2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 CFD Modeling of Heat and Moisture Transfer on a 2-D Model of a Beef Leg Francisco J. Trujillo and Q. Tuan Pham University of New South Wales Sydney 2052, Australia

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Objective 1 To report on problems encountered while trying to model simultaneous heat and mass transfer in both air and solid phases during beef carcass chilling, using the CFD package FLUENT 6.0

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Objective 2 To report preliminary results on a simple shape

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Why we must model heat and mass transfer During chilling of beef carcasses, cooling and evaporation influences – meat temperature – surface water activity – microbial growth – weight loss – tenderness – other quality factors

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Transport equations In the air where  may be 1 (continuity) velocity components u, v turbulent kinetic energy k turbulent dissipation rate  temperature water vapour concentration Change = Convection + Diffusion + Source In the solid where  may be temperature moisture concentration Change = Diffusion

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Turbulence model RNG modification: RNG (Re-Normalization Group) : includes analytical formula for effective viscosity that is valid across the full range of flow conditions, from low to high Reynolds numbers Enhanced wall treatment –Viscosity effects near wall is completely resolved all the way to the viscous sub-layer (y +  1) using very fine mesh. –Enhanced wall functions: smoothly blend linear (laminar) and logarithmic (turbulent) laws-of-the-wall. k-  model: effective viscosity calculated from turbulent intensity k and turbulent dissipation 

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Why is it difficult to model simultaneous heat and mass transfer in meat Moisture diffusion is mostly near the surface, while heat conduction takes place over the whole body Due to steep moisture profile near surface, a very fine mesh is required

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Problem related to FLUENT’s capabilities In the meat, the heat and mass transfer could not be solved using FLUENTs standard heat and mass transfer equations because: –FLUENT cannot solve mass transfer equation in a solid. Thus the meat had to be defined as a “fluid” phase. –Once the meat was defined as a fluid, FLUENT requires the physical properties (c p, , D, k,  ) be the same in all parts of the solution domain (i.e. both air and meat phases are modelled as the same substance). –Since these properties were in fact different in the two phases, new field variables must be defined and solved in the meat. –FLUENT6.0 “ready-made” boundary conditions of the mass transport equations can only be zero flux or a fixed value.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Details of special FLUENT procedures Define new field variables called UDS (User Defined Scalars) which represent the moisture concentration and temperature in the meat. These new fields must be tied to those in the air using equilibrium and conservation laws...

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 UDF1: B.c. for mass transfer in meat: Write user-defined functions in C++ to calculate the boundary condition of mass transfer equation in meat : (the mass flux at the meat surface is calculated from the water vapour gradient in the air cell next to the surface)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 UDF2: B.c. for heat transfer in meat: Write user-defined functions to calculate the boundary condition of heat transfer equation in meat: (the heat flux from the meat at the surface is calculated from the convective, evaporative and radiated heats), where (convective heat flux is calculated from T gradient in the air cell next to the surface.) (evaporative heat flux is calculated from mass flux calculated earlier)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 UDF4: B.c. for mass transfer in air: Write user-defined functions to calculate the boundary condition of mass transfer equation in air: where a w (meat surface) is calculated from meat surface moisture content using measured isotherm on right a w (air surface) is the relative humidity in the air at the interface

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 UDF3: B.c. for heat transfer in air: Write user-defined functions to calculate the boundary condition of heat transfer equation in air:

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Solution procedure for each iteration For each time t Iterate Solve linearized momentum equation in the air phase. Solve energy equation in the air phase. Solve mass equation in the air phase. Solve turbulent kinetic energy in the air phase. Solve eddy dissipation in the air phase. Solve energy equation in the solid phase. Solve mass equation in the meat phase. Update all properties Check convergence. until results have converged (no more change in field variables) Advance to next time t+  t

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 CFD Grid air side: nodes meat side: 6487 nodes

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Velocity profile Bound. layer detachment Impingement (stagnation point) Recirculation

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Result: Temperature profiles at 30 minutes

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Result: Temperature profiles after 5 hours

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Result: Moisture profiles in meat after 5 hours

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Result: Moisture profiles in meat after 20 hours

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Temperature history Comparison of ellipse model with “equivalent” real beef leg:

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Temperature profile along surface Back of ellipse Front of ellipse

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Average surface water activity history Note surface rewetting

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Water activity profile along surface

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Moisture profile in the meat Note diffusion depth  20mm Note surface re-wetting

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results: Variation of htc and mtc along surface

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 How long did it take? CFD calculation is time consuming because Grid is very fine on both sides of interface: –air side (29033 nodes): for enhanced wall treatment –meat side (6487): because of small scale of mass transfer Must iterate at each time interval to satisfy 2 equilibrium equations and 2 flux conservation equations at meat-air interface  160 CPU hours (on a 1.5GHz Pentium 4) to simulate 20 hours of real time.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 CONCLUSIONS Even though FLUENT 6.0 is a very powerful CFD software, special techniques have to be used to deal with simultaneous heat and mass transfer in two different phases Solution is very time-consuming - not yet practical for routine industrial calculations. The model gave reasonable predictions of local variations in temperature, moisture and water activity

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 ACKNOWLEDGMENTS This work was carried out under a grant by the Australian Government via the Australian research Council. Francisco Trujillo is supported by a Postgraduate scholarship funded by the grant.