WHAT IS THE PRANDTL NUMBER? (VALUES ​​IN GASES AND LIQUIDS) - PHYSICAL - 2025

The Prandtl number, abbreviated Pr, is a dimensionless quantity that relates the diffusivity of the momentum, through the kinematic viscosity ν (Greek letter that is read “nu”) of a fluid, with its thermal diffusivity α in the form of quotient:

Figure 1. The German engineer Ludwig Prandtl in his Hannover laboratory in 1904. Source: Wikimedia Commons.

In terms of the fluid viscosity coefficient or dynamic viscosity μ, the specific heat of the fluid C _p and its coefficient of thermal conductivity K, the Prandtl number is also expressed mathematically as follows:

This quantity is named for the German scientist Ludwig Prandtl (1875–1953), who made great contributions to fluid mechanics. The Prandtl number is one of the important numbers for modeling the flow of fluids and in particular the way heat is transferred in them by convection.

From the definition given, it follows that the Prandtl number is a characteristic of the fluid, since it depends on its properties. Through this value, the ability of the fluid to transfer momentum and heat can be compared.

Natural and forced convection in fluids

Heat is transmitted through a medium by various mechanisms: convection, conduction, and radiation. When there is movement at the macroscopic level of the fluid, that is, there is massive movement of the fluid, the heat is quickly transmitted in it through the convection mechanism.

On the other hand, when the preponderant mechanism is conduction, the movement of the fluid occurs at the microscopic level, either atomic or molecular, depending on the type of fluid, but always more slowly than by convection.

The speed of the fluid and the flow regime it has - laminar or turbulent - also influences this, because the faster it moves, the faster the heat transfer is also.

Convection occurs naturally when fluid moves due to a difference in temperature, for example when a mass of hot air rises and another of cold air descends. In this case we speak of natural convection.

But convection can also be forced by using a fan to force the air to flow, or a pump to set the water in motion.

As for the fluid, it can circulate through a closed tube (confined fluid), an open tube (such as a channel for example) or an open surface.

In all of these situations, the Prandtl number can be used to model heat transfer, along with other important numbers in fluid mechanics, such as Reynolds number, Mach number, Grashoff number, number of Nusselt, the roughness or roughness of the pipe and more.

Important definitions in heat transfer in a fluid

In addition to the properties of the fluid, the geometry of the surface also intervenes in the transport of heat, as well as the type of flow: laminar or turbulent. Since the Prandtl number involves numerous definitions, here is a brief summary of the most important ones:

Dynamic viscosity

It is the natural resistance of a fluid to flow, due to the different interactions between its molecules. It is denoted μ and its units in the International System (SI) are Ns / m ² (newton x second / square meter) or Pa.s (pascal x second), called poise. It is much higher in liquids than in gases and depends on the temperature of the fluid.

Kinematic viscosity

It is denoted as ν (Greek letter that is read “nu”) and is defined as the ratio between the dynamic viscosity μ and the density ρ of a fluid:

Its units are m ² / s.

Thermal conductivity

It is defined as the ability of materials to conduct heat through them. It is a positive quantity and its units are Wm / K (watt x meter / kelvin).

Specific heat

Amount of heat that must be added to 1 kilogram of substance to raise its temperature by 1 ºC.

Thermal diffusivity

Is defined as:

The units for thermal diffusivity are the same as those for kinematic viscosity: m ² / s.

Mathematical description of heat transfer

There is a mathematical equation that models the transmission of heat through the fluid, considering that its properties such as viscosity, density and others remain constant:

T is the temperature, a function of time t and of the position vector r, while α is the aforementioned thermal diffusivity and Δ is the Laplacian operator. In Cartesian coordinates it would look like this:

Rugosity

Roughness and irregularities on the surface through which the fluid circulates, for example on the internal face of the pipe through which the water circulates.

Laminar flow

It refers to a fluid that flows in layers, in a smooth and orderly manner. The layers do not intermingle and the fluid moves along so-called streamlines.

Figure 2. The column of smoke has a laminar regime at the beginning, but then volutes indicative of a turbulent regime appear. Source: Pixabay.

Turbulent flow

In this case the fluid moves in a disorderly way and its particles form eddies.

Prandtl number values ​​in gases and liquids

In gases, the order of magnitude of both kinematic viscosity and thermal diffusivity is given by the product of the average velocity of the particles and the average free path. The latter is the value of the average distance traveled by a gas molecule between two collisions.

Both values ​​are very similar, therefore the number of Prandtl Pr is close to 1. For example, for air Pr = 0.7. This means that both momentum and heat are transmitted approximately equally quickly in gases.

In liquid metals, however, Pr is less than 1, since free electrons conduct heat much better than momentum. In this case ν is less than α and Pr <1. A good example is liquid sodium, used as a coolant in nuclear reactors.

Water is a less efficient conductor of heat, with Pr = 7, as well as viscous oils, whose Prandtl number is much higher, and can reach 100,000 for heavy oils, which means that heat is transmitted in them with very slow, compared to momentum.

Table 1. Order of magnitude of the Prandtl number for different fluids

Fluid	ν (m 2 / s)	α (m 2 / s)	Pr
Terrestrial mantle	10 17	10 -6	10 23
Inner layers of the Sun	10 -2	10 2	10 -4
Atmosphere of earth	10 -5	10 -5	one
Ocean	10 -6	10 -7	10

Example

The thermal diffusivities of water and air at 20 ºC are respectively 0.00142 and 0.208 cm ² / s. Find the Prandtl numbers for water and air.

Solution

The definition given at the beginning applies, since the statement gives the values ​​of α:

And as for the values ​​of ν, they can be found in a table of properties of fluids, yes, we must be careful that ν is in the same units of α and that they are valid at 20 ºC:

ν _air = 1.51x 10 ^-5 m ² / s = 0.151 cm ² / s; ν _water = 1.02 x 10 ^-6 m ² / s = 0.0102 cm ² / s

Thus:

Pr (air) = 0.151 / 0.208 = 0.726; Pr (water) = 0.0102 / 0.00142 = 7.18

References

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5. UNET. Transportation phenomena. Recovered from: unet.edu.ve.
6. Wikipedia. Prandtl number. Recovered from: en.wikipedia.org.
7. Wikipedia. Thermal conductivity. Recovered from: en.wikipedia.org.
8. Wikipedia. Viscosity. Recovered from: es.wikipedia.org.