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Thermal Conductivity & Estimated Convection Coefficient:
Dried Banana & Dried Zucchini

Brad Kairdolf & Julianne Forman

BE 4352:  Transport Phenomena

May 7, 2002

Introduction

The properties of different materials determine the rate of heat transferred by conduction and convection.  Fourier’s Law describes heat transfer due to conduction, and Newton’s Law of Cooling describes heat transfer due to convection.  By quantifying heat transfer by conduction and convection, these equations can be applied to determine material properties.  Thermal conductivity is a measurement of a material’s ability to transfer or conduct heat.  Various factors can affect the thermal conductivity of a material, such as water content, electrical conductivity, state of material, and porosity of material.  Steady-state conditions are reached when there is no change in thermal energy stored in the system with respect to time; therefore, when the temperature gradient within the system is not changing with time and the rate of heat transfer into the system equals the rate of heat transfer out of the system, the heat transfer rate will be a constant value.  Allowing heat flow by conduction to equal heat flow by convection, the estimated convection coefficient could be calculated and then used to calculate the thermal conductivity for each food product.

The thermal conductivity and the estimated convection coefficient of dried banana and dried zucchini were determined experimentally using analytical calculation methods.  Numerical methods were used to determine the steady-state temperature profile for each food product by using FEM Lab.  Also, thermal conductivity values were compared with reported k values as found in literature.

Materials and Methods

·        Zucchini

·        Banana

·        Calipers

·        Thermocouple

·        CR10x Datalogger

·        Hot plate

In order to experimentally determine the convection coefficient and thermal conductivity of dried food products, uniform slices were heated on a hot plate (Fig. 1).  The experimental design included the hot plate, food products, and thermocouple.  A thermocouple was used to determine when the temperature on the hot plate and surface of the food product had each reached steady-state conditions.  Once steady-state was reached and surface and hot plate temperatures were collected and recorded, the parameters were calculated analytically.

Using temperature values and known properties of the material, we were able to calculate estimated convection coefficient values for free convection.  For free convection, there is no movement due to bulk movement.  The movement is a result of temperature differences.  Since hot air rises, the air at the surface of a hot plate will rise and be replaced by cooler air, causing circulation.  Because of this, Reynolds number is not used in the calculation of the convection coefficient.  Instead, the Grashof number, Rayleigh number, Prandtl number, and Nusselt number were calculated to determine the estimated convection coefficient.  The established relationships between the average Nusselt value as a function of Grashof and Prandtl numbers were used to determine the convection coefficient for “horizontal flat plate, top surface of flat hot plate conditions.”

Fourier’s Law of Heat Conduction and Newton’s Law of Cooling were applied to determine the thermal conductivity for each dried food product.  By allowing heat flow by conduction to equal heat flow by convection, the estimated convection coefficient was used to calculate the thermal conductivity for each product.  Results

From our calculations, the average convection coefficient for the banana was 2.8376 W/m2K, and the average h value for the zucchini in our experiment was 2.9053 W/m2K.  The thermal conductivity of the banana was 0.061902 W/mK, and the thermal conductivity for the zucchini was 0.0141594 W/mK .  The thermal conductivity values for the dried food products were compared with the reported thermal conductivity value range for dried fruits, k = 0.2-0.4 W/mK.  FEM Lab was used to find the linear temperature profile for each food product at steady-state conditions.

Discussion

Based on the analytical and numerical results of the experiment, the convection coefficient and thermal conductivity of the food products were obtained.  Possible deviations in the thermal conductivity values with known reported values could result from non-uniformity of material thickness, minute thickness, measurement errors with calipers, fluctuating temperatures caused by cycling of the hot plate, and possible wire or contact errors of thermocouple.  Despite these errors, the calculated convection coefficient and thermal conductivity values were found to be reasonable estimates.

Appendix

A. Calculations

Data Collected

 Area Ts T∞ B D Perim. L ba1 0.00057 339.8 296 0.00315 0.027 0.0848 0.0067 ba2 0.00057 345 296 0.00312 0.027 0.0848 0.0067 ba3 0.00045 345 296 0.00312 0.024 0.0754 0.006 zu1 0.00031 339.5 296 0.00315 0.02 0.0628 0.005 zu2 0.00057 348 296 0.00311 0.027 0.0848 0.0067 zu3 0.00053 346.1 296 0.00311 0.026 0.0817 0.0065
 Gr Pr Ra Nu h Avg h Thot plate ∆x k Avg k ba1 1292.2626 1.1351 1467 3.341849 2.7601 2.837552 354 0.006 0.051082 0.061902 ba2 1433.9541 1.1351 1628 3.429911 2.8329 356.5 0.006 0.072422 ba3 1010.1189 1.1351 1147 3.142261 2.9197 356.5 0.005 0.062202 zu1 522.60439 1.1351 593.2 2.66497 2.9714 2.905349 351.2 0.001 0.011048 0.0141594 zu2 1514.6583 1.1351 1719 3.477185 2.8719 357 0.001 0.016593 zu3 1303.8962 1.1351 1480 3.349345 2.8727 355.8 0.001 0.014837

h  values:

Where:

Coefficient of Thermal Expansion:                            Bgas = 1/((Ts + T )/2)

Length = Area/Perimeter

Prandtl number:                                                Pr = kinematic viscosity/diffusivity

Grashof number:                                                     Gr = gB(Ts - Tinf )L32

Rayleigh number:                                                            Ra = (Gr)(Pr)

(h for horizontal flat plate, top surface of hot flat plate)

Nusselt number:                                                             Nu = .54(Ra1/4)                               for 104<Ra<107

h = (Nu)(k)/D

From the table above, the average h values were calculated:

Banana:  2.8376 W/m2K          Zucchini:  2.9053 W/m2K

## k values: From the table above, the average k values were calculated:

Banana:  0.061902 W/mK                    Zucchini:  0.0141594 W/mK

B.  FEM Lab        #### Reference

Drapcho, Dr. Caye M.  BE 4352 Transport Phenomena in Biological Engineering notes, Spring

2002.