TP1:Example Problem 24

Cooking Times

Solving unsteady state transport problems can be extremely tedious as most of the equations are non-linear partial differential equations. This is because in real life, heat and mass transfer depend on a lot of factors. In previous modules, many problems were solved using transport equations in a semi-infinite medium: from -∞ to 0 or from 0 to ∞. How nothing in real life exists in a semi-infinite medium. Real life problems are bounded: -∞ to +∞. However, because they are bounded, real life transport problems can get extremely complicated to solve analytically.

For example, Figure 1 shows the solutions for one-dimensional transient conduction. As one can see, these solutions are quite complex and include the use of eigenvalues and Bessel functions. These solutions get more complex as we look at two-dimensional and three-dimensional conduction.

Luckily there is an easy way!!

M.P. Heisler solved those equations and organized them graphically in his Heisler charts. In this example problem, we will use his charts to solve an unsteady state problem to find the time required to cook a meatball.

File:Summary for the solutions of one-dimensional transient conduction.png

Figure 1: Summary for solutions of one-dimensional transient conduction.

Learning Outcome

Calculate the Biot number and know when to use the Lumped-Sum model - Calculating The Biot Number
Calculate the Nusselt number and understand how it relates to the Biot number - Calculating The Nusselt Number
Use Heisler charts and the Biot number to solve unsteady state transport problems

Example Question

A meatball at an Italian restaurant is kept in the refrigerator at 250 K. It is then baked in the oven at at 415 K. Calculate the time for the center of the meat to reach 370 K if it is spherical in shape with a diameter of 4 cm.

The properties of the meatball are the following:

Thermal conductivity: 0.6 W m-1K-1

Density: 1000 kg m-3

Heat capacity: 3750 J kg-1 K-1

Convective heat transfer coefficient of the oven: 115 W m-2 K-1.

Solution

Given:

T0 (Initial) = 250K

Tinf (Bake temp) = 415K

T (Final) = 370K

K (Thermal Conductivity) = 0.6 W m^-1 K^-1

$ρ$ (Density) = 1000 kg m^-3

Cp (Heat Capacity) = 3750 J kg-1 K-1

h (Heat Transfer Coefficient) = 115 W m^-2 K^-1

Step 1: Convert the diameter 4cm to radius in meters

r = 2 cm x 1m/10cm = 0.02 m

Step 2: Find $α$

α = K / ρ / C p

(1)

Given values plugged in:

$α = (0.6 W * m^{- 1} * K^{- 1}) / (1000 k g * m^{- 3}) / (3750 J * k g^{- 1} * K^{- 1})$

$α =$ 1.6E-07

Step 3: Find the Inverse Biot Number

B i^{- 1} = K / h * r_{o}

(2)

Given values plugged in:

$B i^{- 1} = \frac{0.6 W * m^{- 1} * K^{- 1}}{(115 W / m^{- 2} * K^{- 1}) * (0.02 m)}$

$B i^{- 1} = 0.2609$

Step 4: Find the Theta Value:

θ_{o}^{*} = (\frac{θ_{o}}{θ_{i}}) = (\frac{T (t) - T i n f}{T i - T i n f})

(3)

Given values plugged in:

$θ_{o}^{*} = \frac{370 K - 415 K}{250 K - 415 K}$

$θ_{o}^{*} = 0.2727273$

Step 5: Find the $F_{o}$ Value using the Heisler Charts

F o = t^{*} = (α * t / r_{o}^{2})

(4)

Use the Heisler Charts and the $B i^{- 1}$ and $θ_{o}^{*}$ values to find $F_{o}$ .

Estimated to be:

$F_{o} = 0.5$

Step 6: Find the time required for the center of the meat to reach 370 K

Rearrange the $F_{o}$ equation to solve for time (t)

t = r^{2} * F_{o} / α / 60 s e c

(5)

$t = (0.0004) * (0.5) * \frac{1}{1.6 E - 07 * 60}$

Answer:

t = 20.8333 min

Things to keep in mind

If the Biot number is less than 0.1, the thermal gradients in the mass are negligible, and therefore the Lumped-Sum analysis should be used as a model instead
BOTH the Nusselt number and the Biot number have the same form, h*L/K. However, the Nusselt number is the ratio of convective to conductive heat transfer in the same medium. The Biot number refers to the conductive to convective heat transfer between a solid mass and the surrounding fluid
When Bi = 0, the Lumped-Sum analysis is exact. When the Biot number exceeds 0.1, however, you should only use the Lumped-Sum analysis.

The Lumped-Sum analysis is good to use for small bodies with high thermal conductivity