# Unsteady State Heat Transfer

Before a system is able to reach steady state there must be some startup condition which is dependant on time. During this transient peroid unsteady state models must be used to describe the system. Such models are, by nature, quite complicated, and engineers have devised means of simplifying them when conditions warrant. One such simplification is the lumped thermal capacity model.

In order to determine the accuracy of the lumped thermal capacity model, one must start off by calculating the **Bi** number for the body going through heat transfer. Lumped Capacity model leads to huge errors if the Biot Number exceeds 0.1. The lumped thermal capacity model assumes that the system being heated or cooled does not have temperature gradients. This means that no matter where a measurement is taken in the system the temperature will be the same in all places and one instant in time. This assumption falls apart at Biot numbers greater than 0.1 because temperature gradients develop inside the body. The lumped capacity model also holds true for systems such as well mixed fluids, since there are negligible spacial variations of any quantity in a well mixed system.

When mentioning a small Biot Number, the conceptual way of stating this idea is that the resistance to heat transfer within the system (Conduction) is negligible when compared with the resistance to heat transfer with the surroundings (Convection).

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Integrating from to , and to , gives the following.

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The lumped thermal capacity model is a very useful simplification for systems that have low resistance to internal heat transfer relative to external heat resistance. Knowing when this assumption can be made is important when using this model. To quantify the internal heat resistance to the surface heat resistance use the Biot number. When the lumped thermal capacity model is valid.^{[1]}

If the system that is being evaluated does not have a low internal heat resistance relative to surface heat resistance other correlations or modeling software are commonly used. A useful correlation is the Gurney-Lurie charts for getting temperature at any position in a solid plate, cylinder, or sphere. If the center of a cylinder, plate, or sphere is the point of interest, Heisler charts can be used.

These charts use the following equations to correlate the heat transfer and allow for modeling of a transient system.

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, where is the thermal diffusivity, is the distance from the outer surface and is time. Note that can be put in terms of radius for spheres and cylinders.
^{[1]} Thermal diffusivity can be found using the relationship .

, where is the thermal conductivity of the system and is the convective heat transfer coefficient of the surroundings.
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Often situations are encountered where heat transfer is not one dimensional. To model heat transfer that is two or three dimensional the use of superposition can be used with the before mentioned charts. This method is done by solving for each , then using those values find . ^{[1]}

## Cooking a Turkey Example Problem

From real life experience, we know that a turkey is fully cooked when it reaches 170°F. We will use a 20 pound turkey for this example. After consulting the Butterball Hotline (1-800-BUTTERBALL) it was recommended to cook the 20 pound turkey for 5 hours with the oven set to 325°F.
**Was the recommended time by the Butterball Hotline correct or not?**

##### Variables Needed

- Heat Capacity = 3100 J/Kg K
- Density = 1040 Kg/m
^{3} - Thermal Conductivity = 0.45 W/m K
- Mass = 20 lb = 9 kg
- Radius = 0.13 m
- Turkey Starting Temperature = 45°F
- Heat transfer around the turkey = 20

##### Solution

With the Heisler chart, we can use the Biot number, Θ and τ to find time.

**0.17**

**0.55**

**4.7 hours**

Unsteady state heat transfer can be defined as a process of transfer of heat in which the
rate of heat transfer is changing with the change in time i.e. the rate of transfer of heat is
not constant. This is due to the temperature gradient between the two bodies involved in
the process (Gvozdenac, 2014) . <ref name="one" [ Gvozdenac, D. (2014). An unsteady-state method for determining overall coefficient of
heat transfer (k-value) of insulated bodies at variable external temperatures. Heat and
Mass Transfer , 171 - 180.</ref>

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}^{1.7}Geankoplis, Christie John. Transport Processes and Separation Process Principles. Pearson Education Inc, 2015. Book.