Heat transfer is a very important topic in mechanical engineering. There are various approaches to heat transfer that are very useful in such engineering calculations. One of these approaches is the lumped system analysis. Also in lumped system analysis, the Biot number is a very important value. You can find detailed information about the Biot number and its significance in terms of heat transfer here.

## What is Biot Number in Heat Transfer?

Before the Biot number, we need to discuss the lumped system analysis in heat transfer. In heat transfer, there are different heat transfer mechanisms such as conduction, convection, and radiation. And we are making the calculations of these heat transfer mechanisms with lots of assumptions. These assumptions make the calculations more convenient but they are taking away from the real values. Lumped system analysis moves in here.

In the Lumped system analysis, we calculate both conduction and convection heat transfers of bodies. But the temperature distribution in the body must be even. The calculation of the lumped system analysis will give accurate results if the temperature distribution over the body is even.

But in real life, there is no practical system like this. For example, think that you are making bread in an oven. After taking the baked bread from the oven and if you cut it in half, you will notice that the internal sections are not hot as the external sections. So, it is very hard to apply the lumped system analysis heat transfer to this bread.

But in some systems, lumped system analysis gives approximate heat transfer results. So, we need to calculate the Biot number for the systems to see whether we apply lumped system analyses for a system.

### How to Calculate Biot Number?

Biot number is a tool to see the applicability of lumped system analysis to a system. The Biot number is the ratio of conduction resistance within a body to convection resistance between the surface and the environment. So, the Biot number must be small for the application of lumped system analysis.

In the Biot number formula, ‘Cl’ is the characteristic length of the body. For simple geometries, the characteristic length is;

- For a long cylinder; = r/2
- Plane wall: Thickness/2
- Sphere: r/3

Also, you can calculate the characteristic length of bodies by diving the volume of the body with the surface area of the body.

‘k’ is the thermal conduction constant of the material of the body and ‘h’ is the convection of the body.

So if we talk about the Biot number, we prefer smaller thermal conduction resistance. With the smaller conduction resistances, the body will be a more uniform temperature gradient. We are diving the characteristic length of the shape of the body to the thermal resistance constant. So, with the smaller characteristic length which represents the dimensions of the body, the Biot number will be smaller.

Smaller bodies with high thermal conductivities are more appropriate for the application of lumped system analysis in heat transfer.

### How Much Should This Number Be?

So with the specific value of the Biot number, we can apply the lumped system analysis. A degree of accuracy is valid for heat transfer analyses which are generally %10-15. So with this level of accuracy, the general upper limit value of the Biot number is accepted as 0.1. In other words, we can apply lumped system analysis with the Biot numbers smaller than 0.1.

## What is Lumped System Analysis?

Lumped system analysis is a very convenient way to calculate the heat transfer rate between bodies and the environment. With this approach, you can calculate the total heat transfer time and the temperature of the body in a specific time with only one equation. This equation is;

In this formula, ‘b’ is the time constant which you can calculate with this formula;

Lumped system analysis formula;

- ‘Tambient’ is the ambient temperature,
- ‘Tbody’ is the uniform temperature of the body,
- ‘T(t)’ is the temperature of the body in the desired time.
- ‘t’ is the time in seconds.

Equation of ‘b’;

- ‘h’ is the comvection constant of the environment and body surface (W/(m2°C) or Btu/(hr-ft2°F)),
- ‘As’ is the surface area of the body(ft2 or m2),
- ‘ρ’ is the density of the body(kg/m3 or lb/ft3),
- ‘V’ is the volume of the body(m3 or ft3),
- ‘cp’ specific heat of the body(kJ/kgoC or Btu/lboF),

The ‘b’ is called the time constant and with the increasing ‘b’, the temperature change of the body is rapid and the total time of heat transfer is small.

## Conclusion

Above all, the general explanation of lumped system analysis and the calculation of Biot number is like above.

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