#### Introduction

In the analysis of heat transfer processes, some bodies can be considered to behave as a unit, with an internal temperature essentially uniform (although not constant) at all times during the heat transfer process.

In these situations, the temperature can be considered exclusively as a function of time, T (t). The analysis of heat transfer using this idealisation is called **Lumped System Analysis (LSA)**, or the method of uniform temperature, and provides a great simplification in certain kinds of heat transfer problems without significant losses in the accuracy of the results.

Unfortunately, for engineers this simplification is not applicable to all bodies and conditions, and a correct test application must be found.

#### Examples

Let’s consider two extreme cases.

Temperature measurements of a small steel ball being heated in an oven show the temperature increases with residence time in the oven; however, there are practically no variations with the position on the ball at any given time. Thus, the temperature of the ball remains almost uniform at all times, and we can talk about the temperature of the ball without reference to a specific location on it.

If the steel ball is changed for a lamb leg with potatoes, after a while the skin surface of the lamb begins to show signs of reaching a significant crispy temperature, which is much higher than the central part of the leg. Although warming has been extremely slow and careful (to provide a good meal) significant differences in temperature are unavoidable. If the oven is at full power, the result will be a lamb leg burnt on the outside and raw on the inside.

Even the potatoes will have a similar “burnt/raw” appearance, with very different temperatures being reached in the centre and on the surface.

Fig.1 Temperature distribution for (a) a steel ball and (b) a leg of lamb and potatoes |

The temperature distribution in Fig 1 is as follows.

**T1**– Air temperature in the oven**Tsc**– Body interior temperature**Tss**– Body surface temperature

Fort the Steel ball (a) Tsc and Tss are very similar, while for the lamb leg (b) the difference between the two is significant

We can illustrate the criterion of appropriateness of applying the method of uniform temperature with another example:

Consider the body as an island; with external transport systems (ships and aircraft) to reach the island; while there is also the inland transport network on the island of buses, trains and private cars, etc.

If the frequency and capacity of the external transport exceeds the available internal transport network, people will accumulate in port and airport terminals, as it will be difficult to transport them to other parts of the island. Therefore, the population density on the island will not be uniform, with a large concentration of people at specific points, e.g. ports and airports.

If, however, the inland transport network is effective, people will be distributed throughout the island and there may be a uniform population density throughout.

Thus, a relationship between the capabilities of the exterior systems and interior island transport systems will be a solid basis for considering if the population density on the island is uniform or not..

#### Application Examples

This model can also be used for heat transfer cases.

If a body is surrounded by a fluid, energy is initially exchanged between the fluid and the body by convection (ships and aircraft) and, subsequently by internal conduction to/from all points of the body (the inland transport network on the island).

## The Biot number is a dimensionless number whose value can be considered as a criterion of applicability for the Lumped System Analysis (LSA).

It is defined as:

### Ratio of transfer between the fluid and body (convection) and within the body (conduction).

This applies equally for heating or cooling conditions in the body.

Considering the above definition a little more:

## What is conduction?

It is heat transferred from a higher temperature region to a lower temperature region when there is a temperature gradient in a body in a direction x. The heat transferred by conduction Qk is proportional to the temperature gradient dT and the distance X between the gradient points.

The heat transferred depends on a property of the body material, its thermal conductivity k, which represents the ability of the substance to conduct heat, thus causing the consequent temperature change.

Equation (1) thus becomes

where, k is the thermal conductivity coefficient and A is the exchange cross-sectional area (A = X x L)

Equation (2) is known as **Fourier’s Law**

## What is convection?

This is the heat transfer mechanism by mass movement. It can be produced naturally only by differences in the density of the substance, or forced when the substance is made to move from one place to another (e.g. air moved by a fan, or water by a pump). It only occurs in liquids and gases where the atoms and molecules are free to move within the body. The heat exchanged between a fluid and a body is determined by the expression:

where α is the convection coefficient

A is the exchange surface between body and fluid

T_{1} is the fluid temperature

Tss is the body surface temperature.

Equation (3) is known as * Newton’s Law of Cooling*,

### Therefore the Biot number (Bi) is:

Because the Biot number is the ratio between the external resistance of a body to convection energy exchange and internal resistance to transmitting energy by conduction, a small Biot number implies low resistance to transmission by conduction, and therefore very low temperature gradients within the body; thus leading to the Lumped Analysis System, assuming a uniform temperature distribution throughout the body.

An example of temperature distribution in a solid depending on time t and the Biot number value is shown in Fig. 2. This is a cooling system.

The image shows a body with a low Biot number (Bi ≤ 1) at t=0, when the body temperature is uniform, Ts1. A flowing fluid at a lower temperature will progressively cool the body, but the thermal gradient between the body surface temperature and its centre is always very low, dT ≈ cte.

The other two images show a temperature distribution in solids with a higher Biot number. As the cooling process continues, the temperature gradient tends to decrease (dT1 ≥ dT2). These values are always unacceptable for the application of the simplified method of uniform temperature.

In the first image, the temperature can therefore be considered almost exclusively as a function of time, while in the other two cases, it is also a function of the position x.

Fig. 2. Differences in the temperature distribution within solids according to the Biot number, dT the temperature gradient, t time and x length/thickness. |

#### Conclusions

How much accuracy are we willing to sacrifice to implement this simplification and ease that applying the method of uniform temperature involves?

An uncertainty of up to 15% in the coefficient of heat transfer by convection α is considered acceptable. This coefficient depends on both the fluid composition, the temperature in the process and velocity in the exchange phase.

Thus, assuming a constant and uniform α, the approximation is also questionable, especially for irregular geometries or variable flows. In the absence of sufficient experimental data to accurately evaluate the process under consideration, the results will not be accurate to ± 15%, even when Bi = 0.

In short, introducing another source of uncertainty to the problem will not have much effect on the overall uncertainty, providing this new uncertainty is less than the existing one.

As a basic rule, the simplification representing acceptable application of the Lumped Analysis System is when Bi ≤ 0.1. In most cases, this represents a temperature difference between the body surface Tss and the most distant point within the body Tsc of < 5%.

More specifically, and based on experimental results, the following have been set as standard values, depending on the shape of the body:

Plates | Bi < 0.1 |

Cylinders | Bi < 0.05 |

Spheres | Bi < 0.03 |

Although advances in information technology have resulted in sophisticated computer programs that solve complicated problems of heat transfer quickly and accurately, using the uniform temperature method with acceptable criteria based on the above Biot number values is a useful tool in everyday calculations.