scheme solves the set of equations in various initial conditions. The variation of temperature, volume fraction of brine and salinity of brine pockets are calculated numerically. Experimental samples of brine-spongy ice are examined under transient conditions and their surface temperatures are captured using an infrared thermal camera. The numerical results, which are for various overall salinities, are closely aligned with the measured surface temperatures.

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Abbreviations

\( {\text{c}}_{\text{b}} \) :

Specific heat capacity of brine (J/kg K)

\( {\text{c}}_{\text{e}} \) :

Equivalent specific heat capacity of brine-spongy ice (J/kg K)

\( {\text{c}}_{\text{i}} \) :

Specific heat capacity of ice (J/kg K)

\( {\text{c}}_{\text{s}} \) :

Specific heat capacity of brine-spongy ice (J/kg K)

\( {\text{k}}_{\text{b}} \) :

Thermal conductivity of brine (W/m K)

\( {\text{k}}_{\text{i}} \) :

Thermal conductivity of ice (W/m K)

\( {\text{k}}_{\text{s}} \) :

Thermal conductivity of brine-spongy ice (W/m K)

\( {\text{L}} \) :

Ice sample length (m)

\( {\text{L}}_{\text{Hb}} \) :

Latent heat of fusion of brine (J/kg)

\( {\text{Q}}_{\text{i}} \) :

Input heat flux (J/s)

\( {\text{Q}}_{\text{o}} \) :

Output heat flux (J/s)

\( {\text{Q}}_{\text{g}}^{\prime \prime \prime } \) :

Volumetric heat generation (J/m3)

\( {\dot{\text{Q}}}_{\text{g}}^{\prime \prime \prime } \) :

Rate of volumetric heat generation (J/m3 s)

\( {\text{S}} \) :

Overall salinity (‰)

\( {\text{S}}_{\text{b}} \) :

Salinity of brine pockets (‰)

\( {\text{T}} \) :

Temperature (oC)

\( {\text{T}}_{\text{L}} \) :

Temperature at \( x = L \) (oC)

\( {\text{T}}_{\text{i}} \) :

Initial temperature (oC)

\( {\text{T}}_{\text{s}} \) :

Contact temperature (oC)

\( {\text{t}} \) :

Time (s)

\( {\text{t}}_{0} \) :

Initial time (s)

\( {\text{V}}_{\text{b}} \) :

Volume of brine pocket in the element of brine-spongy ice (m3)

\( {\text{V}}_{\text{Fb}} \) :

Volume fraction of brine (—)

\( {\text{V}}_{\text{i}} \) :

Volume of ice in the element of brine-spongy ice (m3)

\( {\text{V}}_{\text{s}} \) :

Volume of the element of brine-spongy ice (m3)

\( \upalpha_{\text{cb}} \) :

Coefficient of the equation of specific heat capacity of brine (—)

\( \upalpha_{\text{ci}} \) :

Coefficient of the equation of specific heat capacity of ice (—)

\( \upalpha_{\text{kb}} \) :

Coefficient of the equation of thermal conductivity of brine (—)

\( \upalpha_{\text{ki}} \) :

Coefficient of the equation of thermal conductivity of ice (—)

\( \upalpha_{\text{LHb}} \) :

Coefficient of the equation of latent heat of fusion of brine (—)

\( \upalpha_{\text{sb}} \) :

Coefficient of the equation of salinity of brine (—)

\( \upalpha_{{\uprho{\text{b}}}} \) :

Coefficient of the equation of density of brine (—)

\( \upalpha_{{\uprho{\text{i}}}} \) :

Coefficient of the equation of density of ice (—)

\( \updelta{\text{Q}}_{\text{g}}^{\prime \prime \prime } \) :

Variation of volumetric heat generation (J/m3)

\( \updelta{\text{t}} \) :

Time interval (s)

\( \updelta{\text{V}}_{\text{Fb}} \) :

Variation of volume fraction of brine (—)

\( \updelta{\text{V}}_{\text{i}} \) :

Variation of volume of ice (m3)

\( \uprho_{\text{b}} \) :

Density of brine (kg/m3)

\( \uprho_{\text{e}} \) :

Equivalent density of brine-spongy ice (kg/m3)

\( \uprho_{\text{i}} \) :

Density of ice (kg/m3)

\( \uprho_{\text{s}} \) :

Density of brine-spongy ice (kg/m3)

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Acknowledgements

The authors gratefully acknowledge the financial support of Statoil (Norway), MITACS, and Petroleum Research Newfoundland and Labrador (PRNL) (IT03198) for this research.

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  1. Department of Mechanical Engineering, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, A1B 3X5, Canada

    S. R. Dehghani, G. F. Naterer & Y. S. Muzychka

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  1. S. R. Dehghani
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Correspondence to S. R. Dehghani.

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Dehghani, S.R., Naterer, G.F. & Muzychka, Y.S. Transient heat conduction through a substrate of brine-spongy ice. Heat Mass Transfer 53, 2719–2729 (2017). https://doi.org/10.1007/s00231-017-2016-5

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