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.
Steadystate 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 steadystate 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 steadystate conditions.
Once steadystate 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/m^{2}K, and the average h value for the zucchini in our
experiment was 2.9053 W/m^{2}K.
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
= 0.20.4 W/mK. FEM Lab was
used to find the linear temperature profile for each food product at
steadystate 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 nonuniformity 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 
T_{s} 
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 
T_{hot
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: B_{gas} = 1/((T_{s} + T_{∞ })/2)
Length = Area/Perimeter
Prandtl number: Pr = kinematic viscosity/diffusivity
Grashof number: Gr = gB(T_{s}  T_{inf })L^{3}/ν^{2}
Rayleigh number: Ra = (Gr)(Pr)
(h for horizontal flat plate, top surface of hot flat plate)
Nusselt number: Nu = .54(Ra^{1/4}) for 10^{4}<Ra<10^{7}
h = (Nu)(k)/D
From the table above, the average h values were calculated:
Banana:
2.8376 W/m^{2}K
Zucchini: 2.9053
W/m^{2}K
From the table above, the average k values were calculated:
Banana:
0.061902
W/mK
Zucchini: 0.0141594 W/mK
B. FEM Lab
Drapcho, Dr. Caye M. BE 4352 Transport Phenomena in Biological Engineering notes, Spring
2002.