# 6.2 Long-time: examples

Course subject(s) 6. Unsteady conduction and diffusion

## EXAMPLE 6.2A: COOLING A ‘PRESSE PAPIER’ (ELEMENTARY)

A modern “presse papier” is made of Perspex. It is, in essence, a sphere of 10 cm diameter that can be put on a pile of paper sheets, so they don’t get blown away by wind. It is made in a production line at a temperature of 100°C. At the end of the production line, it still has a uniform temperature of 70°C.

It is placed in the open air (of 20°C to further cool down to an average temperature of 30 °C, before further handling can take place. If the resistance to heat is completely inside the sphere:

1. Find the cooling time required;

2. Determine the temperature of the center at the moment the average temperature is 30°C.

Data perspex:

– density = 1200 kg/m3

– specific heat = 1466 J/kgK

– conductivity = 0.1 W/mK

## Example 6.2A: Cooling a 'presse papier' (elementary)

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## EXAMPLE 6.2B: COOLING A ‘PRESSE PAPIER’ WITH OUTSIDE RESISTANCE (MEDIUM)

A modern “presse papier” is made of Perspex. It is, in essence, a sphere of 10cm diameter that can be put on a pile of paper sheets, so they don’t get blown away by wind. It is made in a production line at a temperature of 100°C. At the end of the production line, it still has a uniform temperature of 70°C. It is placed in the open air (of 20°C) to further cool down to an average temperature of 30°C before further handling can take place. In practice, there is heat resistance inside and outside the sphere. For the outside resistance in air, it is given that Nu=20.
Find the cooling time required.

Data perspex:

– density = 1200 kg/m3

– specific heat = 1466 J/kgK

– conductivity = 0.2 W/mK (other value than used in example A!)

Data air

– conductivity = 0.026

Subtitles (captions) in other languages than provided can be viewed at YouTube. Select your language in the CC-button of YouTube.

## EXAMPLE 6.2C: COOLING A ‘PRESSE PAPIER’ WITH EXTERNAL AIR COOLING (ADVANCED)

A modern “presse papier” is made of Perspex. It is, in essence, a sphere of 10 cm diameter that can be put on a pile of paper sheets, so they don’t get blown away by wind. It is made in a production line at a temperature of 100°C. At the end of the production line, it still has a uniform temperature of 70°C
It is placed in the open air (of 20°C) to further cool down to an average temperature of 30°C, before further handling can take place.
In practice, there is heat resistance inside and outside the sphere. On the outside, air of 20°C is blown around the sphere to help it cool down faster. If the cooling time needs to be less then 2 hours, find the required air velocity.

Data perspex:

– density = 1200 kg/m3

– specific heat = 1466 J/kgK

– conductivity = 0.2 W/mK

Data air:

– density = 1.2 kg/m3

– viscosity = 2 10-5 Pas

– specific heat = 1000 J/kgK

– conductivity = 0.026 W/mK

Forced flow around a sphere

Subtitles (captions) in other languages than provided can be viewed at YouTube. Select your language in the CC-button of YouTube.

## BONUS: EXAMPLE 6.2D: COOLING OF A DUTCH CROQUETTE (MEDIUM)

In this example, we are going to consider the cooling of a Dutch croquette. This is, in essence, a sphere of 3cm, that is cooked in hot oil and consumed at parties with drinks.
How long do you have to wait before you can safely put it in your mouth and bite on it?
Data: temperature directly after boiling: 170 ºC; temperature of surroundings: 20 ºC; properties of Ducth croquette: take water values and you will be sufficiently close.

1. How long do you have to wait before the average temperature of the croquette is 40 ºC?
2. What is than the center temperature of the croquette?

You my assume that the resistance to heat is completely inside the croquette.